AN ELEMENT HARMONIC BALANCE METHOD
一种单元谐波平衡法

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.