线载荷作用下面内运动正交各向异性板的亚谐波共振

Subharmonic resonance for in-plane motion of orthotropic plates under linear loads

研究面内运动正交各向异性薄板在线载荷作用下的亚谐波共振问题。给出面内运动正交各向异性薄板的动能和势能表达式,并推得几何非线性下正交各向异性条形板的非线性振动方程。针对简支边界约束情况,考虑三阶模态并运用伽辽金积分法,推得关于时间变量的无量纲化达芬型非线性振动微分方程组。应用多尺度法对非线性系统的亚谐波共振问题进行求解,得到了稳态运动下关于不同阶模态的共振幅值响应方程。应用李雅普诺夫稳定性理论,对解的稳定性进行分析,得到了稳态解的稳定性判别式。通过数值算例,得到了振幅特性变化曲线图,分析了速度、线载荷、材料属性等参量对系统共振特性的影响,结果表明,系统呈现较为明显的非线性共振特征。

Here, subharmonic resonance problems for in-plane motion of orthotropic plates under linear loads were studied. The kinetic and potential energy expressions for in-plane motion of an orthogonal plate were derived and nonlinear vibration equations of an orthotropic strip-shaped plate with geometric nonlinearity were deduced. Under simply supported boundary conditions, considering the first three order modes and using Galerkin integral method, a non-dimensional Duffing nonlinear vibration differential equation system with respect to time variables was deduced. The subharmonic resonance problem of this nonlinear system was solved using the multi-scale method to acquire resonant amplitude equations for different order modes of steady-state response solution. Lyapunov stability theory was applied to analyze solution stability, and obtain the steady-state solution’s stability discriminant. The amplitude characteristics variation curves were obtained with numerical examples. Effects of parameters, such as, velocity, linear load and material properties on the system’s resonance characteristics were analyzed. The results showed that the system reveals more obvious nonlinear resonance characteristics.