Accurate Modeling Of Geometrical Nonlinear Problems In Compliant Mechanisms

Compliant mechanisms exploit deflections of flexible members to transfer or transform motion,force or energy.Advantages of compliant mechanisms include high precision motion,low friction,and compactness,to name a few.Thus,compliant mechanisms have been playing increasingly important roles in high precision instruments,minimally invasive surgeries,etc.In this dissertation,only planar straight beams are studied considering that they are the most commonly used flexible segments in compliant mechanisms due to their simplicity as well as their binary constraint property.The distributed-compliance property qualifies planar beams a wider motion range,while it gives rise to nonlinear elastokinematic effects,which fails the traditional linear models.Therefor,modeling the geometrical nonlinearities is the most fundamental problems in the research community.In terms of this issue,the dissertation puts forward several work include:(1)Based on the beam constraint model(BCM)that was developed for intermediate deflections,the nondimensional bound of the axial force that beam constraint model takes is mathematically determined.The upper bound of the axial force is determined by the condition that the strain energy expression of the BCM is of a positive definite quadratic form,and the lower bound is determined by the buckling condition relate to compressing axial force.Besides,the expressions for the deflected configurations of planar beams and the stresses on the deflected beam are presented.Three examples are analyzed to demonstrate the effects of the axial force on the modeling errors of BCM.(2)Based on Timoshenko beam theory,the formulations of the angle of the cross area rotation and the deflection along the beam are decoupled and derived separately.Meanwhile,the approximating linear expression for curvature and accurate nonlinear expression for load equilibrium are utilized to obtain a shear affection considered model named Timoshenko beam constraint model.The accurate model for a tree-segment fully-compliant bistable mechanism are established based on Timoshenko beam constraint model,and the corresponding kinetostatic behaviors are obtained.The kinetostatic results are verified by NFEA models and experimental results.(3)The beam constraint model may be inappropriate or inexact when planar beams experience large deflections.In order to deal with planar beams undergoing large deflection,based on beam constraint model,a new method named chained beam constraint model is proposed,in which a flexible beam is divided into a few elements and each element is modeled by beam constraint model.Local coordinate is constructed on the split points to ensure the approximate linear expression for curvature is correct.Integrating the load equilibrium equations at every node and the geometric constraint equations construct the chained beam constraint model.Several typical examples were analyzed and the results show CBCM's capabilities of modeling various large deflections of flexible beams in compliant mechanisms.Then,based on the available references and our own experiences,we offer a rough comparison of these methods which might be useful for users to select appropriate methods for their problems.(4)A closed-form deflection model named Bi-BCM is proposed to deal with fixed-guided beams.The model gives the load-displacement relationship of fixed-guided beams analytically.Meanwhile,the boundary line between the first and the second mode bending of fixedguided beams can be easily obtained using a closed-form equation.Furthermore,there are two possible solutions(corresponding to two different deflected shapes)for a fixed-guided beam deflected in its second buckling mode,Bi-BCM gives both solutions in closed-form.Different examples are analyzed to illustrate the application and verify the effectiveness of the Bi-BCM.(5)Chained beam constraint model is tailored for planar ground structure optimization.The primary topology optimization is done which lay the foundation for the following work with large deflection in consideration.