Winkler地基上四边自由矩形薄板的1/3次亚谐共振与混沌分析

Study on 1/3 Subharmonic Resonance and Chaos of a Rectangular Thin Plate with Four Sides Free on the Winkler Foundation

研究Winkler地基上四边自由矩形薄板的复杂运动,按照弹性力学理论建立Winkler地基上四边自由受简谐激励作用矩形薄板的动力学方程;利用Galerkin方法将其转化为非线性振动方程;应用非线性振动的多尺度法求得了系统满足1/3次亚谐共振情况时的一次近似解,并进行数值计算;分析激励、调谐值、阻尼等对系统响应曲线的影响;应用Floquet理论分析了系统的稳定性问题;应用Melnikov方法得到了系统可能产生混沌运动的条件。

Based on elastic mechanics theory and study on complex motion of a rectangular thin plate with four sides free on the Winkler foundation, nonlinear dynamical equation of a rectangular thin plate with four sides free on the Winkler foundation subjected to harmonic excitation is established. A nonlinear vibration equation is obtained by Galerkin＇s method. Applying the method of multiple scales of the nonlinear vibration, the first approximation solution of 1/3 subharmonic resonance of the system is acquired. Numerical analysis on the influence of excitation, detuning and damping on the system is carried out. By means of Floquet theory and Melnikov function, problems of bifurcation and chaos of the system are studied. Conditions of stabilities and chaos are obtained.