Nonlinear forced vibration of functionally graded graphene platelet-reinforced metal foam cylindrical shells: internal resonances

Abstract

In the present study, we analyze the nonlinear forced vibration of thin-walled metal foam cylindrical shells reinforced with functionally graded graphene platelets. Attention is focused on the 1:1:1:2 internal resonances, which is detected to exist in this novel nanocomposite structure. Three kinds of porosity distribution and different kinds of graphene platelet distribution are considered. The equations of motion and the compatibility equation are deduced according to the Donnell’s nonlinear shell theory. The stress function is introduced, and then, the four-degree-of-freedom nonlinear ordinary differential equations (ODEs) are obtained via the Galerkin method. The numerical analysis of nonlinear forced vibration responses is presented by using the pseudo-arclength continuation technique. The present results are validated by comparison with those in existing literature for special cases. Results demonstrate that the amplitude–frequency relations of the system are very complex due to the 1:1:1:2 internal resonances. Porosity distribution and graphene platelet (GPL) distribution influence obviously the nonlinear behavior of the shells. We also found that the inclusion of graphene platelets in the shells weakens the nonlinear coupling effect. Moreover, the effects of the porosity coefficient and GPL weight fraction on the nonlinear dynamical response are strongly related to the porosity distribution as well as graphene platelet distribution.

Appendix

Function_i (i = 1, 2, 3, 4) in Eqs. (48–51) is:

$$ \begin{gathered} {\text{Function}}_{1} \left[ {A_{1,n} ,B_{1,n}^{{}} ,A_{1,0} ,A_{3,0} } \right] = p_{1} A_{1,n}^{3} + p_{2} A_{1,0}^{2} A_{1,n} + p_{3} A_{1,0} A_{1,n} A_{3,0} + p_{4} A_{1,n}^{{}} B_{1,n}^{2} + p_{5} A_{1,n} A_{3,0}^{2} \hfill \\ + p_{6} A_{1,0} A_{1,n} + p_{7} A_{1,n} A_{3,0}^{{}} , \hfill \\ {\text{Function}}_{2} \left[ {A_{1,n} ,B_{1,n}^{{}} ,A_{1,0} ,A_{3,0} } \right] = p_{8} B_{1,n}^{3} + p_{9} A_{1,0}^{2} B_{1,n} + p_{10} A_{1,n}^{2} B_{1,n}^{{}} + p_{11} A_{1,0} B_{1,n} A_{3,0} \hfill \\ + p_{12} B_{1,n} A_{3,0}^{2} + p_{13} A_{1,0} B_{1,n} + p_{14} B_{1,n} A_{3,0}^{{}} , \hfill \\ {\text{Function}}_{3} \left[ {A_{1,n} ,B_{1,n}^{{}} ,A_{1,0} ,A_{3,0} } \right] = p_{15} A_{1,0}^{2} + p_{16} A_{3,0}^{2} + p_{17} A_{1,n}^{2} + p_{18} B_{1,n}^{2} + p_{19} A_{1,0}^{{}} A_{1,n}^{2} \hfill \\ + p_{20} A_{1,n}^{2} A_{3,0}^{{}} + p_{21} A_{1,0}^{{}} B_{1,n}^{2} + p_{22} A_{3,0}^{{}} B_{1,n}^{2} + p_{23} A_{3,0}^{{}} , \hfill \\ {\text{Function}}_{4} \left[ {A_{1,n} ,B_{1,n}^{{}} ,A_{1,0} ,A_{3,0} } \right] = p_{{{24}}} A_{1,0}^{{}} A_{1,n}^{2} + p_{{{25}}} A_{3,0}^{{}} A_{1,n}^{2} + p_{{{26}}} A_{1,0}^{{}} B_{1,n}^{2} + p_{{{27}}} A_{3,0}^{{}} B_{1,n}^{2} \hfill \\ + p_{{{28}}} A_{1,0}^{2} + p_{{{29}}} A_{1,n}^{2} + p_{{{30}}} A_{3,0}^{2} + p_{{{31}}} B_{1,n}^{2} + p_{{{32}}} A_{1,0}^{{}} . \hfill \\ \end{gathered} $$

where p_i (i = 1, 2, …, 32) are constant coefficients.