摘要
为解决圆柱壳在工作状态中由几何大变形而引起的弱非线性振动问题,将渐近摄动法引入求解考虑几何非线性的薄壁圆柱壳振动频率。首先,应用Donnell’s简化壳理论获得了考虑几何大变形情况下具有位移三次项的非线性频率方程,把位移及频率以非线性参数的幂级数形式展开,并令同次幂的非线性项系数相等,由此得到非线性频率一次近似值与初始振幅的一系列耦合代数方程,引入Galerkin’s方法对非线性频率方程进行解耦正交并忽略其中的永年项,考虑了对应实数根,各阶频率对应的振幅间不存在相互耦合的内共振现象,最终在引入小参数后用摄动法求出了非线性频率的一次近似解。计算结果表明,几何非线性使薄壁圆柱壳产生硬化,其非线性频率升高,并同时讨论了线性、非线性频率与节径数及初始位移之间的关系。
To solve weak-nonlinear vibration problems of cylindrical shells with large deformation, an asymptotic approach to solution for natural frequencies of a thin cylindrical shell with geometric nonlinearities is proposed. Starting with the Donnel's shallow-shell theory, its nonlinear frequency equations with cubic of displacement are obtained. Its displacement and frequency are expanded in a power series of a small nonlinear parameter. The coefficient of the nonlinear item with the same order is equal. It leads to a set of coupled nonlinear algebraic equations about the first order approximations of nonlinear frequencies and initial amplitudes. The frequency equations are orthogonal and uncoupled by Galerkin's method and their secular terms are eliminated. Considering real solutions and no internal resonance between amplitudes of different modes, the perturbation method with small parameter is used and the first order approximations of nonlinear frequencies are finally acquired. Results show that nonlinear frequencies increase with effect of large geometric deformation taken Relationships among linear frequencies, nonlinear frequencies and initial displacements are discussed.
出处
《振动与冲击》
EI
CSCD
北大核心
2007年第3期42-44,67,共4页
Journal of Vibration and Shock
基金
教育部重大基础研究前期研究专项(2003CCA03900)
关键词
几何非线性
渐近法
摄动法
非线性频率
内共振
geometric nonlinearities, asymptotic approach, perturbation method, nonlinear frequencies, internal resonance
