The non-linear vibrations of rotating functionally graded cylindrical shells

Abstract

This paper reports the result of an investigate on the non-linear vibrations of rotating functionally graded cylindrical shell in thermal environment, based on Hamilton’s principle, von Kármán non-linear theory and the first-order shear deformation theory. The formulation includes the initial hoop tension, the centrifugal and Coriolis forces due to rotation of the shell. The effects of in-plane and rotary inertia are taken into account in the equations of motion. Galerkin’s method is utilised to convert the governing partial differential equations to non-linear ordinary differential equations. A reduction in the model is presented to investigate non-linear dynamics, including primary resonance responses, quasi-periodic and chaotic responses to harmonic transverse external forces. The modal coefficients of quadratic and cubic nonlinearities are calculated by Galerkin integration and superimposed on the linear part of equation to establish the non-linear reduction equation. To validate the approach proposed in this paper, a series of comparison are performed and the investigations demonstrate good reliability and low computational cost of the present approach.

Appendices

Appendix 1

$$\begin{aligned} \mathbf{C}(z)= & {} \left[ {{\begin{array}{lllll} {C_{11} }&{} {C_{12} }&{} 0&{} 0&{} 0 \\ {C_{21} }&{} {C_{22} }&{} 0&{} 0&{} 0 \\ 0&{} 0&{} {C_{33} }&{} 0&{} 0 \\ 0&{} 0&{} 0&{} {C_{44} }&{} 0 \\ 0&{} 0&{} 0&{} 0&{} {C_{55} } \\ \end{array} }} \right] ,\nonumber \\ {{\varvec{\upalpha }} }= & {} \left[ {{\begin{array}{lllll} {\alpha _{xxe} }&{} {\alpha _{\theta \theta e} }&{} 0&{} 0&{} 0 \\ \end{array} }} \right] ^{T}. \end{aligned}$$

The effective thermal expansion coefficients ($\alpha _{xxe}$, $\alpha _{\theta \theta e}$) in the two principle directions (x, $\theta$) are equal due to in-plane uniform distribution of the functionally graded material properties ($\alpha _{xxe} =\alpha _{\theta \theta e} =\alpha _\mathrm{{eff}})$. The stiffness coefficients are defined according to

$$\begin{aligned} C_{11}= & {} \frac{E_\mathrm{{eff}}}{1-\nu _\mathrm{{eff}}^{2}},\\ C_{12}= & {} \frac{\nu _\mathrm{{eff}} E_\mathrm{{eff}}}{1-\nu _\mathrm{{eff}}^{2}},\nonumber \\ C_{21}= & {} \frac{\nu _\mathrm{{eff}} E_\mathrm{{eff}}}{1-\nu _\mathrm{{eff}}^{2}},\\ C_{22}= & {} \frac{E_\mathrm{{eff}}}{1-\nu _\mathrm{{eff}}^{2}},\\ C_{33}= & {} C_{44} =C_{55} =\frac{E_\mathrm{{eff}}}{2(1+\nu _\mathrm{{eff}})}. \end{aligned}$$

For a given volume fraction exponent $\varPhi$, the effective Young’s modulus $E_\mathrm{{eff}}$, the effective Poisson’s ratio $\nu _\mathrm{{eff}}$ and the effective thermal expansion coefficients $\alpha _\mathrm{{eff}}$ can be obtained according to Eq. (4).

Appendix 2

$$\begin{aligned}&\left\{ {{\begin{array}{ll} {N_x^{T}}&{} {M_x^{T}} \\ {N_\theta ^{T}}&{} {M_\theta ^{T}} \\ {N_{x\theta }^{T}}&{} {M_{x\theta }^{T}} \\ \end{array} }} \right\} =\int _{-\frac{h}{2}}^{\frac{h}{2}}\\&\quad \times {\left\{ {{\begin{array}{l} {Q_{11} (z)\alpha _{xxe} (z)+Q_{12} (z)\alpha _{\theta \theta e} (z)} \\ {Q_{12} (z)\alpha _{xxe} (z)+Q_{22} (z)\alpha _{\theta \theta e} (z)} \\ 0 \\ \end{array} }} \right\} \Delta T(z)\left( {{\begin{array}{ll} 1&{} z \\ \end{array} }} \right) \hbox {d}z},\\&Q_x^{T}=Q_\theta ^{T}=0.,\\&\left\{ {\begin{array}{l} N_x \\ N_\theta \\ N_{x\theta } \\ M_x \\ M_\theta \\ M_{x\theta } \\ \end{array}} \right\} \quad =\left[ {{\begin{array}{llllll} {A_{11} }&{} {A_{12} }&{} 0&{} {B_{11} }&{} {B_{12} }&{} 0 \\ {A_{21} }&{} {A_{22} }&{} 0&{} {B_{21} }&{} {B_{22} }&{} 0 \\ 0&{} 0&{} {A_{66} }&{} 0&{} 0&{} {B_{66} } \\ {B_{11} }&{} {B_{12} }&{} 0&{} {D_{11} }&{} {D_{12} }&{} 0 \\ {B_{21} }&{} {B_{22} }&{} 0&{} {D_{21} }&{} {D_{22} }&{} 0 \\ 0&{} 0&{} {B_{66} }&{} 0&{} 0&{} {D_{66} } \\ \end{array} }} \right] \left\{ {\begin{array}{l} \varepsilon _x \\ \varepsilon _\theta \\ \gamma _{x\theta } \\ \kappa _x \\ \kappa _\theta \\ \kappa _{x\theta } \\ \end{array}} \right\} , \quad \left\{ {\begin{array}{l} Q_x \\ Q_\theta \\ \end{array}} \right\} \\&\quad =\left[ {{\begin{array}{ll} {E_{44} }&{} 0 \\ 0&{} {E_{55} } \\ \end{array} }} \right] \left\{ {{\begin{array}{l} {\gamma _{xz} } \\ {\gamma _{\theta z} } \\ \end{array} }} \right\} ,\\&(A_{ij} , B_{ij} , D_{ij} )\!=\!\int _{-\frac{h}{2}}^{\frac{h}{2}} {Q_{ij} } (1,z,z^{2})dz\, (i,j\!=\!1,2,6), E_{44}\\&\quad =\int _{-\frac{h}{2}}^{\frac{h}{2}} {Q_{\hbox {55}} } dz , E_{55} =\int _{-\frac{h}{2}}^{\frac{h}{2}} {Q_{\hbox {44}} } dz, \end{aligned}$$

where the effective elasticity coefficients $Q_{ij} (z)$ of the FG cylindrical shell are given

$$\begin{aligned} Q_{11}= & {} C_{11}, \quad Q_{12} =C_{12} /A, \quad Q_{21} =C_{21}, \quad Q_{22}\\&=C_{22} /A, \quad Q_{66} =C_{55} /A,\\ Q_{44}= & {} \kappa _G C_{44},\quad Q_{55} =Q_{44} /A, \quad A=1+z/R. \end{aligned}$$

A shear correction factor ($\kappa _G )$ of $\frac{5}{6}$ is used during the evaluation of $E_{44}$ and $E_{55}$ [34].

Appendix 3

$$\begin{aligned} L_{11}= & {} A_{11} \frac{\partial ^{2}}{\partial x^{2}}+\frac{1}{R^{2}}A_{66} \frac{\partial ^{2}}{\partial \theta ^{2}},\\ L_{12}= & {} \frac{A_{12} +A_{66} }{R}\frac{\partial ^{2}}{\partial x\partial \theta },\quad L_{13} =\frac{A_{12} }{R}\frac{\partial }{\partial x},\\ L_{16}= & {} L_{11},\quad L_{14} =B_{11} \frac{\partial ^{2}}{\partial x^{2}}+\frac{B_{66} }{R^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}, \\ L_{15}= & {} \frac{B_{12} +B_{66} }{R}\frac{\partial ^{2}}{\partial x\partial \theta },\\ L_{17}= & {} \frac{L_{12} }{R}, \quad L_{21} =\frac{A_{66} +A_{12} }{R}\frac{\partial ^{2}}{\partial x\partial \theta },\\ L_{22}= & {} A_{66} \frac{\partial ^{2}}{\partial x^{2}}+\frac{A_{22} }{R^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}-\frac{E_{55} }{R^{2}},\\ L_{23}= & {} \frac{A_{22} +E_{55} }{R^{2}}\frac{\partial }{\partial \theta },\\ L_{24}= & {} \frac{B_{66} +B_{12}}{R}\frac{\partial ^{2}}{\partial x\partial \theta }, \\ L_{25}= & {} B_{66} \frac{\partial ^{2}}{\partial x^{2}}+\frac{B_{22} }{R^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}+\frac{E_{55} }{R},\\ L_{26}= & {} L_{21},\quad L_{27} =\frac{A_{66} }{R}\frac{\partial ^{2}}{\partial x^{2}}+\frac{A_{22} }{R^{3}}\frac{\partial ^{2}}{\partial \theta ^{2}},\\ L_{31}= & {} -\frac{A_{21} }{R}\frac{\partial }{\partial x},\\ L_{32}= & {} -\frac{E_{55} }{R^{2}}\frac{\partial }{\partial \theta }-\frac{A_{22} }{R^{2}}\frac{\partial }{\partial \theta },\\ L_{33}= & {} E_{44} \frac{\partial ^{2}}{\partial x^{2}}+\frac{E_{55} }{R^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}-\frac{A_{22} }{R^{2}},\\ L_{34}= & {} E_{44} \frac{\partial }{\partial x}-\frac{B_{21} }{R}\frac{\partial }{\partial x}, \\ L_{35}= & {} \frac{E_{55} }{R}\frac{\partial }{\partial \theta }-\frac{B_{22} }{R^{2}}\frac{\partial }{\partial \theta },\\ L_{36}= & {} -\frac{A_{21} }{2R}\frac{\partial }{\partial x}, \quad L_{37} =-\frac{A_{22} }{2R^{3}}\frac{\partial }{\partial \theta },\\ L_{41}= & {} B_{11} \frac{\partial ^{2}}{\partial x^{2}}+\frac{B_{66} }{R^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}, \quad L_{42} =\frac{B_{12} +B_{66} }{R}\frac{\partial ^{2}}{\partial x\partial \theta },\\ L_{43}= & {} \left( \frac{B_{12} }{R}-E_{44} \right) \frac{\partial }{\partial x},\\ L_{44}= & {} D_{11} \frac{\partial ^{2}}{\partial x^{2}}+\frac{D_{66} }{R^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}-E_{44}, \\ L_{45}= & {} \frac{D_{12} +D_{66} }{R}\frac{\partial ^{2}}{\partial x\partial \theta },\\ L_{46}= & {} L_{41} , L_{47} =\frac{L_{42} }{R},\\ L_{51}= & {} \frac{B_{66} +B_{12} }{R}\frac{\partial ^{2}}{\partial x\partial \theta },\\ L_{52}= & {} B_{66} \frac{\partial ^{2}}{\partial x^{2}}+\frac{B_{22} }{R^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}+\frac{E_{55} }{R},\\ L_{53}= & {} \frac{B_{22} -E_{55} R}{R^{2}}\frac{\partial }{\partial \theta }, \\ L_{54}= & {} \frac{D_{12} +D_{66} }{R}\frac{\partial ^{2}}{\partial x\partial \theta }, \quad L_{55} =\frac{D_{22} }{R^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}\\&\,-E_{55} +D_{66} \frac{\partial ^{2}}{\partial x^{2}},\\ L_{56}= & {} L_{51},\quad L_{57} =\frac{B_{66} }{R}\frac{\partial ^{2}}{\partial x^{2}}+\frac{B_{22} }{R^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}. \end{aligned}$$