Vibration characteristics of moving sigmoid functionally graded plates containing porosities

Abstract

This study investigates vibration characteristics of longitudinally moving sigmoid functionally graded material (S-FGM) plates containing porosities. Two types of porosity distribution, i.e., the even and uneven distributions, are taken into account. In accordance with the sigmoid distribution rule, the material properties of porous S-FGM plates vary smoothly along the plate thickness direction. The nonlinear geometrical relations are adopted by using the von Kármán non-linear plate theory. Based on the d’Alembert’s principle, the nonlinear governing equation of the system is derived. Then, the governing equation is discretized to a set of ordinary differential equations via the Galerkin method. These discretized equations are subsequently solved by using the method of harmonic balance. Analytical solutions are verified with the aid of the adaptive step-size fourth-order Runge–Kutta method. By using the perturbation technique, the stability of the steady-state response is highlighted. Finally, both natural frequencies and nonlinear forced responses of moving porous S-FGM plates are examined. Results demonstrate that the moving porous S-FGM plates exhibit hardening spring characteristics in the nonlinear frequency response. Moreover, it is shown that the type of porosity distribution, moving speed, porosity volume fraction, constituent volume fraction and in-plane pretension all have significant influence on the nonlinear forced responses of moving porous S-FGM plates.

Appendix

The coefficient formulations in Eq. (33) are as follows

$$ \begin{aligned} \bar{J}_{1} = 1 \hfill \\ \bar{J}_{2} = \frac{c}{{\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} }} \hfill \\ \bar{J}_{3} = - \frac{16V}{3a} \hfill \\ \bar{J}_{4} = \frac{{\pi^{4} D_{11} }}{{a^{4} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} + \frac{{2\pi^{4} D_{12} }}{{a^{2} b^{2} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} + \hfill \\ \quad \quad {\kern 1pt} \frac{{4\pi^{4} D_{66} }}{{a^{2} b^{2} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} + \frac{{\pi^{2} N_{0} }}{{a^{2} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ {\kern 1pt} \quad \quad - \frac{{\pi^{2} V^{2} }}{{a^{2} }} + \frac{{\pi^{4} D_{22} }}{{b^{4} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{J}_{5} = - \frac{8cV}{{3a\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{J}_{6} = {{\left( {\frac{{3\pi^{4} A_{11} }}{{32a^{4} }} + \frac{{3\pi^{4} A_{12} }}{{16a^{2} b^{2} }} - \frac{{\pi^{4} A_{66} }}{{8a^{2} b^{2} }} + \frac{{3\pi^{4} A_{22} }}{{32b^{4} }}} \right)} \mathord{\left/ {\vphantom {{\left( {\frac{{3\pi^{4} A_{11} }}{{32a^{4} }} + \frac{{3\pi^{4} A_{12} }}{{16a^{2} b^{2} }} - \frac{{\pi^{4} A_{66} }}{{8a^{2} b^{2} }} + \frac{{3\pi^{4} A_{22} }}{{32b^{4} }}} \right)} {\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}}} \right. \kern-0pt} {\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{J}_{7} = {{\left( {\frac{{3\pi^{4} A_{11} }}{{4a^{4} }} + \frac{{21\pi^{4} A_{12} }}{{16a^{2} b^{2} }} - \frac{{\pi^{4} A_{66} }}{{a^{2} b^{2} }} + \frac{{3\pi^{4} A_{22} }}{{16b^{4} }}} \right)} \mathord{\left/ {\vphantom {{\left( {\frac{{3\pi^{4} A_{11} }}{{4a^{4} }} + \frac{{21\pi^{4} A_{12} }}{{16a^{2} b^{2} }} - \frac{{\pi^{4} A_{66} }}{{a^{2} b^{2} }} + \frac{{3\pi^{4} A_{22} }}{{16b^{4} }}} \right)} {\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}}} \right. \kern-0pt} {\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{J}_{8} = {{\left( {\frac{{32\pi^{2} B_{11} }}{{9a^{4} }} + \frac{{160\pi^{2} B_{12} }}{{9a^{2} b^{2} }} + \frac{{32\pi^{2} B_{66} }}{{9a^{2} b^{2} }} + \frac{{32\pi^{2} B_{22} }}{{9b^{4} }}} \right)} \mathord{\left/ {\vphantom {{\left( {\frac{{32\pi^{2} B_{11} }}{{9a^{4} }} + \frac{{160\pi^{2} B_{12} }}{{9a^{2} b^{2} }} + \frac{{32\pi^{2} B_{66} }}{{9a^{2} b^{2} }} + \frac{{32\pi^{2} B_{22} }}{{9b^{4} }}} \right)} {\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}}} \right. \kern-0pt} {\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{J}_{9} = {{\left( {\frac{{3584\pi^{2} B_{11} }}{{45a^{4} }} + \frac{{512\pi^{2} B_{12} }}{{9a^{2} b^{2} }} + \frac{{1664\pi^{2} B_{66} }}{{45a^{2} b^{2} }} + \frac{{128\pi^{2} B_{22} }}{{45b^{4} }}} \right)} \mathord{\left/ {\vphantom {{\left( {\frac{{3584\pi^{2} B_{11} }}{{45a^{4} }} + \frac{{512\pi^{2} B_{12} }}{{9a^{2} b^{2} }} + \frac{{1664\pi^{2} B_{66} }}{{45a^{2} b^{2} }} + \frac{{128\pi^{2} B_{22} }}{{45b^{4} }}} \right)} {\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}}} \right. \kern-0pt} {\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{J}_{10} = \frac{{4F_{0} }}{{ab\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{-\frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}}\sin \left( {\frac{{\pi x_{0} }}{a}} \right)\sin \left( {\frac{{\pi y_{0} }}{b}} \right) \hfill \\ \end{aligned} $$

(47)

$$ \begin{aligned} \bar{K}_{1} { = }1 \hfill \\ \bar{K}_{2} { = }\frac{16V}{3a} \hfill \\ \bar{K}_{3} { = }\frac{c}{{\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{K}_{4} { = }\frac{8cV}{{3a\left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{K}_{5} { = }\frac{1}{{a^{4} b^{4} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}}(\pi^{4} a^{4} D_{22} - 4\pi^{2} a^{2} b^{4} V^{2} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right) \hfill \\ {\kern 1pt} \quad \quad + 4\pi^{2} a^{2} b^{4} N_{0}^{{}} + 8\pi^{4} a^{2} b^{2} D_{12} + 16\pi^{4} a^{2} b^{2} D_{66} + 16\pi^{4} b^{4} D_{11} ) \hfill \\ \bar{K}_{6} { = }\frac{{\pi^{4} (3a^{4} A_{22} + 24a^{2} A_{12} b^{2} - 16a^{2} A_{66} b^{2} + 48A_{11} b^{4} )}}{{32a^{4} b^{4} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{K}_{7} { = }\frac{{\pi^{4} (3a^{4} A_{22} + 9a^{2} A_{12} b^{2} - 4a^{2} A_{66} b^{2} + 12A_{11} b^{4} )}}{{16a^{4} b^{4} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \bar{K}_{8} { = }\frac{{128\pi^{2} (2a^{4} B_{22} + 25a^{2} b^{2} B_{12} - 4a^{2} b^{2} B_{66} + 5b^{4} B_{11} )}}{{45a^{4} b^{4} \left( {\int_{0}^{{\frac{h}{2}}} {\rho_{1} (z){\text{d}}z} + \int_{{ - \frac{h}{2}}}^{0} {\rho_{2} (z){\text{d}}z} } \right)}} \hfill \\ \end{aligned} $$

(48)