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

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.