rm”.-. RRRRRRRR lllllllllllllll H 530 mm llllllllllll iiIGIII .y-IIIIIQ University This is to certify that the thesis entitled An Analytical And Experimental Investigation Of The Elastodynamic Response Of A Class 6f Intelligent Machinery presented by Vasudivan Sunappan has been accepted towards fulfillment of the requirements for __M_._S_._degree in Mechanical Engineering ?%£:c./g (712:9,?/%L£h\ / Major rrp rofessor Date my Mi? 0-7639 MS U i: an Aflirnlan'w Action/Equal Opportunity Institution \l __ .._.._.__ r: I MSU RETURNING MATERIALS: Place in book drop to LJBRARJES remove this checkout from —_ your record. FINES will be charged if book is returned after the date stamped below. ’T CX A 052 , 7L? 2 9 3.1;!) 0x 121 AN ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF THE ELASTODYNAMIC RESPONSE OF A CLASS OF INTELLIGENT MACHINERY BY Vasudivan Sunappan A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1987 ABSTRACT AN ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF THE ELASTODYNAMIC RESPONSE OF A CLASS or INTELLIGENT MACHINERY By VASUDIVAN SUNAPPAN A methodology is proposed, herein, for reducing the elastodynamic response of high-speed machine systems by introducing an additional perturbational input which is computer-controlled. The work presented here describes a comprehensive analytical and experimental investigation on a bread-board model of the concept, in which the objective is to reduce the vibrational response at the midspan of a flexible rocker link of a "retrofitted" four-bar linkage. A variational formulation is employed to develop the equation governing the response of the flexible member to the combined parametric and forcing functions. This equation is then solved numerically and the viability of the model is validated by comparing the analytical results with response-data from a complementary experimental program. The stability of motion is examined using the above equation. ACKNOWLEDGMENTS I wish to express my deepest gratitude to my advisor Dr. Brian S. Thompson for his guidance and assistance throughout this study and in preparation of this manuscript. His academic excellence and research philosophy have provided a constant source of encouragement. I also wish to thank the members of my committee Dr. B. Fallahi and Dr. Steven W. Shaw for their helpful suggestions. The efforts and assistance of Dr. C. K. Sung and K. M. Soong at Michigan State University are sincerely appreciated. Finally, I would like to thank my parents for their encouragement and financial support that help to complete this work. ii TABLE OF CONTENTS page List of Figures ..................................................... v Chapter 1. Introduction ............................................ 1 1-1 Current Trends in Industrial Machinery .............. 1 1-2 Intelligent Machine Systems ......................... 3 Chapter 2. Kinematic Analysis of The Modified Linkage .............. 6 2-1 Introduction ........................................ 6 2-2 Kinematics .......................................... 7 2-3 Dynamics ............................................ 12 Chapter 3. Variational Theorem ..................................... 14 3-1 Theoretical Development ............................. 14 3-2 Approximations For The Variables .................... 20 3-3 Construction of The Problem Definition .............. 25 3-4 An Approximate Solution.. ........................... 30 Chapter 4. Solution to Hill's Equation ............................. 35 4-1 Quasi-Static Response ............................... 35 4-1 Elastodynamic Response .............................. 36 Chapter 5. Experimental Investigation .............................. 40 5-1 Experimental Apparatus .............................. 40 5-2 Instrumentation ..................................... 44 5-3 Experimental Procedure .............................. 49 Chapter 6. Stability Analysis ...................................... 53 6-1 Stability of Motion ................................. 53 iii Chapter 7. Elastodynamic Response Of The Unmodified Linkage and The Synthesis Of The Shaker/Slider Motion ............... 56 7-1 Response Curves for The Flexible Four-Bar Linkage...56 7-2 Curve Synthesis for Shaker/Slider Motion ............ 60 Chapter 8. Experimental and Computational Results .................. 72 8-1 Response Data and Absolute Acceleration of The Flexible Link ........ .... ............................... 72 8-2 Kinematic Results of The Flexible Link .............. 81 Chapter 9. Discussion of Results ................................... 89 Chapter 10. Conclusions ............................................. 93 Chapter 11. Future Work ............................................. 94 Bibliography ........................................................ 95 iv LIST OF FIGURES page Intelligent Mechanismzthe kinematic chain ..................... 6 Vector-Loop Diagram ........................................... 7 Component Pin Forces .......................................... 11 Definition of axis systems and position vectors ............... 14 The modelling of an Intelligent mechanism ..................... 21 Forces exerted at the rocker-slider pin joint ................. 24 Experimental Intelligent Mechanism ............................ 41 Electrodynamic Shaker-Slider Assembly ......................... 41 Schematic Diagram of the Instrumentation ...................... 45 Assembly of Crank and Airpax pickup ........................... 46 Data-Acquisition System and Equipment ......................... 46 Transient Response of Rocker Link. ............................ 49 Frequency-response curves for the four-bar linkage ............ 51 Link Calibration-Fixation...... ............................... 52 Analytical Frequency-response curves .......................... 55 Elastodynamic Response of the Four-bar Linkage at 148 rpm ..... 57 Elastodynamic Response of the Four-bar Linkage at 170 rpm ..... 58 Elastodynamic Response of the Four-bar Linkage at 200 rpm ..... 58 Elastodynamic Response of the Four-bar Linkage at 215 rpm ..... 59 Quasi-static Response of the Four-bar Linkage at 200 rpm ...... 59 Absolute Acceleration of The Rocker Link ...................... 63 Suggested shaker/slider waveform proportional to quasi-static response ......................................... 65 Absolute acceleration of the rocker link at 200 rpm given shaker/slider waveform shown in Fig. 7.7 ................ 65 7.9 oooooooooo \l U§WNH season ‘0 .10 on .11 Analytical elastodynamic response at 200 rpm given shaker/slider waveform shown in Fig. 7.7 ................ 66 Experimental elastodynamic response at 200 rpm given shaker/slider waveform shown in Fig. 7.7 ................ 66 Angular acceleration of the rocker link at 200 rpm given shaker/slider waveform shown in Fig. 7.7 ................ 67 Suggested shaker/slider waveform inverse of quasi-static response ......................................... 67 Absolute acceleration of the rocker link at 200 rpm given shaker/slider waveform shown in Fig. 7.12 ............... 68 Analytical elastodynamic response at 200 rpm given shaker/slider waveform shown in Fig. 7.12 ............... 68 Experimental elastodynamic response at 200 rpm given shaker/slider waveform shown in Fig. 7.12 ............... 69 Angular acceleration of the rocker link at 200 rpm given shaker/slider waveform shown in Fig. 7.12 ............... 69 Synthesis of shaker/slider waveform ........................... 71 Experimental elastodynamic response at 148 rpm ................ 73 Experimental quasi-static response at 148 rpm ................. 73 Analytical elastodynamic response at 148 rpm .................. 74 Analytical quasi-static response at 148 rpm ................... 74 Absolute acceleration of the rocker link at 148 rpm ........... 75 Experimental elastodynamic response at 170 rpm ................ 75 Analytical elastodynamic response at 170 rpm .................. 76 Absolute acceleration of the rocker link at 170 rpm ........... 76 Experimental elastodynamic response at 200 rpm ................ 77 Analytical elastodynamic response at 200 rpm .................. 77 Absolute acceleration of the rocker link at 200 rpm ........... 78 Analytical and experimental results for the intelligent mechanism at 200 rpm .............................. 78 Experimental elastodynamic response at 215 rpm ................ 79 Experimental quasi-static response at 215 rpm ................. 79 Analytical elastodynamic response at 215 rpm .................. 80 vi oooooooooooooooo QQQGQQGGOQ .16 .17 .18 .19 .20 .21 .22 .23 .24 .25 .26- .27 .28 .29 .30 .31 .32 .33 .34 .35 .36 Absolute acceleration of the rocker link at 215 rpm ........... 80 Shaker/slider displacement at 148 rpm ......................... 82 Shaker/slider displacement at 170 rpm ......................... 82 Shaker/slider displacement at 200 rpm ......................... 82 Shaker/slider displacement at 215 rpm ......................... 83 Shaker/slider velocity at 148 rpm ............................. 83 Shaker/slider velocity at 170 rpm ............................. 83 Shaker/slider velocity at 200 rpm ............................. 84 Shaker/slider velocity at 215 rpm ............................. 84 Shaker/slider acceleration at 148 rpm ......................... 84 Shaker/slider acceleration at 170 rpm ......................... 85 Shaker/slider acceleration at 200 rpm ......................... 85 Shaker/slider acceleration at 215 rpm ......................... 85 Force exerted by the shaker on the slider at 148 rpm .......... 86 Force exerted by the shaker on the slider at 170 rpm .......... 86 Force exerted by the shaker on the slider at 200 rpm .......... 86 Force exerted by the shaker on the slider at 215 rpm .......... 87 Angular acceleration of the rocker link at 148 rpm ............ 87 Angular acceleration of the rocker link at 170 rpm ............ 87 Angular acceleration of the rocker link at 200 rpm ............ 88 Angular acceleration of the rocker link at 215 rpm ............ 88 vii CHAPTER 1 INTRODUCTION 1-1 CURRENT TRENDS IN IRDUSTRIAL.HACHINERY The intense competition in the international marketplace for robots and machine systems which significantly enhance manufacturing productivity by operating at high speeds has resulted in the evolution of a new frontier in machine design. Under these more stringent operating conditions, the traditional design methodologies are unable to adequately predict a machine's performance because elastodynamic phenomena are stimulated due to the inherent flexibility of the moving parts. The traditional design methodologies are based on dynamic analyses wherein all mechanism members are treated as rigid-bodies. When operating in a high-speed mode, vibrations and dynamic stresses in the members of a mechanism can drastically modify the performance characteristics, and the fatigue-life of parts becomes a significant design consideration. Furthermore, the radial clearances in sleeve bearings in mechanical and electro-mechanical systems, which are essential for the operation of these joints, will generally result in excessive stresses and impact loads, and these loads can generate more severe problems such as wear, loss of performance, increased levels of noise and vibration. The trend towards ultra-high operating speeds as a method for increasing productivity has exacerbated the design of modern machine systems and as a consequence of these stimuli, the research community has responded by developing more sophisticated modeling techniques [1,2,3]. In addition to these enhanced predictive capabilities, described in the above review articles the academic community has also developed two design methodologies for reducing the elastodynamic response of linkage machinery. (Vibrations or deflections in a mechanism with elastic links under dynamic conditions is called elastodynamic response.) The first advocates that the articulating members of these mechanical systems should be designed with optimal cross-sectional geometries and fabricated from commercial metals [4,5,6,7,8]. The second advocates that the articulating members should be designed with polymeric composite laminates which are optimally-tailored in order to synthesis the material properties for the specific application [9,10,11,12]. When implemented, the foregoing methodologies achieve a reduction in the levels of the dynamic response by increasing the stiffness/weight ratios of the members by tailoring both the geometrical and material properties of the articulating members. The rationale can be clarified by examining the terms in the approximate linear finite element equations governing the elastodynamic response of a general machine system presented below, [Ml’llxlwi + [Ml’ltcm'n + [111133 - -[I]{R} (1.1) where [M], [K] and [C] are the global mass, stiffness and damping matrices respectively of the system, [I] is the identity matrix, {U} represents the discretized deformation field, the overdot denotes the time derivative and {R} represents the discretized rigid-body acceleration field. Thus for a prescribed machine system operating at a prescribed speed the elastodynamic response is principally dictated by the stiffness-to-mass ratio of the links. In contrast to the foregoing methodologies which focus on the mass and stiffness terms on the left-hand-side of equation (1.1), the methodology proposed herein focuses on the right-hand-side of equation (1.1); namely the kinematic variables which provide the principal system excitation in a high-speed mode of operation. Thus it is argued, if the magnitude of these variables, and hence the inertial loading, can be reduced, then so too will the elastodynamic response. This translates into a reduction in the dynamic deflections of articulating members, a reduction in the associated dynamic stresses, and also a reduction in the severity of the fatigue enviroment in which the members must operate. 1-2 INTELLIGENT MACHINE SYSTEMS Intelligent machine systems have been the subject of numerous publications, principally in the context of robotic manipulators [13,14], but of course this terminology really pertains to a very broad spectrum of machine products which incorporates sensors and computers. The research reported, herein addresses intelligent machinery in this broader context, by proposing a general methodology for controlling the elastodynamic response of linkages machinery. The current generation of robotic and intelligent machine systems are capable of undertaking many diverse tasks, and implicit in this statement is embedded a spectrum of diverse technologies from several different disciplines, such as instrumentation, controls, computing, artificial intelligence, materials and machine dynamics. In contrast to these robotic-orientated publications, herein, the focus of attention is the attenuation of the elastodynamic response of flexible linkage machinery which feature in a very broad range of both industrial and commercial machinery and equipment. These classes of mechanism systems are prone to generate a vibrational response when operating in a high-speed mode, which is attributed to the time- dependent inertial loading imposed upon the members of the articulating system, since these members are inherently an assemblage of flexible, not rigid bodies. Typical contributions to this field include references [15]. The objective, herein, is to develop and then experimentally test the viability of a methodology for reducing the response of flexible members of linkage mechanisms by introducing an additional input in the kinematic chain. The concept is evaluated by studying a bread-board model of the proposed approach which comprises a retrofitted planar four-bar linkage with a flexible rocker link. The retrofitment involves modifying the normally stationary ground-link/rocker-link revolute joint to incorporate an additional prismatic joint whose perturbatational motion is controlled by an electrodynamic shaker which is driven by an arbitrary waveform/function generator. The linkage can function as a classical four-bar mechanism by locking the prismatic joint, or alternatively, as a five-bar linkage by releasing this locking device. In this latter mode, the system has two nominally rigid-body degrees of freedom with inputs provided by the crank rotation and also the motion of the shaker. The idea was first reported in a theoretical publication [16] prior to presenting some preliminary experimental work in reference [17]. The equations of motion and the relevant boundary conditions for the flexible rocker link are derived using a variational technique, and the steady-state response is obtained using an approach developed by Hsu [18]. An analysis of the corresponding homogeneous equation is developed by means of which the stability of motion are examined. The computational results correlate favorably with experimental data from a complementary experimental investigation. CHAPTER 2 KINEMATIC.AIALISIS OF THE.NODIFIED LINKAGE 2-1 INTRODUCTION The kinematic characteristics of the rocker link need to be evaluated prior to determining the midspan transverse deflection. The kinematics of a four-bar linkage have been investigated widely, but for the intelligent mechanism, the kinematic analysis of a four-bar linkage 'with two-degrees of freedom is necessary. The kinematics of this linkage are evaluated assuming the links are rigid, and the effects of friction and clearance at the bearings are neglected. The kinematic analysis of four-bar linkage found in reference [19] is modified in order to analyze the proposed class of intelligent mechanism. A 1‘ Fig. 2.1 : Intelligent Mechanismzthe kinematic chain The linkage under consideration is shown in Fig. 2.1, where it is the length of ground link which is variable with time and £2, 1, and 2, are the lengths of the crank AB, coupler BC and rocker CD respectively. The angles 02, 0,, and 0‘ are the angles of each link. The corresponding angular velocities and angular accelerations are denoted by 02, w,, w, and a2, a3, a, respectively. For the intelligent mechanism, w, is constant and hence a2 - 0. 2-2 KINEMATICS 2-2.1 VECTOR ANALYSIS If It’ 12, I, and I, represent the vectors AD, AB, BC and CD respectively, it follows from Fig. 2.2 that 1, +1, +1, - I (2.1) t Fig. 2.2 : Vector-Loop Diagram 10 and by writing Ir - ire r (r - l,2,3,4) in complex number notation, equation (2.1) becomes 10, 10, 10‘ 101 Ize + lse + i‘e - 2 e (2.2) where 01 - 0 from Fig. 2.2 2-2.2 POSITION ANALYSIS The Euler equation (e10 - cosfi + isinfi) is substituted into eq. (2.2) prior to separating the real and imaginary parts. This process yield equations for the angular positions of the coupler and rocker links in terms of the input angle of the crank. They are lgcosoz + lscoso3 + £,cosfl‘ - It (2.3a) 2,31no, + lasinfl3 + 1‘sin0‘ - 0 (2.3b) Solving equation (2.3) for cases, yields the expression 2 2 2 1/2 coso3 - A? i [ [A%] - E——§—§ ] (2.4) D D D where A - 213(22c0302 - it) 2 2 2 2 B - 1‘ - it - £2 - £3 + 2£t£2coso2 D - (.2 . .211” Once 0, has been determined from equation (2.4), the value of 0‘ follows from eq. (2.3) 2-2.3 ANGULAR VELOCITIES Upon differentiating (2.2) with respect to time and separating real and imaginary parts and solving for w, and w, yields £2w231n(02 ' 0‘) + Itcoso‘ “8 ' ' 1,31n(0,- 0,) (2.5a) 2,w,sin(0,- 0,) + itcoso, “4 ' ' 2,sin(0,- 0,) (2.5b) 2-2.4 ANGULAR ACCELERATIONS The angular acceleration of the coupler and the rocker may be obtained by differentiating (2.2) twice with respect to time and separating the real and imaginary parts. The results can be expressed in the form 10 2 2 2 .. lzwzcos(0‘-02) + lawzcos(0,-0,) + 24w, + itcosfl, “3 ' ' 2,s1n(0,- 0,) (2.6a) 2 2 2 -- lzwzcos(0,-02) + 14w‘cos(03-0‘) + 13w, + Itcosos l,sin(0;- 0,) (2.6b) a‘ - - 2-2.5 LINEAR ACCELERATIONS Expressions for the linear accelerations of the centers of mass Ga, 6,, and G. of the moving links follow from the preceding equations. The expressions are written according to the coordinate axes shown in Fig. 2.3. Thus, if AG,- r2, BC,— rs and DG‘- r,, the component accelerations are written as (azx, azy), (a3x, 83y) and (a4x’ 84y) and can be expressed as a2x - -r2(w:c0302 + azsinoz) (2.7a) a2y - -r2(w:sin0, - 02C0802) (2.7b) a3x - (.22/r2)a2x - r,(w:cos03 + casinos) (2.7c) a3y - (.22/r2)a2y - r3(w:sin0s - ascosas) (2.7d) ahx - 2; - r,(a,s1n0; + wicoso;) (2.7e) a4y - -r,(o:sin0, - a,cosa;) (2.7f) where 0; - (0,- n) a-< '21 ll 21 Fig. 2.3 : Component Pin Forces 12 2-3 DYNAMICS The forces acting at the joints of the mechanism is shown in Fig. 2.3, where, for example, X1, represents the component of the force exerted by link 1 on link 4. Thus X1, - -){‘1 and etc. N is the normal force exerted by the slider guide on the slider and the force exerted by the shaker is denoted by K. The slider acceleration 2t is in the OX direction. The notations m2, m, and m‘ are the masses of the links and IZA’ 13B and 140 are the corresponding moments of inertia about axes perpendicular to the plane of motion and passing through points A, B and D respectively. Then, the equations of motion necessary to evaluate the forces acting on the rocker link are Y2: " Ys‘ - msa3y (2.8b) 2 13 Solving the above equations for the pin forces yields the following expressions, 0 2 0 IBBasl‘coso‘ +I4Da‘23coso8 -w2m,r,lzl,coso.sin(02-03) X3. , (2.9a) 0 2 0 IaBasl‘sino‘ +14Da‘lssin03 -w2m3r812£‘sin0,sin(02-0,) Y34 - . (2.9b) 131‘81n(03-0‘) X‘I "' Xa‘ ' m‘a4x (2.9C) The force exerted by the slider guide and the shaker is given by N - -Y,, (2.10s) s t (2.10b) where Ms is the mass of the slider. VARIATIONAL.TREOREM The objective of this chapter is to develop the equations of motion governing the elastic rocker link. An accurate mathematical model is required which will accomodate the inherent elastic deformations of the link. The variational theorem forming the kernel of this theoretical study was originally developed in the doctoral dissertation of B.S. Thompson [20]. It incorporates the geometrically non-linear form of the field equations so that it may provide a basis for analysing linkages susceptible to dynamic instabilities and also mechanism systems that must be analysed using higher-order theories. 3-1 THEORETICAL DEVELOPMENT The theoretical development follows the approach of reference [21]. The dynamical problem of an elastic body describing a general X Fig. 3.1 : Definition of axis systems and position vectors l4 15 spatial motion relative to an inertial reference frame is considered. In Fig. 3.1, axes OXYZ define the inertial reference frame, while oxyz are Lagrangian coordinates fixed in the body in a refernce state containing zero stresses and strains. Employing an indicial notation (i-x,y,z), then at time t, a general point P in the continuum has the position vector r1, which is defined as r1 - roi + rR1 + u1 (3.1) where roi the component relative to o-x-y-z frame of the position vector of the origin of the body axes relative to the origin of the inertial frame rR1 the position vector of point P in the reference state relative to the origin of the body axes u1 : the deformation displacement vector The non-linear form of the field equations necessary for a body describing a general spatial motion relative to OXYZ frame are as follows: (1) p1 - r01 + 61 + eijkaj(rok + rRk + uk) (3.2) This equation is obtained by differentiating eq. (3.1) where p1 : the velocity associated with the time rate of r1 16 eijk : alternating tensor 31 : component of angular velocity vector for the moving axes or Lagrangian frame (~) : the time rate of change with respect to the moving frame (0) : the absolute rate of change with respect to time (ii) The boundary conditions for prescribed surface tractions and deformation displacements are written as E, on S. (3.3) u1 - ui on S, where 81 : surface on which the prescribed tractions are imposed S2 : surface on which the prescribed displacements are imposed (iii) Non-linear strain-displacement relations 71J'12‘(u1.3+ 113,1 + uk,iuk,j) (3.4) where 113 : Lagrangian strain component (,) : denotes spatial differentiation The kinetic energy density T, of the system is defined as 17 T - %p61jpipj (3.5) where p : the mass density of the material 61J : the kronecker delta Further, defining W as the strain-energy density and X1 as the body forces per unit volume, the principle of virtual work leads to the problem of determining the stationary conditions of the functional t1 G - I {£ [T(p1) - W(1ij) + X1r1]dV + g gir1 d8} dt (3.6) to 1 subject to the auxiliary conditions through the volume and over the surface 82, the prescribed deformation displacement condituion. The volume integral extends over the entire volume V of the elastic body and sum of S1 and 8, defines the total surface area S of the continum. A free variation problem without auxiliary conditions may be constructed by the Lagrange multiplier method which incorporates the constraints in the functional. The first variation of the modified functional enables these undetermined multipliers to be expressed in terms of the system parameters [21]. To obtain this expression it is necessary to use Gauss' theorem relating surface and volume integrals, and to perform integration by parts. Furthermore no variation in the system configuration is permitted at to and t1. 18 The variations in the position vector r1 is obtained by permitting the variations to coincide with the actual displacement that occur during the time interval dt. The following relationship may be proved 6ri - 8ro1 + eijk6¢j(rok + rRk + uk) (3.7) The functional becomes t1 J " {{{lT(P1) ' 110111) + xiri + '11[111-% 24 Over the remainder of the section at the ends of the link, prescribed surface tractions are imposed. At the rocker-pin end(x-0) external forces P(t) and Q(t) are assumed to be uniformly distributed, so that m (3.19) At the rocker-slider joint, the expressions must incorporate the dynamic effects of the slider on the link and the axial loading again has uniform components. As shown in Fig. 3.3, the slider has normal force N(t) exerted by the slider guide and a force exerted by the shaker, K(t) - 4 9,, M!) Fig. 3.3 : Forces exerted at the rocker-slider pin joint K(t) The surface tractions are 3 £121 at x - L, 2% - A sino. - KLEI cosa‘ - (3.20) J 0 A (px)x-L 25 M Hifil _§ - A (pz)x-L c0304 - sinfi‘ - 09 I N I I ? A where M8 is the mass of slider. The flanks of the flexible link are assumed to be unloaded. Hence Ex - gy _ gz - o (3.21) 3-3 “INSTRUCTION OF 1113 PROBLEM DEFINITION The equations of motion and boundary conditions are constructed by substituting the approximating statements (3.11) to (3.21) into the variational eq. (3.10) prior to using the arbitrary independent nature of the variations in the system parameters. Upon subtitution into the volume integrals of eq. (3.10) yields I{-Ipss[8-(itcoso,-zt0,sin0, +uo-zw'x+5,(£tsin0‘ +z +w)}]dA x A -Ip6C[C-l2tsin0‘ +£t94cosa‘ +9 -5,(£tcoso4 -L +x +uo-zw’x)}]dA A +I[(8so-zss,)(ro-zr,— 3% +u(%% +g§ )) + 632(2T2-4682)]dA A xx yy 22 2 2 +I(6To-26T1)lso'251' %{2(Uo,x'zw,xx) +w,x +("‘o,3:'zw,xx) }]dA A +I6T2[S2 - %(uo’x-zw,xx)(-w’x)]dA A 26 +I(8uo-26w’x)[rxj’1 + ux’Jkrkj +“x,k'kj,j' ppx]dA A +£SWIrzj’j +uz,jk'kj + uz,krkj,j -ppz]dA}dx - 0 (3.22) The first two integrals yield the velocity statements B - itcoso, + 60 + ww, (3.23) C - Itsino‘ + 6 - w‘(x + uo-L) which may be differentiated to give the absolute aacelerations - s . s s 2 B - £tcoso‘ +2ww‘ +wa‘ +uo -w4(x +uo -£,) (3.24) 2 C - itsino‘ -2u°w‘ + w -ww‘ -a‘(x +uo -£4) By substituting a strain energy function W for an isotropic homogenous material into the third integral of expression (3.22), one obtains where E is the Young's modulus and G is the modulus of rigidity. The fourth and fifth integrals give o o,x 2 o,x 2 ,x 2A ,xx S, - w,xx(1 + u0,x) (3.26) S, - ' 2“o,x w,x where I is the area moment of inertia of the section. The final pair of integrals in eq. (3.22) must be integrated by parts to give the equations of motion, Isuolfxx,x +("1x,x'xx),x +(ux,z'xz),x - prldv v +ISWI’ ‘k x ])(1 +uo,x) +AGu w - EI(w,xxx[1 +uo ] -w [w o,x ,x -u ,x ,xx ,xxx o,xx - EI(w [u + 1u2 + §w2x + 1-w2 ]) - 0 (3.32c) ,xx o,x 2 o,x 2A ,xx ,x These equations simply complete the problem-definition, since the values of the pin forces P and Q cannot be determined until the problem has been solved for the deformation displacements. On the beam flanks, the prescribed surface traction condition must be combined with the natural boundary terms in eq. (3.28) to be evaluated at constant y. The result is identically zero. The statement for the prescribed surface tractions which are imposed on the major portion of the beam section at the rocker-slider pin must be combined with the natural boundary term and this yields, upon integration, at x - L, 30 , 2 2 N(t)sin0, -K(t)coso, -Ms(px)x_L 'EIw,xx(1 + uo,x) -AGuo xw x 2 2 2 - EA(1 +uo’x)(uo’ + i‘uo’x + 2",); + 21".“) - 0 (3.33s) 1:1 1 3 1‘12 10:2 + I—w2 0 (3 33b) w,xx[ + uo,x + 2 o,x + 2 ,x 2A ,xx] ' N(t)coso, - K(t)sin0, 'Ms(pz)x-L + %u2 + 1w2 + l-w2 ) o,x 2 ,x 2A ,xx EAw (u ,x o,x + EI(w,xx[1 +uo ] -w [w ,x ,xx ,xxx-“o,xXJH1 +uo ) ’x EI(w [u +1112 +1012 +1—w2 ])x-0 (3.336) ,xx o,x 2 o,x 2 ,x 2A ,xx + 3-4.AN'APTROXIMATE SOUUTION The problem definition is now complete since equations of motion and the corresponding boundary conditions have been constructed. Thus the objective is to simplify the equations sufficiently to permit the use of standard solution procedures. The forcing frequency of the rocker link is much lower than the first natural frequency in the axial mode, and hence the axial vibrations may be neglected [23]. If second and higher-order terms are assumed to be small in eq. (3.29) and (3.33a), then the integration of eq. (3.29) with respect to x gives a general statement for the axial force in the link: 31 .. 2 2 2 EAuo x - pAltcos9,(x - L) - pAw,[%x - Lx + L ] (3.34) + N(t)sin0, - K(t)cos0. - “3(Px)x-L Attention is now focused on eq. (3.30) which governs the flexural vibrations of the link. Upon neglecting the shear term, second and higher-order terms in the coefficient of E1 and third-order terms in the coefficient of EA, this equation reduces to EAu w - EAu w + pAC - 0 (3.35) ,xxxx o,x ,xx o,xx ,x The axial force statement (3.34) may be substituted into eq. (3.35) to give 0 s 2 2 . EIw xx-[prltcos 0, -pAw,(%x -Lx) +Nsin0, -Kcoso, -M8(Px) 9 XX x-L so so 2 - mltcoso, -%me,]w +[-pA1tcoso, ~l-pr..>,(x-L)]w’x ,xx so so 2 + pA[£tsin0, + w -ww, -a,(x-L)] - 0 (3.36) where m(m-pAL) is the mass of the flexible link A solution procedure in which normal modes are used will be adopted for this non-linear differential equation. Since any deflection of the beam may be expressed as the sum of deflections in the various principal modes [24], the standard solution can be written as 32 n w(x,t) - 2 W (t)¢ (x) (3.37) n n npl where Wh(t) are unknowns and ¢n(x) are the mode shapes. In accordance with the standard approach, the derivatives of eq. (3.37) are substituted into eq. (3.36) and the orthogonality condition utilized by multiplying the resulting equation by ¢m(x) before integrating each term over the length of the rocker link. This gives an equation of the form so EIC1 c2 + C5 2 C3 + l/ZC. ' C7 2 C‘ W(t) + W(t) pAc, - Tho, + c, w, + pAc1(Kcosfl, 2 c, c2 + c5 M c, .. _ __ - _______ _§__ Nsino, + %me,) + c7L c, + pAc7]ltc°so‘] c, - c.L c. .. - -—-;:—- a, - z: itsino, (3.38) where the constants c,(i - 1,2,....9) are given by L L c, - {¢n¢n,xxxx c2 - {¢n¢n,xdx L L c, - Jm¢n.¢n,xxcix c, - f¢n¢n,xxdx L L 2 c5 - {¢n¢n’xxxdx c, - {¢n¢n,xxx dx (3.39) L 2 L 33 L Equation (3.38) is an inhomogeneous Hill's equation and the first term in the coefficient of W yields the natural frequency of the stationary link. If wn denotes the natural frequency of the link, eq. (3.38) can be written in simpler form as am.) + mow: + r(t)] - K(t) (3.40) For the pinned-pinned beam the form of the principal modes are sine curves [24]. Hence the characteristic function is ¢n(X) - sinnfé (3.41) while the value of the nth principal co-ordinate Wu as that which gives unit amplitude to the sine wave. Upon substitution of eq. (3.41) into eq. (3.39) and integrating them, the o1 values may be found directly. Since the transverse deflection at the midspan of the rocker link is of interest, the mode shape ¢n(x) appearing in (3.37) may be evaluated at x - L/2, hence nfi ¢n(x - L/2) - sin-5 (3.42) 34 The next step is to solve the Hill's eq. (3.38) for the time dependent amplitude W(t) by a viable method. The midspan deflection of the link may then be evaluated from (3.37) by summing over the desired number of terms. CHAPTER 4 SOLUTION TO HILL'S EQUATION The Hill's equation (3.38) obtained in the preceding chapter is linear inhomogeneous differential equation with periodic coefficients and the system is subjected to forcing excitations of periodic nature. This equation can be solved for both elastodynamic and quasi-static responses, since the kinematic characteristics of the intelligent mechanism are known. Only the steady-state solution is considered here, which implies the system is stable and this will be verified in chapter 6. The steady-state response would the amplitude of deflection of the rocker link. 4-1 QDASI-STAIIC RESPONSE A straight-forward method of solving the Hill's equation is to neglect the dynamic term(W ) and obtaining the solution directly, which is termed as quasi-static solution. Upon neglecting the dynamic term, the Hill's equation becomes 2 W(t)[wn + r(t)] - R(t) (4.1) Hence the amplitude of deflection may be evaluated on a digital computer .from the equation W(t) - 23“) (4.2) [wn + r(t)] 35 36 4-2 ELASTODYNAHIC RESPONSE The steady-state solution to Hill's equation is termed as elastodynamic response. An explicit expression for the steady-state periodic response is possible, but for the system under investigation an analytical determination is not feasible. A straightforward numerical procedure developed by Hsu [18] is used here. 4-2.1 MATRIX NOTATIONS The Hill's equation under periodic forcing can be written in standard state-variable form as i(t) - C(t)x(t) + f(t) (4.3) where mt) W(t) 2° 1 ‘(t) ' x2(t) ' fi ' C(t) ' -[wn + r(t)] o ' f(t:) - {32”} In the above equation x is a vector, C is real coefficient matrix periodic in t with period To and f a vector also periodic in t but with period rf. For the intelligent mechanism, to would be the period of <2rank rotation and rf as the period of shaker excitation. 37 4-2.2 NUMERICAL PROCEDURE Let Q(t) with 0(0) - I be the fundamental matrix for the solution of the homogeneous equation(assuming f(t)-O) corresponding to (4.3). Then the solution of (4.3) is given by x - ¢ - (I - Hflng} + xfuz). (4.4) where the first term is the solution to homogeneous equation and xf(t) is steady-state response and is given by t xf(t) - Q(t){ Ie’1(a)£(s)ds + (I - m‘lng} (4.5) 0 t The quantities I§-1(s)f(s)ds and g appearing in (4.5) are 0 evaluated by simply using the Trapezoidal rule of numerical integration. In this manner from Hsu, the final form of steady-state solution is given by -l xf(t~) - xk + Q(t‘)(I - H) xk, x - O,l,2,....,K (4.6) where Xo'o 38 x-A‘%;¢(A)fo+; ;p1(A)f1_1+12‘f} (4.7) " -1 1 1—2 1—2 " and K H - LII p (A)] (4.8) -1 J 9(0) - I, identity matrix K Q(t ) - II p (A), ac - 1,2,....,K (4.9) n 3'1 j - 4.10 fJ f(tj) ( ) The time step A is obtained by dividing the crank period '0 by a large positive integer K, i.e. ro A - E— (4.11) Note that the equations are written by assuming similar frequencies for the crank and shaker excitation. To evaluate numerically the periodic steady-state response, xf(t) will be evaluated at the discrete station points t - tn, where t - nA, x - 0,1,2,....,K (4.12) 39 An efficient expression for ¢3(A) appearing in the above equations may be evaluated using Scheme I of Hsu [18], and it is given by 91(1)) ~ exv(CJA) (4.13) By using the coefficient matrix C and the methodology found in reference [25], eq. (4.13) can be written as cosw A sinw A ‘A 3 j 3 (4.14) -w sinw A cosij J J '1 (A) - e where w - [mi + r (t)]1/2 J J Upon substituting equations (4.7) to (4.12) and (4.14) into eq. (4.6), the steady-state response or the amplitude of deflection of the rocker link may be evaluated at discrete time step. CHAPTER 5 EXPERIMENTAL INVESTIGATION 5-1 EXPERIMENTAL.APPARATUS A photograph of the experimental intelligent mechanism is presented in Fig. 5.1. The rigid coupler link is manufactured from aluminum and the flexible rocker link is from steel. At the end of each link, two clearance holes were drilled. These holes accommodated socket screws which clamped each specimen to the bearing housing. Identical ball bearings of type R4 DB 12 instrument ball bearings supplied by FAG Bearing Limited were used at crank-ground pin and at rocker-slider end. The rocker pin was mounted on R32 ball bearing and a cleavage design joint was used for coupler-rocker end as shown at the top of Fig. 5.1. Each bearing housing in the mechanism was preloaded using a Dresser Industries torque limiting screw driver calibrated to :1 in-lbf to eliminate radial clearence, otherwise the impact loading associated with bearing clearances would cause the links to have larger deflections. One end of the rocker link was connected to a Microslides Inc. 2020 cross-roller slide assembly which permitted translational motion as shown in Fig. 5.2. This precision assembly was then bolted to a force transducer, which permitted the force exerted by the shaker on the slider to be carefully monitored. An accelerometer was mounted on the slide assembly parallel to the translational motion to measure the kinematic characteristics imposed by the vibrator, as shown in Fig. 5.2. 40 41 Fig. 5.1 : Experimental Intelligent Mechanism =Fig. 5.2 : Electrodynamic Shaker-Slider Assembly 42 The mechanism was bolted to a large cast-iron test stand which was bolted to the floor and also to the wall of the laborotary to provide a substantial rigid foundation. A 0.75 h.p. Dayton variable speed d.c. electric motor which was bolted to the test stand, powered the linkage through a 19.05 mm diameter shaft supported on a pair of Timken tapered roller bearings type TS4A-6. A 100 mm (4 in) diameter flywheel (see Fig. 5.1) was keyed to the shaft thereby providing a large inertia to ensure a constant crank frequency, when operating in unison with the motor's speed controller. The data for the mechanism is shown in Table 5.1. The initial ground link length was 381 mm and upon the shaker excitation, it is variable with time. The net mass of the flexible link was 22.6 gramms. This data will be used for computer simulation in the following chapters. 43 Crank Coupler tziocker (rigid) (rigid) (fleXIble) Link length, pin to pin 50 305 293.6 mm Mass,gm (including bearings * 157 l46.7 and joints) Depth,mm *. l9.43 no.2 (in the plane perpen. to the mechanism) Hidth,mm (in the plane of * 6.25 0.838 mechanism) 'i’mters 0.0 0.175 0.192 (center of mass, see . F1023) Mass moment of inergia, - Kg-m , (see Fig. 2.3) * 0.005235 0.00722 Modulus of elasticity for rocker link, ;E=207 Gpa. Mass density of rocker link. P= 9294.3 Kg/m3 Area moment of inertia, I=3.9588Xl0 Mass of slide assembly, "5 = 0.4l5 Kg *Flywheel was used to provide large inertia to insure a constant crank frequency Table 5.1 : Data for Intelligent Mechanism -l3 m4 44 5-2 INSTRUMENTATION A schematic diagram of the instrumentation employed in the experimental investigation is shown in Fig. 5.3. The rated speed of the electric motor was measured in revolutions per minute(rpm) by a Hewlett Packard 5314A Universal Counter which was activated by an electro- magnetic pickup model 58423, manufactured by Electro Corp., sensing a sixty tooth spur gear mounted on the drive shaft of the rig. The gear is shown in Fig. 5.4. The arrangement provided visual feedback to the operator by adjusting the speed controller of the motor in order achieve the desired crank speed. The experimental result presents the variation of link deflections with crank angle. Strain gages were bonded to the midspan of the rocker link and the midspan deflections were monitored by a Micromeasurements Group Inc., strain gage conditioner/amplifier system type 2100. In order to relate the strain gage signal to the configuration of the experimental mechanism, another transducer arrangement was established. An Airpax type 14-0001 zero velocity digital pickup was employed to sense the bolt-head at the end of the crank, when the mechanism is in the position of zero-degree crank angle. This long hexagonal transducer is shown in Fig. 5.4 In order to excite the shaker simultaneously with crank motion, the signal from Airpax pickup at zero-degree crank angle configuration will be fed to wave generator. A Wavetek digital arbitrary function generator permits arbitrary waveforms to be generated for a wide range of frequencies. A desired wave function is programmed into the Wavetek 45 Function Generator Visual monitoring of crank speed(RPM) V7 3 Power HP Universal Amplifier Counter V! A Ling Electra-Corp. Dynamic System pickup V4ll Vibrator A Four-Bar 60 tooth D.C. ‘** r7 Linkage spur gear Motor )7 i ' a a K "Links - Speed Charge strain gaged Controlle Amplifier Strain Gage zero , Conditioner and crank-angle Amplifier system gapggguration . *hccelerometer dynamic ”Force strains Transducer 05c Post-processed' PDP ll/03 ‘_ experimental microcomputer ’ results Fig. 5.3 : Schematic Diagram of the Instrumentation 46 Fig. 5.4 : Assembly of Crank and Airpax Pickup Fig. 5.5 : Data-Acquisition System and Equipment 47 and the signal is amplified by Hafler power amplifier model P500, prior to feeding it to the shaker. The shaker is an air-cooled Ling Dynamic Systems type V411 vibrator, which upon converting electrical current into mechanical force will permit translational motion to the slide- assembly. The precision assembly which was bolted to a Bruel & Kjaer force transducer, type 8200, which permitted the force exerted by the shaker on the slider to be carefully monitored. The acceleration imposed by the shaker upon the slide-assembly was monitored by a Bruel & Kjaer accelerometer type 4731, mounted on the slide—assembly. The output from the two transducers were fed to a Bruel & Kjaer charge amplifier type 2635, which enabled the force and displacement, velocity and acceleration characteristics of the slide-assembly to be monitored. The mechanism configuration signal, the output from the gages and transducers were either fed to an oscilloscope or to a Digital Equipment Corp. PDP 11/03 microcomputer. A photograph of the data-acquisition system and the equipment used for this experimental investigation is shown in Fig. 5.5. The oscilloscope with a C-SC camera attachment was used for photographic recording and preliminary evaluation of the response data. The BNC cables from the experimental apparatus were connected to an input-output module, bolted to the cabinet of the computer. This device had 16 analog-digital channels, 4 digital-analog channels and two schmidt triggers. Using the code developed for digital data-acquisition, the kinematic output and flexural response signal was recorded from the 48 zero crank angle position through 360 degrees by firing one of the Schmidt triggers. 49 5-3 EXPERIMENTAL PROCEDURE The experimental procedure involved first determining the natural frequency of the stationary rocker link, so that the mechanism can be operated away from the undesirable vibrations. The mechanism under stationary condition, the rocker link was deflected and released and the transient vibration was recorded on the oscilloscope prior to photographically recording it on the camera. The result is presented in Fig. 5.6. The mechanism was then operated at constant crank frequency by locking the cross-roller slide assembly in order to create a classical four-bar linkage. The preliminary experimental work was to obtain a frequency-response curve, so that desirable operating speed could be chosen. Fig. 5.6 : Transient Response of Rocker Link (wn- 18.75 Hz) 50 The frequency response of the flexible rocker link is investigated for a wide range of operating speeds. The theoretical elastodynamic response is obtained by solving the Hill's equation. The experimental results were obtained by operating the four-bar linkage through a range of crank speeds and recording the resulting midspan amplitude of the rocker link. The theoretical natural frequency of the rocker link was computed from the Hill's eq. and it is compared with experimentally obtained result below, n,th - 19.32 Hz (1159 rpm) wn,ex - 18.75 Hz (1125 rpm) (5.1) Since the Hill's equation does not incorporate damping, the theoretical natural frequency was found to be slightly higher than the experimental result. Fig. 5.7 shows the frequency response curve from analytical and experimental investigation. The response regime shows the resonance condition when the ratio of natural frequency(experimental result) to operating frequency is an integer. Both curves show good correlation. Having defined the frequency response data, operating speeds can be chosen remote from the resonance condition. 51 . — ANALYTICAL -- EXPERIMENTAL C .1 PAN AM PUTUDE (mm) kflD 0 7 Ti I Y I I I i 1 I I I I W I I r W I I I I I If 100 135 170 _ 265 240 w. CRANK FREQUENCY (rpm) 4 fi lo 6 h 5 8 , 3 w. /w, FREQUENCY RATIO Fig. 5.7 : Frequency-response curves for the four-bar linkage The classical four-bar linkage was then operated at the chosen speed to obtain the transverse deflection curves upon which the response curves with the shaker excited will be compared. The data was then digitally filtered with low-bound frequency set at approximately 30 Hz and 100 Hz to yield quasi-static and elastodynamic response respectively. The digital filtering also removed electromagnetic and other noise sources which may have occured during the course of the experiment. In order to excite the shaker, the prismatic joint was unlocked prior to programming a waveform into the function generator. The displacement of the slider was carefully monitored by controlling the amplifier gain. Upon maintaining a constant crank speed, the mechanism was operated for at least two minutes in order to allow the link 52 deflections to settle down to a periodic nature. This was visually monitored on the oscilloscope along with the displacement, velocity and acceleration of the slider. Experimental results were obtained by implementing the following procedure. The signal from the strain gage instrumentation was fed to the PDP microcomputer and the results were digitally filtered. The data was then post-processed by multiplying the digitized response by a strain-deflection calibration factor for the rocker link specimen. This factor was obtained by supporting the link at the ends on knife edges in a calibration fixture shown in Fig. 5.8, prior to subjecting the midspan to a series of known monotonically increasing transverse deflections which were imposed and measured by a micrometer attachement on the fixture. The corresponding voltages from the strain gages bonded to the link were also recorded to relate strain magnitude to midspan deflection. Fig. 5.8 : Link Calibration-Fixation 'CHAPTER 6 STABILITY'ANALYSIS 6-1 STABILITY OF MOTION It is well known that for systems with time-periodic coefficients which are free from external forcing, the behavior of the system can be investigated according to Floquet theory [26]. When the system is subjected to periodic-forcing and when the homogeneous equation is stable or asymptotically stable, then a key result one will be interested in is the steady-state response of the system under that forcing. Hence, the stability of the mechanism is investigated by means of Hill’s homogeneous equation. The Hill's homogeneous equation is given by 0(t) + W(t)[w; + r(t)] - 0 (6.1) and the corresponding solution to (6.2) is reproduced from eq. (4.4) of chapter 4, th - 900mm - (I - m'lug) (4.4) It is argued that the above equation is stable only if Q(t) is stable at the system's period, i.e. Q(t-r). The stability of this matrix is determined by performing an eigenvalue analysis. The general procedure for eigenvalue analysis is as follows: 53 54 1) Compute eigenvalues of Q(r) let A1,....,An be the eigenvalues 2) Choose An - 1, one eigenvalue will always be equal to l 3) The remainder must lie inside of the unit circle in the complex plane for stability. i.e. system is stable if [All < l i-l, ...... ,n-l system is unstable if [Ail > 1 for all i Fig. 6.1 shows the analytically obtained frequency response curves for the four-bar linkage and intelligent mechanism, where stability analysis has been performed for a wide range of operating speeds. The shaker displacement for the modified linkage was chosen in the range 0 to -l.35 mm (The viable waveform will be verified in chapter 7). The discontinuos portion of the curves are parts where the steady-state response has no meaning because the homogeneous system is unstable. Note that the instability occurs at the same region for both mechanisms. In the following chapters, steady-state solution is utilized by operating in the stable region. 55 200 —. wm-com' SHAKER ‘ —- WITH SHAKER ”E 1501 .3- E, 4 a . [ 0 4 a . . i 2 100- 3 f i .. . S 1 muhle/ o ,ffi '; , . . v 100 130 220' 230 0' ”460' '40 w. CRANK FREQUENCY (rpm) Fig. 6.2 : Analytical Frequency-response curves CHAPTER 7 ELASTODYNAMIC RESPONSE Of THE UNMODIFIED LINKAGE AND THE SYNTHESIS Of THE SHAKER/SLIDER.MDTION 7-1 RESPONSE CURVES FOR.THE FLEXIBLE FOUR-BAR.LINKAGE The first set of results of interest is the midspan transverse deflection of the flexible link of a classical four-bar linkage. The analytical result is the steady-state solution obtained numerically from eq. (4.6) by setting the shaker input to zero (i.e. lt-O). These analytical results are then compared with experimentally obtained elastodynamic deflection in order to verify the correlation between them. A good correlation of these results will allow to make better prediction of the response with the shaker in motion. The main assumptions made in formulating the computer model for simulations are: (1) All bearings were considered frictionless and without clearance. (2) The crank speed was assumed constant. (3) The kinematic analysis of the flexible link was based on the assumption that it was a rigid body (4) Damping was neglected in the derivation of the equations of motion. Figures 7.1, 7.2, 7.3 and 7.4 presents a comparison of the elastodynamic response of the rocker link from analytical and experimental investigations at crank frequencies of 148, 170, 200 and 215 rpm respectively. These curves comprises two components: the quasi- 56 57 static response at the mechanism's operating frequency and a high- frequency component at approximately the link's fundamental natural frequency in flexure. The quasi-static response of the rocker link obtained at 200 rpm is compared in Fig. 7.5 (LSD J -—- ANALYHCAL . -- Exrmweum. 13 (125? j; J g 1 (LOO~ 5 1 a J IL] 4 a 4.251 E . 0- J 0) Q 4 2 “(J-5°] J -O.75 tr T T I f 1' 1 T’ j' 1’ 1 O 90 180 270 ' 360 CRANKIANGLE GMESREES) Fig. 7.1 : Elastodynamic Response of the Four-bar Linkage at 148 rpm 58 ‘ —— ANAerICAL . —- EXPERIMENTAL ,‘,_..4-\ MIDSPAN OEFLECTION (mm) —1.5 vs .7 I . . l , 1 O 90 180 270 360 CRANK ANGLE (DEGREES) Fig. 7.2 : Elastodynamic Response of the Four-bar Linkage at 170 rpm 1.0 ~———— "u . —- ANALYTICAL l -- EXPERIMENTAL I E I E, 6" 8' ‘3 u. u: c: . l U) E; . .. . g f ! “1051‘“?"Wfi 1" T T— 'T'“"‘—I"'" j— T J o 90 1'30 270 3'60 CRANK ANGLE (DEGREES) Fig. 7.3 : Elastodynamic Response of the Four-bar Linkage at 200 rpm FLECTEOl: (mm) IDE \ 4 MIDSPA' MIDSPAN DEFLECTION (mm) 59 -- ANALYTICAL -- EXPERIMENTAL H‘r’ r r r fi— "r T 1* fl“ 0 90 180 270 360 CRANK ANGLE (DEGREES) Fig. 7.4 : Elastodynamic Response of the Four-bar linkage at 215 rpm 1.0 -—--~- -— —- ANALYIICAL 1 -- EXPERIMENTAL (15*+ i 0.0% x , ' . J / . i i 'l/ \ l J /l , \\ F ".0-5"J I"// - \! .’ I' k \I I I I -°105 ' .T.“"T'" — r "‘--T—'" "‘7'"- —r "'-"‘f'-""-T”" *' T"-_"T"'.."T-"'"_4 O 90 180 270 .360 CRANK ANGLE (DEGREES) Fig. 7.5 : Quasi-static Response of the Four-bar Linkage at 200 rpm 60 7-2 CURVE SYNTHESIS FOR.SHAKER/SLIDER.MOTION 7-2.l JUSTIFICATION OF WAVEFORM The current mode of operation of the four-bar linkage may not satisfy the prescribed design specification; namely the amplitude of deflection may be greater than a prescribed quantity. Under these circumstances, a linkage with two-degrees of freedom may provide a solution whereby the motion of the slider would be controlled in a prescribed manner in order to reduce the deflections of the rocker link. This philosophy requires the ability to synthesize a suitable waveform for the shaker/slider motion. A general waveform of sine function is considered here. it - Assin(wst + ¢8), (7.1) where A8 is the amplitude of slider motion, we is the frequency of excitation and d8 is the phase between crank motion and shaker excitation. Hence, there are 3 independent variables that need to be evaluated such that when eq. (7.1) is substituted into the Hill's eq. (3.38), the amplitude of deflection is reduced in comparison to classical four-bar linkage (i.e. 2;— 0). The proposed waveform becomes more complicated if 2t is assumed to be function of sine and cosine terms. A less cumbersome methodology is to excite the shaker using a ‘waveform based on the profile of the link response. This waveform 61 requires a theoretical justification to insure the vibrational response does not exceed the prescribed quantity. For the purpose of discussion, the Hill's equation (3.38) is reproduced here. The methodology proposed herein focuses on the right- hand-side of the equation; namely the kinematic variables which provide the principal system excitation in a high-speed mode of operation. Since the shaker inertial term (2;) appears on both sides of the equation, reducing the angular acceleration of the rocker link (a‘) by means of shaker excitation does not guarantee a reduction in the elastodynamic response. .. EIc2 C2 + cs 2 c8 + l/2c, - c7 2 c‘ W(t) + W(t) pAc, - c7 Lw‘ + c, w‘ + pAc1(Kcos0‘ 2 C‘ 02 + Cg M c‘ so - Nsino‘ + %me4) + E—L - -————-— + -§—-]ltcos0‘] 1 c7 pAc, C9 ° Cg L Cg .- - c, a‘ - c7 ltsin04 (3.38) The equation is analysed by neglecting the dynamic term and solving for the quasi-static response using the following equation, f,(c)a, + f2(t)2; W(t) - (7.2) 2 e a mu + f3(t) + £,(c)2t where f1(t) and f2(t) are the coefficients of a4 and I; on the right hand-side of eq. (3.38), and f2(t) is the last term in the coefficient 62 of W(t).The quasi-static response is reduced by either decreasing the numerator or increasing the denominator by shaker excitation. However a decrease in the numerator does not necessarily increase the denominator because of the coupling of shaker inertial terms. The flexural midspan deflection are measured perpendicular to the link in the plane of motion, the forcing fields imposed in this plane must be continously modified in order to reduce the amplitude of deflection. This requires the absolute acceleration perpendicular to the link to be reduced. Fig. 7.6 shows the absolute acceleration a4a’ measured perpendicular to the rocker link in the plane of deflection; where it is a function of the inertial loading of the link as shown in eq. (7.3). Thus it is argued, if the magnitude of the absolute acceleration can be reduced by a viable shaker motion, then so too will the elastodynamic response, since the absolute acceleration terms provide the principal forcing functions. a - a c030 4a 4xy 2 o 0 a48 - f(ra‘, rw‘, 2t)coso (7.3) 63 Fig. 7.6 : Absolute Acceleration of The Rocker Link 7-2.2 SYNTHESIS OF SHAKER WAVEFORM As discussed earlier in section 7-2.l, the profile of the link response will serve as a basis for the slider motion. Fig. 7.7 shows the analytical quasi-static response obtained at 200 rpm. Superimposed on it is the proposed shaker waveform, the shape being proportional to the deflection curve. Fig. 7.8 shows the computer simulation of the absolute acceleration of the rocker link corresponding to the proposed shaker waveform. It shows a decrease in magnitude in the first half-cycle and an increase in the second half-cycle, so does the elastodynamic response as shown in Fig. 7.9. An experimental investigation was carried out by feeding the experimental quasi-static response curve into the shaker simultaneously with crank motion. The result is shown in Fig. 7.10, 64 which correlates with the analytical result. Fig. 7.11 shows the angular acceleration of the rocker link. Fig. 7.12 shows the inverse of the quasi-static response as being the proposed shaker waveform. The results corresponding to this waveform are shown in Figures 7.13, 7.14, 7.15 and 7.16. The results obtained for two different sets of shaker waveform are summarized in Table 7.1. i Analytical Experimental absolute angular deflection deflection acceleration acceleration of of a link link I .. i .. shaker st nd st nd st nd waveform 1 half 2 lhalf 2 I hale proportional to link deflection T l I, shaker ; waveform I . ,is inverse of 'link deflection 'Iincrease in magnitude ldecrease in magnitude Table 7.1 : Variation of link response with shaker at 200 rpm MILUMETERS ACCELERATION (II/$2) 65 — QUASl—STA'IIC MDSPAN DEFLECTION —- SHAKER DISPLACEMENT 90' '100' '2720' '360 CRANK ANGLE (DEGREES) Fig. 7.7 : Suggested shaker/slider waveform proportional 20 Fig. 7.8 to quasi- static response —- WITHOUT SHAKER -- WITH SHAKER 90 ' ' 100 ' 270. 350 CRANK ANGLE (DEGREES) given shaker/slider waveform shown in Fig. 7.7 : Absolute acceleration of the rocker link at 200 rpm MIOSPAN DEFLECTION (mm) MIDSPAN DEFLECTION (mm) 66 — WITI-DUT SHAKER -- WITH SAHKER 0 "9'0 ' '130' '270,' f360 CRANK ANGLE (DEGREES) Fig. 7.9 : Analytical elastodynamic response at 200 rpm given shaker/slider waveform shown in Fig. 7.7 1.5‘ + — WlTl-DUT SHAKER 1.01 -- WITH SHAKER ,\ I 0.51 f 0.09 i —O.51 -1.0{ -4.51 I -‘-2.0‘..,.. ..,.. 0 90 180 . 270 , 360 CRANK ANGLE (DEGREES) Fig. 7.10 : Experimental elastodynamic response at 200 rpm given shaker/slider waveform shown in Fig. 7.7 ACCELERATION (RAD/$2) -100' .r I | MILLIMETERS 67 — WITHOUT SHAKER 1 -- WITH SHAKER 0 90 ' ' 1:50 ' ' 270 , ' 360 CRANK ANGLE (DEGREES) Fig. 7.11 : Angular acceleration of the rocker link at 200 rpm given shaker/slider waveform shown in Fig. 7.7 2.5 : — QUASI-STATIC MIDSPAN DEFLECTION 2.0; —- SHAKER DISPLACEMENT CRANK ANGLE (DEGREES) Fig. 7.12 : Suggested shaker/slider waveform inverse of quasi-static response ACCELERATION (II/31) MIDSPAN DEFLECTION (mm) 68 I -- WITHOUT SHAKER -- WITH SHAKER . . j . . I . . O 90 180 270 . 360 CRANK ANGLE (DEGREES) Fig. 7.13 : Absolute acceleration of rocker link at 200 rpm given shaker/slider waveform shown in Fig. 7.12 —- WITl-DUT SHAKER -- WITH SHAKER 4 -200 I I I I I ' I I I I r 0 90 180 270 . 360 CRANK ANGLE (DEGREES) Fig. 7.14 : Analytical elastodynamic response at 200 rpm given shaker/slider waveform shown in Fig. 7.12 69 — WITHOUT SHAKER ; -- WITH SHAKER MIDSPAN DEFLECTION (mm) -2.0 A. . . . .fi . . , . . 0 90 130 270. 360 CRANK ANGLE (DEGREES) Fig. 7.15 : Experimental elastodynamic response at 200 rpm given shaker/slider waveform shown in Fig. 7.12 150 -— WITI‘DUT SHAKER -- WITH SHAKER NW g . Z E i U 3 _100 I r ' I v I I v ' I I 0 90 180 270 360 CRANK ANGLE (DEGREES) Fig. 7.16 : Angular acceleration of the rocker link at 200 rpm given shaker/slider waveform shown in Fig. 7.16 70 Both experimental and analytical link deflections are dependent on the absolute acceleration of the link. Note that a decrease in angular acceleration of the link does not necessarily reduce the deflection. The data from Table 7.1 implies that the peak response in both halves of the cycle may be reduced by having a shaker waveform proportional to link deflection in the first half and a waveform inverse of the link deflection in the second half. This notion is graphically shown in Fig. 7.17, where discontinuities on the curve occur due to the inversion process. From the experimental point of view, excitation of the shaker with a discontinous waveform may result in erroneous slider motion (displacement with spikes), which is not recommended [27]. This can be avoided by smoothing the discontinuities as shown by the modified curve. This general approach is accomplished for any operating speed by feeding the modified experimental quasi-static response data to the arbitrary waveform generator which provides the excitation for the shaker. The shaker must be carefully orchestrated with the primary forcing function provided by the crank motion in order to reduce the elastodynamic response of the rocker link. This waveform will serve as a possible optimum curve for the slider motion. In this fashion, the shaker frequency would be the same as crank frequency and the phase would be zero. The only unknown is the slider amplitude, which has a constraint of i8 mm. If this waveform can reduce the absolute acceleration of the rocker link, then it may be justified that it will in turn reduce the link deflection. As it turns out, it does behave accordingly and the results are shown in the next chapter. DISPLACEMENT (mm) 71 1.5 I — TYPICAL DEFLECTION CURVE j -- DESIRED SHAKER FUNCTION 1.0-4 ““ MODIFIED SHAKER FUNCTION 1 O 51 4 0.01 I -o.5{ l J —1.0-: .1 " “1gu 1 f T Tr fl” r I r r I f O 90 180 270 360 CRANK ANGLE (DEGREES) Fig. 7.17 : Synthesis of Shaker/slider Waveform CHAPTERS EXPERIMENTAL.& COMPUTATIONAL.RESUETS In chapter 7, section 7-1, the analytical deflection of the rocker link of a classical four-bar linkage was shown to correlate with the experimental results. Having defined the shaker waveform, computer simulation and experimental investigation were carried out by exciting the shaker simultaneously with crank motion. The results are presented for crank frequency of 148, 170, 200 and 215 rpm. 8-1 RESPONSE DATA & ABSOLUTE ACCELERATION OF THE FLEXIBLE LINE The experimental elastodynamic and quasi-static midspan deflection at 148 rpm are shown in Figs. 8.1 and 8.2 respectively, followed by analytical results in Figs. 8.3 and 8.4. The absolute acceleration at 148 rpm is shown in Fig. 8.5. The experimental and analytical response at 170 rpm is shown in Figs. 8.6 and 8.7 respectively, followed by absolute acceleration in Fig. 8.8. The results at 200 rpm are shown in Figs. 8.9 to 8.11. A comparison of experimental and analytical result for the intelligent mechanism at 200 rpm is shown in Fig. 8.12. Figs. 8.13 to 8.16 present the results at 215 rpm 72 IIIDSPAN DEFLECTION (mm) MIDSPAN DEFLECTION (mm) 73 0.50 CRANK ANGLE (DEGREES) Fig. 8.1 : Experimental elastodynamic response at 148 rpm _OO 75 f I T I r fi’ fi t I I T 0 90 180 270 360 CRANK ANGLE (DEGREES) Fig. 8.2 : Experimental quasi-static response at 148 rpm MIDSPAN DEFLECTION (mm) MIDSPAN DEFLECTION (mm) 74 T r ”T O 90 180 270 380 (HUUfl