Harmonic resonances of graphene-reinforced nonlinear cylindrical shells: effects of spinning motion and thermal environment

Abstract

This work investigates nonlinear harmonic resonance behaviors of graded graphene-reinforced composite spinning thin cylindrical shells subjected to a thermal load and an external excitation. The volume fraction of graphene platelets varies continuously in the shell’s thickness direction, which generates position-dependent useful material properties. Natural frequencies of shell traveling waves are derived by considering influences of the initial hoop tension, centrifugal and Coriolis forces, thermal expansion deformation, and thermal conductivity. A new Airy stress function is introduced. Harmonic resonance behaviors and their stable solutions for the spinning cylindrical shell are analyzed based on an equation of motion which is established by adopting Donnell’s nonlinear shell theory. The necessary and sufficient conditions for the existence of the subharmonic resonance of the spinning composite cylindrical shell are given. Besides the shell’s intrinsic structural damping, the Coriolis effect due to the spinning motion has a contribution to the damping terms of the system as well. Comparisons between the present analytical results and those in other papers are made to validate the existing solutions. Influences of main factors on vibration characteristics, primary resonance, and subharmonic resonance behaviors of the novel composite cylindrical shell are discussed. Furthermore, the mechanism of how the spinning motion affects the amplitude–frequency curves of harmonic resonances of the cylindrical shell is analyzed.

Appendices

Appendix A

Expressions of the differential operators in Eq. (21) are given as

$$\begin{aligned} \begin{aligned} L_{11}&=A_{11}^{*} ()_{,xx} +\left( {r^{-2}A_{44}^{*} +I_0 \Omega ^{2}} \right) ()_{,\theta \theta } , \\ L_{12}&=r^{-1}\left( {A_{12}^{*} +A_{44}^{*} } \right) ()_{,x\theta } , \\ L_{13}&=r^{-1}A_{12}^{*} ()_{,x} -B_{11}^{*} ()_{,xxx}\\&\quad -r^{-2}\left( {B_{12}^{*} +2B_{44}^{*} } \right) ()_{,x\theta \theta } , \\ L_{21}&=L_{12} , \\ L_{22}&=A_{44}^{*} ()_{,xx} +\left( {r^{-2}A_{11}^{*} +I_0 \Omega ^{2}} \right) ()_{,\theta \theta } , \\ L_{23}&=\left( {r^{-2}A_{11}^{*} +2I_0 \Omega ^{2}} \right) ()_{,\theta }\\&\quad -r^{-3}B_{11}^{*} ()_{,\theta \theta \theta } -r^{-1}\left( {B_{12}^{*} +2B_{44}^{*} } \right) ()_{,xx\theta } , \\ L_{31}&=-L_{13} ,\,\,L_{32} =-L_{23} , \\ L_{33}&=-r^{-2}A_{11}^*()+\left( {2r^{-1}B_{12}^*-N^{T}} \right) ()_{,xx} \\&\quad +\left( {2r^{-3}B_{11}^*+I_0 \Omega ^{2}-r^{-2}N^{T}} \right) ()_{,\theta \theta } \\&\quad -D_{11}^*()_{,xxxx} -r^{-4}D_{11}^*()_{,\theta \theta \theta \theta } \\&\quad -2r^{-2}\left( {D_{12}^*+2D_{44}^*} \right) ()_{,xx\theta \theta } . \\ \end{aligned} \end{aligned}$$

(A.1)

Appendix B

Elements of the determinant of matrix (22) have the following forms

$$\begin{aligned} \Gamma _{11}^{*}= & {} A_{\mathrm{11}}^{*} \frac{\Lambda _{\mathrm{2}} }{\Lambda _{\mathrm{1}} }-n^{2}\left( {r^{-2}A_{\mathrm{44}}^{*} +I_0 \Omega ^{2}} \right) ,\nonumber \\ \Gamma _{12}^{*}= & {} nr^{-1}\left( {A_{\mathrm{12}}^{*} +A_{\mathrm{44}}^{*} } \right) ,\nonumber \\ \Gamma _{13}^{*}= & {} r^{-1}A_{\mathrm{12}}^{*} -B_{\mathrm{11}}^{*} \frac{\Lambda _{\mathrm{2}} }{\Lambda _{\mathrm{1}} }+n^{2}r^{-2}\left( {B_{\mathrm{12}}^{*} +2B_{44}^{*} } \right) , \nonumber \\ \Gamma _{21}^{*}= & {} -\,nr^{-1}\left( {A_{\mathrm{12}}^{*} +A_{44}^{*} } \right) \frac{\Lambda _4 }{\Lambda _3},\nonumber \\ \Gamma _{22}^{*}= & {} A_{44}^{*} \frac{\Lambda _4 }{\Lambda _3 }-n^{2}\left( {r^{-2}A_{\mathrm{11}}^{*} +I_0 \Omega ^{2}} \right) ,\nonumber \\ \Gamma _{23}^{*}= & {} -\,n\left( {r^{-2}A_{\mathrm{11}}^{*} +2I_0 \Omega ^{2}} \right) -n^{3}r^{-3}B_{\mathrm{11}}^{*} \nonumber \\&\quad +\,nr^{-1}\left( {B_{\mathrm{12}}^{*} +2B_{44}^{*} } \right) \frac{\Lambda _4 }{\Lambda _3 }, \nonumber \\ \Gamma _{31}^{*}= & {} -\,r^{-1}A_{\mathrm{12}}^{*} \frac{\Lambda _4 }{\Lambda _3 }+B_{\mathrm{11}}^{*} \frac{\Lambda _5 }{\Lambda _3}\nonumber \\&\quad -\,n^{2}r^{-2}\left( {B_{\mathrm{12}}^{*} +2B_{44}^{*} } \right) \frac{\Lambda _4 }{\Lambda _3 }, \nonumber \\ \Gamma _{32}^{*}= & {} \Gamma _{23} , \nonumber \\ \Gamma _{33}^{*}= & {} -\,r^{-2}A_{\mathrm{11}}^{*} +\left( {2r^{-1}B_{12}^*-N^{T}} \right) \frac{\Lambda _4 }{\Lambda _3 }\nonumber \\&\quad -\,n^{2}\left( {2r^{-3}B_{11}^*+I_0 \Omega ^{2}-r^{-2}N^{T}} \right) \nonumber \\&\quad -\,D_{\mathrm{11}}^{*} \frac{\Lambda _{\mathrm{5}} }{\Lambda _3 }-n^{4}r^{-4}D_{\mathrm{11}}^{*} \nonumber \\&\quad +\,2n^{2}r^{-2}\left( {D_{\mathrm{12}}^{*} +2D_{44}^{*} } \right) \frac{\Lambda _4 }{\Lambda _3 }, \end{aligned}$$

(B.1)

and

$$\begin{aligned} \Gamma _{11} =\frac{\Gamma _{12}^{*} \Gamma _{23}^{*} -\Gamma _{22}^{*} \Gamma _{13}^{*} }{\Gamma _{11}^{*} \Gamma _{22}^{*} -\Gamma _{12}^{*} \Gamma _{21}^{*} },\,\,\Gamma _{12} =\frac{\Gamma _{21}^{*} \Gamma _{13}^{*} -\Gamma _{11}^{*} \Gamma _{23}^{*} }{\Gamma _{11}^{*} \Gamma _{22}^{*} -\Gamma _{12}^{*} \Gamma _{21}^{*} },\nonumber \\ \end{aligned}$$

(B.2)

where the coefficients $\varLambda _\mathrm{i} \left( {i=1,\,2,\,3,\,4,\,5} \right)$ related to the assumed shell models can be calculated from

$$\begin{aligned}&\left\{ {{\begin{array}{ccccc} {\Lambda _1 }&{} {\Lambda _2 }&{} {\Lambda _3 }&{}{\Lambda _4 }&{} {\Lambda _5 } \\ \end{array} }} \right\} \\&\quad =\int _0^L {\left\{ {{\begin{array}{ccccc} {\left( {\frac{\hbox {d}\phi }{\hbox {d}x}} \right) ^{2}}&{} {\frac{\hbox {d}^{3}\phi }{\hbox {d}x^{3}}\frac{\hbox {d}\phi }{\hbox {d}x}}&{} {\phi ^{\mathrm{2}}}&{} {\phi \frac{\hbox {d}^{\mathrm{2}}\phi }{\hbox {d}x^{\mathrm{2}}}}&{} {\phi \frac{\hbox {d}^{4}\phi }{\hbox {d}x^{4}}} \\ \end{array} }} \right\} } \hbox {d}x. \\ \end{aligned}$$