携带集中质量的矩形薄板面外非线性动力失稳研究

ANALYTICAL STUDY ON THE OUT-OF-PLANE NONLINEAR DYNAMIC INSTABILITY OF A THIN RECTANGULAR PLATE WITH A CONCENTRATED MASS

对于面内对边周期荷载作用下携带集中质量的矩形薄板,当周期荷载的激振频率在板的两倍自振频率附近时,板发生面外参数共振失稳。该文基于薄板大挠度理论,运用伽辽金法推导出携带集中质量的矩形薄板非线性动力失稳的Mathieu-Hill方程,进而求解得到板发生面外参数共振失稳时周期荷载的临界激振频率域以及非线性动力失稳曲线。运用有限元软件进行瞬态分析得到不同激振幅值作用下板发生面外参数共振失稳时周期荷载的最小与最大临界激振频率值,通过与解析解进行对比,验证了计算结果的正确性。研究结果表明:随着集中质量的增加,参数共振失稳的临界激振频率及其不稳定域的宽度逐渐减小,不稳定域的位置逐渐向低激振频率的方向移动;随着集中质量的增加,面外参数共振失稳域的临界激励幅值逐渐增加;随着集中质量所处位置的模态位移增加,不稳定域的宽度减小。

For a thin rectangular plate with a concentrated mass subjected to a periodic load along its two opposite edges,the out-of-plane parametric resonance instability of the plate occurs when the excitation frequency of the periodic load is about twice the natural frequency of the plate.Based on the large deflection theory of thin plate,the Mathieu-Hill equation of nonlinear dynamic instability for a plate with a concentrated mass is derived by using Galerkin method,and then the analytical solutions of the region of critical excitation frequencies of the periodic load and the nonlinear dynamic instability curves for the out-of-plane parametric resonance instability of the plate with a concentrated masses are solved.The lowest and highest critical excitation frequencies of periodic loads of the plate under different excitation amplitudes are obtained by transient analysis with FEM,and the calculation results are further verified against the analytical solutions.It is found that:as the weight of the concentrated mass increases,the critical excitation frequency and the width of parametric resonance instability regions reduces,while the positions of instability regions move gradually to the lower excitation frequency;the excitation amplitude of dynamic instability region for out-of-plane parametric resonance instability of the plate increases with the increase of the concentrated masses;the width of the instability region decreases with the increase of the modal displacement at the position where the concentrated mass is located.