含集中质量矩形薄板的动力学建模与质量调幅分析

Dynamic modeling of rectangular plates with a lumped mass and the amplitude modification effect of mass

基于经典薄板理论和von Kármán非线性理论,采用Hamilton原理,首先建立了含集中质量矩形薄板的动力学方程,并结合Galerkin法进行模态截断,获得双模态非线性动力学控制方程组;随后,运用多尺度法,引入频率失调参数,重点研究了主共振与1∶3内共振联合作用下系统的非线性动力学特性,获得了幅-频特性表达形式。根据振动同步性以及幅-频曲线的极值特性,在分析参数范围内,发现当集中质量为0.06kg时,第一阶幅频曲线非线性现象衰减,且第二阶幅频曲线取到最小值。最后实例以集中质量作为调谐参数,分析了质量变化对振幅的影响,计算结果表明,质量在某一范围内对主共振与1∶3内共振联合下的第一阶模态振幅影响较小,但对第二阶模态振幅有较强的调制作用。

Based on the classic thin plate theory and von-Kármán nonlinear theory, the dynamic partial differential equations of rectangular plates with a lumped mass are presented by the Hamilton principle. Then, a two-model truncation is carried out by Galerkin technique to obtain the nonlinear ordinary differential equations. The method of multi-scale is used to determine a uniform first-order expansion of the solution. According to solvability conditions, the nonlinear modulation equations are obtained for the combination of primary resonance and 1︰3 internal resonance. Using the vibration synchronization and the features of extreme points on amplitude-frequency curves, the mechanism analysis for amplitude modification arising from parameters is investigated. It is found that when M=0.06 kg, the first-order amplitude curve is suppressed and the second-order amplitude reaches minimum value. Finally, a rectangular thin plate with a lumped mass and four simply supported edges as an example is numerically carried out, where, the mass as the modifying parameter is taken into account. Results show that the influence of the mass on the amplitude for the first model is very weak, whereas for the second model the amplitude modification effect due to the mass is obvious.