杆、梁有限元模型的模态的振荡性质

Oscillation property of modes for FE models of bars and beams

杆、弦、梁等常见一维连续体的固有模态具有振荡性质。一维连续体进行离散后的固有模态是否仍具有振荡性质,表征着数值计算是否真实反映了原问题。业已通过化刚度矩阵为三对角矩阵的乘积的方法证明了:常见支承条件下的有限差分梁、杆以及采用集中质量矩阵的有限元杆、弦的模态具有振荡性质。在有限元计算中,Euler梁通常采用带转角变量的Hermite三次插值函数进行离散,目前尚未见到此种离散梁的模态是否具有振荡性质的论述。从连续杆、弦、梁的振荡性质出发,结合有限元解的特性,指出在集中质量矩阵的条件下,如果离散模型在结点集中力作用下,节点位移与解析解相等,则此离散模型的模态具有振荡性质;具体说来,杆、弦的有限元模型模态具有振荡性质,从最小余能原理构造的梁有限元模型模态具有振荡性质;对于Hermite三次插值函数的位移Euler梁单元,若截面参数在单元内取常数,模态也具有此性质;但是,若截面参数在单元内不为常数,模态未必具有振荡性质。

The modes of continuous bars,strings and Euler beams constrained only at their 2 ends have an important qualitative property called oscillation property.Appropriate discrete models of bars,strings and Euler beams are expected to reflect the oscillation property in a discrete form.With help of an algebraic approach involving tri-and penta-diagonal matrix,the discrete modes of bars,strings and beams obtained with finite difference method were proved to have the oscillation property invariably,regardless of grid and mass distribution.Here,it was proved that if the modes of a finite element model are equal to the modes of an analytic model at nodes,then the modes of finite element model have the oscillation property.Furthermore,the oscillation property also existed for the modes of bars and strings discretized by 2-node finite elements with a lumped mass matrix.However,an Euler beam was meshed with 2-node cubic Hermitian elements,the stiffness matrices were no longer tri-or penta-diagonal and the algebraic approach may become complicated.The discrete oscillation property of bars,strings and Euler beam meshed with finite elements was discussed here with a new approach,i.e.,checking an equivalent condition of the oscillation property.It was proved that the modes of a FE discrete Euler beam with lumped mass have oscillation property invariably,if stiffness matrices are derived with flexibility approach;the 2-node cubic Hermitian element may lead to failure of oscillation property,if the parameters of the beam cross-sections are not constants in elements.Numerical examples supported this conclusion.