多层复合壳体三维振动分析的谱——微分求积混合法

对于较厚的多层复合壳体,其振动位移沿厚度方向呈锯齿形变化且层间剪切和拉、压应力呈三维耦合状态,采用传统的等效单层理论分析已不能满足精度要求. 建立不受结构厚度、铺层材料性质和铺层方式限制的三维分析方法具有重要的研究价值. 本文以独立铺层为建模对象,结合广义谱方法与微分求积技术建立了一种适用一般边界条件和铺层方式的多层复合壳体三维分析新方法——谱--微分求积混合法. 该方法应用三维弹性理论对独立铺层进行精确建模,有效克服了二维简化理论对横向变形以及层间应力估计不确切的缺点;引入微分求积技术对铺层进行数值离散,将三维偏微分问题转化为二维偏微分问题,降低了求解维度和难度;应用广义谱方法近似地表述离散计算面上的场变量,将获取的二维偏微分方程转化为以场变量谱展开系数为未知量的线性代数方程组,避免了对超越方程的求解. 数值验证结果表明该方法收敛性好,计算精度高. 

A SPECTRAL-DIFFERENTIAL QUADRATURE METHOD FOR 3-D VIBRATION ANALYSIS OF MULTILAYERED SHELLS

The equivalent single layer (ESL) theories can be grossly in error for predicting vibration characteristics of thick multilayered shells because the vibration displacement and stress field of such shells under vibration are in full 3-D coupling condition. It is necessary to develop more accurate and efficient methods which are capable of dealing with multilayered structures with different boundary conditions, general laminations as well as arbitrary thickness universally. In order to overcome the drawback of the existing three-dimensional methods that are only confined for very limited cases such as cross-ply laminated rectangular plates under simply-supported boundary conditions, a general spectral-differential quadrature method is proposed. This method is undertaken by the exact 3-D elasticity theory so that it’s able to study very well the dynamic behavior of thick multilayered structures which cannot be provided by the 2-D ESL theories. In each individual layer, the transverse domain is discretized by the differential quadrature technique. The displacement fields of the discretized surfaces are selected as fundamental unknowns. Then, each fundamental unknown is invariantly expanded by the general spectral method as a series of complete, orthogonal polynomials. The problems are stated in a variational form by the aid of penalty parameters which provides complete flexibilities to describe any prescribed boundary conditions. The current method can successfully avoid solving a highly nonlinear transcendental equation that is rely on roots-locating numerical method and all the modal information can be obtained just by solving linear algebraic equation systems. Numerical verification shows that the proposed method has high calculation precision. The method can be directly extend to the static and dynamic analysis of multilayered shells as well.