力热载荷下的一般模态方程及热模态计算

摘要

Structural vibration problem under static, thermal and dynamic load is common in engineering whose hypostasis is nonlinear problem. When the vibration level is small enough , nonlinear problem can be predigested to linear problem that produces concept of preload mode and thermal mode. In this article, the general expression of stiffness matrix and mass matrix for preload mode and thermal mode is derived based on continuous medium mechanics and dynamical Lagrange equation. Modal stiffness matrix and mass matrix derivation of specific issues can be simplified such as preload mode and thermal mode of bar and plate in particular case. The cause and restriction of preload mode and thermal mode simulation in commercial finite element software is also revealed.%工程中往往遇到静/热/动载荷联合作用下结构的振动问题，其本质为非线性问题，但当振动量级较小时，可以将非线性振动问题简化为线性问题进行处理，从而产生了静载模态和热模态的概念。本文基于连续介质力学及动力学拉格朗日方程，推导了静/热载模态刚度阵和质量阵的一般表达式，一方面简化了具体问题模态质量阵和刚度阵的推导，如梁、板等具体问题静/热载模态只是一般表达式的特例，另一方揭示了商用有限元软件静/热载模态计算错误的原因和其使用的限制。