一种单元谐波平衡法

AN ELEMENT HARMONIC BALANCE METHOD

基于有限元离散，对于工程中的非线性响应问题，提出一种单元谐波平衡法．与常规的谐波平衡法不同，本文将谐波平衡方程建立在有限元素上，从而兼顾了有限元素法和常规谐波平衡法两大优势．有限元技术的应用能使得求解问题的范围扩大到复杂工程结构，而谐波平衡概念的使用将使得含有复杂变形和复杂本构关系的动力学响应问题得到有效解决．所提方法能适用于工程结构中具有复杂非线性关系的动力学响应问题．由于谐波平衡法的实施依赖于谐波系数方程及其切线刚度矩阵的解析推导，尽管已经局限到有限元素上，但对于较为复杂一些的本构关系，推导仍非易事．为解决这些问题，放弃通常对于变形梯度和应变张量所作的向量假设，而是从连续介质力学中基本的几何关系入手，提出一种矩阵分解形式．通过利用张量的内蕴导数定义以及关于迹函数的有关性质，给出应力增量的一种新的表现形式．当它与变形梯度的矩阵分解相结合时，使得切线刚度矩阵的导出变得十分简单，而且所得计算形式也比通常紧凑和方便许多．

An element harmonic balance method based on finite element method is proposedin this paper and is used to solve nonlinear dynamic response problems. Different from generalharmonic balance method, such harmonic balance equations airs established in finite element level,making the method advantageous in both finite element method and general harmonic balancemethod. The applications of finite element method can extend the solving range to complex engi-neering structures, and the application of harmonic balance concept will make the dynamic responseproblems with complex deformation and complex constitutive relationship be solved efficiently. So,the method in this paper is suitable for solving dynamic response problem in engineering structureswith complex nonlinear relationship.Although formulation deductions are localized to the finite element level, it is not easy forthe more complex constitutive relationships, because the execution of harmonic balance method1) The project supported by the National Natural Science FOundation of China and Aeronautical Science Foundation.depends on analytical deduction of both harmonic coefficient equations and tangent stiffness ma-trix. In order to solve these problems, starting with geometry relationship in continuous mediummechanics, a matrix decompose technology is proposed to replace the general vector assumption ofdeformation gradient. In terms of using tensor intrinsic derivative and property of trace function,new forms of stress increment are given. Combined it with the matrix decomposition of deforma-tion, the tangent stiffness matrix can be derived fast, and the obtained calculation form is morecompact and more convenient.As application, an elastomeric lag damper of modern helicopter was used to elucidate the har-monic response analysis. Because the material property of elastomer is nonlinear and incompress-ible, and has the characteristic of irreversible thermal dynamics and fading memory, it consumesenergy and reduces vibration level in helicopter rotor system. Its mathematical model will resultin infinite delay kind of nonlinear functional differential equations with incompressible restriction.This example shows that the method presented can solve a large kind of nonlinear dynamic re-sponse problems in engineering. According to the structure of elastomeric lag damper, plane straindeformation is adopted. The material property of elastomer is taken as closing to CN-1 materialmade in China. Because solving curve in Newton-Raphon method can not be traced at more valueseffectively, the curve continuation method is used in calculation. The calculation results show thatthe method of this paper is very successful.