薄板3倍超谐振动的分析与试验

基于Karman方程的动态比拟，运用Galerkin法，选用合适的正交函数将控制薄板振动的偏微分方程离散化为常微分方程，得到一带有平方和平方非线性的参数激励和外激励联合作用的非线性动力学系统。由于立方非经线性对系统的调节，系统存在出现3倍超谐振动的参数域。在出现3倍超谐共振的频率附近，系统的响应为主振动响应与3倍超谐振动响应共同组成的稳定的周期振动。理论分析和仿真计算及试验研究表明，参数激励简支屈曲薄板振动系统在一定的参数条件下将出现3倍超谐振动。当激励幅值不变、激励频率逐渐接近3倍超谐共振频率点时，3倍超谐振动成分对系统响应的影响逐渐增加，这表明立方非线性对系统的调节作用越来越强。

Analysis and Experiment on 3-Times Superharmonic Resonance of Thin-Plate

Nonlinear vibration occurred in the large amplitude vibration of simply supported rectangle buckled thin plate subjected to parametric excitation and its effect on the dynamic performance of the buckled plate were investigated. Based on the dynamic analog of Karman plate equation, the partial differential equations which control the plate's vibration is discretized into ordinary differential equations via the Galerkin procedure and selecting suitable expanding functions. The nonlinear dynamic system with quadratic and cubic nonlinearities is obtained by applying mathematical transformation to the equation. By using the perturbation analysis procedure, it has been provided that, due to the regulating by the cubic nonlinearities, some parametric ranges in which 3 times superharmonic vibration arises exist in the system. The fact is provided by simulation that, when the excitation frequency is close to the 3 times superharmonic resonance frequency, the system response consisted of principal response and 3 times superharmonic resonance is stable and periodic. It has been shown by experiments that 3 times superharmonic resonance will arise in this buckled plate dynamic system in some parametric ranges. If the excitation amplitude is constant and the excitation frequency is close to 3 times resonance frequency, the effect of the 3 times superharmonic resonance on the system response increases greatly. This fact shows that regulating action of the cubic nonlinearities on the vibration system is more strongly. The experimental result has a good agreement with that from theoretical analysis and simulation.