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A survey of finite element techniques for mechanism design
Mechanism and Machine Theoo' Vol. 21, No. 4, pp. 351-359, 1986 Printed in Great Britain. All rights reserved
0094-114X/86 $3.00 + 0.00 Copyright ~ 1986 Pergamon Journals Ltd
A SURVEY OF FINITE ELEMENT TECHNIQUES FOR MECHANISM DESIGN B. S. T H O M P S O N a n d C. K. S U N G Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, U.S.A. (Received 5 September 1984) Abstract--During the creation of a modern machine system, the designer must generally develop a mathematical model to investigate the levels of vibrational activity, dynamic stresses, bearing loads and often the acoustic radiation associated with the system. Since all of these phenomena may be represented by field theories whose governing equations of motion may be stated as either differential, integral or integro-differentialequations, they are all amenable to solution by the single most powerful computational tool available to the analyst: the finite element method. Mechanism design methodologies based on this versatile technique are reviewed herein, their different characteristics are highlighted and future trends indicated. INTRODUCTION In 1875, Reuleaux [1] defined a mechanism to be an assemblage of resistant (rigid) bodies, connected by movable joints, to form a closed kinematic chain with one link fixed and having the purpose of transforming motion. This fundamental assumption was incorporated into mechanism design methodologies for almost the next 90 years, and these techniques were employed in the successful development of a very broad class of machinery products. In the early 1960s, however, marketplace interactions and commercial stimuli for increased productivity were responsible for the evolution of a new class of mechanisms. The performance specifications associated with these mechanical systems generally required operating speeds to be increased while imposing additional criteria on design procedures. Typically, these criteria called for one or more of the following: a reduction of power consumption, an increase of external loading, the reduction of acoustical radiation and the generation of more accurate output characteristics. Under these stringent operating conditions, elastodynamic phenomena, that are of little consequence in the operation of low-speed mechanisms, have a significant detrimental affect on the performance characteristics of this new generation of high-speed machinery. The kernel of the problem is that the links vibrate due to their inherent flexibility and the more severe force fields which are being imposed upon them. Consequently, the traditional rigid-linked analyses are inadequate for the design of these machine systems because the methematical models, by definition, are unable to represent the phenomena associated with these assemblages of interconnected flexible bodies. The response of the academic and industrial communities to this major limitation in mechanism design methodologies is documented in two comprehensive survey papers published in 1972 [2, 3]. It is clear from MM~[
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these publications that a wide variety of analytical and computational techniques were proposed, but with such a complex set of dynamic boundary-value problems to be solved in the design of this class of mechanisms, closed-form analytical solutions are unattainable, and the designer clearly must resort to employing computers for modeling and simulation purposes. While several numerical techniques continued to be developed further after 1972, it soon became apparent that the finite element method was evolving as the most popular approach for analyzing flexible mechanisms. The reasons for this popularity are numerous, and will not be repeated here. However, the interested reader is referred to Refs [4-9] which are reviews of finite element methods by authorities in the field. This paper presents a reasonably complete list of references on finite element methods for the design of planar mechanisms. This body of literature is reviewed herein, by highlighting the phenomena being studied, the design function being undertaken and the different aspects of finite element analyses, such as element selection, formulation strategy and the procedure employed to solve the equations of motion.
FINITE ELEMENT FORMULATIONS 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 the 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 all of the analyses to date assume that joints are without clearance, thereby greatly simplifying the complexity of modeling impactive bearing forces. Furthermore, most analyses,
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B.S. THOMPSONand C. K. Slry~i
with the exception of Refs [10] and [l 1], 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 element in an undeformed reference state. Local element matrices [M], [C] and [K] are generally established to model the mass, damping and stiffness characteristics of each link before pre- and postmultiplying 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. 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 for a high speed linkage are written [Mc] {U} + [C6] {I~T}+ [K(;] {U} :
- [M~]{ii} + {x} + {r},
where {U} is the column vector of deformation displacements, ( . ) is the time derivative, [Mz], [CG] and [K~] are the global mass, damping and stiffness matrices, respectively, which may assume many different formats depending upon the mathematical assumptions, {X} are body forces, and {T} are surface tractions. The nodal absolute acceleration terms, derived from a rigid-body kinematic analysis of the mechanism, are contained in the vector {~}, 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 the model in order to avoid singular matrices. Winfrey[12] accomplished this by directly applying the principle of conservation of momentum to the complete mechanism. Imam et al.[13] considered the crank as a cantilever beam to avoid this complication, and this approach was also employed by Midha et al.[14], and Gandhi and Thompson[15]. Nath and Ghosh[16] 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 the equations o1' motion. Winfrey[12,17] 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 yields directly the nodal displacements and requires displacement compatibility on interelement boundaries. Erdman el al.[18,19] employed the equilibrium finite element method (flexibility method of structural analysis) to study flexible mechanisms. With this approach, adjacent elements have equilibrating stress distributions on interelement 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, and mechanism design is no exception. Midha et al.[14, 20- 23] 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 [24-26]. Bagci et a/.[27-35] 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 a/.[15, 36-50] 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 [47, 48, 50] and also the acoustic radiation from linkage machinery [43, 45]. This latter work involved modeling the operation of mechanisms submerged in a perfect fluid as a fluid-structural interaction problem based on interacting continua. ELEMENT SELECTION 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. Will information on the axial and flexural deformations suffice, or is the torsional deformation field also important? Should a linear or a geometrically
A survey of finite element techniques for mechanism design
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nonlinear model be developed? What are the consti- equations. Generally, in the mode superposition approach, the damping is assumed to be uncoupled and tutive equations of the link material? And so on. is given in each mode as a percentage of the critical Generally, these decisions will be guided by endamping. However, when adopting the step-by-step gineering intuition or experimental evidence. approach of numerical integration, the complete A large number of papers [10, 12, 13, 21, 34, 38, damping properties of the mechanism must be estab50-60] have been devoted to studies of the planar lished, and this may be a difficult task. elastodynamic response of flexible linkages by modIn the appendix of Ref. [21], Midha et al. discussed eling the links using finite elements with only one the different damping matrices in the context of spatial variable. The axial response is generally modmodal superposition solutions. Winfrey[12] assumed eled by a linear interpolation function and the a damping matrix proportional to a linear combinaflexural response by a cubic interpolating polynomial tion of the mass and stiffness matrices (Rayleigh using elements originally derived for structural applidamping). In contrast to this, Alexander and cations. Bahgat and Willmert[53], in one of the Lawrence[51] assumed the damping coefficient for pioneering works in this field, considered both the each o r t h ~ coordinate to be defined by axial and flexural responses of a wide variety of flexible linkages using a finite element formulation Cii = 2 ~ix/ K , / M i i . The energy-dissipation characteristics of a mechemploying higher-order hermite polynomial approxianism are dependent upon both the constitutive mations for the deformation fields. The same quintic equations of the link material and also the characterelement was also employed by Hossne et a/.[58] and istics of the joints of the mechanism. In recent also by Cleghorn et al.[55-57]. In [56], a comparison was undertaken between finite elements utilizing combined experimental and finite-element-based investigations [26, 39, 44, 46, 49], logarithmic decrequintic polynomial approximations and those based ment transient response studies have been undertaken on cubic polynomials. These results not only sugfor a number of mechanism configurations and the gested that axial deformations may be neglected in experimental data employed to establish semithe analysis of some flexible mechanisms, but also empirical damping matrices for the damping in the that fewer of the higher-order elements are required mechanism. Bagci and Kalaycioglu[28] have underto generate a specified solution accuracy when comtaken preliminary work on coulomb damping in pared with the results from a model incorporating a joints, which they then combine with the model for larger number of lower-order cubic elements. The finite-line-element nomenclature employed by material damping employed in [51], thereby employBagci et a1.[27-35] is an alternative terminology for ing individual mathematical models for both mechthe standard rod and beam elements of structural anism joints and also the link materials. mechanics. Naganathan and Willmert[62] proposed that two special elements should be employed for quasistatic analyses, because they argued, the cubic SOLUTION PROCEDURES beam element does not yield exact results for the Two computational methods are generally emanalysis of mechanisms. Khan and Willmert[63] developed an element based on harmonic functions for ployed to solve the equations of motion for both linear and geometrically nonlinear vibrational anaperforming vibrational analysis of linkages. The two common formulations for mass matrices, lyses. These are the modal superposition approach namely the consistent mass matrix [14-16, 18-23, [12, 22, 26, 59] and the use of direct integration meth36-39,41,44,51,60] and the lumped mass matrix ods [10, 15, 16, 51], such as Runge-Kutta algorithms [27-35] have both appeared in the mechanism design or the Newmark method. literature. Tong et al.[65] demonstrated that while the The former approach is based on the assumption lumped mass approach will not suffer any loss in the that the displacement vector may be expressed as a rate of convergence when utilizing simple rod ele- linear combination of the vibrational mode shapes. ments, a consistent mass formulation is to be pre- This solution strategy is most efficient if the essential ferred when using higher-order elements such as dynamic response of the mechanism is contained in beam elements. the first few modal combinations. Hence it is most Having discussed the mass and stiffness matrices useful for studying the steady-state response of employed to model mechanisms, attention is now systems operating at constant crank frequencies. focused on damping matrices. Damping in materials However, additional complexities arise when develis a complex phenomenon [66] which probably re- oping geometrically nonlinear elastodynamic requires the development of thermomechanical models sponses [25]. The approach is not to be recommended in order that it be fully understood. The interested for linkages fabricated with journal bearings because reader is referred to a comprehensive work on the the inherent clearance needed for the operation of these joints creates impact loading which is often subject [67]. The assumptions employed to develop damping characterized by high-frequency components. This matrices are partially governed by the solution tech- generally requires a solution based on many modes in nique to be adopted for solving the finite element order to predict the response of the mechanism.
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B.S. Ft~O.MPSONand C K. SUNtl
Numerical integration algorithms employ step-bystep 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 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 may be applied to accurately model systems subjected to complex loading, or loads containing significant high frequency components. Imam et al.[13] 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 et al.[22] developed 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 [14], and in addition, a numerical algorithm for performing transient analyses of linkages [23]. While the above procedures are dedicated to determining the elastodynamic response of linkages, some authors have advocated that a quasistatic analysis of a linkage mechanism is quite adequate for most design purposes in an industrial environment [62, 68], 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 nonhomogeneous 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 o1"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 Willmert[68]. This philosophy involves condensing all internal degrees of freedom of each link to create a superelement with only the principal degrees of freedom retained, and has been used extensively in commercial codes tot structural dynamics problems. The authors reduced the number of system equations by 50%, so that the computational effort, which is proportional to the cube of the number of equations, was considerably reduced. As they rightly indicated in [68], this approach is especially useful when the designer is
searching for an optimal solution which generally involves many iterative analyses. ANALYSIS OF FLEXIBLE MECHANISMS The vast majority of the papers oll 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 { T} in equation (I) are often neglected. This approach generally yields a transverse ftexural 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 [14, 53, 60] and also transient responses [23] have been obtained. Since these classes of solution can be computationally time consuming, some authors [62, 68] have advocated that a quasistatic analysis is adequate for the analysis of most industrial machinery. This approach neglects the terms [M~;]{I~I} and [(~;]~ " ~" Uj in equation (1), thereby considering the mechanism to be a statically loaded structure. This assumption greatly simplifies tile con> putational aspects of the finite element analysis (see the previous section) 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 featured in some analyses [16, 21,24, 25, 49, 55]. 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. The higher-order theories modeling the elastic motion of links employ geometrically nonlinear analyses which retain the terms in the strain-displacement equations that couple the axial and flexural deformations [16, 24, 25, 49, 60, 71 73]. 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 to model each flexible link [12, 51]. 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[51, 52, 74, 75] in pioneering work on combined experimental and theoretical investigations of flexible four bar linkages. Midha et al.[20] reinforced these conclusions by demonstrating the effects of multielement idealizations on the response of a structure. They showed that a 15% error existed in
A survey of finite element techniques for mechanism design the second mode frequency and a 400% error in the third mode when a single element was employed. Similar errors were demonstrated by Gamache and Thompson[36] 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 coplanar 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[35]. Although the natural frequencies of a mechanism are continually changing during the operating cycle, as the stiffness and mass characteristics change relative to a fixed reference frame, nevertheless, critical speed ranges must be identified and avoided in practice. This class of problem has been investigated by Bagci et al.[29, 31, 34] using finite line elements and lumped mass matrices to study slider crank mechanisms, four-bar linkages and also six-bar mechanisms. Solutions were generated by an eigenvalue algorithm and compared with experimental data.
SYNTHESIS OF FLEXIBLE MECHANISMS
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 [76]. 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 because the analysis and synthesis procedures cannot be conveniently decoupled, as occurs in the design of rigid-linked mechanisms. While 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 [54, 59, 77-83], and they are all based on finite element methods. Before discussing the different solution strategies, it is appropriate to again review equation (1), which provides the key to understanding the proposed methodologies. Generally, when a mechanism is operating in a high-speed mode, the body forces and surface tractions may be neglected in comparison with the inertial loading; and hence {X} and {T} disappear. Premultiplying all the terms by the inverse of the mass matrix [M~] -~ yields [/] {1~} + [ M a ] - ' [ C a ] {(J} + [Me]-' [Ka] {U} = - [I]{I~}.
(2)
Thus it is evident that for a given mechanism operating at a given speed, the elastodynamic response is
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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 [54, 59, 77-83] involves designing links in the commercial metals while optimizing the crosssectional geometry of the members. The second [41,69] 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.[13, 18, 19, 59, 77] developed a general method of kineto-elastodynamic analysis and synthesis based on the equilibrium finite element method. The link geometries were synthesized in aluminum alloys using optimization techniques, which incorporated stress and deflection constraints, to develop special cross-sectional shapes and tapered members. Willmert et aL[78-81] 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. The approach is applicable to all finite element analyses [81], and the authors advocate employing the "rigidbody stress" as the initial searching point. Cleghorn et aL[54] employ the same algorithm as in Khan et aL[79] with the modification that each iteration 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 iterations needed to achieve the optimum solution when employing the finite element formulation presented in [55]. Zhang and Grandin[82, 83] 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% of that obtained in [54] when addressing the same synthesis problem. In contrast to the previous approaches which are all concerned with homogeneous isotropic materials, Thompson et al.[38, 40, 41, 47] have proposed 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 [40, 41,46, 47]. 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
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B.S. THOMI'SONand ('. K. St~y~i
materials. The models have been verified in combined experimental and computational studies [44, 46] and the superior response demonstrated in experimental comparative studies [40]. A third approach for synthesizing flexible linkages [69, 70] 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. DIRECTIONS OF FUTURE RESEARCH The success of the finite element method in both the industrial environment and the academic community may be attributed to its generality, versatility and its ability to analyze field problems with complex boundary conditions. To be effective, this new methodology requires interdisciplinary contributions from the fields of theoretical mechanics, approximation theory, numerical analysis and of course computer science. While industry craves for finite element methods that are more efficient, have user-friendly pre- and post-processors and which exploit the capabilities of the large supercomputers, this concluding section will focus instead on topics of fundamental research in mechanism design that have emerged from this review of the current literature. The maturing of research on the elastodynamic response of mechanisms is clearly evident when the prediction of finite element codes are validated by experimental data from the laboratory [24-26, 35, 39, 40, 44, 46, 49]. It is anticipated that this mode of research, based on the scientific method, will continue as investigators probe damping phenomena, geometrically nonlinear elastodynamic effects and the significance of inertial coupling terms [16, 24, 25, 49, 60, 71 73]. These terms will probably assume a significant role in the operation of ultra high-speed machinery. Under these high-speed operating conditions, outof-balance loading and the consequences of foundation excitation must also be modeled to establish a valid research program because these phenomena cannot be investigated in isolation in the laboratory. While these studies are limited to responses in the plane of the mechanism, further work is clearly needed on out-of-plane coupling as dictated by the coplanar nature of a large class of industrial machinery. Three-dimensional elements must not only be developed to model the out-of-plane responses of these systems, but also the dynamic behavior of robotic manipulators. Preliminary finite element work on analyzing this important class of mechanisms has already been published [33, 47, 84, 85]. Probably the two largest voids in the finite element literature on flexible mechanisms concern the modeling of joints and also the synthesis of these systems. All mechanisms contain joints which may incorpo-
rate either frictionless bearings, as in precision products, o1 alternatively plain bearings or bushes. These machine elements require a running fit in order that they function properly and this radial clearance results in impactive loads being generated in the joint, Considerable work has already been undertaken on modeling joints with clearance, and this is reviewed in a comprehensive survey paper by Haines[64], but the results of this research has not yet been incorporated into finite element codes. Contact-impact phenomena are difficult to model and some survey papers are presented in a book edited by Cheng and Keer[86]. The major challenges in this area are the modeling of the surface tractions and solving the associated equations of motion for which hybrid finite elements or global-local finite elements are probably required. One of the consequences of the impactive loading at the joints of a mechanism is the excitation of the vibrational response of the adjacent links which can generate noise radiation. If industrial machinery is to be designed to comply with the federal legislation restricting the noise levels to which employees may be exposed, then the designer requires a computer-based predictive capability for assessing a design prior to its manufacture. The finite element method is an obvious tool for this fluid-structural analysis, but only preliminary studies has been accomplished to date [43, 45], although there has been some success with an alternative approach [87]. Prerequisites for this class of work includes a good model of joints with radial clearance and also the ability to model unbounded fluid domains using infinite finite elements, or else global-local finite elements. Research on synthesizing flexible mechanisms requires the integration of a finite element algorithm with an optimization algorithm for both the elastodynamic effects and also the kinematic parameters of the mechanisms. Since these methods are of an interactive nature, further work is needed to reduce the time spent in the finite elemenl routine, by developing superior solution techniques and also generating superelements with less degrees of freedom. The models for mechanisms fabricated in composite laminates are of a first-order, using a continuum geometrically weighted averages concept to represent the materials [41,44,47, 50]. Clearly this can be extended to include volume fraction, stacking sequences, etc., to further extend the ability to model these complex materials. However, more demanding work needs to be undertaken to more closely model the constitutive behavior of these polymeric materials when subjected to variations in temperature and moisture. Again variational methods may provide a useful foundation for finite element formulations to this class of problem [61].
Acknowledgement---This material is based upon work supported by the National Science foundation under grant MEA-8216777. This funding is gratefully acknowledged.
A survey of finite element techniques for mechanism design REFERENCES 1. F. Reuleaux, Theoretische Kinematik: Grundziige einer Theorie des maschinenwesens. Vieweg, Brunswick, Deutschland (1875). English translation by A. B. W. Kennedy, Kinematics of Machinery. Macmillan, London (1876). Reprinted by Dover Publications, New York (1963). 2. A. G. Erdman and G. N. Sandor, Kineto-elastodynamics--a review of state of art and trends. Mech. Mach. Theory' 7, 19-33 (1972). 3. G. G. Lowen and W. G. Jandrasits, Survey of investigations into the dynamic behavior of mechanisms containing links with distributed mass and elasticity. Mech. Mach. Theory 7, 3-17 (1972). 4. R. W. Clough, The finite element method after twentyfive years: a personal view, Comput. Struct 12, 361-370 (1980). 5. T. J. R. Hughes, Some current trends in finite element research. Appl. Mech. Rev. 33, 1467-1477 (1980). 6. A. K. Noor and W. D. Pilkey (Eds), State-of-the-Art Surveys on Finite Element Technology. Applied Mechanics Division, ASME, New York (1983). 7. J. T. Oden and K. J. Bathe, A commentary on computational mechanics. Appl. Mech. Rev. 31, 1053-1058 (1978). 8. O. C. Zienkiewicz, The finite element method: from intuition to generality. Appl. Mech, Rev. 23, 249-256 (1970). 9. O. C. Zienkiewicz, The generalized finite element method--state of the art and future directions. A S M E Jl Appl. Mech. 50, 1210-1217 (1983). 10. T. E. Blejwas, The simulation of elastic mechanisms using kinematic constraints and Lagrange multipliers. Mech. Mach. Theory 10, 441-445 (1981). 11. J. O. Song and E. J. Haug, Dynamic analysis of planar flexible mechanisms. Comput. Meth. Appl. Mech. Engng 24, 358-381 (1980). 12. R. C. Winfrey, Elastic link mechanisms dynamics. A S M E Jl Engng Ind. 93, 268-272 (1971). 13. I. Imam, G. N. Sandor and S. N. Kramer, Deflection and stress analysis in high speed planar mechanisms with elastic links. A S M E Jl Engng Ind. 95B, 541-548 (1973). 14. A. Midha, A. G. Erdman and D. A. Frohrib, An approximate method for the dynamic analysis of elastic linkages. A S M E Jl Engng Ind. 99B, 449-455 (1977). 15. M. V. Gandhi and B. S. Thompson, The finite element analysis of flexible components of mechanical systems using a mixed variational principle. ASME Paper No. 80-DET-64. 16. P. K. Nath and A. Ghosh, Kineto-elastodynamic analysis of mechanisms by finite element method. Mech. Mach. Theory 15, 179-197 (1980). 17. R. C. Winfrey, Dynamic analysis of elastic link mechanisms by reduction of coordinates. A S M E Jl Engng Ind. 94B, 577-582. (1972). 18. A. G. Erdman, I. Imam and G. N. Sandor, Applied kineto-elasto-dynamics. Proceedings of the 2nd OSU Applied Mechanisms Conference, Stillwater, Okla, Paper No. 21 (1971). 19. A. G. Erdman, G. N. Sandor and R. G. Oakberg, A general method for kineto-elastodynamic analysis of mechanisms. A S M E Jl Engng Ind. 94B, 1193-1205 (1972). 20. A. Midha, M. L. Badlani and A. G. Erdman, A note of the effects of multi-element idealization of planarelastic-linkage members. Proceedings of the 5th OSU Applied Mechanisms Conference, Stillwater, Okla, pp. 27.1-17.11 (1977). 21. A. Midha, A. G. Erdman and D. A. Frofirib, Finite element approach to mathematical modeling of high speed elastic linkages. Mech. Mach. Theory 13, 603~518 (1978).
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22. A. Midha, A. G. Erdman and D. A. Frohrib, A closed-form numerical algorithm for the periodic response of high-speed elastic linkages. A S M E JI Mech. Design 101, 154-162 (1979). 23. A. Midha, A. G. Erdman and D. A. Frohrib, A computationally efficient numerical algorithm for the transient response of high-speed elastic linkages. A S M E Jl Mech. Design 101, 138-148 (1979). 24. D. A. Turcic and A. Midha, Modeling of high speed elastic mechanisms for vibration response. Proceedings of 2nd International Computer Conference and Exhibition, Chicago, Vol. 2 (1983). 25. D. A. Turcic and A. Midha, Dynamic analysis of elastic mechanism systems, part I: applications. A S M E Jl Dynam. Syst. Meas. Control. 106, 249-254 (1984). 26. D. A. Turcic, A. Midha and J. R. Bosnik, Dynamic analysis of elastic mechanism systems, part II: experimental results. A S M E Jl Dynam. Syst. Meas. Control. 106, 255-260 (1984). 27. C. Bagci, Actual versus artificial finite line elements in force-displacement related problems of mechanics. Mech. Engng News, ASEE 14, No. 4, 35-47 (1977). 28. C. Bagci and S. Kalaycioglu, Elastodynamic of planar mechanisms using planar actual finite line elements, lumped mass systems, matrix-exponential method and the method of critical-geometry-kineto-elasto-statics. A S M E Jl Mech. Design 101,417-427 (1979). 29. C. Bagci, Critical operating speeds of constrained space linkages using spatial finite line element method and lumped mass system. ASME Paper No. 78-DET-37. 30. C. Bagci, Elastic stability and buckling loads of 3-D systems using spatial finite line elements. Comput. Struct. 12, 731 743 (1979). 31. C. Bagci and D. R. Falconer, Control speeds of multithrow crankshafts using spatial line element method. Shock Vibrat. Bull. 51, 25-41 (1981). 32. C. Bagci and J. A. Abounassif, Computer aided dynamic force, stress and gross-motion response analyses of planar mechanisms using finite line element technique. ASME Paper No. 82-DET-I1. 33. C. Bagci, Methods of elastodynamics of mechanical, structural and industrial robotic systems using finite line elements. ASME Paper No. 83-WA/DSC-7. 34. S. Kalaycioglu and C. Bagci, Determination of the critical operating speeds of planar mechanisms by the finite element method using planar actual line elements and lumped mass systems. A S M E Jl Mech. Design 101, 210-223 (1979). 35. F. R. Stamps and C. Bagci, Dynamics of planar, elastic, high-speed mechanisms considering three-dimensional offset geometry: analytical and experimental investigations. ASME Paper No. 82-DET-34. 36. D. Gamache and B. S. Thompson, The finite element design of linkages--a comparison of Timoshenko and Euler-Bernoulli elements. ASME Paper No. 8 I-DET- 109. 37. C. K. Sung and B. S. Thompson, A note on the effect of foundation motion upon the response of flexible linkages. ASME Paper No. 82-DET-26. 38. C. K. Sung and B. S. Thompson, Material-selection: an important parameter in the design of high-speed linkages. Mech. Mach. Theory. 19, 389-396 (1984). 39. C. K. Sung, B. S. Thompson, T. M. Xing and C. H. Wang, An experimental study on the nonlinear elastodynamic response of linkage mechanisms. Mech. Mach. Theory 21, 121-133 (1986). 40. C. K. Sung, B. S. Thompson, P. Crowley and J. Cuccio, An experimental study to demonstrate the superior response characteristics of mechanisms constructed with composite laminates. Mech. Mach. Theory 21, 103-119 (1986). 41. B. S. Thompson and M. V. Gandhi, The finite element analysis of mechanism components made from
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fiber-reinforced composite materials. ASME Paper No. 80-DET-63. 42. B. S Thompson, A variational formulation for the finite element analysis of high-speed machinery. ASME Paper No. 80-WA/DSC-38. 43. B. S. Thompson, Variational formulations for the finite element analysis of the noise radiation from high-speed machinery. A S M E Jl Engng Ind. 103, 385--391 (1981). 44. B. S. Thompson, D. Zuccaro, D. Gamache and M. V. Gandhi. An experimental and analytical study of a four bar mechanism with links fabricated from a fiberreinforced composite material. Mech. Mach. Theory 18, 165 171 (1983). 45. B. S. Thompson, A variational formulation for the finite element analysis of the vibratory and acoustical response of high speed machinery. J. Sound Vibr. 89, 7-15 (1983). 46. B. S. Thompson, D. Zuccaro, D. Gamache and M. V. Gandhi, An experimental and analytical study of the dynamic response of a linkage fabricated from a unidirectional fibre-reinforced composite laminate. A S M E Jl Mech. Transmiss. Automat. Design. 105, 528-536 (1983). 47. B. S. Thompson and C. K. Sung, A variational formulation for the dynamic viscoelastic finite element analysis of robotic manipulators constructed from composite materials. A S M E Jl Mech. Transmiss. Automat. Design 106, 183 190 (1984). 48. B. S. Thompson and C. K. Sung, The design of robots and intelligent manipulators using modern composite materials. Mech. Mach. Theory 20, 471-482 (1985). 49. B. S. Thompson and C. K. Sung, A variational formulation for the nonlinear finite element analysis of flexible linkages: theory, implementation and experimental results. A S M E Jl mech. Transmiss. Automat. Design. 106, 482-488 (1984). 50. B. S. Thompson and C. K. Sung, A variational theorem for the viscoelastodynamic analysis of high-speed linkage machinery fabricated from composite materials. ASME Paper No. 83-DET-6. 51. R. M. Alexander and K. L. Lawrence, An experimental investigation of the dynamic response of an elastic mechanism. A S M E Jl Engng Ind. 96, 268-274 (1974). 52. R. M. Alexander and K. L. Lawrence, Experimentally determined strains in an elastic mechanism. A S M E Jl Engng Ind. 97, 791-794 (1975). 53. B. M, Bahgat and K. D. Willmert, Finite element vibrational analysis of planar mechanisms. Mech. Mach. Theory 11, 47-71 (1976). 54. W. L. Cleghorn, R. G. Fenton and B. Tabarrok, Optimimum design of high-speed flexible mechanisms. Mech. Mach. Theory 16, 399--406 (198t). 55. W. L. Cleghorn, R. G. Fenton and B. Tabarrok, Finite element analysis of high-speed flexible mechanisms. Mech. Math. Theoo' 16, 407M-24 11981). 56. W. L. Cleghorn and C. L. Konzelman, Comparative analysis of finite element types used in flexible mechanism models. Proceedings of the 8th OSU Applied Mechanisms Conference, St Louis, Mo., pp. 30-1-30-7 (1983). 57. W. L. Cleghorn, R. G. Fenton and B. Tabarrok, Beam finite element approximations of structures undergoing free vibration. Proceedings of the Ninth Canadian Congress of Applied Mechanics, Saskatoon, Saskachewan, pp. 199 200 (1983). 58. A. J. Hossne, A. H. Soni and J. W. Harvey, Kinetoelasto-dynamics analysis of mechanisms by finite element formulation. ASME Paper No. 80-DET-4. 59. I. Imam and G. N. Sandor, High speed mechanism design---a general analytical approach. A S M E Jl Engng Ind. 97, 609~28 (1975). 60. P. K. Nath and A. Ghosh, Steady-state response of
mechanisms with elastic links by finite elcmcm mcth,,d~ Mech. Mach. Theory 15, 199 2ll (1980). 61. C. K. Sung, B. S. Thompson and J. J. McGrath. A variational principle t~)r the Iinear coupled thermoelastodynamic analysis of mechanism systems. ASME Jl Mech. Transnmw. Automat. Design 106, 291 296 (1984). 62. G. Naganathan and K. D. Willmert, Nc~ finite elements for quasi-static deformations and stresses m mechanisms. ASME paper No. 80-WA/DSC-35. 63. M. R. Khan and K. G. Willmert, Vibrational analysis of mechanisms using constant length linite elements. ASME Paper No. 76-WA/DE-21. 64. R. S. Haines, Survey: 2 dimensional motion and impact at revolute joints. Mech. Math. TheotT 15, 361 370 (1980). 65. P. Tong, T. H. H. Pain and L. L. Bucciarelli, Mode shapes and frequencies by finite element method using consistent and lumped m a s s e s . Compu[. .S'lruel. 1, 623~638 (1971). 66. C. W. Bert, Material damping: an introductory review of mathematical models, measures and experimental techniques. J. Sound Vibr. 29, 129-153 (1973). 67. P. S. Torvik, Damping Applications Jbr Vibration Control. ASME book, AMD, Vol. 38, New York (1980). 68. M. R. Khan and K. D. Willmert, Finite element quasi-static deformation analysis of planar mechanisms with external loads using static condensation. ASME Paper No. 81-DET-104. 69. J. H. Oliver, D. A. Wysocki and B. S. Thompson, Quasi-static deflection balancing -a new approach to synthesizing flexible linkage mechanisms. Mech. Mach. Theoo' 20, 103 114 11985). 70. J. H. Oliver, D. A. Wysocki and B. S. Thompson, The synthesis of flexible linkage mechanisms by quasi-static balancing. ASME Paper No. 83-WA/DSC-37. 71. O. 1. Sivertsen and A. O. Waloen, Non-linear finite element formulations for dynamic analysis of mechanisms with elastic components. ASME Paper No. 82-DET- 102. 72. K. Van der Werff, Kinematics of coplanar mechanisms by digital computation, Proceedings of the Fourth World Congress on Theory of Machines and Mechanisms, Newcastle upon Tyne, England, Vol. 3, pp. 685 690 (1975). 73. K. Van der Werff, A finite element approach to kinematics and dynamics of mechanisms. Proceedings ~)/'the F(fth Worm Congress on Theoo' of Machines and Mechanisms, Montreal, Canada, Vol. 1, pp. 587 590 (1976). 74. R. M. Alexander and K. L. Lawrence, Dynamic strains in a four-bar mechanism. Proceedings qf ttt(, Third Applied Mechanisms Conference, Stillwater, Okla, Paper No. 26 11973). 75. K. L. Lawrence and R. M. Alexander, Elastic mechanism dynamic strains. Proceedings of Fourth World Congress on the Theory of Machines and Mechanisms, Vol. 1, pp. 133 137 11975). 76. B. S. Thompson, A survey of analytical path-synthesis techniques for plane mechanisms. Mech. Math. Theory 10, 197 205 (1975). 77. I. Iman and G. N. Sandor, High speed mechanism design--a general analytical approach. A S M E Jl Engng Ind. 97, 609-628 11975). 78. M. R. Khan, K. D. Willmert and W. A. Thornton, Automated analysis and design of high speed planar mechanisms. Proceedings of the 5th OSU Applied mechanisms ConJerence, Oklahoma City, Okla (1977). 79. M. R. Khan, W. A. Thornton and K. D. Willmert, Optimality criterion techniques applied to mechanism design. A S M E Jl Mech. Design 100, 319 327 (1978). 80. W. A. Thornton, K. D. Willmert and M. R. Khan, Mechanism optimization via optimality criterion techniques. A S M E JI Mech. Design 101, 392 397 (1979).
A survey of finite element techniques for mechanism design 81. K. D. Willmert, W. A. Thornton and M. R. Khan, A hierarchy of methods for analysis of elastic mechanisms with design appliations. ASME Paper No. 78-DET-56. 82. Ce. Zhang and H. T. Grandin, Kinematical refinement technique in optimum design of flexible mechanisms with external loads using static condensation. ASME paper No. 8I-DET-104. 83. Ce. Zhang and H. T. Grandin, Optimum design of flexible mechanisms. A S M E Jl Mech. Transmiss. Automat. Design 105, 267 272 (1983). 84. W. H. Sunada and S. Dubowsky, On the dynamic analysis and behavior of industrial robotic manipu-
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lators with elastic members. A S M E Jl Mech. Transmis. Automat. Design. 105, 42 51 (1983). 85. W. Sunada and S. Dubowsky, The application of finite element methods to the dynamic analysis of flexible spatial and co-planar linkage systems, A S M E Jl Mech. Design 103, 6434551 (1981). 86. H. S. Cheng and L. M. Keer, (Eds) Solid Contact and Lubrication. ASME book AMD, Vol. 39. New York (1980). 87. N. D. Perreira and S. Dubowsky, Analytical method to predict noise radiation from vibrating machine system. J. Acoust. Soc. Am. 67, 551-563 (1980).
UBERBLICK UBER FINITE-ELEMENT-METHODEN KONSTRUKTIONEN VON VECHANISMEN
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Zusammenfassung--Bei der Entwicklung moderner Maschinensysteme mug der Konstrukteur im allgemeinen ein mathematisches Modell aufstellen, um den Grad der Schwingungsaktivit/it, der dynamischen Beanspruchungen, der Lagerbelastungen und oft auch der akustischen Ausstrahlung des Maschinensystems untersuchen zu k6nnen. Da solche Erscheinungen durch Theorien repr~isentiert werden, deren maBgebende Bewegungsgleichungen entweder als Differential-Integral- oder Integro-Differentialgleichungen darstellbar sind, k6nnen sie alle mit einem einzigen sehr wirkungsvollen Computer-Mittel, das jedem fiir diese Analyse zug/inglich ist, einer L6sung zugefiihrt werden: mit der Finite-Element-Methode. Es werden die Methodologien fiir Mechanismenkonstruktionen, die auf dieser vielseitigen Technik basieren, besprochen, ihr unterschiedlicher Charakter beleuchtet und zukiinftige Trends angegeben.
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0094-114X/86 $3.00 + 0.00 Copyright ~ 1986 Pergamon Journals Ltd
A SURVEY OF FINITE ELEMENT TECHNIQUES FOR MECHANISM DESIGN B. S. T H O M P S O N a n d C. K. S U N G Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, U.S.A. (Received 5 September 1984) Abstract--During the creation of a modern machine system, the designer must generally develop a mathematical model to investigate the levels of vibrational activity, dynamic stresses, bearing loads and often the acoustic radiation associated with the system. Since all of these phenomena may be represented by field theories whose governing equations of motion may be stated as either differential, integral or integro-differentialequations, they are all amenable to solution by the single most powerful computational tool available to the analyst: the finite element method. Mechanism design methodologies based on this versatile technique are reviewed herein, their different characteristics are highlighted and future trends indicated. INTRODUCTION In 1875, Reuleaux [1] defined a mechanism to be an assemblage of resistant (rigid) bodies, connected by movable joints, to form a closed kinematic chain with one link fixed and having the purpose of transforming motion. This fundamental assumption was incorporated into mechanism design methodologies for almost the next 90 years, and these techniques were employed in the successful development of a very broad class of machinery products. In the early 1960s, however, marketplace interactions and commercial stimuli for increased productivity were responsible for the evolution of a new class of mechanisms. The performance specifications associated with these mechanical systems generally required operating speeds to be increased while imposing additional criteria on design procedures. Typically, these criteria called for one or more of the following: a reduction of power consumption, an increase of external loading, the reduction of acoustical radiation and the generation of more accurate output characteristics. Under these stringent operating conditions, elastodynamic phenomena, that are of little consequence in the operation of low-speed mechanisms, have a significant detrimental affect on the performance characteristics of this new generation of high-speed machinery. The kernel of the problem is that the links vibrate due to their inherent flexibility and the more severe force fields which are being imposed upon them. Consequently, the traditional rigid-linked analyses are inadequate for the design of these machine systems because the methematical models, by definition, are unable to represent the phenomena associated with these assemblages of interconnected flexible bodies. The response of the academic and industrial communities to this major limitation in mechanism design methodologies is documented in two comprehensive survey papers published in 1972 [2, 3]. It is clear from MM~[
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these publications that a wide variety of analytical and computational techniques were proposed, but with such a complex set of dynamic boundary-value problems to be solved in the design of this class of mechanisms, closed-form analytical solutions are unattainable, and the designer clearly must resort to employing computers for modeling and simulation purposes. While several numerical techniques continued to be developed further after 1972, it soon became apparent that the finite element method was evolving as the most popular approach for analyzing flexible mechanisms. The reasons for this popularity are numerous, and will not be repeated here. However, the interested reader is referred to Refs [4-9] which are reviews of finite element methods by authorities in the field. This paper presents a reasonably complete list of references on finite element methods for the design of planar mechanisms. This body of literature is reviewed herein, by highlighting the phenomena being studied, the design function being undertaken and the different aspects of finite element analyses, such as element selection, formulation strategy and the procedure employed to solve the equations of motion.
FINITE ELEMENT FORMULATIONS 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 the 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 all of the analyses to date assume that joints are without clearance, thereby greatly simplifying the complexity of modeling impactive bearing forces. Furthermore, most analyses,
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with the exception of Refs [10] and [l 1], 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 element in an undeformed reference state. Local element matrices [M], [C] and [K] are generally established to model the mass, damping and stiffness characteristics of each link before pre- and postmultiplying 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. 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 for a high speed linkage are written [Mc] {U} + [C6] {I~T}+ [K(;] {U} :
- [M~]{ii} + {x} + {r},
where {U} is the column vector of deformation displacements, ( . ) is the time derivative, [Mz], [CG] and [K~] are the global mass, damping and stiffness matrices, respectively, which may assume many different formats depending upon the mathematical assumptions, {X} are body forces, and {T} are surface tractions. The nodal absolute acceleration terms, derived from a rigid-body kinematic analysis of the mechanism, are contained in the vector {~}, 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 the model in order to avoid singular matrices. Winfrey[12] accomplished this by directly applying the principle of conservation of momentum to the complete mechanism. Imam et al.[13] considered the crank as a cantilever beam to avoid this complication, and this approach was also employed by Midha et al.[14], and Gandhi and Thompson[15]. Nath and Ghosh[16] 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 the equations o1' motion. Winfrey[12,17] 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 yields directly the nodal displacements and requires displacement compatibility on interelement boundaries. Erdman el al.[18,19] employed the equilibrium finite element method (flexibility method of structural analysis) to study flexible mechanisms. With this approach, adjacent elements have equilibrating stress distributions on interelement 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, and mechanism design is no exception. Midha et al.[14, 20- 23] 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 [24-26]. Bagci et a/.[27-35] 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 a/.[15, 36-50] 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 [47, 48, 50] and also the acoustic radiation from linkage machinery [43, 45]. This latter work involved modeling the operation of mechanisms submerged in a perfect fluid as a fluid-structural interaction problem based on interacting continua. ELEMENT SELECTION 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. Will information on the axial and flexural deformations suffice, or is the torsional deformation field also important? Should a linear or a geometrically
A survey of finite element techniques for mechanism design
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nonlinear model be developed? What are the consti- equations. Generally, in the mode superposition approach, the damping is assumed to be uncoupled and tutive equations of the link material? And so on. is given in each mode as a percentage of the critical Generally, these decisions will be guided by endamping. However, when adopting the step-by-step gineering intuition or experimental evidence. approach of numerical integration, the complete A large number of papers [10, 12, 13, 21, 34, 38, damping properties of the mechanism must be estab50-60] have been devoted to studies of the planar lished, and this may be a difficult task. elastodynamic response of flexible linkages by modIn the appendix of Ref. [21], Midha et al. discussed eling the links using finite elements with only one the different damping matrices in the context of spatial variable. The axial response is generally modmodal superposition solutions. Winfrey[12] assumed eled by a linear interpolation function and the a damping matrix proportional to a linear combinaflexural response by a cubic interpolating polynomial tion of the mass and stiffness matrices (Rayleigh using elements originally derived for structural applidamping). In contrast to this, Alexander and cations. Bahgat and Willmert[53], in one of the Lawrence[51] assumed the damping coefficient for pioneering works in this field, considered both the each o r t h ~ coordinate to be defined by axial and flexural responses of a wide variety of flexible linkages using a finite element formulation Cii = 2 ~ix/ K , / M i i . The energy-dissipation characteristics of a mechemploying higher-order hermite polynomial approxianism are dependent upon both the constitutive mations for the deformation fields. The same quintic equations of the link material and also the characterelement was also employed by Hossne et a/.[58] and istics of the joints of the mechanism. In recent also by Cleghorn et al.[55-57]. In [56], a comparison was undertaken between finite elements utilizing combined experimental and finite-element-based investigations [26, 39, 44, 46, 49], logarithmic decrequintic polynomial approximations and those based ment transient response studies have been undertaken on cubic polynomials. These results not only sugfor a number of mechanism configurations and the gested that axial deformations may be neglected in experimental data employed to establish semithe analysis of some flexible mechanisms, but also empirical damping matrices for the damping in the that fewer of the higher-order elements are required mechanism. Bagci and Kalaycioglu[28] have underto generate a specified solution accuracy when comtaken preliminary work on coulomb damping in pared with the results from a model incorporating a joints, which they then combine with the model for larger number of lower-order cubic elements. The finite-line-element nomenclature employed by material damping employed in [51], thereby employBagci et a1.[27-35] is an alternative terminology for ing individual mathematical models for both mechthe standard rod and beam elements of structural anism joints and also the link materials. mechanics. Naganathan and Willmert[62] proposed that two special elements should be employed for quasistatic analyses, because they argued, the cubic SOLUTION PROCEDURES beam element does not yield exact results for the Two computational methods are generally emanalysis of mechanisms. Khan and Willmert[63] developed an element based on harmonic functions for ployed to solve the equations of motion for both linear and geometrically nonlinear vibrational anaperforming vibrational analysis of linkages. The two common formulations for mass matrices, lyses. These are the modal superposition approach namely the consistent mass matrix [14-16, 18-23, [12, 22, 26, 59] and the use of direct integration meth36-39,41,44,51,60] and the lumped mass matrix ods [10, 15, 16, 51], such as Runge-Kutta algorithms [27-35] have both appeared in the mechanism design or the Newmark method. literature. Tong et al.[65] demonstrated that while the The former approach is based on the assumption lumped mass approach will not suffer any loss in the that the displacement vector may be expressed as a rate of convergence when utilizing simple rod ele- linear combination of the vibrational mode shapes. ments, a consistent mass formulation is to be pre- This solution strategy is most efficient if the essential ferred when using higher-order elements such as dynamic response of the mechanism is contained in beam elements. the first few modal combinations. Hence it is most Having discussed the mass and stiffness matrices useful for studying the steady-state response of employed to model mechanisms, attention is now systems operating at constant crank frequencies. focused on damping matrices. Damping in materials However, additional complexities arise when develis a complex phenomenon [66] which probably re- oping geometrically nonlinear elastodynamic requires the development of thermomechanical models sponses [25]. The approach is not to be recommended in order that it be fully understood. The interested for linkages fabricated with journal bearings because reader is referred to a comprehensive work on the the inherent clearance needed for the operation of these joints creates impact loading which is often subject [67]. The assumptions employed to develop damping characterized by high-frequency components. This matrices are partially governed by the solution tech- generally requires a solution based on many modes in nique to be adopted for solving the finite element order to predict the response of the mechanism.
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Numerical integration algorithms employ step-bystep 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 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 may be applied to accurately model systems subjected to complex loading, or loads containing significant high frequency components. Imam et al.[13] 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 et al.[22] developed 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 [14], and in addition, a numerical algorithm for performing transient analyses of linkages [23]. While the above procedures are dedicated to determining the elastodynamic response of linkages, some authors have advocated that a quasistatic analysis of a linkage mechanism is quite adequate for most design purposes in an industrial environment [62, 68], 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 nonhomogeneous 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 o1"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 Willmert[68]. This philosophy involves condensing all internal degrees of freedom of each link to create a superelement with only the principal degrees of freedom retained, and has been used extensively in commercial codes tot structural dynamics problems. The authors reduced the number of system equations by 50%, so that the computational effort, which is proportional to the cube of the number of equations, was considerably reduced. As they rightly indicated in [68], this approach is especially useful when the designer is
searching for an optimal solution which generally involves many iterative analyses. ANALYSIS OF FLEXIBLE MECHANISMS The vast majority of the papers oll 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 { T} in equation (I) are often neglected. This approach generally yields a transverse ftexural 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 [14, 53, 60] and also transient responses [23] have been obtained. Since these classes of solution can be computationally time consuming, some authors [62, 68] have advocated that a quasistatic analysis is adequate for the analysis of most industrial machinery. This approach neglects the terms [M~;]{I~I} and [(~;]~ " ~" Uj in equation (1), thereby considering the mechanism to be a statically loaded structure. This assumption greatly simplifies tile con> putational aspects of the finite element analysis (see the previous section) 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 featured in some analyses [16, 21,24, 25, 49, 55]. 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. The higher-order theories modeling the elastic motion of links employ geometrically nonlinear analyses which retain the terms in the strain-displacement equations that couple the axial and flexural deformations [16, 24, 25, 49, 60, 71 73]. 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 to model each flexible link [12, 51]. 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[51, 52, 74, 75] in pioneering work on combined experimental and theoretical investigations of flexible four bar linkages. Midha et al.[20] reinforced these conclusions by demonstrating the effects of multielement idealizations on the response of a structure. They showed that a 15% error existed in
A survey of finite element techniques for mechanism design the second mode frequency and a 400% error in the third mode when a single element was employed. Similar errors were demonstrated by Gamache and Thompson[36] 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 coplanar 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[35]. Although the natural frequencies of a mechanism are continually changing during the operating cycle, as the stiffness and mass characteristics change relative to a fixed reference frame, nevertheless, critical speed ranges must be identified and avoided in practice. This class of problem has been investigated by Bagci et al.[29, 31, 34] using finite line elements and lumped mass matrices to study slider crank mechanisms, four-bar linkages and also six-bar mechanisms. Solutions were generated by an eigenvalue algorithm and compared with experimental data.
SYNTHESIS OF FLEXIBLE MECHANISMS
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 [76]. 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 because the analysis and synthesis procedures cannot be conveniently decoupled, as occurs in the design of rigid-linked mechanisms. While 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 [54, 59, 77-83], and they are all based on finite element methods. Before discussing the different solution strategies, it is appropriate to again review equation (1), which provides the key to understanding the proposed methodologies. Generally, when a mechanism is operating in a high-speed mode, the body forces and surface tractions may be neglected in comparison with the inertial loading; and hence {X} and {T} disappear. Premultiplying all the terms by the inverse of the mass matrix [M~] -~ yields [/] {1~} + [ M a ] - ' [ C a ] {(J} + [Me]-' [Ka] {U} = - [I]{I~}.
(2)
Thus it is evident that for a given mechanism operating at a given speed, the elastodynamic response is
355
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 [54, 59, 77-83] involves designing links in the commercial metals while optimizing the crosssectional geometry of the members. The second [41,69] 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.[13, 18, 19, 59, 77] developed a general method of kineto-elastodynamic analysis and synthesis based on the equilibrium finite element method. The link geometries were synthesized in aluminum alloys using optimization techniques, which incorporated stress and deflection constraints, to develop special cross-sectional shapes and tapered members. Willmert et aL[78-81] 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. The approach is applicable to all finite element analyses [81], and the authors advocate employing the "rigidbody stress" as the initial searching point. Cleghorn et aL[54] employ the same algorithm as in Khan et aL[79] with the modification that each iteration 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 iterations needed to achieve the optimum solution when employing the finite element formulation presented in [55]. Zhang and Grandin[82, 83] 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% of that obtained in [54] when addressing the same synthesis problem. In contrast to the previous approaches which are all concerned with homogeneous isotropic materials, Thompson et al.[38, 40, 41, 47] have proposed 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 [40, 41,46, 47]. 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
356
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materials. The models have been verified in combined experimental and computational studies [44, 46] and the superior response demonstrated in experimental comparative studies [40]. A third approach for synthesizing flexible linkages [69, 70] 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. DIRECTIONS OF FUTURE RESEARCH The success of the finite element method in both the industrial environment and the academic community may be attributed to its generality, versatility and its ability to analyze field problems with complex boundary conditions. To be effective, this new methodology requires interdisciplinary contributions from the fields of theoretical mechanics, approximation theory, numerical analysis and of course computer science. While industry craves for finite element methods that are more efficient, have user-friendly pre- and post-processors and which exploit the capabilities of the large supercomputers, this concluding section will focus instead on topics of fundamental research in mechanism design that have emerged from this review of the current literature. The maturing of research on the elastodynamic response of mechanisms is clearly evident when the prediction of finite element codes are validated by experimental data from the laboratory [24-26, 35, 39, 40, 44, 46, 49]. It is anticipated that this mode of research, based on the scientific method, will continue as investigators probe damping phenomena, geometrically nonlinear elastodynamic effects and the significance of inertial coupling terms [16, 24, 25, 49, 60, 71 73]. These terms will probably assume a significant role in the operation of ultra high-speed machinery. Under these high-speed operating conditions, outof-balance loading and the consequences of foundation excitation must also be modeled to establish a valid research program because these phenomena cannot be investigated in isolation in the laboratory. While these studies are limited to responses in the plane of the mechanism, further work is clearly needed on out-of-plane coupling as dictated by the coplanar nature of a large class of industrial machinery. Three-dimensional elements must not only be developed to model the out-of-plane responses of these systems, but also the dynamic behavior of robotic manipulators. Preliminary finite element work on analyzing this important class of mechanisms has already been published [33, 47, 84, 85]. Probably the two largest voids in the finite element literature on flexible mechanisms concern the modeling of joints and also the synthesis of these systems. All mechanisms contain joints which may incorpo-
rate either frictionless bearings, as in precision products, o1 alternatively plain bearings or bushes. These machine elements require a running fit in order that they function properly and this radial clearance results in impactive loads being generated in the joint, Considerable work has already been undertaken on modeling joints with clearance, and this is reviewed in a comprehensive survey paper by Haines[64], but the results of this research has not yet been incorporated into finite element codes. Contact-impact phenomena are difficult to model and some survey papers are presented in a book edited by Cheng and Keer[86]. The major challenges in this area are the modeling of the surface tractions and solving the associated equations of motion for which hybrid finite elements or global-local finite elements are probably required. One of the consequences of the impactive loading at the joints of a mechanism is the excitation of the vibrational response of the adjacent links which can generate noise radiation. If industrial machinery is to be designed to comply with the federal legislation restricting the noise levels to which employees may be exposed, then the designer requires a computer-based predictive capability for assessing a design prior to its manufacture. The finite element method is an obvious tool for this fluid-structural analysis, but only preliminary studies has been accomplished to date [43, 45], although there has been some success with an alternative approach [87]. Prerequisites for this class of work includes a good model of joints with radial clearance and also the ability to model unbounded fluid domains using infinite finite elements, or else global-local finite elements. Research on synthesizing flexible mechanisms requires the integration of a finite element algorithm with an optimization algorithm for both the elastodynamic effects and also the kinematic parameters of the mechanisms. Since these methods are of an interactive nature, further work is needed to reduce the time spent in the finite elemenl routine, by developing superior solution techniques and also generating superelements with less degrees of freedom. The models for mechanisms fabricated in composite laminates are of a first-order, using a continuum geometrically weighted averages concept to represent the materials [41,44,47, 50]. Clearly this can be extended to include volume fraction, stacking sequences, etc., to further extend the ability to model these complex materials. However, more demanding work needs to be undertaken to more closely model the constitutive behavior of these polymeric materials when subjected to variations in temperature and moisture. Again variational methods may provide a useful foundation for finite element formulations to this class of problem [61].
Acknowledgement---This material is based upon work supported by the National Science foundation under grant MEA-8216777. This funding is gratefully acknowledged.
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mechanisms with elastic links by finite elcmcm mcth,,d~ Mech. Mach. Theory 15, 199 2ll (1980). 61. C. K. Sung, B. S. Thompson and J. J. McGrath. A variational principle t~)r the Iinear coupled thermoelastodynamic analysis of mechanism systems. ASME Jl Mech. Transnmw. Automat. Design 106, 291 296 (1984). 62. G. Naganathan and K. D. Willmert, Nc~ finite elements for quasi-static deformations and stresses m mechanisms. ASME paper No. 80-WA/DSC-35. 63. M. R. Khan and K. G. Willmert, Vibrational analysis of mechanisms using constant length linite elements. ASME Paper No. 76-WA/DE-21. 64. R. S. Haines, Survey: 2 dimensional motion and impact at revolute joints. Mech. Math. TheotT 15, 361 370 (1980). 65. P. Tong, T. H. H. Pain and L. L. Bucciarelli, Mode shapes and frequencies by finite element method using consistent and lumped m a s s e s . Compu[. .S'lruel. 1, 623~638 (1971). 66. C. W. Bert, Material damping: an introductory review of mathematical models, measures and experimental techniques. J. Sound Vibr. 29, 129-153 (1973). 67. P. S. Torvik, Damping Applications Jbr Vibration Control. ASME book, AMD, Vol. 38, New York (1980). 68. M. R. Khan and K. D. Willmert, Finite element quasi-static deformation analysis of planar mechanisms with external loads using static condensation. ASME Paper No. 81-DET-104. 69. J. H. Oliver, D. A. Wysocki and B. S. Thompson, Quasi-static deflection balancing -a new approach to synthesizing flexible linkage mechanisms. Mech. Mach. Theoo' 20, 103 114 11985). 70. J. H. Oliver, D. A. Wysocki and B. S. Thompson, The synthesis of flexible linkage mechanisms by quasi-static balancing. ASME Paper No. 83-WA/DSC-37. 71. O. 1. Sivertsen and A. O. Waloen, Non-linear finite element formulations for dynamic analysis of mechanisms with elastic components. ASME Paper No. 82-DET- 102. 72. K. Van der Werff, Kinematics of coplanar mechanisms by digital computation, Proceedings of the Fourth World Congress on Theory of Machines and Mechanisms, Newcastle upon Tyne, England, Vol. 3, pp. 685 690 (1975). 73. K. Van der Werff, A finite element approach to kinematics and dynamics of mechanisms. Proceedings ~)/'the F(fth Worm Congress on Theoo' of Machines and Mechanisms, Montreal, Canada, Vol. 1, pp. 587 590 (1976). 74. R. M. Alexander and K. L. Lawrence, Dynamic strains in a four-bar mechanism. Proceedings qf ttt(, Third Applied Mechanisms Conference, Stillwater, Okla, Paper No. 26 11973). 75. K. L. Lawrence and R. M. Alexander, Elastic mechanism dynamic strains. Proceedings of Fourth World Congress on the Theory of Machines and Mechanisms, Vol. 1, pp. 133 137 11975). 76. B. S. Thompson, A survey of analytical path-synthesis techniques for plane mechanisms. Mech. Math. Theory 10, 197 205 (1975). 77. I. Iman and G. N. Sandor, High speed mechanism design--a general analytical approach. A S M E Jl Engng Ind. 97, 609-628 11975). 78. M. R. Khan, K. D. Willmert and W. A. Thornton, Automated analysis and design of high speed planar mechanisms. Proceedings of the 5th OSU Applied mechanisms ConJerence, Oklahoma City, Okla (1977). 79. M. R. Khan, W. A. Thornton and K. D. Willmert, Optimality criterion techniques applied to mechanism design. A S M E Jl Mech. Design 100, 319 327 (1978). 80. W. A. Thornton, K. D. Willmert and M. R. Khan, Mechanism optimization via optimality criterion techniques. A S M E JI Mech. Design 101, 392 397 (1979).
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lators with elastic members. A S M E Jl Mech. Transmis. Automat. Design. 105, 42 51 (1983). 85. W. Sunada and S. Dubowsky, The application of finite element methods to the dynamic analysis of flexible spatial and co-planar linkage systems, A S M E Jl Mech. Design 103, 6434551 (1981). 86. H. S. Cheng and L. M. Keer, (Eds) Solid Contact and Lubrication. ASME book AMD, Vol. 39. New York (1980). 87. N. D. Perreira and S. Dubowsky, Analytical method to predict noise radiation from vibrating machine system. J. Acoust. Soc. Am. 67, 551-563 (1980).
UBERBLICK UBER FINITE-ELEMENT-METHODEN KONSTRUKTIONEN VON VECHANISMEN
FOR
DIE
Zusammenfassung--Bei der Entwicklung moderner Maschinensysteme mug der Konstrukteur im allgemeinen ein mathematisches Modell aufstellen, um den Grad der Schwingungsaktivit/it, der dynamischen Beanspruchungen, der Lagerbelastungen und oft auch der akustischen Ausstrahlung des Maschinensystems untersuchen zu k6nnen. Da solche Erscheinungen durch Theorien repr~isentiert werden, deren maBgebende Bewegungsgleichungen entweder als Differential-Integral- oder Integro-Differentialgleichungen darstellbar sind, k6nnen sie alle mit einem einzigen sehr wirkungsvollen Computer-Mittel, das jedem fiir diese Analyse zug/inglich ist, einer L6sung zugefiihrt werden: mit der Finite-Element-Methode. Es werden die Methodologien fiir Mechanismenkonstruktionen, die auf dieser vielseitigen Technik basieren, besprochen, ihr unterschiedlicher Charakter beleuchtet und zukiinftige Trends angegeben.
M M T 21i4--F