Comprehensive Solution To Nonlinear Large-delfection Problems Of Planar Beams And Accurate Modeling Of Compliant Mechanisms

A compliant mechanism is a device which achieves some or all of its motion throughthe defections of fexible segments. A compliant mechanism exhibits many advantagesover its rigid-body counterpart, such as increased precision, decreased manufacturing,and reduced number of parts. Therefore, it attracts wide attention from both academiaand industry. Because the motion of a compliant mechanism is always accompanied bycomplex large defections of its compliant members, modeling the large defections, soas to accurately model the whole mechanism, has been one of the most fundamentalproblems in the research community of compliant mechanisms.Based on the Bernoulli-Euler beam theory, this dissertation presents a general gov-erning equation for the large defection of thin beams. By explicitly incorporating thenumber of infection points and the sign of the end-moment load in the derivation, acomprehensive solution based on the elliptic integrals is derived for solving large de-fection problems. The comprehensive solution is capable of solving large defectionsof thin beams with multiple infection points and subject to any kinds of load cases.The use of the comprehensive solution is discussed for the cases that pure force, puremoment, or combined force and moment are applied at the beam tip. Through a fewtypical examples, the capacities of the comprehensive elliptic integral solution in solvinglarge defections with multiple infection points and subject to various complex loadsare demonstrated.With regard to the problem of multiple solutions in solving the large defectionproblems, a judging method based on the principle of least action for determining thereasonable solution among the multiple solutions is discussed. Based on the basic strain-stress relationship of mechanics of materials, the comprehensive elliptic integral solutionof the strain energy stored in defected beams is obtained. By taking the fxed-guidedbeam and the arc-guided compliant mechanism as examples, the actual motion pathsare determined using the comprehensive strain energy solution and the principle of leastaction, which are verifed by the corresponding experimental results.The critical buckling loads and the defection modes during the post-buckling for in-clined beams of diferent constraint conditions are analyzed based on the comprehensiveelliptic integral solution. The comprehensive solution can be directly simplifed to theformulas for the critical buckling loads under certain conditions, which can be further re-duced to Euler’s column formula for critical loads. The examples of inclined beams showgraceful and diversifed defection curves during post-buckling. The load-displacement characteristics of inclined beams during the post-buckling phase are analyzed.The accurate models for cross-spring pivots and the LITF pivots are presented basedon the comprehensive elliptic integral solution. The analytical results of cross-springpivots show that the stifness nonlinearity becomes remarkable during large angles, andpredict that infection points may occur in the defection confgurations of the compliantbeams. The maximum defection angles and the maximum allowable loads of the cross-spring pivots are also obtained. The NFEA results validate the accuracy of the model.The experimental results concur the bulking confgurations of the cross-spring pivotpredicted by the comprehensive solution. By analyzing a positive and a negative LITFpivots, the maximum defection angles and associated allowable loads are obtained. TheNFEA results validate of the model for LITF pivots. A modeling method of LITF pivotsfor buckling analysis using NFEA software is given and validated by examples.Finally, the accurate models for a partial compliant four-bar mechanism, a coupler-curve guided compliant mechanism, and a three-segment fully-compliant bistable mech-anism are established based on the comprehensive elliptic integral solution, and thecorresponding kinetostatic behaviors are obtained. The kinetostatic results are veri-fed by the NFEA and experimental results. By comparing the time consumption ofthe NFEA models and the comprehensive-solution models, it is concluded that thecomprehensive-solution models are much more efcient.In general, due to lack of efective modeling methods, the analysis and design (es-pecially in the later stage of design) of compliant mechanisms have long been relying onNFEA softwares. However, NFEA softwares are less efcient and often yield solutionsthat do not match with the actual in engineering. This situation severely restricts thedevelopment and application of compliant mechanisms. The comprehensive elliptic inte-gral solution presented in this dissertation exhibits good performance (both in efciencyand accuracy) in modeling compliant mechanisms. As a future work, we are going todevelop the comprehensive solution into one of the most important tools for accuratemodeling compliant mechanisms.