平面柔顺机构大挠度弹性静力学分析的低维参数化曲率模型  

Curvature-based low-dimensional parametric model for the elastostatics analysis of planar compliant mechanisms under large deflection

作　　者：黄勇刚[1] 谢丹 HUANG YongGang;XIE Dan(Chongqing Municipal Key Laboratory of Mechanism Design and Control for Manufacturing Equipment,Chongqing Technology and Business University,Chongqing 400067,China;College of Engineering and Technology,Southwest University,Chongqing 400715,China)

机构地区：[1]重庆工商大学制造装备机构设计与控制重庆市重点实验室,重庆400067 [2]西南大学工程技术学院,重庆400715

出　　处：《中国科学：技术科学》2022年第4期565-584,共20页Scientia Sinica(Technologica)

基　　金：国家自然科学基金(批准号:51805448)资助项目。

摘　　要：针对以均质细长弹性梁为柔性构件的平面柔顺机构,通过采用Bernstein多项式逼近柔性梁弯曲曲率,提出了一种适用于大挠度强几何非线性弹性静力学分析的低维参数化曲率模型,并给出了基于高斯积分和牛顿拉弗森迭代的数值求解算法.该建模方法的特点是直接以柔性梁弯曲应变即曲率为基本未知场函数,以Bernstein多项式试函数的待定系数为基本未知量,基于最小总势能原理建立系统的几何非线性静力平衡方程.因而避免了传统基于位移的建模求解算法通过求导计算弯曲应变能引入的数值误差,并降低了试函数的光滑性要求,建模精度高、数值计算收敛速度快且直曲梁通用.同时所求参数化曲率解与坐标系无关,具有丰富的几何和力学意义,既能实现刚体位移和柔体变形的统一建模,又能高效呈现细长梁大挠度弯曲复杂、丰富的变形信息.通过多个算例的建模分析,数值计算结果充分证明了本文方法的有效性和优越性.Based on the approximation of the bending curvature by Bernstein polynomials,a novel parametric model suitable for large deflection analysis is proposed for studying planar compliant mechanisms with uniform slender beam members.The numerical formulation is also presented based on Gaussian quadrature and Newton-Raphson iterations.By considering the bending curvature and coefficients of the Bernstein polynomial as the deformation field and unknown parameters,respectively,the equilibrium equations of the compliant mechanisms with geometric nonlinearity were established based on the principle of the minimum total potential energy.Further,the proposed method effectively avoided the numerical error introduced by using the deformation displacement to derive and calculate strain energy and greatly reduced the smoothness requirement of the trial function.Hence,the novel model possesses the benefits of high modeling accuracy,fast convergence speed in the numerical calculation,and versatility for application to straight and initially curved beams.Meanwhile,the parameterized curvature solutions have great geometric and mechanical significance that allows not only to realize the unified kinematic modeling of rigid and flexible bodies but also to efficiently present the complex and rich deformation information of slender beams under large deflections.The improved performance of the developed models was demonstrated through some classic numerical examples.

关 键 词：柔顺机构 几何非线性 BERNSTEIN多项式 曲率逼近 参数化模型 

分 类 号：TH112[机械工程—机械设计及理论]