Free vibration analysis of FG‐GPL and FG‐CNT hybrid laminated nano composite truncated conical shells using systematic differential quadrature method

Abstract

In the present study, the free vibration of functionally graded graphene platelet‐reinforced (FG‐GPLs) and functionally graded carbon nanotube‐reinforced (FG‐CNTs) hybrid laminated nanocomposite truncated conical shells and panels are analyzed. Multi‐layers truncated conical shell and panel of pure FG‐CNTs, pure FG‐GPLs and hybrid CNTs‐GPLs reinforcement were evaluated. In light of its high accuracy in the calculation of thin and thick shells, a third‐order shear deformation theory is adopted. The governing equations and boundary conditions is derived using Hamilton's principle and is solved numerically using the systematic differential quadrature method (DQM) which uses Kronecker delta function. The effective mechanical properties of the CNT‐reinforced nanocomposite layers are estimated using the rule of mixtures, whereas those of the GPL‐reinforced nanocomposite layers is calculated using the Halpin‐Tsai micromechanical model. Convergence and accuracy evaluation of the presented study are confirmed and a number of parameters, including the CNTS volume fraction, GPLs mass fraction, distribution patterns (i.e., Uniform distribution (UD), Functionally graded O‐distribution (FG‐O), Functionally graded X‐distribution (FG‐X), Functionally graded V‐distribution (FG‐V) and FG‐A), different boundary conditions and the vertex angle of the cone, are investigated. The results obtained in this article for the combination of two different materials, GPLs and CNTs, showed that in controlling the natural frequency of the system, without changing the percentage of fiber and only by changing the arrangement, very wonderful results can be achieved.

Vibration characteristics of FG saturated porous annular plates integrated by piezoelectric patches on visco‐Pasternak foundation

“…Therefore, the derivative of a function in a specific point is approximated to the linear summation of weighting coefficients and the value of the function in that point and others. Thus, if the function f be a function of r , for the m ‐th order derivative of it, the following relation may be used [53, 54]: $$$$\begin{equation}{f}^{\left( m \right)}\left( {{x}_i} \right) = \sum_{j = 1}^N {C_{ij}^{\left( m \right)}f({x}_j)} ,\quad \quad i = 1,2, \ldots ,N\end{equation}$$$$in which C ij are weighting coefficients of GDQ and N is the number of grid points. To determine the first‐order weighting coefficient which is related to the first‐order derivative can be written as [55, 56]: $$$$\begin{equation}C_{ij}^{(1)} = \left\{ { \def\eqcellsep{&}\begin{array}{@{}*{2}{l}@{}} {g_j^{\left( 1 \right)}\left( {{x}_i} \right)}&{j \ne i}\\ { - \sum_{j = 1,j \ne i}^N {C_{ij}^{(1)}} }&{j = i} \end{array} } \right.\quad \quad i,j = 1,2, \ldots ,N\end{equation}$$$$…”