四边简支功能梯度圆锥壳的非线性振动分析

研究了四边简支条件下功能梯度圆锥壳的非线性自由振动。首先,通过Voigt模型和幂律分布模型描述了功能梯度材料的物理属性。然后,考虑von-Karman几何非线性建立了功能梯度圆锥壳的能量表达式,利用Hamilton原理推出圆锥壳的运动方程。在此基础上,采用Galerkin法,只考虑横向振动,功能梯度圆锥壳运动方程可简化为单自由度非线性振动微分方程。最后,通过改进的L-P法和Runge-Kutta法求解非线性振动方程,讨论功能梯度圆锥壳的非线性振动响应,分析几何参数和陶瓷体积分数指数对圆锥壳非线性频率响应的影响。结果表明,几何参数对非线性频率和响应的影响相较于陶瓷体积分数指数更明显;圆锥壳的几何参数和陶瓷体积分数指数通过改变非线性频率影响振动响应;功能梯度圆锥壳呈弹簧渐硬非线性振动特性。

Analysis of nonlinear vibration of a functionally graded conical shell with simply support at four edges

The purpose of this article is to study the nonlinear free vibration of a functionally graded conical shell simply supported at its four edges.Firstly,the Voigt model and the power-law distribution function were employed to describe the physical properties of the FGMs.Subsequently,the energy expression of the functionally graded conical shell considering von-Karman geometric nonlinearity was established,and the equations of motion were derived by the Hamilton principle.Thereafter,the equations of motion of the conical shell were simplified into a single degree of freedom nonlinear differential equation by using the Galerkin method and only considering only the transverse vibration.In the end,the nonlinear vibration equation was solved by the improved L-P method and the Runge-Kutta method.And the nonlinear vibration responses of the shell were analysed and the effects of geometric parameters and ceramic volume fraction exponents on the nonlinear frequency responses of the shell were discussed.Results indicate that the geometric parameters have more effects on the nonlinear frequency and the vibration response than the ceramic volume fraction exponents;the geometric parameters and the ceramic volume fraction exponents affect the vibration response by affecting the frequency;the conical shell presents the nonlinear characteristics of a hardening spring.