S I. 3 :3 A . 4...... . tr. .9 1 IVvu. 7.. t. > 95.. 3. .7...) .. . ‘r . 211:! UV, mmIHMMHIIIHIWWII/1W 2 "z 3 570 0 93 300577 0023 L..v’6‘a-*\D‘ i.- Michigan Stat. University This is to certify that the dissertation entitled AN ANALYTICAL AND EXPERIMENTAL STUDY OF THE ELASTODYNAMIC RESPONSE CHARACTERISTICS OF PLANAR LINKAGE MECHANISM WITH BEARING CLEARANCES presented by KUEIMING SOONG has been accepted towards fulfillment of the requirements for PHoD degreein MoE. /%~J7Z~W Major professor Date SRD-1?,1988 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FWES will be charged if book is returned after the date stamped below. AN ANALYTICAL AND EXPERIMENTAL STUDY OF THE ELASTODYNAMIC RESPONSE CHARACTERISTICS OF PLANAR LINKAGE MECHANISM WITH BEARING CLEARANCES By Kueiming Soong A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1988 ABSTRACT AN ANALYTICAL AND EXPERIMENTAL STUDY OF THE ELASTODYNAMIC RESPONSE CHARACTERISTICS OF PLANAR LINKAGE MECHANISM WITH BEARING CLEARANCES BY Kueiming Soong A comprehensive investigation of the dynamic response characteristics of a general planar linkage mechanism featuring both rigid and flexible links with and without radial clearance in the revolute joints is presented. In the coherent combination of analytical, computational, and experimental work undertaken herein, the equations Soverning the dynamical behavior of this general mechanical system are established by incorporating a four-mode model of the phenomenological behavior of the principal elements of each revolute joint into the generalized form of Lagrange's equations. A displacement finite element formulation is developed for undertaking the nonlinear vibrational analysis of general flexible linkage mechanisms. This formulation incorporates elastic geometrical nonlinearities and also terms coupling the rigid-body kinematics of the linkage mechanism and the elastic deformations. The equations of motion for this class of mechanical l (i J system are composed of a mixed set of ordinary differential equations and algebraic constraint equations. These governing equations are solved by employing a quasi-static procedure, Adams numerical integration scheme and the mode superposition approach of vibrational analysis. Four different illustrative examples are considered and the analytical and experimental results show impressive correlation. To my parents and to my wife Jui—Mei ii ACKNOWLEDGMENTS I would like to express my deepest gratitude to my advisor, Professor Brian S. Thompson for his help and support during all stages of this research program. Grateful thanks are extended to the other members of the guidance committee, Professors M.V. Gandhi, 0.8. Yen and C.Y. Vang. Also I would like to thank all the students in Machinery Elastodynamics Laboratory for their valuable suggestions and assistance. The author wishes to gratefully acknowledge the four-year financial support of the Chung-Cheng Institute of Technology, Taiwan, which has enabled him to undertaken this research. A debt of thankfulness is owed to my lovely wife, parents, parents-in-law and my two children for their encouragements and supports, without which my research could have not been finished. iii TABLE OF CONTENTS Page LIST op FIGURES ..................................... .. .......... viii NOMENCLATURE .................................................... xvi CHAPTER I. INTRODUCTION . ........................................... 1 1,1 Motivation ..... ......................................... 1 1. 2 Literature I'CVIEV ....................... . e e .......... . . . 5 1.2.1 Mechanisms with rigid-links and featuring bearings without clearances .............................................. 6 1.2.2 Mechanisms with rigid-links and featuring bearing clearances ...... . ............ .... ...... ........... ..... 7 1.2.3 Flexible-links in mechanical systems without bearing clearances ..... . ......... . ..................... . ........ 14 1.2.4 Flexible-links in mechanical systems with bearing clearances ........................ . ..................... 19 1.3 Thesis outline ...... .................................... 22 1,4 Principal contributions ...... ... ........... ...... ....... 25 II. GENERAL PLANAR MECHANISMS FEATURING RIGID-LINKS VITH RADIAL CLEARANCES IN THE REVOLUTE JOINTS °°°°°°° .. ------- 27 2.1 Introduction ..... ....... . ....... . ....................... 27 iv 2.2 2.3 2.4 2.5 2.6 2.7 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5 2.7.6 2.7.7 2.8 Model of a general planar rigid-linked mechanism with hearing clearances ................................. ..... (1) Contact-mode-o-°----° ........ ............ ....... .... (11) ptcc_flignt-nodc ............ . ............. ......... (iii)Impact-mode---'--° ........................... . ...... (iv) Transient-mode ... ..... .............. ........ .. ..... Governing equations for the contact-mode ---°-°~°~------- Governing equations for the free-flight-mode - ----------- Governing equations for the impact—mode ........ -------- - Strategy for a numerical solution ....................... Illustrative Example - Dynamical response model of a slider-crank mechanism with clearance in the gudgeon-pin joint ................................................... Equations of motion for the contact-mode ................ Equations of motion for the free-flight-mode ------------ Equations of motion for the impact-mode ................. Equations of motion for the transient-mode -------------- Simulation protocols .................................... (1) Contact loss .................... ..... .... ...... .... (ii) Termination of the Free-flight-mode ................ (iii)Restoration of the contact-mode --------------- ..... (iv) Solution procedure ......... ...... .. ...... . ......... Experimental slider-crank mechanism, associated instrumentation, and experimental procedure ............. Comparison of experimental and numerical results -------- Conclusions of the chapter ................. ............. 28 32 32 33 33 34 40 42 45 49 49 51 52 54 54 54 56 57 57 6O 67 76 III. GENERAL PLANAR MECHANISMS FEATURING FLEXIBLE—LINKS AND BEARINGS VITHOUT RADIAL CLEARANCE °°°°°°°°°°°°°°° ° °°°°°°° 77 3.1 IntrOduction ......OOOOOOOOO0............OOOOOOOOOOOOOOOO 77 3.2 Equations of motion of a flexible body describing a general planar motion ....... ......... .... ...... ................. 73 3.3 Solution procedure ...................................... 39 3.4 Illustrative Example - Elastodynamic response of a slider- crank mechanism with a flexible connecting rod ----- -°-°' 93 3.4.1 Equations of motion eeeeeeeeeeeeeeoeeeeeeeeemeeeeeemoeeee 97 3.4.2 Some remarks on the elastodynamic behavior of a flexible connecting rod eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 101 3.4.3 Comparison of experimental and numerical results -------- 109 3.5 Conclusions of the chapter eeeeeeeeeeeeeeeeeeeeeeeee eeeee 110 IV. ELASTODYNAMIC ANALYSIS OF A GENERAL PLANAR MECHANISMS FEATURING FLEXIBLE-LINKS WITH RADIAL CLEARANCES IN THE REVOLUTE JOINTS ......................................... 117 4.1 Introduction .......... ......... . ..... ................... 117 4.2 Model of a general planar mechanism with flexible-links and featuring radial clearances in the revolute joints °- 118 (i) Contact-mode . ..... ................................ 118 (ii) Free-flight—mode .......................... ... ..... 119 (iii) Impact-mode .... ................................... 119 (1V) Transient-mode .................................... 120 4.3 Equations of motion ..................................... 120 4.3.1 Equations of motion for the contact-mode ---------------- 129 4.3.2 Equations of motion for the free-flight-mode ------------ 132 4.3.3 Equations of motion for the impact-mode -°'°-- °°°°°°°°°°° 133 vi 4.4 Illustrative Example 1 - Dynamical behavior of a compound pendulum subjected to impact ....... ...... ............... 4.4.1 Equations of motion for the free-flight-mode ----° ------- 4.4.2 Equations of motion for the contact-mode --- ------------- 4.4.3 Equations of motion for the impact-mode - ---------------- 4,4,4 Solution procedures .................... .......... ....... 4.4.5 Comparison of experimental and theoretical results ------ 4.5 Illustrative Example 2 - A slider-crank mechanism with a flexible connecting-rod featuring radial clearance in the gudgeon-pin joint ....................................... 4.5.1 Equations of motion for the contact-mode ---------------- 4.5.2 Equations of motion for the free-flight-mode ------------ 4.5.3 Equations of motion for the impact-mode - ---------------- 4.5.4 Equations of motion for the transient-mode ---°°-0 ------- 4,5,5 Simulation protocols .......... ....... ........ ........... 4.5.6 Comparison of experimental and theoretical results ------ 4.6 Conclusions of the chapter .......... . ........ . .......... V. CONCLUSIONS AND RECOMMENDATIONS ------------------------- APPENDIX ......o ................................................. List of publications .................................... BIBLIOGRAPHY eeeeee ...................................... eeeeeeee vii 138 139 142 142 143 147 152 154 156 158 161 161 167 174 175 181 181 239 Figures Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 W Figure 2.1 Figure 2.2 LIST OF FIGURES Classical design of linkage mechanism ............ First generation of analysis techniques for elastodynamic design of flexible mechanisms. Two-pag. method .................................. New analysis techniques for elastodynsmic design of flexible mechanisms. One-pass method .......... Mlins .tr‘tggy eemeeeeeeeeeeeeeeeeeeeeemmeeeeee Radial clearance in the revolute joint in a planar ‘ech‘nig- system eeeeeeeeeeeeeeeeeeeeeeeeeeeemeeee Definition of the position vectors and reference frames modeling a planar mechanism system with “firing clgar‘nceg eeeeeeeeeeeeeeeeeeeeeeeeeeeeeee viii Page 11 12 13 25 29 29 Figure 2.3 Schematic diagram presenting the four-mode describing the phenomenological behavior of a revolute joint with clearance .................... 31 Figure 2.4 (a) The slider-crank mechanism showing dominant clearance at the gudgeon-pin bearing (b) Vector-loop model of the mechanism ........... 48 Figure 2.5 Free-body diagram of the slider-crank mechanism in an. 1.po¢t-.od. .................................. 53 Figure 2.6 Flow chart for the computer protocol ............. 59 Figure 2.7 Photograph of experimental planar slider-crank ”Ch‘nisl eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 62 Figure 2.8 Schematic of experimental apparatus and associated injtrumentation eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 63 Figure 2.9 Photograph of accelerometer configuration at the gudgeon-pin bearing of the experimental slider-crank ..ch.nisn ........................................ 54 Figure 2.10 Photograph of experimental apparatus for measuring the impactive force in the gudgeon-pin bearing and determining the associated coefficient of restitution between the pin and the journal of the bearing ... 66 Figure 2.11 Schematic diagram of the experimental apparatus and instrumentation for determining the coefficient of restitution of the gudgeon-pin bearing ----------- 68 Figure 2.12 Gudgeon-pin bearing characteristics: Velocity characteristics of the impactive behavior between pin and journal at a pin excitation frequency of 20 a; eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 69 ix Figure 2.13 Gudgeon-pin bearing characteristics: Excitation force characteristics and associated velocity characteristics of the pin impacting the journal .t . frequency of 20 Hz .......................... 69 Figure 2.14 Connecting-rod tangential acceleration. Mechanism opgrotins speed 200 rpm .......................... 70 Figure 2.15 Connecting-rod tangential acceleration. Mechanism operating speed 300 rpm .......................... 70 Figure 2.16 Slider acceleration. Mechanism operating speed 200 rp- eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 71 Figure 2.17 Slider acceleration. Mechanism operating speed 300 rpm .............................................. 71 Figure 2.18 Connecting-rod longitudinal acceleration. Mechanism operating speed 200 rpm .......................... 72 Figure 2.19 The theoretical prediction of the longitudinal acceleration of the connectingorod. Mechanism operating 300 rpm ................................ 72 Figure 2.20 The maximum impactive force between the pin and journal. The radial clearance was 0.000127 meters- 73 Figure 2.21 The maximum impactive force between the pin and journal. Mechanism operating speed 300 rpm ....... 73 W Figure 3.1 Generalized coordinate of two adjacent links ..... 80 Figure 3.2 The nodal displacement of each element ----------- 80 Figure 3.3 Axial force is represented as a linear function .. 85 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure .10 .11 .12 .13 .14 .15 .16 .17 Quasi-static solution procedures ................. (a)The slider-crank mechanism with flexible connecting rod (b)Vector-loop model of the mechanism ............ The six finite elements of the connecting rod .... Photograph of the flexible connecting rod with strain gauge. .................................... The relationship between the natural frequency and the “18]. force eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee The relationship between the damping factor and the ‘xial force eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee Simulation of a simply-supported beam by applying an external tensile force eeeeeeeeeeeeeeeeeeeeeeee Simulation of a simply-supported beam by applying an external compressive force .................... The mode shape of the first three modes .......... The variation of the natural frequency: the first Ede ......OOOOOOOOO......OOOOOOOO00.00.000.000... The variation of the natural frequency: the second .ode eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee The variation of the natural frequency: the third ”d3 0.00.0.0...O0.0....O0.0.0..........OOOOOOOOOO The mode shape of the first-mode at 0, 45, 180, and 225 degree of crank angle. Mechanism operating speed 185 rpm eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee The mid-span deflection of the connecting rod. Mechanism operating speed 185 rpm ................ xi 93 95 96 96 104 105 105 106 106 107 107 108 108 111 _._ tilt. l i I Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure .18 .19 .20 .21 .22 .23 .24 .25 .26 .27 The mid-span deflection of the connecting rod. Mechanism operating speed 225 rpm ................ The mid-span deflection of the connecting rod. Mechanism operating speed 258 rpm ................ The acceleration of the slider. Mechanism operating spggd 258 rpm .................................... The acceleration of the slider. Mechanism operating .pged 185 rpg eeeeeeeeeeeeeeeeeeeeeeeeeeeemeeeeeee The variation of the acceleration of the slider between the rigid-link and flexible-link of a slider -crank mechanism. Mechanism operating speed 185 rpm The variation of the acceleration of the slider between the rigid-link and flexible-link of a slider ocrank mechanism. Mechanism operating speed 225 rpm The variation of the acceleration of the slider between the rigid-link and flexible-link of a slider -crank mechanism. Mechanism operating speed 258 rpm The magnitude of the pin force in gudgeon-pin bearing. Mechanism operating speed 185 rpm ....... The force angle of the pin force in gudgeon-pin bearing. Mechanism operating speed 185 rpm ....... The variation of the pin force in the gudgeon-pin bearing between the rigid-link and flexible-link of a slider-crank mechanism. Mechanism operating speed 185 rp. 000......O0.000000............OOOOCOOOOOOO xii 111 112 112 113 113 114 114 115 115 116 Figure 3.28 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 The variation of the force angle in the gudgeon-pin bearing between the rigid-link and flexible-link of a slider-crank mechanism. Mechanism operating speed 185 [pl eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 116 Generalized coordinate of two adjacent links ..... 122 A schematic diagram of a compound articulating under gravitational loading prior to impating with a fixed mechanical stop .................................. 139 Photograph of the compound pendulum and stationary mechanical stop .................................. 145 Photograph of experimental compound pendulum showing the pin and the bearing .......................... 145 The experimental mid-span deformation of the compound pendulum. The initial angular position 60 degrees ..........................o............... 143 The theoretical mid-span deformation of the compound pendulum. The initial angular position 60 degrees 149 The theoretical angular velocity of the compound pendulum. The initial angular position 60 degrees 149 The theoretical angular position of the compound pendulum. The initial angular position 60 degrees 150 The experimental mid-span deformation of the compound pendulum. The initial angular position 30 “gr... eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 150 xiii Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 The theoretical mid-span deformation of the compound pendulum. The initial angular position 30 degrees The theoretical angular velocity of the compound pendulum. The initial angular position 30 degrees The theoretical angular position of the compound pendulum. The initial angular position 30 degrees (a)The slider-crank mechanism with flexible connecting rod showing dominant clearance at the gudgeon-pin bearing (b)Vector-loop model of the mechanism ............ Flow-chart for the computer simulation ........... Slider acceleration. Mechanism operating speed 185 rp- ......OOOOOOOOO...O.......OOOCOOOCCOOOOOOI.... Slider acceleration. Mechanism operating speed 199 mu ......IOOOOOOO0.00.000............OOOOOOOOOOOO Slider acceleration. Mechanism operating speed 225 rp- 00.000.00.000.0.00.00.00.00......OOOOOOOOIOOOO Slider acceleration. Mechanism operating speed 258 rpm .............................................. Connecting-rod tangential acceleration. Mechanism operating speed 135 rpm .......................... Connecting-rod tangential acceleration. Mechanism opgrating speed 199 rpm eeeeeeeeeeeeeeeeeeeeeeeeee Connecting-rod tangential acceleration. Mechanism operating speed 225 rpm .......................... Connecting-rod tangential acceleration. Mechanism operating speed 258 rpm eeeeeeeeeeeeeeeeeeeeeeeeee xiv 151 151 152 153 166 169 169 170 170 171 171 172 172 Figure 4.23 The mid-span deflection of the connecting rod. Mechanism operating speed 185 rpm oo-o------------ 173 Figure 4.24 The mid-span deflection of the connecting rod. Mechanism operating speed 199 rpm ................ 173 Figure 4.25 The mid-span deflection of the connecting rod. Mechanism operating speed 258 rp ---------------- 174 NOMENCLATURE External force imposed upon the system External force imposed on the j-th link Force imposed by link i on link j Impulsive force Length of the i-th link Mass of the i-th link Generalized mass The normal direction of the contact surface Generalized coordinates Generalized force associated with coordinate qk, including both conservative and nonconservative effects Generalized external impact force associated with the generalized coordinate qk xvi Generalized impulse associated with coordinate qk k 1': Position vector defining any point p of the kth link relative to the origin of the inertial reference frame OXY rQa Position vector of any contact point Q of the link a relative to the origin of the inertial reference frame OXY Ifi/p Position vector defining the relative location of point a and point p on link 1 T Total kinetic energy of the system m3 Angular acceleration of link j ‘p Holonomic or nonholonomic constraint equations AP p-th Lagrange multiplier (0) Time derivative 'k Position vector defining the center of mass of the kth link relative to the origin of the inertial reference frame * Denotes post-collision parameter or characteristic c Coefficient of restitution uh Angular velocity of link j W F12 Resultant force at crankshaft bearing I62 Mass moment of inertia of the crank about the gravitational center xvii Mass moment of inertia of the connecting rod about the G3 gravitational center m2 Haas of the crank m3 Mass of the connecting rod (including the mass of the bearings and the mass of the accelerometers) ms Mass of the slider r2 Length of the crank r3 Length of the connecting rod r4 Radial clearance at the Gudgeon-pin joint rG2 Length between the axis of rotation of the crank shaft and the mass center of the crank rGa Length between the crank-pin and the mass center of the connecting rod Tor Impulsive torque imposed on the crank during the impact-mode 02 Argument of the crank 03 Argument of the connecting rod 04 Argument of the vector representing the bearing clearance u Frictional coefficient between the slider and the guide W W xviii F(x) The axial force at any point of the link 1 The length of an element Mn" The mass matrix related to the inertia coordinate Man The mass matrix related to the inertia and elastic coordinate Huu The mass matrix related to the elastic coordinate ‘n The direction of impactive force Qk Generalized force associated with cooedinate qk, including both conservative and nonconservative effects r1 The posotion vector of any point p in the deformed body 1:01 The components measured in the moving frame of the position vector of the origin of the body axes relative to the origin of the inertial frame rki The position vector of point p in the reference state relative to the origin of the body axes Te The kinetic energy of any element uk The generalized elastic coordinates ux The axial deformation in the ox direction uy The transverse deformation in the oy direction {U} The nodal displacement of a element Va The potential energy of any element XA The component in OX direction of point A relative to the inertia frame YA The component in OY direction of point A relative to the inertia xxii [R] External forces on the elastic link To The kinetic energy of any element u1 The deformation displacement vector u1 The generalized elastic coordinates ux The axial deformation in the ox direction uy The transverse deformation in the oy direction {U} The nodal displacement of a element Ve The potential energy of any element 6 x/l p. The mass density of any element (') The spatial derivative with respect to x (') The second spatial derivative with respect to x ‘1 The ith generalized constraint equations (0) The first time derivative (') The second time derivative A The Lagrange multipliers conservative and nonconservative effects w The natural frequencies of the link The generalized coordinates in modal transformation Wale E The modulus of elasticity of the connecting rod p The mass density of the connecting rod The length of the crank shaft The length of the connecting rod The acceleration of the slider The nodal displacements The angular position of the compound pendulum in inertia frame The angular velocity of the compound pendulum in inertia frame Denotes a row matrix or a column matrix The crank angle The argument of the connecting rod The crank angular velocity The angular velocity of the connecting rod The crank angular acceleration The angular acceleration of the connecting rod The mass density of the connecting rod xxiv Chapter I INTRODUCTION 1 . 1 MOTIVATION Mechanisms and mechanical systems with discontinuities in their kinematic chains have made significant contributions to many diverse fields of engineering, ranging from the military and aerospace environments to computer-integrated-manufacturing environments. Typically these machine systems incorporate gears, geneva mechanisms, linkages, articulating levers, ratchets, and bearings with radial clearances. While these systems are frequently incorporated in many products in these diverse marketplaces, the dynamic phenomena associated with the discontinuities in these kinematic chains cannot-be predicted using classical design methodologies. This observation, coupled with the dearth of design guidelines of this field, and also the importance of these mechanical systems have motivated the research program reported in this thesis. Upon reviewing the frequency with which these mechanical systems and kinematic pairs with discontinuities are featured in a broad range of mechanical systems in the diverse military, advanced manufacturing, automotive and industrial products, it is clearly evident that articulating systems with bearing clearances are the most frequently encountered class of mechanical system. Bearing clearances are attributed to wear in the elements of anti-friction rolling contact multig-element precision bearings, or due to the prescribed functional clearance in bushings and journal bearings. The magnitude of this radial clearance is typically of the order of a few thousandth of inches. Hence, in general, the displacement characteristics of a mechanism system with bearing clearances differ from those developed by the same mechanism system without clearances by only a small percentage of the total path generated by a specific trace point on a link. Of course this may render the output totally unacceptable in the case of a precision mechanism, and mandate the deployment of expensive precision preloaded bearings'which do not featuring clearance. However, the associated velocity and acceleration characteristics, and also the associated dynamical effects created by the presence of bearing clearances, are generally dramatically different from the zero-clearance case, consequently the performance of the machine is generally drastically modified by the presence of clearance in the joints of the system. The most important phenomena are generally those associated with impact forces in the joints and also dynamical stresses in the structural members, which are generally of a much greater magnitude than those of a system without clearances. The large impulsive forces developed in these systems with discontinuities result in the accelerated wear of the bearings and also the development of a significant vibrational response of the mechanism which can result in high levels of noise radiation and the link members are exposed to a more severe fatigue environment. Moreover, these vibratory motions-of the links generally contaminates the kinematic output-characteristics of the mechanism.which is especially undesirable when these characteristics are the principal output of the system. Recent reports from NSF-sponsored studies on the state-of-the- art of 11.8. research on mechanical systems have clearly identified the significant void in the field of mechanical systems with bearing clearances for coherent analytical, computational, and experimental investigations on the dynamic response of these systems [3,6]. The comprehensive work presented herein is directed towards partially filling this void in the literature, by proposing a phenomenological model of a revolute joint with radial clearance which captures the four principal behavioral modes of the dynamical response of these systems prior to evaluating the predictive capabilities of this model by comparing the theoretical results with experimental results from a complementary laboratory study. These modes are classified, herein, as the contact-mode, the free-flight-mode, the impact-mode, and the transient-mode. By capturing these behavioral modes in a mathematical framework, the dynamical response of generally mechanical systems with bearing clearances can be accurately-predicted. Traditionally, mechanism design has been based on rigid-body analyses in which all mechanism members are treated as rigid bodies [66,74]. Stresses in the mechanism members due to inertia forces and external loading are then calculated in a sequential manners using quasi-static analyses. In the past two decades, mechanisms have been designed to operate at higher and higher speeds in order to increase industrial productivity. A consequence 'of this trend is that the members of these systems can no longer be treated as rigid bodies since initial 4 leads are dominant at high speeds [26.52.593.89]. In order to reduce the inertial loading imposed on the machine, the weight of mechanism members need to be. made as light as possible without sacrificing structural integrity. However the lighter members are more likely to deform and vibrate due to the inertial and external forces. The inaccuracy of the position, velocity, etc. due to thelarge elastic deformation and occasionally due to the resonant. condition will cause the design unacceptable. Hence the elastodynamic analysis of a mechanical system becomes necessary and important. On the other hand, if the links were fabricated with lightweight and high-stiffness materials, the vibrational motion of links can be reduced. Some pioneering research [82,90,911 has been undertaken on fabricating mechanism links in composite materials. Furthermore, if clearances are presented in the bearings of the mechanism, then the impact forces in the joints will increase thereby increasing the acoustical radiation from the machine and also reduce the life of parts which are subjected to a more severe fatigue environment [20,87]. . The dearth of experimental investigations in the field of mechanism systems incorporating joints with radial clearance, and also the significant void in the literature for publications presenting coherent combinations of theoretical and experimental work 'in the field, have motivated the comprehensive investigation reported herein. This investigation presents a methodology for analyzing the elastodynamical response of planar linkage systems with bearing clearances in the joints. A four-mode model is employed to model the dynamic behavior in each. joint with clearance and a multi-step Gear algorithm is employed in the computer simulations. A complementary experimental investigation of a slider-crank mechanism with prescribed radial clearances in the gudgeon-pin bearing is undertaken in order to furnish viable laboratory data with which to evaluate the predictive capability of the proposed approach. 1.2 LITERATURE REVIEW The branch of scientific analysis which deals with motions, time, and forces is called mechanics. and is made up of two parts, statics and dynamics. Statics deals with the analysis of stationary systems, i.e'., those in which time is not a factor, and dynamics deals with systems which change with time. Dynamics is also made up of two major disciplines, kinematics and kinetics. Kinematics is the study of the relative motion of machine parts in which the displacement, velocity, and acceleration are considered while kinetics deals with the forces acting on the parts of a machine and the motions resulting from these forces. I A kinematic chain is a system of links, which are either joined together or are in contact with one another in a manner that permits them to move relative to one another. If one of the links is fixed and the movement of any other link to a new position will cause each of the other links to move to definite‘predictable positions, which generate a desired function, the system is a constrainedkinematic chain. A linkage mechanism is a constrained kinematic chain. A linkage mechanism is called planar linkage if all the intended motions are parallel to a single plane. The following literature review can be divided into four distinct sections in which different model classifications are proposed to analyze this class of mechanical system. These classifications are dependent upon whether the link members are assumed to be either rigid or flexible bodies, and whether or not the joints are assumed to contain clearance at the pin- journal interface. 1.2.1 Mechanisms with rigid-links and featuring beari'ngs without clearances Traditionally [55.66.74.100] the kinematic analysis of linkage mechanisms was undertaken by assuming that the links of linkage mechanisms were treated as rigid-bodies and without considering the effects of bearing clearances. The kinematic solutions were obtained by using the graphical method. and then analytical method for simple. cases. The forces. torques on. links were then obtained based upon the kinematic data. Since the traditional graphical and analytical methods are only available for simple cases of linkage mechanisms and because of the broad usage of computers, numerical solutions for linkage mechanisms were then developed. Chace. et.al. [6.7.8] made significant contributions to the development of viable methodologies for predicting the dynamic response of mechanical systems by employing Lagrange's equations to simulate the dynamical behavior of highly constrained. multi-degree-of-freedom'mechanical systems without considering bearing clearances. This pioneering research resulted in the development of a commercially-available simulation software tool. DRAM which is used extensively throughout U.S. industry and to a less extent Europe [11]. Uicker, et al. [73] developed a general purpose program, IMP. which gained wide acceptance in the academic community. but failed to be marketed commercially. Haug. et al. [63]~developed‘a package similar to DRAM entitled DADS, which is beginning to emerge in the commercial marketplace. Paul [64] presented an overview of the various principles and techniques proposed for formulating the equations of motion of mechanical systems. prior to seeking the appropriate numerical schemes for solving this class of dynamical problems. Kane and Levinson [Ali] addressed the complexity of multibody dynamics problems. in which the equations of motion were formulated by, both Lagrange's equations and the principle of virtual work. The results of these two methods were then compared. They concluded that the latter formulation was more laborious than the former formulation, and also lead to equations having a superior form. for purposes of numerical integration. In fact. to choose a viable method for the formulation~and rapid solution of the equations of motion of mechanical systems is not an easy undertaking since is is determined by seeking minimal computing time. complexity of the equations of motion. desired characteristics. . . . . etc. . 1.2.2 Mechanisms with rigid-links and featuring bearing clearances The serious undesirable effects of bearing clearances on the dynamic behavior of high-speed mechanical systems. have motivated numerous theoretical investigations on the different dynamical phenomena associated with the behavior of revolute joints [37]. However. in sharp contrast to the large number of analytical and computational studies in this area,‘ the academic community has not developed experimental programs to further motivate these purely theoretical studies. and only in a limited number of cases have combined theoretical and experimental studies been. undertaken. The field is. therefore. somewhat, i-ature and not well developed. Garrett and Hall [30] investigated the effect of tolerance and clearance in linkage design. in which the effects of these tolerances and clearances are presented in the form of mobility bands for the linkages. Kolhatkar and Yajnik [50] analyzed the errors caused by play in the revolute joints of a precision function generating linkage. in which the error occurring in the output angular displacement of a four- bar linkage is shown to be maximum for a given angular input when the vectors representing the clearances are parallel to the coupler. Earles and Wu [23] who employed a modified Lagrange's equation approach in which constraints were incorporated using Lagrange multipliers in order to predict [the behavior of rigid-link mechanism with clearance in a bearing. The clearance in the bearing was modeled by a massless imaginary link. but the simulation was restricted to only predicting when contact between the pin [and the journal was terminated. Later Earles and Wu investigated the prediction of contact-loss in a bearing of a linkage mechanism [24,25,105] in which they proposed that contact- loss would not occur if Q/R < 1, where R is the magnitude of minimum bearing reaction force and :1 is the rate of change of R which were obtained from a zero-clearance analysis of the mechanism. Wilson and Fawcett [102] assumed clearance exist in the sliding bearing in a slider-crank mechanism in which a theoretical investigation into the effects of parameters. such as the geometry. speed and mass distribution of the mechanism. upon the transverse motion of the slider was reported. In their simulation. the theoretical trajectories of slider corners were evaluated. Eventhough the experimental results were not shown in the paper. they reported that 0.4 of the coefficient of restitution was chosen for the simulation. since the theoretical results agreed best with. the experimental results on this choice. The dynamic effects of bearing clearances in mechanisms can cause [increased bearing load and noise [13.37.80.87] . These undesirable effects occur when contact is remade in a bearing after relative motion of the two parts of the bearing within the clearance space. Thus if loss of contact could be prevented, the impulsive loading which would normally occur would be eliminated. Under this concept. Grant and Fawcett proposed a method to prevent contact loss between the pin and journal of a revolute joint with clearance [35]. Their experimental results verified the approach for a limited class ofsystems [36]. but the method is not universal [37] . Mansour and Townsend [54,92] modeled a four—bar crank-rocker mechanism with clearance as two sets of compound pendulums in a theoretical study. They ignored the motion in contact- mode entirely, a close succession of small pseudo-impacts were assumed for the simulation. Subsequently. Miedema and Mansour [59] extended their previous two-mode model. for the free flight and impact modes. to a three-mode model in which a following mode was proposed. In their numerical simulations the following-mode was always assumed to occur innnediately after the impact-mode. however, this is frequently not observed in practice [37]. Dubowsky and Freudenstein . instead of the use of coefficient of restitution. formulated an impact. pair model to 10. predict the dynamic response of an elastic mechanical joint with clearances [12.13]. In their model. springs and dashpots were arranged as Relvin-Voigt model [33]. This work was subsequently extended and presented in references [14-16.21]. Recent work by Haines [38-40] and Bengisu'. et al.[3] has emphasized the prediction of contact-loss in mechanisms with clearances. Haines constructed a design chart from the ‘numerical solutions of the equations of motion with the assumption that during the noncontact phase the locus of the contact force is a straight line. but this assumption is not universally viable as shown by Bengisu, et al. [3]. Bengisu. et al. developed a separation parameter for a four- bar linkage mechanism which was based on a zero-clearance analysis. The theoretical results are compared with the experimental results and showed qualitatively agreement. However. the generality of their formulation is currently unknown. ll [ start [ design specifications: stresses. velocities accelerations. desired coupler curve. ... etc. I type synthesis] l kinematic analysi;] flint-id selec th fit" of “Eh-- pin forces and stress analysis unsatisfied compare the analytics results with the desi specifications satisfied [bearing selectieh] [fanufacturing processesl ® Figure 1.1 Classical design of linkage mechanisms 12 start design specifications: stresses. velocities accelerations. desired coupler curve.... etc. [type synthesie1 kinematic analysis (rigidolinkad_system) obtain ( R) elastodynamic analysis based upon the finite element equations [Kliul + [61(6) + (K](u) - [MllRi F—-{material selectien}—~i - pin forces and stress analysis unsatisfied unsatisfied 4———{sh£pe of link;}-——_.. compare the analytica results with the design specifications satisfied [bearing selectioh‘] lmanufacturing processee] @ Figure 1.2 First generation of analysis techniques for elastodynamic design of flexible mechanisms. Two-pass method l3 [ It!!! ] design specifications: stresses, velocities accelerations. paths [type synthesie] elastodynamic analysis based upon the coupled equations of motion .——:Lma‘terial selection}——q -——{thp9 of 11E]..— pin forces and stress analysis unsatisfied compare the analytics results with the design specifications unsatisfied satisfied [bearing selectioh] [manufacturing processes] Figure 1.3 New analysis techniques for elastodynamic design of flexible mechanisms. One-pass method 14 1.2.3 Flexible-links in mechanical systems without bearing clearances Traditionally, mechanisms have been designed on the basis of the assumption that the members in mechanism are rigid-bodies. Thus, only a geometrical relationship between the input and the output is required and. therefore. once the input is specified the output is completely determined. However the linkages may undergo severe elastic deformations due to its own inertia when operating at a high-speed condition and due to the external loads. In order to achieve higher degree of accuracy of the performance. the elastic deformation of the meflers in a mechanism have to be taken into consideration. The methodologies proposed for analyzing flexible-linkage mechanisms may be classified into two major groups: the one-pass and two-pass methods. The one-pass method models the inertia motions. which is relative to the inertial frame, and the elastic motion. which is relative to the moving frame. of the} link simultaneously. Based on this method, a system of coupled differential equations are obtained. The two-pass method needs two sets of analytical systems. Before analyzing the vibrational response of flexible links. the mechanism system is analyzed as if it were rigid and thus yielding the inertial characteristics. These inertial characteristics are then become the external and inertia forces of the flexible links, upon which the responses of the flexible links are estimated. The design procedures of linkage mechanisms can be clearly shown by the following three schematic figures. Figure 1.1 presents the classical design procedures of linkage mechanisms. Design 15 specifications. such as maximum working stresses, velocities. accelerations, desired paths. . . . etc. , are requested. Accordingly, a designer has to synthesis the proper linkage mechanism. and then rigidw ‘body analysis is executed.to find the kinematic properties. Upon selecting the proper material and shape of links. the pin forces in joints and stresses in the link can be analyzed. By comparing the analytical results with the expected specifications. the material and shape of links may be modified to improve the design. Finally; the design procedures are fulfilled by choosing an economic and efficient manufacturing processes. Figure 1.2 presents the first generation of analysis techniques for elastodynamic design of flexible mechanisms. i.e. . the two-pass method. According to the design specifications. the designer has to synthesis a linkage mechanism to generate the desired paths and functions. Kinematic analysis of this linkage mechanism is done as if it were rigid. Upon selecting the proper material and shape of links. the elastodynamic analysis of the links is accomplished by incorporating the kinematic properties which already known in the previous procedure. The global characteristics of the linkage mechanism are obtained by superposing the elastic characteristics on the inertial characteristics. Based on the global characteristics, the pin forces and stresses of links are calculated. By comparing the designed characteristics with the design specifications, the design is completed by choosing a proper manufacturing processes. Figure 1.3 presents the procedures of one-pass method. Again, according to the design specifications, the designer has to synthesis a linkage mechanism to generate the desired paths and functions. A system of coupled differential equations is formulated. The global characteristics of the l6 linkage mechanism are obtained by solving the coupled governing equations. The pin forces and stresses of links are calculated. The design problem ending with a proper choice of manufacturing processes. The following literature review can be divided into two categories according to two different approaches. two-pass method and one -pass method. W Neubauer. Cohen and Hall [62] investigated the transverse vibrational characteristics of the connecting rod in a slider-crank mechanism by neglecting the axial vibrations and friction. the Coriolis component of acceleration, the additional tangential acceleration and normal acceleration due to the transverse deflection of the connecting red but external loading on the piston was considered. Winfrey [103] investigated the elastic behavior of flexible link in a mechanism by using the assumption of superposition of gross rigid-body motion and a small elastic deformation. An investigation of the vibratory bending responses of the elastic connecting rod of a slider-crank mechanism was studied by Viscomi and Ayre [97] . in which the longitudinal vibration of the link was neglected and the nonlinear equations of motion was solved analytically. In his study. he concluded that within the frequency range of interest. the second mode is relatively unimportant and the response is closely approximated by the first mode; At low frequency ratios. where the response is small, the linear and nonlinear equations gave closely similar results. However. in regions of large response. the solution are quite different. Erdman. Sandor and Oakberg [26] employed . ' '.:l1l'm. l7 kineto-elastodynamic method for. the analysis and synthesis of mechanisms. in which the effects of elastic deformation upon the inertia forces are included. Sadler and Sander [68.69] developed a method of kineto-elastodynamic analysis by employing lumped parameter model for simulating moving mechanism components subject to elastic bending vibration. In their studies. the higher order acceleration terms, such as the Coriolis. normal and tangential accelerations. were taken into consideration but buckling effect of the rigid-body axial forces. Chu and Pan ’[9] studied the transient dynamic response of an elastic connecting rod in the slider-crank mechanism in both the transverse and longitudial direction on the basis of the ratio of the length of the crank and length of the connecting rod. crank speeds, viscous damping and natural frequencies. In their results, the solutions were compared by using a Runge-Kutta method and a Piecewise Polynomial method. Midha . Erdman and Frohrib [56] employed displacement finite element method to develop the mass and stiffness properties of an elastic link. They derived the nodal displacements and accelerations by including the terms coupling the elastic and rigid-body motions. However they ignored the Coriolis. normal and tangential acceleration terms when formulating kinetic energy and no buckling effect was considered. In their another study [57], a nmericsl procedure based on an iterative scheme was employed for obtaining an approximate particular solution from equations of motion of an elastic link with small damping and at subresonant speeds. Hath and Ghosh [60] employed finite element method to_analyse general flexible mechanisms in which the Coriolis. tangential and normal components of elastic acceleration had been derived for a moving link and also included the geometric effects of axial forces. 18 Turcic and Midha [93] developed a generalized equations of motion for an elastic__mechanism system by employing displacement finite element method, in which none of the higher order acceleration terms were neglected. By improving their equations of motion. a geometric stiffness matrix was added to the stiffness matrix of the equations of motion by sealing a constant axial force [9b]. By comparing the theoretical results and experimental results. the improved equations of motion showed better results [95]. Thompson and Barr [84] employed a variational principle to systematically develop equations of motion of members of elastic mechanisms. Thompson and Ashworth investigated the resonance in planar mechanisms mounted on vibrating foundations [83,85] .. in which an average perturbation method was employed. Gandhi and Thompson [29] formulated the finite element equations of motion by using a mixed-variational principle. prior to solving the equations of motion by employing the Hewmark method. Based on a previous variational principle [81] . a nonlinear finite element analysis of flexible linkages was derived by Thompson and Sung [88]. Having reviewed the literature on two-pass method, the attention now focuses upon reviewing the one-pass method. W The one-pass method accounts for the coupling of elastic and inertial characteristics. which is analytically precise and, of course, the equations of motion are more coqlicated than the two-pass method. Song and Haug [78] treated each member in a mechanical system as being 19 flexible and combined the kinematic constraint equations and also the equations of motion to obtain a coupled system of equations prior to solving the problem using an implicit numerical integration method. The effect of the geometrical nonlinearity was not included in their formulation. neither was the material damping. The authors. analyzed a slider-crank mechanism with a flexible'connecting rod. All the members were assumed rigid. This single flexible link was treated by modeling it using one. two and three finite elements using a displacement model. Their results were not verified by, experimental data. Indeed, a principal void in this area is the shortage of experimental work to evaluate the predictive capabilities of the analytical and numerical formulations developed. Only the following reports have been found cowering their numerical results with the experimental data : Alexander and Lawrence [1], Thompson-Gandhi group [86.88] and mom, a: .1. [95]. 1.2.10 Flexible-links in mechanical systems with bearing clearances Intermittent motion occurs in many practical mechanisms. in which jump discontinuities manifest themselves in the kinematic chains. Some mechanisms are for manufacturing equipment in what the primary function is for punching. cutting, milling, . ... etc.. Under these conditions intermittent motion. or impact. will routinely occur during their normal usage. Some examples such as rachets. geneva mechanisms. the touch down of airplanes, the gap between two matched gears, and clearances in joints will cause intermittent motion. A typical change for the mechanism in the intermittent motion occurs in the mathematical 20 model of these systems when there is a change in the constraints and iqulsive forces may occur as well. Dubowsky and Gardner [17] investigated the effects of clearances andlink flexibility on the stresses in joints of high-speed mechanisms. They extended their former impact pair [15] to impact beam model (IIM) and solved‘the problem by employing displacement functions which is the normal mode shape of vibration. Letter on, they studied the dynamic behavior of general planar mechanism with elastic links and multiple clearance connections by using a perturbation coordinate approach [18] . The bearing forces in a scotch yoke mechanism was studied. Dubowsky and Moening [l9] experimentally verified the usefulness of their impact beam model. Deck and Costello joined Dubowsky's work and extended his model from planar mechanism to spatial mechanical system [22]. in which finite element technique was incorporated. In recent years. the. intermittent motion has been studiedpby several researchers. Wehage and Haug [99] studied the dynamic analysis ofsystems with impulsive forces. impact. discontinuous constraints. and discontinuous velocities. in which momentum-impulse relations were obtained by integrated .the Lagrange's equations with Lagrange multipliers [33]. and the coefficient of restitution was incorporated in the equations of motion. A weapon mechanism and a trip—plow were simulated while assuming all links were rigid. They also presented a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for mechanical systems [98] . The dynamic analysis of spatial substructures .with large angular rotations has been studied by Shabana and Wehage [71] . The mode 21 superposition [method was employed for the solution and the eigenvalue problem was solved only once even though the resulting set of differential equations is highly nonlinear. In their studies [71, 98. 99]. the geometrical nonlinearity has‘never been considered, thus. the natural frequency is constant during the simulation. which is a contradiction to the author's experimental results presented in Chapter Three. Dy using the same formulation. Shabana and Wehage [70.72] studied the dynamic response of inertia-veriant flexible mechanical systems. The same technique was applied on the transient dynamic analysis of a vehicle system which was composed of interconnected rigid and flexible bodies in a paper by Kim. Shabana and Haug [48]. Khulief and Shabana [66,47] investigated the impactive responses“ of constrained systems of rigid and flexible bodies by employing the Rayleigh-Ritz method. in which a planar slider-crank mechanism and a crank-rocker mechanism are studied. Hone of the material damping and the effect of the geometric nonlinearity were considered. The effect of geometric nonlinearity is first incorporated into the equations of motion by Bakr and Shabana in 1987 [2]., in which its effect on the impact response of- flexible multi- body systems was studied. However. . they treated the axial force as a constant. Haug. Wu and Yang presented a uniformed formulation of the equations of motion for constrained multi-body system with Coulmfl: friction. stiction. impact and constraint addition-deletion [£1,106,107]. None of the theoretical results prescribed in publications by Haug or Shabana are verified by experimental data. Hence a question of credibility exists when modeling such complex phenomenological systems . ' 22 The field of research on mechanical systems with flexible links and incorporating bearings with radial clearances has been reviewed. it is clear that in order to predict the elastodynamic behavior of limge mechanisms. the higher-order coupling terms like Coriolis acceleration. normal and tangential accelerations (he to the transverse deflection of the link. cannot be ignored. Furthermore. it is hypothesized that the geometric nonlinearity. or [buckling effect, must be consideredin the formulation. Humorous research studies have been done on the field of mechanical systems with clearances using analytical methods. finite element methods..veriational principles. or perturbation method. Moreover. it is clearly evident from this literature review that name of the current techniques are able to model all of the phenomena while manifest themselves in the behavior of mechanism systems featuring bearings-with radial clearances. In addition. with the emphasis on purely theoretical studies in the literature. it is clearly evident that a coherent codination of analytical, computational. and'experimental work be undertaken in order to. establish a viable design guidelines and direction for subsequent studies in this important field. 1.3 THESIS OUTLINE The modeling strategy. which is an iterative process [33.76] , presented in this thesis can be represented in the block diagram of Figure 1.4. The start is the recognition of the problem and the reviewing of the existing researches on this problem. a new mathematical model for this problem is, then proposed. By evaluating the existing and expecting equipments, the test problem is defined. The numerical .results 23 of the proposed mathematical model are then compared with the experimental data. The mathematical model is adjusted by adding more asstqtions until the cowarison is satisfied. I One-pass method and the finite element method are eqloyed to formulate the highly coupled nonlinear equations. of motion by incorporating the Lagrange's equations and Lagrange multipliers. All of the higher-order coupling terms are included in the equations of'motion, the axial force for the geometric matrix is linearly assmed. which is different from other reports. thus, generates two geometric matrices. During intermittent motion in a mechanical system the mathematical formulation mandates the-incorporation of different constraints to model the complex phenomenal behavior of the elements. The linkbeing treated as pin-free. beam during free-flight-mode and as simply-supported beam during contact-mode. ‘ In Chapter Two, model of a general planar rigidolinked mechanism with hearing clearances are derived. A four-mode model is proposed to describe the dynamical response of the mechanism. The criteria for the contact loss. termination of the‘fre‘e-flight-mode and restoration of the contact-mode. are also defined. Finally. a slider- cramk mechanism with clearance in the gudgeon-pin joint is considered as an illustrative example. Thehtheoretical and experimental results are then compared. Correlation is demonstrated very favorable between the two sets of results In Chapter Three, .the governing equations for moving links are derived by using a displacement finite element method which combines the kinematic constraint equations and the equations. of motion to obtain a coupled system of equations including both the rigid-body motion and the ‘¥—#—_7— 24. elastodynamic responses of flexible links. The higher-order coupling acceleration terms such as the Coriolis acceleration. normal and tangential accelerations due to the. transverse deflection of the link hrs included. The buckling effect of the axial force is included in the geometrically-nonlinear stiffness matrix. The eigen values. or natural frequencies. are found to vary with time and this is verified experimentally. Finally. a slider-crank mechanism with a flexible connecting rod without bearing clearances is considered. The theoretical responses of the link are compared with the experimental results. The variation of the dynamical characteristics between rigid link and flexible link are also calculated. . - In Chapter Four. mechanical systems with flexible links and joints featuring bearing clearances are studied. The mathematical model cofiines. the modelling techniques which presented in Chapter Two and in Chapter Three. and therefore a modified governing equation is obtained. The criteria and the equations of motion for four-mode model are dependent upon the deformation of the links. Two examples 'are studied.- The first example features a compound pendulm. which incorporates a flexible link. whose tip impacts against a stationary mechanical step during its motion. The theoretical elastodynamic responses of the mid-span of the link and the angular position are compared with the‘ experimental results. The second example features a. slider-crank mechanismwith flexible connecting rod incorporatingclearance in‘the gudgeon-pin joint. The experimental and theoretical results are also cowared. , - ;:- O .‘l e“ I '50:: {Hg .. ‘ :.E;c 25 ( start ) lhypothesisl [define test problea }————— ‘ codify ‘ predict theoretical experiaental hypothesis results results not good coapare acceptable Figure 1.4 Modeling strategy 1 . 4 PRINCIPAL CONTRIBUTIONS 1. An analytical model was dynamic developed for predict the response of mechanical systems featuring both rigid and flexible links with joints incorporating clearances. 2. Experimental works have been done on a mechanical system featuring both rigid and flexible links with and without radial clearance in the reVOlute joint. These works not only partially fill the void of the field, but motivate and guide the development of theoretical models of 26 with prescribed radial clearance in the. joints of the systems, but these results provide lotivation and guidance in the future development of theoretical models of mechanical systems featuring flexible links with radial clearances. 3. A criterion has been established to predict the restoration of the contact-mode. This has not been addressed before in the literature. A four-mode model was developed in order to predict the behavior of a joint with radial clearance. 4. The experimental investigation indicates that the damping ratio in a link is dependent not only on the. material, but also on the axial forces imposed on the link. ‘ 5. A quasi-static procedure to solve the coupled system of equations is proposed. ' 6. When the geometrical effect was added, the axial force was modelled as a linear function of displacement. This generated an extra geometrical matrix, K62 in equation (3.48), which differs from the traditional constant axial force assumption. 7. The initial values for simulating a moving flexible link are determined by the first mode of the link, and the coefficients for the displacement, velocity and acceleration are found from experimental results. This procedure is quite different from previous theoretical publications where uo - {to - uo - 0 were given. Chapter II GENERAL PLANAR MECHANISMS FEATURING RIGID-LINKS WITH RADIAL CLEARANCES IN THE REVOLUTE JOINTS 2 . 1 INTRODUCTION Bearing clearances cannot be avoided in mechanical systems unless extremely stringent, and hence expensive, criteria are imposed upon the design, engineering and manufacture of the joints of these systems tolerances and also upon the wear in these kinematic pairs. Bearing clearances are, therefore, a common feature in every large percentage of articulating mechanical systems developed for the industrial and commercial marketplaces. Even though the displacement characteristics of these mechanism systems with bearing clearances frequently differ from the. displacement characteristics of the zero clearance system by only a small percentage of the overall characteristic, some other kinematic characteristics, such as the velocity, acceleration and jerk profiles, are often dramatically different from the zero-clearance case and these characteristics frequently results in the development of unacceptable products. In this chapter, the equations of motion for a general planar mechanical system 27, 28 with hearing clearances are developed and a four-mode model is proposed and developed to describe the dynamic response of bearings with radial clearance. A slider-crank mechanism with radial clearance in the gudgeon-pin joint is employed as an illustrative example to demonstrate the proposed methodology . 2.2 WEI. OF A GENERAL PLANAR MECHANISM WITH BEARING CLEARANCES The dynamical response of a general planar mechanism system incorporating revolute joints which feature a distinct radial clearance, as shown in Figure 2.1, is considered herein by modeling the system 'as an assemblage of interconnected rigid bodies by employing the mathematical model presented in Figure 2.2. The position vectors shown in Figure 2.2 not only define the position and orientation of the bodies but they are also employed to describe the behavior of the joints in the system. The pin and journal is assumed well-arranged in the same plane, and thus no out-of—plane motion is concerned. Collectively, these vectors impose kinematic, or geometrical, constraints on the motion of the bodies. Utilizing the classical notation for generalized constraint equations between system coordinates, such as those presented in references [42] and [51] for example, then a set of i generalised constraint equations ‘1 may be written ‘1 - ‘(qlo an°nuo (IN) " 0 (2.1) where there are N generalized coordinates qN. 29 Figure 2.1 Radial clearances in the revolute joint in a planar mechanism system Figure 2.2 Definition of the position vectors and reference frames modeling a planar mechanism system with hearing clearances _' “L. L equ and pr: as: N 5°‘ C01 “1' in fo in 30 In general, the minimization of functionals with constraint equations can be achieved by employing the Lagrange multiplier method and the penalty function method. [65]. The penalty function method involves the reduction of conditional extremum problems to extremum problems without constraints by the introduction of a penalty function associated with the constraints, while the method of Lagrange multipliers offers a viable approach to developing the equations governing the response of a general mechanical system when the i constraint equations cannot be conveniently manipulated to express 1 of the generalized coordinates in terms of the (N-i) remaining generalized coordinates. This classical method [43,52] involves modifying the Lagrangian equations of motion for an unconstrained system of bodies, by introducing additional terms which are the product of a Lagrange multiplier and the derivative of the associated constraint equation. By incorporating the second time derivative of equation (2.1),. the Lagrange multiplier approach yields a set of algebraic and differential equations governing the dynamical response of the system. These equations are generally cast in a matrix format prior to solving for the Lagrange multipliers and the acceleration terms which are subsequently integrated to obtain the velocities and displacements of the system. The Lagrange multiplier method readily enable the joint forces of the system to be calculated and which is an essential ingredient of any methodology for predicting the dynamical behavior of mechanical systems with hearing clearances. 31 free-flight mode impact mode transient mode Figure 2.3 Schematic diagram presenting the four-modes describing the phenomenological behavior of a revolute joint with clearance A four-mode model is proposed herein for predicting the dynamical response of planar articulating mechanical systems with bearing clearances. The form of this model is motivated by experimental dynamical response-data presented herein in a subsequent section. Upon reviewing these experimental results, it is clearly evident that four distinct phenomena manifest themselves in the planar motion of these dynamical systems. These four modes can be conveniently classified as the contact mode, the free-flight-mode, the impact-mode and the transient-mode. These modes are schematically presented in Figure 2.3 which illustrates the typical dynamical behavior exhibited by a revolute joint with clearance. 32 (i) Contact-Node: In the contact-mode, the pin and the journal are in contact and the relative motion between these members of the joint is assumed to described by a sliding motion. Experimental evidence suggests that rolling does not occur and hence it is not considered in the modeling process. Furthermore, the contact surface is assumed to be smooth and frictionless, thus there is no energy loss in this mode. Since the principal mefl>ers of the revolute joint, the pin and the journal, cannot transmit a tensile force across the pin-journal interface in the contact-mode, this mode is assumed to terminate when the contact force between the pin and the journal becomes tensile in the computer simulation. Clearly in practice this mode is terminated at the instant when the pin and journal separate, however the computer simulation is based on a time-step philosophy, where the state of the system is only known at discrete times, therefore, in order to determine the precise configuration in which the contact force is zero, this time instant must be known prior to initiating the simulation. As a compromise, in order to avoid utilizing even smaller step sizes, the simulation software package was formulated to trigger the change from the contact-mode to the free-flight- mode when the contact force in a joint reached a small fictitious tensile state. (ii) Free-Flight-Hode: In the free-flight-mode, the pin and the journal in a specific joint are not in contact, hence there are no reaction forces between these two members. Under these conditions, a complex multi-element system can be treated as several compound subsystems of articulating 33 bodies which do not interact with each other and whose initial dynamical state is obtained from the simulation results at the termination of the previous mode. Thus for exaqle, if a slider-crank mechanism is assumed to feature significant clearance in the gudgeon-pin bearing, the system would be treated as two independent dynamical systems during the free- flight-mode. One system would comprise two compound pendulms and the other system would comprise-a translating block. This mode would be terminated when a geometric constraint mandated the triggering of a different mode. (iii) Impact-Diode: The sudden contact of the pin and the journal which generally occurs at the termination of the free-flight-mode triggers significant impactive behavior. This behavior is the basis for the impact-mode. This contact, which occurs in an extremely short time interval, is characterized by a discontinuity in the kinematic and kinetic characteristics, and a significant exchange of momentum occurs between the two members. Herein, this phenomenon is modeled by invoking a momentum-exchange approach which incorporates the coefficient of restitution. This coefficient, which captures the material properties of the impacting bodies, is dependent upon the stiffness of the contacting materials, the roughness of the contact surfaces, the geometrical shape of the impacting bodies, and also the velocities of the bodies prior to impact. The value of the coefficient of restitution is generally determined experimentally. (iv) Transient-lode: The transient-mode begins at the first impact between the pin and the journal in a specific joint, ,and ends with the restoration of 34 contact between these members. Thus this modegenerally comprises a sequence of several free-flight-modes of progressively smaller time duration and also several impact-modes. Herein the simulation software requires the pin and journal to always separate and enter the free- flight-mode after completing the impact-mode unless the time duration of the free-flight-mode is less than a prescribed extremely small time- interval and also the original point of contact differs from the subsequent point of contact by a very small prescribed quantity. Upon satisfying these criteria, control in the algorithm then transfers to the contact-mode. Although the duration of this mode is relatively small compared with the overall cycle of the pin-journal interactions, this intermittent motion behavior is clearly evident from the experimental results presented herein. Having described the phenomenological aspects of the problem, attention now focuses upon capturing this behavior in a mathematical framework prior solving the governing equations. The subsequent sections present the mathematical apparatus for modeling each mode. 2.3 GOVERNING EQUATIONS FOR CONTACT-MODE The equations of motion of mechanical systems with constraints and bearing clearances are formulated herein by employing the generalized Lagrange's equations in which the Lagrangian multipliers are utilized in order to incorporate the system constraints into the equations of motion. In this case the radial clearances in the revolute joints are represented by position vectors, and only compressive forces 35 are permitted to develop at the pin and journal interfaces in these joints. If a dynamical system is characterized by s holonomic constraint equations, then these constraints can be incorporated within the generalized Lagrange's equations and written as s 3‘ sL £1. £1. E _2 dt [8;1k aqk k 1 p aqk where ‘p represents the p-th holonomic constraint equation. The term A J g: in equation (2.2) is a generalized force which imposes the j-th constraint which is associated with the k-th coordinate. Each constraint thus contributes an additional generalized force to each of the equations of motion. If r clearances are present in a general dynamical system with 1 links, then at any time instant, t, the position vector of the mass center of the arbitrary k-th link can be represented by the position VOC COT pk 'k - ~(q1.q2,000,qn), where N-l + r. (2-3) Thus, the velocity of the mass center of the k-th link may be written 36 N 1:: ° -E a, (2.4) ’1: 1—1 sq, 1 The kinetic energy of the mechanical system may, therefore, be written 1 n. N . . l O C T ' 2 E '1'1"1 ' 2 E E '11 “1‘13 (2'5) 1-1 1—1 5-1 3p I —-1 —‘-. . (2.6) Since the vectors defined representing the joint clearances in Figure 2.2 are massless quantities, the generalized mass can be rewritten as 3r I —1-—1. (2.7) Furthermore, the corresponding first and second time-derivatives of the constraint equations, equation (2.1), are respectively 4 —“ q, (2.8) 37 - N a s . . 3:2 - 2 9 ‘p '2 E aqiaq ‘th *2 «(‘1' ‘ ° ’ 1 j 1 3 1 Utilizing equation (2.2), the dynamical equations of motion of the k-th link of the general kinematic chain can bewritten N _ N N 8 ' N N a E wu} E mum-2‘} Emu-um. 1-1 1-1 3-1 j-l 1-1 5 6r 3 £1 .. 1‘1. 39k +§ X1 aqk (2.10) i-l i-l . Equations (2.9) and (2.10) constitute the equations of motion governing the behavior of a general constrained planar mechanical system with bearing clearances in the contact mode. These equations can be reformulated in a matrix format and can be compactly written as It)”: 111-[:1 - where 38 (;)-(;, ....... gm)" ' ( A ] - [x1 ........ A.]T [SI-[81 eeeeeeee gul’r [13]...”1 ........ fslT all N N a in 513$ '1'aqk’2 E fihflqfifid i-l -l - j- k-l, 2, .... N N N 3 0‘ fp' '21} 3‘11an Qqu P'l: 2 - J- m11 m12 .......... m1N ‘ m m .......... m [ N l _ 21 22 2N (2.12) N a E 53; ['11] “qu 1 1-1 (2.13) s (2.14) (2.15) 39 ‘mm 15‘ 341 391 ....... aql ,, 211+; 22. _‘ aq aq ....... aq [ aq ] 2 2 2 (2.15) 3‘1 3‘2 3‘ L an an an J In equation (2.11) the differential equations are represented as a set of simultaneous, linear algebraic equations in the variables q1 and AP, and the coefficient matrix can be conveniently partitioned into four submatrices. The upper-left submatrix is symmetric and contains the generalized mass coefficients .ij’ which are the coefficients of the second-order acceleration terms in the Lagrange's equations. The main diagonal coefficients of this matrix are constants. The upper-right submatrix, which is the system Jacobian, is the transpose of the lower~ left submatrix. The lower-left submatrix is formed directly from the second-order terms of the constraint equations, while the upper-right submatrix is composed of the Lagrange multiplier coefficients. The column vector of dependent variables includes the second- order derivatives q in the upper sub-vector, while the lower sub-vector, -A, includes the Lagrange multiplier toms, which originated in the generalized Lagrange's equations. On the right-hand side of equation (2.11) the column vector contains displacement and velocity-dependent forces, g(q,q), pertaining to the particular Lagrange's equation, or the constraint equation associated with the row in which they appear. ‘The terms gk include the effect of all external forces, as well as the 40 centripetal and Coriolis components, while the terms fp contain the centripetal and Coriolis terms from the loop-closure equation for the mechanical system. 2.4 GOVERNING EQUATIONS FOR FREE-FLIGHT-MODE During the free-flight-mode, in a specific revolute joint with clearance, the constraints imposed at the joint during the contact mode are not enforced during the short time duration in which relative motion between the pin and the journal occurs as the pin moves across the bearing clearance. Consequently the Lagrangian multipliers imposing the constraints at these joints disappear for those joints in which contact has been lost and the free-flight-mode initiated. The equations of motion for the free-flight-mode assume the same general format as equation (2.11). Thus they may be obtained by deleting the columns and rows associated with those Lagrangian multipliers which are zero in order to relax the associated constraints. If there are 'a" joints in the phase of free-flight-mode, the equations of motion .become r f ”f. —t‘ 'f f H fr aq q 8 (2 17) [2f] o f ff ' ' a - q L J where (:5) - [gf .1; T 41 [.f) - [,5 ‘54. ]T (2.18) [8f] - [8f eeeeeeee sg"].r [ff] — [ff eeeeeeee ff-2a]T N-a N-a N-a N-a 8r 0 a f . __1 _ ___’ .f.f 1 .f.f 3k '5 ’1 aqf E aqf [mm] “1‘11 + 2 E Ea qf ['13] qjqi 1-1 k 1-1 3-1 j 3-1 -13 k-l, 2, 000-, N-a (2.19) f N-a N-a a 3‘f . fp - - a fa f qu1 , p - 1, 2, 000-, s- -2a (2.20) 1-1 1-1 ‘11 qj m11 m12 .......... m1d ‘ m m .......... m [ Hf] - 21 22 2d d - N-a (2.21) L -d1 -d2 .......... Ide r f f f . 31% fig ....... 3:423 aql aql aql f f f aof 211: 31% ....... egg-=23 [ ;-£] - aq2 aqz aq2 d - N-a (2.22) q eeeeeeeeeeeeeeeeeee an? if; Mfg-“ L sq: aqfi ....... aqg J 42 2.5 GOVERNING EQUATIONS FOR THE IMPACT-MODE The discontinuities associated with mechanical systems with hearing clearances, create impulsive phenomena when the two principal members of a revolute joint suddenly contact. This behavior is modeled in the impact-mode of the proposed model by employing a formulation developed in Goldsmith's classical text on impact phenomena [33] which is based on the form of Lagrange's equations presented by equation (2.2). Integration of equation (2.2) over the interval t to t+7, if r is assumed arbitrarily small, yields the equation 8T * 3T . ' . as —.' - '7 Wu) * —n (“3) aq aq p6 p-l “k where 81' N . ‘3— ' E 'kiqi (2.24) aqk 1-1 A t+f I A fit - ‘ - —l . Qk It QR (1!: § 1F] 3‘1]; nj (2.25) and F is defined as the impulse in the n direction. In order to solve equations (2.23), (2.24) and (2.25) for the A A unknown quantities (6:, A1, F1), a velocity constraint relationship must 43 be developed and also a relationship for the kinematic behavior before and after the impact must be formulated. The first requirement is satisfied by employing equation (2.8) and the second requirement is satisfied by developing an expression incorporating the coefficient of restitution c at each joint. Equation.(2.8) may be applied to model the impact phenomenon by utilizing the expression a» * e* e 3, - 3r -) 4an (<11 - <11) - o. (2.26) i—l The coefficient of restitution is defined as e* e* N (r - r )on . a: . e - - —Q"—Qb—- , where r -E — q (2.27) . . . 6q 1 where the relative velocity after impact is (1":- Q #1,) while (. Q ’Qa' 'Qb’ is the relative velocity before impact. Since 1': and °r are both Qa Qb dependent on in”, ('11), equation (2.27) can be rewritten as N 25k e* . aqi qi uk 6 - - 1-1 V v 2: 2k . . aq1 ‘11 “1: 1-1 (2.28) Equations (2.23), 44 (2.26) and (2.28) are the equations governing the impact-mode. These expressions can be easily expressed in matrix form as where as as an .3q as 5 an E 1:0 }. generalized mass matrix as as (2.29) system Jacobian matrix associated with the Lagrangian multipliers system Jacobian matrix associated with the impulse 9* [q1. [91' e* ‘12: m) 2’ V) 2! e* T eeee’ qN] A T eeeee' F p] A T eeeee’ As] T 00..., pH] (2.30) 4S [ s l - [31. 32. ----°. 331T [ 151-(fl. f if 2! N pk -§ Ina, k - 1, 2, N (2.31) k-l n a¢ - —1 . - O... 31 E aqi ql 9 J 19 2o o 3 (2.32) 1-1 at k-l This completes the development of the equations of motion for study the dynamic phenomena in revolute joint with radial clearance. Attention now focuses on establishing a viable numerical algorithm for solving the system of algebraic-differential equations which govern this class of dynamic problems. 2.6 STRATEGY FOR A NUMERICAL SOLUTION The equations of motion, equations (2.11) and (2.17), governing the dynamical behavior of a planar mechanical system with bearing clearances are written as a mixed set of differential and algebraic equations which can be expressed in the matrix form [A](X}-{B) (2.34) 46 where the coefficients in matrix [A] and vector {B} are dependent upon the displacements and velocities of the kinematic chain. Thus upon selecting the initial values for those kinematic parameters, [A] and (B) can be determined using a step-by-step procedure which discretizes the time domain prior to solving equation (2.34) for {X} using PLU- factorization in conjunction with forward and backward substitution [79) . The column vector {X} includes the second-order derivatives, qk, and also the Lagrangian multipliers A. The displacements and the velocities at the next time step are determined by subsequently integrating the accelerations qk. This integration process was accomplished by employing the subroutine, DGEAR, in the IHSL subroutine library which featured both a fifth-order Gear algorithm for stiff systems, which is referred to as any initial-value problem in which the complete solution consists of fast and slow components [31,32,43,63,81], and also a twelfth-order implicit Adams method. Thenumerical results presented herein were obtained by invoking the Adams method, since for systems where stiffness is not a problem, the Adams method uses fewer time-steps and is more accurate than the Gear algorithm [31]. In order to initiate the simulation software, a set of initial conditions are required, and the selection of the appropriate conditions is crucial in the prediction of the dynamical behavior of this class of nonlinear mechanism systems. The initial (conditions for the methodology proposed herein are based upon the results of kinematic and dynamic analyses of the mechanism system in which all of the joints are assumed to be characterized by zero clearance. The orientations of the position vectors representing the clearance in each revolute joint of a mechanism system with bearing clearances are assumed to coincide with the 47 direction of the vector representing the resultant force at each respective joint, and the angular velocity of these position vectors are assmed to be equal to the rate of change of the angular orientation of these resultant force vectors. The subsequent initial conditions for each time-step in the simulation are obtained in the usual manner from the final conditions of the previous time—step. Industrial and couercial kinematic chains featuring revolute joints with bearing clearances exhibit discontinuities in their dynamical response behavior which invariably occur in the transition from one mode to the subsequent mode. A typical exmle is the distinct change in velocity between the contact-mode and the free-flight-mode. In the simulation of these phenomena, these discontinuities can be amplified to yield erroneous results. In order to minimize these effects, the critical point in switching from one mode to the next is carefully predicted using the bisectional method [81] in which smaller and smaller time-steps are employed in the mode—transfer time-zone. The mathematical apparatus is now complete for predicting the dynamical response of general planar mechanism systems incorporating radial clearance in the revolute joints. The equations governing this class of nonlinear systems have been described and a four-mode model proposed for predicting the complex phenomenological behavior of these systems. Furthermore, the subtleties of simulating these systems have also been enunciated. Attention now focuses upon applying this proposed approach to a planar slider-crank mechanism with radial clearance at the gudgeon-pin bearing, in order to illustrate the approach, prior to comparing the theoretical results with experimental data from a complementary study . '11 48 (b) Figure 2.4 (a) The slider-crank mechanism showing dominant clearance at the gudgeon-pin bearing (b) Vector-loop model of the mechanism 49 2.7 THEORETICAL MODEL OF SLIDER-CRANK MECHANISM WITH CLEARANCE IN THE GUDGEON- PIN JOINT A planar slider-crank mechanism with a prescribed non-zero radial clearance at the gudgeon-pin joint between the connecting-rod and the slider was analyzed using the proposed methodology prior to comparing the theoretical results with experimental response data from an accompanying experimental investigation. The friction between the pin and journal is ignored. A schematic of the mechanism is presented in Figure 2.4(a) and the relevant kinematic parameters are presented in Figure 2.4(b). The subsequent paragraphs systematically describe the equations governing the dynamical response of the linkage when the pin and journal of the gudgeon-pin are in the contact-mode, the free-flight- mode, the impact-mode, and finally the transient-mode prior to describing the complementary experimental program. 2.7.1 Equations of motion for the contact-mode The dynamical behavior of the slider-crank mechanism in the contact-mode is governed by a set of independent simultaneous second- order differential equations which are established using equations (2.ll)-(2.16). The final equations can be expressed in matrix form, as [A1] {XI} - I 31} (2.35) 50 where '-l m3r2rc3cos(O3-02) O 0 rzsin 02 -r2cos 02 ‘ 0 I + m r2 0 0 r sin 0 -r cos 0 G3 3 G3 3 3 3 3 I ‘1] _ 0 0 0 0 -r4sin 04 racos 04 0 0 0 m l 0 s O r3sin 03 -r4sin 04 l 0 0 L O -r3cos 03 racos '4 O O 0 4 (2.36) and, T - - - l X,} - [ A3 03 '4 s 21 l2 ] (2.37) while, r 2 1 m3 r2 r6383 sin(03-02) -m r r ’2 sin(I -0 ) 3 2 G3 2 3 2 { 31’ - 0 (2.38) 4831:)» H, s 2 2 2 -r2’2cos 02 - r3’3cos 03 + raincos 04 2 2 2 L -r2’Zsin 02 - r3’3sin 03 + raiasin 04 The crank is assumed to operate with a constant angular velocity 3 thereby requiring the angular acceleration of the crank—shaft to be zero. This situation mandates the introduction of the Lagrange 51 mltiplier A, which represents the torque required to maintain this constant crank frequency. The Lagrangian multipliers A, and A, in equation (2.37) represent, F. the force imposed on link 3 by the 3x' slider in the ox direction, and, F. the force imposed on link 3 by 3y’ the slider in the oy direction respectively. 2.7.2 Equations of motion for the free-flight-mode The loss of contact betwoen the connecting rod and the slider in the free-flight-mode mandates that the mechanism to be modeled as two systems. One system is the double-pendulums arrangement of the crank and the connecting rod which moves independently of the second system which comprises the slider. Thus, the pin force at the gudgeon-pin joint is zero, since there is no contact between the pin and journal of this bearing. Utilizing the constrained Lagrange's equation again, only three second-order differential equations govern this mode and they can be written, in matrix form, as [A211X21-132) (2.39) where -l m3r2r63cosU3 - 02) O 1 2 [ A2 1 - 0 163+ m3rc3 o (2.40) 0 0 M 1 8 J (x1T-[13s] (241) 2 1 3 ° and, ‘32)- r 12 ”3 2 G3 3 I3:21:63 2 - «mm; L sin(03 2sin(03 02) 02) S2 1 (2.42) where p is the coefficient of friction between the slider and the guide. 2.7.3 Equations of motion for the impact-mode With reference to the free-body diagram presented as Figure 2.5, the equations of motion governing the impact behavior between the pin and the journal in the gudgeon-pin bearing are established using equations (2.29)-(2.33). Upon applying these general equations to the planar slider-crank mechanism, the following equations emerge where [A3]- and, -l m3r 2 r63cos(03 2 0 m3rG3 + I O O 1 0 -r3sin(03 A * e* A [Tor ’3 s F ] G3 02) 4) (2.43) -r2s1n(02 - 04) ‘ -r3sin(l3 - '4) , (2.44) -cos 04 o J (2.45) S3 ' m3r2rc3’3 cos(l3 - 02) (“3‘63 + Isa) ’3 l 3: l - vi,- (2.46) {s {rzizsinu2 - 04) + r3’3sinu3 so“) .+ scos 04) L +r2’2sin(02 - 04)} J Figure 2.5 Free-body diagram of the slider-crank mechanism in the impact-mode The coefficient of restitution, e, which is an energy- dissipation parameter was experimentally determined to be 0.46 for the gudgeon-pin bearing by dynamically stimulating the bearing in a carefully controlled environment in order to replicate the impactive dynamical environment imposed on the journal and pin by the articulating mechanism. This experimental procedure is described in section 2.7.6. 54 2.7.4 Equations of motion for the transient-mode The transient-mode, which is clearly evident in the experimental response profiles furnished by the transducers from the complementary laboratory investigation, comprises a series of imact and free-flight-modes which are terminated when the contact-mode is initiated. The governing equations have therefore, been documented in the previous sections . 2 . 7 . 5 Simulation protocols This section of the thesis documents the mathematical formulations which define the bounds of the different behavioral modes described previously for revolute joints with radial clearance, and also the subtleties incorporated into the simulation software. (1) Contact Loss Tvro criteria were employed in order to trigger the change from the contact—mode to the free-flight-mode. First, in every time-step of the simulation, the resultant contact force between the pin and journal at the gudgeon-pin joint was calculated using the expression, F. - A1 cos 0‘. + 12 sin 04 (2.47) The physical system remains in the contact-mode provided that the resultant contact force between the pin and the journal, F is '9 compressive. Thus from a mathematical standpoint, where compressive forces are assumed to be negative, this notation mandates that Ps will 55 always be less than zero when the pin and journal are in the contact- mode. Experience accrued in the complementary experimental investigation clearly indicates that this criterion is only valid at crank speeds below 100 rpm for the specific slider-crank mechanism under investigation. The generality of this observation is unknown. Other experimental mechanisms with different kinematic and dynamic characteristics would need to be designed, engineered and fabricated in order to investigate this observation. A second criterion was, therefore, introduced in order to overcome the lack of generality associated with the first criterion. This second criterion states that a loss of contact will occur when F. assumes a minimum value and, in addition, the rate-of-change of the pressure angle is high relative to other values of this parameter. The pressure angle is defined as the angle of the resultant force vector F‘ relative to the inertial reference frame OXY. This criterion is the result of both theoretical and experimental observations. If only the pin force criterion alone is employed to predict the loss of contact, then the simulation software will predict that the pin and journal remain in the contact-mode provided that F. is negative, no matter how rapidly the pressure angle changes with time. However, experimental results both herein, and also in references [25,37,38], clearly contradict this statement since contact loss is normally characterized by a high rate of change of pressure angle. Consequently, the simulation software package predicts that contact between the pin and the journal of a revolute joint will be lost when the contact force at the joint is small and also the angular rate-of-change of the force vector is high. These criteria capture the instability characteristics 56 associated with this dynamical state. (ii) Termination of the Free-Flight-Mode The pin of a revolute joint is assumed to be in the free- flight-mode relative to the journal until the following geometrical inequality criterion is violated whereupon the impact-mode is initiated, o < (sz + Ay2)1/2 - {. (2.48) where € denotes the radial clearance. This constraint relates the locations of the geometrical centers of the pin and the journal relative to the radial clearance in the joint. The terms Ax and Ay are obtained from the loop-closure constraint. Namely, Ax - rzcoso2 + racosl3 - racosla - s (2.49) Ay - r5 + rzsino2 + r3sin03 - rhsinoa (2.50) Ideally, if (Ax2 + Ay2)]’/2 - 6, then the system will be in the impact-mode, since this mode is assumed to follow the free-flight-mode. However, in all digital simulation work, computational round-off errors accumulate and a tolerance must inevitably be introduced in order to accomodate these small inaccuracies in the numerical results. Thus, the system was assumed to be in the impact-mode if the following inequality was satisfied 57 |1.o x 10'8l s [ (sz + 113,3)“2 - e 1 (2.51) (iii) Restoration of the Contact-Mode The software was written to incorporate the constraint that the motion of the pin and journal is governed by the free-flight-mode following the impact-mode provided that the duration of the flight-time of the pin and journal between contact-loss and the re-establishment of contact subtends more than a twentieth of a degree of crank rotation. If contact between the pin and journal is re-established within a twentieth of a degree of crank rotation and there is also an insignificant difference between the original contact point and the new contact point, then the simulation control switches to the contact-mode. (iv) Solution Procedure A flow chart documenting the logic of the simulation software for predicting the dynamic response of planar mechanisms incorporating bearing clearances is presented in Figure 2.6. The initial conditions for simulating the dynamical behavior of the slider-crank mechanism were based upon the assumption that the position vector representing the clearance in the gudgeon-pin joint had the same direction as the angle of the resultant force in the bearing. The magnitude of this angle was obtained from an analysis of the mechanism incorporating a gudgeon-pin bearing with zero clearance, and in addition, the angular velocity of the clearance-vector was assumed to equal the rate-of-change of the angle of the resultant force. The step-by-step simulation, therefore, began in the contact-mode, and the equations of motion were expressed in matrix form as presented by equation (2.35). Upon determining the appropriate initial values for the system, the coefficients of, [A1] and 58 {81} were then evaluated prior to calculating the vector (x1) which included the terms 03, 04, s, Al, 12, and A3. FLU-factorization was employed in conjunction with forward and backward substitution in order to undertake this task. Subsequently, the second-order variables 03, 04, and s were transformed into a set of first-order equations, and a subroutine 'DGEAR“ in the IMSL subroutine library was employed to integrate these second-order derivatives and obtain values of 31, 01, s, and s. The joint force and the rate-of-change of the angle of the resultant force were also calculated at each time-step. Simulation control switched to the free-flight-mode once the criteria for contact loss were satisfied, and equation (2.39) was then solved. The equation governing the impact-mode, equation (2.43) was solved, once the inequality relationship (2.51) was satisfied. After the first impactive motion, the simulation entered the transient-mode, and following a series of relatively. brief free-flight and impactive-mode states, the control returned to the contact-mode again provided the criteria for the restoration of the contact-mode were satisfied. The average CPU-time for one cycle was 2 minutes and 50 seconds on a Vax-ll/750 computer. Having described the equations of motion governing the dynamical response of a planar slider-crank mechanism with radial clearance at the gudgeon-pin joint, and discussed the salient features of the computational scheme for solving these equations, attention now focuses on describing a complementary experimental program which furnished experimental response data for evaluating the predictive capabilities of the proposed methodology. S9 Prescribe the crank angle, crank frequency, total simulation time (T F) Calculate tha initial values 01. 01. S. S r 4—5i;sontact mode__}=7* imulation time .GE.T NO Separation Free - flight mode imulation time .GE.T Contact restored Impact mode Calculate time in free—flight mode and contact position Restoration of contact Figure 2.6 Flow chart for the computer protocol 60 2.7.6 Experimental apparatus and procedures The comprehensive experimental study presented herein, on the dynamical response of a planar slider-crank mechanism featuring a bearing with a prescribed radial clearance in one joint. is motivated by the dearth of experimental studies in the literature on this subject. The literature reporting on experimental studies on planar slider-crank mechanisms with rigid links and bearing clearances is confined to only two publications [28,75]. Funabashi, et al. [28] investigated the input torque and the relative displacement of the pin and journal where the clearance occurred in the crank bearing in a planar slider-crank mechanism. Shimoj ima, et al. [75] measured the relative displacement of the pin and journal where the clearance presented in the crank bearing in a planar slider-crank mechanism by employing an optical method. Thus there is a dearth of experimental work in this important field of predicting the dynamical behavior of planar mechanisms with radial clearances. The principal subassembly in this experimental investigation was a planar slider-crank mechanism incorporating substantially rigid links which articulated in a plane perpendicular to the gravitational field. The essential features of the mechanism are presented in Figure 2.7. The crank, which had a length, r,, of 48 mm, was fabricated as an integral part of a 146 mm diameter flywheel. This heavy circular disc was keyed to a 19.05 mm diameter drive-shaft, which was supported on a matched pair of Timken type TS4A-6 tapered-roller bearings, and was powered by a 0.56kw Dayton variable-speed dc motor, model 22846A. The slider of the classical slider-crank mechanism was represented by the 61 translating table of a Micro Slides,Inc. , type 2050-RW-118-5 linear crossed-roller slide assembly, which is a precision preloaded system without clearance. The mass of this sliding element was 0.63607 kg. The connecting-rod assembly of the classical slider-crank linkage comprised three components: a uniform steel bar with dimensions 267x6x19 was, a crank-pin bearing, and a gudgeon-pin bearing. The total length of this link member was 293 - between bearing centerlines, the total mass of the link and joints, m3, was 0.21435 kg, and the moment of inertia I was 0.003883 kg-mz. The crank-pin bearing was designed to CB incorporate a matched pair of instrument ball bearings, type R4DBl2, supplied by FAG Bearings Limited. The spindle diameter was 6.35 n. This bearing assembly was carefully preloaded using a Dresser Industries torque-limiting screw driver. This procedure enabled bearing clearance to be removed from the assembly, since the vibrational response associated with this clearance at the crank-pin would have contaminated the experimental data from the transducers monitoring the dynamic response of the system at the gudgeon-pin. The gudgeon-pin bearing was fabricated as a journal bearing with a prescribed radial clearance. This was achieved by employing a steel pin of diameter 6.35 mm which was supported in a threaded hole in the reciprocating portion of the crossed-roller slide assembly. The journal element of the bearing was provided by a brass bushing with an internal diameter of 6.604 mm which when assembled with the pin provided a bearing with a radial clearance of 0.127 m. In addition, a second bearing assembly was designed and developed for the gudgeon-pin in order to develop a joint with zero clearance. This action was motivated by the necessity to generate a 62 reference set of experimental data for comparison with the results from the linkage with the prescribe radial clearance at the gudgeon-pin bearing. This precision bearing assembly without radial clearance incorporated a matched pair of preloaded instrument ball bearings, type R40812, supplied by FAG Bearings Limited. The spindle diameter was 6.35 m. The experimental apparatus and the associated instrumentation is schematically represented in Figure 2.8. The operating speed of the mechanism was measured by a Hewlett Packard 5314A universal counter which was activated by an Electro Corporation digital-magnetic pickup, model 58423, which sensed a 60-tooth spur gear mounted on the crank shaft. This gear and the associated pickup are clearly evident in Figure 2.7. Figure 2.7 Photograph of experimental planar slider-crank mechanism 63 Speed of Hechanism ‘ at 5314A Universal Counter Gudgeoo-Pin Bearing . . r___..,, Electro Corp. 12 volts pick up 0.6. Journal Pin ‘ (connecting rod) (slider) Accelerometer Accelerometer Accelerometer Eaperimenta 60 Tooth 0.6. B 6 K 4371 B 6 R 4371 B 6 K 4371 mechanism W gear Hotor I I,I l— : .4 Charge Amplifie rge Amplifie rg. A-Plifie Air Fax p. as: Type 2635 as Type 2635 as: Type 2635 "°‘:‘_'Pr_h “Mallet <94 Hawetek Dual 111/w Filter Trigger nodal 432 DEC POP 11/03 Hicrocomputer Experimental Results Figure 2.8 Schematic of experimental apparatus and associated instrumentation 64 Figure 2.9 Photograph of accelerometer configuration a the gudgeon-pin bearing of the experimental slider-crank mechanism The dynamic behavior of the slider and the connecting rod of the experimental slider-crank mechanism were monitored using three Bruel and Kjaer piezoelectric accelerometers, type 4371, which were each connected to Bruel and Kjaer charge amplifiers, type 2635. These accelerometers were positioned at the gudgeon-pin bearing in order to measure the dynamic response of the reciprocating slider and also the longitudinal and lateral motions of the connecting rod. This cluster of accelerometers are clearly evident in the right-hand-side of Figure 2.9. The signals from these accelerometers were fed to a Wavetek HI/ID analog filter, model 432, prior to being recorded on a PDPll-based digital- data-acquisition system. In order to clarify the response profiles, frequencies above 60 hz were removed by analog filtering. This cut-off frequency was selected by studying the experimental response-data from the accelerometers clustered at the gudgeon-pin bearing when this hearing featured the zero-clearance FAG instrument ball bearings. The 65 kinematic characteristics furnished by these transducers not only exhibited the anticipated low-frequency responses but a much higher frequency low amplitude ripple was clearly evident in the signal. This component was subsequently removed by analog filtering. The response profiles measured by the accelerometers were related to the angle of the crankshaft using a third transducer arrangement, which featured an Airpax zero-velocity digital pickup, 'type lit-0001. This transducer was employed to sense the bolt head protruding from the flywheel when the crank was in the conventional zero crank- angle position, prior to firing the Schmitt triggers on the input/output panel of the data-acquisition system. The highest sampling rate employed in this study was 2,500 samples per second. In addition to the apparatus assembled to measure the dynamic response of the laboratory slider-crank mechanism with controlled radial clearance at the gudgeon-pin joint, a second apparatus was designed, engineered and fabricated in order to evaluate the dynamic behavior of the gudgeon-pin joint in a carefully controlled environment similar to the enviroment experienced by the joint during the articulation of the mechanism. The objective of this investigation was to excite an impactive motion in the gudgeon-pin joint and then measure the velocities of both the pin and the journal before and after impact in order to distill an empirical value for the coefficient of restitution from the experimental response-data. The value of this coefficient would then be employed in the complementary computer simulations for predicting the dynamical response of the experimental system. The experimental procedure involved exciting the Microslides table, and hence the pin of the gudgeon-pin bearing, by a Ling Dynamic Systems vibrator, model 411, which was driven by Wavetek function generator, 66 model 275, and a Hafler power amplifier, model P500. These tests were performed with the connecting-rod disconnected from the crank in order to isolate the motion between the two impacting machine elements at the revolute joint with prescribed clearance. The connecting rod was configured so that the longitudinal axis of the member was colinear with the axis of translation of the slider as shown in Figure 2.10, and very soft rubber springs were connected to it in order to ensure continual co-axial alignment during the experiments. Figure 2.10 Photograph of experimental apparatus for measuring the impactive force in the gudgeon-pin bearing and determining the associated coefficient of restitution between the pin and the journal of the bearing A Bruel and Kjaer force transducer, model 8200, was incorporated in the fixture between the shaker head and the sliding table in order to continuously monitor the forcing function imposed upon the gudgeon-pin bearing. The dynamical motion of the journal and also the pin were both carefully monitored by strategically locating Bruel 67 and Xjaer piezoelectric accelerometers, type 4371, on the bearing assembly. These transducers were operated in conjunction with Bruel and Kjaer , charge amplifiers, type 2635, which contain built-in integrators that permit velocity and displacement histories to be measured. Figure 2.10 presents a photograph of the experimental arrangement, and Figure 2.11 presents a schematic diagram of the apparatus and the associated instrumentation. 2.7.7 Experimental and numerical results Figure 2.12 presents a typical set of experimental response data for the velocities of the pin and the journal of the gudgeon-pin bearing when the pin of the gudgeon-pin bearing was excited by an electrodynamic shaker at 20 Hz. The coefficient of restitution was calculated to be 0.46 using these response data since that the relative velocity of the pin and journal before impact was 5.81 cm/sec and the relative velocity after impact was 2.67 cm/sec. Figure 2.13 shows the corresponding impact force, which was monitored at the. head of the shaker by the force transducer. The duration of the impact was measured to be 0.0012 seconds. The frictional coefficient, u,-between the slider and guide was calculated to be 0.0224 by employing p - ws/Fd’ where U. is the weight of the slider while F d is the total countweight' which caused the slider began to slide. These data were employed in the model of the slider-cram mechanism in order to predict the dynamical behavior of the system. 68 Uhvetek Function “"' Lin i S Generator Node]. 275 ‘—" Roller 31 5 ”M c ytteme Alplifier “ctIOdynamic Shaker M1 P500 “04.1 ‘11 '_, Gudgeon-Pin Bearing Trigger journal pin 5 g Transducer Signal (connecting rod)“ (3114.2) “13‘0011400 cat; K 8200 ' ' Reciprocating 3 6 K #371 3 6 K 6371 Accelerometer Accelerometer Table HDd.1 2050 I B & K 2635 B 8 K 2635 B & K 2635 DEC Charge Amplifer Charge Amplifier Charge Amplifier POP 11/03 Hicrocomputer v7 Havetek Dual HI/LO Filter Nodal A32 Experimental Results Figure 2.11 Schematic diagram of the experimental apparatus and instrumentation for determining the coefficient of restitution of the gudgeon-pin bearing 69 7.0a.--e-,.,.- . 7 2 2 i 1‘ A Ir' . l u 3.00: i 0 4 . m «4 '. j \ I '1 . E 1 g 8’ -1.00-: E : 4 o 1 E 3 _] m § -5-001 , , .33 3 Journal 2! -9.oc‘ '7'“? . s . , . . . r . t! 0.00 0.06 0.12 TlME(Seconds) Figure 2.12 Gudgeon-pin bearing characteristics: Velocity characteristics of the impactive behavior between pin and journal at a pin excitation frequency of 20 Hz 10.00 e v - - r t i r . 7.004 f, S g .1 A * : . ?8 q 0: :: J C 0) 4.001 .— 3\ + .-" '9." g E JR 1 at} 1.00-1 .. 3‘3 ‘ ~ 0 4 00:3 -2.00-< 1 LLLJJ d > + -'5eOO""'i a .3 O1. . . : - force (shot-é) 3": “ -a.oc ‘2 ”“3"" (3%"- e r - , e . . A 0.130 ' 0.06 0.12 TIME(Seconds) Figure 2.13 Gudgeon-pin bearing characteristics: Excitation force characteristics and associated velocity characteristics of the pin impacting the journal at a frequency of 20 Hz 6&3 I I Y t—T T I I I lytico! IIIII ACCELERATION(m/s/s) ' f1'20.‘0' ' T303 ' 340.0 ' """ CRANK ANGLE(Degrees) Figure 2.14 Connecting-rod tangential acceleration. Mechanism operating speed 200 rpm 1' r I I f] I l’ r l — analytics: ~ - -- experimental ACCELERATION(m/s/s) YYYYYYYYYY ' 750.0 ' 340.01 Wt CRANK ANGLE(Degrees) I I T r T 300.0 360.0 Figure 2.15 Connecting-rod tangential acceleration. Mechanism operating speed 300 rpm 71 30.0 TYVYTI:UUITITVfVT‘IIirYtr — analytnca - experimental 10.0- —10.0- ACCELERATION(m/s/s) -3000 T T v r v 1 vvvvv 1- T u 1 1 r T vvvvv 1 rrrrr 1 7 T v u u 0.0 60.0 120.0 180.0 240.0 300.0 360.0 CRANK ANGLE(Degrees) Figure 2.16 Slider acceleration. Mechanism operating speed 200 rpm 5000 T 5 T ‘ r I F Y I I I i FFFFF I r TT’I I I FFFFF 1’ 11111 1 — analytical .. 1 a experimental 1 ”a? 26.0- ‘1 \ m ., .l \ ; E. 2.0- l 2 ! 9 . l 1— < 0: -22.0- a . O 2 -46.0d -70.0 ....,.r...,...-.r.....,efi..l ,,,,, 0.0 60.0 120.0 180.0 240.0 300.0 360 O CRANK ANGLE(Degrees) ‘ Figure 2.17 Slider acceleration. Mechanism operating speed 300 rpm 71 30.0 1 I V T r r l l f I Y 1 ''''' I f Y fir T I I I I I —— anolytnco - experimental 10.0- -10.0- ACCELERATION(m/s/s) -30.0 ..... , ..... , ..... ,...-.,..T..,.-... 0.0 60.0 120.0 180.0 240.0 300.0 360.0 CRANK ANGLE(Degrees) Figure 2.16 Slider acceleration. Mechanism operating speed 200 rpm 50.0 I t r r 1 """ i """" T 1' tfw u — analytical ,. ,_ - experimental ACCELERATION(m/s/s) TTTTT CRANK ANGLE(Degrees) . Figure 2.17 Slider acceleration. Mechanism operating speed 300 rpm 72 @100 '7. _-,- .-. ~~~~~ . ~ ' -, .-- -,, ..... 1.? 3 - \ 40.00- - In 1 . \ .. E 20.00- _ z . . 9 E ‘ ‘ a: 0.004 - Lu 1 1 _l m a Q ‘ ‘ 2 -20.00- - d 4 -40.00 hfhpfifi. ..... ,.--T-, vvvvv r vvvvv 0 1 20 240 360 4:10 600 720 CRANK ANGLE(Degrees) Figure 2.18 Connecting-rod longitudinal acceleration. Mechanism operating speed 200 rpm vvvvvvvvv ACCELERATION(m / s/ s) J4_1_a.1_1 J.‘ .J...a_..1. :— 1.4.1.1-]. .1- 0 1&0 ' ' '240 330' #7150 600 720 CRANK ANGLE(Degrees) Figure 2.19 The theoretical prediction of the longitudinal acceleration of the connecting-rod. Mechanism operating speed 300 rpm 73 sci fi 1 1 i 1 i ... . 2 40-4 - o ! 3 0 5 + O U L 2 a, '1 ._>. ; 1'3 0 Q _' E 100 200 300 400 500 600 700 Crank soeeclrom) Figure 2.20 The maximum impactive force between the pin and journal. . The radial clearance was 0.000127 meters 4’0 1 1 r T 1 T 1 A 2 i .9 304 4 s , . 0 C v 8 E 20-1 _ _e a: 1 .2 i “g . ‘1 10 i E 1 O I l l j— r T 1 1 1 2 3 4 5 6 7 8 Radio! clea’rcnce(l O ’3 inches) Figure 2.21 The maximum impactive force between the pin and journal. Mechanism operating speed 300 rpm 74 Figures 2.14 - 2.17 present experimental and theoretical response data for the slider-crank mechanism operating at crank frequencies of 200 and 300 rpm. The radial clearance is 0.127 n in the gudgeon-pin bearing for all response profiles. Figures 2.14 and 2.15 present the tangential accelerations of the connecting rod at 200 and 300 rpm respectively while Figures 2.16 and 2.17 present the corresponding accelerations of the slider. In order to develop a viable environment for comparing the analytical and experimental results, the numerical results were subjected to digital filtering, using LSPSFILTER_NONREC, which is a nonrecursive, finite impulse response, filtering software on a DEC MicroVAX II/GFX workstation [96]. This was undertaken in order to remove the fictitious very high frequency components from the theoretical results which were attributed to the very small time-steps employed in the computational algorithm. I Upon reviewing the acceleration profiles presented in Figures 2.14 - 2.19, it is clearly evident that there is excellent correlation between the theoretical and experimental results for a broad range of operating conditions. Thus, the proposed four-mode model for predicting the complex phenomenological behavior of planar mechanism systems incorporating a distinct radial clearance in the revolute joints is able to accurately predict the dynamical response of these systems in the contact-mode, the free-flight-mode, the impact-mode, and the transient- mode which comprise the global response regime. Moreover, it is also clearly evident that the tangential acceleration characteristics of the connecting rod are much more sensitive to clearance in the gudgeon-pin bearing than the longitudinal acceleration characteristic. A consequence in practice of this distinct sensitivity of the lateral 75 acceleration component to bearing clearance manifests itself as high levels of acoustical radiation in these systems because the vigorous lateral motion excites the adjacent fluid medium [87] . Moreover, upon reviewing Figures 2.14 - 2.17, it is clearly evident that the pin and journal loose contact and undergo major excursions relative to each other twice during each revolution of the crank shaft, and the restoration of contact occurs at the end of the transient-mode. The first separation generally occurred at approximately 80 degrees of crank rotation and the second separation occurred at approximately 280 degrees of crank rotation. Figures 2.16 and 2.17, indicate that the most severe impact conditions occur at 98 degrees and 295 degrees of crank rotation, while the experimental results indicate that the most serious impact occurs at 102 degrees and 295 degrees of crank rotation. This is an impressive predictive capability for such a complex nonlinear mechanical system. Figures 2.14 and 2.15 indicate that at approximately 80 degrees of crank rotation the first impact creates a more severe vibrational response of the transverse acceleration characteristic of the connecting-rod in the transient-mode than the second impact which occurs at approximately 290 degrees of crank rotation. This phenomenon has not been reported before in the literature. Figures 2.18 presents the analytical and experimental longitudinal acceleration characteristics of the connecting rod for two revolutions of the crank shaft at 300 rpm for the case where the gudgeon-pin has zero clearance. These profiles present a null check in the evaluation of the software and they also provide a measure of the experimental behavior of the system under these classical operating conditions. Figure 2.19 presents the numerical predictions for the 76 longitudinal acceleration of the connecting rod at 300 rpm with 0.127 m clearance in the gudgeon-pin bearing. Upon examining these two curves it is clearly evident that bearing clearance in the gudgeon-pin has very little affect on the longitudinal dynamic behavior of the connecting rod. The maximum impactive force between the pin and journal in the gudgeon-pin bearing was also calculated during the simulation. Figure 2.20 presents the predict results which shows that the impactive force increases proportionally with the crank speed. The radial clearance was 0.000127 meters. Figure 2.21 shows the relationship between the maximum impactive force and the magnitude of the radial clearance at the gudgeon-pin bearing while the crank speed was 300 rpm. Since the measurement of the impactive force between the pin and the journal is not available, only simulation results are presented. 2.8 CONCLUSIONS OF THE CHAPTER Analytical and computational models of general planar mechanisms with rigid links and bearing clearances have been developed. A four-mode model of the complex phenomenological behavior of revolute joints with radial clearance has been proposed and incorporated in a methodology for predicting the dynamic response of these mechanical systems. As an illustrative example of the approach, a slider-crank mechanism with radial clearance in the gudgeon-pin bearing was studied in a comprehensive analytical, computational and experimental investigation and the numerical results are in qualitative agreement with the results from a complementary experimental program. Chapter III GENERAL PLANAR MECHANISMS FEATURING FLEXIBLE-LINKS AND BEARINGS WITHOUT RADIAL CLEARANCE 3 . 1 INTRODUCTION Traditionally, mechanism design has been based on rigid-body analyses in which all mechanism members are treated as rigid bodies in order to study the kinematic and kinetic characteristics of these articulating mechanical systems [55,66,74,100]. The stresses in the mechanism members due to inertia forces and external loading are then calculated in a sequential algorithm in which kinematic, kinetic and elastic considerations are all decoupled. More recently, the operating speeds of mechanisms are being continually increased to increase industrial productivity. In order to reduce the inertia force of the machine and driving torque of the motor, the weight of mechanism members need to be made as light as possible. However the lighter members are more likely to deform and vibrate due to the inertia and external forces. The position inaccuracy and occasionally the resonant condition 1will cause the design unacceptable. Hence the elastodynamic analysis of a mechanical system becomes necessary and important. 77 78 In this chapter, a coupled system of equations is developed by combining the second-order constraint equations and the equations of motion for a general planar linkage mechanism featuring flexible links and bearings without radial clearance. This coupled system of equations are then divided into two sets of equations in the basis of inertial coordinates and elastic coordinates, respectively. The flexible links are assumed homogeneous, isotropic and have a small deflection. Since the thickness-to-length ratio of the links considered in this study is 0.00433, the effects of shear deformation and rotary inertia shall be ignored. Each link is modelled by employing the Euler-Bernoulli beam theory which is incorporated in a geometrically nonlinear formulation for predicting the coupled axial and flexible responses of each link. Finally, a slider-crank mechanism with a flexible connecting rod is analyzed in an illustrative example. The planar linkage system incorporating bearings without clearance is analyzed using the general formulation developed herein prior to evaluating the predictive capabilities of the methodology by comparing the theoretical results with results from a complementary experimental investigation. 3.2 EQUATIONS OF MOTION OF A FLEXIBLE BODY DESCRIBING A GENERAL PLANAR MOTION Consider the elastodynamic motion of a general planar linkage mechanism moving relative to an inertia reference frame OXY. Each link has a Lagrangian coordinate oxy fixed in it. Figure 3.1 shows two arbitrary adjacent links, 1 and j of the mechanism. OXY denotes the 79 inertial reference frame, while oixiyi denote the Lagrangian coordinates fixed in the body in an undeformed reference state. At any instant t, a 0 general point in the ith-link was denoted as p1 and p1.1 before and after deformation. Therefore, the general point p after deformation can be expressed as +I' +0 ‘ 11.1 1 1 - r”1 (3.1) where 1':01 are the components measured in the oixiyi frame of the position vector of the origin of the body axes relative to the origin of the inertial frame. Similarly, rR1 denotes the position vector of point p in the reference state relative to the origin of the body axes and n1 is the deformation displacement vector, may be expressed as uxe1 + uyez, where ux is the axial deformation in the ox direction and uy is the transverse deformation in the oy direction. Terms a1 and a2 are the unit vectors in the ox and oy direction measured relative to the moving reference frame, respectively. Since the finite element method is employed for the derivation of the equations of motion, the link may be divided into many elements. The elastic deformation of each element can be described by six nodal displacements, ul - u6, illustrated in Figure 3.2, 80 Figure 3.1 Generalized coordinate of two adjacent links 0 Figure 3.2 The nodal displacement of each element 81 From Figure 3.2, a general point p in the element can be expressed in the fixed reference frame in the deformed state It - [XA+(x+ux)cosO - uysin0]i + [YA+(x+ux)sin0 - uycos0]j (3.2) where XAi + YAJ is the position vector of point A relative to the inertial frame. Differentiating equation(3.2) with respect to time, yields the velocity *1 - [EA + uxcoso - (x+ux)0sin0 - uysino + uy’coslli + [TA + uxsino + uxdcoso + uycosfi - uy’sinllj (3.3) Equations (3.2) and (3.3) can be expressed relative to the rotating frame, r1 - (XAcosO + YAsino + x + ux)el + (-XAsin0 + YAcosl + uy)e2 (3.4) i, - (EAcosO + TAsino - uyé + 1.1x)e1 + {-iAsina + TAcoso + (x+ux)i + ay1e2 (3.5) The displacement at any point in the element can be expressed in terms of the nodal displacements as u(X.t) - [ N] l U i (3.6) where N is the shape function and is dependent upon the position only, and u is function of time. 82 The deformation of the flexible link in el and e, directions can now be expressed as “x' [RX] 101 (3.7) and, “y' [NY] {U} (3.8) where [xx] - [N1 0 0 N o 0] 4 [N ] - [0 N2 N 0 N 3 5 N6] (U) - (u1 u2 u3 “4 u5 u6} N - l - 5 2 8 N - 1 - 35 + 26 2 I 2 3 3 1(€ - 26 + e > N - E 2 3 N - 3e - 25 Z I 2 s 6 no +€L€-XN Equation(3.5) can be rewritten in the following form by introducing equation (3.7) and (3.8) into equation(3.5), i - (iA- [Ny11u1é + [qufine1 + (§A+ x6 + [Nx11u13 + [Ny]1&))e2 * (3.9) where fo EAcosl + TAsina, and yA- -XAsin0 + YAcosO 83 For convenience equation(3.7) can be rewritten in matrix form as Ex -[IAB] i; (3.10) . i r u y where 1-[10],A- ~1Ny11u1 ,B_ 1le ’ 0 1 “[le In] [NY] and i.- [iA] (3.11) 5'. If the link is assumed to be homogeneous with constant cross sectional area (A.) and mass density (pe)’ the kinetic energy of an element can be expressed as 1 1 O 0 Te- 2 PGA‘Jo r1 . ‘1 dx (3.12) By using equation(3.10), the kinetic energy can be rewritten as 1 1 eT e I v To- 2 p.A. [p i u] [AT] [1 A 3] dx (3.13) 0 3T ceQeiue l I A. B - 1 “11.1 {33 a ]{ AT ATA ATB dx} (3.16) 2 0 3T BTA 8TB cow-e- The kinetic energy of the link , therefore, is n T1 ..E T. , n is the number of elements in each link (3.15) e-l 84 In equation(3.l4), defining q -[p 0], then a - [i 3], where q denotes the corresponding generalized coordinate. Therefore, equation(3.l4) can be expressed as - 1% [<1 1'1] [M] [‘3] _ (3.16) where N )1 [H] _ qq Q“ T (3.17) M qu um I I .A qu - peAe AT AFA dx (3.18) o 2 B Mqu - peAeJ‘o ATB dx (3.19) I T . Muu - p3A‘Io[ B B‘) dx (3.20) The strain energy of the element can be expressed as 1 82 2 l a 2 u u v--1- sI——;de+—1—EA—3‘dx e 2 0 8x 2 0 8x 2 t Bu + +I0F(x)[ a—xy] dx (3.21) The first and the second terms in equation(3.21) are the strain energy due to the transverse and axial deformation of the link while the the third term is the work done by the axial force F(x) due to the change in the horizontal projection of an element. 85 The introduction of equations (3.7) and (3.8) into equation (3.18) permits the strain energy to be written as v .. J- ! snuirm' ]T[N" ](U)dx + J I EAIU)T[N']T[N']lU}dx e 2 o y y 2 o x x +411 F(xumrlu'lrw'lwm 2 o ' Y Y I ' F - -% 1011' [K] 101 + -%I F(x){U}T[N flu ](U)dx (3.22) o y y where [K] is known as stiffness matrix which equals 1 I T I 1 l T 0 21 N N dx + EA N N dx 3.23 L 1,11,] L 1,11,] ( > where (') denotes the second spatial derivative with respective to x, while (') denotes the spatial derivative with respective to x. Figure 3.3 Axial force is represented as a linear function The axial force at any point in the link may be expressed as, refers to Figure 3.3, .15 F(x) - 111 + (1‘2 - F1) 1 - F1 + AF-f (3.24) Substituting equation(3.23) into equation(3.21), the strain energy of the link can be rewritten as _ .1 T J. ‘1‘ .1 I v0 2 (U) [K] (U) + 2 Flu” “(611101 + 2 (WW) [KGZHUl 86 (3.25) where l , T , 1x611 - [0111,] my] dx (3.26) 1 [K02] - 1;} [N;]T [N;] dx (3.27) The total kinetic energy of the system is nl n T - E ( E Te )1 (3.28) i-l e-l where n1 is the number of links of the mechanical system, and n is the number of elements in a link. Similarly, the total strain energy of the system is nl n V - E ( E Ve )1 (3.29) i-l e-l The ith generalized constraint equations may be written ‘1 - ¢- 2 fi 8 0 -1304 < -230 ,,,,-'. ..... ,m.---., r...., ..... , ..... 0.0 60.0 120.0, 180.0 240.0 300.0 360.0 CRANK ANGLE(Degrees) Figure 3.21 The acceleration of the slider. Mechanism operating speed 185 rpm (102 .. -..-r.. .-. , ..... , ..... , ..... , . r ’5? 0.014 .. \\ m \\ ~£5 (1004 . Z 9 22 g -0.01-‘ ' - LIJ ° 1 U U U 2 -0.02-» U U - -0.03 ..-..Ire...,.....rr.ee.. ..... .....+ 0.0 60.0 120.0 180.0 240.0 300.0 360.0 CRANK ANGLE(Degrees) Figure 3.22 The variation of the acceleration of the slider between the rigid-link and flexible-link of a slider- crank mechanism. Mechanism operating speed 185 rpm 0.03 .....I.....' ..... (...... ..... I ..... A m \ 0.01- - m \ E V 5 F: -0601“ < 05 1.1.1 ..J 31‘ < U U -0005 YYYYY I VVVV I YYYY r ‘ rT T f 0.0 50.0 120.0 180.0 240.0 ' 306.0 ' 060.0 CRANK ANGLE(Degrees) Figure 3.23 The variation of the acceleration of the slider between the rigid-link and flexible-link of a slider- crank mechanism. Mechanism operating speed 225 rpm 0.04 *H'rr ..... I ..... I ..... T ..... ,s f. 6 a..- U \ m \ E 0.00- .1 Z 9. g 0 02 11.1 - . 4 1 ..J LIJ 0 2 -0.04-‘ 1 -0.06 .r...,r..rr, ..... l.r.T., ..... T.... 0.0 50.0 120.0 . 180.0 240.0 300.0 350.0 CRANK ANGLE(Degrees) Figure 3.24 The variation of the acceleration of the slider between the rigid-link and flexible-link of a slider- crank mechanism. Mechanism operating speed 258 rpm 115 15.0 Trrvfr vvvvv rrfi‘f'vl’tvtfi'rrw #— YY—‘r' L 11.2661; -- rigid A U) 5 1 10.0 .1 "i a Z v 11.1 0 a: O u- 5.0- 4 Z O. 0.0 .r. . . ..... . e. r 0.0 60.0 120.0 "180.0 7240.0 300.07' 7360.0 CRANK ANGLE(Degrees) Figure 3.25 The magnitude of the pin force in gugeon-pin bearing Mechanism operating speed 185 rpm 360.0-1 -....Irr... — flexible -- n'gid A U) Q) 0 a 240 0“ d o O O v 12.1 ..I 0 Z < LIJ 120.0- - O 0: 0 L1. 0.0 Irw‘vlrIY—rvltIIWIIfirvttl IIIII IIerr 0.0 60.0 120.0 . 180.0 240.0 300.0 3610.0 CRANK ANGLE(Degrees) Figure 3.26 The force angle of the pin force in gugeon-pin bearing Mechanism operating speed 185 rpm 116 0.04.5 ..... ..... ,-....,--...,.-..., ..... 1? _ c 0.015- a Jul 3 o 5 ' LU -00015‘ —i 0 O: O u. z - a: -0.045 . -0.075 1...... . . . ..... 0.0 60.0 020.0 000.0 340.0 000.0 360.0 CRANK ANGLE(Degrees) Figure 3.27 The variation of the pin force in the gudgeon-pin bearing between the rigid-link and flexible-link slider-crank mechanism. Mechanism operating speed 185 rpm 14 ,-....., ..... ,..fieT-....,..-.-T....fi A ‘0’ m 0.7- —. L C5 Q) O V “ii 00 o '7 Z < LIJ 8 07 O . L1. -104 ''''' I rrrrr thfTrr YYYY ITrTfiVI IIII 0.0 50.0 120.0 180.0 240.0 300.0 360.0 CRANK ANGLE(Degrees) Figure 3.28 The variation of the force angle- in the gudgeon-pin bearing between the rigid-link and flexible-link slider-crank mechanism. Mechanism operating speed 185 rpm Chapter IV ELASTODYNANIC ANALYSIS OF A GENERAL PLANAR MECHANISM FEATURING FLEXIBLE- LINKS WITH RADIAL CLEARANCES IN THE REVOLUTE JOINTS 4 . 1 INTRODUCTION The major purpose of this chapter is to formulate a general analytical model to predict the vibration of the links and the dynamical characteristics of a general planar mechanical system featuring flexible-links and bearing clearances. Subsequent to the development of this general methodology, two illustrative examples are presented in order to predict the elastodynamic response of these systens and compare these theoretical results with experimental results obtained from complementary experimental programs. The first example is a compound pendulum with prescribed initial dynamical conditions which is subjected to a sudden impact from a fixed mechanical stop. The second example is a slider-crank mechanism with a flexible connecting rod and featuring a prescribed radial clearance in the gudgeon-pin bearing. 117 118 4.2 ”DEL OF A GENERAL PLANAR MECHANISM FEATURING FLEXIBLE-LINKS AND REVOLUTE JOINTS WITH BEARING CLEARANCES A general planar mechanism system incorporating joints with bearings with radial clearances undergoes several different phenomenological responses which are associated with the relative motion of the pin and journal in each joint. The four-mode model, as proposed in Chapter Two, was extended in order to incorporate elastodynamic effects and employed to described the elastodynamical response of this general planar mechanical system. These four modes are classified as the contact-mode, the free-flight-mode, the impact-mode and the transient- mode, and these modes were schematically presented in Figure 2.3. The mathematical descriptions of these four modes for a rigid- linked system are described in Chapter Two. In order that these formulations are applicable to flexible-linked systems several additional constraints must be appended to these definitions in order to acco-odate flexibility effects. The four-mode model is described as follows: (i) Contact-Mode: In this mode, the pin and the journal are in contact and a sliding motion relative to each other is assumed to exist. The contact surface is assumed to be rigid, smooth and frictionless, thus there is no energy loss in this mode. Two criteria must be satisfied by the pin and journal of a joint in order that the bearing behavior be described by the contact- mode: firstly, the kinematic constraint equations for the contact-mode 119 must be satisfied. Secondly, the contact force between the pin and the journal must be compressive force. The second criterion is one of the two criteria which dictate the termination of the contact-mode. The other criterion for initiating the free-flight-mode is the rate of change of the force angle. Note that the termination of the contact-mode always leads to the simulation control initiating the free-flight-mode. If all of the joints in a mechanism in which the behavior of the joints are governed by the contact-mode, than each flexible-link of the system would be treated as a Euler-Bernoulli beam with the assumption that the thickness-to-length ratio is very small. Contact stresses at the pin- journal interface are not included in the formulation. (ii) Free-Flight-Mode: In the free-flight-mode, the pin and the journal in a specific joint are not in contact, hence there are no reaction forces between these two members. Thus a constraint equations different from those employed for the contact-mode must be developed. During the free-flight- mode, the flexible-link will be treated as a pin-free beam. The tip position of the link is calculated and continuously monitored during the simulation process in order to determine when the free-flight-mode is terminated. This event occurs when the tip of the link, which is the location of the pin, again contacts the inner surface of the journal. The duration of the free—flight-mode is governed by the flexibility and dynamical behavior of the kinematic chain subjected to analysis. (iii) Impact-Mode: A significant impactive force is developed in a hearing when the pin and the journal contact suddenly. This behavior is the basis for 120 the impact-lode, and this phenomenon is modeled by invoking a momentum- exchange approach which incorporates the coefficient of restitution. During the impact duration, the joint is treated as being in the contact-mode, thus the constraint equations governing the contact-mode must be satisfied. The coefficient of restitution, which captures the material properties of the impacting bodies, are dependent upon the stiffness of the contacting materials, the roughness of the contact surfaces, the geometrical shape of the impacting bodies, and also the velocities of the bodies prior to impact. The value of the coefficient of restitution is generally determined experimentally. (iv) Transient-Mode: The transient-mode begins at the first impact between the pin and the journal in a specific joint, and ends with the restoration of contact between these members. Thus this mode generally comprises a sequence of several free-flight-modes of progressively smaller duration and also several impact-modes. This progressively attenuating vibrational response is also governed by the flexibility of the links of the system. The subsequent sections present the mathematical apparatus for modeling each mode. 4. 3 GOVERNING EQUATIONS OF THE SYSTEM Two adjacent links, 1 and j, in a general planar mechanical system with flexible-links incorporating revolute joints which featuring a distinct radial clearance can be modeled as shown in Figure 4.1. Axes 121 OXY denotes the inertial reference frame, while oixiy1 denote' the moving frame fixed in the body in an undeformed reference state. At any instant 0 t, a general point in the ith link was denoted as p1 and p:l before and after deformation. Therefore, the general point p after deformation can be expressed as r1 - r01 + rR1 + “i (4.1) where' :01 are the components measured in the 01x1y1 frame of the position vector of the origin of the body axes relative to the origin of the inertial frame. Similarly, denotes the position vector of point ’31 p in the reference state relative to the origin of the body axes and “i is the deformation displacement vector, may be expressed as uxe1 + uyez, where ux is the axial deformation in the ox direction and uy is the transverse deformation in the oy direction. Terms e, and e, are the unit vectors in the ox and oy direction measured relative to the moving reference frame, respectively. The constraint equation can be obtained from the closed-loop V96 tor roi+tai+ui+£ij+roj+tkj+“j'o (4.2) where £11 denotes the clearance vector in the revolute joint which connecting the ith link and jth link. Note that the clearance vector is a position vector which is zero if the bearing is assumed to be devoid of clearance. 122 YA Figure 4.1 Generalized coordinate of two adjacent links In general, the constraint equation can be expressed as ‘1 ' “‘11'32' "'91' 51' 52”” 5w '“1'“2"""“3(N+1)) (4.3) where there are l inertial generalized coordinates, m inertial clearanced coordinates and 3(N+l) elastic generalized coordinates, N is the number of finite elements employed to model the mechanism system. For convenience, q 1 and 5k can be represented by a set of inertial generalized coordinates, n1. Thus the general constraint equation can be rewritten as ‘1 - ¢("11 "21 .... "pt “11 u21°°°°o “3(N+1)) (404) wherep-l+m 123 The finite element method is incorporated in the derivation of the equations of motion of each link which is assumed herein to be modeled as a beam rather the shell or plate. Each link can be divided into many finite elements and the elastic deformation of each element described by six nodal displacements, u1 - “6’ which represent the axial, transverse and rotating deformation at each node as illustrated in Figure 3.2. The deformed state of a general point p in the element can be expressed relative to the inertial reference frame by the expression [1 - [xA + (x+ux)cos0 - uysinHi + ”A + (x+ux)sin0 - uycosllj (4.5) where XAi + YAj is the position vector of point A relative to the inertial frame . The corresponding velocity, thus, is i1 - [RA + fixcosl - (x-l-uxflsinl - uysino + uyacosflu + [YA + uxsinl + uxicosl + uycosd - uy’sinou (4.6) Equations (4.5) and (4.6) can be expressed in the rotating frame, r1 - (XAcosl + Y sin! 4» x + ux)e1 + (-X sin! + Y cos! + uy)e2 A A A (4.7) 2'1 - (RAcosD + TAsinl - uy’ + 1.1x)e1 + [-RAsin0 + TAcosl + (x+ux)’ + {nyle2 . (4.8) The displacement at any point in the element can be expressed in terms of the nodal displacements as u(X.t) - [ N] 1 U i (4.9) 124 where [N] is the shape function and is dependent upon the position only, while (U) is the column vector of nodal displacement which is function of time. The deformation in the e, and a, directions can now be expressed as ux- [Nx] 1U) (4.10) and, - N U 4.11 “y [ y] 1 l ( ) where [N ] - [N1 0 0 N4 0 0] [N ] - [0 N2 N 0 N 3 5 N6] (U) - {u1 u2 u3 114 u5 u6) N - l - C 2 a 3g + 2; 2 I H I 2 s N - 1(C ' 2C + C ) N-: 2 3 11 -3g -2; 2 I 2 s 5 1(‘C + C ). C ' X/l Equation(4.8) can be expressed in the following form by introducing equation (4.10) and (4.11), 125 i - (3c,- [Nyluni + [letfiml + (5'; xi + [lewii + [Nyltfii)e2 (4.12) where iAf XAcosl + TAsinl, and yAf ~2Asinl + IACOSO For convenience, equation(4.10) can be rewritten in matrix form as [rx] - [ I A.) ] 5 1-[10],A- ~[gum].._ [g] ’ 01 xqgnm [g] and i - [ iA ] i. If the link is assumed to be homogeneous with constant cross sectional (4.13) 5'.“ '0' where area (A.) and mass density (Pe)’ the kinetic energy of an element can be expressed as or e I [p Iu][J]HAB] fie ‘e‘e 2 (4.14) 2 T T T 1 I A a - 1 ,.AG[ i?) a ]{ 1? 132.233 dx } , a sans 33° 0". The kinetic energy of each link, therefore, is 126 n Ti- E Te , where n is the number of elements in each link e-l (4.15) In equation(4.l4), defining 6 - [i 3], where n denotes the generalized coordinate. Therefore, equation (4.14) can be expressed as '1' 11° '1 “an ”on 5 - ‘1 ° 2 n a: n [ 0.] nu uu -§ 1561 In] ['1] (4.15) u where n u [a] - g" "“ (4.17) M nu uu '1‘ ‘1 an" - peAe ‘1 ‘1A_ dx (4.18) 0 1 a nflu - peAéIo 1?: dx (4.19) .. 412.)... .20 uu - Pe e o ( ' ) The strain energy of the element can be expressed as l 82 2 l 8 2 u u v - -1jI EI -;y dx + -1jI EA -—4x dx 5 2 o ax 2 0 3X 2 2 l a u + -%-Ior(x)[ -3;1 dx ’ (4.21) 127 The first and the second terns in equation(4.2l) are the strain energy due to the transverse and axial deformations of the link respectively, while the the third tern is the work done by the axial force F(x) due to the change in the horizontal projection of an elenent. Introduction of equations (4.10) and (4.11) into equation (4.21), the strain energy in each finite element can be written as 2 . . 2 . . ve - % Io mmTINlelNyHundx + 4% Io “(a)T[NxITINXIde +-1 I! F T N' T N' dx 2 0 (mu) [,1 [ylhn 1 , , " "g' {MT [K] {11} + -%I F(x){u}T[N ]T[N ]{u)dx (4.22) 0 Y Y where [K] is the stiffness natrix which is defined by z . .r . 1 . T . EI N N dx + EA N N dx 4.23 I. 1,11,] L 1,11,] < > Note that (') denotes the spatial derivative with respect to x, while (') represents the second spatial derivative with respect to x. The axial force at any point in the link nay be expressed as, F(x) - 1?1 + (F2 - r - F1 + 49-} (4.24) .8 1) 1 where the terns are defined relevant to Figure 3.3. Redefining equation(4.24) to incorporate equation(4.23), the strain energy of the link can be rewritten as ve - -% (Mr [K] (u) + 4,- rltufixcluu) + -% AFluirlkczltu) (4.25) where 128 2 .T . [x611 - IolNy] my] «I: (4.26) 2x ..r . [KGZ] '10: [Hy] [Ny] dx (4.27) The total kinetic energy of the 'systen nay be written T - E T (4.28) where n1 is the nu-ber of links of the mechanical systen. Similarly, the total strain energy of the system may be written nl n V -E ( E Va )1 (4.29) i-l e-l The first and the second tine derivative of the constraint equations, equation(4.4), may be written 3(N+1) u - u - 3 - S a,” at + E auj uJ (4.30) 1-1 3-1 . 3(N+l) 3(N+l) , ' _ _fl_L. ' 0 it " _a__§_ . . ‘ S S 3013’!“ "ink + S 3191 "1 + E E aujau‘ ujul 1-1 k. - J- - 3(N+1) Q1 .. + E auj uj (4.31) 129 4.3.1 Equations of motion for the contact-mode The equations of motion for a mechanical system in the contact-mode are developed herein by utilizing the generalized Lagrange's equation and expressing all terms relative to the inertial generalized coordinates . The Lagrange's equation is employed in the elastic generalized equation. These two equations are nc 3‘ ( . ) - + - E A - Q (4.32) dt 3,1 an, 801 j-l j a», i _d £1. 21. i1. ( ) - + - Q (4.33) dt 3. auk auk k where nc denotes the number of constraint equations Employing equations(4.32) and (4.33) and introducing the second time derivative constraint equation(4.31), yields the equations of motion for the adjacent flexible links where the pin and journal as the kinematic pair are describing the contact mode r 31 1 r _) r 1 r a [Man] [an] [Man] a o o 4 [3‘9 0 0 d + 4 " r .. 0 [RR] Mu] ° ”ml. ‘2 L . 1 u. r 1 (g) - (f) (4.34) (h) where [xx] - [K] + [K0,] + [xc21 (4.35) 130 nf a r . 3k - E r, -'3;1 - (in. flan] [ f ] ,k-l,2,--,p (4.36) 1_1 k u where nf denotes the number of external forces ’ 3(N+1) 3(N+1) ' _S 5 up. H... fk aqiaqk '1'k + aujaul “j“: i-l k-l j-l l-l 3(N+1) _ + E 3‘ u , kp1,2,---,nc (4.37) uJ j 1-1 - where no denotes the number of constraint equations nf at an 0 e hQ-E '1'71+-%[561[T] ’0’ -[firuiuu] ? 1-1 “Q “Q “ ' “ .Q-1.2.°°'.3(N+1) (4.38) In equation (4.34) the differential equations are represented as a set of simultaneous, linear algebraic equations in the variables 9'1, A p and u k and the coefficient matrix can be conveniently partitioned into four submatrices. The upper-left submatrix is syusetric and can be divided into four more submatrices, [MW] , (3': l, and its 1' transpose [3:] , and null-matrix. Submatrix [MM] is sy—etric but no longer constant, unlike in the analysis of rigid-linked mechanisms, which are the coefficients of the second-order inertial acceleration terms in the Lagrange's equations, while [3:] is the system Jacobian which is formed directly from the second-order terms of the constraint equation, while its transpose is the coefficient of the Lagrange 131 multipliers. Notice that the constraint equations are influenced by the deformation of the links. The matrix [Haul is a coefficient matrix describing the non-linear coupling of the inertial motion associated with the articulating system and elastic deformation associated with the flexible member. Submatrix [ "qu is the mass matrix and need only be calculated once in the procedure of computer simulation. The coefficient submatrix [KR] in equation (4.34) is the stiffness matrix which is frequently encountered in solving problems of structural dynamics and is dependent upon the deformation of the links and the axial force in the element. The second-order column vector contains the second-order derivatives of q and u, and also the Lagrange multipliers which originated in the generalized Lagrange's equations. On the right-hand side of equation (4.34) the sub-column vector contains the displacement and velocity-dependent forces, g(q,:p,u,u), pertaining to the particular Lagrange's equations, or the constraint equation associated with the row in which they appear. The terms gk and hQ include the effect of all external forces, as well as the centripetal and Coriolis components, while the terms fk contain the centripetal and Coriolis terms from the loop-closure equation for the general mechanical system being analyzed . 132 4.3.2 Equations of motion. for the free-flight-mode During the free-flight-mode, in a specific revolute joint with clearance, the constraints imposed at the joint during the contact mode are not enforced during the short time duration in which relative motion between the pin and the journal occurs as the pin moves across the bearing clearance. Consequently the Lagrange multipliers, imposing the constraints at these joints, disappear for those joints in which contact has been lost and the free-flight-mode initiated. The equations of motion for the free-flight-mode assume the same general format as equation (4.34). Thus they may be obtained by deleting the columns and rows associated with those Lagrange multipliers which are zero in order to relax the associated constraints. If there are several joints, 'a", which are governed by the free-flight-mode, the equations of motion become r Hf “f Hf 1 r . f1 r 1 r f 1 I M] [3,, l I 0“] a O 0 a lull. 0 0 -1f + .)f a; 1- f "f 0 “(Hf f L [unu] 0 [nun] J b n J b J L u J r ‘ (51f - (f)f (4.39) (mf L J where [mf- mf+ [x6.1f+ “(calf (4.40) 133 nf 8r 0 - O —L - f .f n - .° ' 3k 1?1 anf [ "q nan] [ a ] ,k 1,2, ,p a (4.41) 1-1 k p a -a az‘f 3(N+l) 3(N+1) 83‘f a fa f "1'3 a fa f 3 1 1-1 j 1 "1 '1 3-1 1—1 “3 “1 3(N+l) f _ + E iii a: , k.1,2,...,nc (4.42) J-1 “J nf f a: an -£ 1 of f _1 _1 of -f __ a f f a hQ -§ ’1 . a f + 2 I" u l I a f] .f - [fiqu fiuu] [ e J 1-1 “Q “Q “ u ,Q—1,2,---,3(N+1) (4.43) where the superscript "f' denotes the characteristics in free-flight- mode, and there are "m-a" joints which are governed by contact, transient or impact conditions. 4.3.3 Equations of motion for the impact-mode The discontinuities associated with mechanical systems with bearing clearances, create impulsive phenomena when the two principal members of a revolute joint suddenly contact. The equations of motion governing the impact-mode can be obtained by integrating equations(4.32) and (4.33) over an arbitrarily small time interval 7, ar * ar . “° . a4 —.— - T—Qz+§ 1_n (4.44) an as p 8» and 134 5—: 1 {:—:- * where the asterisk, (*), denotes the post-collision property, and 3T M . —.' " 4" 1 wk) (4.46) fink liq“ 8T u . -.— - fl { “1} ((6.47) a“! nun nf A t+f A at a - -n - 4 O - 0.. Qk It Qk dt E j 6"}: nj , k 1,2, ,p (4.48) j-l A t+f nf A at Q‘; -I (2‘; dt -§ 1' a—J- . n , 1-1,2,o--,3(u+1) (4.49) and Fj is defined as the impulse in the nJ direction. In order to develop sufficient equations for determining the 41*“ unknown quantities (1):, u1,)11, F1), a velocity constraint relationship must be developed and also a relationship for the kinematic behavior before and after the impact must also be formulated. The first requirement is satisfied by employing equation (4.30) and the second requirement is satisfied by developing an expression incorporating the 135 coefficient of restitution c at each joint. Equation (4.30) may be applied to model the impact phenomenon by utilizing the expression * S 2‘ 0* e 3§N+1)8_¢I e* e 3, - 3, - 3"1 ('11 - 4,) + “5 (m3 - uj) - o. (4.50) 1-1 1-1 which describes the velocity constraint relationship before and after impact . The general expression defining the coefficient of restitution is (is - EE])-n . 8r . 3 "+1’ar ‘--(E -. )"l ,wherer- aziq1+ 8'"qu1 Qa Qb i-l i-l (4.51) where the relative velocity after impact is (r3; fab) while (EQ; i’Qb) is the relative velocity before impact. The subscript Q denotes the impact point while a and b denote the impacting bodies. Since EQ‘ and in are both dependent on r011, :11, ui, 1°11), equation (4.51) can be rewritten as (4.47) S 3‘1 . 3(N+1) a‘k . 331 "1 ' “k + 333 “j ' "k 1-1 1-1 vhi 501 136 which expresses the energy dissipation coefficient, c, in terms of the relevant system parameters. Equations (4.44), (4.50) and (4.52) are the equations governing the impact-mode in a revolute joint with radial clearance. These expressions can be conveniently expressed in matrix form as ' as an 1 r .H r 1 "an an an "an " ‘°’ 30 T as . '5'; O O ”a: -X "' if} (4°53) an T as . 3; o o 3; -F {g} T 33.]T .* -—' h L “W“ 0 6“ ou L n J L{ )J where as - ] - system Jacobian matrix associated with the Lagrange multipliers 8R 5; ] - system Jacobian matrix associated with the impulse formulation e* e* e* 0* T t n I {01. 02. . up} A A A A T 1 F ) - [F19 F2: .....o anl {A } "' {X19 X29 .....o A 1 (4°54) 137 l e 1 {el' 32’ eeeee’ .p} (fl-(f 1’ 2’ ( g ) (31. 32. °°°°°. gnf} { e I In,” 1 l n 1 + [KW] ( u 1 (4.55) fJ a“ :11 3‘11 u1 , j - l, 2, one, nc (4.56) i-l i-l 3(N+1) a: - _ (.1. (_k 3“ ‘ {51991 nk) 51+ E Hank) J’} 1'1 1'1 k - 1, 2, coco, nf (4.57) l h } - [“3“ l l n } + [Mu u] i u 1 (4.58) Equation.(4~58) completes the development of the mathematical apparatus necessary for predicting the elastodynamic behavior of a four- mode model of a kinematic pair comprising flexible links and a revolute joints with radial clearance. The equations of motion of a planar general mechanism system with flexible links and bearing clearances have been developed. Attention now focuses upon applying this proposed approach to predict the dynamical response of two mechanical systems in order to illustrate the approach . 138 4.4 ILLUSTRATIVE EXAMPLE 1 - DYNAMIC BEHAVIOR OF A COMPOUND PENDULUH SUBJECTED TO IMPACT A general mathematical model for predicting the elastodynamic behavior of impacting bodies has been developed in the previous sections of this chapter. In order to evaluate the predictive capabilities of this model, two illustrative examples are considered in which the theoretical results are compared with experimental results from a complementary experimental investigation. The first example examines the elastodynamic behavior of a compound pendulum in which a beam accelerates from rest under gravitational loading prior to impacting a stationary mechanical stop. A schematic diagram of the principal elements of this system is presented in Figure 4.2. This simple test was undertaken to verify the four-mode model. The four-mode model can be applied to any kind of mechanism describing intermittent motion. In this illustrative example, the free-flight mode describes the major motion of the compound pendulum as it rotates from the initial point I to impulsively contact the stationary mechanical stop. The impact-mode and the contact-mode exist only for a very short time period during each impact while the transient-mode is initiated by the first impact until the equilibrium position is attained. 139 Figure 4.2 A schematic diagram of a compound pendulum articulating under gravitational loading prior to impacting with a fixed mechanical stop 4.4.1 Equations of motion for the free-flight-mode The dynamical behavior of the pendulum in the free-flight-mode can be obtained by using equations(4.l4) through (4.34), in which the frictional torque of the bearing is neglected. Constraint equation (4.4) is not employed in the derivation since the angular position 0 is the major factor for determining the termination of the free-flight-mode rather than the pin force‘which occurs when 0 - 0. The pendulum was divided into six elements as before. The final set of equations of motion in the inertial frame is (m 12(n2+n+l)+{u)T[M] {u}+2m[H](u))8- i 3 e i G -mig(n1 + % 2)sin0 - 29{u}T[M]{u) - 2m18[HG]{u} - mi[EG]{u) + milu)T[FG]{u} (4.59) 140 while the set of equations of motion in the elastic frame is where [u1(&) - [c1(a) + ([61 + [k6,] - I’Iul)(u) - -m1;[FG]{u) - 2m17IFbllu) - n17(EG)T + m182{HG}T (4.60) m1 - PeAel l — the length of one finite element n - N-l N - the number of finite elements 0 - the angle between the center-line of the pendulum and the vertical direction 8 - the angular velocity of the pendulum - the angular acceleration of the pendulum 2 g - gravitational force, assumed to be 9.8 m/sec ' 140 o 0 70 o o ‘ o 156 221 o 54 -131 2 2 [M] _ g g 0 221 41 0 131 -31 i 420 70 0 0 140 0 O O 54 131 0 156 ~22! 2 2 . o -131 -31 o -221 41 J mmi-u(%k+%)o 0(%k+§) 00},bbl 2 2 2 (26),— (o < 35 + 3% ) ( 30 + §§- ) 0 (£3 + ké > < 3% 12-12 )1 ,k-i-l [Feli- [x6111- [KG2]1- [K11 - L ' :41 o o -EA1 0 O —1 - 20 ° .1 ° 20 .1 ° 30 -1 - 30 ° .1 ° 20 ..1 ° 20 o o .1 10 ° 21 15 ° 0 o .1 10 ° -.1 30 ° 0 o .1 10 ° .1 30 ° 0 o _.1 10 ° .1 ’60 0 o o 1221 61EI 2 61EI 41 21 o o -1221 -6121 2 élEI 21 E1 0 2 EA! 0 GlEI 2 21 E1 -6lEI 2 41 BI 142 [C] - the damping matrix, which is obtained from experimental results 4.4.2 Equations of motion of the contact-mode The tip of the pendulum is assumed to have the same kinematic characteristics as the stationary stop at the moment of impact. Therefore, the angular position, angular velocity and angular acceleration are all zero at the initiation of the contact-mode. The compound pendulum can be treated as a simply-supported beam in this condition. Hence, the equation of motion of the pendulum in inertial frame is 3 - o (4.61) and the equations of motion in the elastic frame are same as equation (4.60). Note that while the global form of the equations of motion are the same in both the free-flight-mode and the contact-mode, the elements of these matrices are different. The elastic equations of motion will yield a solution which will be different in each mode, since different boundary conditions will be applied in each different mode. 4.4.3 Equations of motion in impact-mode By employing equations (4.44) - (4.53), the equations of motion for the impact-mode is obtained 143 A11 A12 A13 1* 31 A22 A23 Q* ' B2 (“-62) syuetric A33 F 83 where 2 2 1 T An - ”it (n + n + 3 ) + (u) [M] (u) + 2m1[HG](u} A - n (E 1 - n (a)T[r 1 12 i G i G A13 ' r3 ' 1 ‘“’T[Kcll‘“’ * “L 622 - [M] 1* - the rebound angular velocity after impact a - the velocity characteristics of the link after impact 5' - impactive force between the pendulum and the block 31 ' [A11] ’3 + [‘12] ‘a’ B2 ' [A21] ’3 + [‘22] ‘8’ B - -¢ 1 A13 “L - axial displacement at the tip node of the link Note that the under-line may denotes a submatrix, a sub-row vector or a sub-column vector. 4 . 4 . 4 Solution procedures Equations of motion for the compound pendulum subjected to impact have been developed. This section describes the solution procedures of the equations of motion. 144 Equation (4.59) can be rewritten as [A] [31 - [B] (4.63) in which 2 2 1 T [A] - (1|1 1 (n + n + 3 ) + (u) [H] (u) + 2m1[HG]{u}) and [a] - -m1g(n1 + % l)sin8 - 21(61rtn1(u1 - 2n11[ucl[&1 - m1[EG][u] + nilulTlFG][;] Thus, [A] is function of the beam inertia and also u, the deformation of the pendulum ,and [B] is function of 0, 1, u, 6, u. Similarly, equation (4.60) can be expressed in the standard form as [M1191 + {011%} + [KK](u) - (a) (4.64) where 2 [KR] - [K] + [k - i [u] G2] {R} - ‘fl1;[Fh]{u) ' 2-1’[Fb]{a} ' n1;[EG]T + .1’QIH61T To solve the equations of motion (4.63) and (4.64), the following procedures are employed: 1. The initial values, 1, u, u, and u associated with the initial angle 0 of the pendulum are set equal to zero since the pendulum is stationary prior to initiating the experiment. 2. In equation(4.63), the angular acceleration, 0, is calculated, and the angular velocity and angular displacement of the pendulum are obtained by integrating 0. 145 3. At the same time, equation(4.64) is solved by employing the mode- superposition method. The first three modes are chosen for the solution. 4. The angular displacement 0 is calculated and monitored continuously during the simulation. If the angular displacement is found to equal zero, the simulation switches to the impact-mode. 5. During the impact-mode, equation(4.62) is solved and the impactive velocity characteristics, 1* and (1*, are calculated in addition to impulse F. 6. The simulation is switched to the contact-mode imediately after the completion of the impact-mode. An impact duration of 0.008 seconds, which is the total time duration of the contact-mode, was obtained from the experimental results, and used for the simulation of the contact- mode. 7. The final calculated-values of the previous time-step are employed as the initial conditions for calculating the values of the subsequent time-step. Figure 4.3 presents the experimental equipment employed in this study, in which the compound pendulum is clamped in a test fixture by using a precision bearing, which is the same bearing described in Chapter Two. The compound pendulma was initially released from a given angular positions, 0, of 30 degrees and 60 degrees. The tip of the compound pendulum articulated through these angles prior to impacting a stationary block. This impact phase followed by a transient response phase in which the pendulum repeatedly impacted and rebounded from the mechanical stop prior to attaining equilibrium. 146 Figure 4.3 Photograph of the compound pendulum and stationary mechanical stop Figure 4.4 Photograph of experimental compound pendulum showing the pin and the bearing 147 The pendulum was fabricated from a low carbon steel with the s dimensions 0.29318 x 0.01905 x 0.00127 m and with a mass density, p, of 7860 kg/m. and a modulus of elasticity, E, of 200 GPa. The strain gauges were mounted on the mid-span the flexible link in a bending half -bridge configuration. The foil gauges, type CEA-13-250W-350, were supplied by the Micromeasurements Group of Vishay Inc. and the signal was fed into Vishay 2120 portable strain conditioner and then into a Vavetek 111/LO analog filter, model 432, prior to being recorded on a HicroVax-based digital-data-acquisition system. The cut-off frequency was 1000 Hz. 4.4.5 Experimental and theoretical results Figure 4.5 shows the experimental results for the transverse deformation of the mid-span of the pendulum for a two seconds duration when the initial angular position was 60 degrees. Four impacts are clearly evident from this experimental data and 22 cycles of vibration were observed between the two impacts which indicates a damped natural frequency of 52.63 Hz. Figures 4.6 - 4.8 show the corresponding simulation results. Figure 4.6 shows the mid-span transverse deformation of the pendulum during the two second period following the initiation of the free-flight-mode. Four impacts are observed and 22 cycles of vibration betwaen two impacts is also observed. The deformation of the pendulum, the vibration of the pendulum and the period between two impacts show excellent correlation with the experimental results in Figure 4.5. Figure 4.7 presented the corresponding angular velocity of the pendulum during the same time period and Figure 4.8 presents the angular displacement of the pendulum. 148 When the initial angular position of the pendulum was set to 30 degrees prior to releasing the pendulum, the subsequent experimental mid-span deformation of the pendulum following impact are presented in Figure 4.9. Figures 4.10 - 4.12 present the corresponding simulation results. Figure 4.10 shows the calculated mid-span deformation of the pendulum, Figure 4.11 represents the angular velocity and Figure 4.12 shows the angular position of the pendulum during the first two seconds time period of the simulation. Again, the results show an impressive predictive capability for the proposed mathematical model. The conclusion to be drawn from this first illustrative example is that the mathematical formulation developed herein can accurately predict the elastodynamic response of a compound pendulum released from rest and accelerating under a gravitational field prior to impacting a stationary mechanical stop. 5.0 , ,r‘ 2.0" -1 E E V z 8 -1.0- _ 0 1.11 ..l L1. L11 O -4.0-« -7.0 0 1 2 TlME(Seconds) Figure 4.5 The experimental mid-span deformation of the pendulum. The intial angular position 60 degrees 149 5.0 i A 2.0- - E E V <25 : -1.0-1 '1 U 'uJ .J U. LIJ ‘3 ‘-¢0« « -7.0 l O 1 2 TlME(Seconds) Figure 4.6 The theoretical mid-span deformation of the pendulum. The intial angular position 60 degrees 4.5 1 ANGULAR VELOCITY(rod/sec) —7.5 0 1 2 T|ME(Seconds) Figure 4.7 The theoretical angular velocity of the pendulum. The initial angular position 60 degrees 150 60.0 . 20.0- _ 0.0 I /\ O 1 2 TlME(Seconds) ROTATING ANGLE(Degrees) Figure 4.8 The theoretical angular position of the pendulum. The initial angular position 60 degrees 2.5 I 9 U 1 DEFLECTION(mm) J. 9' -15 I 0 1 2 TlME(Seconds) Figure 4.9 The experimental mid-span deformation of the pendulum. The intial angular position 30 degrees 151 2.5 , WM") 7 -3.5 r 1 .0 U‘ 1 DEFLECTION(mm) 1. (1| 1 [G TlME(Seconds) Figure 4.10 The theoretical mid-span deformation of the pendulum. The intial angular position 30 degrees 3.5 . ’3 Q) U, \ U o L. E 8 -0 5-4 \I ..J L1J > ‘3‘ 3 (.2 :f -4.5 0 1 2 TlME(Seconds) Figure 4.11 The theoretical angular velocity of the pendulum. The initial angular position 30 degrees 152 30.0 I 20.0-1 10.0- .1 ROTATlNG ANGLE(Degrees) 0.0 , 0 1 2 TlME(Seconds) Figure 4.12 The theoretical angular position of the pendulum. The initial angular position 30 degrees 4.5 ILLUSTRATIVE EXAMPLE 2 - A slider-crank mechanism with flexible connecting-rod featuring radial clearance in the gudgeon-pin joint A planar slider-crank mechanism with flexible connecting-rod and featuring radial clearance in the gudgeon pin joint was analyzed using the proposed methodology prior to comparing the theoretical results with experimental results. A schematic of the mechanism is presented in Figure 4.13(a) and the relevant kinematic parameters are presented in Figure 4.13(b). The characteristics of the mechanism are shown in table 3.1. 153 (0) Y1} (b) Figure 4.13 (a) The slider-crank mechanism with flexible connecting rod showing dominant clearance at the gudgeon-pin bearing (b) Vector-loop model of the mechanism 154 4.5.1 Equations of motion for the contact-mode The dynamical behavior of the slider-crank mechanism in the contact-mode is governed by a set of independent simultaneous second- order differential equations which are established using equations (4.14)-(4.34). Two set of equations were obtained which are in inertial and elastic bases respectively; and " 2 ["qu m + ([K]+F1[KG;]+AF[l 15.00- \ E V 5 E -5em-J 0 -25.00-1 2 -45.00 .-.rr .mr .r. +.-r- -..r. ..- 6 a? 12'? Foo 210 360 no CRANK ANGLE(Degrees) Figure 4.18 Slider acceleration. Mechanism operating speed 258 rpm 171 8.00 Tffi'ri"f"l**"'l """ 1'w'rvuffivv ; I U } 3.00- - E 1| I v I ‘I‘ l | s E 000 I I" ‘ ” . \ ‘ Q E ‘ l' I \ d '0' \ ' ‘-’, I o -3001 ' d 2 I ' - enndmmmm -s.oo ”rhfihrq...“ ..... ,T.T.Wr 0 so 120 150 240 300 300 CRANK ANCLE(Degrees) Figure 4.19 Connecting-rod tangential acceleration. Mechanism operating speed 185 rpm 8.00 **r' r """ r 'w *r fi rr' fl uvvrw " i < ' . 4.m-I ‘ ' d \ I E I, ’ E 1"“ " \‘ I = Oemq ' \ I fl \ \ I g I [I J I "" 0 «$00 I o D d I - < - W —s.oo .-...rfi..- n... W..- “W 0 so 150 150 230 360mm C5UU“(IMWGLEKDmgnees) Figure 4.20 Connecting-rod tangential acceleration. Mechanism operating speed 199 rpm 172 ‘UV'fiV'VfVV' C ”ox V'rVVtVVVfiVI‘TTV‘FV’IVY‘V'V A m \ < 400 E O V Z O I: g -1Cm‘ 8 ( ‘tm V V V r—‘r forV ' I I V fr 0 ' 'er ”Tgfrfjsohnzio 300 300 CRANK ANCLE(Degrees) Figure 4.21 Connecting-rod tangential acceleration. Mechanism operating speed 225 rpm 1 0.00 A n \ Sew "‘ I \ E V 5 g Dem - 3 Q -5.00 - < -10.m CRANK ANCLE(Degrees) Figure 4.22 Connecting-rod tangential acceleration. Mechanism Operating speed 258 rpm 173 o.” wwwww rfwwvv" wwwww IrwwrvI wwwww 'wvvww 0.00 -I DEFLEC'flON(mm) -0.00 ...n ..... fl.-- ............... 0.0 06.0 120.0 100.0 240.0 305.0 300.0 CRANK murmurs”) Figure 4.23 The mid-span deflection of the connecting-rod. Mechanism operating speed 185 rpm 0'” 'r"Ir“"'T ''''' I"'Tri'ffr'3'r'r - W A ‘1 E . E V 2 E om-I 0 1 O O a . . O -0.80 -fir .. 0.0 06.0 ' ' ' Irzbbfi 7100.0 ' 340.0" (300.0 300.0 CRANK ANGLE(Degrees) Figure 4.24 The mid-span deflection of the connecting-rod. Mechanism operating speed 199 rpm 174 a.” 'thiwwwv'rwfff'vwTww’rvffww'w —thesreiled -- W DEFLECTION(mm) O 8 l 40m ‘I—VTV‘ rT—T'VY—r r‘r'T'TrY—TTVYrY VVT 0.0 60. 0 120.0 1M.0 240.0 300. 0 300.0 CRANK ANCLE(Degrees) Figure 4. 25 The mid-span deflection of the connecting- -ro.d Mechanism operating speed 258 rpm 4.6 CONCLUSIONS OF THE CHAPTER Analytical and computational models of general planar mechanisms with flexible links and bearing clearances have been developed. The dynamical characteristics of the mechanisms have been described by a proposed four-mode model. The nonlinear equations of motion can be SOIVed by a quasi-static procedure. As illustrative examples of the approach, a compound pendulum subjected to impact and a slider-crank mechanism with flexible connecting rod with radial clearance in the gudgeon-pin bearing were studied in a comprehensive analytical, computational and experimental program and the numerical results are in qualitative agreement with the results from a complementary experimental program. CHAPTER V CONCLUSIONS AND RECOMMENDATIONS IMPACT ON THE TECHNOLOGY—BASE AND THE SCIENCE-BASE The fundamental research prosecuted in this document has been undertaken in response to the insatiable demand in the international marketplace for viable design tools for predicting the dynamical and elastodynamical responses of articulating mechanical systems with radial clearances in the revolute joints of these systems. Such systems are extremely common and are numerous in the defense, automotive, aerospace and advanced manufacturing segments of the economy. The design tools developed herein can be employed in an industrial setting to predict the dynamic and elastodynamic responses of general planar mechanical systems for systems with one revolute joint with clearance and one flexible link. Futher experimental work is needed in order evaluate the predictive capabilities of this methodology developed in this thesis, when applied to more demanding systems with multiple flexible links and also several links with bearing clearances. Naturally, further work is required in order to bring a software package 175 176 to the marketplace, but this research program has undoubtably enhanced the technology-base in this field. In order to extend the technology-base, the research reported herein has involved extending the science-base. The science-base has been extended by the development of a four-mode model of the dynamically behavior of revolute joint with clearance which has been evaluated in several examples by comparing the theoretical results with experimental data. This work established a mathematical apparatus for each dynamical mode of the joint behavior, and also developed a viable simulation protocol for attaining a viable solution. Finally, new experimental evidence has been obtained by the monitoring of the dynamical and elastodynamical responses of planar mechanical systems with bearing clearances. These results should not only provide the basis for evaluating future mathematical models, but also motivate further theoretical studies on planar mechanisms with prescribed radial clearances in the joints. 5.1 CONCLUSIONS OF THE THESIS 5.1.1. The equations of motion of a general planar linkage mechanism with rigid links, or flexible links, featuring revolute joints with, or without bearing clearances have been developed by employing a general Lagrange's equations formulation. This formulation results in the establishment of a mixed set of partial differential equations and algebraic equations which must be solved in order to predict the dynamical behavior of the rigid—linked system. If the system comprises an assemblage of flexible members then the elastodynamic behavior of the 177 system is obtained by developing a mixed set of ordinary differential equations, using the displacement finite element method, and also algebraic constraint equations. These sets of equations are solved using the Adams numerical integration scheme. 5.1.2. A four-mode model has been proposed to describe the phenomenological behavior of a revolute joint with a prescribed radial clearance between the pin and the journal. Such a model is relevant to antifriction bearings which are worn, journal bearings, plain bearings and bushings which require a prescribed radial clearance in order to function correctly. The four modes are the contact-mode, the free- flight-mode, the impact—mode and the transient-mode. The four-mode formulation describes the dynamical response of every kind of intermittent motion. For example , the formulation can be applied to predict the dynamical response of an aircraft landing on an airport runway. The aircraft is in the free-flight-mode when flying in the air and then switches to the impact-mode when touching the ground. The simulation control transfers to the transient-mode when the aircraft bounces back into the air and then touch down on the runway again and again until finally this bouncing phenomenon ceases. Finally the simulation is in the contact-mode when the aircraft cruises down the runway. 5.1.3. The criteria for switching the modes have been carefully defined. The termination of the contact-mode is controlled by contact loss which is determined by the contact force between the pin and the journal in each joint. In addition, the contact loss is also determined 178 by the magnitude of the contact force and the rate of change of the force angle. Physically, the higher the rate of change of the force angle posseses bigger turbulent energy and contact loss happens when the contact force is small and can not maintain the contact any more. The termination of the free-flight—mode is controlled by the geometrical constraints of the mechanism from which two contact points in each members are at the same position. The contact restoration is determined by the fact that the contact point has not change very much even a free- flight-mode is assumed. 5.1.4. The initial values for the inertial generalized coordinate are determined by performing a dynamical analysis of the general planar mechanism without bearing clearance. For example, in slider-crank mechanism with bearing clearance in gudgeon-pin joint, the angular velocity of the position vector defining the clearance in the revolute joint with clearance is assumed to equal the rate of change of the angle of the contact force vector in that joint. 5.1.5. The initial values for the elastic generalized coordinates are determined by employing the first vibrational mode of the flexible link. The mode shapes of the flexible link be compatible with the boundary conditions of that member. For example, in the contact-mode, the flexible connecting rod of a slider-crank mechanism is modeled as simply-supported beam while in the free-flight-mode the link is modeled as pin—free beam. 179 5.1.6. In order to model the continuously varying natural frequency during the operating cycle of the mechanism, a geometrical nonlinearity formulation had to be incorporated into the formulation. 5.1.7. A quasi-static solution procedure has been employed to solve the inertial equations of motion and the elastic equations of motion in order to reduce the dimension of the equations of motion, and also the computing time. The error can be neglected when the time step is small enough. 5.2 RECOMMENDATIONS FOR FUTURE WORK 5.2.1. The methodology derived in this doctoral work can be generalized from planar two-dimensional systems to more complex spatial three-dimensional mechanical systems relevant to robot arms design and automotive suspension systems. 5.2.2. The plain journal bearing which was studied herein can be replaced by other kinds of bearings such as ball bearings, hydraulic thrust bearings, --,etc., in which the interaction between pin and journal is more complicated. 5.2.3. The contact stress in the bearing, fatigue life, failure criteria and the acoustic radiation resulting from a bearing with the clearance, are worth developing. However, the phenomena is quite complicated. 180 5.2.4. If the bearing is not well—aligned, an out-of—plane motion may occur. Thus it is necessary to extend the planar bearing analysis to study the three-dimensional spatial response of these bearings. 5.2.5. Throughout the theoretical work reported herein, the coefficient of restitution is experimentally determined and employed in the equations of motion. Thus the impact duration can not be entirely predicted theoretically. A more detailed formulation, such as the Kelvin-Voigt model may be included in the impact-mode to improve the analysis, but the stiffness constant and damping coefficient still have to be determined experimentally. 5.2.6. In this thesis the surface of the bearing was assumed smooth. Friction may be added in the formulation to improve the model if the effects of the roughness of the surface become serious. 5.2.7. In the illustrative examples, the mechanical system featured only one bearing with clearance. The research program may, therefore, be extended to more complicated mechanism systems featuring mechanisms with multiple bearing clearances and multiple flexible links. Again, a coherent program of analytical, computational and experimental work is needed. 5.2.8. The pin force and the relative motion between the pin and journal have not been measured in this study due to technical problems. A special design of the kinematic pair is needed, and high speed cinematography may be necessary. APPENDIX LIST OF PUBLICATIONS Liao, D.X., Sung, C.K., Thompson, B.S. and Soong K. "An Experimental Study to Document the Response Regimes of a Class of Flexible Four—bar Linkages." Ninth A lied Mechanisms Conference, Oklahoma State University Held In Kansas city, Oct.,1985. pp.III-l - III-6 Thompson, B.S., Soong, K. and Sung, C.K. ”An Experimental Bread-Board Model for a Class of Intelligent High—Speed Machinery: Some Preliminary Results" Ninth Applied Mechanisms Conference, Oklahoma State University Held in Kansas city, Oct.,1985. pp.IV—l - IV-4 Liao, D.X., Sung, C.K., Thompson, 8.8. and Soong, K. "A Note On The Quasi—Static Response, Dynamic Responses, and The Super—harmonic Resonances of Flexible Linkage: Some Experimental Results" ASME Paper No. 86—DET-146, ASME Conference, Columbus, OH, Oct. 1986. Soong, K., Sunappan, D. and Thompson, 8.8. ”The Elastodynamic Response Of A Class Of Intelligent Machinery, Part I, Theory" ASME Design Technology_ponferences, 1987. Held in Boston, Sep. - , DE-Vol.8, ADVANCED TOPICS IN VIBRATIONS, pp.147 - 154 Sunappan, D., Soong, K. and Thompson, 8.8. "The Elastodynamic Response Of A Class Of Intelligent Machinery, Part II, Computational and Experimental Results" ASME Desi Technology Conferences, 1987. Held in Boston, Sep. 27-30, DE-Vol. 8, ADVANCED TOPICS IN VIBRATIONS, pp.155 -160 Soong, K. and Thompson, 3.8. "An Experimental Investigation Of The Dynamic Response Of A Mechanical System With Bearing Clearance" ASME Desi n Technolo Conferences, 1987. Held in Boston, Sep. 27-30, DE-Vol. 10- ,ADVANCES IN DESIGN AUTOMATION -1987, Volume Two, Robotics, Mechanisms, and Machine Systems. pp.411 - 419 181 10. 11. 12. 182 Soong, K. and Thompson, 8.8. "An Experimental and Analytical Investigation of The Dynamic Response Of A Slider-Crank Mechanism Vith Controlled Radial Clearance At The Gudgeon-pin Joint." 10th Applied Mechanisms Conference. Held in New Orleans, Louisiana. Dec. 649, 1987, Vol. 3, pp. 9A-III.1 - 9A-III.9 Brown, J.M.B., Soong, K., Borthwick, V.K.D. Gandhi, M.V. and Thompson, 8.5. "Effects Of Intermittent Contact In Mechanisms Of The Performance Of Numerical Integration Methods." 10th Applied Mechanisms Conference. Held in New Orleans, Louisiana. Dec. 6—9, 1987, Vol. 3, pp. 8C-IV.1 - 8C—IV.7 Soong, K., Sunappan, D. and Thompson, 8.8. "The Elastodynamic Response Of A Class Of Intelligent Machinery, Part I, Theory" ASME Journal of Vibration, Acoustics, Stress and Reliability in Design, in press. Sunappan, D., Soong, K. and Thompson, 8.8. ”The Elastodynamic Response Of A Class Of Intelligent Machinery, Part II, Computational and Experimental Results" ASME Journal of Vibration, Acoustics, Stress and Reliability in Design, in press. Brown, J.M.B., Soong, K., Borthwick, V.K.D. Gandhi, M.V. and Thompson, 8.8. "The Effects Of Intermitten Contact In Flexible Mechanism Systems On the Performance Of Numerical Integration Methods." Mechanism and Machine Theory, under review. Soong, K. and Thompson, 8.8. "A Theoretical and Experimental Investigation of the Dynamic Response of Planar Mechanisms with Radial Clearances in the Revolute Joints." (Accepted by ASME Mechanism Conference. To be held in Orlando, Florida, Sep. 25 - 28, 1988) AN EXPERIMENTAL STUDY T0 DOCUMENT THE RESPONSE REGIMES OF A CLASS OF FLEXIBLE FOUR-BAR LINKAGES 1(33 ummmmmmmrmm GAMWWm-MILIMH + 0.x. Lino'. 0.x. Sung . 0.0. run-0000', and r. soong“ Department of lbchanical Engineering Michigan State University last Lansing. Mi 48824-1226 .Visiting Scholar and Associate Professor, haahomg University of Science and Toclulology. Uahsn, Chins *Craduate Iasearcb Assistant ‘Associate Professor *Graduate Student 52222255 This investigation focuses on the response of flexible four bar linkages coqrising two flexible links, the coupler link and the rocker link. The crank and ground link are both taken to be rigid bodies. An experimental apparatus and the asso- ciated instrusntation is described for studying the midspan transverse deflections of flexible coupler and rocker links of four bar mechanisms for a wide range of operating speeds and link length combina- tions. hence a range of forcing frequencies and natural frequencies were studied. The results are documented and presented in this paper which demonstrates that the vibrational characteristics may be grouped into quasi-static. and dynamic regimes. and also conditions of resonance. issssssssisn A debate has raged in academic circles for some years concerning the modeling of flexible linkage mechanisms. The focus has primarily concerned the nature of the flexural response of the members. and to some extent a secondary debate has centered on the frequency response of those flexible arti- culating systems. Since much of the discussion has been based on theoretical presentations. the authors have attempted herein. to furnish the participants in the debating chamber with some experimental results in order to provide an unbiased foundation for further interactions. The main theme of the debate has centered on whether the analyst must undertake an elastodynamic analysis or a quasi-static analysis for predicting the response of a flexible mechanism “-31. The first class of analyses are computationally more time-consuming, and hence expensive. than the second group of techniques but they provide more accurate results from a mathematical standpoint. The difference in the two formulations may be illustrated by considering the finite element equations for a general flexible linkage mechanism. which may be written as. [Kliul . IclIfiI . IHIIUI - IQI (I) In equation (I). [I]. [C] and [M] are the global stiffness. daqing and mass matrices, column vector {0} represents the loading imposed on the system. {0} represents the discretized deformation dis- placement fiold. and the overdot denotes the time derivative. Equation (1) is generally solved in its entirety by numerical integration or nodal super- position techniquea to yield the elastodynamic III response of the system. In contrast to this approach. a quasi-static solution may be sought by siqlifying equation (I). This approximate solution is sought by_ first neglecting the terms characterised by {U} and W}. prior to solving the equation [KIWI ' {01 (Z) by Gaussian elimination or some other technique in linear algebra. This philosophy models a dynamic rystom as a static system, and hence is charac- torised by the term "quasi-static.” The floxural response of a flexible linkage mechanism is generally the dominate mode of vibra- tion. The waveform cmrises a periodic response at the operating frequency of the mechanism, upon which is superimposed a higher frequency component. From a mathematical standpoint, a quasi-static analysis will only predict the periodic response at the lower frequency, while an elastodynamic response will predict both the low-frequency component and also the high-frequency response superimposed upon it. Hence when the amplitude of the high-frequency component is small, the elastodynamic response and the quasi-static response are nominally the same. Under those conditions the designer can legitimately siqlify the mathematical model and hence reduce the computational costs associated with the analysis. The fundamental question, the kernel of the debate. centers upon the conditions under which this aiqlification is admissible. Raving described the salient features of the debate on the elastodynamic and quasi-static responses of flexible mechanisms. attention now focussos on the second, somewhat less volatile topic for discussion which centers on the frequencies at which flexible mechanisms respond. Modeling the frequency response of a flexible multi-link mech- anism involves considerations of the forcing frequency (.the crank speed) and the natural fro- quencios of the individual links in their anti- cipated response mode (floxural. torsional. etc.) and the natural frequencies of the assemblage of flexible bodies. The situation is colpllcmlwd somewhat by the periodically changing configuration of the mechanism as the assemblage articulates. unci this presents one of the fundamental difforomes between a structure. such as a portal frame, and a linkage mechanism. The final complication is associated with tho exaggerated flexibIlILy ul‘ Ina-"Iv laboratory systems which introduces additional nonlinear phenomena l4-6]. The purpose of this paper is In prusvm experimental data relevant to the turn Ila-oat... described above. This is accomplished Iw ”pa-rating 184 a class of flexible four bar linkages through a range of crank speeds and recording the resulting floxural responses of the members. Specifically. the investigation focuses on the response of flexible four bar linkages with flexible coupler and rocker links. The midspan transvarse deflections of these flexible links are presented for a wide range of operating speeds and link length combinations. Thus a range of forcing frequencies and natural frequencies are investigated. Experimental Apparatus A photograph of the experimental four bar linkage apparatus used in this study is presented in figure l. from which it is clear that it incor- porated two flexible members, the coupler and the rocker links. This apparatus was designed to accommodate a wide range of link lengths for the coupler and rocker links. but the crank and ground link were always held constant at 63.5mm and “2.8mm. respectively. The crank was assumed to be rigid since it was manufactured from steel bar stock of cross-sectional dimension-Ila 25.4- x 25.4-. Fig. 1 Photograph of Experimental four-bar Linkage Three classes of mechanism were examined in this study and those kinematic chains are defined in Table l. The rocker and coupler links were both manufactured from steel strip material and link specimens were interchangeable so that a variety of kinematic chains could be studied. Two classes of steel strip were employed with the following length, width and depth dimensions: 268.0 - x l9.05 - x I.“ m and 165.0 - x 25.4 - x l.14 -. The coupler and rocker links of the experi- mental four-bar linkage apparatus were both manu- factured from strip steel material and the ends of each specimen were clamped to the respective bearing housings by two socket screws. The clamping loads were distributed over the ends by flat plates which are clearly visible on the coupler link in Figure l. These small clamping plates were found to be essential components of the mechanism since they ensured a smooth load transfer between the three principal components of each link. fable II Llsaege Classlllsat Ion I I I I I I I unless I Cree-l “at (-l I Creek (.0 I temples (.0 I lea-v I-) l .floeslflrauom I I I I I I “or I I I I I I I I l I I I I I I I I I I I I I I , I . IIIJ . 01.! . Ios.s . l9.) . I I I I I I I I I I I I I I I I I I I I I I I I , I . “2.0 . ”.0 . me.o . 200.0 . l I I l I I I I I I I I I l I I I I I I I I I I . I . "1.0 . 00.5 . some . Isms . I I I I l I I I I I I I Identical aluminum bearing housings were manufactured for the rocker link where it was retained on the ground-linklrocker joint, and also for the coupler where it was retained on the crank pin joint. The dimension of the housings in the longitudinal direction of their respective links were 38.]- from the end of the housing adjacent to the threaded holes for the socket screws and the centerline of the bearings. The masses of these components including bearings, socket screws and clamping plates was 0.05kg. The coupler and rocker links were supported on matched pairs of 6.4mm bore K3 DB 312 instrument ball bearings supplied by PAC Bearings Limited. Each bearing housing in the mechanism was preloaded using a Dresser Industries torque limiting screw driver calibrated to 20.113 M-m which permitted bearing clearance to be eliminated since the impactive loading associated with hearing clearances would have resulted in larger link deflections. Conversely, if the bearings were subjected to large axial preloads, then the deflections would be attenuated. The two flexible links articulated in the same plane in order to eliminate the complications of nonlinear torsional coupling terms which charac- terize co-planar flexible linkages. This was accomplished using a cleavage bearing design that is clearly visible in the top center of Figure l. The housings were fabricated in a aluminum alloy and the longitudinal dimension of the housing comprising part of the coupler was 38.lmm from the bearing centerlino to the end of the housing adjacent to the threaded holes used to clamp the link specimen. The mass of this assembly, including bearings. socket screws and clamping plate was 0.052 kg. The cloavagod component of the coupler-rocker joint was bolted to the rocker link. The axial dimension of this part from the bearing conterline was 44.5 mm and the mass was 0.063 kg including the spindle. washers and nuts. The mechanism was bolted to a large cast-iron test stand which was bolted to the floor and also to the wall of the laboratory to provide a substantial rigid foundation. A 0.75 h.p. Dayton variable speed D-C electric motor (model 22846) which was bolted to the test stand, powered the linkage through a [5.9 mm diameter shaft supported by Fafnir pillow block bearings. A 100 mm diameter flywheel was keyed to the shaft thereby providing a large inertia to ensure a constant crank frequency, when operating in III 2 unison with the motor's speed controller. 185 lgtmtation A schematic of the instrumentation used in the experimental investigation is presented in figure 2. The rated speed of the electric motor was 2500 rpm and this was directly measured in revolutions per minute by a Hewlett Packard 53le universal counter which was activated by a digital-magnetic pickup. model ”‘23. manufactured by Electro Corporation sensing a sixty tooth spur gear mounted on the drive shaft of each rig. This large gear is clearly visible in figure l. The aforementioned arrangement provided visual feedback to the operator when the speed controller of the motor was being adjusted to establish a desired speed. “males-Mia's fig. 2 Schematic Diagram of Instrumentation The floxural deflections at the midspan of each link specimen were monitored by Iii... -..":- Gl'oups Inc. type EA-06-125A0-120 strain gages bonded to each link at the midspsn location. bending hIlf‘bridge configurations were adopted. using one use on each side of the specimen. and they were used in conjunction with a hicromeasurementa Group Inc” strain gage conditionerlamplifier system. I”. 2100. In addition. the deflection at the ”IO-quarter span location was also monitored by "rain gauges.‘ This transducer arrangement was c‘I'Ieidered important in order to monitor potential 'Ocond-mode floxural responses of the links. In order to relate the strain gage signal to “I. configuration of the particular mechanism being "Udied. a third transducer arrangement was esta- bl‘mhed. An Airpax type 16-0001 zero velocity d1 81tal pickup was employed to sense the bolt head at the end of the crank when the mechanism was in ‘1‘- convectionel taro-degree crank angle position. 11‘1- long hexagonal transducer is again clearly H.1.ble in figure 1. This mechanism configuration signal and the °‘"’- Put from the gages were fed to a Digital Equip- "“t Corporation PDPll/03 microcomputer with a 1.3! “’23 processor. This digital-data-acquisition '7' tom featured 256 kl of memory for post-processing d". and also a 5 ml hard disk for storage. A dual 9"" floppy disk system was also available. The Digital Equipment Corporation VTlOO terminal with rotrogrsphics enhancmt. The IIC cables from each experimental apparatus were connected to a input-output module. bolted to the cabinet of the coquter. which was built by the Electronic and Cowuter Services Department at ISU. This instrument featured 16 analog-digital channels. 6 digital-analog channels and two Schmidt triggers. Using software developed specifically for digital data acquisition purposes. the flexural response signal was recorded from the sore crank angle position through 360 degrees by firing one of the Schmidt triggers. In order to activate the trigger. a 100 u! capacity was used to modify the square-wave output from the Airpax pickup. Experimental results were obtained by inle- menting the following procedure. The signal from the strain gage instrumentation was passed through a first-order analog low-pass filter in order to remove the electrically or mechanically inducted noise prior to being digitised by the analog-to- digital converter and recorded by the PO? 11103 microcomputer. This low-pass filter also prevented aliasing problems when employing a rectangular data window. and a sampling rate of (257!) seconds. which was considered to be at least twice the highest frequency component of the pure elasto- . dynamic response signal. Results The experimental results contained in Figures 3-0 present the flexural response of the coupler and rocker links over a wide range of crank frequencies for a 'vsristy of link-length combinations. Upon undertaking a cursory review of these results it is evident that a broad spectrum of response histories are presented. and clearly some classification is desirable in order to aid the design engineer. A classification is defined herein in an arbitrary manner to characterise each response regime. leagues Classi ficat ion The floxural response of four-bar linksges may be classified by three response regimes. and these are now defined (albeit. somewhat arbitrarily) for the purposes of this study. Dynamic Response This response history is charac- terised by two waveforms comprising a low-frequency periodic waveform (at the crank frequency) upon which is superimposed a high-frequency sinusoidal waveform at spproaimately the same frequency of the link being monitored. The amplitude of the high frequency component is generally of the same order of magnitude as the amplitude of the low-frequencx component. Quasi-Static Rospgpse This response regime is only characterised by the low-frequency periodic waveform at the crank frequency. for the purposes of this study. responses were classified as "quasi-static" when the amplitude of any high-frequency ripple superimposed upon the lowrfrequency carrier was less than 10 percent of the amplitude of the carrier response. Resonance Conditions of resonance were defined bv the amplitude of the response at a specific crank frequency being significantly large: then the amplitude of the response at adjacent crank fre- quencies. while this appears to be a somewhat unscientific and an ill-defined stntuneni, in -practics it was quite appropriate for the phenomena “PQrimental response curves were displsyrd 0“ J 1"’obaurved in thin study. Misses Transverse Deflection in) Mdspea I “Harm in; d 4 asanorIJuI I (Q‘s-lemmas) m 9.: 5.1 1 E‘s : . . "‘8'“ V 33 : t -d 1 ..e:‘1—fwv VVVfi'UTfiTVY i Ii. ass I. II Cnstlngleiml Fig. 5 Class 2 Mechanism; Coupler and Rocker JALA Quasi-static (at as ml aha s s 3. 1136 I V r Crass hale (ms) Fig. 3 Class I nechanism; Rocker Link Hidspsn Transverse Deflection: at Three Operating Speeds "Fm Dynamic Cassi-static ‘ (Kt 2“ ml (at ass I") (at Ill rpm) . a . l A // Ii In! I 1 \"H ":4 V V U V TV V i 2;“.-.1..r..~-..-”--fi“ Crank Angle (Degrees) Fig. 4 Class 2 Mechanism; Coupler Link Hidspan Transverse Dcflsctions at Three Operating Hidspan Transverse Deflections at 268 rpn 111.4 1 Coupler Response lsspoess (Quasi-Static) 3". (Issuance) 3.. ,- 1 ’ i i... I V i -"". 38 .-‘ 3. “'1 5' j‘L-A ""3 .3 5 "~. ’ / U 35 1i ..-' a f 5 2.» - 13 s a A \J E? 8 -e.: - . - “ 4 ' V 'w' i . q 1 'Iv: v v r v v v v v r v v v s *7 V u we ass as «Is Crass Angle (Decrees) Fig. 6 Class 1 Mechanism: Coupler and Rocker Hidspan Transverse Daflsctions at 32l rpa I.: Coupler Response locker Issassse j (must-static) (quasi-static) d 3‘; ‘ L g m 8"” ' ‘ i3 . and a d ....4 3‘; 4 “x f ‘-. 8 - a .. ‘ "’ 3 °" \ F I \oc‘x ' sue a «Is casvxnnsh-innrvss) Fig. 7 Class l Hochanise; Coupler and Rocker Hidspan Transverse Daflsctions at 357 rpm L: Coupler Response locker Response 1 \(ressnsacs) / (quasi-static) 0.: E? 52’ 3... £5 a I I V V V Y T V r V VTrr I00 Crass Angle (0090.!) Fig. 8 Class 3 Hechanisn; Coupler and Rocker Hid- span Transverse Deflections at ZBl rpm 187 Discussion Figures 3-11 present only a glimpse of the data collected during this experimental investigation. but nevertheless these response histories permit some interesting observations to be recorded. For example. Figure 3 presents the midspan floxural responses of the rocker link of a class 1 mechanism operating at 23‘. 2” and 320 rpm. Two classes of response are presented. with the amplitude of the response condition at 320 rpm (a condition of resonance) being greater than that at 23‘ rpm (a second state of resonance). Figure 6 presents a similar set of data for the midspan flexural response of the coupler link of a class 2 mechanism. Three classes of response regime are presented in Figure I with the resonance condition at 264 rpm dominating. The amplitude of the vibrations at ass rpm are larger than the amplitude of the dynamic response at 25‘ rpm and the quasi-static response at 300 rpm. Figures 5.0 each present simultaneously the response histories of both the coupler and rocker links of a variety of mechanisms operating at different creek frequencies in order to demonstrate the variety of response regimes developed by this class of four-bar linkage. Thus for example. Figure 5 presents the response of a Class 1 of mechanism in which both the coupler and rocker simultaneously develop a dynamic response. Figure 6 records an interesting condition whereby at 321 rpm a Class 1 mechanism exhibits a resonant condition of the rocker link while the coupler responds quasi- statically. At 357 r1- the same mechanism develops a quasi-static rocker response and also a quasi- static coupler response. and these are presented in Figure 7. Figure 0 presents the response of a class 1 mechanism operating at 201 rpm where the rocker exhibits a quasi-static response while the coupler is in a state of resonance. Thus. it is clearly evident the loperating frequency significantly affects the response behavior. Figures 9-11 present the response regimes of some of the links studied in this investigation. For the mechanisms systems tested. the flexural response was generally of a dynamic nature with resonance conditions observed at the indicated speeds. Quasi-static regimes were generally confined to narrow bands. but this was not univer- sally the case as indicated by Figure 9 where this link did not develop a quasi-static response. The observed resonance conditions are also indicated. I“ ‘0’“ 0.0““ Fig. 9 Class 3 Hechanisn; Coupler Response Regimes 111.5 m to“ g ".l I-W“ a: s a l s'] ' n-msmmmm III! ms:7-— In“. Im In as uses.I 1. “mm Fig. 10 Class 2 Mechanism; Coupler Response Isgiees I.“ Comet-us . ' I . ll .i . - - - -..-.:... ssasas as a I I ' Iii? / I Hfl r ’ emma _“r‘ fl I humid Fig. 11 Class 1 Hechanisn: Rocker Response Regimes Conclusions An experimental study of flexible four-bar linkages has been presented which documents the different classes of floxural midspan elastodynamic responses of the coupler and rocker links. it is evident that these responses comprise three types: quasi-static. dynamic and resonant. Moreover. it was also observed that adjacent links may be responding quite differently. Thus. for example. the response of a coupler link may be classified as quasi-static response. while at the same speed the adjacent rocker link may be in a state of resonance. The development of theoretical tools to explain these nonlinear vibrational phenomena appear to be quite challenging. References 1. Hillmert. K.D.. Thornton. H.A. and Khan. H.k.. ”A Hierarchy of hethods for Analysis of Elastic Mechanism with Design Application.” ASME Paper No. 78-021-56. 2. Sanders J.R.. and Tesar. 0.. "The Analytical and Experimental Evaluation of Viblutorv Oscil- lations in Realistically Propnriinned Mech- anisms." ASHE, Journal of Hechanic~u_i__1)_e_s~1_gn. Vol. 100. p. 762 (1978). 188 3. lisines. 1.8.. ”On Predicting Vibrations in Realistically Froportionsd Linkage Mechanisms.” ASME. Journal of Mechanical Design. Vol. 103. p. 706 (1901). ' 6. Thompson. I.8.. and Sung. C.E.. "A Variational Formulation for the lonlinsar Finite Element Analysis sf Flexible Linkages: Theory. lwlsmen- tation. and Experimental Insults.“ ASME. Journal of Mechanisms. Transmissions. and Automation in Design. Vol. 106. p. “2 (190A). 3. Sung. C.K.. Thompson. 2.3.. Xing. T.M.. and Hang. c.n.. "Am Experimental Study on the lonlinsar Elastodynamic Response of Linkage Mechanisms.” Mechanism and Machine Theorz (in press) (1905). 6. Turcic. D.A.. and Midha. A.. "Dye-ic Analysis of Elastic Mechanism Systems. Fart I: Applica- tisn. and Fart 1!: Experimental lesult.” ASME. f stem Measurements and Control. Vol. 106. p. 2‘9 (190‘). 111.6 AN EXPERIMENTAL BREAD—BOARD MODEL FOR A CLASS OF INTELLIGENT HIGH-SPEED MACHINERY: SOME PRELIMINARY RESULTS 189 “MMMWMMAMOFWM MICE-”ED W: 3.8. Thoqson’. E. soong*. and C.E. Sung' Department of Mechanical Engineering Michigan State University East Lansing. MT m2b-l226 *Aesociate Professor *Oraduate Student ‘Oraduate Research Assistant am; The elastodynamic response of high-speed machinery is governed by the mass. stiffness and damping characteristics of the links. and also the inertial loading inoeed upon these members. For a specific mechanism. the physical link character- istics are of course invariant and the response is governed by the kinematic variables upon which the inertial loading depends. heroin. preliminary work is described which concerns the control of the inertial loading imposed on flexible mechanisms in order to achieve a desired response. The philosophy is directed towards developing counter-controlled intelligent machinery which are capable of both reducing link deflections and also operating with more versatile path-generation capabilities. Introduct ion The vast majority of industrial machine systems have one rigid-body degree of freedom and one power source which are responsible for creating a single distinct output characteristic. This output characteristic will generally satisfy the criteria imposed on the performance of the machine. but the machine will be generally dedicated to performing a limited class of operations. This paper reports on some preliminary work directed towards stimulating the evolution of a new generation of computer-controlled intelligent machine systems which are more versatile. more adaptable and more cost-effective than the current generation of machine systems available in the marketplace. This new generation of products will enable single machines to generate a number of different classes of performance characteristics that currently must be undertaken by several individual machines. An example of this is the variety of different coupler curves generated by the tracer points of several different four-bar linkages. The work described herein is directed towards replacing these different classes of four-bar linkages by a reduced number of more versatile mechanisms which can generate all of the coupler curves of the separate four-bar mechanisms. A second class of problem concerns the synthe- sis of high-speed machinery. These systems develop elastodynamic effects which may contaminate the performance characteristics and prevent the machine from satisfying prescribed design specifications. Alternatively. bearing degradation may result in a machine operating outside the perfornsnce envelope after operating for a period of time. The work reported herein is directed towards resolving theseIV_1 situations by developing a methodology for designing intelligent machines. The methodology requires the machine's performance to be continually monitored by sensors on the principal elements of the system. with the response data being fed to a controller or microprocessor which has the capability of modifying the machine's performance by activating a secondary force input to the system. Thus the output charac- teristics would then be the result of a primary and a secondary excitation. Dy this means it is anticipated that the desired response would be achieved. While this approach to adaptive control is the goal of this project. currently an open-loop mode of operation is employed rather than the closed-loop philosophy proposed above. Morein the current status of the project is documented and some preliminary results presented. Frmaed Methodology The proposed methodology is based on the approach described in reference [1]. which utilises a four-bar linkage in which the constraints imposed at the rocker/ground-link joint are relaxed. This relaxation involves replacing this normally sta- tionary revolute joint by a sliding joint which permits translational motion to occur in the horirontal direction 0! as shown schematically in Fig. 1. This modification creates a linkage with two rigid-body degreesv-of-freedom and hence provides the option for two inputs to the mechanism: one due to the crank rotation and one due to a prescribed motion at the sliding joint. The nature of those excitations govern the performance characteristics of the linkage. Fig. 1 Intelligent Mechanism: Chain Basic Kincnatic PatheCeneration Capabilities Consider. for example. the operation of a four-bar linkage in a path-generation mode where the tracer-point is located at the midspan of the coupler link. A single coupler curve is generated by this nechanism system. However. if the length of the ground link is either lengthened or shortened to create a nueber of discrete ground link lengths: then a number of coupler curves may be developed as indicated in Fig. 2. Moreover. by employing the single system described schematically in Fig. 1. then any of the these curves may be generated by first changing the ground link length to a pre- scribed value and then activating the mechanism. The proposed system has the additional advantage of permitting any point to within the work-envelope presented in Fig. 2 to be accessed by the tracer- point. This is achieved by coordinating the motions of the slider and crank members of the system described in Fig. 1. lot only can these points be accessed. but the two degrees of freedom permit a variety of coupler-curves within that work- envalope to be generated. These inputs may. of course. be computer-controlled. Fig. 2 Family of Coupler Curves Developed by the Intelligent Mechanism .- J --..-gol Pan-hi l 1 time The motion of the tracer-point of a high-speed. or flexible. linkage will not generate the smooth coupler curves shown in Fig. 2. Instead. an elasto- dynamic deflection will be imposed on the curves. as indicated in Fig. 3. It is clearly plausible for this deflection and the associated stresses to be unacceptable. because of potential fatigue failure or inaccurate coupler-curve generation. Under these circumstances. the linkage system proposed in Fig. 1 may provide a solution whereby the motion of the slider would be computer-controlled in a prescribed manner in order to reduce the deflections of the coupler midspan. Thus the forcing function exciting the response of the link would be fine-tuned through a careful orchestration of the crank motion and the slider motion. From a matheeetical standpoint the boundary conditions and force fields imposed upon the assemblage of articulating beam-shaped bodies are being continuously modified. Thus a variety of dynamical mixed boundary-value problems are being solved to create a suite of response character- istics. 190 Iat- Pres-use s. hm use-s lI—tlt hairs-t bun-Mails lea—as laser-Cam h Itete-u-ad-y “ . Hansen-is Isa-ass mm b Clgla LII-es Des.- IPig. 3 Coupler Curve Components for a Flexible Four-Bar Linkage This philosophy is examined herein by studying the response of an experimental four-bar linkage The midspan.tranaverse deflection of the coupler is first recorded for the system operating in a Then the shaker is excited. using a waveform based on the quasi-static response of the coupler. in order to attenuate the with a flexible coupler link (see Fig. A). conventional four-bar mode. amplitude of this deflection. Fig. l Photograph of Intelligent Mechanism Model. Hhilo the mechanism was designed to accommodate this investigation was undertaken with an initial ground link length of 381 mm. a crank length of 50 mm. a coupler of length a range of link lengths. - A s . and ' ‘ The proposed class of adaptive presented in Figs. h and S. Figur project. but at this time the system runs IV.Z of the mechanism. four—bar linkages was studied by constructing the system a 5 schematically presents the final system configuration for the open-loop. with the controller module being the investigator. Gravitational loading is perpendicular to the plane 191 293.6 - and a rocker of length ”3 -. The coupler was a very flexible link of cross-sectional dimen- sions 0.030 - x 10.2 - while the rocker was a nth more substantial member of cross-sectional dimen- sions 6.23 n x 19.63 -. Elva-sued *I H Fig. 3 Final Syst- Configuration The bearings of the system were carefully preloaded to eliminate radial clearance and the couplerlrocker joint was a cleavage design. One end of the rocker link was connected to a Microslides Inc.. type A92020 cross-roller slide assembly which permitted translational motion in the ox direction in Fig. l to occur. This precision assembly was then bolted to a Iruol and Ejaer force transducer. type 8200. which permitted the force exerted by the electrodynamic shaker on the mechanism to be carefully monitored. This shaker was an air-cooled Ling Dynamic Systems. type V611 sxcitsr. The acceleration imposed by the exciter upon the slide-assembly was monitored by a Iruel and Ejaer accelerometer. type 6731. mounted on the slide-assembly. The output from this transducer was fed to a Iruel and Kjaer conditioning anlifier. type 2633. which enabled the velocity and displacement characteristics of the slide-assembly to be monitored. The output from the transducers were either fed to an oscilloscope for preliminary evaluation of the response data. or fed to a DEC PDF lll03 micro- computer for storage and post-processing using FFT algorithms. 1V.3 E_xmrimsntal Procedure, Eesults and Discussion The experimental procedure involved first operating the mechanism at a constant crank fre- quency with the cross-rolled slide assembly clamped in order to create a classical four-bar linkage system. Strain gauges mounted at the midspan of the coupler monitored the lateral deflections of the link and the data was fed through an analog filter prior to storing in the PDF 11103 microcomter. Dpen coepleting a digital filtering operation and multiplying the resulting data by a link calibration factor relating the transverse deflection and the corresponding voltage. the resulting response history for the mechanism operating at 169 rpm is presented in Fig. 6. The waveform comprises two components: the quasi-static response at the nechanism's operating frequency and a high-frequency component at approximately the link's fundamental natural frequency in flexure. It may be classified as an elastodynamic response. l.- d I ..- I a l 4 {ms-1 of I j . ’4.“ :1 -'e: as?‘ Vrrr Vtefi‘vvva u In no .1- «s celluloid-sun! Fig. 6 Coupler Midspan Transverse Deflection of Four-bar Linkage In this current open-loop mode of operation. this data was then post-processed again by simu- lating a low-pass digital filter in order to yield the quasi-static response of the coupler at 189 rpn prior to progra-ing the data into a Vavetek digital arbitrary waveform generator. model 275. This device permits arbitrary waveforms to be generated for a wide range of frequencies. For this study the generator was progra-ed to synthesise the inverse of the quasi-static response of the coupler midspan transverse deflection. This waveform was proposed as being a viable motion with which to excite the shaker in this inverse force-deflection problem. although there is little juatification other than the algorithm proposed in reference [1]. Thus. using the waveform described by the inverse of the quasi-static deflection. the shaker was then excited at the crank frequency (189 1pm). Thus the coupler link was being excited by an inertial force field. due to the notion of the articulating mechanism. and also a second input due to the force field imposed by the shaker. These combined excitations. in addition to the different boundary conditions at the ground-link/rocker joint which create a kinematic chain with a time-dependent configuration. yield the results presented in Fig. 7. This is a quasi-static response. 192 \ / llll Alli .1 1". ‘\] ass sis «a 'hmaamhiamem) All] Fig. 7 Coupler Midspan Transverse Deflection of "Four-Isr Linkage” with Shaker Activated This response history is for the coupler midspan transverse deflection at 189 rpm. The high-frequency component is absent and the maximum deflection has been reduced by 7 percent through the shaker excitation of the modified linkage. The resultant force at the shaker/slider connection. as monitored by the Druel and Ejaer force transducer. is presented in Fig. 8. This force profile is comprised of a force component due to the arti- culating motion of the mechanism and also the force component generated by exciting the shaker. The corresponding displacement profile of the crossed- roller slide assembly developed during one revolu- tion of the crank at 3.13 Mr (169 rpm) is presented in Fig. 9. I‘ll hueflhlmu) -.~_~ < pr"""”"r / N \8 / Gillflhiflilfl Fig. 8 Force lesponse at the Shaker/Mechanic Interface IV.‘ These preliminary experimental results provide a boost in confidence for the original proposition that a more versatile class of machines can be developed. based on the four-bar linkage chain with an additional input at the normally fixed ground- linklrocker joint. Clearly more work must be undertaken to document the control algorithm for the active control of these distributed parameter systems since the input from the shaker must be carefully orchestrated with the primary forcing function provided by the crank motion. in order to avoid combination resonances and other undesirable elasuuhnuedc phenomena. D lJ \_/\\ ' V -6.: . . .~ . . . . u~ r . . . . . . .7 0 ID. 200 300 608 amelmbimmen) hawtflflu-mthfl O 2. ALAJ .4- < < ALIA Fig. 9 Displac-ant Iesponso of the Reciprocating Slide Asa-bly Conclusions An experimental investigation into a class of adaptive flexible linked mochanises with two forcing inputs has been presented. by operating in an open-loop mode. this bread-board model for a class of intelligent machines has demonstrated that link deflections may be reduced by controlling the force. excitation of the linkage system using an electro- dynamic shaker. Acknowledggggnts The authors wish to acknowledge with thanks the financial support for this project in the Machinery Elastodynamics Laboratory that has been provided by the Division of Engineering Research at Michigan State University. References 1. Oliver. J.M.. Vysocki. D.A.. and Thompson. 8.8.. "The Synthesis of Flexible Linkage by Balancing the Tracer Point Quasi-Static Deflection’s using Microprocessor and Advanced Materials Tech- nologiea.” Mechanism and Machine Theory. Vol. 20. Mo. 2. pp. 103-116. 1983. A NOTE ON THE QUASI-STATE RESPONSE, DYNAMIC RESPONSES, AND THE SUPER-HARMONIC RESONANCES OF FLEXIBLE LINKAGE: SOME EXPERIMENTAL RESULTS 193 :IvM ' UWuMIBW ‘ “bimbo-Immune” , t p. . - ‘rlinW—‘ 3L.“ rliv ', .. A Note. on the Quasi-Static Responses. Dynamic Responses, andlhe’Super-Harmonic Resonances of Flexible Linkages: Some Experimental Results 0. X. LIAO. C. K. 30MB. 0. S. THOMPSON. Ind K. SWIG ol new Slim Michigan sun u ' ' East Lansing. MI 43324-1226 ABSTRACA This paper reports on sevsral experimentally- observed vibration phenomena in flexible four-bar linkages that have not appeared before in the liter- modela. will. it is anticipated. trigger renewed theo- retical activities in this aspect of linkage design. The investigation focuses on the response of flexible four-bar linkages coeprising two flexible links. the coupler link and the rocker link. An experimental approach is described for measuring the midspan transverse deflections of these links for a wide range of operating speeds and different link length “ ‘ The . ‘ ' response data presented herein demonstrates that the vibrational characteristics at the link midspans may be grouped into quasi-static and dynamic regimes. and also conditions of super-harmonic resonance. INTRODUCTION A debate has raged in academic circles for some years ‘ _ t e . ‘ ‘ ‘ ‘ a paratus for modeling flexible linkage aechanisms. The focus has primarily concerned the nature of the floxural response of the members. and to some extent a secondary debate has centered on the frequency response of these flexible articulating systems. re much of the discussion has been based on theoretical presentations. the authors mental results which should provide an unbiased foundation for further academic interactions. The focus of the debate has centered on whether the analyst must undertake an elastodynamic analysis. or a quasi-static analysis. for predicting the response of a flexible nechanism [1-3). class of analyses are computationally more time- consuming. and hence expensive, than the second group of techniques but they provide more accurate results from a mathematical standpoint. The difference in the two formulations say be illustrated by considering the finite element equations of motion for a general flexible linkage mechanism. which may be written as (Kill!) e [CNN 1» [111(0) - (Q) . (1) In equation (1). [I]. [C] and [M] are the global stiffness. damping and mass matrices respectively. column vector (Q) represents the loading imposed on the system. (0) , the di ‘ ‘ ‘ ‘ displacement field. and the overdot denotes the time derivative. Equation (1) is generally solved in its entirety by numerical integration or model super- position tschniquss to yield the elastodynamic response of the system. In contrast to this approach. a quasi-static solution may be sought by simplifying equation (1). This approximate solution is sought by first neglecting the terms characterized by (U) and (U). prior to solving the equation [KHW ' (Q) (2) by Gaussian elimination or some other technique in linear algebra. This philosophy models a dynamic systen as a static system. and hence is characterized by the term "quasi-static. Th floxural response of a flexible linkage mechanism is the dominant mode of vibration. The waveform often comprises a periodic response at the operating frequency of the mechanism. upon which is superimposed a higher frequency coeponent. From a mathematical standpoint. a quasi-static analysis will only predict the periodic response at the lower frequency. while an elastodynamic response will predict both the low-frequency component and also the high- frequency response superimposed upon it. Hence when the amplitude of the high-frequency component is snsll. the elastodynamic response and the quasi-static response are noeinally the same. Under these condi- tions the designer can legitimately simplify the mathematical model and hence reduce the coeputational costs associated with the analysis. The fundanental Presented at the Design Engineering Tecnnlcm Conlevsnco Colunvous. onlo — October so I“. 196 question. the kernel of the debate. centers upon the conditions under which this siwlificstion is admis- sible. At this time. there are no design guidelines to provide assistance with this decision. llaving described the salient features of the of flexible nechanisms. attention now f second. somewhat less volatile topic for discussion which centers on critical operating speeds and the frequencies at which flexible nechanisns respond. Modeling the frequency response of a flexible multi- link mechanism involves considerations ,of the forcing frequency (the crank speed). the natural frequencies of the individual links in their anticipated response mode- (flsxural. torsional. etc. ). the net tmral frequen- cies of the assemblage of flexible bodies. The situation is complicated somewhat by the periodically changing configuration of the mechanism as the assen- blage articulates. and this presents one of the funda- mental differences between a structure. such as a portal frame. and a linkage mechanism. The early work in this area reported on linkages with only one flexible menber (6-6]. The approaches generally involved transforming the governing linkage equations into a set of uncoupled liathisu-I'lill's equations. This class of equations have received considerable attention in the literature. leferencss [7] and [0) presented and used Nathieu-Iiill equations and (9) used floquet theory for the solution of the liill's equations resulting free the analysis of a four-bar mechanism with an elastic rocker link. Subsequently finite element analyses have been devel- oped for predicting the critical running speeds and dynanic instabilities in linkages conprising several flexible links [lo-ll). While these mathematical nodels are indeed valuable contributions to the nechanisna literature. they are unable to predict the experimentally-observed nonlinear phenomena reported herein. it is therefore anticipated that these experimental results will hopefully catalyse further theoretical studies in this area. The experimental results were obtained by opera- ting a class of flexible four-bar linkages through a range of crank speeds and recording the resulting flexural responses of the members. Specifically. the investigation focuses on the response of four-bar linkages with flexible coupler and rocker links. The midspan transverse deflections of these flexible links are presented for a wide range of operating speeds and link-length combinations in order to denonstrate the generality of the observed phenomena and also provide some classification of the responses. Three classes of link configuration have been studied: class 1 has the same link characteristics for both coupler and rocker links while classes 2 and 3 possess different link characteristics. The reason for employing two dif- ferent cross-sectional links is to observe the possible mode interaction between two links. “Palm“. APPARATUS A photograph of the experimental four bar linkage apparatus used in this study is presented in Figure 1. from which it is clear that it incorporated two flexible members. the coupler and the rocker links. This apparatus was designed to scco-odata a wide range of coupler and rocker links lengths. but the crank and ground link were always held constant at 63.5. and 612. 8mm. respectively. The crank was assumed to be rigid since it was aanufactured from steel bar stock of cross-sectional dimensions 25. 4-x 25.6 Fig. l Photograph of lxperinental Four-her Linkage Three classes of four-bar mechanism were exaained in this study and these kinematic chains are defined in Table l. The rocker and coupler links were both manufactured from steel strip material. and link eer so that a variety of kinematic chains could be studied. Two classes of steel strip were employed with the following length. width and depth dimensions: (268.0 mm x l9.05 - x 1.27 -) and (165.0 - x 25.6 - x l.lh -). Tabla II Limb classifieauam E I l I teristics I I I I I cass- 0|— List I Crass I “Islet lesser l I I I I I I I I I I I I | All-I I 6).! I ”I. I ”.3 I I I I I I I I I I I I I l I I Ingtl (-l I 2 I "1.! I 63.3 I ”.0 I 23.! I I I I I I I I I I I I I I I l I 3 I AILI I “.3 I 20,-. I ”9.! I I I I I I I I I I I I I I I I I l I I lLNIlI.‘ I [Loki-17 Imus-1.11. I I I I I I I s-swet I I I I II :7.- I -I.‘ I I I I “Juli-h I "-ohl." ITSJIIJI I I I I I I I : I I I I I I I I 3 I I fl-Isfl-I I 35-1-1.“ II9.oIII.n. I I I I I I I I I I I I I I I I I “-5 I 27.0 I I I I I I I I I lateral I I I I I I I lruqmmmey (ll) I I I I I III! I ”.2 I I I I I I I I I I I I I I I I I I I I I ”.e I 16.! I I I I I I I I The coupler and rocker links of the experimental four-bar linkage apparatus were both nanufsctured fro- strip steel material and the ends of each specimen were clasped to the respective bearing housings by two socket screws. The clamping loads were distributed over the ends by flat plates which are clearly visible on the coupler link in figure I. mechanism since they ensured a smooth load transfer between the three principal components of each link 195 Idontical aluoinuo boaring hooainga uoro oonnfac- turod for tho rockor link whoro it uoa rotainod on tho groond-linklrockor joint. and aloo for tho couplor whoro it uaa rotainod on tho crank pin joint. Tho dioonaion of tho houainga in tho longitudinal diroction oi thoir roopoctivo links uoro 30.1.. tron tho ond of tho houaing adjacont to tho throadod holoa for tho aockot acrowo and tho contorlino of tho boaringa. Tho oaaaoa of thoao coopooonta including booringo. aockot acrowo and claoping platoa woo 0.05kg. Tho cooplor and rockor liaka woro aopportod on aatchod poira of o.loo born [3 DD 112 inatruoont ball boaringa ouppliod by 7A0 Doaringa Lioitod. lach boaring houaing in tho oochoniao uoo proloadad uaing a Droaaor Induatrioo torquo lioiting acrow drivor calibratod to 20.113 l-o which pornittad boaring cloaranca to bo olioinatod aixco tho iopactivo looding aaaociatod with boaring cloarancoa would havo roaultod in largor link dolloctiono. Convoraoly. it tho boaringa woro aubjoctod to largo axial proloada. thon tho dofloctiona uoold bo attonuatod. Tho two floxiblo linko articulatod in tho ao-o piano in ordor to olioinato tho cooplicationa of nonlinaar toraional coupling tor-a which charactorito co-planar floxiblo linkagoa. Thia waa accoopliabod uaing a cloavago boaring doaign that in cloarly viaiblo in tho top contor oi Yiguro l. Tho houainga uoro fobricatod in a aluoinuo alloy and tho longitudinal dioonaion of tho houaing coopriaing part of tho couplor waa 38.1Io tron tho boaring cantorlino to tho ond of tho houaing adjacont to tho throadod holoa uaod to ,claop tho link apocioon. Tho our of thia aaaonbly. including boaringo. oockot acrowa and claoping plato waa 0.052 kg. Tho cloavagod cooponont of tho couplor- rockor joint waa boltod to tho rockor link. Tho axial dioonaion of thin part froo tho boaring contorlino wao tb.5 on and tho Ioao waa 0.063 kg including tho opindlo. woahoro and nuta. In thia invaatigation. tho driving froquoncioo aro rathor aoall cooparing to tho natural fronuoncioo of both linka. thoroforo. tho dofloction ia nainly dooinant by tho firat oodo of vibration. In addition. tho boaringa and tho link- boaring connoctiono aro placod on both onda of tho linko. on tho influonco on tho natural troouoncioa of tho individual linka in ooall. Tho oochaniao waa boltod to a largo cant-iron toat atand which won boltod to tho floor and aloo to tho wall of tho laboratory to provido a aubatantial rigid foundation. A 0.75 h.p. Dayton variablo apood D-C oloctric ootor (oodol 22066) uhich uaa boltod to tho toat atand. poworod tho linkago through a 15.9 on diaootor ahai’t aupportod by Tainir pillow block boarinoa. A 100 on diaootor flywhool waa koyod to tho ahaft thoroby providing a largo inortia to onauro a conotant crank froquoncy. whon oporating in uniaon with tho ontor'a opood controllor. IISTIUHIITATIOI A achonatic of tho inatruoontation uaod in tho oxporioontal invoatigation ia proaontod in Figuro 2. Tho ratod apood of tho oloctric ootor waa 2500 rpo and thia waa diroctly ooaaurod in rovolutiona por oinuto by a flowlott Packard 531bA univoraal countor which wao activatod by a digital-nagnotic pickup. oodol 58‘23. nanufacturod by lloctro Corporation oonaing a oixty tooth apur goar oountod on tho drivo ahaft of oach rig. Ihia largo goar ia cloarly viaiblo in Figuro l. Tho aforooontionod arrangonont prowidod viaual foodback to tho oporator whon tho apood controller of tho ontor waa boing adjuatod to oatabliah a doairod apood. s —_ apood ...-- [mu ”pom unu- OJ. "Monitor-ciao ‘ I'll/'3 til Tig. 2 Schoootic Diagrao of Inatruoontation Tho floxural doiloctiona at tho nidopan of oach link apocioon woro oonitorod by Hicronoaaurooonta Groupa Inc. typo lA-06-125AD-120 atrnin gagoa bondod to oach link at tho oidapan location. Bonding half-bridgo configurationa woro adoptod. uaing ono gago on oach aido of tho apocioon. and thoy woro uaod in conjunction with a hicroooaaurooonta Group Inc.. atrain gago con- ditionor/aoplifior ayatoo. typo 2100. In addition. tho doiloction at tho ono-quartor apan location waa alao oonitorod by atrain gaugoa. Thia tranaducor arrango- oont uaa conaidorod ioportant in ordor to oonitor potontial aocond-oodo floxural roaponaoa of tho linka. In ordor to rolato tho atrain gago aignal to tho configuration of tho particular oochanioo boing atudiod. a third tranaducor arrangooont waa oata- bliahod. An Airpax typo 16-0001 toro volocity digital pickup uaa ooployod to aonao tho bolt hood at tho ond of tho crank whon tho oochaniao waa in tho convantional toro-dogroo crank anglo poaition. Thin long hoxagonal tranaducor ia again cloarly viaiblo in Figuro l. Thia oochaniao configuration aignal and tho output froo tho gagoo woro fad to a Digital Bouipoont Corpor- ation P0711103 oicrocooputor uith a LSI 11I23 procoa- aor. Thia digital-data-acquiaition ayatoo foaturod 256 kl of oooory for poat-procoaoing data and alao a 5 on hard diak for atorago. A dual port floppy diak ayatoo waa alao availablo. Tho oxporioontal roaponao curvoa woro diaplayod on a Digital Equipoont Corporation V1100 torninal with rotrographica onhanconont. ' Tho DNC cabloa froo oach oxporioontal apparatu- woro connoctod to a input-output oodulo. boltod to tho 196 cabinot of tho cooputor. which won built by tho llactronic and Conputor Sorvicon Dopartnont at MS“. Thia inatrunont foaturod 16 analog-digital channola. 5 digital-analog channola and two Schnidt triggora. Uning noftwaro dnvolopod npocifically for digital data acquioition purponon. tho floxural roaponao nignal waa rocordod Tron tho aoro crank anglo ponition through 360 dogrooa by firing ono oi tho schoidt triggora. In ordor to activato tho triggor. a 100 u! capacity uaa uaod to nodin tho oqoaro-wavo output tron tho Airpax pickup. lxporinontal rnnulto woro obtainod by inplanonting tho following procoduro. Tho aignnl fro- tho atrain gago inatrunontation uaa paaaod through an analog variablo low-poan filtor nnnufacturod by Havotok Inc.. typo 632. in ordor to ronovo tho oloctrically or nochanically inductnd noioo prior to boing digitinod by tho analog-to-digital convortor and rocordod by tho PD! lll03 nicroconputor. Thin low-pana filtor alao provontod alianing problona whon onploying a roctanggf lar data window. and a aaopling rato o! (2570) aocondn. which wan conaidorod to bo at loaat twico tho highont Iroquoncy cooponont of tho puro olaatodynonic roaponao nignal. 8!?! GI Tho oxporinontal rooulta containod in Tiguroa 3-0 proaont tho floxural roaponaoa of tho couplor and rockor linka ovor a wido rnngo of crank Iroquoncioa for a varioty of link-longth conbinationn. Upon undortaking a curnory raviow of thoao ronulta it in ovidont that a brood apoctrun of rooponno hiatorioa aro proaontod. and cloarly oono clnoaiiication in donirablo for diacuanion purpoooa. A claaaitication in dotinod horoin in a nonowhat arbitrary nonnor to charactorino oach roaponao rogino. Iongggno Claaaification Tho floxural roaponao of four-bar linkagoa nay bo clanaifiod into throo roaponao roginon. and thoao aro now dotinod (alboit. oonowhat arbitrarily) for tho purpooon of thia invoatigation. Dynnnic lonponao Thin roaponao hiatory ia charactorinod by two uavofornn conpriaing a low-lroquoncy poriodic wavoiorn (at tho crank froquoncy) upon which in auporiopoaod a high-froquoncy ainuaoidal wavaforo at approxinntoly tho naoo froquoncy of tho link boing oonitorod. Tho anplitudo of tho high froquoncy couponont in gonorally of tho aano ordor of nagnitudo aa tho anplitudo of tho low-iroqooncy cooponont. Quaai-Static looponoo Thin roaponao rogino in only charactorinod by tho low-froquoncy poriodic wavoiorn at tho crank fronuoncy. Tor tho porpoaon of thin otudy. roaponaoa woro claaai- find an 'quaai-otatic" whon tho anplitudo of any high-iroquoncy ripplo auporiopoaod upon tho low- Iroquoncy carrior wavoforn wan loan than 10 porcont of tho anplitudo of tho carrior roaponao. Ioaonanco Conditiona oi roaonanco woro doiinod by tho anplitudo of tho roaponao at a apocific crank froquoncy boing aignificantly largor than tho anplitudo of tho roaponao at adjacont crank iroquoncioa. Hhilo thin appoara to ho a aonouhat unnciontific and an ill- dolinod atatonnnt. in practico it wan quito appropriato for tho phononona oboorvod in thia ntudy. 21mm! Tiguroa 3-11 pronont axporinontal roaponao hiatorioa noaaurod during thin laboratory invoati- gation. Upon undortaking a curaory roviow oi thoao roaponaoa. it in innodiatoly ovidont that thorn aro a varioty of difforont wavofornn. For oxnnplo. Tiguro 3 proaonta tho nidapan floxural roaponaoa of tho rockor link of a clnan l nochanian oporating at 23‘. 299 and 320 rpn. Two claaaon of roaponao aro proaontod. with tho anplitudo of tho ranonanco condition at 320 rpn boing groator than tho rononant roaponao at 234 rpn. Tiguro 6 proaonta a ninilar not of data for tho nidapan floxural roaponao of tho couplor link of a cloaa 2 nochanion. Throo clanaoa of tonponao rogino aro proaontod in Piguro ‘ with tho rononanco condition at 2“ rpn doninating. In fact. tho anplitudo of tho vibrationa at 2“ rpn aro largor thnn tho anplitudo of tho dynanic ronponao at 256 rpo and tho quaai-atatic roaponao at 300 rpn. It in cloar. tharoforo. that tho roaponao profilo. tho atrooo-lovola and honco tho iatiguo lifo aro all dopondont upon tho oporating apood of tho nochaninn. Quasi-static (at 299 no) .-“-. at T20 m) ALLA poo. nmm Transvoru Dofloction (an) o. \t '1 . '0' '7‘. for? W‘U’t V‘V" 0 ma too an «n Crank Anglo (Door-nos) Claaa l Hochanian; Rockor Link hidapan Trans- worao Dolloctionn at Throo Oporating Spooda Iig. 3 having provioualy Iocuaod on tho rooponnoa of a particular link at n varioty of oporating apooda. attontion now focuaoa on tho roaponaoa of both tho couplor and rockor linka of a apocific nochanian oporating at noon apocific crank froquoncion. Thono oxporinontal roaulta aro proaontod in figuron 5-8 which dononntrato tho varioty of ronponao roginoa dovolopod by throo dinoinilar claanon of four-bar linkago oporating at diffaront apooda. It in cloarly ovidont Iron thoao roaulta. that whilo tho iloxiblo couplor and rockor linkn havo a con-on joint. which nay inply nono connonality of roaponao for-at. that thoao adjacont linka can roopond quito difforontly. koaultn donon- atrating thoao offactn hawo not prowioualy appoarod in tho litoratura. Figuro 5 proaonta tho roaponao of a Claan l of nochanian in which both tho couplor and rockor ainul- 197 ...-I (c «a an) (a'c 2543—) 733323? Dofloctlon (an) llidtpnn Ironsvorto V V I Y wvo' "‘1' v I Y‘arv V Imf‘r' 1” «' CnuUIAnghI(0qpuos) Fig. A Claan 2 hochanian: Couplor Link hidapan Tranavorao Dofloctiona at Throo Oporating Spoodn ; s ‘ \ A- ] loo Anononno' %=::o::5tat:::’. If3} (losonanco) . c 4 1L :0 l I a} T) '3.) Aldsnan lransworso cofloctlon (loo 1 l .-~_-~. ”0"" “,. ' 1 : / V l I ‘-. 1 ‘ \ ~o.: qlg. a”; ‘K’ a} '00: 0 I v v 1‘.1 I v vmfi v 1 out I 1 pm Crank Anglo (nonrans) fig. 6 Claao 1 hochonion: Couplor and Iockor hidopan Tranavorao Dofloctiona at 321 rpn tanooonly dovolop a dynanic roaponao at 260 rpn. Piguro 6. houovor. rocordo an intoronting condition whoroby at 321 rpn tho rockor link of a Clana 1 nochanino vibratoa vigorouoly in a atato of rononanco. whilo tho adjacont couplor link ronponda in a quaai-atatic nannor with niniual vibrational bohavior. Stato-of-tho-art conputor nodola aro unablo to prodict thia phonooonon. In aharp contraat to thin. at 357 rpn tho aano oochanian dovolopa a quaai-ntatic rockor roaponno and alao a quaai-ntatic couplor rooponao. Thooo dofornation hiatorioa aro proaontod in Figuro 7. If donign guidolinon oxiotod for prodicting auch a ronponao profilo. thon thia roaponao could havo boon prodictod by undortaking a quaai-atatic finito olonont nnalynia. prior to tho fabrication of tho oxporinontal nochanian. Such a guidolino dooa not oxiat at thia tino. ‘A I- l 1 Counlor Link Aockor Link ‘ (dynuaflo:ronnonsol ( \ I Aidsnan Tranovorso unfloctlan (an) O h. It‘ll V f ' Y I I V i too 400 A -"' W V V V I T j V o no CM Mol- (00m) Tig. 5 Claaa 1 hachonian: Couplor and lockor hidapan Tranavorao Dofloctiono at 260 rpn 1.: . Couplor Antponao lockor Aooponna . (quasi-static) (quasi-static) go... if ‘ .2: ‘ 3» ‘ -+ g: ‘ ‘ ”NJ ‘1. .. E ...-II ;‘”"- T. I, ‘ 3' ‘k ‘\\ I \a”.‘\. I ...: l V V I 1 f1 V I fl f U T V ‘U 1 I a too too too ‘wu Crank Anglo (Dooroot) Fig. 7 Claaa l hochaninn: Couplor and lockor hidapan Tranavorno Dofloctiona at 357 rpn Tiguro 0 proaonta tho roaponoo hiatorioa of a claan 3 nochnninn oporating at 201 rpn. Tho rockor link roaponao in quaai-ntatic in naturo whilo tho adjacont couplor link vibratoa in a atato of roaonanca. Again. donign guidolinon do not oxiat for prodicting ouch a roaponao. Thua. it in cloarly ovidont fron thia colprohonaivo oxporinontal invoatigation that tho froquoncy at which tho nochaninn oporaton nignificantly affocta tho roaponao bohavior. At thin tino. tho donign onginoor lacka analytical toola for prodicting whothor a dynaoic finito olonont oodol of a nochaninn ahould bo utilitod. or whothor a ainplifiod quani-ntatic anolynia in juatifiod prior to tho nanufacturo and tonting of a product. Furthor- noro. tho ability to prodict whothor a npocific link 198 lockor no sauna / (quasi-static) Dailoctiona (-l hidspan'hnuuvorso I I I I I I I I I Ifi r I I I I 1| l zoo son «to Crank Anglo (Dogma) Pig. 0 Claaa 3 hochonian: Couplar and Iockor Hidapan Tranavorao Dofloctionn at 201 rpn ll - Rosonanco 05 - Quasi-static Rasponso Dynuic Rosponso Lianlaaanaan “Mir-l Pig. 9 Claaa 3 Hochanian: Couplor Raaponao loginoa as m 05 as ‘15 OS ‘35 Dynanic OS 05 05 “080003. R R R My \ \\ \ X \ Link Rosponso 92 107125 152 90 "5135 161 100 202210234 250279 299 320 7 210 22] 2“ 33344357 270 200 Crank Froquoncy (rpn) Pig. [0 Claan 2 Hochanian; Couplor Roaponao Roginoa will vibrato in a quaai-atatic rogioo. a dynaoic rogino. or in a atato of nupor-harnonic roaonanco in alao lacking..‘ Cloarly donign guidolinon nuat bo dovolopod to addrona thia aignificant void in tho litoraturo. . Pigoroo 9-11 proaant tho global roaponao roginon of aono of tho linka of tho nochaninna atudiod in thin invontigation. Tho conplox naturo of tho roaponaoa. which han boon dononatratod in tho provioua figuroa. in again cloarly ovidont upon raviowing thoao throo aota of data. For tho nochaniaoo ayatooa tontod. tho floxural roaponao wan gonorally of a dynanic natutg with roaonanco conditiona obaorvod at tho indicatod apooda. A typical not of roaponao-rogino data in proaontod in Tiguro 9. Quaai-atatic roginoa woro gonorally confinod to narrow banda an dononatratod in Piguron 10 and 11. but thin waa not univoraally tho caao aa indicatod by Figuro 9 whoro thia link did not dovolop a quani-atatic roaponao for tho apood rango tontod. Tho obaorvod roaonanco conditiona aro alao indicatod. Tho thoorotical onplanationa for thoao porploxing aituationa undoubtodly raquiroo a dotailod nonlinoar vibrational analyaia. 199 05 k-Iunnuuo 05 '3 05 00 - Quasi-static Inopnnno 03 II R I I <3. onnou 1 unwou:luwuuo humoua d —I——‘ 132 1'1 2h. 2U! 21!!! Iflifll 162 an ill tron rm (no) Fig. 11 Claaa 1 Hochanian: lockor chponno Roginoa M An oxporinontal study of floxiblo four-bar linkagon haa boon proaontod which docunonta tho difforont claaaon of floxural nidapan olaatodynanic roaponaoa of tho couplor and rockor linka. an boing quaai-atatic. dynaoic and roaonant. whilo thia torninology han boon onployod in tho floxiblo nocha- niann for aono tino. thin invontigation haa uncovorod for tho firat tino a wido varioty of roaponao con- ditiona which havo not appoarod in tho litoraturo boforo. It in anticipatod that thin oxporinontal ovidonco will atinulato ronowod thoorotical activitioa in thin aron. fron which donign guidolinon will bo diatillod for practicing onginoora. W 1. Hillnort. K.D.. Thornton. V.A. and Khan. H.l.. ”A lliorarchy of Hothoda for Analyaia of llantic Hochanian with Doaign Application." ASH! anor No. 70-DtT-56. 1970. 2. Sandora J.k.. and Toaar. 0.. "Tho Analytical and Exporinontal Evaluation of Vibratory Oacillntiona in lonliatically Froportionod Hochaninnn.” ASHE. Jgurnal of Hochanical Doaign. Vol. 100. 1970. pp. 762-760. 3. Hainoa. 1.3.. ”On Prodicting Vibrationa in loaliatic- ally Proportionod Linkago Nochaniann.‘ ASHE, Journal of Hochanical Doaign. Vol. 103. 1901. pp. 706-711. 6. Thonpnon. 0.5. and Aahworth. 3.9.. ”Rononanco in Planar Linkaga Hochaniana Mounted on Vibrating Foundationa." Journal of Sound and Vibration. Vol. 69. No. 3. 1976. (b03-616). N 10 11. Snith. H.R. and Haundor. L.. ”Stability of a Four-bar Linkago with Floxiblo Couplor." Journal of Nochanical gnginooring ficionco. Vol. 13. lo. a, 1971. PP. 237-262. Dadlani. H. and Kloinhonn. 9.. "Dynanic Stability of Zlaatic Hochaniana.” ASH! Journal of Kochanical Doaign. Vol. 101. lo. 1. 1979. pp. 169-153. Dolotin. V.V.. Tho Dynanic Stability of Elantic Sygtona. Roldan-Day. San Franciaco. 1966. HcLachlan. N.H.. Thoorv and Application of Hathiou Function. Dovor. Now York. 1966. Jandraaitn. H.6. and Lowon. 0.0.. ”Tho Elaatic- dynanic Dohavior of a Countorwoightod Rockor Link with an Ovorhanging Cndnaaa in a Four-bar Linkage." Part I: Thoory; Part 11: Application and Export- nont. ASH! Journal of Hochanical Doaign. Vol. 101. "on 1. 1979. ”a 77-78o Kalaycioglu S. and Dagci. C.. ”Dotornination of tho Critical Oporating Spoodn of Planar Hochaniana by tho Actual Lino Elononta and Lunpod Hana Syatona." ASH! Journal of Hochanicnl Doaign. Vol. 101. No. 2. 1979. pp. 210-223. Cloghorn. U.L.. Tabarrok. 0.. and Fonton. A.6.. ”Critical Running Spoodn and Stability of high- Spood Floxiblo Hochaniana." Hochaninn and Hachino Thoorv. Vol. 19. No. 3. 1906. pp. 307-317. THE ELASTODYNAHIC RESPONSE OF A CLASS OF INTELLIGENT MACHINERY, PART I, THEORY 200 THE 2mm nannor 0' A CL“ OF wrotuoom WRY — rawrrn:11nannnr AISTIACT A nothodology in proponod. horoin. for roducing tho olantodynaoic roaponao of high-apood nochinn nyntonn by introducing a ratrofitnant nchano which porIita thoao ayntaoa to bo nubjoctod to an additional dynanical porturbational input which in conputor- controllod. Thin papor in first of two coopanion paporn which doacribo a cooprahonaivo analytical and oxporinontal invontigation on a broad-board nodal of tho concopt. in which tho objactivo in to roduca tho vibrational roaponao at tho aidnpan of a floxiblo rockor link of a four-bar nochanian. A variational for-ulation in ooployod to dovolop tho aouationn of notion govorning tho roaponao of tho floxibla nonbor which in nubjoctod to coobinod paraootric and forcing functionn. oquationa aro nolvod nulorically and tho viability of tho approach in volidatod by conparing tho analytical prodictiona with roaponao data fron a conplonontary oxporinontal prograa. lntalligont nachina ayntann havo boon tho nubjoct of nuonroua publicationa. principally in tho contaxt of robotic nanipulatorn. but of couraa thin toroinology roally portainn to a vary broad claan of nochinory productn which incorporata aonnorn and cooputara. Tho ronoarch roportod horoin. addronnon tho donign of intalligant nochinory in thin broadar contoxt by proponing a gonoral nathodology for auppraaaing tho olantodynaoic roaponao of linkaga nachinory. Tho currant gonoration of robotic nyatano aro capablo of undortaking nany divorao taako. and thoao nignificant capabilitian aro obtainod by intograting a broad a npoctrua of divorao tachnologion ranging froo inatruoontation. controln. cooputing and artificial intolliganco to natariala and nachino dynanicn [1.2]. In contraat to thoao robotics-oriontatod publicationn. horoin. tho focuo of attontion in tho raduction of tho olantodynaoic roaponao of flaxibla linkago nachinory. Thin clann of nachinory foaturon in a vory broad rango of both induatrial and coo-orcial nachinaa and aquipnont. and thoao nochaninn ayatona aro prono to In Part II of thoao coopanion papara. thoao‘ 147 gonorata a vibrational roaponao whon oporating in a high-apood onda. Thin roaponao charactoriatic in attributad to tho tinn-dopondnnt inortial loading iopoood upon tho nonhorn of tho articulating ayaton. which aro inhorontly an anaonblago of floxiblo. not rigid. bodion. Typical contributiona to thin fiald includa roiorancan [3-0]. Tho objactivo. horoin. in to dovolop and than oxparinontally toot tho viability of a propoood nothodology for roducing tho olantodynaaic roaponao of floxiblo noobarn of linkago nochaninoa through tho introduction of an additional oicroprocoanor-controllod input into tho original kinooatic chain. Tho concopt in ovnluatad by ntudying a broad-board nodal of tho propoaod nathodology which conprinan a rotrofittod planar fourobar linkaga with a floxibla rockor link. Tho ratrofitnnnt lnwolvna nodifying tho nor-ally atationary ground-link/rockar-link ravoluta joint to incorporata an additional priaootic joint whono porturbatational notion in controllod by an oloctrodynanic nhakar which in drivon by an nicroprocanaor-controllod arbitrary wavoforI/function gonorator. Tho ronulting linkago can function on a clanaical four-bar nachanian by locking tho prinnatic joint. or altarnativaly. an a fivo-bar linkago whon thia locking dovico in rolaanod. In thin lattor oodo. tho nyotan haa two noninally rigid-body dogrooa of froodon with ona input providod by tho crank rotation and tho aocond input providad by tho notion of tho nicroprocoaaorocontrollod ahakar. Tho aquationn of notion and tho ralovant boundary conditiona for tho flaxibla rockor link aro darivod. horoin. uning a variational tochniquo. and tho ataady-atato roaponao 1. obtainod uning an approach dovolopod in rafaranca [9]. W Elaatodynaoic dofloctionn. dynaoic atronaon, and fatigun conaidorationa aro all ioportant donign factora which aunt bo carofully accountod for in tho dovolop-ant of tho noxt ganoratlon of nochinory to oporata in tho intornational narkotplaca.’ Tho trond towardn ultra-high oporating npooda an a nothod for lncroaaing productivity haa oxacorbatod tho donign oi oodorn nachino nyntooa and an a conaaquonca of thoao 201 atiauli. tho ronoarch coo-unity haa ranpondod by dovoloping aoro aaphioticatod nodaling tachniquoa [10.11]. In addition to tho onhancod prodictivo capabilitiaa. daacribod in tho ahown raviow articloo. tho acadoaic coo-unity hon alao dovolopod two donign nothodologian for roducing tho olantodynaaic roaponao of linkngo aachinory. Tho firnt advocaton that tho articulating anaborn of thoao nochanical ayatoao nhould bo donignod with optiaal cronn-aoctional gooantrioa and fabricatod in tho coaaorcial antala [6.12.13]. Tho aacond advocatao that tho articulating anabora nhould bo donignnd with polyaoric coapooitn laainaton which aro optianlly-tailorad for tho npocific application (lo-l6). Uhon iaploaontod. tho forogoing anthodologioo achiavo a raduction in tho lovoln of tho dynanic roaponao by incrang tho atiffnonn-toowoight ration of tho nonhora by tailoring tho gooaotrical and aatarial proportion of tho articulating nonbora. Tho rationala can bo clarifiad by axaaining tho tarao in tho approxinata linaar finita olonont aquationn of a gonoral nochino ayatoa proaontod bolow. m‘1mm o m'cho; 0 [min - «mi» . (1) In aquation (l). [I]. [I] and [C] aro tho global noon. ntiffnaaa and daaping aatricoa ranpoctivoly of tho ayatoa. [I] in tho idontity natrix. (Di raproaantn tho dincratixod daforaatign fiold. tho ovordot donotan tho tiao darivativo and ill rapranantn tho dincratixod rigid-body accoloration fiold. Thun for a proacribod nochino ayatoa oporating at a proacribod apood tho olantodynaaic roaponao in principally dictatod by tho atiffnaanoto-aona ratio of tho linka. In contraat to tho forogoing nothodologian [6.12-16] which focun on lncroaaing tho atiffnana-to- nonn ratio tora on tho loft-hand-aido of aquation (l). tho nothodology proponod horoin focuaaa on tho right- handoaida of aquation (l); naaoly tho kinnaatic variablao ill which provido tho principal ayatoa oncitation in a high-apood aodo of oporation. Thun it in arguod. if tho aagnitudo of thoao variablao can bo roducad. than tha nagnitudn of tho forcing function will alao bo roducad. with a corronponding raduction in tho olantodynoaic roaponao. Thin raduction in tho alantodynoaic roaponao trannlataa into a raduction in tho dynaaic dofloctionn of tho articulating nonbora. a raduction in tho anaociatad dynaaic atraaaon and alao a raduction in tho aavority of tho fatigua anvironaont in which tho anabora aunt oporata. Tho idaa wan firnt roportod in a thoorotical publication [17) and aono proliainary oxporiaontal work wan proaontod in rafaranco [ll]. Tho proponod anthodology would bo iaplanontod in practico by invoking tho following procoduro. Tho nochino ayatoa undor raviow would bo oporatod at a proacribod apood and tho alantodynaaic roaponao of a npocific aoahor. ouch on tho output link. would bo carofully aonitorod by atratagically ponitioning aonoora on tho aaabor. Tho nannor output would than bo aaaplod. ntorad. and poat-procaaaod by a aicroprocoanor prior to utilizing tho profila of tho dynnaic doflaction to auppronn tho action of tho vibrating link poat-procanaod. Thin gonaral vibration-nuppranaion aothodology would bo iaploaontod at any oporating apood by fanding tho axporiaontal roaponao-data to an arbitrary wavoforn function gonorator which would autoaatically nynthaaixa tho appropriata oxcitation voltaga for an oloctrodynaaic ahakor. Thoaa lattar oporationn would bo porforaod by a aicroprocannor. honco. tho torn 'intolligont aachino.‘ Tho nhakor would than provido anothor dynaaical input axcitation to tho output link via a rotrofitaont faaturo of tho 148 nochino ayatoa. Thin rotrofit nchano would involvo roplacing a noraally fixod ravoluto joint by a prinaatic joint. tharaby poraitting tho introduction of a parturbational input to tho ayatoa. Thun a nochino ayatoa with n dogrooa of froodoa would bo rotrofittod to gnnorata (n+1) dogrooa of froodoa. Tho porturbational input of tho rotrofittod ayatoa aunt bo carofully orchantratod rolativo to tho inputn of tho original ayatoa in ordor to achiovo tho doairod offoct of roducing tho alantodynaaic roaponao of tho anabar tarnattad for roaponao tailoring. ' Tho nacnaaity for roaponao tailoring. or activoly nuppranaing tho olantodynaaic roaponao of a nochino ayatoa. ara nuaoroun and includo tho dooiro to roducn tho aovoro alantodynaaic roaponaoa anaociatad with boaring dagradation and dynaaic inntabilitiaa ronulting froa tho paranitic axcitation of nochino foundationn; tho noad to control nochino outputn in hootila hygrothoraol anvironaontn; and to natinfy tho doaand for tho dovolop-ant of a now gonnration of anchinon which gonorata aoro accurata output choractoriaticn with aoro vornatila porforaonca onooiopnn. W Tho viability of tho forogoing proponod ganaral aothodology for activoly nuppraaaing tho olantodynaaic roaponao of dintributadoparaaotor linkago nyataan wan ovaluatnd both analytically and axporiaontally by invontigating tho roaponao of a broad-board laboratory aodol which oporatod opon-loop. Thin nodal coaprinod a planar four-bar linknga with a flaxiblo rockor link which won configurod to principally vibrato in tho piano of tho aochaniaa. Thin clanaical ainglo-input linkngo ayatoa wan rotrofittod by introducing a nacond input at tho ravoluto joint connocting tho rockor-link and tho -link. through tho introduction of a prinaatic joint. Thin prinaatic joint poraitn tho nor-ally fixod ground-link langth to bo variad in a proacribod faahion whon tho fivo-bar linkago ayatoa oporaton in tho coaputor-controllod aodo. and in thin aodo. tho ayatoa in tarand an intolligont nochaniaa. Tho linkago ayatoa in ahown achoaatically in Figuro l. o) . t a rigid 11‘ " floniblo 1 Lab \ .1 ‘ ' -- 0 ..."... ‘ 1 fr- Figurn l. Schoaatic of Intnlllgont flochanina. 202 An analytical nodal of tho dynanical notion of tho floxibla rockor link of tho intolligant nochino ayntan. which in nubjoctad to both poriodic and paranotric forcing fron tho crank axcitation and tho parturbationnl axcitation of tho nicroprocaaaoro controllod ahakor. wan dovolopod uning a vnriationnl tochniquo [l9.201. Tho tochniquo in banod on tho introduction of approxinating atatoaontn into a variational aquation of action prior to pornitting arbitrary indopondant variationn of tho ayatoa variablao which yialdn. an atationory conditiona. tho fiald aquationn and tho anaociatad boundary conditiona for tho particular boundary-valuo probloa boing invontigatod. Tho ralovant variational aquation of notion [20] in * i 8.) - I: { J" [ "”il’i - "Oifli”ijk‘j“0kfllkfik)” T ‘1LJI'LJ. ("7’711’1 l ‘ "11"11 ' 2 (“1.J“J.x“k.1“k.1” * “tux"u.j“1.kj'jk*"l.i'1k.3'"tl"V ’ Ll “'1 (Ei"i°ui.k‘kldsl . Is (‘It T ‘“i.k‘klui,k“kl [at-0‘] ‘32} dc - o 2 (2) In aquation (2). V dofinon tho voluno of an olaatic body doncribing a gonoral apatial notion rolativo to an inortial fraao. whilo 0 and S nun to aqual tho total nurfaca aroa. S. of tho flaxibla body. Tho anon donnity in raproaantod by p and p in tho tino rata of chango of a poaition voctor of a lanoral point in tho body. Thin poaition vactor hon coaponontn r . which aro tho coaponontn rolativo to tho aoving boa; axon onyx of tho poaition vactor of tho origin of thoao body axon rolativo to tho origin of tho inortial fraao. .R which ropronont tho poaition vactor of a gonoral poi 2 in tho body in a rafaranco atato rolativo to tho body axon and u1. tho dofornation dinplaconont vactor. Tho atrain onorgy donnity in raproaantod by U. 1 aro tho Lagrangian atrain conponnnta. r aro tho aifonnon and Xl rapranont tho body forcon p6} unit voluao. Froncribod nurfaco tractionn E aro iapoood on nurfnco aroan 81 and proacribod nurfaco dinplacoanntn u on nurfncn aroa 0 ._ Tho tiaa rato of chango of a qdantity rolativo to tho aoving axon oxyx in raproaantod by (*), ( ) donotoa tho abnoluto rato of chango and (.) dofinon apatial difforontiation. Coaponontn of tho angula volocity vactor of tho frano oxyx aro raproaantod by 1. and oUh in tho altornating tannor. Thin goonotrically-nonlinoar fornulation haa boon aoloctod for analyzing thin clana of floxiblo laboratory linkago bocauna proviouo conbinod thoorotical and oxporinontal ntudioa [o.0| havo cloarly doaonntratod tho nuporior prodictivo capabilitioa of tho nonlinoar nodal. 149 W Approxinoting atatoaontn for tho ayatoa variablao in aquation (2) aunt bo fornulatod no that it any bo oaploynd to conntruct tho appropriata problon dofinition. Tho nochaninn proaontod in l in anauaod to havo a rigid crank Al of longth i which rotaton at conntant apood A . a rigid couplof ac of longth l and olaatic rockor link 00 of undofornod longth 1 vi a conntant crooa-noctionnl aroa A. Tho longth 3f tho ground link. 2 . in tiaa-dopondont whon tho aicroproconaor activatoa tho ahakor. Tho ayatoa in anauaod to bo fabricatod with nanoth pin jointn and tho alidor at point D in nan-ad to tranalata in a frictionlaaa guido. Coordinatoa oxyx aro Lagrangian anon fixod in tho rockor link in an undnfornod rofaronco atato which in dafinod by tho anglo 0‘ rolativo to tho inortial frano 0:12. Dofornationa of tho rockor link aro roatrictod to aiapla axial and floxural conponontn in tho plano of tho nochanina. Tho link in conoidorod to bo alandor and tho dafornationa anall. honca tho lular-lornoulli boon thoory any bo aaployod. Tho nnnuaptionn for tho olaatic dofornotionn aro u‘ - u°(x.t) - x w “(n.t) u - 0 - v(8.t) (3) Tho longitudinal ntronn r in anauaod to connint of two coaponontn with tho fornot on tho axial dinplacoannt. All othor ntroanan aro aoro. oxcapt for 'u. MO. r - T°(x.t) - le(x.t) - T2(x.t) -7 I: f (9) I - 0 ‘7 -7 C? xx xx 77 Y! Sinilarly. for tho atraina. 1“ - Soix.t) - :81(x.t) 1yy - 71x - " 1xx (5) .- n o 7:! '3" 1n - 82(x.t) whoro w in Foinaon'n ratio. Tho volocity foraatn aro anauaod to bo p‘ - 0(x.t) v -0 <6) 7 P, - C(x.t) Tho abnoluta angular vnlocity of tho rotating frano oxyx rolativo to tho inortial fraao 0!!! in (7) Tho poaition vactor doncribing tho location of tho origin of tho aoving axon oxyr rolativo to tho inortial frnan OXYZ haa coapononta 203 8 r0. - it nin 0‘ (a) noanurad rolativo to onyx. and a gonoral point P in tho undnfornod rockor link nay bo dofinod by tho conponontn 'lx - x r.’ - y rn - x (9) Fronoribod dofornation dinplacoanntn aro to bo iapoood on a naall rogion aurrounding tho cantroid of tho boon aaction at both tho rockor-couplor joint and alao tho rockor-alidor joint. Tho flaxiblo link in cnnaidorad to ho a pinnod-pinnod boon. with tho dinplncnaont conntraintn u: - 0 at x - 0. a; - -rw.‘ (10) and at x - 2‘ I; (u‘ nind‘ - ug coal‘) d03 - 0. 3 Thin conntraint aquation dofinoa tho horixontal ntraight-lino notion of tho joint drivon by tho oloctronochanical ahakor. and thin conntraint in iapoood by tho Lagrango aultipliar conl‘. Froncribod nurfaco tractionn aro nod ovor tho roaaindor of tho aontionn at tho onda of tho link. Thuo. ..‘ .c ‘ - ’ . n‘ - F(t)/A . gy - 0 . n3 - Q(t)/A 6 I: - k [ h(t) nin 0‘ - F.(t) con 0‘ - H. 5‘] (11) i: - -k n o; -o. 1: a - o; - -1. it a < o DIIIIIIIQII The dynanic phenonena evident in the response - behavior of a revolute joint with radial clearance say be captured by considering four different behavioral nodes. These nodes are: (i) The contact node: The pin and the journal are in contact together and a sliding notion relative to each other is asaunad to exist. The sliding surface is asaunad to be snooth and thus no energy is dis- sipated. Since the clearance connection cannot trananit a tensile force in the contact node at the pin-journal interface. this node is terninsted when the contact force between the pin and the journal baconas tensile [5]. (ii) The free-flight nods: The pin and the journal loose contact in this node. hence there are no constraints between these nenhers. Under these conditions. the crank and connecting rod are treated as two conpound pendula. Thus the systen nay be considered to be two indo- pendant dynanical systens. until a goonstric constraint nandates the triggering of a different node. (iii) The inpect node: At the terninstion of the free-flight node. inpactive behavior is trig- gered by the sudden contact of the pin and the journal. An exchange'of nonentun occurs between the two nonbora. (iv) The transient phase: This phase begins at the first inpect and ends with the restoration of the contact node. The fundaaontal question associated with this transient node is whether the systen will return to the contact node or enter the free-flight soda. after the terninstion of the inpact node. Herein the pin and the journal always asaunad to separate after the inpact node. Thus the systen is asaunad to enter the free-flight node. If the pin and journal re-establish contact in a “very short tine“ near the original contact point than the systen is asaunad to be in the contact node. The negnitude of this “very short tine“ wont be carefully quantified. Hence. this was ascertained froa axporinontal data. 9IIIIIIA.IIIIIII_IHI_IDDIR This section of the nanuacript docunants the nathonatical fornulations which define the different behavioral nodes described previously for revolute joints with radial clearance. QRRERSS_LRIS Two criteria were enployed in order to sinulate the change froa the contact node to the free-flight node. First. in every step. the contact force between the pin and journal was calculated using the expression. F. - 1, cos 0. + A, sin 0. (6) 225 ThesystasisinthacantactnedeifF.so. Rnperience accrued in the coqlansotary axporinontal investigation shows that this criterion is only valid at speeds below 100 rpn. The second criterion is defined as follows; the contact loss occurs when F; is the nininun and the rate of the pressure angle changes rapidly. This enpirical criterion is the result of both theoretical and axporinontal observa- tions. Theoretical work suggests that F. will always be greater than zero. This inplies that contact will never be lost. however. axporinontal results clearly contradict this observation. In order to reconcile these conflicting states. the above criterion was established. Iarsinaticn.cf.tha_lrsa;zlisht.lcda The pin is asaunad to be in free-flight soda until the following geonatrical inaquali is violated c, o < (Ax, + 45ya )V' - c (7) where Ax - t, cos 0, + r. cos I. - r. cos 0. - s (8) 0y - r. + r.sin d. + r. sin I, - r. sin 0. (9) s i Ideally. if (Ar. + 6y ) I, - C the systen is in the inpect soda. in digital sisulation work. a tolerance is inevitably introduced in order to acco-odate coqutational rand-off errors. Thus. the systen is asaunad to be in the inpact soda if the following inequality is satisfied I 1.0 x 10"l s 1 (11' + Ay' )'/' - e 1 (10) mm As nentionad before. the systen is asaunad to be in free-flight soda after the inpact node. the pin and journal sake contact within a flight tiaa which subtends sore than a twentieth of a degree of crank rotation. than this assunption is valid. if contact between the pin and journal is re-estahlishad within a twentieth of a degree of crank rotation and there is an insignificancs dif- ference between the original contact point and the new contact point. the sisulation switches to the contact node. The subsequent sections present the equations of action for the different behavioral nodes of the nochanian. l0IOIl0IS_Qf_IOIlfll.1fl.1fll.§0fl10£1;lnfll 8y equations (3) - (5). a set of independent sinultanaous second order differential equations can be obtained. which can be expressed. in nstrix fora. as If I All I 3:] r I 3:1 (11) where 3. 0 t..8. .. 0'.“ O" -l n.r.rh,sea(d.-d.) 0 O '03. a. ti, 0 8 r.sin d. -r.aes 0. Mel ' 0 9 0 8 -r.ain d. r.eoa 0. (12) 0 8 0 a5 1 0 . 'g.“ .. 9'..8. '. 8 0 0 , 0 or.aos 0. r.cos d. 8 0 0 r .e as .9 ( ‘gl - I A. '. '. . Ag ‘.1 (13) 0 1 a. r. r.,l. sin(8.-0.) “I. 'g Tu’: 08“....g) "sl ' 0 (1‘0) «Us I. d -r.eos d. 8: - r.aoe d. 8: e r.cos d. 8: .-r.sin l, 6: - r.sin 0. 8: e r.sin 0. 8: The Lagrangian aultipliar A. equals F‘3‘(tha force isposes on link 3 by the slider in x - direction) and A, equals F.3,. while A. is the torque inposed I on the crank. A free-body diagran of the systen can be enployed to calculate the bearing forces. shown as Figure 2. Thus. ’31:' ‘1 ’ “' “03x ‘1’) '32y- A, - s. ‘G3y (16) ’12:‘ '* “02x‘ ’32: ‘17) r (18) 12y. .’ ‘GZy' r32y 4'”. (03 ill (assume. is) Fig. 2 Free-body diagran of the systen in contact nodo. Zilév IflIhI1III.DI_lEIIQI.LI.£III;£L1§HI_IDDI in this node. we treat the crank and the coupler as a set of double pendula. Therefore. the constraint force at the end of-the coupler disappers. A torque is inosed on the crank to keep a constant angular velocity. by applying the constrained Lagrange equation again. only three second-order differential equations ransin. which on be written. in astrix fore. as (Mild-[Ia] (19) where .1 g, r, rhjaeei0. - 0,) o “,3 . 0 ifle a, r; (20) O O I T ' n I ‘1} ‘ “a 'e ' l (21) .e ‘s '33.: ‘l'i'e ' ’9) "pl ‘ "s 's I” : “'“e ‘ '0’ (22) -«asnulg,e IflIAI1flII_9ILIHIIQI.1I.IHI.LIIAGI_HDDI if the inact Marion. 7. is seemed to be arbitrarily ensll. the lagrange euetions of notion say be adapted . [19]. and written weal-nuns. the kinetic energy before and after iapact can be espressed as (23) r - § (a. 232+ ‘02) l: 9 % ..(r: I: + ‘ca': 2 + 2 r, ‘63" 0,cos(0, - 0,)) + % 163 I, + g n. 3’ (25) g I t. 3 t 1' -é (a. :62 + I“) I, + 12' a,(r, I. s a, a a + ‘63,. * 2:.‘63’. ’.C°.('. ' '3), a, a, + g [63 f, + g I. 3 (25) respectively. leferring to the free-body diagran presented as figure 3 and applying the lagrange equations of notion. (equation 23). three equations are obtained a, r, '63”: - ’,)cos(0, - 0,) - T” + ; r, sin(0, - o.) (26) (ID) (ID) Pig. 3 Free-body diagran of the systen in inapct 3 a (.g r“: * 163) (’: ' i.) - P t, sin(0, - 0.) (17) A I(s.-;)-Pcos0. (28) This there are four unknowns. tor‘ P. 5:. and 31' but only three equations. hence an extra equation is required in order to detarnine a unique solution. The coefficient of restitution nay be eqloyed to provide the addition infornation. The resulting anpression is (V' . v )-3 ¢ - . M7. (29) (V0, - VQ.)-n which say be written. r, I, sin(0, - o.) + r, l: mu. - o.) g ... r, I, sin(0, . o.) + r, l. sin(0, - o.) + 5* cos 0. . (30) e s cos 0. Again. the above equations of notion can be expressed. in natria fore. as t‘sl [‘el ‘ I 3s) (31) where -l s, r, ra cee(0, - 0,) 0 -r, sis(0, - 0.) “.1 . o .. 3;, a [6" 0 -|'g 0"“. ‘ 'e) (32) g o e. -ces 0. O -r. sini', . 0,) ocos 0. 0 r f of . [ x.] - [for l. s r 1 (33) 227 le 's 'a ’s “."s ' ‘0) I lie] - (I. '6: ° la) 0. (u) n. i s u. I, we“. - o.) e r. l. uaa. - a.) . i «a 0.) or. Luau. . o.) The principal eubassadly in this enperinen- tal investigation was a planar slider-crank nechanisn incorporating adutantially rigid links which articulated in a plane perpendicular to the grnvitationed field. The crank. which had a length of 50 -. was fabricated as an integral part of a 166 - di-etar flywheel. This heavy circular disc was keyed to a 19.05 - dianeter drivs- shaft. which wasrtsuppo ted on a natchad pair of Tinken tapered- roller bearings. and was powered by a 0. 56K" (0. 75 hp) Dayton variable- -speed dc notor. nodal zzuu. The slider of the slider-crank .chanisn was the translating table elenent of a llicro Slides. lnc.. type ZOSO-W-lll-S linear crossed-roller slide easedly which is a precision preloaded systen without clearance. The conecting rod assedaly coqrised three conpononts: a steel bar with dinsnaionn 2671:6119 - .a crank- -pin bearing. and a gudge on-pin hearing. The total length ofth sinkl nenbar was 293- between bearing centerlines. The crank- -pin hearing was deniped to inc orporsre a notched of instn-ent ball bearings supplied by PM: bearings3 ted. mm: Thes spindle diaeter was -. This shearing easedly was carefully preloaded3 using a Dresser industries torque- ~tlini ing screw driver. .This pr rocedure enabled bearing clearance to be r drf ron the ass efily. since the vibrational response .associatad with this clearance at the crank pin would contsninated the axporinontal data fro- the transducers nonitoring the dynanic response of the systen at the gudgeon pin. The gudgeon~pin bearing was fabricated as journal bearing with variable redial clearance. This was achieved by aqloying steel pins of disaster 6.35 - and 6.572 - which were each sup- ported in a threaded hole in the reciprocating portion of the crossed-roller slide essafily. The journal elenent of the hearing was provided by a series of interchangeable brass bushings with income dis-stars of 6. A- h -. 6. 606 - which pornittad a variety of joints to be assdled with radial cala arances of 1. 01.6., 0.127 -, 0.0762 - and 0.0256 - respectively. In addition. a second bearing assenbly at nwas developed in order to design a joint with zero clearance. This would provide a reference set of data for cowarison with the results of the axporinonts with non-zero clearance. This precision bearing sass-bl y incorporated natchad pair of inst ttn-en ball bearings supplied by FAG Bearings Linitsd. type “0312. The spindle dianntor was 6. 35- The sxperinentel apparatus and the associated instn-entation is schensticelly represented in Figure lo and Figure 5. The dynanic behavior of the slider and the connecting rod were Fig. A Thotogreph of kperinental Apparatus hauled “- I—Ini—IAI lee-its Pig. 6 Schenatic of Experinental Appretus and Associated instrunentation. nonitored using three bruel and Kjaer piezoelectric acceleronetere. type 6371, which were each connected to a bruel and Kjaer charge aqlifier. type2 These acceleroneters were positioned at the gudgeon- pin bearing in order to neasure the dynanic response of the reciprocating slider and alsoth longitudinal and lateral notions of the connecting !rod. The signals fron these acceleronetsrs were fed to a Vavstek III/m analog filter. nodal 432. prior to being recorded on e PDTll- based digital- -data- acquisition systen The cut- off frequency enployed in this study was 60 lit The response profiles neasursd by the scceleroneters were related to the angle of the crankshaft using a third transducer arrange-ant, which featured an Airpsx zero-velocity digital pickup. type Ila-0001. This was enployed to sense the bolt head protruding fron the flywheel when the 228 crank was in the conventional sero crank-angle position. prior to firing the Schnitt triggers an the iqut/output panel of the data-acquisition systen. The highest eaqling rate eqloyed in this study was 2.500 eagles per second. W A slider-crank nechanisn with r,- 0.066 n. r,- 0.293 n. n,- l.0 lg. n,- 0.21635 kg. a; 0.63607 lg. ‘02- 0.1 lg-n' . Tc,— 0.003603 kg-n. and a - 0.5 was used in this study. The initial conditions for the toquter si-lstiens were based won the scantion that the ficticious clearance link has the sane direction as the pressure angle. The negnitude of this angle was obtained fron an analysis of the nechanisn incor- porating a gudgeon pin bearing with zero clearance. in addition. the angular velocity of the clear-Ice- linkwssaaauedteequalthe rate ofchsnge ofthe pressure angle. As nentioned in the previous section. the equations of notion were expressed in the netrix fotn l A“ l :1] - l 'i" if the proper initial values are given. then I ‘i' and [ I.) can be calculated in order to solve for l ‘il' nn- facterisstien was eqloyed together with forward and bachrard substitution in order to achieve this task. in this mic. [ ‘il includes sene second-order terns. suchss 0",. 0. m3. and Lagrange nultipliers. by converting the second-order equations into a first-order systen. a sitroutine moon- in the Ml. subroutine library was eqloyed to integrate the ascend-order derivatives. Thus 0,. at. i. one s were obtained. The initial condition for each tine-step in the toquter simlations were obtained froa the previous tine-step. The average CfD-tine for one cycle was 2 situates and 50 seconds on a st-ll/750 coquter. Space linitstions only pernit sons of the experinental and analytical results to be presented herein. figures 6-9 present experinental response histories fron this experinental investigation for 200 and 300 rpn. The radial clearance is 0.127 n _ at the opin bearing for all response profiles. figures 6 and 7 present the tangential accelerations of the comecting rod at 200 and 300 rpn respectively while figures 0 and 9 present the corresponding acceleration of the slider. figures 10-13 present the analytical results for the tangen- tial accelerations of the coinecting rod. and also the accelerations of the slider when the crank rotates at 200 and 300 rpn respectively and the gudgeon-pin bearing is 0.127 -. Upon viewing figures 6-9 it is clear that the pin and journal loose contact. or separate. twice during each revolution of the crank ahaft as evidenced by the nejor excursions fron the snooth curves of the taro-clearance nechanisn. The first separation generally occurred at approxinately 00 degrees of crank rotation and the second separation occurred at approxinately 260 degrees of crank rotation. Upon reviewing the acceleration profiles . 6. P Ai-1- WWII/SIS) ii - l A J - I - 15o sic do Wffi‘mo cam mcLEwEGS) fig. 6 prerinental tangential acceleration of the couecting rod. liechanisn operating speed 200 rpn. radial bearing clearance . 0.000127 n. Y 0 '20 WWII/SIS) fish etc etc cum meanness) 25o fig. 1 lxperinental tangential acceleration of the cementing rod. nochanian operating speed 300 rpn. radial bearing clearance 0.000l27 n. of the analytical predictions presented. figures 10- ll. it is evident that there is excellent correla- tion between the theoretical and experinental results. These two sets of results could have been si-iltsneously plotted on the cue diagran. however it is extrenely difficult to differentiate between the two sets of results. figures 10 and ll. indicate that the ncst serious inset occurred at 95 degrees and 296 degrees of crank rotation. while the experinental results. presented in figures 6 and 7. indicate that the first inset occurs between 90 and 96 degrees of the crank rotation. and the second inact occurs between 290 and 296 degrees of crank rotation. figures 16 presents the longitudinal acceleration of the contacting rod at 300 rpn. in which the solid line is the experinental curve and the dotted line is the theoretical curve with xero clearance. figure 15 shows the in-ericsl predic- tions for the longitudinal acceleration of the cor-acting rod st 300 rpn with 0.127 - clearance. Upon exanining these two curves it is evident that the bearing clearance has very little affection in t2: longitudinal dyn-ic behavior of the contacting r . 229 *K'Tficie 25* etc cum mom) I ah fig. 6 Inperinental slider acceleration. hechanisn epersting speed 200 rpn. redial bearing clearance 0.000127 n. v‘vv WWII/SIS) A4lalilal-l-l-1-1-{ 413 r - ct i do :30 in do tie mmwtiz) fig. 9 lxperinental slider acceleration. operating speed 300 rpn. radial bearing clearance 0.000127 n. da—L—‘ uLo—“~J mmu/s/s) J oce- 4 -|£In i ‘ 1 -m .1 J -m ] ""KETWK'HHHH'SH'WHWic CRANK ANGLEfleruesl fig. 10 Theoretical tangential acceleration of the connecting rod. hechanisn operating speed 200 rpn. radial bearing clearance 0.000127 n. figures 6 and 7 indicate that the first inpsct. at approxinately 95 degrees crank rotation. crests a sore severe vibrational response than the second inpsct at approxinately 300 degrees of crank hechanisn 7. fig. w" Y V V Y ‘ 1 j . A .9” fl d i J 4 Q 1 d 1 Q 2.”: 1 1 1 cred . 1 . -t W 1 i 4 1 1 4 ‘ O‘md- al ‘ 1 4 1 1 4 -. M f ' f vTv—j v V I) l 10 360 3.0 6” I“ 710 CthK.ANOL20hnreee) ll Theoretical tangential acceleration of the connecting rod. Ischanian operating speed soc rpn. radial bearing clearance 0.000l27 n. as 9‘ LA AAAAAA ACCflERAWM/S/S) - -§ 4-1;--AAAAAA1 u ""15""zic' sec do CRANK.ANGLEahnuees) fig. 12 Theoretical slider acceleration. Mechanisn operating speed 200 rpn. radial bearing clearance 0.000127 n. V N O AOCELIRAW/S/S) I 1 c l I . § 3 § § § :tiv § g g 3 8 8 8 8 8 8 8 8 oakLLLA—b-L‘J‘LLLLLAL 1 1 1{//,,,;’-——1:::://::)l J L—e—Le—J—h‘fiJ—a LaL—a—‘a 6‘] fig. 13 Theoretical slider acceleration. llechanise operating speed 300 rpn. radial bearing clearance 0.000127 n. rotation which present the tangential acceleration profile of the connecting rod. The predictions in figures 10 and ll show the sane results. 230 L L LAAL‘ mama/5m § § I N ‘ 6 LA I! AAAJALAAAAAA ‘3 W?“ 1'“ ’“ a & * cum mwtos) fi . 16 Lengitudial acceleration of contacting rod. ‘ nechanisn operating speed 300 rpn. Solid line presents the experinental result with 0.000127 a radial bearing clearance. The distiinaous line presents the theoretical result without clearance. soc: . - - . 3 acocn * \ m \ 3. 2c.oc< .. S 4 b & °.m4 < ‘3 8 . 8 -zo.co-< " 1 -“.= v r v v V ‘ u no no can eec soc no CRANK ANGLE(Ocqreel) fig. 13 Theoretical longitudial acceleration of contacting rod. nechanisn operating speed 300 rpn. radial bearing clearance 0.000127 n. Since the coefficient of restitution was eqloyed in nodeling the dynanic response of this class of nechanisn systen and the insct duration was seemed to be arbitrary snall in the innericsl sisulation. the inact force tends to infinity. The spikes in figure 10 and 11 were truncated in order to facilitate ccqarision with the experinental curves. In fact the insct duration is finite. in the order of 10" second. and the frequencies above 60 hr were filtered while recording the experinental response data. m The nathenstic nodal and internittent behavior of a pic of a slider-crank nechanisn joint with radial clearance have been studied. The ni-ericsl results are in qualitative agreenent with the experinental results. Thvo loss- ofoccntact situations occur during each nechanisn cycle and the loss of contact have been accurately predicted along with the associated dynanic response 3;. behavior . The first author wishes to gratefully acknowledge the financial support of the ~Cheng Institute of Technology. Taiwan. which has enabled his to undertake this research. 1. Kobrinskii. A.I.. . fllffo look Ltd.. London. 1969. 2. Hitteworg. J.. m. 6.6. Teubner. Stuttgart. 1977. 3. rate. r.s. and taut-on. D.A.. 'lhaltibody Dru-lean A831! We. Vol. 30. 1976. pp. 1071-1076. 6. haines. 6.6.. 'Survey: 2-0ineuional lotion and lqsct at levoluta Joints.‘ W. W. Vol. 13. 1960. pp. 361-370. 3. talker. J..). 'A Survey of the llechanics of ,, Contact Ietwoan Solid 6cdies.‘ . 2A!!! 37. T3 - T17 (1977) 6. lhulief. Y.A. and Shabana. A.A.. 'lqact leoponses of lhaltibody Systens with Consistent and M liaases.‘ mm. Vol. 106. 1966. pp. 167-207. 7. longisu. II.T’.. hidayetoglu. T.. and Akay. A.. 'A Theoretical and lxperinental Investigation of Contact less in the Clearances of a four-6st Linksge.‘ AS!!! . Vol. 106. 1966. pp. 237-266. 6. Dubowsky. 8.. Dock. J.f.. Costello. 8.. 'The Dynanic nodoling of flexible Spatial liachine 6ysten with Clearance Connections.‘ anus faper llo. 66-DlT-62. and m W m. (to pro-I). 9. flaines. 6.3.. 'An lnperinentsl Investigation into the Dyn-ic behavior of Revolute Joints with Varying Degrees of Clearance.‘I W W. Vol. 20. 1963. pp. 221-231. Dubowaky. S.. and Hooning. li.f.. 'An lxperinental and Analytical Study of lnsct forces in Slastic Systens with Clasrances.‘ . Vol. 13. 1976. pp. 651-663. 10. 11. Grant. SJ. and fawcett. 3.6.. 'Iffecta cf Clearance at Coupler-locker bearing of a four- 6st Linksge.‘ Vol. 16. 1976. pp. 99-110. 12. Shinjina. IL. Ogswa. K., and llatsincto. K., 'Dynsnic Characteristics of flanar nechanisn with Cloarances.‘ . Vol. 21. 1976. pp. 303- 306. 13. Wilson. 6.. and fawcett. J.l.. 'Dyn-ics of the Slider-Crank nechanisn with Clearance in the “hunt Retina.“ W. 1976. Vol. 9. pp. 61-60. 16. 13. 16. 17. 16. 19. 231 lliedena. 6. and m. 9.11.. mechanical Joints with Clearace: A Three-lode llcdel.‘ ASIII . Vol. 96. 1976. pp. 1319-1323. 0snen. li.0.li.. 6ahgat. 6.11.. ad Setter. T.S.. ‘on as fredictien of Journal-6earing Separation in high-Speed hechaisns with Clearsncae.‘ A6!!! paper 60-ilA/IISC-36. 1960. llasour. 9.6.. and Towuend. I.A.. 'laact Spectra and intensities for high-Speed lechenisns.‘ ASIII m. Vol. 97. 1973. pp. 367-333. hut. 6.6.. ad Crossley. f.6.6.. 'Coefficient of Iestitution interpreted as Ming in "but-8:3 All! W m. Vol. 97. 1973. pp. 660-663. Tatara. 1.. 'lffects of Internal force on Contacting Tinea ad Coefficients of Restitution in a feriodic Collision.‘ ASlil m W. Vol. 66. 1977. pp. 773-776.- Coldanith. 9.. Tendon. 1960. Soong. K., ad Thoaeon 6.S.. 'An lxperinental investigation of the Dynaic lesponse of a lechaicsl Systsn with 6earing Clearance.‘ an Design Technolog Conference. 1967. 06- “1.10.2. W 31“.. pp. 611-619 . . ldward Arnold. EFFECTS OF INTERMITTENT CONTACT IN MECHANISMS OF THE PERFORMANCE OF NUMERICAL INTEGRATION METHODS 232 IITICTS Of IITIIMITTIIT CONTACT ll MECHANISMS 0. TI! PIIFUIMAICI Of IUMIIICAL INTICIAIIOI METHODS J.li.6. arm'. I. Soong’. w.r.o. comment". M.V. Mi’ and 3.3. rho-poo»+ * University of Dundee. Scotland. U.6.; Visiting frofessor. Michigan State University + . hachinery [lastedynanics Laboratory. Hechanical lngineering Departnent. lichigan State University. USA x leriot-Uatt University. ldinburgh. Scotland. 0.6. high-speed nechanisns cannot be accurately stinulstod. and hence designed. without incorporat- ing link elasticity and joint-clearance effects in the nathonatical nodels. lunerical sinulations based on these nathonatical nodels often incorporate algerithns of the lungs-6utta type. The perfornance of this classical nethod applied to a sinple oscil- lator is reviewed herein. and the error-perfornance is shown to collapse to first-order when the algo- rithn is applied to a spring/neon systen with a stiffness discontinuity. Discrete energy changes are found to occur at discontinuity traversals with gains for hardening spring-systens and losses for softening spring-systens. These energy changes will occur each tine contact is gained or lost when shaulating the behavior of a flexible nechanisn with internittent contacts. In the last two decades connercial pressures have stinulated the developnent of nechanisns capable of higher operating speeds while requiring inprcvenents in output characteristics and reduc- tions in power consunption and acoustic pollution. Consequently. it has been necessary to develop nore detailed theoretical nodels of nechanisns by incor- porating elasticity and danping of links and nountings. and by also including the effects of clearances in the joints. Sons of these theoretical developnents are reviewed in references [1] and [2]. Even in the sinplost of nechanisns the equations of notion obtained when the rigid- link/perfect-joint nodal is abandoned are of such conplexity that the associated initial/boundary value problens cannot be solved analytically in closed-fore. frogress in the study of these effects can thus only be node by the application of various nunorical sinulation techniques available with digital conputers. in conjunction with experinental and/or analog investigations. luerical simulation requires the nodelling of the distribution of nose and elastic stiffness. and the tracing of the tenporal evolution fron sons initial conditions. While lunped-paraneter nodels of seas and stiffness have been utilized [3] it is now generally accepted that the finite elenent approach [2) is sore suitable for this spatial problon. Recent developnents have included non- linear effects [6] and the use of ccnposite naterisls [5]. There are two basic approaches to the sisulation cf the tenporal evolution of the nechanisn fron a prescribed initial state; these are nodal superposition. and direct nunorical integra- tion in tine. - Nodal superposition uses a linear conbina- tion of the vibrational node shapes. in nonlinear systens the approach nay require frequent redotar- nination. furthernore. in nechanisns which have internittent contacts whether by design or due to clearances. it is likely that nany of the higher nodes of vibration will be excited. Under such conditions nodal superposition is likely to be sore expensive than direct nunorical integration [2]. Direct nunorical integration procedures utilise a step-by-stap approach in which. starting fron known initial conditions. the equations of notion are effectively solved only at a series of discrete tine points. The classical nethoda fall into two nain groups [6.7). the 6unge-6utta algo- rithns. and nulti-step algorithns. both are widely used in nechanisn sinulstions: the forner. for exanple. as the classic fourth-order lungs-Kutta. [6]. or lungs-Kutta-neson [9]; the latter. in various types of predictor-corrector nethoda [10]. These nethods are fornulated for systens of first- order ordinary differential equations. Sons nethods applicable directly to equations of notion have been developed by structural dynanicists [11]. and a forn of one of these. the lewnsrk nethod. has been used to analyse nechanisns (3]. A conncn characteristic of internittent contact nechanisns [12] and nechanisns with bearing clearances [13]. is the occurrence of discon- tinuities of the systen paranotara. in tine. Loss of contact is followed by a phase of independent response by two or nore parts of the nechanisn. This phase is terninated by a resunption of contact usually involving inpact. 6ecause of these repeated changes in connectivity there are corresponding changes in paraneters and restraint conditions leading to. for exanple. repeated changes in the underlying natural nodes of vibration. Recent research [16.15] indicates that special difficulties nay be presented by the application of nunorical integration algorithns to systens which experience repeated. sudden. changes of systen paranotara. The purpose of this paper is to describe the application of the classical lunge-Kutta nethod to a test problon exhibiting sudden changes in systen paranotara. This widely used and usually very accurate nethod is shown to behave very badly when applied to a systen of this type. 2233 W The classical fourth-order echoes. often referred to as ‘the 6unge-6utts nethod'. is fornulated for a systen of first- order ordinary differential equations. ‘flherefore the equation of action of a single oscillator d.c...-rm (l) oust be ro-exprassed as the systen 1' - 2‘ (2a.b) 2x - (f(t) - c 2x - k lxl/n In equations (1) and (2) k ad c are stiff- ness and daing coefficients; f(t) is the applied forcing faction: s - ‘x is the displacesent. and s w 2x is the velocity. of the ness n. The huge-hurts nethod applied to a single equation y ‘ f(y.t) (3) generates a sequence of appreciation y“ where n - 0.1.2.... to the exact values y(ta) where tu - iii and h is the tine-step. The algoritl'. has the fore. for a single equation. ’nel - ’n e (k. e 2k1 + 2k2 e k3)h/6 (6a) where k. - f(yu.tu) (6b) "1 - a,“ e 12“., :;u + h/2) (be) u, - mu + gray ta + h/2) (6d) tau-f(yuehkz. tueh) (‘0) The errors generated by such a nethod are of order 6‘ [6.7]. provided that the function of equation (3) seats certain continuity conditions. This theoreti- cal error-perfornence can be confireed by the application of equations (6) to the undaaed free vibration problon nt+ka-0 (50) r-s°;x-0 act-0 (55.6) kh-6s2 (5c) The displecenents obtained for the first few oscil- lations using a tine-step h - 0.03 (equal to l/20th of the true period of the oscillator (3)) are shown in figure 1. The un-erical results are close to the fore of the true solution x - x. cos 2ot. moor-u- -——-——-—-- :‘ \j/\ Geo 1.3a 15c fo—o {on tho TmflSeconcs) WHENNmm) EAL. ..a..._L__a_. ..L_..a_...l. . ...—c 1.30 S ring/ness systen displacenent with p 2 _ “2 [J..] constant stiffness; e 2 . x. 1.3-;x x‘-1-;x. -0-/sec: cubic spline on displecenents by 6th-erder hage-6utts: exact solution points indistinguishable on scale of figure. TAILIl Displacenent error ncrn (equation 6) for the oscillator (equation 3) using 6unge-6utta nethod with ties steps b - 2'k h rrtot mot/h“ 2’2 0.27066307 69.2 2'3 1.711667366-2 70.1 2" 1.006309766-3 66.1 2" 6.233667796-3 63.6 2" 6. 766166616-6 79.9 Repetition of the analysis using aiforn tine-steps h - 2'k where k - 2.3....6 resulted in error perfornance as shown in Table l. The error shown in coli- 2 is ”'n ' ago)“. - 8.5 1.2....6 where N - 1711 is the nu-ber of tine-steps in one exact period of oscillation. In colu-i 3 the error divided by the fourth power of h is seen to be approxinately constant for k - 2... .6. The data were obtained using only single precision aritl-etic and so the sgreenent could be inproved using double precision. Ian 1 6(tn)| (6) Thus the error-order h‘ for the Imge-6utta nethod (6) has been verified for the sinple oscil- lator (3). The actual errors are very snail. for exsqle. usingh-0.03Tas in figure 1. II - II as defined by equation (6) is 601037666-6, 234 cerrdkponding to s naxinun error in displacenent m the first cycle of oscillation equal to Sinilsrly. the error in the energy of the oscillator i.e.. In-gksgelz‘ns: (7) is found to be 0.0266 after one cycle of oscillation usingthe tine staph-1720. adteincrease proportionately for the first few cycles. Since the exact solution of equation (3) is x-s. cos at. it corresponds to a pair of exponen- tials with the cealex conjugate. purely inaginery. enpenents ti wt. Thus by reference to putlished stability diagrans (6.7] the oscillation will be stale provided «a < 13 auto 6’— k/n. The lungs-nutta nethod applied to problens such as defined by equations (1) and (3). having well-behaved functional forns is thus highly accurate; though. of course. it is only condition- ally stable. It has the further advantage of enplicit fornulation. 6.IISI.IIflILII_EDI—IIIIIIIIIIII.£HIIAEI The spring-ness arrangenent shown in figure 2 has been introduced in [13] as a test problon for nochanical systens experiencing sudden changes in paranotara. The case n is attached to ground through the spring of stiffness k1. for snall excursions the nose nay oscillate within the r -x‘ s s s 'd where 'd > 0. for displecenents It > ad the ness is asaunad to contact (without loss of energy ) the springs of stiffness k2 which are preconpressed such that the force/displecenent characteristic is as shown in figure 3. There is thus a sudden change of stiffness at [2' - ‘d' and in addition. a sudden change of force. \\\i '//// VIII \\ \\\\\\\ : I C F“ H "' OI. : nqpen N figure 2 Spring/ness systens with clearance 24 and pro-loaded springs to obtain stiffness 3, discontinuity. force/displecenent characteristics obtained iron the arrangenent of figure 2. figure 3 figure 6 four bar linkage nechanisn. The srrangenent described above contains essential features of nechanisns with internittent contact. for exanple. consider a 6-bsr chain with clearance at the coupler/follower bearing as sketched in figure 6. When contact exists at C the stiffness natrices of the two subsystens ADC and CD are linked. "hen contact is lost at C these two subsystens becone independent; there is then a sudden change of elastic stiffness as seen by each subsysten. Contact-loss is analogous to passage frcn It] > cd to Icl < rd; when contact is re- established. this is equivalent tc crossing fron the inner zone to the outer zone of figure 3. Thus the application of a nunorical integra- tion algorithn to this test systen should indicate the error perfornance which can be expected when the algorithn is applied to a whole nechanisn. for a conplete assessnent of error the exact response in free vibration is required for the systen. As is shown in reference [13] the exact solution consists of a chain of linear initial-value-problens with 235 change points each tine the displacenent Isl crosses fron one side of x‘ to the other. In the case of initial conditions a - a. and s - 0 at the first of the chained solution analyses the period of oscilla- tion is shown in reference [13] to be r - 6(cws.1(6‘/l°)/ua. e tsn.t(-xdus,/zc)/UII (6a) where ;c - - '6'5 siniooe'1(s‘/ao)) (lb) and (see figure 3) as: - (k1 + k,)/n: a; - “1" (can!) fron equations (6) it is seen that the period depends on the relative incursion into the outer stiffness sons i.e.. on so/x‘. and on the ratio of the stiffness kb and k1 of the outer and inner tones respectively. i.e.. on r - Rafi, fer exanple. taking kI - 6a2 the period is 1.0 for r - 1. the ordinary linear oscillator; however. if xo/x‘l - 1.3 the period is 0.737631. for r - 2 and 0.333661 for r - 6. Uhen the initial incursion is reduced to xa/rd - 1.1 the period for a stiffness ratio r - 6 is 0.666692 which is loss than that for r - 2 with xc/x‘ - 1.3. Thus at larger stiffness ratios the period is sensitive to even snall incursions into the outer zone. This property is advantageous in testing algorithn perfornance at internittent contacts. Should the outer zone be less stiff than the inner (the ‘softening' systen) the period is greater than for the ‘hardening' spring systen. Thus for 'ol'd - 1.3 and r - 0.3 the period is 1.331332. W (9) The classical Iunge-Kutta nethod of equations (6) was applied to the initial value problon on + ks - 0 (100) k-k1-6s2 (ISIS!) (106) d k - k. - 6szr (Isl > ad) (10¢) a - s.. a - 0 when t - 0 (10d) described in the previous section. In particular choosing r - 2. 'o"d - 1.3. ad - 1 the result obtained using a tins step h - 0.03 is the connected line shown in figure 3. The anplitudo of the nunorically obtained oscillation is seen to grow rapidly. The exact solution to the problon (10) 6. obtained as described in reference [13] is plotted at significant tine points and of course naintains constant anplitudo in the absence of any physical danping. X a I I d I {awn-u f'-—r "" r -2oc _.M e oases“ “coo coo L'oo do too 2.30 3.6a sec 6.00 nME(Seconda) N éJMow—a.‘ $.64. figure 3 Spring/ness systen displacenent as in figure 2 with (k1 + k2)/k1 - 2; k1/n - 2.1.. . .' 6a “2].xo-l.3-.xd-l-.xo- 0 nn/sec: displecenents by 6th order lungs-Kutta connected by cubic splines: enact solution unconnected squares. A possible explanation for this behavior night lie in the stability condition h s li/o (11) nentioned earlier for this algorithn. however. even in the stiff zone a - J(k'/n) - sJi. Hence. condi- tion (9) requires h < l/s which is certainly satisfied. Thus al the -6utta nethod is being applied correctly (according to the definition for continuous functions) it is unable to perforn to its design order when applied to a systen with a functional discontinuity (equation 10). When r < l in equation (10c) the stiffness of the outer zone is less than that of the inner tone. figure 6 illustrates the nunorical oscilla- tion obtained when xo/rd - 1.3. xd - 1 (as in figure 3) but r - 0.3 (conpared to r - 2 for figure 3). The oscillation now decreases steadily in anplitudo and after about 3 conplete oscillations is con- plately contained within the inner stiffness tone. Since no further discontinuity traverssla occur the subsequent behavior is that expected fron the nethod; nanely. a very gradual loss of anplitudo. aecurning to systens of the 'hardoning' type. figures 3 and 7 show the effects of increasing the stiffness ratio fron 2 to 6 for the sane initial conditions. The frequency of the oscillation is larger in figure 7 because of the greater stiffness in the outer zone than in the systen of figure 3. The growth of anplitudo/cycle is nore rapid in figure 7. attaining 3.3 in 6 cycles conpared with 6 cycles in figure 3. It nay also be noted that after an anplitudo of 6 is attained in figure 7 the oscillation appears to beccne steady. This occurs because the energy of the oscillator is now so great 236 .thatthenesspsesesfrcntheeuterxone (s>sd> 70)tetheeutersone(s<-s‘<0) inonetine-stop (and vice verse). Thus at each tine point the function evaluations of equations (6) are all node in the sac stiffness sons. The iaer tone is effectively hidden frcn the nunorical process. no diacentimities are traversed. ad the usual perfornace of the algoritl- is raevered. D. 8J—.—L—‘- ‘ .e / <1 cause-Ia u.cc IE :5 L'soSEo 3.30 «cc 6. nuaSeoonde) -W—2- cue figure 6 Spring/ness systen displace-ant with outer cone stiffness half the iinuer cone stiffness (which 16 6oz); n - 1 kg; other data as for figure 3. fi I f T ‘77 U: M I Q 1 ’5‘“: /\ I ‘ I ces- " a F\J/\\/ \ «col 4 «at a 4.5.4. 6.64.an ncsn'aubujetbozjoafiaicchcc'cc “HEGecoHdfl figure 7 Spring/ness systen displacenent as for figure 3. but (k1 e k2)/k1 - 6; other infornetion as in figure 3. icon of figures 7 and 6 shows. for the sac stiffness and initial conditions. the effect of halving the tine-step. The anplitudo of oscillation grows ncre slowly in figure 6 requiring about 6 cycles to reach 3.3 conpsred with 6 cycles in the case of figure 7. Thus the iarovenent in perfor- naca due to halving the tine step is roughly proportional to the chaga of the step. This contrasts with the expectation of h‘ dependence which was confirned for a sinple oscillator in Table 1. Thus the effect of the discontinuity traverssls has been to degrade the error perfornance of the classical lungs-Kutta nethod froa order h“ to order h. These findings are consistent with a result proved in a slightly different context by feldstein and Cocdnan [l6] . S. The effect of the initial incursion into the outer (stiff) acne is shown by figures 7 and 9 which were obtained for the sac stiffness ratio (r - 6). tine step (b - 0.03) and discontinuity position 'd - 1. In both cases 6 cycles are required for alitude growth to nagnitudo 6 at which the iaer sane beccnee ‘hidden' fra the nu-erical nethod. Thus the phenonenon depends on the discontinuity traversal itself. not on the extent of imrsion into the outer cone. 6.”! 1 f T— —r— t v v I V Y 1 Loc- 4 J 1 1N4 4 \ 4 4 - moo-1 -... coo c.'ac omm(m) ’- A_A a 1 a a a LA+J ‘.‘_h.‘—J 1b 1.313 {no fan 3.30 3.30 sic a.» maflSeconds) figure 6 Spring/ness systen displecenent as for figure 7. but u - 0.023. ‘7 f V V ‘ V V 1 1 u .' " 2.N-4 ' E . V 1 1 coo-4 1 1 1 ~2.N< 1 4 ..aJ—a—a—e—J ...:e T V T v T—W— * cm can nco vac z.oc zoo 3.oo 3.:c .37.]... nuc(3eccnde) figure 9 Spring/ness systen displecenent as for figure 7. but with x0 - 1.1 a. 3.“ , .- a uI-2.0T F ' ' .1 a cut. 2.” . 0.0., cases ] 2.20-3 1 O ‘ scenes. 1 L“ 1 ocean ‘g ‘ I.” so 5 ? sense 1 moi 4 T . see a ‘ ‘.m-i-0.I'Iggg :::::::seeeaeeaeeeaeeeeeaaeeeeeeeeeseee ua‘ v v Y Y v v r v f v ' :e:eee:el 0.1” 0.30 I.” 1.30 2.” I.” TlME(Soconds) figure 10 Relative energy of the spring ness systens of figures 1.3.6 corresponding to stiffness ratios r - 1. r - 2. r - 0.3; xo - 1.3 -; xd - l -; xo - 0 a/sec; h - 0.03. 237 The changes of the energy of the oscillator are illustrated in figure 10 obtained using a tine- step h - 0.03. and to]:‘ - 1.3. r‘ - l. x. - 0. The points for r - 1 correspond to s sinple linear oscillator and show constant energy within the scale of this figure (actual loss. 0.0636 at tine 2.3). The energy in the ‘hardening' case (r - 2) is seen to jun uupwsrds each tine a discontinuity traversal occurs. The period is about 0.76. and the 6 traver- sala of each ca be followed: n-ely. outer o to iasr e. issuer - to outer - . outer - to iiuuer -. iuuer o to outer e. Thus energy increases at each traversal regardless of whether the crossing is fron the inner to the outer zone or fron the outer to the iaer sons. The actual sire of the jugs appears to vary substaintially and without obvious pattern. An explanation of these phenonena say be found in reference [13]. Although not visible at the scale of figure 10 the scary decreases slightly during the ties- steps between discontirauity traversala in accordance with the expectation of the lungs-6utta nethod applied to an oscillator with constant stiffness. Thisisthecaseforbothr-2sndr-0.3. Inthe latter case the energy jugs at discontinuity traversala are always seen to be to lower levels. as anticipated fren figure 6. The energy ultinately becane approxinately that for an initial displace- nent so - 'd in sgreenent with the discussion of figure 6. 96691631063 The perfornance of lungs-Kutta nethods which are widely used in nechaisn simulation has been confirned to be order h‘ when applied to a systen of constant stiffness. There is a snail decay of energy at a steady progression resulting in a slow decrease of alitude. however. when the lunge-6utta nethod is applied to a oscillator with a discontinuous stiffness characteristic the error perfornance degrades to order h. for ‘hardening' systens the energ ju-s to a higher level at each traversal of the discontinuity regardless of the direction of crossing. Conversely with ‘softening' systens the energy jugs to lower levels at each discontinuity traversal. These energy juns eight be order 20 percent. are of widely varying nagnitudo. without obvious pattern. and are generally enorncus coaared to the natural loss (< 0.1 percent per cycle typically). The energy jinn were not affected by the extent of the initial incursion into the outer zone. Thus it is clearly evident that the application of this type of numerical integration nethod to flexible systens in which internittent contacts occur could give rise to large spurious energy ju-a. In a large nechanisn simulation these jugs would be disguised eonewhat. losses being attributed to insct for oxale. and gains to resonances. further investigations are being carried out to establish nethods of reducing or olininating the energy jugs at discontinuities for lunge-Kutts nethods. ad other co-only used nethods such as the lot-ark fanily of integration algo- ritlms. 10. ll. 12. 13. 6.S. Ilaines. 'Survey: 2-dinensional notion and inpact at revolute jointe.’ . 1.1. 361-370. 1960. 6.S. Thcqson and C.6. Sung. 'A survey of finite elenent teclnuiqueo for nechanisn doairt.‘ . . 331-339. 1966. J.f. Sadler and 0.6. Sandor. 'A lumped paraeter approach to vibration and stress analysis of elastic 1inkages.‘ ASHE . l . 369-337. 1973. u 6.S. Thoqson and C.6. Sunug. 'A variational formulation for the nonlinear finite olonont analysis of flexible linkages: theory. inplenentation. and experinental results. Iracsniaaicnai_acd_6utasaticn_in_nasisn. m. 662-666 . 1966. 6.S. Thoason and C.6. Sung. 'An aalyticsl and experinental investigation of high-speed nechanisns fabricated with coacsite lainstes. ' librarian. Ill. 399-626. 1966. 3.0. Lanbort. ccscutaticcalgaatbcda_in_ardinarx_ diffarontial_cauationa. H1107. Chichost-r. 1973. C.6. Cear. . Prentice- liall. Englewood Cliffs. no. 1971. 0.6. Sandor and A.C. Erdnan. W Prentice-Hall. Inglewood Cliffs. 6.7. 1966. 6. Uilson and JJI. fawcett. 'Dynanics of the slider-crank nechanisn with clearance in the sliding bearing.‘ We 20 ‘1'”) 197‘- S. Dubowsky and S.C. Young. 'An experinental ad analytical study of connection forces in higu-speed nechanisns.‘ 6866 m 1973 . 913. 1160-1176. K.-J. lathe. . Prentice-hall. Inglewood Cliffs. ID. 1962. T.U. lee and A.C. Hang. 'On the dynanics of internittent-notion nechanisns: fart I. Dynaic nodal and response.‘ ASHE faper 62-DET-66 presented at the Design and Production Engineering Technical morons. Usahington. D.C.. Sept. 12-13. S. Mowaky. J.f. Deck and ll." Costello. 'The dynaic nodelling of spatial nechine systens with clearance countections.‘ ASME fsper 66-DET-62. presented at the Design Engineering Technical Conference. Coluaus. Ohio. October 3-6. 1966. 238 1a. "3.0. Isrthich. 'Tbe mmsrical solution of discontimuous struuctural systens.‘ W W. odo- I- her: and I.f. Wolfe. University of mm. hglnd. 307-316. 1906. 13. J.I.I. brown and 91.0. Wok. 'A test problon for structural systens with discontirnuties.' (to appear) M. WM. 16. A. feldatain and E. Coohan. 'Iumerical solution of ordinary and retarded differential equations with disosntimous brimim.‘ W. 21. l-13.. 1973. BIBLIOGRAPHY 10. BIBLIOGRAPHY Alexander, R.M. and Lawrence, K.L., "An Experimental Investigation of the Dynamic Response of an Elastic Mechanism," ASME Journal of Eggineerigg for Industry, 1974, Vol.968, pp. 268—274. Bakr, E. M. and Shabana A. A., "Effect of Geometric Elastic Non- Linearities on the Impact Response of Flexible Multi-Body Systems," Journal of Sound and Vibration, 1987, Vol. 112, No. 3, pp. 415-432. Bengisu, M.T., Hidayetoglu, T. and Akay. A.. "A Theoretical and Experimental Investigation of Contact Loss in the Clearances of a Four-Bar Mechanism,"ASME Journal of Mechanisms, Transmissions, and Automation in Desigg, Vol. 108, 1986, pp. 237—244. Bolotin, V.V., The Dynamic Stability of Elastic System, Holden-Dsy, San Francisco, 1964. Brovn, J.M.B., Soong, K., Borthwick, V.K.D., Gandhi, M.V. and Thompson, 3.8., "Effects of Intermittent Contact in Mechanisms of the Performance of Numerical Integration Methods," 10th Applied Mechanisms Conference, Dec. 1987, New Orleans, Louisiana. Chace. M.A., "Analysis of the Time-Dependence of Multi-Freedom Mechanical Systems in Relative Coordinates,"ASME Journal of Engineering for Industry, Feb. 1967, Vol.89B, pp. 119-125. Chace, M.A. and Bayazitoglu, Y.0., "Development and Application of a Generalized D'Alembert Force for Multifreedom Mechanical system, "ASME Journal of Engineeringg for Industry, Vol.9BB,Feb. 1971, pp.317-327. Chace. M.A. and Donald, A.S., "DAMN - Digital Computer program for the Dynamic Analysis of Generalized Mechanical Systems,”§A§ Transactions, Vol. 80, pp. 969—983, 1971. Chu, S.C. and Pan K.C., "Dynamic Response of a High-Speed Slider- Crank Mechanism Vith an Elastic Connecting Rod," ASME Journal of Eggineering for Industry, 1975, Vol. 978, pp. 542-550. Cleghorn, V. L., Fenton, R. G. and Tabarrok, 3., "Finite Element Analysis of High-Speed Flexible Mechanisms," Mechanism and Machine Theory, 1981, Vol. 16, No.4, pp. 407—424. 239 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 240 DRAM - Users Guide to Dram, Mechanical Dynamics, Inc.. Ann Arbor, Michigan 48164, Dec. 1979. Dubowsky, S. and Freudenstein, F., "Dynamic Analysis of Mechanical Systems with Clearances, Part 1: Formulation of Dynamic Model,"ASME Journal of Engineering for Industry, 1971, Vol. 938, pp. 305—309. Dubowsky, S. and Freudenstein, F., "Dynamic Analysis of Mechanical Systems with Clearances, Part 2: Dynamic Response,"ASME Journal of Engineeringgfor Industry, 1971, Vol. 938, pp. 310-316. Dubowsky, 8., "0n Predicting the Dynamic Effects of Clearances in Planar Mechanisms,"ASME Journal of Engineering for Industry, 1974, Vol. 968, pp. 317-323. Dubowsky, S.. "0n Predicting the Dynamic Effects of Clearances in One—dimensional Closed Loop Systems,"ASME Journal of Engineeringgfor Industry, 1974, Vol.968, pp. 324—329. Dubowsky, 8., Young, S.C., "An Experimental and Analytical Study of Connection Forces in High-Speed Mechanisms,”ASME Journal of Eggineering for Industry, 1975, Vol. 978, pp. 1166-1176. Dubowsky, S. and Gardner T. M., "Dynamic Interactions of Link Elasticity and Clearance Connections in Planar Mechanical Systems," ASME Journal of Engineering for Industry, 1975, Vol. 978, PP.652— 661. Dubowsky, S. and Gardner T.N., "Design and Analysis of Multilink Flexible Mechanisms With Multiple Clearance Connections," ASME Journal of Engineering for Industry, 1977, Vol. 998, PP. 88-96. Dubowsky, S. and Moening M.F., "An Experimental and Analysis Study of Impact Forces In Elastic Mechanical Systems with Clearances," Mechanism and Machine Theory, 1978, Vol.13, PP. 451-465. Dubowsky, S. and Morris, T.L., "An Analytical and Experimental Study of the Acoustical Noise Produced by Machine Links," ASME Journal of Vibration, Acoustics, Stress, and Reliability in Desigg, 1983, Val. 105, pp.393-401. Dubowsky, S.. Norris, M., Aloni, E., Tamir, A.. "An Analytical and Experimental Study of the Prediction of Impact in Planar Mechanical Systems with Clearances,"ASME Journal of Mechanisms, Transmissions, and Automation in Desigg, Vol. 106, 1983, pp. 554—651. Dubowsky, S.. Deck, J.F. and Costello 8., "The Dynamic Modeling of Flexible Spatial Machine Systems With Clearance Connections", ASME Journal of Mechanisms, Transmissions, and Automation in Design: I987, v01. 169’ ppe 87-9de 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 241 Earles, S.V.E. and Vu, C.L.S., "Motion Analysis of a Rigid-link Mechanism with Clearance at a Bearing, Using Lagrangian Mechanics and Digital Computation,"Mechanisms, Institute of Mechanical Engineers, London, England(1973), pp.83-89. Earles, S.V.E., "Predicting the Occurrence of Contact Loss and Impact at a Bearing from a Zero-clearance Analysis,"Proc. of the IFToMM Fourth World Con ress on the Theory of Machines and MechaniSms, Newcastle upon ¥yne,EngIand,1975, pp. 1013-1018. Earles, S.V.E. and Kilicay, O., "Predicting Impact Conditions due to Bearing Clearances in Linkage Mechanisms,"Proceedings of the Fifth World Con ress on Theory of Machines and MechanISms, 1979, Montreal, Canada, pp. 078—1081. Erdman, A.C., Sandor, G.N. and Oakberg, R.G., ”A General Method for Kineto-Elastodynamic Analysis and Synthesis fo Mechanism," ASME Journal of Engineering for Industry, 1972, Vol. 948, pp. 1193—1265. Erdman A.G. and Sandor G.N., "Kineto-elastodynamics - a review of state of art and trends," Mechanism and Machine Theory, 1972, vol 7, pp. 19-33. Funabashi M., Ogawa K., and Horie M., "A Dynamic Analysis of Mechanisms with Clearances", Bulletin of the JSME, Vol. 21, No. 161, Nov. 1978, Paper No. 161-14 Gandhi, M.V. and Thompson, 8.8., "The Finite Element Analysis of Flexible Components of Mechanical Systems Using a Mixed Variational Principle,” ASME Paper No. 80-DET-64. Garrett, R.E. and Hall, A.S., "Effect of Tolerance and Clearance in Linkage Design,”ASME Journal of Engineerigg for Industry, 1969. Vol.918 , pp. 1981262. Gear, 0.9., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs. Gear, C.V., "Simultaneous Numerical Solution of Differential- Algebraic Equations,"IEEE Transactions on Circuit Theory, Vol. Ct- 18, No. 1, Jan. 1971, pp.89¥95: Gibson, J.E., Introduction to Engineering Desigg, 1968, Holt, Rinehart and Winston, Inc.. New YorE. Goldsmith, V., Impact, the Theory and Physical Behavior of Colliding solids, London, E. Arnold, 1960. Grant, S.J. and Fawcett, J.N., "Control of Clearance Effects in Mechanisms,"ASME Journal of Mechanical Desigg, Oct. 1978, Vol. 100, pp. 728—731. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 242 Grant, S.J. and Fawcett, J.N., "Effects of Clearance at the Coupler- Rocker Bearing of a 4-Bar Linkage,"Mechanism and Machine Theory, 1979, v01. 14’ ppe 99-110e Baines, R.S., "Survey: 2-Dimensiona1 Motion and Impact at Revolute Joints.”Mechanism and Machine Theory, 1980, Vol. 15, pp. 361-370. Haines, R.S. "A Theory of Contact Loss at Revolute Joints with Clearance,"Journa1 of Mechanical Engineering Science, Vol. 22, No. 3, 1980, pp. 129-136. Haines, R.S., "Contact Loss at Revolute Kinematic Joints: a 'Ludodynamic' Theory Tested,”Institution of Mechanical Engineers, VOle 199, "Go C3, ppe 173-180. Baines, R.S., "An Experimental Investigation Into The Dynamic Behaviour of Revolute Joints Vith Varying Degrees of C1earance,”Mechanism and Machine Theory, Vol. 20, 1985. No. 3, pp. 221-231. Haug, E.J., Vu, S.C. and Yang, S.M., "Dynamics of Mechanical Systems With Coulomb Friction, Stiction, Impact and Constraint Addition- Deletion - I," Mechanism and machine Theory, 1986, Vol. 21, No. 5, pp. 401-406. Hildebrand, F.B., Methods of Applied Mathematics, second edition, 1962 Prentice-Hall, Inc.. Englewood Cliffs. iMSL Library Reference Manual, 1982, published by IMSL, Houston, Texas. Kane, T.R. and Levinson, D.A., ”Multibody Dynamics,” Journal of Applied Mechanics, 1983, Vol. 50, pp.1071-1078. Keller, J.E., "Impact with Friction," Journal of Applied Mechanics, 1986, Vol. 53, pp. 1-4. Khulief Y. A. and Shabana A. A., "Impact Responses of Multi-Body Systems With Consistent and Lumped Masses," Journal of Sound and Vibration, 1986, Vol. 104, No. 2, pp. 187-207. Khulief Y. A. and Shabana A.A., "Dynamic Analysis of Constrained System of Rigid and Flexible Bodies with Intermittent Motion," ASME Journal of Mechanisms, Transmissions, and Automation in Design: 1986, Vol. 108, pp.38-45} Kim, S.S., Shabana. A.A. and Haug, E.J., "Automated Vehicle Dynamic Analysis with Flexible Components,” ASME Journal of Mechanisms, Transmissions, and Automation in Design, 1984, Vol. 106, pp. 127: 132. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 243 Kobrinskii, Dynamics of Mechanisms with Elastic Connections and Im act S stnn, English translation By R. Lennox-napier, London, lLIFFE Books LTD., 1969. Kolhatkar, S.A. and Yajnik, K.S., ”The Effects of Play in the Joints of a Function - Generating Mechanism,"Journal of Mechanisms, Vol. 5, pp. 521-532, 1969. Lanczos, C.. Ine Variational Piinciples of Mechanics, third edition, University of Toronto Press, Toronto, 1966. Lowen, G.G. and Jandrasits, V.G., "Survey of Investigations into the Dynamic Behavior of Mechanisms Containing Links with Distributed Mass and Elasticity," Mechanism and Machine Theory, 1972, Vol. 7, No. 1, pp. 3-17. Lowen, G.G. and Chassapis, C., "The Elastic Behavior of Linkages: An Update,” Mechanism and Machine Theory, 1986, Vol. 21, No. 1, pp. 33-42. Mansour, V.M. and Townsend, M.A., "Impact Spectra and Intensities for High-speed Mechanisms,"ASME Journal of Engineering for Industry ,1975, Vol. 97B, pp. 347-353. Martin, G.N., Kinematics and Dynamics of Machines, McGraw-Hill Book Company, New York, 1969} Midha A., Erdman A.G. and Frohrib, D.A., "Finite Element Approach to Mathematical Modeling of High-Speed Elastic Linkages,” Mechanism and Machine Theory, 1978, Vol. 13, pp. 603-618. Midha A., Erdman A.G. and Frohrib D.A., "An Approximate Method for the Dynamic Analysis of Elastic Linkages," ASME Journal of Engineering for Industry, 1977, Vol. 97B, pp. 449-455. Miedema. B. and Mansour, V.M., ”Mechanical Joints with Clearance: a Three-Mode Model,"ASME Journal of Engineeringiifor Industry, 1976, VOI. 988’ ppo 1319-1323s Nath P.K. and Ghosh A., "Kineto-Elastodynamic Analysis of Mechanisms by Finite Element Method,” Mechanism and Machine Theory, 1980, Vol. 15, pp. 179-197. Nath P.K. and Ghosh A., "Steady State Response of Mechanisms with Elastic Links by Finite Element Method," Mechanism and Machine Theory, 1980, Vol. 15, pp. 199-211. Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillations, 1979 Neubauer, A.H., Jr., Cohen, R. and Hall, A.S.,Jr., "An Analytical Study of the Dynamics of an Elastic Linkage," ASME Journal of Engineering for Industry, 1966, Vol. 888, pp.311-317. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 244 Nikravesh, P.E., Computer-Aided Analysis of Mechanical Systems, Prentice Hall, Englewood Cliffs, NJ, 1988. Paul, 8., "Analytical Dynamics of Mechanisms - a Computer Oriented Overview,"Mechanism and Machine Theony, 1975, Vol. 10, pp. 481-507. Reddy, J.N., Energy and Variational Methods in Applied Mechanics yith an iintroduction to the finite element method, John Wiley & Sons, New York, 1984. '7 Reuleaux, F., "Theoretische Kinematik: Grundzuge einer Theorie des maschinenwesens. Vieweg, Brunswick, Deutschland (1875). English translation by A. B. V. Kennedy, Kinematics of Machinery, Macmillan, London(1876). Reprinted by Dover Publications, New York (1963). Reid, K.N., Cohen, R., Garrett, R.E., Rabins, M.J., Richardson, 8.8., Viner, V.O., Research Needs in Mechanical Systems, prepared by the Select Panel on Research Goals and Priorities in Mechanical Systems, for the National Science Foundation, ASME book I00178, 1984 Sadler J. P. and Sandor G. N., "A Lumped Parameter Approach to Vibration and Stress Analysis of Elastic Linkages," ASME Journal of Engineering for Industr , 1973, Vol. 958, pp. 549-557. Sadler, J.P. and Sandor G.N., ”Nonlinear Vibration Analysis of Elastic Four-Bar Linkages," ASME Journal of Engineering for Industry , 1974, VOle 96B, ppe 411’41ge Shabana A.A. and Wehage R.A., "Variable Degree-of-Freedom Component Mode Analysis of Inertia Variant Flexible Mechanical Systems," ASME Journal of Mechanisms, Transmissions, and Automation in Design, I983 , Vol. 105, pp.371-378. Shabana, A.A. and Wehage, R.A., ”A Coordinate REduction Technique for Dynamic Analysis of Spatial Substructures with Large Angular Rotations," Journal of Struct Mechanics, 1983, Vol. 11, No.3, pp.401-431. Shabana, A. A. and Wehage R. A., "Spatial Transient Analysis of inertia-Variant Flexble Mechanical Systems," Asme Journal of Mechanismsm, Transmissions, and Automation in Design, 1984, Volfl06, pp. 172-178. Sheth, P.N. and Uicker, J.A. Jr., "IMP : A Computer-aided Design Analysis System for Mechanisms and Linkages," ASME Journal of Engineering for Industry, 1972, Vol. 94B, pp. 454-464. Shigley, J. E. and Uicker J. J.. Jr., Theory of Machines and Mechanisms, 1980, McGraw-Hill Book Company, N.Y. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 245 Shimojma 8., Ogawa K., and Matsumoto K., "Dynamic Characteristics of Planar Mechanisms with Clearances", Bulletin of th JSME, Vol. 21, No. 152, Feb. 1978, Paper No. 152-18 Simon, R.A., A Student's Introduction to Engineering Design, 1975, Pergamon Press Inc., New York. Smith, M.R. and Maunder L., "Inertia forces in a four-bar linkage,” Journal of Mechanical Engineeringi§cience, 1967, Vol. 9, No. 3, pp. 218-225. Song, J.O. and Haug, E.J., "Dynamic Analysis of Planar Flexible Mechanisms,” Computer Methods in Applied Mechanics and Engineering, 1980, Vol. 24, pp. 359-381. Soni, A.H., Huston, 8., Paul, B., Haug, E., Valdron, K. and Bejczy, T., Draft: Strategic Research in Dynamic Systems and Control, Resenrch Needs and Opportunities in Maching Dynamics, prepared by the Select Panel on Research Goals and Priorities in Machine Dynamics, for the National Science Foundation, Mechanical Engineering and Applied Mechanics Division. Soong, K. and Thompson, 8. S., "An Experimental and Analytical Investigation of the Dynamic Response of a Slider-crank Mechanism with Controlled Radial Clearance at the Gudgeon-pin Joint,” 10th Applied Mechanisms Conference, Dec. 1987, New Orleans, Louisiana. Stoer, J. and Bulirsch, R., Introduction to Numerical Analysis, Springer-Verlag, New York Inc., 1980} Sung, C. K., Thompson, B. S.. Crowley, P. and Cuccio, J., "An Experimental Study to Demonstrate the Superior Response Charaterictics of Mechanisms Constructed with Composite Laminates,” Mechanism and Machine Theory, 1986, Vol. 21, No. 2, pp. 103-119. Thompson, 8.8. and Ashworth, R.P., "Resonance in Planar Mechanisms Mounted on Vibrating Foundations," Journal of Sound and Vibration, 1976, Vol. 49, No. 3, pp. 403-414. Thompson, 8.5. and Barr, A.D.S., "A Variational Principle for the Elastodynamic Motion of Planar Linkages," ASME Journal of Engineering for Industry, Vol. 98, Nov. 1976, PP.1306-13l2} Thompson, 8.8., "The Analysis of an elastic Four-Bar Linkage on a Vibrating foundation using a Variational Method," ASME Journal of Mechanical Design, 1980, Vol. 102, No. 2, pp. 320-328. Thompson, B.S., Zuccaro, D., Gamache, D. and Gandhi, M.V., "An Experimental and Analytical Study of the Dynamic Response of a Linkage Fabricated From a Unidirectional Fiber-Reinforced Composite Laminate," Journal of Mechanical Design, Paper No. 82-DET-67. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 246 Thompson, 8.8., "A Variational Formulation for the Finite Element Analysis of the Vibrating and Acoustical Response of High-Speed Machinery,"Journal of Sound and Vibration, Vol. 89, No. 1, 1983, pp. 7-15. Thompson, 8.5. and Sung, C.K., "A Variational Formulation for the Nonlinear Finite Element Analysis of Flexible Linkages: Theory, Implementation, and Experimental Results,"ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 106, No. 4; Dec. 1984, pp.482-488. Thompson, 8.8. and Sung, C.K., "A Survey of Finite Element Technique in Mechanism Design," Mechanism and Machine Theory, 1986, Vol. 21, NOe 4, ppe351’359e Thompson, 8.8. and Sung, C.K., "A Theoretical and Experimental Study of the Response of Linkage Mechanisms Fabricated with Composite Laminates," Journal of Sound and Vibration, 1986, Vol. 111, No. 2, pp. 399-428. Thompson, 8. 8., "Composite Laminate Components for Robotic and Machine Systems: Research issues in Design," npplied Mechanics Review, 1987, Vol. 40, No. 11, pp. 1545-1552. Townsend, M. A. and Mansour, V. M., "A Pendulating Model for Mechanisms with Clearances in the Revolutes,"ASME Journal of Engineering for Industry, 1975, Vol. 978, pp. 354-358. Turcic, D.A. and Midha A., "Generalized Equations of Motion for the Dynamic Analysis of Elastic Mechanism Systems," Journal of Dynamic Systems,Measurement, and Control, 1984, Vol. 106, pp. 243-248. Turcic, D.A. and Midha A., "Dynamic Analysis of Elastic Mechanism Systems. Part I: Applications," Journal of Dynamic Systems, Measurement, and Control, 1984, Vol. 106, pp. 249-254. Turcic, D. A. and Midha A., "Dynamic Analysis of Elastic Mechanism Systems. Part II: Experimental Results," Journal of Dynamic Systems, Measurement, and Control, 1984, Vol. 106, pp. 255-260. VaxLab/Lantar Prngrammer's Guide, AA-HX04B-TE, Digital Software, Digital Equipment Corporation, Maynard, Massachusetts. Viscomi, B.V. and Ayre R.S., "Nonlinear Dynamic Response of Elastic Slider-Crank Mechanism," ASME Journal of Engineering for Industry, 1971, Vol. 938, pp. 251-262. Wehage, R.A. and Haug, E.J., "Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems," Journal of Mechanical Design, 1982, Vol. 104, pp. 247-255. 247 99. Vehage, R.A. and Haug, E.J., "Dynamic Analysis of Mechanical Systems with Intermittent Motion," ASME Journal of Mechanical Design, 1982, Vol. 104, pp. 778-784. 100.Vilson, C.E. Jr. and Michels, V.J., Mechanism, Desi n-Oriented Kinematics, American Technical Society,’Chicago, I1 inois. 1969. 101.Vi1son, E. L. and Penzien, J., "Evaluation of Orthogonal Damping Matrices," International Journal for Numerical Methods in Engineering, 1972, Vol. 4, pp. 5-10} 102.Vilson, R. and Fawcett, J.N., "Dynamics of the Slider-Crank Mechanism with Clearance in the Sliding Bearing,"Mechanism and Machine Theory, 1974, Vol. 9, pp. 61—80. 103.Vinfrey, R.C., ”Elastic Link Mechanism Dynamics," ASME Journal of Engineering for Industry, 1971, Vol. 938, pp. 268-272. 104.Vinfrey, R.C., Anderson R.V. and Gnilka C.V., ”Analysis of Elastic Machinery Vith Clearances," ASME Journal of Engineering for Industry 1973, Vol. 958, pp. 695-703. 105.Vu, C.L.S. and Earles, S.V.E., "A Determination of Contact-loss at a Bearing of a Linkage Mechanism,"ASME Journal of Engineering, for Industry, Vol.99B, May 1977, pp. 375-380. 106.Vu, S.C. and Yang, S.M.,Baug, E.J., ”Dynamics of Mechanical Systems With Coulomb Friction, Stiction, Impact and Constraint Addition- Deletion - II," Mechanism and machine Theory, 1986, Vol. 21, No. 5, pp. 407-416. 107.Vu, S.C. and Yang, S.M.,Haug, E.J., "Dynamics of Mechanical Systems Vith Coulomb Friction, Stiction, Impact and Constraint Addition- Deletion - III," Mechanism and machine Theory, 1986, Vol. 21, No. 5, pp. 417-425.