黏弹性文克尔地基修正Timoshenko梁横向振动复模态分析

利用修正Timoshenko梁理论,考虑地基黏性阻尼,建立黏弹性文克尔地基修正Timoshenko梁动力方程,解决了黏弹性地基经典深梁(Timoshenko)两频谱矛盾。利用复模态分析方法导出微分方程在多种约束条件下的复频率求解超越方程及其振型函数。当不计梁剪切转动惯量的影响时,方程可退化为黏弹性文克尔地基经典深梁计算模型;当不计梁剪切变形的影响及地基黏滞阻尼时,方程退化为常见的弹性地基Euler梁振动模型,其计算模型较为通用。分析了黏弹性地基梁在不同理论下的计算差别,结果表明:黏弹性地基Euler梁在高频段及梁长较短的低频段具有很大误差,黏弹性文克尔地基经典Timoshenko梁相对于Euler梁误差有所减小,但随阶次的增加相对误差逐次增大,在较高频段亦产生不可忽略的影响,证明了地基黏滞阻尼对地基梁振动低频段有较大影响。

Complex Modal Analysis of Transverse Vibration of Modified Timoshenko Beam on Viscoelastic Winkler Foundation

In the light of the modified Timoshenko beam theory, a vibration-control differential equation of the modified Timoshenko beam on viscoelastic Winkler foundation is established in consideration of the viscous damping of foundation. The present equation overcomes the shortcoming of the classical Timoshenko beam on viscoelastic foundation that “one mode has two frequencies”. The frequency solution to the transcendental equation and its modal function under various boundary conditions are derived by complex modal analysis. When the influence of shear inertia caused by shear deformation of beams is not taken into account, the equation degrades to the classical Timoshenko beam model of viscoelastic Winkler foundation; when the shear deformation of beams and the viscous damping of foundation are not taken into account, the equation decays to the common Euler beam vibration model of elastic foundation. The computational model in this paper is a more general one. The calculation differences of viscoelastic foundation beams under different theories are analyzed. The results show that Euler beams on viscoelastic foundation have great errors in high frequency band as well as in low frequency band with short beams. The error of classical Timoshenko beams on viscoelastic Winkler foundation is smaller than that of Euler beams; but the relative error increases gradually with the increase of order, and the influence cannot be ignored in high frequency band. The research finding proves that the viscous damping of foundation has great influence on the low frequency vibration of foundation beam.