Shooting method for free vibration of FGM Reissner-Mindlin circular plates resting on elastic foundation in thermal environments

Abstract

This paper presents a free vibration analysis of functionally graded Reissner-Mindlin circular plates with various supported boundaries in thermal environments. A FGM consisting of metal and ceramic was considered in the study. Based on the geometric equation, physical equation and equilibrium equation of thick plate, taking into account the transverse shearing deformation, the free vibration equation of the axisymmetric FGM moderately thick circular plates was derived in terms of the middle surface angles of rotation and lateral displacement. The material properties of the plate were assumed to vary continuously in the thickness direction according a power law. By using shooting method to solve the coupled ordinary differential equations with different boundary conditions, the natural frequencies of FGM thick circular plates were obtained numerically. The effects of material gradient property, thickness ratio and boundary conditions on the natural frequencies were discussed in detail.

The influence of surface effects on axisymmetric vibration of graded porous Mindlin circular nanoplate in a temperature field

“…The specific operation process of shooting method can be referred to in the literature [44][45][46].…”

Free vibration analysis of graded porous circular micro/nanoplates with various boundary conditions based on the nonlocal elasticity theory

“…By assuming the vibration response is harmonic, the solution of the circular nanoplate with axisymmetric can be expressed by [40]: $$$$\begin{equation}\text{ }\!\!\{\!\!\text{ }w(r,t)\psi (r,t)\text{ }\!\!\}\!\!\text{ }=\text{ }\!\!\{\!\!\text{ }\tilde{w}(r)\tilde{\psi }(r)\text{ }\!\!\}\!\!\text{ }\cos \Omega t\end{equation}$$$$where $$\tilde w(r)$$, $$\tilde \psi (r)$$ represent the shape functions of the porous circular nanoplate, and Ω is its natural frequency. $$$$\begin{equation}D\left( {\frac{{{\partial ^2}\tilde \psi }}{{\partial {r^2}}} + \frac{1}{r}\frac{{\partial \tilde \psi }}{{\partial r}} - \frac{{\tilde \psi }}{{{r^2}}}} \right) - {\kappa _s}S\left( {...$$…”

“…By assuming the vibration response is harmonic, the solution of the circular nanoplate with axisymmetric can be expressed by [40]: {𝑤(𝑟, 𝑡)𝜓(𝑟, 𝑡)} = { w(𝑟) ψ(𝑟)}cos Ω𝑡 (30) where w(𝑟), ψ(𝑟) represent the shape functions of the porous circular nanoplate, and Ω is its natural frequency.…”

Free vibration analysis of a composite elliptical plate made of a porous core and two piezoelectric layers

“…Feyzi and Khorshidvand 17 studied the symmetrical axial postbuckling behavior of porous circular plates with the simply supported and clamped boundary conditions and solved the governing equations by the shooting method. 18 They also examined the effects of porosity coefficient, pores' distribution and fluid compression, thickness change, and boundary conditions on the plate post-buckling behavior.…”