MSU LIBRARIES m RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped be10w. germ? A UMETICAL IND EXPERIE’ITAL INVBTIGATION OF THE DYNAIIC RESPONSE OF FLEXIBLE EGANISI SYSTEMS FABRICATBD FRO! FIBROUS (DIPOSITB IATERIALS By Chung-Inc Sun; A DISSETATION Submitted to lichigan Stat. Univaraity in partial fulfill-out of tho roqnira-onta for tho don-u of DOCIOR OF PHILOSOPHY Dapatt-ant of lochanioal 8331110021113 1986 v ABS‘IRACT A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF TEE DYNAIIIC RESPONSE OF FLEXIBLE NECEANISN SYSTBIS FABRICA‘I‘ED non FIBROUS mmSITE lATERIALS By Cheng-Kuo Sung The articulating members of linkage machinery must be designed vith high stiffness-to-eeight rstics in order that these machine systems operate successfully in a high-speed mode. One approach to satisfying this criterion is to exploit the high specific stiffnesscs of polymeric fibrous composite laminates. This ecrk is divided into two parts. First. the mechanism systems are operating under isothermal conditions. Candidate materials are subjected to mechanical testing and their constitutive behavior classified. A variational theorem is then derived to obtain the governing equation and the essociated boundary conditions. A finite element formulation is also developed based on the variational equation of motion. The predictive cepability of this analytical approach is evaluated by simuleting the vibrationnl response of both experimental four-bur linkages and also slider-crank mechanisms prior to comparing the computer results with experimental dete. Secondly. the same mechanism systems are operating under adverse environmental conditions. The constitutive behevior of some of these composite materiels is. however. dependent upon the ambient environmental conditions. and hence models must be developed in order to predict the response of mechanism systems fabricated in this cless of materials. A variational principle is presented that may be employed for systematically establishing the equations governing the dynamic response of planar flexible linhege mechanisms simulteneously subjected to both mechanical and hygrothermal loadings. As an illustrative example. the equations of energy belance and mass balance are validated by comparing the theoretical simulation with the experimental results performed by Browning and 'hitney. ACKNOWLEDGEMENTS 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. Bis academic excellence and research philosophy have provided a constant source of encoursgement. I also wish to thank the members of my committee Drs. Rohan Abeyaratne. John J. IcGrath and David You. This research was finnncially supported through the effort of Dr. Brian S. Thompson by the Nationnl Science Foundation under grants CHE-192124 and HEA98216777. and by Division of Engineering Research at Iichigan State University. This assistance is warmly acknowledged. Finally, I would like to thank my parents for their tremendus support. my wife and two sons for their loving support and willingness to share their time with my work. 11 TABLE OF CONTENTS LIST OF FIGURESeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee Chapter 1. Chapter 2. mnonualm.OOOOOOOOOOOOOOOOO0.0.0.000...0....0...... 1.1 Current Trends in Industrial Nachinery............ 1.2 Scope of This Investigation....................... 1.2.1 Relationship between the Theories Developed herein and Other Theories......... 1.2.2 Review of this Thesis....................... 1.3 Literature Survey for Isothermal Elastodynamic An.ly.i.oOI.0.0...OCOOOOOOOOOOOOO0.00.00.00.00...O A THEORETICAL AND EXPERIMENTAL INVESTIGATION INTO TEE DTNAIIC RESPONSE OF FLEXIBLE NECEANISNS NADE ERON ELASTIC MATERIALS AND OPERATING UNDER ISOTEERIAL CONDITIONS............. 2.1 A‘Variational Formulation for the Geometrically Nonlinear Finite Element Analysis of Flexible Linkages lede from Elastic Iaterials end Operating under Isothermal Conditions......... 2.2 Exp.ri..nt.l studyOOOOOOOOOOOCOOOOOOOIOOOOOOOOOOIO 2.2.1 Experimental Apparatus: Four-bar Linkages.. 2.2.2 Experimental Apparatus: Slider-crank .‘ch.n‘“.IOOCOIOOOCOOOOOIOOOOOOOOOOOCOOOOO. 2.2.3 In.tru.nt.t‘on0.000.000.0000000000000000000 2.3 capot.‘ 8i.ul.tion‘o....IOOOOOIOOOOOOOOOOO0.0.... 2.4 R..nlt‘ .nd ”1.0“..‘on.OOOOOOOOOOOOOOOO00.0.00...O iii Page vi II 27 27 45 46 50 52 60 64 Chapter 3. AN EXPERIMENTAL STUDY TO COMPARE THE RESPONSE CHARACTERISTICS OF MECHANISMS FABRICATED 'IT‘H COMPOSITE MATERIALS AND SIMILAR MECHANISMS CONSTRUCI'ED MITH TWO COMMERCIAL METALS................ 76 3.1 Theoretical Motivation for this Experimental stndYOOOOOOOOOOOOI0.0.00.0...OOOOOOOOOOOOOOOOOOOOO 76 3.2 Objectives and Material Characterization.......... 77 3.3 Results end Discussion............................ 93 Chapter 4. A THWRETICAL AND EXPERIMENTAL INVESTIGATION (N THE DYNAMIC RESPONSE OF FLEXIBLE MECHANISMS MADE FROM mMPOSITE MATERIALS OPERATING UNDER lsommy‘ui CWDITIONSO00.000.000.000...OOOOOOOOOO...O. 101 4.1 Intro‘nctiODOOOOOO00.0.0.0...OCOOOOOOOOOOOOOOOO0.. 101 4.2 Material Characterization study................... 102 4.3 Variational Principle............................. 109 4.4 Finite Element Formulation........................ 116 4.5 Comparison between Theoretical and £3p°rin°nt.l Re‘nlt‘oO...I0.0000000000000000000000 122 Chapter 5. A THEORETICAL ANALYSIS OF EYGROTEERMOVISCOELASTICITY.. 131 5.1 Thermoviscoelssticity: A Background Review........ 131 5.2 Problem Definition................................ 133 5.3 Conservation Lews................................. 136 5.3.1 Conservation of Nass........................ 136 5.3.2 Conservation of Momentum and Energy......... 139 5.4 Entrapy Balance and Entrepy Production............ 142 5.5 The PhenomenOIOgical Equations and the on‘.‘er Principle..OOOOOOIOOOOOOOOOCOOOOOOOOOOOOOO 147 Chapter 6. A THEORETICAL INVESTIGATION ON THE LINEAR COUFLED HYGROTHERMOELASTODYNAMIC ANALYSIS OF EGMISH SYSEMSeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee 158 6.1B.ct.r°nnd0OOOOOOOOIOOOOOOOOOO0......OOOOOOOOOO... 158 6.2 Variationel Principle............................. 164 iv 6.3 Finite Element Formuletion............ 6.4 Parameters Definition................. 6.5 Illu‘tr.tiv. Ex.npl...............O... 6.6 Results and Discussion....... Ch.pt.r 7. DlstSIm' WEBSIWO0....OOOOOOOOOOOOOO0.0.0000...O BIBLIOGRAPBYOOIOOOOOO 178 185 192 I98 208 217 Figure 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 LIST OF FIGURES Definition of Axis Systems and Position Vectors.......... Experimentel Four-Ear Linknge Mechanism.................. Experimental Slider-Crank Mechenism...................... Slider Assembly for Experimental Slider-Crank Mechanism.. Schematic of Experimental Apparatus and Instrumentation.. The Digital Dita-Acquisition System in the Machinery Bl.‘t0dymic L‘bor.tory.........0...COCOCOOOOOOODOOOOCOO Four-Bar Linkage: Rocker Midspan Transverse Deflection st 254 rpm before Digital Filtering...................... Four-Bar Linkage: Frequency Spectrum of Rocker Midspan rt.n".r‘. n.‘1.°ti°n ‘t 2,4 t".eeeeeeeeeeeeeeeeeeeeeeee Four-Ear Linkage: Rocker Midspan Transverse Deflection 0f 25‘ r” .ft.t Dilit.l Filt.tin‘lOOOOOOOOOOOOOOOOOOOOOO Finite Element Model of Experimentel Four-Dar Linkage.... Four-Bar Linkage: Coupler Midspan Transverse Deflection at 342 rpm. Integration TIme-Step 0.00048733 Seconds..... Four-Ear Linknge: Coupler Midspan Transverse Deflection st 193 rpm. Integration Time-Step 0.00086356 Seconds..... Four-Ear Linknge: Rocker Midspan Tremsverse Deflection at 254 rpm. Integration Time-Step 0.0006561? Seconds..... Four-Bar Linkage: Rocker Midspan Transverse Deflection at 205 rpm. Integretion Time-Step 0.00081301 Seconds..... Four-Ear Linkage: Rocker Midspen Transverse Deflection at.290 rpm, Integration TIme-Step 0.00057471 Seconds..... Slider-Crank Mechanism: Connecting-Rod Midspan Transverse Deflection at 235 rpm. Integration Ti..-St.p 0.0007m22 8.00“.eeeeeeeeeeeeeeeeeeeeeeeeeeeee vi Page 30 47 51 51 53 54 57 58 62 65 66 67 68 71 2.17 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 4.1 4.2 4.3 4.4 Slider-Crank Mechanism: Connecting-Rod Midspan Transverse Deflection at 274 rpm. Integration Ti'.-st°p 0000060827 sacond..OOOOOOOOOOOOOOOOOOOOOO00.... Alignment Fixture and Mechanical Arrangement for TG'tinB Gr.phit./Epoxy Sp.°in.n‘.....OOOOOOOOO...0.00.... The Mechanical Testing of Graphite/Epoxy Laminates in.mT°.tin‘ u‘chin..................CICCCCOOOOO0.... Dynamic Test Results for Unidirectional “-4/3501-6 Luin.te00.0.0.0000...OOOOOOOOOOIOOOOOOOO0.0. Dynamic Test Results for [tssl‘ AS-4/3501-6 Laminate..... Creep Response for Unidirectional AS-4/3501-6 Laminate... Creep Response for [:45], As-4/3501-6 Laminate........... Link Stiffness Characteristics........................... Steel Specimen Transient Response. Horizontal Scale ZOIS/diV. VertICII Selle OeIVIdiVeeeeeeeeeeeeeeeeeeeeeeee Aluminum Specimen Transient Response. Horizontal Scale SOUS/div. v.ILSCMl sc.1° OOIVIdiVOOOOOOOOOOOOOOOOOOOOO... [t451‘ Specimen Transient Response. Horizontal Scale 20mS/div. Vertival Scale 0.1V/div........................ [0] Specimen Transient Response. Horizontal Scale 20mS/div. Vertical Scale 50mV/div........................ Coupler Midspan Bending Dcflections: 280 rpm............. Coupler Midspan Bending Dcflections: 198 rpm............. Rocker Midspan Bending Deflections....................... Connecting-Rod Midspan Bending Deflections............... Link Calibration-Fixture................................. Method of Obtaining the Stress Relaxation Function....... Stress Relaxation Function Obtained from I.t.ri.1Te'tin‘.O'COOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO Standard Linear Solid Model.............................. Stress Relaxation Function Obtained from Curve-Fitting... The Deformation of a Beam Element........................ vii 72 81 81 83 84 86 87 89 91 91 92 92 94 95 96 97 98 104 106 107 108 116 4.6 Slider-Crank Mechanism: [£45]. Connecting-rod Midspgn Trans. Deflection at 154 rpm. time-step 1.80375x10’ s... 124 4.7 Slider-Crank Mechanism: [1451‘ Connecting-rod Midsp n Trans. Deflection at 212 rpm. time-step 1.31027x10’ s... 125 4.8 Four-Bar Linkage: Unidirectional Coupler Link Midspan Trans. Deflection at 212 rpm. time-step 1.31027x10‘55... 126 4u9 Four-Bar Linkage: Unidirectional Coupler Link Mids an Trans. Deflection at 278 rpm. time-step 0.9992x10" s.... 127 4.10 Four-Bar Linkage: Unidirectional Rocker Link Midspan Trans. Deflection at 187 rpm. time-step 1.48544x10’5s... 128 4-11 Four-Bar LilleO= [£45] Rocker Link Midspag Transverse Deflection at 223 rpm. tIme-step 1.24564x10' s........... 18 5.1 Definition of Axes System and Position Vectors........... 135 6.1 Definition of Axes System and Position Vectors........... 165 6.2 Plot of Experimental Data versus Fick's L... for N..t Epoxy B..in.OOOOOOOOOOOOOOOOOOOOOO000...... 195 6.3 Master Plot of Experimental Data versus Fick's Law. for Unidirectional Graphite/Epoxy Composite......... 196 6.4 Master Plot of Experimental Data versus Fick's Law. for Bidirectional Graphite/Epoxy Laminate........... 197 6.5 Comparison of Data. the Upper Bound and the Lower Bound for G (Hygroelastic Modulus) Based on the Modified Fick's Law. for Bidirectional Graphite/Epoxy Composite................................................ 205 6.6 Comparison of Data. Fick's Law and Modified Fick's Law. for Bidirectional Graphite/Epoxy Composite... 206 viii CHAPTER 1 INTRODUCTION 1.1 Current Trends in Industrial Machinery 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 the 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. A force and stress analysis of the members is undertaken subsequently. Mhen 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 may result in excessive stresses and impactive loads. and these loads can generate more severe 1 2 problems such as wear. loss of performance. reduced stability and more critical levels of noise and vibration. Link flexibility has become an active field of research since the late 1960's and this work is documented in two comprehensive survey papers I 52.95 1. Upon reviewing these papers it is evident that for high-speed operation. mechanism links should be designed with high stiffness-to-weight ratios in order to reduce link deflections and power consumption. In all of the publications cited in'references I 52 l and I 95 l. the members were fabricated in the traditional steel and aluminum alloys. and the desired lightweight form-designs developed by using optimization software packages. An alternative design methodology has recently been proposed by Thompson et al I 158.160.173.178 1. This methodology advocates that composite materials should be employed to reduce the elastodynamic phenomena. such as link deflections and dynamic stresses. As is well known. these materials have superior strength and stiffness-to-weight ratios than the commercial metals. Consequently. they offer the designer reduced inertial loading at specific speeds. or else higher speeds of operation because of smaller deflections and superior dynamic response characteristics. 1.2 Scepe of This Investigation In this section. a comparison between the theories developed in this thesis and other theories is undertaken in sub-section 1.2.1. such as the methodologies of developing the equations of motion which govern the dynamic behaviors of the mechanism systems. and the hygrothermal analysis of the linkage mechanisms fabricated from composite materials. An overview of this thesis is presented in sub-section 1.2.2 which outlines the content of each chapter. 1.2.1 Relationship between the Theories Developed Herein and Other Theories The Legrenge's equation was employed by a few researchers in the field of machine dynamics in order to deve10p the equations of motion which govern the dynamic behaviors of the mechanisms. B.S. Thompson and A.D. Barr I 172 ] prOposed a mixed variational approach which incorporates auxiliary conditions such as the strain-displacement equation. constitutive equations and geometrical boundary conditions into the Hamilton's principle. So the functional depends upon displacements. stresses and strains. Moreover. the stationary conditions are the governing equations in primitive form: the kinematical. dynamical and constitutive relationship. More insights may be gained from this mixed variational principle via a finite element formulation which discretizes a continuous medium into several finite elements. As the basis of an elementary approximation. each field can be independently selected to achieve the discrete model. namely. the 4 approximating function of stresses may be different from that of the strains. The discrete counterparts of the displacement. stresses and strains are governed by algebraic equations: kinematical. dynamical and constitutive. This approach offers an effective mean to achieve simple. efficient models that possess the desirable attributes of those derived by approximating the displacement in the potential. The high-order theories modeling the elastic motion of links are developed in this investigation. These theories employ geometrically nonlinear analyses which retain the terms in the strain-displacement equations that couple the axial and flexural deformations I 107.108.147.180.185 ]. These additional terms are readily handled by the numerical integration solution philosophy. but they present additional complications if the model superposition approach is employed. A composite material is constructed by combining two or more materials on a macroscopic scale to form a useful material since the best qualities of the constituents are often significantly exposed in accordance to the intention of a designer. For example. if attention is focused on the first-ply failure of a composite material. then the first ply has to be designed strong enough by either selecting an appropriate fiber material or rearranging the stacking sequence in order to keep the first ply from being damaged. Some. but not all. of the polymeric materials are very sensitive to both the environmental temperature and relative humidity fluctuations. 5 the materials absorb moisture cause swelling and micro-crack formation under extreme conditions. These micro-cracks. in turn increase the diffusivity of the composite and this characteristic can be increased further by the dimensional changes associated with elevated temperatures. Consequently. the mechanical properties degrade and cause the structural components to fail. The classical Fick's law is employed to model the diffusive process when the composite material is immersed in humid air or a liquid. The most important assumption invoked in the develcpment of Fick's law is that the solid (in this case. the composite) is assumed to be rigid I 145 1. Several other assumptions have also been made on the nature of the kinematical quantities such as : ( 1 ) The motion takes place under isothermal conditions. ( 2 ) The velocity components are sufficiently small. therefore. the kinetic energy of the diffusing masses can be neglected. ( 3 ) Body forces are absent. Hence. it is not surprising that the Fickian model of diffusion proves to be inadequate for predicting the response of materials exposed to normal environmental conditions because of the complex interaction of a composite material and the diffusing moisture. There are two theories developed in this thesis in order to closely 6 describe the complex physical phenomena. One is based upon the first and second laws of thermodynamics. non-equilibrium thermodynamics and classical continuum mechanics. Iith the application of classical thermodynamics. the response of the materials which are exposed to the environmental conditions may be completely described by the following equations: the balance equations of mass. momentum and energy. constitutive equations and the equations of entropy inequality. The other is to derive the governing equations by developing a variational principle. Euler equations for the variational principle are the field equations of motion. heat conduction and mass diffusion. and the strain-displacement equations in terms of primary field variables. This principle also yields prescribed boundary conditions on heat flux. mass flux and surface traction vectors. In most theoretical research publications on modeling the hygrothermal response of composites. the heat conduction and mass diffusion equations are generally decoupled in order to establish a mathematically tractable problem. but this approach is not always appropriate. For example. if a material is simultaneously subjected to both hygrothermal and mechanical loadings. and the rate of change of the material structure is of the same order of magnitude as the rate of change of temperature and moisture concentration. then the general equations governing the cross-effects. i.e. the coupling terms between the stress and temperature. or between the temperature and moisture concentration. are necessary for analyzing this physical problem. For example. a polymeric material which demostrates a strong viscoelastic characteristic is subjected to a prescribed strain. the relaxation of 7 the stress inside the material will persist for a long time. In this case the rate of change of the stress relaxation of the material structure is of the same order of magnitude as the rate of change of temperature and moisture concentration. therefore. the cross-effects are significant and should be taken into account. 1.2.2 Review of This Thesis The analysis of high-speed linkage mechanisms fabricated from composite materials is a very difficult task because of the complex constitutive behaviors. A fundamental investigation into the dynamic response of four-bar linkages and a slider-crank mechanisms made from a well-known elastic material is undertaken both theoretically and experimentally and is documented in chapter 2 in order to ensure that one has the ability to analyse the more complex mechanism systems. The links of both mechanisms are made from elastic metals. A variational theorem is develcped based on extended Hamilton's Principle to establish the equations governing the geometrically nonlinear deformation of an elastic continuous medium subjected to the dynamical loading conditions. The geometrical nonlinearity is defined under the assumption that the deformation is small and so the strain is also small. However. the rotational terms in the strain-displacement equation have the same order of magnitude as the linear terms in the expression. A displacement finite element formulation for a single degree of freedom beam element is presented and all of the analyses performed in this thesis assume that joints are without clearance. thereby greatly simplifying the complexity of modeling the impactive bearing forces. The computer 8 simulation results are compared with experimental results and some design guidelines are drawn from this study. In chapter 3. an experimental study to demonstrate the superior dynamic response characteristics of mechanisms constructed with composite material is described. This work was performed following a sequence of theoretical research presented in references I 158.173.176.178.179 ]. Mechanical material tests were performed to determine the constitutive behavior of unidirectional and [t45]. angle-ply composite laminates. The dimensions of each link are determined by synthesizing links with a constant flexural rigidity in order to compare the dynamic response of each link when it is incorporated on the mechanism. The damping coefficient of each link is measured by the method of logarithmic decrement. The experimental results clearly demonstrate the superiorities of constructing flexible mechanisms from these two composite materials because of lower stress levels and small deflections. In chapter 4. the work of chapter 3 is extended in order to study flexible mechanism systems fabricated from composite laminates. The material characteristics from the experimental program were carefully examined and the unidirectional laminate was modelled as elastic material. because of the time-independent behavior. and the [1:45]8 composite laminate is modelled as viscoelastic material since the material behavior is time-dependent. A mixed variational principle which incorporated auxilary conditions such as the constitutive equation. strain-displacement equation and geometrical boundary 9 conditions into Hamilton's Principle is presented using the Stieltjes convolution integral notation. This variational principle also provides a basis for the finite element formulation which is employed in order to obtain the numerical results. The discrepancy between the experimental results and computer simulations is 105 which may suggest the need for an improved modelling of the constitutive equation. Composite materials are often sensitive to the environmental conditions such as the changes of temperature and moisture. and the degradation of the material properties are presented in a variety of references including I 215,216,217 ]. A fundamental phenomenological study of the dynamic response of composite materials to a wide range of both mechanical and hygrothermal loadings is proposed and is presented in chapter 5. This study is based upon the first and second laws of thermodynamics. non-equilibrium thermodynamics and the classical continuum mechanics. The application of classical thermodynamics. permits the moving continuous media modeling a complex physical phenomenon. such as changing temperature and moist environment. to be completely described by the following basic equations: the balance of mass. the balance of momentum. the balance of energy and the entropy inequality. In chapter 6. a variational principle is developed to obtain the linear coupled hygrothermoelastic response of flexible mechanism systems. The primary field variables are taken to be the displacement of a material point and two vector field variables called the entropy displacement and the flow potential displacement which is analogous to 10 entrapy displacement. Euler equations for the variational principle are the field equations of motion. heat conduction and mass diffusion. and the strain-displacement equations in terms of primary field variables. This principle also yields prescribed boundary conditions on heat flux vector. mass flux vector and surface traction vector. temperature. material displacement and moisture concentration. A finite element formulation is also performed in order to obtain a numerical mean to render a tractable solution to these complicate equations. An observation may be drawn from the final equations that the cross-effects may occur in the equations of motion. energy balance and mass balance. since a sudden change of temperature. moisture or finite strain may significantly result in the changes of stresses. heat and mass fluxes. An example in which moisture diffuses into neat epoxy resin and graphite/epoxy composites was investigated by J. M. Ihitney and C. E. Browning [ 217 l. is examined analytically and the non-Fickian diffusion in bidirectional composite laminates is also studied. The experimental data presented in Figure 3 of reference I 217 I showed a significant descrepancy .from classical Fick's law i.e. the test data approach equilibrium at a slower rate than that predicted by Fick's law. lhitney and Browning postulated that the large in-plane tensile residual stresses which resulted from environmental change increase the initial through-the-thickness diffusivity; then as swelling relieves the residual stresses. the diffusivity decreases. The diffusion coefficient approaches the diffusivity of a unidirectional composite as the residual stresses are completely relieved. Direct proof of a stress-dependent diffusion process. however. requires a measurement of the diffusion 11 coefficient under various constant-stress conditions. As an alternative solution philosophy. a modified Fick's law is proposed herein by incorporating a stress-dependent term in the diffusion equation; the results are compared with experimental data and after a parametrical study is completed.the computer results and the model are discussed. Finally. the extension of current work is proposed at chapter 7. 1.3 Literature Survey for Isothermal Elastodynamic Analyses The mathematical model of a flexible linkage mechanism must generally capture the mass. stiffness and damping characteristics of the links. the external loading. the equation of closure for kinematic chain. the principal characteristics of the joints of the mechanism. the behavior of the drive shaft and the kinematics of the machine foundation. Finite elements from the structural mechanics literature have generally been employed to model the material properties of the links. and most analyses. with the exception of I 29 l and [ 148 l. , assume that the absolute motion of each link may be decomposed into a rigid-body displacement upon which is superimposed a deformation displacement measured relative to a moving coordinate system fixed in the link in an undeformed reference state. Local element matrices [M]. [C] and [R] are generally established to model the mass. damping and stiffness characteristics of each link before pre- and post-multiplying by transformation matrices relating the local and global (inertially fixed) reference frames. The trigonometrical functions contained in these transformation matrices are calculated from kinematic analyses of the rigid-linked system in a large number of different configurations. 12 This manipulation enables the global characteristics of the mechanism to be established by formulating the global matrices prior to incorporating the relevant boundary conditions. Typically the global equations of motion I 174 I for a high-speed linkage are written [-61th + [Collin + [scum - - [sent] + m + m (1.1) where {U} is the column vector of deformation displacements. (') is the absolute time derivative. [MG]. [CG] and [16] are the global mass. damping and stiffness matrices respectively. I!) are body forces and IT} are surface tractions. The nodal absolute acceleration terms. derived from a rigid-body kinematic analysis of the mechanism. are contained in the vector Ill]. and when multiplied by the mass matrix yields the nodal inertial loading. The equations are then solved for the global degrees of freedom from which link deflections. vibrational response and dynamic stresses may be obtained. In order that the finite element method be applied to a linkage mechanism. the articulating system is generally modeled as a series of instantaneous structures. Thus the continuous motion of the system is replaced by a sequence of structures at discrete crank angle configurations upon which is imposed the relevant inertial loading. In order that these structures may be solved by the finite element method. the rigid-body degrees of freedom of the linkage must be removed from 13 the model in order to avoid singular matrices. The method of removing the rigid-body degrees of freedom is to assume that the crank may be modelled as a cantilevel beam at each time instant. therefore. the deformation at the build-in end is zero. Iinfrey I 196 ] accomplished this by directly applying the principle of conservation of momentum to the complete mechanism. Imam. Sandor and Kramer I ‘D I considered the crank as a cantilever beam to avoid this complication. and this approach was also employed by Midha et al I 103 l. and Gandhi and Thompson I 58 ]. Nath and Ghosh I 107.108 ] removed the rigid-body degrees of freedom from the global matrices using a matrix decomposition approach. Having previously reviewed the basic assumptions and nomenclature of finite element methods. attention is now focused on formulating these equations of motion. Iinfrey [196.197 ] employed the displacement finite element method (the stiffness technique of structural analysis) to study the elastic motion of planar mechanisms in some pioneering publications. This popular finite element approach solves for the nodal displacements and requires displacement compatibility on inter-element boundaries. Ekdman et al I 53 I employed the equilibrium finite element method (flexibility method of structural analysis) to study flexible mechanisms. Iith this approach. adjacent elements have equilibrating stress distributions on inter-element boundaries and the global degrees of freedom are the stress components. This approach is not as popular as the displacement formulation in most fields of finite element work since the additional compatibility equations have to be added in the I4 equilibrium formulation. and mechanism design is no exception. Midha et al I 102-106 1 were responsible for developing linear and geometrically nonlinear finite element formulations for linkage mechanisms based on Lagrange's equation. The results of this latter work. with Turcic. were validated experimentally I 184.185 ]. Bagci et al I 13,14,154 ] have written a suite of publications on the dynamic response of flexible mechanisms using the linear theory of elasticity. These formulations are based on the stiffness technique of structural analysis. Thompson et al I 57.58.157-161.171-181 ) developed variational principles as the foundations for studying the linear and geometrically nonlinear elastodynamic responses of linkages. Unlike other methodologies. this class of formulation explicitly presents the boundary conditions and also the governing equations of motion in a single mathematical expression. The analyst then has the freedom to established displacement, equilibrium, mixed or hybrid finite element models from these general variational statements. Variational methods have also been employed to investigate linkages fabricated from light-weight viscoelastic composite materials I 178,179,181 ] and also the acoustic radiation from linkage machinery I 175 I. This latter work involved modeling the operation of mechanisms submerged in a perfect fluid as a fluid-structural interaction problem based on interacting continua. 15 Having established the governing equations of motion, boundary conditions and the initial conditions for a particular design task, the engineer must then select the appropriate finite elements for modeling the physical phenomena to be investigated. and a number of questions need to be answered carefully. For example, in a study of the vibrational behavior of a linkage. the designer must decide whether the in-plane and also out-of-plane responses are relevant. Mill information on the axial and flexural deformations suffice, or is the torsional deformation field also important? Should a linear or a geometrically nonlinear model be developed? And so on. Generally. these decisions will be guided by engineering intuition or experimental evidence. A large number of papers I 8-10,13-15.20,44-46,53,57,58,79-82.88. 102-108.153.161.177.181 ] have been devoted to studies of the planar elastodynamic response of flexible linkages by modeling the links using finite elements with only one spatial variable. The axial response is generally modeled by a linear interpolation function and the flexural response by a cubic interpolating polynomial using elements originally derived for structural applications. Bahgat and Millmert I 15 l. in one of the pioneering works in this field. considered both the axial and flexural responses of a wide variety of flexible linkages using a finite element formulation employing higher-order hermite polynomial approximations for the deformation fields. The same quintic element was also employed by Cleghorn et al I 44-46 I. In I 46 l. a comparison was undertaken between finite elements utilizing quintic polynomial approximations and those based on cubic polynomials. These results not only suggested that axial deformations may be neglected in the analysis 16 of some flexible mechanisms. but also that fewer of the higher-order elements are required to generate a specified solution accuracy when compared with the results from a model incorporating a larger number of lower-order cubic elements. The finite-line-element nomenclature employed by Bagci et al I 13,14,154 ] is an alternative terminology for the standard rod and beam elements of structural mechanics. The two common formulations for mass matrices. namely the consistent mass matrix I 45.53,57,58,102,108,157,158,173,178,179 ] and the lumped mass matrix I 13,14,154 ] have both appeared in the mechanism design literature. Tong. Finn and Bueciarelli I 182 ] demonstrated that while the lumped mass approach will not suffer any loss in the rate of convergence when utilizing simple rod elements, a consistent mass formulation is to be preferred when using higher-order elements such as been elements. Having discussed the mess and stiffness matrices employed to model mechanisms. attention is now focussed on damping matrices. Damping in materials is a complex phenomenon I 23 l which probably requires the development of thermo-mechanical models in order that it be fully understood. The assumptions employed to develop dumping matrices are partially governed by the solution technique to be adopted for solving the finite element equations. Generally in the mode superposition approach, the damping is assumed to be uncoupled and is given in each mode as a percentage of the critical damping. However, when adopting the 17 step-by-step approach of numerical integration the complete damping properties of the mechanism must be established and this may be a difficult task. In the appendix of reference I 104 I. Midha, Erdman and Frohrib discussed the different damping matrices in the context of modal superposition solutions. Minfrey I 196 I assumed a damping matrix prOportional to a linear combination of the mass and stiffness matrices (Rayleigh damping). In contrast to this. Alexander and Lawrence I 8 I IIIUIOG th' 44-91“! coefficient. C11. for each orthogonal coordinate to be defined by Cii . ZSII‘iiluii)1/2' where {1 is the damping ratio. R11 is the stiffloti matrix and .ii is the mass matrix. Furthermore. (xii/'ii)1/2 are the natural frequencies of the mechanism system. The energy-dissipation characteristics of a mechanism are dependent upon both the constitutive equations of the link material and also the characteristics of the joints of the mechanism. In recent combined experimental and finite-element-based investigations I 159.166.176.177.180,185 ]. logarithmic decrement transient response studies have been undertaken for a number of mechanism configurations and the experimental data employed to establish semi-empirical damping matrices for the damping in the mechanism. Bagci and Stamps I 154 l have undertaken preliminary work on coulomb damping in joints, which they then combine with the model for material damping employed in I 8 1. thereby employing individual mathematical models for both mechanism joins and also the link materials. 18 Tho computational methods are generally employed to solve the equations of motion for both linear and geometrically nonlinear vibrational analyses. These are the modal superposition approach I 81.105.185.196 1 and the use of direct integration methods I 8.29.58.107 I. such as Runge-Kutta algorithms or the Newmark method. The former approach is based on the assumption that the displacement vector may be expressed as a linear combination of the vibrational mode shapes. This solution strategy is most efficient if the essential dynamic response of the mechanism is contained in the first few modal combinations. Hence it is most useful for studying the steady-state response of systems operating at constant crank frequencies. However. additional complexities arise when developing geometrically nonlinear elastodynamic responses I 185 l. The approach is not to be recommended for linkages fabricated with journal bearings because the inherent clearance needed for the operation of these joints creates impact loading which is often characterized by high frequency components necessitating a solution based on many modes in order to predict the response of the mechanism. Numerical integration algorithms employ steprby-step procedures. Instead of attempting to solve the equations of motion at any time t. they only solve the equations at discrete time intervals At apart. Although, of course. this time interval may be made extremely small to closely approximate a continuous function. Furthermore. these algorithms assume a definite variation of the displacement. velocity and accelerations within each time interval. and this has a major effect on 19 the stability. accuracy and cost of the solution. These approaches can be readily applied to determine both linear and geometrically nonlinear elastodynamic responses, and furthermore. they any be applied to accurately model systems subjected to complex loading, or loads Icontaining significant high frequency components. Imam. Sandor and Kramer [79] applied the rate of change of eigenvalue-vector method to mechanisms, in order to undertake deflection and stress analyses. This approach eliminated the need for an eigenvalue solution at all the mechanism configurations subjected to analysis. Midha. Erdman and Frohrib I 105 ] develped a technique for directly determining the steady-state solution of differential equations with time-periodic coefficients, which govern the elastodynamic motion of high-speed linkages. The same authors were also responsible for developing an alternative solution strategy to the same class of problem I 103 l. and in addition, a numerical algorithm for performing transient analyses of linkages I 106 ]. Ihile the above procedures are dedicated to determine the elastodynamic response of linkages, some authors have advocated that a quasi-static analysis of a linkage mechanism is quite adequate for most design purposes in an industrial environment I 84 I. Since most of the complications associated with an elastodynamic analysis are avoided by adopting this philosophy, the results are less costly, but they are also less accurate. This approach requires the solution of a set of 20 nonhomogenecus algebraic equations to be obtained. and this is readily accomplished using one of the Gaussian elimination family of algorithms. All of the mathematical models for flexible linkages involve a large number of degrees of freedom and hence a large number of equations of motion must be solved. This is computationally inefficient and hence expensive. However. this may be overcome by using static condensation techniques and an approach was developed by Khan and Millmert I 84 I. This philosophy involves condensing all internal degrees of freedom of each link to create a super element with only the principal degrees of freedom retained, and has been used extensively in commercial codes for structural dynamics problems. The authors reduced the number of system equations by 50 percent. so that the computational effort. which is proportional to the cube of the number of equations, was considerably reduced. As they rightly indicated in I 84 I, this approach is especially useful when the designer is searching for an optimal solution which generally involves many iterative analyses. The vast majority of the papers on flexible mechanisms present vibrational analyses of slider-crank or four-bar linkages comprising one or more elastic members and sited on a stationary rigid foundation. Deformations are generally restricted to axial and flexural modes in the plane of the mechanism and column vectors IX] and IT} in equation (1.1) are often neglected. This approach yields. for example. a transverse flexural vibration comprising a periodic response upon which is superimposed a high frequency waveform near the fundamental natural frequency in flexure of the link being studied. Steady state responses 21 I 15.45.103.108 ] and also transient responses I 106 I have been obtained. Since these classes of solution can be computationally time consuming, some authors I 84.125 ] have advocated that a quasi-static analysis is adequate for the analysis of most industrial machinery. This approach neglects the terms [M6][U] and ICGIIU] in equations (1.1). thereby considering the mechanism to be a statically loaded structure. This assumption greatly simplifies the computational aspects of the finite element analysis and the response comprises only the periodic waveform component of the vibrational response cited earlier. Inertial terms coupling the rigid-body kinematics and the elastodynamic response have been featured in some analyses I 45,104,107,180,185 ]. However, while these analyses present a more accurate mathematical model of the mechanism. these terms have. so far. been found to have a negligible effect on the elastodynamic response of the experimental mechanisms investigated in the laboratory I 1$ .184,185 I. The higher-order theories modeling the elastic motion of links employ geometrically nonlinear analyses which retain the terms in the strainrdisplacement equations that couple the axial and flexural deformations I 107.108.147.180.185 ]. These additional terms are readily handled by the numerical integration solution philosophy. but they present additional complications if the modal superposition approach is employed. The early work on finite element analyses employed only one element 22 to model each flexible link I 8 l and I 196 I. This naturally produced inaccurate results. since one of the fundamentals of finite element techniques is the assurance of solution convergence as the number of elements in a model is increased. This was originally highlighted by Alexander and Lawrence [7-9 I in pioneering work on combined experimental and computational investigations of flexible four-bar linkages. Midha, Badlani and Erdman I 102 ] reinforced these conclusions by demonstrating the effects of multi-element idealizations on the response of a structure. They showed that a 15 percent error existed in the second mode frequency and a 400 percent error in the third mode when a single element was employed. Similar errors were demonstrated by Gamache and Thompson I 57 l as part of a comparative study on modeling the flexural response of four-bar linkages using Timoshenko and Euler-Bernoulli beam elements. Most of the work to date on planar flexible mechanisms has concentrated on the planar response, thereby neglecting the co-planar motion of industrially realistic systems in which torsional effects. due to the offset of the joint members. may be significant. Preliminary work on this subject has been undertaken by Stamps and Bagci I 154 1. Although the natural frequencies of a mechanism are continually changing during the operating cycle, as the stiffness and mass characteristics relative to a fixed reference frame change. nevertheless. critical speed ranges must be identified and avoided in practice. This class of problem has been investigated by Bagci et al I 13.14 I using finite line elements and lumped mass matrices to study 23 slider-crank mechanisms. four-bar linkages and also six-bar mechanisms. Solutions were generated by an eigenvalue algorithm and compared with experimental data. The synthesis of rigid-linked mechanisms may be accomplished by either precision-point procedures or else optimization techniques which impose constraints on the minimization of an objective-function I 170 I. In order that these latter modern techniques be applied to synthesize flexible mechanisms, where deflections and dynamic stress levels are also introduced as constraints. the synthesis package must iteratively interact with software for analyzing flexible mechanisms. This is Ibecause the analysis and synthesis procedures cannot be conveniently decoupled. as occurs in the design of rigid-linked mechanisms. Mhile a considerable number of papers have been published on the analysis of flexible mechanisms. only a very small number have been written on the synthesis of these systems I 44.80.81.84,125.202.203 ] and they are all based on finite element methods. Before discussing the different solution strategies. it is appropriate to again review equation (1.1). which provides the key to understanding the proposed methodologies. Generally, when a mechanism is operating in a high-speed mode the body force and surface tractions may be neglected in comparison with the inertial loading, hence IX] and IT] disappear. Premultiplying all the terms by the inverse of the mass matrix IMGJ'1 yields [IIIUI + [IGI'IICGIIU] + [MGI'IIKGIIUI - - [IIIRI (1.2) 24 Thus it is evident that for a given mechanism operating at a given speed. the elastodynamic response is governed by the energy dissipation per unit mass and also the stiffness-to-mass ratio of the mechanism links. In other words. a high stiffness-to-weight ratio will result in small deflections and stresses. This observation has spawned two design philosophies. The first I 44.80,81.84,202.203 ] involves designing links in the commercial metals while optimizing the cross-sectional geometry of the members. The second I 125 ] advocates that modern fiberous composite laminates should be employed because of their inherent superior damping. and higher stiffness-to-weight ratios. Erdman. Sandor et al I 52.53.79-81 ] developed a general method of kineto-elastodynamic analysis and synthesis in aluminum alloys using optimization techniques. which incorporated stress and deflection constraints, to develop special cross-sectional shapes and tapered members. Millmcrt et al I 84 I developed the Optimality criterion optimization technique for mechanism design. This is based on the Kuhn-Tucker conditions of optimality for the minimum-weight design of mechanisms subjected to stress limits with the variables being the cross-sectional geometry of the links. Cleghorn, Fenton and Tabarrok I 44 I employ the same algorithm as in I 84 l with the modification that each interation incorporates the effect of the inertial loading of one link upon the stress levels as all the other links. This modification substantially reduced the number of 25 interations needed to achieve the optimum solution when employing the finite element formulation presented in I 45 J. Zhang and Grandin I 202,203 I developed a novel approach which combines the previous Optimality criterion technique with a kinematic refinement technique to achieve an optimal solution. This marriage involves a finite element analysis, a modern optimization algorithm and also a rigid-linked mechanism synthesis procedure for adjusting link lengths. location of fixed pivots. etc.. in a unified approach. The authors achieve considerable success with this method, recording a design weighing only 27 percent of that obtained in I 44 I when addressing the same synthesis problem. In contrast to the prevous approaches which are all concerned with homogeneous isotropic materials. Thompson et al I 158.160.173.178 ] have preposed that material selection should enter the design process. No longer should the search for appropriate materials be restricted to the metals but it should also include composite materials which offer much more desirable properties I 173,176-179,181 ]. Finite element models have been developed by extending the standard rod and beam elements to model the effect of ply angles upon link stiffnesses and to also represent viscoelastic materials. The models have been verified in combined experimental and computational studies [SS-60.63.66.681 and the superior response demonstrated in experimental comparative studies I 160 l. A third approach for synthesizing flexible linkages I 125 ] 26 proposes that a microprocessor-controlled actuator should be introduced into the original mechanism in order to modify the inertial loading and hence stress levels in the system. The analysis phase of the synthesis algorithm is undertaken using standard finite element theory. A systematic study is undertaken in the subsequent chapters for obtaining the dynamic response of high-speed linkage mechanisms fabricated from composite materials operating under both isothermal and extreme environmental conditions. A theoretical and experimental investigation of the dynamic response of flexible linkage mechanisms constructed from a well-known elastic material operating under isothermal conditions is presented in chapter 2 in order to ensure that one has the capability to analyse the more complex mechanism systems. CHAPTER 2 A THEORETICAL AND EXPERIMENTAL INVESTIGATION INTO THE DYNAMIC RESPONSE OF FLEXIBLE MECHANISMS MADE ROM ELASTIC MATERIALS AND OPERATING UNDER ISOTHERMAL CONDITIONS The analysis of high-speed linkages mechanism fabricated from composite materials is a very difficult problem because of the complex constitutive behaviors. The ability to predict the dynamic response of systems constructed from a well-known elastic materials is an essential step in order to ensure that one has the capability to analyse the more complex mechanism systems. This is what is undertaken herein. 2.1 A Variational Formulation for The Geometrically Nonlinear Finite Element Analysis of Flexible Linkages Made from Elastic Materials Operating under Isothermal Condition The Lagrange's equation was emplOyed by a few researchers in the field of machine dynamics in order to develop the equations of motion which govern the dynamic behaviors of the mechanisms. B.S. Thompson and A.D.S. Barr I 172 ] proposed a mixed variational approach which incorporates auxiliary conditions such as the strain-displacement equation. constitutive equations and geometrical boundary conditions 27 28 into the Hamilton's principle. So the functional depends upon displacement. stresses and strains. Moreover. the stationary conditions are the governing equations in primitive form: the kinematical. dynamical and constitutive relationship. More insights may be drawn from this mixed variational principle via a finite element formulation which discretizes a continuous medium into several finite elements. As the basis of an elementary approximation, each field can be independently selected to achieve the discrete model. Namely, the approximating function of stresses may be different from that of the strains. The discrete counterparts of the displacement. stresses and strains are governed by algebraic equations: kinematical, dynamical and constitutive. This approach offers an effective mean to achieve simple. efficient models that possess the desirable attributes of those derived by approximating the displacement in the potential. The variational theorem forming the kernel of this theoretical study was originally developed in the doctoral thesis of 8.8. Thompson I 220 I. It incorporates the geometrically nonlinear form of the field equations so that it may provide a basis for analyzing linkages susceptible to dynamic initabilities. and also mechanism systems that must be analyzed using higher-order theories. The deformation displacements are assumed to be small. and the strains are also assumed to be small. however. the rotational terms in the strain-displacement equations are assumed to be of the same order of magnitude as the linear terms in these expressions I 116 l. The variational equation of motion governing the motion of a 29 continuum of volume V and surface area S representing a portion of a link incorporating a revolute joint and also a sliding joint may be written using a general tensor notation I 220 l as 5’ ‘ ° ‘ $5 I Iv 57iJI°ij “3"3711] “V + . Iv 6u1[ xi + Oij'j + ui.kjojk + ui.kcjk'j - pp‘ ] dV + are, [ j§ x1 av +Ijsliid81 - IQpB. av] + 5’1 I IV'ijkin tok*tgg+nk ) dV + I52 °ijr31(ror*rnk*nk’ ‘182 + O Iv 'ijkppi(rokI‘Rk+°k) 4V 1 + Iv Soij [ Yij '1/2( ui'J+uj.1+uk.‘uk'j ) ]dV - IV PapiIPI- [zoi+;i+°ijk‘j(r0k+r3k +“k)] ] dV + Isl5ni‘31'li‘erni.k’dsi + [82(6'I+ bgkui.g+gk6ui.k)(ui-ui) dSZ + Is3( 5'z+6'kug,k*lkbnz'k )IcOSG (uxsin0+uzcos0)dS3 ] dt (2.1.1) 30 In Figure (2.1). the link has a Lagrangian axis frame oxyz fixed in the member in a reference state with zero stresses and strains, and the system parameters are defined relative to these coordinates. An arbitrary point P in the link is defined by the position vector ti = ‘oi + ‘Ri + “i' where ’oi are the components of the position vector of the origin of the body axes oxyz relative to the origin of an inertial reference frame OXYZ. ‘Ri reprcgentg the position vector of point P in the undeformed reference state relative to the body axes. and “i is the displacement vector. Figure 2.1 Definition of Axis Systems and Position Vectors 31 The absolute velocity associated with the time rate of change in ri 1' written ‘3 9;. p is the mass density. 11 is the body force per unit volume, I is the strain energy density. 7ij is the nonlinear Lagrangian strain tensor and “ij is the associated stress tensor. Components of the angular velocity vector for the moving Lagrangian frame are represented by .j and °ijk is the alternating tensor. The time rate of change with respect to oxyz is denoted by I”) while the absolute rate of change is denoted by (°). The comma represents the covariant derivative. The definition of this class of mixed boundary value problems is completed by prescribing surface tractions ii on gagion 31- while surface displacements'ii are prescribed on region 82. The surface region 33 has a special constraint imposed upon it which models the kinematic action of a sliding joint. I 172 l and this constraint appears under the final integral in equation ( 2.1.1 ). where 0 is the angle between the adjacent links at the joint. The constraint requires the direction of the resultant deformation vector at the joint to always be coincident with the axis of the guide. Hence. for example, the deformation components of the axial and flexural displacement fields must combine in the prescribed manner to ensure this. The objective that follows is to develop from first principles a displacement finite element formulation for the analysis of the nonlinear vibrational behavior of linkages fabricated with straight slender flexible beam-shaped members. This is accomplished by first assuming that the flexural deformation field being modeled by the 32 element is governed by the classical Bernoulli-Euler hypothesis. The amount of detail sought by different analyses has resulted in a large number of theories being developed for predicting the nonlinear vibrational response of beam systems and many are cited in I 136 1. One of the principal reasons for this large variety of theories is due to the different assumptions employed to describe the elastic geometrical nonlinearity expression. which typically couples the axial and flexural modes of deformation. Other simplifying assumptions are also generally introduced into the equations of motion to make them mathematically tractable, or else reduce the computing effort. by neglecting higher-order terms. The presentation that follows is no exception. The objective, herein, is to develop a displacement formulation for a single one-dimensional finite element with two exterior nodes. each having three nodal degrees of freedom. Tho nodal variables M and 0 describe the flexural displacement and slope respectively, while U describes the longitudinal displacement. Defining the nodal displacement vector for the element by [UJT - to. o, v, v, o. a.) (2.1.2) then the general displacements u(x.t) and w(x.t) at any point in the element may be related to [U] by 33 [u vlT - [NJIU] (2.1.3) where [N] contains the shape functions. The axial displacement is defined by u(x.t) - [Nina] + [:1/2tuflmnlflmnll(an. (2.1.4) where [N1] is the shape function for the linear terms. [anl the shape function associated with the nonlinear terms, and (') denotes spatial differentiation with respect to x. This notation has been adopted to clarify the linear and nonlinear terms in the subsequent mathematical development. The transpose is denoted by T. A more familiar formulation of equation ( 2.1.4 ) may be found in many works on nonlinear vibrations, such as I 34 l, and this is “x . u. - zdw/dx + [fl/2(3w/3x)'dx (2.1.5) where u. is the axial displacement. w(x.t) the transverse displacement. and x is the longitudinal spatial variable. The axial strain corresponding to ( 2.1.4 ) may be written as the sum of the linear and nonlinear terms. Thus. [7,,1 - [1“]1 + [1,,ln1 (2.1.6) 34 01' [1,,1 -= [Elna] + 1/2w1Tmnllenlnm (2.1.7) which may be amplified to explicitly define [N1] and [Na]. 11,,1 = I an. (,2. -.N,,. -.N,,. -.t(,,. -le5 1 I (1,. 0,. v,. v,. 0,. 9,1T + 1/21 0,. u,. (1,. 1),. 9,. 9,1 I 0- 0- "n13o I31.114. Nn15' I“.116 IT I 0- 0' “1113' I“1114' 1“1115- *an 1’ [0,. (1,. v,. v,. 9,. 9, 1" (2.1.8) Assuming that the constitutive equations have a very small dependence upon the rate of loading of the material. then the variational equation of motion ( 2.1.1 ) may be written as 35 5’ ' ° ‘ III [I.za.,,)r<:.,,1—.c1:.,,) - (c11[§,,1)av - Iv plaplT((PJ-(N,1[pRJ-[Nth))av - Iv5°i(xi*°ij.J'“i.kj°jk+“i.k‘jk,j'951)dv + j.ts.,,1T( [7,,1-(N11101 - 1/2IUJTIN,,1T [anjtu] )dv + IntauthNfl til-(.J-(mmm )ds, - js,( (5.1T+ta.)TIN1(01+111TINJ(an) )( [U] -IN]IU] )as, + ISn‘ (5,,1T+Is.1T(m(n,1+t,1Tmmu,1 ) ( (U,1.1n¢—(u,)co.¢ )lcosG as. ]dt (2.1.9) where [C] is the constitutive matrix. [c,] is a damping matrix providing stresses proportional to the strain rate. [NR] contains the ahape functions approximating the rigid-body kinematics (to be described later) while [PR] is the column vector containing the nodal rigid body kinematic degrees of freedom. The nodal absolute velocity components are defined by [P] and the surface tractions by [g]. In addition. the terms [0:] and [0:] are defined by IU,JT=[U, u, o o o 0] (2.1.10) and (ule- [ o o v, v, e, 9,] (2.1.11) 36 Finally. because the rigid-body equations of motion are of no consequence here. they may be removed from the formulation by taking variations [broil and [60,] to be zero. In order to complete the transformation of the variational theorem from an expression written in a Cartesian tensor notation to a matrix relationship appropriate for a finite element analysis, attention is focused upon the equations of equilibrium obtained from the second term in equation (2.1.9). The second term may be written IoISUITINITalcth/axdv (2.1.12) An integration-by-parts over x yields [IAI5UJT["]T[°uldA], - IvlbanlNlTlouldv (2.1.13) The first term in equation (2.1.13) is a boundary term. to be evaluated on the cross-sectional area A. which will be manipulated later. For a homogeneous, isotropic material. the second term becomes. using the first and fourth integral expressions in equation (2.1.9). _ Ivloul'l'mll'rtNlleldv - IvIWlTZINllT IUITINnIITINnIITIUIE/Zdv 37 - [VI/41501711,,1T11,,1(u1mm“;In“... -IVIwJT(N,1T(c,1(Nnb]av ‘ IVI’HWJTWITmn1W...)(c.Itmwlav (2.1.1.) The first term in equation (2.1.14) yields the standard linear stiffness matrix [[1] defined by [3.1 - IvtNllTamlldv (2.1.15) The second term in (2.1.14) defines [[3] which 1. one of several stiffness matrices incorporating elastic geometrical nonlinearity terms and is explicitly defined by [r,) . If [FxlllfinllTIanldx (2.1.16) where L is the length of the finite element and [Fxli 1. the linaag component of the axial force defined by [Ex]l - EAINIJIUx] (2.1.17) since the antisymmetric terms in z disappear upon integration. The third term in equation (2.1.14) is neglected. being of fourth 38 order, while the fourth term is linearized to give a damping matrix [Di] ' IVINIITICIIINlldV‘ (2.1.18) The fifth term is also neglected because of higher-order derivatives. Returning to the third integral expression in equation (2.1.9). the third and fourth terms may be combined as the derivative of a product. and upon subjecting this expression to Gauss'thecrem they yield JVbni(“i,k°kj).jdv ' [IA ‘jk‘jni,k6“idA]a ’ Iv°kjni,35n1,jdv (2.1.19) vhere “j is the unit-vector normal to the surface. The first term on the right-hand-side of (2.1.19) is a boundary term to be considered later. while the second term may be written $150111 (N,)+1/2(N,,JT(N,,(m Pu") -[ (N,J+1/2(01T(N,1)T(N,,J ][(n¢1v (2.1.20) 0903 substituting for [6:31 from the first and fourth integrals in equation (2.1.9). only the following three lower-order terms are retained from the twelve terms obtained from the expansion of (2.1.20) 39 -I801T(x,1 [DI-ISUJTIKJ[DI-[601110,][In (2.1.21) Stiffness matrix [Kg]. depends upon the linear axial force and is defined by It.) - filFxlllNllrlNlldx (2.1.22) Matrix [1.]. however. contains terms modeling the geometrically nonlinear elastic behavior and is defined by ll.) - 2ft 11213,),wJTIN11(Nuleumx (2.1.23) Matrix [0.] is a damping matrix. associated with the nonlinear behavior and is defined by ID.) - I} (2,1,11,1T1q1m113. (2.1.24) This completes the operations on the elastic terms governing the deformation throughout the volume V and attention is now focused on the boundary terms modeling surface tractions. The unit vectors on the ends of a one-dimensional element with plane faces. described relative to a Cartesian reference frame, are orthogonal. Consequently. upon incorporating the direction cosines into the fifth integral expression in equation (2.1.9) and adding it to the first term in equation (2.1.13) 40 and also the boundary term in equation (2.1.19). terms cancel to yield Is: [SUITINITIIIdS, (2.1.25) The manipulations of the elastic terms are now complete and attention is focused on the dynamic behavior of a link which is governed by the second integral expression in equation (2.1.9). This expression must be differentiated with respect to time and substituted into the final term in the second integral in equation (2.1.9). Instead of adopting this direct approach. an alternative method will be employed I 172 l. in an effort to better demonstrate the parallels between classical methodology and a numerically-orientated formulation. Fundamental kinematical theory states that the absolute acceleration of one point. say B, on the link is the acceleration of point P relative to point B. Confining attention to the axial (ox) and flexural (oz) deformations u and w. respectively, the following statement may be readily developed I 172 1: Pp‘ - Bx + H + Uw + 20: - O'(r+u) Pp, - I3, + = - um) - 231'. - 8’. (2.1.26) where r is the length of the position vector defining the position of point P. relative to B in the undeformed configuration. and O and O are the angular velocity and acceleration of the link. It is assumed that 41 ’33 and PB' may be approximated by classical rigid-body kinematic terms alone. but a more refined approach may be developed I 172 l where additional degrees of freedom are used to model the system. The absolute rigid-body acceleration of point B may be written r ’33: ‘ f Bx-t8' W [In] - - -[Nglligl (2.1.27) L9RD: ’ k fiBz". 7 '90:. [N3] contains the shape functions modeling the linear distribution of the absolute rigid-body acceleration within the finite element and is the vector of nodal degrees of freedom. If, by using equations (2.1.10) and (2.1.11), equation (2.1.3) is rewritten as [u] 8 [ux] - "x 0 .[Ux then equation (2.1.26) may be reformulated as r0 N 1 r-"Nx sz~ [I] - [Nnnigl + [NJIU] + 20 [In In] ~ -N o. L-DN -0'NzJ (2.1.29) Substituting (2.1.29) into the final term in the third integral in equation (2.1.9) yields the expression -(su1T(u,1(fi,1 - (suthulth - IOUITIMOORIIU] -(501T(ulc)(u1 (2.1.30) where [lR] is the mass matrix associated with the rigid-body motion and is defined by [.3] '3 ijINITINRIOV (2.1.31) the second matrix. [M]. is defined by 43 [u] - I§ptNJTINlav (2.1.32) the third matrix. [HCORI' is associated with the Coriolis acceleration. and is defined by (“coal = Ivsz av (2.1.33) and the final matrix. [MIC], describes the inertial coupling between the gross rigid-body motion of the mechanism and the elastic deformation kinematic terms. It is defined by . N o 3 T (' -O’Nx -UN 1 [“1c1 . 1&9 av (2.1.34) L 0 NZ’ 3 "Nx “‘3N1J The variational equation of motion may now be written in the final form a: - o - {fl jvtsynl'fi [uni-Ely") ~(c,n;,,) )av 44 - jvptsplT( [Pl-[NRIIPRI-INIIU] )av + (501T( ( It,l+ls,]+[x,l+[r,l+lu1cl )[01 +( (n,1+(n,1+(uc031 )(b)+(u1(n1 - IVINITIXIdV - [alumna-(awful ) (2.1.35) + j§(3a,,JT( (1,,1-(N111u1 - 1/2[U]T[Nn1)T (N,,1(u: )av - f3, ( (3.1T4taalT1N)(01+111TINIISUI )( [U] -IN]IU] )as, + j,,( (3,,1T+(3.1T1N1(u,1+(.1T(N1(50,) ) ( IletanO-IUz] )as, ]a: Equation (2.1.35) contains the field equations and displacement boundary conditions for one finite element and these governing equations may be obtained by taking arbitrary independent variations of the variables in this variational equation of motion. The resulting matrix formulation must be pre- and post-multiplied by standard transformation matrices in order that this general statement he used to develop a finite element model of a specific linkage. The matrices formulating the equations of motion, which are contained in the third term of equation (2.1.35), are presented explicitly in the Appendix of reference I 180 ]. Having established the theory and the finite element formulation which provide a numerical scheme to theoretically predict the dynamic response of the high-speed flexible linkage mechanisms. an experimental 45 study is necessary in order to prove the applicability of the theory. The experimental set-up and the procedures will be described in the following section. 2.2 Experimental Study In the field of research dedicated to flexible mechanism systems there have been a number of combined theoretical and experimental studies. which include references I 51.60.82.137.154,l62,171.184.185 ]. Upon reviewing these publications and others, it is evident that most investigators have focused upon four-bar linkages and only a small number of papers have been dedicated to studying the flexural response of slider crank mechanisms incorporating bearings without clearance. Furthermore, there have been no combined experimental and finite element publications on flexible slider crank mechanisms. even though this is a very common linkage in industry. Herein, the results of a comprehensive experimental study on the dynamic response of slider-crank linkages configured with the plane of the mechanism perpendicular to the gravitational field, and also four-bar linkages with the plane of the mechanism colinear with the gravitational field are presented. These kinematic chains were constructed with several link-length ratios and different link cross-sectional dimensions, and the systems operated over a wide-range of speed to generate a large variety of response histories for evaluating the predictive capabilities of the mathematical model developed above. 46 2.2.1 Experimental apparatus: Four-bar linkages A photograph of the experimental four-bar linkage apparatus used in this study is presented in Figure (2.2) in page 47, from which it is clear that it incorporated two flexible members. the coupler and the rocker linkages. This apparatus was designed to accommodate a wide range of link lengths for the coupler, rocker and ground links. but the crank was always held constant at 63.5mm (2.5 inches). This member was assumed to be rigid since it was manufactured from steel bar stock of cross-sectional dimensions 25.4mm x 25.4mm (1 in. x l in.). The depth of all of the flexible members used in this investigation was less than 25.4mm (l in.) perpendicular to the plane of mechanism. and the thickness of the links in the plane of the mechanism was always less than 2.54mm (0.1 in.). These dimensions are clearly not representive of an industrial mechanism, but they were chosen in order to accentuate the flexural deformations in the plane of articulation. thereby creating the large signal-to-noise ratios desired by all experimentalists in all fields of scientific research. 47 H1111 ninmmuxpummmmuw Figure 2.2 Experimental Four-Bar Linkage Mechanism Naturally, the same signal-to-noise ratios could have been achieved using realistically proportioned links operating at much higher speeds. but this presents additional data-equisition complexities when measuring the response of the links. and in addition. the response data may well be contaminated by accentuated elastodynamic effects at these higher speeds. due to linkage out-of-belance and other more subtle phenomena caused by manufacturing errors. It is hypothesized that if mathematical models are capable of predicting the elastodynamic response of linkages fabricated with slender links operating at several hundred revolutions per minute, then they can also predict the response of industrial, realistically proportioned mechanisms with the same stiffness-tc-lceding ratios operating at higher speeds, because the two response histories are 48 fundamentally governed by the same equations of motion. The coupler and rocker links of the experimental four-bar linkage apparatus were both manufactured 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 (2.2) on page 47. 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. Identical aluminum bearing housings were manufactured for the rocker link where it was retained on the ground-link/rocker 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 was 38.1 mm (1.5 in.) 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 0.05 kg (0.11 lbm). The coupler and rocker links were supported on matched pairs of 6.4 mm (0.25 in.)‘ bore R3 DB R12 instrument ball bearings supplied by FAG Bearings Limited. Each bearing housing in the mechanism was preloaded using a Dresser Industries torque limiting screw driver calibrated to t0.1l3 Nm (*1 in-lbf) which permitted bearing clearance to be eliminated since the impactive loading associated with bearing clearances would have resulted in larger link deflections. Conversely. if the bearings 49 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 characterize co-planar flexible linkages. This was accomplished using a cleavage bearing design that is clearly visible in the top center of Figure (2.2). The housings were fabricated in an aluminum alloy and longitudinal dimension of the housing comprising part of the coupler was 38.1 mm (1.5 in.) from the bearing centerline 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 (0.114 lbm). The cleavaged component of the coupler-rocker joint was bolted to the rocker link. The axial dimension of this part from the bearing centerline was 44.5 mm (1.75 in) and the mass was 0.063 kg (0.138 lbm) 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 15.9 mm (0.625 in) diameter shaft supported by Fafnir pillow block bearings. A 100 mm (4 in) diameter flywheel 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. 50 2.2.2 Experimental apparatus: Slider-crank mechanism The experimental slider-crank linkage used in this study was also bolted to a test stand and a photograph of the apparatus is presented in Figure (2.3) on page 51. The length of the steel connecting rod was 292 mm (11.5 in) and the length of the crank, which was part of the flywheel, was 50 mm (2 in). The slider was a Micro Slides. Inc. type 2050-RM-118-5 linear crossed-roller slide assembly and this precision table was without hearing clearance. A photograph of this arrangement is presented in Figure (2.4) on page 51. A non-rotating spindle was fitted to the translating portion of the slide assembly and this supported the gudgeon pin bearings which were a matched pair of 6.35 mm (0.25 in) R4 DB 12 instrument bearings supplied by FAG Bearings Limited. A similar set of hearings were also used at the crank pin, and both bearing assemblies were preloaded using the technique described earlier. Identical aluminum bearing housings were manufactured for the crank pin and gudgeon pin joints. These subassemblies each had a mass of 0.045 kg (0.098 lbm) including the mass of the socket screws and the clamping plate. The axial dimension of the housing from the bearing centerline to the extremity adjacent to the steel specimen was 31.75 mm (1.25 in). 51 Figure 2.3 Experimental Slider-Crank Mechanism Figure 2.4 Slider Assembly for Experimental Slider-Crank Mechanism 52 The mechanism was again powered by 0.75 h.p. Dayton variable speed d.c. motor through a 19.05 mm (0.75 in) diameter shaft supported on a pair of TImken tapered roller bearings type TS4Ar6. The action of the motor's speed controller was again augmented by a flywheel of 146.05 mm (5.75 in) diameter. Both of the experimental systems were covered by a safety enclosure fabricated from transparent Lexan sheeting of thickness 4.76 mm (0.1875 in). 2.2.3 Instrumentation A schematic diagram of the instrumentation used in both experimental investigations is presented in Figure (2.5) on page 53. The rated speed of the electric motor was 2500 rpm and this was directly measured in revolutions per minute by a Hewlett Packard 5314A universal counter which was activated by an electro-magnetic pickup. model 58423. 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 both Figures (2.2) and (2.3). The aforementioned arrangement provided visual feedback to the operator when the speed controller of the motor was being adjusted to establish a desired speed. The flexural deflections at the midspan of each link specimen were monitored by Micromeasurements Groups Inc. type EArO6-l25AD-l20 strain gages bonded to each link at the midspan location. Bending half-bridge configurations were adopted, using one gage on each side of the specimen, and they were used in conjunction with a Micromeasurements Group Inc.. strain gage conditioner/amplifier system. type 2100. 53 Visual monitoring of crank speec (rpr) __ HP 53l4A universal counter A Electro-Corp. 12 volts. pickup D.C. ) Links Mechanism 60 tooth D.C. strain gaged spur gear motor A v Air Pax speed pickup controller ) l2 volts D.C. zero crank-angle configuration DEC 15 3' DPD ll/03 pu e v microcomputer l Micromeasurements system 2l00 Wheatstone bridge and amplifier system I low-pass dynamic strains '7 analog filter post-processed experimental results I Figure 2.5 Schematic of Experimental Apparatus and Instrumentation. 54 In order to relate the strain gage signal to the configuration of the particular mechanism being studied. a third transducer arrangement was established. An Airpex type 14-0001 zero velocity digital pickup was employed to sense the bolt head at the end of the crank when the mechanism was in the conventional zero-degree crank angle position. This long hexagonal transducer is clearly visible in both Figures (2.2) and (2.3). This mechanism configuration signal and the output from the gages were fed to a Digital Equipment Corporation PDP 11/03 microcomputer with a L81 11/23 processor which is presented in Figure (2.6). Figure 2.6 The Digital Date-Acquisition System in the Machinery Elastodynamic Laboratory 55 This digital-data-acquisition system featured 256 kB of memory for post-processing data and also two 5 m8 hard disks for storage. A dual port floppy disk system was also available. The experimental response curves were displayed on a Digital Equipment Corporation VTIOO terminal with retrographics enhancement. The BNC cables from each experimental apparatus were connected to an input-output module, bolted to the cabinet of the computer, which was built by the Electronic and Computer Services Department at MSU. This instrument featured 16 anaIOg-digital channels. 4 digital—analog channels and two Schmidt triggers. Using software developed specifically for digital data acquisition purposes. the flexural response signal was recorded from the zero crank angle position through 360 degrees by firing one of the Schmidt triggers. In order to activate the trigger, a 4 uF capacitor was used to modify the square-wave output from the Airpax pickup. Experimental results were obtained by implementing the following procedure. The signal from the strain gage instrumentation was passed through a first-order analog low-pass filter with a cut-off frequency of 160 Hz in order to remove the electrically or mechanically induced noise prior to being digitized by the analog to digital converter and recorded by the PDP 11/03 microcomputer. This low-pass filter also prevented aliasing problems when employing a rectangular data window, and a 3‘lplin8 rate of (2578)" seconds, which was considered to be at least twice the highest frequency component of the pure elastodynamic response signal I 35.97.128.135 1 56 The data was then post-processed by first multiplying the digitized response by a strain-deflection calibration factor for each link specimen. These factors were obtained by supporting each specimen at the ends on knife edges in a calibration fixture. prior to subjecting the midspan to a series of known monotonically increasing transverse deflections which were imposed and also measured by a micrometer attachment on the fixture. The bending strain corresponding to each midspan deflection was then recorded. The second post-processing operation involved using a fast fourier transform (FFT) algorithm to convert the time-domain signal to the frequency domain and this was motivated by the need to reinforce the operation of the analog filter by removing induced noise from the strain-gage signal. Using an FFT algorithm. the frequency spectrum of the signal was constructed to determine the frequency range and amplitude of the noise relative to the desired signal. This noise was then removed by simulating a digital low-pass filter prior to transforming the modified date back into the time-domain for presentation of the response on a graphics terminal. Figures (2.7). (2.8). and (2.9) demonstrate the aforementioned post-processing operations. The experimental results are for the midspan transverse deflection of a four bar linkage with a rigid crank length of 63.5mm (2.5 in), a ground link length of 406.4mm (16 in) and the coupler and rocker links were both 304.8mm (12 in) long with a thickenss in the plane of the mechanism of 1.575mm (0.062 in) and a width perpendicular to the mechanism of 19.05mm (0.75 in). The crank 57 manganese _.anamn shaman can emu a. momuoouuo: oumo>amanh sumac“: moxoom "unnamed nominee“ p.~ ounmmu GHERE 522 xzsa 5 (W) (011331130 BSUPASIMII NVdSGIH 58 manna“: women: no amuuuemm monemvoem m? can can an nemuoouuoa oueoeenemh NE hbbpp ”omwxcmq unalenou .NI. >uzm30wmm N s“ :_.b _ _ b a.~ unseen DUJUHCJLLJJ Scenes—am museums “one. can «we so common—won oauoeamauh c2732 non—com "omexnmq men—(unch— QN can»: SEEDS 592 xzéu 'l d e e d M O M I 6.3 (W) IDIDMU NEWWI lIVdSt'IIl! 6O frequency was 254 rpm. Figure (2.7) presents the experimental data following analog filtering. digitization and post-processing for converting the strain gage signal to midspan deflections. The noise content of the signal is clearly evident. Figure (2.8) presents the frequency spectrum and bandwidth of the same signal presented in Figure (2.7). The natural frequency of the rocker link in a simply-supported beam configuration was experimentally measured to be 41.1 Hz and this response is evident in Figure (2.9). This frequency-response data also indicates mains noise at 60 Hz and higher harmonics. The frequency content above 160 Hz is attributed to roll-off in the circuitry of the analog filter. Using an FFT software package the data was then subjected to digital low-pass filtering with a cut-off frequency of 55 Hz to yield the response presented in Figure (2.9). 2.3 Computer simulations Multi-element general purpose finite element programs were written for both of the experimental mechanisms described previously and each code had the versatility of permitting the analyst to arbitrarily specify the number of elements to be employed for modeling each link. A parametric study revealed that a six element model of each link provided adequate convergence capabilities when comparing the computer generated results with the experimental results. The steel specimens were assumed to have a Young's modulus. E. of 201 MPa (30 x 10‘ inf/11') and a density of s310 kglm' (0.3 lbf/in'). 61 The bearing housings, however. were assumed to posses the elastic characteristics of aluminum (Young's modulus of 73.14 MPa (10.6 x 10‘ lbflin') but the density of the assembly was obtained using the mass and dimensional data. The cleavaged sub-assembly of the coupler-rocker joint, which is part of the coupler, was assumed to have a constant second moment of area of 18272.56 mm‘ (0.0439in‘) while the mating assembly, which is part of the rocker link. had a second moment of area of 1194.58 mm‘ (0.00287 in‘). The magnitude of the reciprocating table mass of the slide-crank mechanism, which was required as a input parameter for the simulations. was obtained by configuring the table so that it reciprocated in the vertical direction, prior to resting the reciprocating portion on a laboratory weighing balance while holding the normally fixed portion of the table. The laboratory balance was then activated and the instrument employed to weigh the total mass of the cast-iron table top. the steel spindle and the nuts of the gudgeon-pin assembly. The resulting value was 0.35 kg (0.77 lbm). The complex phenomenon of system damping was treated in an approximate manner. using the philosophy adopted in references I 154,185 I. Namely, transient response studies and logarithmic decrement calculations were undertaken for each mechanism in a large number of different system configurations. An average value of the damping ratio was then determined and then distributed uniformly throughout all of the finite elements modeling the particular‘ mechanism under investigation. A mean value of 0.029 was employed to model the 62 slider crank mechanism and 0.031 for the four bar linkage. The global equations of motion for the two experimental mechanisms were develcped using the local element equations of motion obtained from the variational equation of motion (2.1.35). by taking variations of IbUlT. The model of the four-bar linkage incorporated gravitation loading using IX] and surface tractions I?) were taken to be zero. Figure (2.10) presents the model of the four-bar linkage system and indicates the global degrees of freedom. In this formulation. region S, in the variational equation of motion. was assumed to be zero. l/Q\ \ j Figure 2.10 Finite Element Model of Experimental Four-Bar Linkage 63 This was not the case for the slider-crank model. where region S, was employed to incorporate the rectilinear kinematic constraint at the gudgeon pin. By utilizing this condition, the transverse deformation was replaced by a function of the axial deformation at the end of the connecting rod thereby reducing the nodal degrees of freedom from three to two. The inertial loading imposed upon the flexible link at the gudgeon pin by the reciprocating table was incorporated into the finite element model as a surface traction [3:] in the axial digaction, This was defined by the approximation. (3,) s .(i,, + i,,(.n(—o)1 (2.3.1) where m is the total mass of the reciprocating portion of the sliding table and the bearing assembly, and prx and If: .r. the .b'olnt‘ rigid-body accelerations in the axial and transverse directions respectively at the gudgeon pin. The angle 0 denotes the angle between the centerline of the reciprocating table and the undeformed configuration of the connecting rod. Upon formulating the global equations of motion, these equations were then solved by the Newmark method of direct integration using a step-size governed by the highest frequency anticipated in the response. which in this case was dictated by the axial mode. Several of the matrices modeling nonlinear terms contain an axial force component. For an analysis at time step tn the axial loading at the previous time step, tn-l' gag agployad in the simulation algorithm in order to effectively model this dynamical term. 64 2.4 Results and Discussion Figures (2.11)-(2.14) on pages 65-68 present the results of combined experimental and analytical investigations on a flexible four-bar linkage with a ground link length of 406.4mm (16 in). a crank length of 63.5mm (2.5 in) and both the coupler and rocker links were 304.8mm (12 in) long. The depth of both flexible links in the plane of the mechanism was 1.4mm (0.055 in) and the width perpendicular to the plane of the mechanism was 19 .05- (0.75 in). The cperating speed of the mechanism and also the time-step employed in the simulations are indicated in the legend for each figure. It is evident from an evaluation of the experimental data and the results of the simulation that the mathematically nonlinear formulation has a very favorable predictive cpability for both frequency content and also amplitude response of the strain-gage signals over the experimental speed range fromil93 rpm to 342 rpm. Mhile there are indeed portions of the response where the analytical and experimental results diverge. when this discrepancy is viewed in the context of a peak deflection of only 1.5mm in a very flexible linkage then the results are certainly impressive. Figure (2.15) on page 69 presents the results of a combined analytical and experimental study of a four-bar linkage with different 65 unecoom nmnwvooo.o mommiosmh acmuaumouou .sme men an memuueuuon condemn-uh oenumma ecunnou Homeucmg unmimmom 85%ro 592 View Sm EN 03 8 Hu.~ panama .ZCEEMEVQ \ .. 45. p55,?) (.0 —( I (W) IOIDZ’BBC HENRI/III lh’dSCll)‘ 66 «vacuum ennemooo.o moumloluh momumumoucn .anu mad an unmade—won ounce-meek memupma uo—mmou unmanned meniemou 85:85 59: .4255 8m SH ue.u seamen (1 (a I 4 d e d 4 \l 2:32:35 . (2)1) 8 ._S_ 52% ( D 0*“) 9011331230 BS‘dI-II‘SIWI NVdSUIk' 67 ammooom hammococ.o noumioauh coma-umcumn .anu vnN an nemuoo~uoa manoenmeuh magnum: nouuoz "eunucmq mamiunom nu.« oummam 8“ are? $2 V36 0 a u . q a O.MI Wm . fl 5552/ w a . , \ (om K k m . , . 3.92% . m V ( . m 68 ammooem Homnwcoo.o moumioamh modueumoumn .amu noa um memuooauo: oaueeanceh menace: mexooz “omeumma mamlunou Va." enema“ GHEQBV 592 2255 1 e d e d a a {I d I 4 o. N l A I :3 (W) NOIDH'LBG PRBNSWdl WdSGIU ._Ezw.=zw5 o.~ .eaoaom epvpmooc.c aunm-oase aceuauaosem .aau can S. menace—mo: oaooaaneuh menace: undue: "oueucmq mamiumom ma.u eonuum Ammmzemov msoz< xzm<:<((\\\ . 70 cross-sectional dimensions and link lengths to the system previously investigated, thereby providing a totally different set of test conditions for evaluating the nonlinear formultion developed in I 180 I. The crank length of the coupler was 228.6mm (9 in) and the length of the rocker was 304.8mm (12 in). Furthermore. the depth of the coupler and rocker links in the plane of the mechanism were 1.17mm (0.046 in) and 1.57mm (0.062 in) respectively. while the width of these links in the plane perpendicular to the plane of the mechanism were 25.4mm (1 in) and 19.05mm (0.75 in) respectively. . Figure (2.15) again demonstrates a favorable comparison between the experimental and theoretical responses since the maximum amplitude deviation between the two waveforms is 0.2mm and the frequency correlation is quite good. Figures (2.16) and (2.17) on pages 71 and 72 present the results of combined experimental and analytical studies of the slider crank apparatus photographed in Figure (2.4). Both systems cperated with a constant crank length of 50mm (2 in) and a connecting rod of length 292m (11.5 in). The depth of the steel connecting rod in the plane of the mechanism was 1.4mm (0.055 in) and the width perpendicular to the plane of the mechanism was 3.05m) (0.75 in). Correlation is again very favorable between the two responses considering the maximum deflections are both less than one millimeter. All of the response data presented in Figures (2.11)-(2.l7) were obtained by modeling the mechanisms using the complete formulation developed in reference I 180 ] as part of a mathematical modeling exercise. However. in industry. the designer is often content with 71 avenue» «Nachooo.o neamlolah acmueumeao— .ane nnu an nouuoe—uoc oauoeunenh menace: vomiumuuoonmcu "summoned: acquaiuoum~m ma." enough 35:38 50.2 x55 . 8H o C? 'T‘ (W) (1011331333 ESEB’SIIVSI (kid: IN c4 72 aueouom paaoeooo.o nonmioaeh coma-muoumm .snu vhn an common—mo: euuo>umemh manna“: mouiummuoomnou "mumcemooz menaciuommum mm.“ omnmmm Rammeomov m4c=< x=433 ”2.55.... “use! me. . :l|.: . i- . III— .2~\x» ~85 ~Q\\~\fi 85 second set of tests were initiated to study the creep response of the materials. The results are presented in Figures (3.5) and (3.6) on pages 86 and 87. Time is the abscissa for both plots and the pen on the x-y plotter was programmed for the different speeds indicated. In addition, the pen was programmed to quickly return to the left band edge of the paper in the stand-off mode upon reaching the limit of travel at the right hand edge. This procedure permitted each creep response to be presented in one compact figure rather than several feet of paper from a strip recorder. The response presented in Figure (3.5) verifies that the unidirectional laminate is an elastic material. while the creep data in Figure (3.6) indicates that the It451, composite is viscoelastic. Having completed the dynamic and creep testing of the laminates. the response data presented in Figures (3.3)-(3.6) provides the basis for a number of different investigations. In the context of this paper the effective modulus of the unidirectional laminate can be readily calculated from Figure (3.3). The It45I‘ laminate has a more complex response because the stress-strain history is not linear. and moreover, the gradient depends upon the strain rate. Clearly a wide range of different Young's moduli could be deduced from the experimental data depending upon the assumptions deemed to be relevant by the investigator. Because a link of any mechanism experiences rapidly fluctuating stress levels. the response curve on the extreme left of Figure (3.3) was selected to provide the basis for the Young's modulus calculation because this records the highest strain rate of any 86 oncoming wlnonnxvim< maneuaooummmea ecu announce moonu n.m canned - L... .- -252 «000.0 I. I I I III . T O I I s I I- I. VIII II I I al I e I- III 7 I- 0 II- II. I ' IIILf I IIIIIIII II a. I It I I .s I '3‘. IIIII I IIIIIIII O OIs.II‘ III I II III I. I I a I. .w. 1 . m - I)- --i).l.)) - ) I'M .. u . . -.-o (I) (In) I-.. - -ol ( III (III!) (Ii—(iii) I-.. (I) L I... — — _ . . _. .. . . _ . . . m . m . . q I) . . . . . 1.).) ._ ...._. m u w . m u n . . . u . _ . . . u . — . . . . _ I-..“ m v H . )-)--)i|l.)|ll||.|m).!):- .. . . .- ---)l (I) - .I ( o -I- (lull)! I)- u I. 7 as . 2.9a... e....e-.. m III: a . t p . Is .1 Intel 0 . if. . . . SEE: A. . I.IPI7I.IPIIIIIII.IIII.III (1|-..I I ..II .ltIIIL (II. I. .... . . 2. null III 0 .... H 3-33-83 3595: 35:5.— »xoau-u:.E2 .... f .... . 1. Jr! .— .. fi- _. L r. .... fl Ilgi; . p.....~. “HI . :7 ...x... . w... n.” we; mew- S... H. C ..«m r. m 5.59:5 mow > 1 Ea.\ 5252K < 4 - mé. _.s. _.s mé I-4~Iwbwcrm 96 anemone—won memumom canoe“: moxuom .mounmua. wnoz< uzunu scam mocmamao nuances: commuNuuom amoumw oooH cow coo _uom_ ooq Mc‘?r ooN v.v mummma TIj I Ij ‘1”1 1 Omom CONN ommm comm Omwm comm S S 3 H l S 109 where G(t) 9 0 for t 9 O and 0(t) is continuous for 0 1 t <.. Assuming further that G(t)-0 for t<0. then equation (4.2.7) may be shown to satisfy the commutivity. associativity and distributivity properties of Stieltjes computations [ 62.63.64 I. 4.3 Variational Principle The same methodology is adopted as the variational development in the previous chapters. In this chapter. the variational principle incorporated a variety of auxilary conditions such as the constitutive equation. the strain-displacement equation and the geometrical boundary conditions. Hamilton's Principle is presented using the Stieltjes convolution integral notation. The dynamical problem of a viscoelastic body of volume V and surface area S describing a general spatial motion relative to an inertial frame OXYZ is considered. In Figure (2.1) page 30. oxyz are Lagrangian coordinates fixed in the body in a reference state containing zero deformations. strains and stresses. Furthermore. it is also assumed that these parameters have been zero throughout the previous time history. Employing a Cartesian tensor notation. at time t. a general point P in the continuum has the position vector r1 = roi + rRi + 111 (‘.3.1) where ’oi is the position vector of the origin of the body axes relative to the origin of the inertial frame. rRi rgprgggntg the pagition voctor 110 of point P in the undeformed reference state relative to the origin of thO body 3803 and “i is the deformation displacement vector. Equation (4.3.1) may be differentiated with respect to time to yield the velocity rate-of-change of displacement expression Pi ‘ E01 + 31 + °ijkaj(roi + ‘Rk + uk) (4.3.2) where Pi is the absolute velocity associated with ri. In addition. °ijk 18 thO alternating tensor. 01 is the angular velocity defining the rotation of the Lagrangian frame oxyz. (”) represents the time rate-of-change with respect to the moving frame oxyz and (°) represents the absolute rate-of-change. In order to provide the basis for analyzing flexible linkage mechanisms fabricated with viscoelastic materials. the stationary conditions are sought for the functional F. ‘ Iv [l/ZOiiji.de + Xi‘dri ' 1/2611k1‘drij‘drkl ] dV + Isl'g’isdqu. (4.3.3) subject to the constraints imposed by equations (4.2.6) and (4.3.2). and also the following field equation Vii = 1/2(“1.j + “5.1) (4.3.4) 111 which is the linear strain-displacement relationship. In these equations. a comma denotes spatial differentiation. 5ij is the Kronockar delta. 9 is the mass density of the material. g1 1; the gnxfgca trgction vector and the overbar (') denotes a prescribed quantity. The total surface area of the continuum is denoted by S which is thfi '“fll4t10fl 0f £02103! 3, and S3. Tractions are prescribed on 8,. 'hilO on 3;. the prescribed deformation displacement boundary condition Ta. = u. (4.3.5) is imposed. Equations (4.2.6). (4.3.2). (4.3.4) and (4.3.5) may be incorporated within functional F. using Lagrange multipliers to create a free variational problem defining a new functional F. The first variation is then generated using the standard rules of the variational calculus. and this procedure involves utilizing the divergence theorem and also the symmetric properties of the tensors where appropriate. Upon setting the first variation equal to zero. the Lagrange multipliers may be expressed in terms of the system parameters leading to the following variational equation of motion: 112 5F = 0 ' j§[ (oij-Gijkl‘dykl)‘dbyij *(oij.j+xi-pP1)‘d6ui -(1,j-1/2[u1,3+uj,il)'dsaij -(Pi-p[;°1+ai+eijk.j(rok+rlk+uk)I‘dSPi )]dv + [$1 [(gi-EiPdSuildS1 + Is,[(ui-ui)‘d6gi Ids, + [Iv xidv + ISIEids, - I§p§.dv]tderoi (4.3.6) +[)v °ijkxi"ok*‘ax*“k’dv * )3: 'ijkzj ‘tox*rsk+“k’dsa -1)! .ij kfiip (tok+rnk+nk)dv1.d5aj Independent arbitrary variations of the deformation displacement. strain. stress. absolute velocity. and the kinematic parameters defining the rigid-body motion enable equation (4.3.6) to yield. as Euler equations. the field equations and boundary conditions for this class of dynamic viscoelastic problem. This variational statement represents a generalization to the 'theory of viscoelasticity of the elastodynamic variational theorem presented in chapter 2. In fact if the body being subjected to the analysis were elastic. then the time integrations in equation (4.3.6) could be evaluated and the resulting theorem would distill to the formulation developed in chapter 2. The characteristic equations obtained from equation (4.3.6) precisely define the dynamic viscoelastic problem associated with the analysis of mechanisms fabricated with linear viscoelastic materials. 113 However an exact solution for ethese equations is beyond current mathematical means. and in any case would probably contain too much information to be useful to an industrial design engineer for solving practical problems. Simplifying assumptions are therefore needed. and _ these center upon which model to be adopted to represent the material constitutive relationship. equation (4.2.5). A variety of approaches have been develOped for this task. Numerical viscoelastic analyses have been undertaken by Laplace transform techniques [ 76 l. by numerical integration of the constitutive equations I 37 I. by step-by-step procedures used in conjunction with mechanical models for the constitutive equations I 204 I. by using complex-modulus forms of the relaxation functions [ 201 l. and also by finite-difference formulations [ 98.212 1. Thus. a wide variety of techniques have been proposed. but there is no consensus of opinion as to which is the best approach. In order to simplify this viscoelastic problem. a one-dimensional linear-solid model is assumed herein and a finite-difference approach is adopted I 98.212 1. The constitutive equation (4.2.5) is to be used in a numerical analysis scheme by changing the integration form into a summation of discretized time increments for obtaining the dynamic response of the mechanism systems. For example. the strain-rate §(:) in .qumtion (4.2.5) may be represented by a linear interpolation function over each increment: it.) - $(t-h) + (h-t+t/h)[;(t)-;(t-h)] (4.3.71 114 in which t’h S t‘ t and h represents the integration increments. Separating equation (4.2.3) into two parts. it can be shown that q(t) = exp(-h/p)q(t-h) + )tt-h epr-(t-r)/p];(t)dr (4.3.8) Then substituting equation (4.3.7) into equation (4.3.8) and explicitly carrying out the integration leads to q(t) = but-h) + 1,171.-..) + ifim (4.3.9) in which 1. = pI-B+u/h (1—511 (4.3.10) 1. = u(l-p/h (1-511 (4.3.11) 3 = exp(-h/u) (4.3.12) Equation (4.3.9) is a recurrence formula for the value of the present stress in terms of its value at the previous time step and the value of the strain rate at both the previous and the present time steps; it is used in conjunction with equation (4.2.2) to characterize a linear viscoelastic material. It is seen that this approach permits characterization of the hereditary nature of the material by retaining 115 strain history information at only the immediately proceeding time step. The appearance of both the relaxation moduli of the material. u. and the integration step size. h. in the parameters of the recurrence formula for q indicates an interaction between the mechanism system. the material. and the solution procedure. In general the step size is governed by the highest frequency among the natural frequencies of the system and the dominant forcing frequency. If the step size. h. is determined from the system considerations. an examination of the numerical behavior of the coefficients in equation (4.3w9) as a function of the material parameter. p. it is expected to provide some insight of the system behavior. For small values of the dimensionaless parameter h/u. utilizing (4.3.10). (4.3.11) and (4.3.12) 11 g 1‘ 3 h/2 (4.3.13) 3.. a 11 (4.3.14) In the former case the behavior is primarily that of an elastic element. in the latter that of a viscous element. 116 4.4 Finite Element Formulation The variational equation of motion (4.3.6) may be employed as the basis for a variety of finite‘ element models depending upon the geometrical shape of the body being analyzed. the type of deformation theory assumed to be appropriate. the information sought from the analysis. and the accuracy of the model for the constitutive equations. Herein. a displacement finite element model is developed for analyzing the flexural response of the beam-shaped links of planar linkage mechanisms deforming in the plane of the mechanism. This is accomplished by first assuming that the flexural deformation field is governed by the classical Euler-Bernoulli hypothesis. The constitutive equation is obtained by material testing. and the deformation of the link is assumed to be so small that it is not necessary to model geometrically nonlinear deformations. The objective. herein. is to develop a linear displacement formulation for a single one-dimensional finite element with two exterior nodes. each having three nodal degrees of freedom. Two nodal variable W and 9 describe the flexural displacement and slape respectively. while U describes the longitudinal displacement as shown in Figure 4.5. Figure 4.5 The Deformation of a Beam Element 117 Defining the nodal displacement vector for the element by [U)T= [111 u, w, w, o, 9,] (4.4.1) the general displacements u(x.t) and w(x.t) at any point in the element may be related to [U] by [u wlT . [NIIU] (4.4.2) where [N] contains the shape functions. The axial displacement is defined by Ill 3 no - za'lax (4.4 e3) 'hOtO u°(x.t) is the axial displacement. w(x.t) the transverse displacement and x is the longitudinal spatial variable. The axial strain corresponding to equation (4.4.3) may be written as [13,] - [fillU] (4.4.4) where ('l denotes spatial differentiation with respect to x. 118 The finite element representation of the constitutive equation (4.2.2) of the viscoelastic material is [“xx(t)) = [5,53/(a,+8,)][1,,(t)l +u[exp(-h/p)[q(t-h)] + 4.1%,,(t-h11 + i.[§,,(t)1] (4.4.5) The variational equation of motion (4.3.6) may be written as 5F = 0 - (vtsy,,(t)1T [[a.,(t))-(E.a. 1(e.+s.1)(.,,(.11 O O ’ 0[exp(-h/u)[q(t-h)]+14[1xx(t-h)l+h;[y:x(t)l ] ] dV + )vptap]T[ [Pl-[Nglngl-[NJIU] ]dv - IQIaUJTINJTI alaxx(t)I/8x+[XI-p[P] ]dv + Ivlsex‘lr[ [yxxl-[RIIU] ]dv (4.4.6) + IglbUJTINITHEl-[ghdg ' )s.lbslT([fiJ-[Nllnl)ds, where [NR] contains the shape functions approximating the rigid-body kinematics 'h119 [Pg] is the column vector containing the nodal rigid-body velocities. The nodal absolute velocity components are defined by [P] and the surface tractions by [g]. Finally. because the rigid-body equations of motion are of no consequence here. they may be 119 removed from the formulation by taking variations [broil and [691] to be 19:0. In order to express the finite element formulation in terms of . deformation of the links. attention is now focused upon the equation of equilibrium. The first term may be written as )v [SUJTINJTaIcuuU/ax «N (4.4 .71 An integration-by-parts over x yields [ IAISUITINITIOxx(t)IdA 1x - [VISUITIRIT [ox‘(t)]dv (4.4.8) The first term in equation (4.4.8) is a boundary term. to be evaluated on the cross-sectional area A. at the extremes of the one-dimensional element. and this is identical to the manipulation in chapter 2. Therefore. it will not be discussed in this chapter. The second term becomes. using the first and fourth volume integral expressions in equation (4.4.6). “JVI50]T[N]T[ (HIE,/(EI+E,))[R1[U(t)I + u[ exp(-h/p)[q(t-h)] + 1.1-111111.-..) + “mutual ] ]dV (4.4.9) 120 The first term in equation (4.4.9) yields the standard linear stiffness matrix [K] defined by [Kl ' I§1n1T(e,E./(e.+e.1)tfildv (4.4.10) The second and third terms in equation (4.4.9).which feature the hereditary nature of the material. are defined as [Fh] = IlelTu exp(-h/p)q(t-h)dv + IV [Nlrlg ulN][U(t-h)]dv (4.4.11) FIOI equation (4-4-11) [Pb] is known as a force term which is contributed from the stresses and viscous strains of the previous time step. The fourth term of equation (4.4.11) is recognized as a damping matrix [C] [C] - I. 1N1T1,a1nlav (4.4.121 Substituting equations (4.4.10). (4.4.11) and (4.4.12) into equation (4.4.9) leads to ~1601T[ [kllU] + [eh] + (eltfil ] (4.4.131 121 According to the derivation in chapter 2. the variational equation of motion may now be written in the final form 5F ‘ 0 ij [5731111 [Oxx(tI]’(EgEg/(El +Ez))[7;x] - u( exp(-h/p)[q(t-h)] + A11§x,(t—h)l + 1.1%,, (:11 )dv I§ptsplT1 [Pl-[NRIIPRI—[NJIUJ )dv 4. [solT[ 1 [K][01+[cllbl+[ullvl+[Fh] 1-(§1~171x)av )5. [muting-1113111531) ] + Ivtaa,,1T( [1,,1-(fi11u1 14v (4.4.14) )3.“le [fil—[lel )ds, Ihen independent arbitrary variations of the system variables in equation (4.4.14) are permitted. this equation yields the field equations and displacement boundary conditions for one finite element. This formulation provides the basis for developing a finite element model of a general planar flexible mechanism in order to investigate the dynamic linear viscoelastic response. The experimental apparatus of four-bar linkage. slider-crank mechanism and related intrumentations are the same as those used in chapters 2 and 3. The only difference is the dimension of the linkage specimens employed in this study. These data will be introduced later. 122 4.5 Comparison between Theoretical and Experimental Result A solution for these equations will be sought using the Newmark method of direct integration which uses the following statements to update the kinematic terms as the response is discretized. Ut+At a fit + 111-p10. + BUt+AtIAt (4.5.1) ut+At = ut + UtAt + tun—emut + aflt+A.]At’ (4.5.2) vhorc B 20.5 and u 2 0.25(o.5+a)’ for stability of solution I 21 l. The algorithm models the system by considering the continuous motion of the mechanism to be discretized. Thus. the inertial loading from the rigid-body analysis is used to continually update the force funtion for a series of time intervals. and the response is determined for each time increment to give a set of discrete values for the link deformation. Vith all direct integration methods. solution instability must be considered carefully. and it was avoid here by choosing 8'0.5. Under these conditions the system is unconditional stable. Finally. a numerical value must be assigned to the step-size At which is determined by choosing At 1 T/lO. where T is the smallest period in the response [ 21 1. Thus the step-size is governed by the higher modes of vibration presented in the response and for stiff systems this may necessitate a small step size. 123 Figures (4.6)-(4.11) on pages 124-129 present computer-generated experimental response curves superimposed upon the results of computer simulations for the dynamic response of the midspans of the links of the flexible four-bar linkages and slider crank mechanisms tested. The predictive capabilities of the mathematical models were tested by running the mechanisms at different operating speeds. which inherently imposes different loading on the links; mechanisms with different link-length combinations and a variety link materials. and link cross-sections were also employed in this investigation. Figures (4.6) and (4.7) present the midspan transverse deflections of a 304.8 mm long connecting-rod of the slider-crank mechanism fabricated with 4 [1:45], laminate. The thickness of the link in the plane of the mechanism was 2.34mm. the width perpendicular to the plane of the mechanism was 19.05mm and the length of the composite specimen was 228.6mm. It is evident from the figures that there is reasonably good correlation between both the amplitude and phase components of the response at both operating speeds for these viscoelastic links. Figures (4.8). (4.9) and (4.10) present the midspan transverse deflections of the coupler and rocker links of a four-bar linkage in which both links were fabricated with a unidirectional laminate. and operated at three different crank frequencies. The coupler and rocker links had the same overall length of 304.8mm long. The thickness of the laminate in the plane of the mechanism was 1.93mm and the width perpendicular to the plane of the mechanism was 19.05mm. Again there is good correlation between the theoretical and experimental responses for 124 a icnunbnow.~ moguloamu .amu end «a momuoe—uen .nneuh amuse“: moulunuuoemuou u—mQH. Hanan-moo: uc-u01uomanm .mmmmomov MAUZ< xzJmAJAv+n+e.+nu+x.ri]dv ( ) an -fl .. + ISH (1/p9.) Q GdSB + I86 giridsa + Isn (1/SM.) oa‘“’u dSD]dt (6.2.3) subject to a number of auxiliary conditions or constraints. In equation (6.2.3) the term 0 represents the temperature increment above the tOfGIOBCO t'flPGIFtUIO 9.. s represents the entrapy-density increment above a reference entropy-density s., n 1. the flow-potantial donsity increment above a reference density n., u 1. the .oigtnto concentration increment above a reference concentration Ho, xi ( N/m' ) .2. 1h. body forces per unit volume. U‘fl) ( N-m/m’-sec ) and G(u) ( N/m’-sec ) are the prescribed heat and mass transfer. and ii .3. the pgggcg1bgd .ngfgce tractions. D ( N-m/N ) is the specific chemical potential. The term. I. is hygrothermoelastic potential density. V ( N-mlm' ) defined by + l'./21. (n-aijyij+pe)‘ (6.2.4) Cijkl ( N/m‘ ) being the symmetric tensor of elastic moduli in the isothermal state. c ( N—m/m’-°K ) is the specific heat per unit volume (c'pE. where E is the conventional specific heat with units of N-m/kg-‘H 168 in the reference state; Bij ( N-m/m'-'l ) and “ij ( N-m/m' ) is the symmetric tensors governed by the thermal-expansion and diffusive-expansion properities of the medium. respectively: p ( N-m/m"K ) is the hygrothermal coefficient; b ( N-mlm' ) is the hygroscopic capacity. The function D ( N-m/m' ) is dissipation function due to the irreversibility of heat conduction and mass diffusion in the medium and is defined as n a ( IIZsOOxij )qi(B)qj(H) + ( um...” Mumqjus) (6.2.5) where 3 is the mathematical operator 3 = d/dt. Efij ( N-m/m-sec-‘E ) and 911 ( N/msec ) are the thermal conductivity and mass diffusivity potential tensors. respectively. and quH) ( N--/.‘-g.c ) q1(') ( N/m'-sec ) are defined below. The constraints imposed upon the parameters in equation (6.2.3) may be formulated as the strain displacement equations 1‘3 ' 1/2 ( “i.j+nj.i ) (6.2.6) where comma (.) denotes spatial differentiation: the velocity rate-of-change of position statement 169 Pi ‘ roi + 3i I °1jkaj(fokftgg+nk) (6.2.7) where (”) represents the time rate of change with respect to the moving frame oxyz. (.) defines the absolute rate of change. °ijk 1. the alternating tensor. and components of the angular velocity vector for the moving axes are represented by Oj; gin) " -xue,j (6.2.8) and 9;“) '3 ~015M'1-Guuyud (6.2.9) where Gijkl ( N/m-sec ) is the hygroelastic modulus. These are the phenomenological equations describing the heat and mass transfers. which are also known as Fourier's and modified Fick's equations. respectively I 42.43 I. The mechanical boundary condition on region Sn 11°, 31 " “i . (6.2.10). the thermal boundary condition on region Se ‘19 5 = 0 (6.2.11) and the hygroscopic boundary condition on region SH .re 170 I 8 H (6.2.12) A free variational problem with auxiliary conditions may be . constructed by the Lagrange multiplier method which incorporates the constraints within the original functional after each condition is first multiplied by an undetermined multiplier. 1. with numerical superscript and appropriate tensor indicial subscripts. The functional 6. for the free variational problem becomes the functional 1. where m. = (EH Iv [r-wmennmxirikv + Isaziridsc + Isa (1/39.) (imiidsg + Isa (15“.) (1'6')””)InsD + IV 11%)[111-1/2 (ui'j+uj.i)]dV Iv *éz’himwmaldv Iv 1(3)I31"+”15'.j‘Gijkllkl-J]dv + + + IT 1:4)[Pi‘(;°i*;i+eijkaj (r°k+rgg+nk))]dv + Is“ 1(5) (ni'ai)dsn + 159 A‘s) (0.3)d89 + (S. 1‘7’(u-I)nsu ]3. (6.2.13) The variational equation corresponding to the above equation may be obtained by taking variations of the system variables and setting the resulting expression equal to sero. Hence using the definitions (6.2.2). (6.2.4) and (6.2.5). this is written 171 B - t 5’ 0 I11 I Iv 9915P1'913117215713 9./c(s-Bijyij+ul)(5s-Bijbyij+ubl) u./b(n-oijyij+u9)(Sn-oij671j+u50) + (1’59.X15 )qja’bqin’ + (1/Su.nij ) oqju’aqj" + ‘59 '5' a: '5’ a6" + .5" + Xibti ]dV ‘5' Isa-ii 5rids° + [SE (1159.) EIHnedsn . Isn (1,5...) gymnasn + Iv 5*II)I713'1/2(u.,5+nj,1)]dv + I. hi})[8111-1/2(5ui.j+5nj'i)]dv + Iv51§2)(qffl)+xijq.j)dv + Iv 1:2)(5q‘a)+lij50’j)dv + Iv 8113)(qi')+Di1H.j+Gijklykl.j)dv Iv 1:3) (agil)+ng16“'1+cijk151kl'j)dv + + IV 51:4’[p.-(;°.+3.+..jkdj(.0. +rnk+uk))]dV 4' IV hint: bPi-(6?°i+bii+eijkbaj (r°k+r3k+uk))]dV + Isn115)5uidsn + Ign6115’(§.-n.)dsn + Isa 1(6)80d89 + I59 51(5)(o-U)nse . Is" 1(7’6udsu . ISM 51(7)(n-I)asu 1dr (6.2.14) 172 Utilising Gauss' theorem. the sixteenth product in equation (6.2.14) may be written as -Iv A£})1’2(5Di.j+bnj.1)dv ' -Is xLIME“) ‘5 + Iv 1)}315nidv (6.2.15) where ”j is the unit vector outward normal to S. Similiarly. the eighteenth and the twentieth products in (6.2.14) may be written as Iv 112)K1180.jdV ‘ Is ‘11112)315945 ' Iv lifglijbedv (6.2.16) and V Iv 1§3’[ Dijbfl.j+Gijklbykl.j ]dv - Is[ 0.1213’n16u+cij111§3’n167k1 ]dS ' IVI *I?I°135“+1I?I°11115711 ]dV (6.2.17) The terms under the eleventh integral must be subjected to integration-by-parts over time. This integration procedure is subject to the constraint that variations are not permitted at the extremes of the time interval. Finally. by permitting the variations to coincide 173 with the actual displacements that occur during the time interval dt. then 5’1 ' 5‘oi+5“1+°131503(rog+rgx+nk) (6.2.18) Upon incorporating these re-arrangements into equation (6.2.14). the characteristic equations for this class of mixed dynamical problem in hygrothermoelasticity may be obtained by permitting independent arbitrary variations of the system parameters. and this operation also enables the undetermined multipliers to be expressed as functions of the system variables. These operations yield the following equations. The constitutive equations 1 3 *Ij) ‘ cijkIYkl ’ 51j° ' “13" ' 5111111.I - gij (6.2.19) nia’ - -r.,0,j (6.2.20) and Hi“) ' -Diju.j-Gijk1711.j - (6.2.21) The equations of state are 9.8 ‘ 09 + 9.fl11111 " 9.11" (6.2.22) 174 and “on g I)" + u.aij71j - nope. (6e2e23) Th0 multiplier 114) c pPi. and the equations of motion are written as Oi. J.j . pfii. (6.2.24) +1:i The balance of linear and angular momentum also emerge from the analysis and are written and Iv °1jkxi‘tos+rak+“k’dv + Isa '13231 (’ok+rRk+°k)dso = IV °ijkp§i(tok+‘Hk+“k)dv° (6.2.26) The multiplier 1:2) 1. 4.11..d by 2) ~ H which enables the energy equation to be developed from . . 1M - xijuifg. (6.2.28) 175 “0300: upon substituting for 1(2) and applying the mathematical operator 3. this yields 0.: - O'Hi - gig). (6e2e29) Similarly. the multiplier 113) i. 4,111.4 by li” - -( 1/3M.u.j )nq§“’ (6.2.30) and hence l.§ - “.10 - noif’. (6.2.31) Equations (6.2.6)-(6.2.11) also emerge as characteristic equations. as anticipated. In addition. the following Lagrange multipliers 1(5), 1(6) and 1(7) are obtained from the process of taking arbitrary variations. 1(5) 3 11%)nj (6.2.32) 1(5) 3 -xI2)nj‘ij . ‘1’5°°’QIB)31 (6.2.33) and 1(7) . ’113’n1013 . (1/3u,)gqil)ni (6.2.34) The previous three equations enable the following surface traction 176 boundary conditions to be obtained ‘1 a 01ij (6.2.35) as anticipated. Finally. the heat and mass flux boundary conditions are expressed as 0‘") = qIH’ni (6.2.36) and 0‘") - qi"’ni (6.2.37) Using the above definitions of the Lagrange multipliers. the first variation of functional 1. from which the equations governing the hygrothermoelastodynamical analysis of mechanism systems may be obtained. is written 61 ' 0 ' III I I) 5711(015 -C‘3517k1+fliie * “ij“'1/‘3"0911’Gijk1 anII Idv I Iv 5‘I9‘9o/c (s-Bij71j+flu)]dv I Iv 5nIll-I‘m. (fl'aijyij+p9)]dv + IV 5u1[oij'j+X‘-p§1]dv ‘+ IV 56{s-pl+(1/39.) qigildV 177 + Iv buIfl'WHl/EN.) anflldv + IV 5“ijIYij'1/2(“i,j+uj,1)]dv ' Iv (SqIH)/59.K.k)[q§")+xkje.j]dv - IV (5‘)“)Isu'nis’aniu)*°sj"-j +stk17kl.j]dv ' Iv P5P1[Pi';oi-;5 -eijkaj(rok+r3k+uk)]dv + 6roiIIv 1.6V+I3. iidsc'vafiidv] +6¢JIIV ‘ijkin‘ok*’Rk+nk)dV + ISo °ijk;i(rok+tgk+uk)dv - IV 'ijkp§i(‘0k+‘kk*nk)dv] ' Isa 5“i(li’31)dsa * Is. 581(ni-Ei)dsu * Isa (oqia’n./SG.)(3-§)3se + Isa ‘59/39.)(5(n)-qin’ni)nsn * Is. (bqiu’niliu.)0(u-i)dsM + I30 (SH/SH.)0(§(N)'QII)ni)dSD ]33 (6.2.33) Independent arbitrary variations of the deformation displacement. strain. stress. absolute velocity. and the kinematic parameters defining the rigid-body motion. and the entropy density. flow-potential density. temperature and moisture concentration field. enable equation (6.2.38) to yield. as Euler equations. the field equations and boundary 178 conditions for this class of hygrothermoelastodynamic problem. This variational principle also provides a basis for finite element formulation which will be develcped in next section. 6.3 Finite Element Formulation The objective. herein. is to develop a ”displacement“ finite element formulation for a single one-dimensional finite element with two exterior nodes. each having five nodal degrees of freedom. Since this analysis is to investigate the vibrational behavior of linkages fabricated with straight slender flexible beam-shaped composite laminates. the flexural deformation field be modeled by the elenent is governed by the classical Bernoulli-Euler hypothesis. The nodal variables I and 0 describe the flexural displacement and slope. respectively. U describes the longitudinal displacement. T denotes the temperature field and H is the moisture concentration. Defining the nodal "displacement" vector for the element by [MT = [01,w1,e1,1~1,u1.u2.v2.ez.rz.)s2 1 (6.3.1) and the general solid-body displacement is denoted as u(x.t) - [N‘JIU] (6.3.2) and the temperature field and moisture concentration are expressed by. 179 9(x.t) = [Nhlm (6.3.3) and m(x.t) 8 INEJIH] (6.3.4) where IN‘]. [Nb] and IN") are shape functions which describe the spatial distribution of displacements. temperature and moisture concentration. respectively. throughout the element. These functions are independent of time. However. [U]. [T] and [M] are time dependent. Incorporating equations (6.3.2)-(6.3.4) into equation (6.2.38). then the discretized variational equation may be written as 67 -= 0 " III I Iv [Mali-[[621] -IC]IB']IU]+IB]INhJITI+Ia1[N"]IM] -1/(isu,)[nl'llcloalq"’l/ax ]dV + Iv (3.1111Nh1171—e./. ((.)-(oluxgmmulm )].w + Iv [331T[[nilth-u./e (Isl-[cl[1,,l+u[Nhl[T])]dv + Iv [SUITIN'ITIBchxllaxi'IXI-flP]]dV . IV [sTlTlflh]q[.]-"[lelll +(1/ie.) BIq(a))/ax]dv + IV [gulrINulTIInl-uINhIIT]+(1/5I.) 03[q(‘)l/3x ]dV 130 + IV IfioxxlTllyxxl-[B‘llfll]dv ' Iv (lbum’lTlll'1/Be.)[[q‘fl’hm[3h][31].“; "' Iv (l6q("’lTlDl'1/3I.)o[[q‘"’)+[0][3mnu] +IG] (331111].w ' Iv “[5PITI[Pl-INRIIPRI-IN‘llfilldV - Is6 IbUlTIN‘lTHgI-IEIMSO . ISu [onglN'J (ml-[Unnsu + Isa “want/390mb](In-[Those + Isa (lsrlTlNthl'ien([‘d‘H’J-[ommsa I IS): (lfiqmlT/‘iflnm'lnuul-[B])35“ + [SD (lauleJT/su,m[a‘“’1-[0("’))350 ]dt (6.3.5) where [C] ( N/ma ) is the elastic modulus matrix: [B] ( N-m/m'°E ) and In) ( N-m/n' ) are thermal-expansion and hygroscopic-expansion coefficient matrices: [NH] contain the shape functions approximating the rigid-body kinematics (see chapter 2) while [PE] 1. the column vector containing the nodal rigid-body kinematic degrees of freedom. The nodal absolute velocity components are defined by [P]; the surface tractions. heat and mass fluxes are defined by In]. (G(H)] and IQIMIJ. respectively; [B] is the spatial derivative of shape function IN]. Because the rigid-body equations of motion are of no consequence here. they may be removed from the formulation by taking variations [5‘01] and (591] to be sero. 181 Substituting the constitutive equations of the first integral and the tenth integral into the equations of equilibrium yields the expression -(301T[ [Iv [N‘lTpIN']IU]+[B'JT[C][B'][U] 433mm 1th [Tl-[B'lTlc] mu) m +1/(su,[n)'1(B'JTloloa(q"’)/ax]nv + Iv (N‘JTbINRJIiglav - Iv [N‘ITIdeV - ISclN'lleldso ] (6.3.6) The following definitions are introduced. Ulfil - Ivtn'lTplNRmv (6.3.7) [u5'] - I§[N')Tbln'ldv (6.3.3) IX") - Ivtn'lTlclln'Jnv (6.3.9) (x'h) - “IVIB'JTIBIINhldV (6.3.10) IX") .. -Iv(3'lTlaJ(N'13v (6.3.11) Substituting equations (6.3.7)-(6.3.1l) into (6.3.6) yields 182 -(601T [(u"l[U]+[t“l[U]+[K‘h][TJ+[K'°][u] +II§JIERJ-IVIN'ITIXIdV-Isa[N'ITIgldSa +I§1I(Su.)[n]’1[0][B'lToolq‘u’l/oxdv ] (5.3.12) The second integral expression in equation (6.3.5) may be employed to eliminate a from the energy equation in the fifth integral of (6.3.5). The energy equation may then be written as Iv (311T[ 0.[Nth(B][B'l[(11+cINthlNh][I] “299.[NthlNul[91+[BthlxllBhl[T]]av -ISB (311‘INhJTtrltnhltrlnsa (6.3.13) Upon introducing the following definitions [ch51 - I§1Nh1T0,131[3'13v (6.3.14) [chh] = IvluthcINhldv (6.3.15) [chnl . —Ivln§lT2pe.tn‘lnv (6.3.16) [xhh] . Ivtuthtxlluhldv (6.3.17) (6(3)) = -[r][ahllrl (6.3.18) equat where 183 ion (6.3.13) may then be written as (611T[(ch'1(fil + [09911)] + (ch-1m) + (xhhm) + jsflmthto‘n’lasn] (6.3.19) Similarly. the equation of mass balance may be written as [691T[ [c“][0] + [clhlll] + (cu-1(1) + (1mm +[K'"l[7] + ISDIN“]T0[0("’ldsD] [6“] . IVIN“]TM.[a][B'ldv (cl-h) = -IV(N'1T2..)(.mh13v [cm] - Ivtn'lTbIN'ldv It“) s Ivm-JTomJIB-Iuv It”) = Iv‘3'1T9‘61 (3'13v (0W) - 40113-1131 (6.3.20) (6.3.21) (6.3.22) (6.3.23) (6.3.24) (6.3.25) (6.3.26) 184 The final form of the discretized finite element formulation for the variational equation of motion (6.2.38) may now be written as 6] = 0 - Ifi [Iv (1,,1113n1-(c1t3'1wl +[3][Nh][T]+[aJ[N'l(Ml-11(Bu,)[01’1lcloalq‘“’llax]dv + jv13.1T[mhlm-o./c ((.)-(31(3‘1l01+uIN'l um] 3v + IvtnanIIN'llul-l./b (Isl-[alln‘llUl+ulNhl(Tl) ]nv + (301T[ [11“][01+[t"][U]+[K‘h][1‘]+[t"'l[ll *I'Il[Inl'Ile'lrlx'dv’Is3 [N'JT([El-Igl)3s +IV1/(sumnrllol[351Tnalqm1/axav] + [ETITI tel")(01+(chb)much-Mints“)m +Isu (nh)T((0‘“’1-(0(“’1)3sn] + (39111 [c"l(fil+[c‘hl[hue-'JIAHIK'WJ[Ml +181, [N”]T([0(“’l-[Q"))nsn] + Iv (6.1T[(1)-(3-1101]3v + Iv ((3.‘mlT(xr1/39.)[(q‘mhm(3h):1~)]3v + Iv ([sq("’lT[nl'1/‘§M.)[[q‘u’lflnl[8'][u] +[GJIB'II11 ldv 185 ' Iv PI5PJ1IIPl-[N31IPgl-[N][Ulldv 4. Is. (3.1Tmnum-(umsu 4. I53 ([69‘5’1T/30.)(uh1(III-[Those .1. Is“ (tsq‘l’lT/sN.)(mun-mm“ (6.3.27) Equation (6.3.27) contains the field equations and prescribed displacement. temperature and moisture concentration boundary conditions for one finite element. These governing equations may be obtained by taking arbitrary independent variations of the variables in this variational equation of motion. The resulting matrix formulation must be pre- and post-multiplied by standard transformation matrices in order that this general statement he used to develop a finite element model of a specific linkage. After an apprOpriate modeling of a physical problem by employing these governing equations and boundary conditions. solutions for these equations will be sought by using a proper numerical scheme. 6.4 Parameter Definition For calculating temperature. moisture distribution and the deflection of the composite laminate. a full knowledge of the following parameters is required: 1. geometry: dimensions of the specimens. 2. boundary conditions: ambient temperature and the level 186 of the relative humidity. 3. initial conditions: temperature distribution and moisture concentration inside the material. 4. material preperties: density p. elastic moduli Cijkl' thermal-expansion coefficients Bij' d1ffugion-gxpgnsion coefficients aij' heat capacity c. thermal conductivity Kij- hygrothermal coefficient u. hygroscopic capacity b. moisture diffusivity D. maximum moisture content II, and a relationship between the maximum moisture content and the ambient conditions. The density. thermal-expansion coefficient. diffusion-expansion and specific heat are generally known. However. the determination of the other coefficients are more difficult and will now'be discussed. 6.4.1 Elastic Moduli C”k1 Data are available showing the effects of moisture and temperature on the buckling. tensire. and compressive moduli of various composite laminate I 153.218 ]. The available data on these moduli were compiled by Shen and Springer I 218 l in a manner similar to the tension test data. On the basis of available information. the following major observations can be made. 187 6.4.1.1 Thmperature Effects For 0-degree and n/4 laminates. the temperature (in the range of 200 to 450K) has a negligible effect on the elastic moduli regardless of the moisture content of the material. For 90-degree laminates. an increase in temperature causes a decrease in the elastic moduli in the direction perpendicular to fiber. For example an increase in temperature from 300 to 450 K. the elastic modulus of AS-5/3501 I 221] may decrease by as much as 50 to 90 percent. The decrease in the modulus depends upon both the temperature and moisture content. 6.4.1.2 Hoisture Effects For O-degree and n/4 laminates. there appears to be very little change in the elastic moduli over the entire spectrum of moisture content from dry to fully saturated. This conclusion appears to be valid regardless of temperature in the range 200 to 450 K. For 90-degree laminates. the elastic moduli decrease considerably with increase in the moisture content. The decrease in the value of the modulus of Iodmor II/Narmco 5206 [222] may be as high as 50 to 90 percent in the temperature range of 200 to 400°x, 6.4.2 Thermal Conductivity ‘ij The thermal conductivity of a material is a measure of the speed at which heat is conducted through a material. This material property may also depend on the moisture concentration and on the stress level as 188 well as on temperature I 93.152 1. However. the variations of Kij with both moisture concentration and with stress are not known in detail for most composite material I 146,217 1. Mathematical models are based upon the assumption thlt ‘ij is a function of temperature only I 152 l. The thermal conductivity. xij' can also be approximated from the known fiber and matrix conductivities I 152 I. For many composite materials. the thermal conductivity K13 is 10‘ to 10‘ times larger than mass diffusivity Dij I 93.143 ]. Thus. the temperature equilibrates much faster than the moisture concentration. 6.4.3 lass Diffusivity Dij The mass diffusivity characterizes the speed at which moisture is transported through the material. The value of D depends on the material. on the fluid surrounding the material. on the moisture concentration inside the material. on the stress level inside the material. and on the temperature. In calculating the moisture content inside the material D is assumed to depend only on temperature. In many practical problems this is an adequate approximation. The diffusivity of a composite material may be determined directly from tests I 93.143 1. or may be approximated from the known fiber and matrix properties I 143 1. 6.4.4 HygroscOpic Capacity b The hygroscopic capacity. b. is a measure of absorbed energy carried by the moisture moving through the material and is analogous to 189 the heat capacity. It can be expressed mathematically by I 118 l I) '3 1/I°(an/au)y’9 (6.4.4.1) The hygroscOpic capacity. b. can be represented explicitly by (bBDE) I 42.43 I where D (N—m/N) is the specific chemical potential. and b is generally a material prOperty which may be interpreted as mass of moisture being absorbed per unit volume of the material. The material property 5 is generally known by experimental testing and the definition of specific chemical potential will be described in the following subsection. 6.4.5 Specific Chemical Potential 0 Several assumptions are made in order to find the specific chemical potential in this study: 1. The transferred moisture is considered as saturated vapor and this is further assumed to be an ideal gas. 2. Since the moisture diffusion velocity is assumed to be slow and no elevation in the diffusion process. i.e. no potential energy resulted from gravitational force. the kinetic and potential energies associated with this phenomena are considered to be negligible. 190 The derivation of the specific chemical potential 0 is described briefly as following: The specific chemical potential of a pure substance is equal to its specific Gibbs function g at the same state I 223 ]. therefore. the specific Gibbs function ‘T.P of 1 ‘nb‘t.ncg at temperature T and pressure P is. by definition. ‘T,P = hT.P "' TST'p (6.4.5.1) The enthalpy of an ideal gas at temperature T and pressure P can be written as hT'p = 1% (6.4.5.2) where 3% represents both the enthalpy of formation and the enthalpy difference between 298'! and the specific temperature T. The standard state of an ideal gas is defined at 1 atmosphere and 25'C. and that is symbolized by the superscript '. The absolute entropy. s. at temperature T and pressure P are assumed to be related by ‘T.P ‘ ti - R In? (6.4.5.3) The specific chemical potential of an ideal gas may be obtained by 191 combining equations (6.4.5.1). (6.4.5.2) and (6.4.5.3). and is expressed by 0 ‘ ST'p = g; + RT InP (6.4.5.4) where the unit of the quantity P is in atmospheres. R is the universal gas constant and g' is defined by s' -= 11% - 13% (6.4.5.5) 6.4.6 Maximum Moisture Content MIn 75‘ “9‘33““ ‘0355939 content I. (i) is the moisture level within the material. This can be attained asymptotically under certain conditions. after a prolonged period of exposure to a moist ambient environment at constant temperature and at constant relative humidity. For materials exposed to humid air the maximum moisture content appears to be insensitive to temperature and depends on the relative humidity 0 in the following manner: III a .of I 93.143 1. The terms a and f are constants which depend on material and are determined experimentally. Experimental evidence suggests that the value of f is near unity I 93 l. The above relationship greatly facilitates the calculations. however. it is only an approximation. in fact. I. "3.1-.113; varies .1:]; temperature. Twenty percent variations in H. with temperature .1. common for most graphite/epoxy composite laminate in the temperature 192 range of 300-400.! I 93 I. An advanced theory should be (has not yet been) develcped in order to accurately predict the maximum moisture content after the chemical and mechanical preperties of composite materials being clearly understood experimentally I 145 I. In many situations changes in the value of Mn 51v. bggn obggrvgd after the maximum. asymptotic value of M. h.. bggn ropchgd [ 93 1. Both increases and decreases in M. have been observed [93.143 ]. The increase in M. is hypothesized as being due to cracks developing in the material and also. possibly. to non-Fickian diffusion. A decrease in M. may be caused by loss of material due to leaching or cracking I 215 I. The above relationship M.I - .of is invalid when the composite structure is near failure as manifested by the formation of microcracking or delamination. 6.5 Illustrative Example Having established a theory describing a high-speed flexible linkage mechanism constructed from polymeric fibrous composite laminates operating in an environment characterised by variations in both temperature and humidity. a sample problem will be presented in order to illustrate the applicability of this theory. All of the experimental publications reviewd in section 6.1 of this chapter were focused upon the static analysis of hygrothermal effects on graphite/epoxy composite materials and/or neat epoxy resin. only one paper was devoted to a dynamic analysis I 211 l. The analysis of a 193 composite material under static loading greatly simplifies the investigation since under the static loads the strain rate is sero. Consequently. the coupling effects between the moisture distribution and strain rate in equation (6.2.27) and also between the temperature distribution and the strain rate in the same equation are sero. Furthermore. the external loading imposed upon the composite material are assumed to be limited in order to prevent the formation of micro-cracks which are resulted from the excessive stress levels. And the micro-cracks in the composite material may result in non-Fickian diffusion due to the changes of saturation level. In 1978. Ihitney and Browning I 217 ] experimentally studied the diffusion of AS/3501-5 graphite/epoxy and 3501-5 (Hercules Inc.) neat resin coating. Unidirectional and bidirectional. IOV9OI‘, gpgc1ngng were also included in this investigation. Seven neat epoxy resin specimens used in these experiments were thin plates of dimension 508mm square with a thickness of 3.18mm. All of the composite materials data reported were obtained by testing sixteen 4-ply specimens which machined from large autoclave-cured panels fabricated from the Hercules AS/3501-5 graphite/epoxy prepreg system. These composite specimens were 508mm square and 0.64mm thick. In order to provide an initially dry condition. all absorption specimens were preconditioned in a vacuum oven at 93°C and under full vacuum until a near equilibrium weight was obtained. Dry specimens were placed in an environmental chamber under constant temperature and humidity conditions. Specimens were removed from the chambers at various time 194 intervals for weight measurement (see the scatter symbols on Figures 6.2. 6.3 and 6.4 from pege 195 to'197). Absorption experiments on neat resin specimens were run at 75 percent relative humidity with a temperature of 82°C. and at 95 percent relative humidity with a temperature of 71°C. Unidirectional composite data were also obtained at 95 percent relative humidity with two different temperature conditions of 49°C'and 71'C. while the test conditions for bidirectional composite laminates were at a relative humidity of 95 percent with a temperature of 49'C. and at 95 percent relative humidity and with a temperature of 71°C. In accordance to the experimental work performed by lhitney and Browning I 217 l as mentioned above. the problem definitions for the analytical analyses are to determine the moisture content of three kinds of materials. i.e. neat epoxy resin coatings. unidirectional composites and bidirectional composite laminates. The geometry of the specimens and the boundary conditions of each case are described as following: I. An initially dry neat resin plate with dimension of 508mm x 508mm x 3.18mm is placed in an environmental chamber under two constant temperature and humidity conditions: the first is at 75 percent relative humidity with a temperature of 82'c, a. other .1. .1 95 percent relative humidity with a temperature of 71'C. II. An initially dry unidirectional composites with dimension of 508mm x 508mm x 0.64mm is placed in an environmental chamber which has a constant relative humidity of 95 percent and two different temperature 195 0.0 manna macaw “.02 new .sea ..xoam ounce» nae: ~euneamuemmm we «can «.0 vacuum SE 0.0 v.0 «.0 0.0 b — — p p (90.9 COMuoflmmma 3nd m.xoflm H 3.3%: .532. 5.21 n. 7|m~0 $5qu .533 6.: G . I i \ 35 m m - In, \ (2.0 f H II.“ Woo _ 196 0 cummomaou muomm\cum:aeuc «ecouuueummmna uou .sea ..uomm «ounce .uea neunoaduomum no uoum ocean: 0.0 cannon {E .0 0.0 v.0 «.0 0.0 b — h — b 00.0 COMuomUomm 3mg m.xoflm H 80.? z .0500 6.: B 2.0 wen.Hqu .:m»m0 .000 AV - 197 cueemaea muomm\ouammeu0 unmoauoouqvmm new .wea ..xomm acumen no.0 ueonenmuemum no «can noun-a ..0 .....m file 0.. 0.0 0.0 «.0 «.0 0.0 — — b b p — 00.0 ofiuoflnwmm 3mg "3&0?” l :04"; .533 6.2. U 1 mm 0 30.7; .5500 6.3 4 AV H G 00.0 a m d u d 1 0\ 2.0 q H D 1 . .q f a 0 II) . 00 _ T T Iw/w 198 conditions of 49'C and 71°C. III. An initially dry bidirectional composite laminates [0/90]' with the dimension of 508mm x 508mm x 0.64mm is placed in an environmental chamber with two test conditions: one is at a relative humidity of 95 percent with temperature of 49 'C. the other is at the same relative humidity with a temperature of 71'C. 6.6 Results and Discussion Since several assumptions have been made in the "illustrative example" section (section 6.5) in order to simplify these examples to be mathematically tractable for this priliminary study. the coupling terms between the temperature distribution and the strain rate. and between the moisture concentration and the strain rate in equation (6.3.27) are assumed to be sero. Therefore. by removing the terms associated with the strain rate. the three governing equations (i.e. equations of mass balance. momentum balance and energy balance) may be written as follows: ( 1 ) the equation of mass balance [c'hll‘ll + (cit-1(9) + (rum) - ~IK"]I11 - ISDIN'JTom‘“’ldsD (6.6.1) ( 2 ) the equation of momentum balance [K‘hllTJ + (re-1(9) + (rum) - IVIN‘JTIdeV 199 + ISOm'JTIEsta - Iv1/(;u°)[nl’1[61[B‘lThanIuIJIBde (6.6.2) ( 3 ) the equation of energy balance (chhlttl + [Ch'llfll + (xhhltrl = -ISB[NhJTI0(B)l ass (6.6.3) The above three equations are further simplified by assuming that there is no natural boundary conditions in each equation and the body force may be neglected from equation (6.6.2). therefore. specimens are only subjected to prescribed essential boundary conditions. In addition. the last term on the right-hand side of equation (6.6.2) muy also be neglected because that the deformation resulted from the mass-flux gradient is insignificant I 42.43 I. By assuming the deformation induced by the temperature and moisture is decoupled. therefore. the equation of momentum balance (6.6.2) may be written as IK")IU) - -[x'hl[T] - [I'lllu]. (6.6.4) The equations of mass balance (6.6.1) and the energy balance (6.6.3) coupled by the cross-effect between temperature and moisture may be expressed by (cnhltt) + (cnnltfll + (x-nltu) - -[K“'l[1] (6.6.5) 200 and [chhlli] + [challil + [rhhllrl = 0 (6.6.6) A solution for these equations will be sought using the Crank-Nicolson method I 134 l of direct integration which use the following statements to update the temperature and moisture distribution in the specimens mt+At a [r]. + I9lI1.+3. + (1-0)[il.]33 (6.6.7) and [)(JW“ = [n]t + [etil.+At + (1-0)[il.]3t (6.6.3) where the Crank-Nicolson value 9-0.5 for stability of solution. Knowing the initial conditions. equations (6.6.7) and (6.6.8) may be rearranged. and subjected into equations (6.6.5) and (6.6.6) prior to solving for temperature and moisture distribution in time t+At. After several manipulations. the final form of equations (6.6.5) and (6.6.6) may be expressed by ICnhIITIt+At + IAIIlIg+At ' [cnhllTlt + IBIIMIt - IPIAt (6.6.9) where 201 [A] = IC"]+At0IK'm] (6.6.10) (3) -= IC“)-At(1-0)IK“] (6.6.11) IP] = elxn‘llylt+M+(1-0)[r")[1]t (6.6.12) and [3][T1t+At + (ch-mu)”At = (31(1). + (chumt (6.6.13) where m - (chhmtetxhhl (6.6.14) [8] = [chhl-At(1-0)lrhhl (6.6.15) With all direct integration methods. solution instability must be considered carefully. this problem occurs in earlier time-steps are transmitted with amplification into subsequent steps and it was avoided here by choosing 0=0.5. Under these conditions the system is unconditional stable. Finally. a numerical value must be assigned to the step-size At. Since the thermal conductivity is much larger than the mass diffusion coefficient. the determination of the integration step-sire At must base on the thermal conductivity in order to obtain a stable solution. A stability analysis I 134 ] demostrates that the numerical scheme (6.6.9) and (6.6.13) are stable without oscillations if the minimum eigenvalue 1 of the equation 202 det(IB]-1III) = 0 (6.6.16) is greater than 0 and less than 1. Figure 6.2 shows a comparison between the experimental results and the analytical predictions of the Hercules 3501-5 neat resin exposed to environments definedby a relative humidity of 95 percent with a temperature of 71'C and a relative humidity of 75 percent with a temperature of 82°C. Since the neat resin may be assumed to be an isotrOpic material. the components of mass diffusivity tensor Dfij in any direction are considered to be identical. Good correlations between the experimental data in reference I 217 ] and the computational results by classical Fick's law analyses are obtained. The hygrothermal effects. i.e. temperature changes caused by the rate of change of moisture concentration and moisture concentration changes resulted from the rate of change of temperature distribution. is not significant because the initial temperature distribution of the specimens is at 93°C which is close to the test temperature of the environmental chamber. The unidirectional graphite/epoxy composites. AS/3501-5. were also exposed to two test conditions: one at relative humidity of 95 percent “Id 90099149010 0f 49 °C. the other at relative humidity of 95 percent and temperature of 71°C. However the test environments for bidirectional graphite/epoxy composites. I0/90)‘, ..g. 95 ”(cent relative humidity with a temperature of 49°C ““195 pox-cont relative humidity with a temperature of 71°C. The coup.r1.on. between experimental results and analytical results obtained from classical 203 Fick's law analyses for unidirectional and bidirectional composite laminates are presented in Figure 6.3 and Figure 6.4. respectively. Upon comparing the results presented in Figure 6.3 with Figure 6.4. it is evident that there is superior theoretical/experimental correlation for the unidirectional specimens to the bidirectional specimens. It is evident. therefore. that the mathematical model is more apprOpriate for analysing unidirectional material. Figure 6.4 indicates a rather significant descrepency between the theoretical predictions by a model incorporating the classical Fick's law and the experimental results of the bidirectional composite laminates. The test data approach equilibrium at a slower rate than that predicted by Fick's law. Whitney and Browning postulated I 217 I that large in-plane tensile residual stresses were created which were a consequence of the environmental change increasing the initial through-the-thickness diffusion coefficient: the swelling relieves the residual stresses and in addition. the diffusivity decreases. The diffusion coefficient then approaches the diffusivity of a unidirectional composite as the residual stresses are completely relieved. If swelling continues. the residual stress becomes compressive in nature and a further decrease in the diffusivity may be anticipated. asymptotically approaching a lower limit. These statements are all hypotheses and a direct proof of a stress-dependent diffusion process requires a measurement of the diffusion coefficient under various constant-stress conditions. Alternative approach includes modifying Fick's mass diffusion law by 204 incorporating a stress-dependent term in the diffusion equation. This is presented in equation (6.1.9). and is rewritten here for convenience (M) qi ' -DijM’j-Gijklyk1'j (6.6.17) The units of Gijkl' in equation (6.6.17) are same as the diffusion coefficient 0". therefore. it may be interpreted as the coupling effects between mass flux and elastic deformation gradient. The value of G must be determined experimentally. however. analytical analyses can provide a lower-bound and an upper-bound on its numerical values. Thus upon examining Figure 6.5 on page 205. the lower bound of coefficient G is 9 .013E-10 because if the value is lower than this. the curve of moisture content obtained from modified Fick's law will have no difference from the curve of the classical Fick's law. while the upper-bound being set as 1.27Er8 since if the value exceeds this. the moisture content of this material will be negative which is not acceptable from a physical standpoint Figure 6.6 on page 206 shows a better agreement between the experimental data and the analytical result (solid line) predicted using a model with the modified Fick's law than the analytical result obtained using classical Fick's law. However. these analytical results need to pursue the accurate G value experimentally. 205 h m cummoqeou no m\euummeuo uecOmuoeumvmm mow .ueq ..momu vcmuumox on» no men-m Ann—9902 cauueucouumuv u now venom more; use on. women «can: emu ..u.: no conga-omen 0.0 oueuum 1mm. LWL 1 u— 0— 0 0 e N 0 r — - — — — _ p _ 00.0 o 00 vcsom momma) I . 2.0 o 00 oczom mm3OA ........ .. I 00043:. .530... 6.3. 4 m \\\ x. om.0 o .. . .. 2.0 \ 4.... ( mm. Iw/w 206 eumnomaou muommxoummmeuu unmouuoouummm ecu .ued a.uomm memummoz mm. and ..ucum .euea no con«uemaou v— N— 0— 0 0 0 N b — p — n p r p 0.0 ....mm comuoeuonm 364 6.x0om noeomuoz counuvfimvmum 364 m . x0«..~....--.. «$4M: .5300 .03 G T I I TTfiTI 00... ow... om._0 m~._0 00.— mm.— Iw/w 207 Both equation (5.5.19) and equation (6.6.17) describe the relationship between mass flux and the driving forces. The mass flux in equation (5.5.19) is driven by strain rate rather than driven by the strain gradient as in equation (6.6.17). Equation (5.5.19) is derived from the point of view of irreversible thermodynamics. therefore. the strain rate plays a role as a thermodynamic force and features in an irreversible process. However. equation (6.6.17) indicates that the mass flux is driven by the strain gradient. This hypothesis is motivated specifically by the experimental evidence of bidirectional composite materials I 217 I. In order to relate equation (6.6.17) to Ihitney and Browning's postulation on their experimental investigation. the coefficient G may be divided into two variables such as G-IE where E is the elastic modulus and 1 may be considered to be the moisture diffusion velocity within the continuum and is material-dependent. Therefore. the second term on the right-hand side of (6.6.17) may be interpreted as that the mass flux is generated by the strain gradient along the through-the-thickness direction. This assumption is referred to the gas or liquid diffusion under pressure difference I 145 ]. Again the phenomenological coefficients have to be determined experimentally. CHAPTER 7 DISCUSSION. CONCLUSIONS Theoretical and experimental studies have been undertaken in order to investigate the dynamic response of the high-speed flexible linkages constructed from both commercial metals and graphite-epoxy composite laminates. Variational principles form the kernel of the theoretical studies and are employed to derive the equations of motion which govern the dynamic behavior of the mechanism systems. In addition. the variational principles provide the bases for finite element formulations. This computational approach has been recognised as a very effective numerical tool for solving these complex mathematical problems. In chapter 2. correlations between computer simulations and experimental results are reasonably good in the sense of phase and amplitude content. These have offered some degree of confidence in the mathematical models. variational approaches. finite element formulations. the solution techniques and also the assumptions made herein. A comparative study on the selection of Timoshenko and Euler-Bernoulli beam elements for modeling flexible linkages. and also the number of elements needed to effectively model the systems. was performed by Gamache and Thompson I 57 I. In addition. a study of the element interpolation function and the number of elements required was 208 209 investigated by several researchers I 46.57.102 ] in order to obtain an accurate and efficient solution technique. By observing the degree of agreement between experimental and. theoretical results in Figures (4.6)-(4.10). it is apparent that the mathematical models developed in this research. for analysing the dynamic response of linkages fabricated from commercial metals. is also capable of predicting the response of linkages made from composite materials. In general. good phase agreement between the theoretical and experimental results for four-bar linkage mechanisms was obtained. This was anticipated since the consistent mass matrix formulation utilised in this research does yield a stiffness idealised system I 182 l comparing to the actual system. Furthermore. the damping coefficient of the four-bar linkage mechanism was experimentally determined according to the procedures documented in reference I 154 l and this mechanism was also carefully adjusted in order to avoid bearing clearances and out-of-plane motion. However. the phase descrepencies occured in Figures 4.5 and 4.6 for slider-crank mechanism may be attributed to the viscous behavior and the dynamic friction of the slider assembly. There are amplitude discrepancies in Figures 4.9 and 4.10 between the experimental and computational results. It is postulated that these discrepencies may be attributed to the imperfect modeling of the constitutive equation of the material. This is somewhat nonlinear and in addition the behavior of this polymeric material is dependent upon the environmental conditions. Other possibilities responsible for the amplitude discrepencies include the fluctuation of crank speed. which is 210 assumed to be constant in the computer simulation. The industrial applications of the previously derived theories may be mainly dependent upon the fields of application. For example. safety and reliability are the most important factors in the aerospace industry. therefore. mechanical components of the aircraft may be designed with larger safety factors in order to prevent any possible failure. However. in most industrial applications. the cost of a product plays a crucial role in order to make it more competitive in the marketplace. Therefore. qualitative figures for justifying the degree of accuracy are not available and this parameter from the design specification essentially depends upon the strategies for marketing the product. The design engineers should no doubt comply with the guidances of the strategies. Sanders and Tesar I 137 I suggested that for stiff-designed industrial mechanism systems the deformations are largely due to the quasi-static application of cyclically varing inertial forces acting on the links. Therefore. the correlation of the quasi-static response is certainly important because it has a dominant influence in order to maintain a specific level of precision for its functional operation. The amplitude of vibrational response with higher frequency superimposed upon the quasi-static response are normally small compared with the amplitude of the quasi-static response. Therefore. from the above statement a 10 percent amplitude discrepancy in the sense of relative agreement of dynamic response between computer simulation and experimental results should be acceptable in general industrial 211 applications. Theoretical investigations have been developed in chapters 5 and 6 in order to predict the temperature and the moisture concentration distribution inside the link materials. and the dynamic response of the linkage mechanism subjected to both mechanical and hygrothermal loadings. In chapter 5. irreversible thermodynamics provides a general framework for deriving the governing equations. There are two advantages for this approach. First. this approach has found a great variety of applications in modeling the "real world" phenomena I 49.66 I. For example. in addition to the physical phenomena described in chapter 5. this approach may also be able to model chemical reactions occured within the system. and the aspect when an electromagnetic field acts upon a material system. Secondly. the solutions of the differential equations. such as equations (5.5.29). (5.5.30) and (5.5.31). may be easily sought by using the method of weighted residuals I 55.56 1. However. the physical insight. possessed only in the variational integrals. is not able to obtained explicitly by this method. The disadvantage of this approach is that the domain of validity of this theory is essentially the one for which local equilibrium is attained and the phenomenological equations are linearly defined. Therefore. the total entropy production. which leads to the generation of phenomenological equations. may be formed by summing the entropy production of each irreversible process. 212 There are two advantages documented herein for the mixed variational approach. The first is that it may be applicable to the condition of geometrical nonlinearity and this situation occurs frequently in machinery elastodynamics. Secondly. more insight may be deduced from utilizing this approach via a finite element formulation. The mixed variational approach has incorporated auxiliary conditions such as the strain-displacement equation. phenomenological equations and essential boundary conditions into the Hamilton's principle. So the functional depends upon temperature. moisture concentration. displacement. specific entropy density. flow potential density . strain and stress. As the basis of an elementary approximation. each field can be independently selected to achieve the discrete model. namely. the approximation function for the stresses may be different from that of the strains. The disadvantage of this mixed variational principle is the rather complex mathematics. and it may create some degree of difficulty for the industrial utilisation. In addition. from mathematical point of view this variational principle is somewhat imperfect because of the non self-adjointness existing in the heat conduction and mass diffusion equations I 56.62 ]. Therefore. future work may be based on the notation of Stieltjes convolution integral adopted in the variational principle so as to ensure the existence of the self-adjointness property. The selection of the appropriate approach to be adopted is the key for practising design engineers. The evaluatory criteria are governed 213 by the type of the problem involved and the degree of accuracy required for the particular design purpose. A careful evaluation of the above considerations toward the problem should guide us in search of an appropriate approach. The scientific method requires the interplay of theory and experiment in all research endeavors and this fundamental notion is particularly important in the study of composite materials because the behavior of the constituents is so complex. As an extension of the current work. a systematically experimental program for validating the theory with experimental evidence has been established. It provides a sound basis for testing hypotheses and can also be instrumental in guiding and directing further experimental work. In the first phase of the experimental program. test coupons should be exposed to a variety of hygroscopic loadings while under the conditions of constant temperature and sero mechanical loading and vice verse in order to determine the basic properties of the composites. such as the coefficients of thermal expansion. the diffusivities. thermal conductivities. the coefficients of moisture expansion. and the coefficients of hygrothermal cross-effects. The techniques to be adopted for performing these tests are well documented in the literature. See. for example. reference I 224 l. The second phase of the experimental program should initially involve subjecting composite plate and beam specimens to static mechanical loads in an air environment with static values of temperature 214 and humidity. A variety of response curves should be generated under conditions of variable humidity at constant temperature and vice versa. while maintaining a constant static mechanical loading. These specimens should be housed in an environmental chamber and several fixtures should be employed to impose the mechanical loading using either dead-weights or screw and 'locknut arrangements incorporating load-cells at the fixture-specimen interface. The flexural and uniaxial structural deformations should be monitored by strain gages mounted on the test specimens. and a variety of response characteristics should be generated using a combination of discrete temperature and humidity combinations. The experimental results should then be compared with the predictions of finite element based computer simulations which incorporate the mathematical models from the theoretical develcpment. If the correlation between the theoretical and experimental results is unsatisfactory. then .the models would be modified. but if it is favorable. the test specimens could then be exposed to dynamic hygrothermal loading in conjunction with the same values of the discrete static mechanical loading employed previously. Again the mathematical models would be evaluated. The dynamic hygrothermal excitation could be imposed on the specimens by the environmental chamber using a microprocessor-based programmer/controller. These devices are marketed by the manufacturer of these chambers. and they compute and generate precision time related set point signals for the profile of the desired hygrothermal environment. The third phase of the program would have the same algorithm as the 215 second phase with the exception of dynamical mechanical loading being imposed on the composite specimens. This would be accomplished by locating the specimens in dynamic test-fixtures that incorporate a vibration exciter which can subject the specimen to a broad range of . dynamic loading characteristics. This vibration exciter would be positioned inside the environmental chamber provided that the conditions develcped by the environment chamber are within the exciter specification. Vhen this is not possible. the exciter would then be bolted to the outside of the chamber and a push-rod arrangement employed to excite the specimens through the access port. The excitation of the specimens would be monitored by accelerometers. The response would again be monitored by strain gages. These dynamic tests would include the excitation of double-cantilever arrangements to study the dependence of material damping on temperature and moisture. These results will provide an important data set for the final phase of this research which concerns the study of linkage mechanisms fabricated with composite laminates. By incorporating strain-gaged beam-shaped specimens into both four-bar linkages and also slider-crank mechanisms. these specimens will be exposed to a wide variety of loads at different speeds to effectively test the predictive capabilities of the mathematical models developed in the theoretical investigation. Finally. the contributions achieved in this work to the state of the art may be summarised as follows: 1. An experimental study performed in chapter 3 clearly proves the effectiveness of the new design methodology prOposed by Thompson et a1 216 I 158.160.173.178 ]. This advocates that composite materials should be employed to reduce the elastodynamic phenomena. such as link deflections and dynamic stresses. 2. Theoretical investigations have been develcped by utilising the concept of irreversible thermodynamics. and a mixed variational principle. These provide the capacity for analysing the physical phenomena that govern the response of the linkage mechanisms which are subjected to both mechanical and hygrothermal loadings. 3. An experimental program presented as the future work provides a systematic process for exploring the material properties. the static response of the material and. furthermore. the dynamic response of the linkage mechanisms under varing temperature and humidity environmental conditions. This experimental work essentially supports the theoretical development as being a useful check for comparing the mathematical model with the experimental evidence. This future work is an essential ingredient for developing a viable design methodology based on the design and fabrication of machine systems with composite materials. 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"Coupled Thermoelastic Vibrations of A Simply Supported Beam.” Journal pp Sppnd ppp Vippppio . Vol. 83. No. 3. 1983. pp. 425-429. 102. Midha. A.. Badlani. M. L. and Erdman. A. G.. ”A Note on the Effects of Multi-Element Idealisation of Planar-Elastic-Linkage Members." rogeppingg pp.pp; gpp O,§,Ul ppppp;p Mpchgnismy Qpnfprgnc . Oklahoma City. Oklahoma. Nov. 6-9. 1977. pp. 27-1 to 27-11. 103. Midha. A.. Erdman. A. G. and Frohrib. D. A.. "An Approximate Method for the Dynamic Analysis of Elastic Linkages." ASME Journal 21 Epginepring Lo_r Inppptry. Vol. 998. 1977. pp. 449-455. 104. Midha. A.. Erdman. A. G. and Frohrib. D. A.. "Finite Element Approach to Mathematical Modeling of High Speed Elastic Linkages.” Mechanism ppp Machine Theory. Vol. 13. 1978. pp. 603-618. 105. Midha. A.. Erdman. A. G. and Frohrib. D. A.. "A Closed-Form Numerical Algorithm for the Periodic Response of High-Speed Elastic Linkages." ASME Joppnal pp Mpchgnppal pppign. Vol. 101. No. 1. 1979. pp. 154-162. 225 106. Midha. A.. Erdman. A. G. and Frohrib. D. A.. "A Computationally Efficient Numerical Algorithm for the Transient Response of High-Speed Elastic Linkages .” ASME Journal pp Mechanical erig . Vol. 101. No. l. 1979. pp. 138-148. 107. Nath. P. R. and Ghosh. A.. "Kineto-Elastodynamic Analysis of Mechanisms by Finite Element Method." Meghanism ppp Maphine Theory. Vol. 15. 1980. pp. 179-197. 108. Nath. P. E. and Ghosh. A.. "Steady-State Response of Mechanisms with Elastic Links by Finite Element Methods." Mechanism ppg Maphine Theory. Vol. 15. 1980. pp. 199-211. 109. Nayfeh. A. and Nemat-Nasser. S.. "Thermoelastic Waves in Solids with Thermal Relaxation.” pct; Mpphpnip . Vol. 12. 1971. pp. 53-$ e 110. Newmark. N. M.. "A Method of Computations for Structural Dynamics." Jpprnal pp Epgipepplng Mechapip; DivipipnI ASCE Prpppegipg . Vol. 85. No. BB. 1959. pp. 67-94. 111. Ni. R. G. and Adams. R. D.. "The Damping and Dynamic Moduli of Symmetric Laminated Composite Beams--Theoretical and Experimental Results." Jpprnal pp Qompopite Material . Vol. 18. pp.104-121. 112. Nickell. R. E. and Sackman. J. L.. "Variational Principles for Linear Coupled Thermoelasticity." Qparpprly pp Applipg Maphemapip . Vol. XXVI. No. 1. 1968. pp. 11-26. 113. Nickola. V. E.. "Strain Gage Measurements on Plastic Models.” presented at the Fall Meeting of the Society for Experimental Stress Analysis. Idaho Falls. Idaho. 1975. 114. Niordson. F. I.. "On the Optimal Design of a Vibrating Beam." Qparterly Applieg Mathematics. Vol. 23. No. 1. 1965. pp. 47-53. 115. Noise Control pp Strain Gag; Mpapurementp. Tech-Note TN-501. Micromeasurements Group. Raleign. NC. 116. Novoshilov. V. V.. Foundapions 21 pp; Nonlipppr Thpory pp Elpspipity. Graylock Press. Rochester. N.Y.. 1953. 117. Nowacki. M.. pypamic Problpms pp Thprmoelappipipy. Noordhoff International Publishing. Leyden. The Netherlands. 1975. 118. Nowacki. 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Vol. 5. 1969. pp. 1077-1m3. 123. Oden. J. T.. Finipe Elpmpnts pp Nonlinear Conpinu . McGraw-Hill Book Company. 1972. 124. Oden. J. T. and Raddy. J. N.. !;ri;pional Mephods pp Theoretical Mechanipp. Second Edition. Springer-Verlag. New York. 1983. 125. Oliver. J. B.. Vysocki. D. A. and Thompson. B. S.. "The Synthesis of Flexible Linkages by Balancing the Tracer Point Quasi-Static Deflections using Microprocessor and Advanced Materials Technologies.” Me;h;ni;m ppp Machine Ippory. Vol. 20. No. 2. 1985. pp. 103-145. 126. Pagano. N. J. and Halpin. J. C.. "Influence of End Constraint in the Testing of Anisotropic Bodies." Journal pp Cpmpoplpe Materials. Vol. 2. 1968. pp. 18-31. 127. Paipetis. S. S.. ”Mathematical Modelling of Composites." in pevplopments pp Composite Materials--2 Sprep; Apalypig. G. S. Holister (ed.). Applied Science Publishers Inc.. N.J.. 1981. 128. Papoulis. A.. pp; Eoprier lnpegral ppp lp; Applipation . McGraw-Hill Co.. New York. 1962. 129. Phrkus. B.. Thprmpelpptipity. Blaisdell Publishing Co.. Valtham. Mass.. 1968. 130. Pipes. R. B.. Vinson. J. R. and Chou. T. I.. "On the Hygrothermal Response of Laminated Composite Systems." Jourp;l pp Compopipe Maperi;l . Vol. 10. 1976. pp. 129-148. 131. Provost. J. H. and Tao. D.. "Finite Element Analysis of Dynamic Coupled Thermoelasticity Problems with Relaxation Times.” ASME. Journal pp Applied Mechpnips. Vol. 50. 1983. pp. 817-822. 132. Ray. J. D. and Bert. C. I.. "Nonlinear Vibrations of Beams with Pinned Ends." ASME qurnal pp Engineering pp; lupupppy. Vol. 9. 1969. pp. 977-1004. 133. Reddy. J. N.. ”Modified Gurtin's Variational Principles in IL; 227 the Linear Dynamic Theory of Viscoelasticity." anprpational Joprpal pp Soli s ppg Sprpcpurps. Vol. 12. 1976. pp. 227-235. 134. Raddy. J. N.. 4; Introduction pp 1p; 'ni e em nt Method. McGraw-Hill Inc.. 1984. 135. Robinson. E. A. and Silvia. M. T.. Qigipal Signal Ppopepping ppg Time S;ries Analysis. Chapter 5. Holden-Day Inc.. San Francisco. CA. USA. 1978. 136. Sarma. B. S. and Varadan. T. R.. "Lagrange-Type Formulation for Finite Element Analysis of Non-linear Beam Vibrations." Jopppal pp Sound ppg Vipration. Vol. 86. 1983. pp. 61-70. 137. Sanders. J. R. and Tesar. D.. "The Analytical and Experimental Evaluation of Vibratory Oscillations in Realistically Preportioned Mechanisms.” ASME qurpal pp Mephpnipal Depign. Vol. 100. 1978. pp. 762-768. 138. Schmit. L. A.. Jr. and Farashi. B.. ”Optimum Laminate Design for Strength and Stiffness.” lnpppnational Joypnal pp; Npp;rip;l pppppp; pp Engineering. Vol. 7. 1973. pp. 519-536. 139. Schults. A. B. and Tsai. S.W.. "Dynamic Moduli and Damping Ratios in Fiber-Reinforced Composites.” Journal pp Composip; Maperial . Vol. 2. 1968. pp. 368-379. 140. Schults. A. B. and Tsai. S. M.. "Measurements of Complex Dynamic Moduli for Laminated Fibre-Reinforced Composites.” Journal pp Qomposip; Mat;rlal;. Vol. 3. 1969. pp.434-443. 141. Shapery. R. A.. "On the Time Dependence of Viscoelastic Variational Solutions." Qparterly pp Applppp Math;mapip;. Vol. XXII. No. 3. 1964. pp. 207-215. 142. Sharif-Bakhtiar. M. and Thompson. B. 8.. "Reducing Acoustical Radiation from High-Speed Mechanisms by Judicious Material Selection: An Experimental Study." ASME Paper 84-DET-49. 1984. 143. Shen. C. H. and Springer. G. S.. ”Moisture Absorption and Desorption of Composite Materials.” qurpal pp om o i e M;p;rl;l;. Vol. 10. 1976. pp. 2-20. 144. Shen. C. H. and Springer. G. S.. ”Effects of Moisture and Temperature on the Tensile Strength of Composite Materials." qurnal pp Composite Materipls. Vol. 11. 1977. pp. 2-16. 145. Shi. J. J.. Ueinman. A. S. and RajagOpal. R. R.. "Applications of the Theory of Interacting Continua to the Diffusion of a Fluid Through a Nonlinear Elastic Medium.” luternapional Joyrpal pp Engipeering Spiencg. Vol. 9. 1981. pp. 871-889. 146. Sih. G. C.. Shih. M. T. and Chou. G. C.. "Transient 228 Hygrothermal Stresses in Composites: Coupling of Moisture and Heat with Thmperature Varying Diffisivity." lntpppapional Journal pp Enginepring Splencp. Vol. 18. 1983. pp. 19-42. 147. Sivertsen. O. I. and WalOen. A. 0.. “Non-linear Finite Element Formulations for Dynamic Analysis of Mechanisms with Elastic Components." ASME Paper 82-DET-102. 1982. 148. Song. J. O. and Haug. E. J.. "Dynamic Analysis of Planar Flexible Mechanisms." meputpr Methpds pp Appllpd Mpghapig; ppg Engineerlng. Vol. 24. 1980. pp. 358-381. 149. 'Springer. G. S.. "Moisture Content of Composites Under Transient Conditions." Joprn;l pp gpmpopipe Mpperipl . Vol. 11. 1977. pp. 107-122. 150. Springer. G. 8.. ”Effects of Tbmperatura and Moisture on Sheet Modelling Compounds.” Journal pp Rpinforppd P a ti s ppp Composite . Vol. 2. 1983. pp. 70-89. 151. Springer. G. 8.. "Model for Predicting the Mechanical Properties of Composites at Elevated Temperatures." Japrnal pp R;lnforped Plastips ppp Compositep. Vol. 3. 1984. pp. 85-95. 152. Springer. G. S. and Tsai. S. W.. ”Thermal Conductivities of Unidirectional Materials." Joyrpal pp Compoplp; Mpppylpls. Vol. 1. 1967. pp. 166-173. 153. Springer. G. S.. Environmpntal Effppt; {pp Qomposite Materialp. Vol. I and Vol. II. Technomic Publishing Company. Inc.. 1984. 154. Stamps. F. R. and Bagci. 0.. "Dynamic of Planar. Elastic. High-Speed Mechanisms Considering Three-Dimensional Offset Geometry: Analytical and Experimental Investigations." ASME Paper No. 82-DET-34. 1982. 155. Sunada. W. and Dubowsky. S.. "The Applications of Finite Element Methods to the Dynamic Analysis of Flexible Spatial and Co-planar Linkage Systems." Journal of Mechanical Design. ASME. Vol. 103. No. 3. July 1981. pp. 643-651. 156. Sunada. W. H. and Dubowsky. 8.. "On the Dynamic Analysis and Behavior of Industrial Robotic Manipulators with Elastic Members." ASME Joprnal pp Mephpnisms, Transmissions £29 butomapion pp 2;;ign. Vol. 105. No. 1. March 1983. pp. 42-51. 157. Sung. C. R. and Thompson. B. S.. "A Note on the Effect of foundation Motion upon the Response of Flexible Linkages." ASME Paper No. 82-DET-26. 1982. 158. Sung. C. E. and Thompson. B. S.. "Material-Selection: An Important Parameter in the Design of High-Speed Linkages." ppchanism ppp 229 Maphine Theory. Vol. 19. No. 4/5. 1984. pp. 389-396. 159. Sung. C. R.. Thompson. E. S.. Ring. T. M. and Wang. C. B.. ”An Experimental Study on the Nonlinear Elastodynamic Response of Linkage Mechanisms.” Meph;ni;m ppg Maphppe Theory. in press. May. 1984. .160. Sung. C. R.. Thompson. E. S.. Crowley. P. and Cnccio. I.. ”An Experimental Study to Demonstrate the Superior Response Characteristics Mechanisms Constructed with Composite Laminates." Mephanigm ppg M;phine Theory. in press. May. 1985. 161. Sung. C. R.. Thompson. B. S. and McGrath. J. J.. "A Variational Principle for the Linear Coupled Thermoelastodynamic Analysis of Mechanism systems.” ASME qurnal pp M;chani;msI TransmissionsI ppg utomation pp Deslg . Vol. 106. No. 3. Sept. 1984. pp. 291-296. 162. Sutherland. G.H.. "Analytical and Experimental Investigation of High-Speed Elastic-Membered Linkage." ASME Jpppppl pp ppgpp;;;ppp pp; Industry. Vol. 98. 1976. pp. 788-794. 163. Tajima. Y. A.. ”The Diffusion of Moisture in Graphite Fiber-Reinforced Epoxy Laminates." SAMPE‘Qpppppply. 1980. pp. 1-9. 164. Thmbour. Y.. "On Local Thmperature Overshoots Due to Transport Coupling of Heat and Moisture in Composite Materials." Jpprpal pp Composite Materials. Vol. 18. Sept. 1984. pp. 478-494. 165. Thmbour. Y.. " A Three-Dimensional Field Transformation for AnisotrOpic Coupled Heat and Moisture Transfer in Composite Materials.” to be submitted to the Jourpal pp om osi e Matpripls. 1984. 166. Tao. D. and Prevost. J. B.. "Relaxation Effects on Generalised Thermoelastic Waves.” Jpprn;l pp Thepmal Str;;;e;. Vol. 7. 1984. pp. 79-89. 167. Tauchert. T. R. and Adibhatla. 8.. "Design of Laminated Plates for Maximum Stiffness." Journal 21 Compopit; Materigl . 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"The Finite Element analysis of Mechanism Components made from Fiber-Reinforced Composite Materials." ASME Paper No. 80-DET363. 1980. 174. Thompson. E. S.. "A Variational Formulation for the Finite Element Analysis of High-Speed Machinery." ASME Paper No. 80-IA/DSC-38. 1980. 175. Thompson. E. S.. "Variational Formulations for the Finite Element Analysis of the Noise Radiation from High-Speed Machinery.” ASME Journal 21 Engineering pp; Induspry. Vol. 103. No. 4. Nov. 1981. pp. 385-391. 176. Thompson. B. S.. Zuccaro. D.. Gamache. D. and Gandhi. M. V.. "An Experimental and Analytical Study of a Four Bar Mechanism with Links Fabricated from a Fiber-Reinforced Composite Material." e h ni m ppp Machine Thcopy. Vol. 18. No. 2. 1983. pp. 165-171. 177. Thompson. B. S.. Zuccaro. D.. Gamache. D. and Gandhi. M. V.. "An Experimental and Analytical Study of the dynamic Response of Linkage Fabricated from a Unidirectional Fiber-Reinforced Composite Laminate." ASME Journal pp Megh;ni;msI Transmi;slon;I ppp Aupom;pipn p; Qe;ign. Vol. 105. No. 3. Sept. 1983. pp. 528-536. 178. Thompson. B. S. and Sung. C. E.. "A Variational Formulation for the Dynamic Viscoelastic finite Element analysis of Robotic Manipulators Constructed from composite Materials." ASME Jpprpal pp MpphanismsI Transmipsions, ppg Aptomapion pp Dppign. Vol. 106. No. 2. 1984. pp. 183- 190. 179. Thompson. B. S. and Sung. C. R.. "The Design of Robots and Intelligent Manipulators Using Composite Materials." Mephppism ppp Machine Ippory (in press). 180. Thompson. 8. S. and Sung. C. R.. "A Variational Formulation for the Nonlinear Finite Element Analysis of Flexible Linkages: Theory. Implementation and Experimental Results." ASME Joprnal pp M;ph;ni;m;, Tr;n;mi;;ipns ppg Appom;pion pp Qpppgp. Vol. 106. No. 4. 1984. pp. ‘82-‘88 e . 181. Thompson. B. S. and Sung. C. R.. 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ASME Jpppppl pp Epglpperlng pp; Jppppppy. Vol. 948. No. 2. May 1972. 198. Woinowsky-Rriegar. S.. ”The Effect of an Axial Force on the Vibration of Hinged Bars.” ASME Journ;l pp Applppp Mppppnipp. Vol. 17. 1950. pp.35-36. 199. Wu. C. H. and Tauchert. T. R.. ”Thermoelastic Analysis of Laminated Plates 1: Symmetric Specially Orthotropic Laminates." Jpprnal pg Thermal Sppeps. Vol. 3. 1980. pp. 247-29. 200. Wu. C. H. and Tauchert. T. R.. "Thermoelastic Analysis of Laminated Plates 2: Antisymmetrical Cross-ply and Angle-ply Laminates.” Journ;l pp Thppmal Sprpss. Vol. 3. 1980. pp. 365-378. 201. Yamada. Y.. Takabatake. H. and Sato. T.. ”Effect of Time-Dependent Material Properties on Dynamic Response." nte n ion Jourpal pp; Numerical Mephods 1; Eggpppppppg. Vol. 8. 1974. pp. 202. Zhang. Ca. and Grandin. H. T.. ”Kinematical Refinement Technique in Optimum Design of Flexible Mechanisms with External Loads Using Static Condensation." ASME Paper No. Sl-DET-104. 203. Zhang. Ca. and Grandin. H. T.. "Optimum Design of Flexible Mechanisms." ASME Jpppppl pp M;ph;pi;msI Trpn;mi;sion;I ppp ppppmppipn a 0222‘ a v01. 105a 11138 1983p PP. 267-272e 204. Zienkiewics. O. C.. Watson. M. and King. I. P.. "A Numerical Method of Visco-Elastic Stress Analysis." lpt;rn;plpnal Jpppppl p1 Mpphanippl §pipnp . Vol. 10. 1968. pp. 807-827. 205. Zienkiewics. O. C.. “The Finite Element Method: From Intuition to Generality." Applppp Mephapip; zpview. Vol. 23. No. 3. 206. Zienkiewics. O. C.. Ippgfipppp; Elpmpnp Mephpp. Maidenhead. England: McGraw-Hill Book Company. 1977. 207. Donaa. J.."Thermal Conductivities Based on Variational Principles." Journal pp Qompp;lpe gupppp;l_. Vol. 6. 1972. pp. 262-270. 208. Flaggs. D. L. and Crossman. F. W.. ”Viscoelastic Response of a Bonded Joint due to Transient Hygrothermal Exposure." M ern 233 Developments pp Composite Maparipls ppp Spruppure;. J. R. Vinson. ed. ASME (1979). 209. Flaggs. D. L. and Crossman. F. 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Vol. 11. 1977. pp. 250-264. 219. Gurtin. M. E. and Yatcmi. C.. "On a Model for Tic Phase Diffusion in Composite Materials." Journal pp Qompoplte Maperiplg. Vol. 13. 1979. pp. 126-130. 220. Thompson. B. S.. ”An Elastodynamic Analysis of Planar Linkage Mechanisms." Doctoral Thesis. University of Dundee. 1976. 234 221. Verette. R. M.. "Temperature/Humidity Effects on the Strength of Graphite/Epoxy Laminates.” AIAA Paper No. 75-1011. AIAA 1975 Aircraft Systems and Techn010gy Meeting. Los Angels. California. August 4-7. 1975. 222. Refer. Jr. E.E.. “Larsen. D. and Humphreys. V. E.. ”Development of Engineering Data on the Mechanical and Physical Properties of Advanced Composite materials." Thchanical Report AFML-TR-74-266. Feb.. 1975. Air Force Material Labortory. Wright-Patterson Air Force Base. Dayton. Ohio. 223. Wark. R.. Thermodynamip;I New York. McGraw-Hill Book Company. Third Edition. 1977. 224. Whitney. J. M.. Daniel. I. M. and Pipes. R. B.. 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