大挠度平面柔顺机构高精度建模理论与数值方法研究

Without the use of traditional hinges, compliant mechanisms can realize the transfer of motion or force by elastic deformation. Due to the advantages of simple structure, high accuracy, free maintenance and contact compliance, compliant mechanisms have become a research focus in such fields as MEMS, micro-nano positioning platform, and Tri_Co robots. However, compliant mechanisms are often accompanied by the large deflection which greatly increases the difficulty and complexity of design and analysis. Therefore, the project is proposed to build a high precision model of compliant mechanisms which can reveal the geometric nonlinearity exactly and at the same time can be of formatted numerical expressions. Firstly, the relationship between displacement and strain is fully considered with the high-order strain and coupling effects of longitudinal and transverse deformation; secondly, combined the constitutive relation of material and boundary conditions, the Galerkin weak form about the plane elastic statics is obtained by the principle of minimum potential energy; lastly, an element-free numerical approach is adopted for the spatial discretization of the system based on the generalized moving least squares which are used for the description of the displacements and derivatives. The approved high accuracy model and numerical methods are applied to the partial compliant crank rocker mechanisms and the bistable mechanisms of high-order buckling to explore the transmission mechanism between force and motion, for example, the coupling rules, stiffness effect of loading, fine-grained stiffness characteristics of buckling and post buckling and so on. This study can provide a calculation approach of high precision and high universality for the design and analysis of compliant mechanisms.

利用弹性变形实现运动功能的柔顺机构，具有结构简单、免维护且精度高等优势，是微机电系统、微纳定位平台、共融机器人等领域的研究热点。然而，柔顺机构工作时常伴随着复杂的大挠度变形，给设计和分析带来很大困难。针对平面大挠度的几何非线性特征，在变形位移与应变关系中充分考虑纵横变形的耦合效应并计及高阶应变项，根据弾性静力学的最小势能原理，推导能够准确反映大挠度平面柔顺机构几何非线性效应的高精度伽辽金弱式；基于无网格法，以几何离散点为控制点，应用易于构造高阶形函数的广义移动最小二乘算法拟合变形位移及其一阶导数变量，最终推导出高精度静力学模型的规范离散格式和数值计算流程。对部分柔顺曲柄摇杆机构和具有高阶屈曲变形的双稳态机构进行高精度建模和数值求解，探索其纵横位移耦合规律、载荷刚化效应、近屈曲和后屈曲细粒度刚度变化特性等静态力/运动传递机理，为柔顺机构的设计分析提供高精度和高普适性计算手段。

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方法的有效性和优越性。