平方非线性对屈曲薄板振动性能影响的研究

分析研究参数激励简支矩形薄板非线性振动中的超谐振动。依据Karman方程的动态比拟，运用Galeerkin法将控制薄板振动的微微分方程转化参数激励Duffing型方程。针对该方程进行的变换表明，屈曲薄板振动系统为带有平方和立方非线性的参数激励非线性动力系统。应用摄动分析研究系统中平方非线性因素对系统的调节机理以及平方非线性导致的2倍超谐振动。分析结果表明，由于平方非线性对系统的调节作用，在一定的参数域响应中自由振动项的幅值不会因阻尼的存在而衰减，自由振动以激励频率2倍的频率参与系统的响应。基于理论分析的试验研究证明，所讨论的参数激励屈曲薄板振动系统在一定的参数条件下将出现2倍超谐振动。由2倍超谐振动参与的系统的振动状态是稳定的周期运动。

INVESTIGATE ON THE EFFECT OF QUADRATIC NONLINEARITY ON THE BEHAVIOR OF THE BUCKLED PLATE VIBRATION SYSTEM

Superharmonic resonance arisen in the nonlinear oscillator of simply supported rectangular thin plate subjected to a harmonic parametric excitation is investigated by theoretical analysis and experiments in the present work. Based on the dynamic analog of the Karman plate equations, the partial differential equations governing the vibrations of the thin plate are reduced to a Duffing equation with the term of harmonic parametric excitation. By introducing a mathematical transformation into the Duffing equation, it makes clear that the nonlinear dynamic system of the buckled plate contains terms of quadratic and cubic nonlinearities. The modulating mechanism of the quadratic nonlinearity on the nonlinear system, and the 2 times superharmonic resonance caused by the quadratic nonlinearity, are investigated by means of perturbation technique. The results from the perturbation analysis are that, because of the modulation action of the quadratic nonlinearity, the amplitude of the natural response don't decline with time in some excitation ranges, and the natural response participates in the system response with 2 time of the excitation frequency. It has been verified by experiments based on the theoretic analysis that 2 times superharmonic resonance arises in the nonlinear dynamic system of buckled plate in some parametric regions, and the response participated by 2 times superharmonic resonance is steady and periodic.