RETRACTED ARTICLE: Chaotic simulation of the multi-phase reinforced thermo-elastic disk using GDQM

Abstract

In this research, a mathematical derivation is made to develop a nonlinear dynamic model for the nonlinear frequency and chaotic responses of the multi-scale hybrid nano-composite reinforced disk in the thermal environment and subject to a harmonic external load. Using Hamilton’s principle and the von Karman nonlinear theory, the nonlinear governing equation is derived. For developing an accurate solution approach, generalized differential quadrature method (GDQM) and perturbation approach (PA) are finally employed. Various geometrically parameters are taken into account to investigate the chaotic motion of the viscoelastic disk subject to harmonic excitation. The fundamental and golden results of this paper could be that in the lower value of the external harmonic force, different FG patterns do not have any effects on the motion response of the structure. But, for the higher value of external harmonic force and all FG patterns, the chaos motion could be seen and for the FG-X pattern, the chaosity is more significant than other patterns of the FG. As a practical designing tip, it is recommended to choose plates with lower thickness relative to the outer radius to achieve better vibration performance.

Appendix

In Eqs. (32a–c), L_ij and M_ij are expressed as follows:

$$\begin{gathered} \delta u_{o} : \hfill \\ L_{11} = A_{11} \frac{{\partial^{2} u}}{{\partial R^{2} }} - \frac{{A_{22} }}{{R^{2} }}u, \hfill \\ L_{12} = - D_{11} c_{1} \frac{{\partial^{3} w}}{{\partial R^{3} }} + \frac{{D_{22} c_{1} }}{{R^{2} }}\frac{\partial w}{{\partial R}} \hfill \\ L_{13} = B_{11} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - D_{11} c_{1} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - \frac{{B_{22} }}{{R^{2} }}\phi + \frac{{D_{22} c_{1} }}{{R^{2} }}\phi \hfill \\ M_{11} = I_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }},\,\,M_{12} = - I_{3} c_{1} \frac{{\partial^{3} w}}{{\partial R\partial t^{2} }},\,\,\,M_{13} = \left( {I_{1} - I_{3} c_{1} } \right)\frac{{\partial^{2} \phi }}{{\partial t^{2} }} \hfill \\ \end{gathered}$$

(i)

$$\begin{gathered} \delta w_{0} : \hfill \\ L_{21} = c_{1} D_{11} \frac{{\partial^{3} u}}{{\partial R^{3} }} - \frac{{c_{1} D_{22} }}{{R^{2} }}\frac{\partial u}{{\partial R}}, \hfill \\ L_{22} = - G_{11} c_{1}^{2} \frac{{\partial^{4} w}}{{\partial R^{4} }} + \frac{{G_{22} c_{1}^{2} }}{{R^{2} }}\frac{{\partial^{2} w}}{{\partial R^{2} }} + \left( {A_{55} - 3C_{55} c_{1} } \right)\frac{{\partial^{2} w}}{{\partial R^{2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, - 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\frac{{\partial^{2} w}}{{\partial R^{2} }} - N^{T} \frac{{\partial^{2} w}}{{\partial R^{2} }} \hfill \\ L_{23} = {\text{C}}\frac{\partial w}{{\partial t}},\,\,\,\,\,\,\,\,L_{24} = \frac{3}{2}{\text{A}}_{11} \frac{{\partial^{2} w}}{{\partial R^{2} }}\left( {\frac{\partial w}{{\partial R}}} \right)^{2} \hfill \\ L_{25} = c_{1} E_{11} \frac{{\partial^{3} \phi }}{{\partial R^{3} }} - G_{11} c_{1}^{2} \frac{{\partial^{3} \phi }}{{\partial R^{3} }} - \frac{{c_{1} }}{R}\frac{{E_{22} \partial \phi }}{R\partial R} - \frac{{c_{1} }}{R}\frac{{G_{22} c_{1} }}{R}\frac{\partial \phi }{{\partial R}} \hfill \\ \,\,\,\,\,\,\,\,\,\, + \left( {A_{55} - 3C_{55} c_{1} } \right)\frac{\partial \phi }{{\partial R}} - 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\frac{\partial \phi }{{\partial R}} \hfill \\ M_{21} = c_{1} I_{3} \frac{{\partial^{3} u}}{{\partial R\partial t^{2} }},\,\,M_{22} = I_{0} \frac{{\partial^{2} w}}{{\partial t^{2} }} - c_{1}^{2} I_{6} \frac{{\partial^{4} w}}{{\partial R^{2} \partial t^{2} }}, \hfill \\ M_{23} = \left( {c_{1} I_{4} - c_{1}^{2} I_{6} } \right)\frac{{\partial^{3} \phi }}{{\partial R\partial t^{2} }}, \hfill \\ \, \hfill \\ \end{gathered}$$

(ii)

$$\begin{gathered} \delta \phi : \hfill \\ L_{31} = B_{11} \frac{{\partial^{2} u}}{{\partial R^{2} }} - c_{1} D_{11} \frac{{\partial^{2} u}}{{\partial R^{2} }} - \frac{{B_{22} }}{{R^{2} }}u + \frac{{D_{22} }}{{R^{2} }}u, \hfill \\ L_{32} = - E_{11} c_{1} \frac{{\partial^{3} w}}{{\partial R^{3} }} + G_{11} c_{1}^{2} \frac{{\partial^{3} w}}{{\partial R^{3} }} + \frac{{E_{22} c_{1} }}{{R^{2} }}\frac{\partial w}{{\partial R}} + \frac{{G_{22} c_{1}^{2} }}{{R^{2} }}\frac{\partial w}{{\partial R}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\, - \left( {A_{55} - 3C_{55} c_{1} } \right)\frac{\partial w}{{\partial R}} + 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\frac{\partial w}{{\partial R}} \hfill \\ L_{33} = C_{11} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - E_{11} c_{1} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} - c_{1} E_{11} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} + G_{11} c_{1}^{2} \frac{{\partial^{2} \phi }}{{\partial R^{2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\, - \frac{1}{{R^{2} }}\left\{ {C_{22} - E_{22} c_{1} } \right\}\phi + \frac{{c_{1} }}{{R^{2} }}\left\{ {E_{22} - G_{22} c_{1} } \right\}\phi \hfill \\ \,\,\,\,\,\,\, \, - \left( {A_{55} - 3C_{55} c_{1} } \right)\phi + 3c_{1} \left( {C_{55} - 3E_{55} c_{1} } \right)\phi \hfill \\ M_{31} = \left( {I_{1} - c_{1} I_{3} } \right)\frac{{\partial^{2} u}}{{\partial t^{2} }},\,\,\,M_{32} = \left( {I_{6} c_{1}^{2} - I_{4} c_{1} } \right)\frac{{\partial^{3} w}}{{\partial R\partial t^{2} }}, \hfill \\ M_{33} = \left( {I_{6} c_{1}^{2} - 2c_{1} I_{4} + I_{2} } \right)\frac{{\partial^{2} \phi }}{{\partial t^{2} }}. \hfill \\ \end{gathered}$$

(iii)