矩形厚梁结构弯曲自由振动精确化方程新形式

Refined dynamic equations of thick beam bending

本文基于厚板结构振动精确化方程,应用算子代数及其谱分解理论,采用适当的规范条件和满足板条两侧边界条件,首次给出了更为精确化的厚梁结构弯曲振动支配方程.支配方程的总阶数为4阶,即关于横向位移函数的4阶偏微分方程以及相应的广义位移函数F和剪切变形函数f的表达式.分别基于Euler-Bernoulli梁和Timoshenko梁理论绘出了结构内存在的波模频散关系曲线,并与本文得到的厚梁结构内的波模频散关系做了对比,讨论了本文提出的矩形厚梁弯曲振动精确化方程的正确性和适用条件.本文提出的梁结构振动方程可用于厚梁较高频动力学分析与振动控制以及评价现有工程梁理论的适用条件.

In this paper, based on the theory of refined dynamic equations of thick plates bending, applied the differential operator algebra and decomposition of operator spectra, the refined dynamic equation of the beam flexural motion is first obtained by using proper gauge conditions and satisfying the boundary conditions. The refined equation of beams is a fourth-order equation, which governs the generalized displacement functions W, F and f. The dispersion relations, which are from the given beam theory, Euler-Bernoulli beam and Timoshenko beam, respectively, are compared. The refined equations of thick beams and applicable condition are investigated and discussed. Since derivation of the refined dynamic equation is conducted without any assumptions, so the proposed equation of thick beam bending is exact, that can be used to analyze vibration of thick beams at the high frequency and to evaluate the applicable condition of the engineering beam theory.