复合材料层合板的非线性组合共振特性及分岔

考虑几何非线性项和阻尼的影响, 给出了四边简支的正交各向异性矩形层合板在两项横向简谐激励作用下的非线性振动微分方程, 利用伽辽金法导出了相应的达芬型非线性强迫振动方程。应用多尺度法对组合共振问题进行求解, 得到了系统在稳态运动下的幅频响应方程。基于李雅普诺夫稳定性理论, 得到了解的稳定性判定条件。通过数值算例, 分析了不同参数对系统组合共振及其分岔特性的影响。结果表明, 随着调谐参数、板厚度、阻尼系数以及激励力等参数的改变, 系统存在多幅值现象、滞后现象和跳跃现象, 出现不稳定解, 且在某些参数点处具有运动性态发生变化的分岔特性, 表现出较为复杂的动力学特性。 

Nonlinear combination resonances and bifurcation of orthotropic laminated plates

Considering the effects of geometrical nonlinearity and damping, the vibration differential equation of simply supported rectangular orthotropic laminated plate excited by two-term harmonic forces was established. The non-dimensional Duffing nonlinear forced vibration equation was deduced by using Galerkin method. The amplitude frequency response equation of system steady motion under combination resonance was obtained by the method of multiple scales. Based on Lyapunov stable theory, the critical conditions of steady-state solutionsp stability were got. By some examples, the influence of different parameters on nonlinear combination resonances and bifurcation properties of system was analyzed. The results show that the detuning parameter, thickness of plate, damping and amplitude of excitation have different influences on combination resonance and bifurcation. With the change of parameters, the jump phenomenon, hysteresis phenomenon and unstable solutions will occur. It is also shown that the system presents relatively complicated dynamics behaviors, and there exists multi-valued phenomenon, and the dynamics behaviors will change in some values.