An analysis is conducted in the present investigation to study the free and forced vibration characteristics of composite laminated conical panels. It is assumed that composite panel is reinforced with graphene platelets (GPLs) where the amount of GPLs may be different between the layers which results in a piecewise functionally graded media. In addition, the effect of uniform and non-uniform porosities is also included into the formulation. The elasticity modulus of the media is estimated following the Halpin-Tsai rule while the simple rule of mixtures approach is applied to obtain the mass density and Poisson's ratio on the panel. With the aid of the first order shear deformation theory of shells and Donnell kinematics, definition of strain, kinetic and potential energy of applied loads is given. After that with the general idea of Ritz method suitable for arbitrary types of boundary conditions and Legendre polynomials as the basic functions, the energies are discreted which results in the general form of the equations of motion. While standard eigenvalue solution is applied to obtain the natural frequencies, a Newmark time marching method is implemented to obtain the temporal evolution of displacement of the panel. Results of this study are well-compared with the available data in the open literature and after that novel results on the present study are provided in the form of figures and tables. It is highlighted that type of porosity and graded pattern of GPLs are both important on the frequencies and dynamic deflection of the panel.

中文翻译：

---

使用 Legendre-Ritz 方法研究多孔 GPL 增强复合材料锥形板的自由振动和受迫振动

本研究进行了分析，以研究复合材料叠层锥形板的自由振动和受迫振动特性。假设复合板是用石墨烯片（GPL）增强的，其中各层之间 GPL 的量可能不同，从而产生分段功能梯度介质。此外，均匀和不均匀孔隙率的影响也包含在配方中。介质的弹性模量按照 Halpin-Tsai 规则进行估计，同时应用简单的混合规则方法来获得面板上的质量密度和泊松比。借助壳的一阶剪切变形理论和Donnell运动学，给出了施加载荷的应变、动能和势能的定义。然后利用适用于任意类型边界条件的Ritz方法的总体思想，以Legendre多项式为基本函数，对能量进行离散化，得到运动方程的一般形式。在应用标准特征值解来获得固有频率的同时，采用纽马克时间推进方法来获得面板位移的时间演化。这项研究的结果与公开文献中的可用数据进行了很好的比较，然后以图表的形式提供了本研究的新颖结果。值得注意的是，GPL 的孔隙率类型和分级模式对于面板的频率和动态偏转都很重要。