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Dynamic analysis of a multilayered piezoelectric two-dimensional quasicrystal cylindrical shell filled with compressible fluid using the state-space approach

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Abstract

The state-space approach is developed to analyze the dynamic behaviors of a multilayered two-dimensional piezoelectric quasicrystal circular cylinder filled with the compressible fluid. With simple support at both ends, the hollow cylindrical shell has imperfect bonding between the layers. The analytical solution of a homogeneous cylindrical shell has been derived based on the state equations. The general solution for the corresponding multilayered case is also obtained by utilizing the propagator matrix method. The numerical results present the natural frequencies in free vibration with different length-to-radius and radius-to-thickness ratios. The critical load and dynamic behaviors of the model are exactly predicted in the axial buckling problem. For the impulse case, the influences of the density of the filled fluid and coefficients of interfacial imperfections on the dynamic responses are also discussed.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11972354, 11972365, 51704015 and 11772349) and China Agricultural University Education Foundation (No. 1101-2412001).

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Authors and Affiliations

  1. College of Science, China Agricultural University, Beijing, 100190, China

    Yunzhi Huang, Yang Li, Liangliang Zhang & Yang Gao

  2. College of Engineering and Computer Science, Australian National University, Canberra, ACT, 2601, Australia

    Yang Li

  3. China State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing, 100190, China

    Han Zhang

  4. Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, Beijing, 100190, China

    Han Zhang

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  1. Yunzhi Huang
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  2. Yang Li
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  3. Liangliang Zhang
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  4. Han Zhang
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Correspondence to Yang Gao.

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Appendix

Appendix

The operational matrixes A \((13 \times 13)\) and B\((13 \times 8)\) in Eqs. (4) and (5) are presented as follows:

$$\begin{aligned} {\mathbf{A}}= & {} \left[ {{\begin{array}{ll} {{\mathbf{A}}_{11} } &{}\quad {{\mathbf{A}}_{12} } \\ {{\mathbf{A}}_{21} } &{}\quad {{\mathbf{A}}_{22} } \\ \end{array} }} \right] ,\\ {\mathbf{A}}_{11}= & {} \left[ {{\begin{array}{llllll} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{e_{15} }{\alpha }} \\ 0 &{} {\frac{1}{r}} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ {\frac{\vartheta _{1} }{r}\frac{\partial }{\partial z}} &{} {\frac{\vartheta _{3} }{r^{2}}\frac{\partial }{\partial \theta }} &{} {\frac{\vartheta _{2} }{r^{2}}\frac{\partial }{\partial \theta }} &{} {\frac{\vartheta _{4} }{r}} &{} {\frac{\beta _{5} }{r\beta _{2} }} &{} 0 \\ {\frac{C_{13} \beta _{4} }{r\beta _{2} }\frac{\partial }{\partial z}} &{} {\frac{\vartheta _{2} }{r^{2}}\frac{\partial }{\partial \theta }} &{} {\frac{\vartheta _{5} }{r^{2}}\frac{\partial }{\partial \theta }} &{} {-\frac{\beta _{4} }{r\beta _{2} }} &{} {-\frac{C_{11} \beta _{4} }{rR_{1} \beta _{2} }} &{} 0 \\ {-\left( \frac{e_{24} }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}+\frac{\kappa _{2} }{\beta _{2} }\frac{\partial ^{2}}{\partial z^{2}}\right) } &{} {-\frac{(e_{24} \beta _{2} +\kappa _{1} )}{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} &{} {-\frac{e_{31} \beta _{4} }{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} &{} {\frac{e_{31} K_{1} }{\beta _{2} }\frac{\partial }{\partial z}} &{} {-\frac{e_{31} R_{1} }{\beta _{2} }\frac{\partial }{\partial z}} &{} {-\frac{e_{15}^{2} }{r\alpha }} \\ \end{array} }} \right] ,\\ {\mathbf{A}}_{12}= & {} \left[ {{\begin{array}{lllllll} {\frac{\chi _{11} }{\alpha }} &{} 0 &{} 0 &{} 0 &{} {-\frac{\partial }{\partial z}} &{} 0 &{} 0 \\ 0 &{} {\frac{1}{C_{66} }} &{} {-\beta _{10} } &{} {-\beta _{10} } &{} {-\frac{\partial }{r\partial \theta }} &{} {\frac{2R_{1} }{C_{66} }\frac{\partial }{r\partial \theta }} &{} 0 \\ 0 &{} 0 &{} {\frac{1}{(K_{1} -K_{2} )}} &{} {\frac{1}{(K_{1} -K_{2} )}} &{} 0 &{} {\frac{\partial }{r\partial \theta }} &{} 0 \\ {-\frac{\partial }{\partial z}} &{} {-\frac{\partial }{r\partial \theta }} &{} 0 &{} 0 &{} {(\rho \frac{\partial ^{2}}{\partial t^{2}}+\vartheta _{3} )} &{} {\frac{\vartheta _{2} }{r^{2}}\frac{\partial }{\partial \theta }} &{} {\frac{\kappa _{1} }{r\beta _{2} }\frac{\partial }{\partial z}} \\ 0 &{} 0 &{} 0 &{} 0 &{} {\frac{\vartheta _{2} }{r^{2}}} &{} {\left( \rho \frac{\partial ^{2}}{\partial t^{2}}-K_{4} \frac{\partial ^{2}}{\partial z^{2}}+\frac{\vartheta _{5} }{r^{2}}\frac{\partial }{\partial \theta }\right) } &{} {\frac{e_{31} \beta _{4} }{r\beta _{2} }\frac{\partial }{\partial z}} \\ {-\frac{\chi _{11} e_{15} }{r\alpha }} &{} 0 &{} 0 &{} 0 &{} {-\frac{\kappa _{1} }{r\beta _{2} }\frac{\partial }{\partial z}} &{} {-\frac{e_{31} \beta _{4} }{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} &{} {(\frac{\chi _{22} }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}+\frac{\kappa _{3} }{\beta _{2} }\frac{\partial ^{2}}{\partial z^{2}})} \\ \end{array} }} \right] , \end{aligned}$$
$$\begin{aligned} {\mathbf{A}}_{21}= & {} \left[ {{\begin{array}{ccccccc} {\left( \rho \frac{\partial ^{2}}{\partial t^{2}}-\frac{C_{44} }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}-\frac{\kappa _{4} }{\beta _{2} }\frac{\partial ^{2}}{\partial z^{2}}\right) } &{} {-\frac{(C_{44} \beta _{2} +\kappa _{5} )}{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} &{} {-\frac{\beta _{4} C_{13} }{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} &{} {\frac{K_{1} C_{13} }{\beta _{2} }\frac{\partial }{\partial z}} &{} {-\frac{R_{1} C_{13} }{\beta _{2} }\frac{\partial }{\partial z}} &{} {\frac{e_{24} C_{44} }{r^{2}\alpha }\frac{\partial }{\partial \theta }} \\ {-\frac{(\beta _{2} \kappa _{6} +C_{13} \beta _{3} )}{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} &{} {\left( \rho \frac{\partial ^{2}}{\partial t^{2}}-\frac{\vartheta _{3} }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}-C_{44} \frac{\partial ^{2}}{\partial z^{2}}\right) } &{} {-\frac{\vartheta _{2} }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}} &{} {-\frac{\beta _{3} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {-\frac{\beta _{5} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {\frac{e_{24} C_{44} }{r\alpha }\frac{\partial }{\partial z}} \\ {-\frac{C_{13} \beta _{4} }{r\beta _{1} }\frac{\partial ^{2}}{\partial \theta \partial z}} &{} {-\frac{\vartheta _{2} \beta _{2} }{r^{2}\beta _{1} }\frac{\partial ^{2}}{\partial \theta ^{2}}} &{} {\left( \rho \frac{\partial ^{2}}{\partial t^{2}}+\frac{\vartheta _{5} \beta _{2} }{r^{2}\beta _{1} }\frac{\partial ^{2}}{\partial \theta ^{2}}-K_{4} \frac{\partial ^{2}}{\partial z^{2}}\right) } &{} {\frac{\beta _{4} }{r\beta _{1} }\frac{\partial }{\partial \theta }} &{} {\frac{(2R^{2}-\beta _{2} )}{r\beta _{1} }\frac{\partial }{\partial \theta }} &{} 0 \\ {-\frac{C_{13} \beta _{4} }{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} &{} {-\frac{\vartheta _{2} }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}} &{} {\left( \rho \frac{\partial ^{2}}{\partial t^{2}}-\frac{\vartheta _{5} }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}-K_{4} \frac{\partial ^{2}}{\partial z^{2}}\right) } &{} {\frac{\beta _{4} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {-\frac{\beta _{6} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} 0 \\ {-\frac{K_{1} C_{13} }{\beta _{2} }\frac{\partial }{\partial z}} &{} {\frac{\beta _{3} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {\frac{\beta _{4} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {-\frac{K_{1} }{\beta _{2} }} &{} {\frac{R_{1} }{\beta _{2} }} &{} 0 \\ {-\frac{R_{1} C_{13} }{\beta _{2} }\frac{\partial }{\partial z}} &{} {-\frac{\beta _{5} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {-\frac{\beta _{6} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {\frac{R_{1} }{\beta _{2} }} &{} {\frac{C_{11} }{\beta _{2} }} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {-\frac{C_{44} }{\alpha }} \\ \end{array} }} \right] , \end{aligned}$$
$$\begin{aligned} {\mathbf{A}}_{22}= & {} \left[ {{\begin{array}{ccccccc} {-\left( \frac{1}{r}+\frac{e_{15} e_{24} }{r^{2}\alpha }\frac{\partial }{\partial \theta }\right) } &{} 0 &{} 0 &{} 0 &{} {-\frac{\kappa _{5} }{r\beta _{2} }\frac{\partial }{\partial z}} &{} {-\frac{\beta _{4} C_{13} }{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} &{} {\frac{\kappa _{2} }{\beta _{2} }\frac{\partial ^{2}}{\partial z^{2}}} \\ {-\frac{e_{15} e_{24} }{r\alpha }\frac{\partial }{\partial z}} &{} {-\frac{2}{r}} &{} 0 &{} 0 &{} {-\frac{\vartheta _{3} }{r^{2}}\frac{\partial }{\partial \theta }} &{} {-\frac{\vartheta _{2} }{r^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}} &{} {-\frac{\kappa _{1} }{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} \\ 0 &{} 0 &{} {-\frac{1}{r}} &{} {-\frac{1}{r}} &{} {-\frac{\vartheta _{2} \beta _{2} }{r^{2}\beta _{1} }\frac{\partial }{\partial \theta }} &{} {\frac{\vartheta _{5} \beta _{2} }{r^{2}\beta _{1} }\frac{\partial }{\partial \theta }} &{} {-\frac{\beta _{4} e_{31} }{r\beta _{1} }\frac{\partial ^{2}}{\partial \theta \partial z}} \\ 0 &{} 0 &{} {-\frac{1}{r}} &{} {-\frac{1}{r}} &{} {-\frac{\vartheta _{2} }{r^{2}}\frac{\partial }{\partial \theta }} &{} {-\frac{\vartheta _{5} }{r^{2}}\frac{\partial }{\partial \theta }} &{} {-\frac{\beta _{4} e_{31} }{r\beta _{2} }\frac{\partial ^{2}}{\partial \theta \partial z}} \\ 0 &{} 0 &{} 0 &{} 0 &{} {\frac{\beta _{3} }{r\beta _{2} }} &{} {\frac{\beta _{4} }{r\beta _{2} }} &{} {\frac{e_{31} K_{1} }{\beta _{2} }\frac{\partial }{\partial z}} \\ 0 &{} 0 &{} 0 &{} 0 &{} {-\frac{\beta _{5} }{r\beta _{2} }} &{} {-\frac{\beta _{6} }{r\beta _{2} }} &{} {-\frac{e_{31} R_{1} }{\beta _{2} }\frac{\partial }{\partial z}} \\ {\frac{e_{15} }{\alpha }} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ \end{array} }} \right] .\\ {\mathbf{B}}= & {} \left[ {{\begin{array}{ccccccccccccc} {\frac{\kappa _{4} }{\beta _{2} }\frac{\partial }{\partial z}} &{} {\frac{\vartheta _{1} }{r}\frac{\partial }{\partial \theta }} &{} {\frac{\beta _{4} C_{13} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {-\frac{K_{1} C_{13} }{\beta _{2} }} &{} {\frac{R_{1} C_{13} }{\beta _{2} }} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{\vartheta _{1} }{r}} &{} {\frac{\beta _{4} C_{13} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {\frac{\kappa _{2} }{\beta _{2} }\frac{\partial }{\partial z}} \\ {\vartheta _{1} \frac{\partial }{\partial z}} &{} {\frac{\vartheta _{3} }{r}\frac{\partial }{\partial \theta }} &{} {\frac{\vartheta _{2} }{r}\frac{\partial }{\partial \theta }} &{} {-\frac{\beta _{3} }{\beta _{2} }} &{} {\frac{\beta _{5} }{\beta _{2} }} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{\vartheta _{3} }{r}} &{} {\frac{\vartheta _{2} }{r}\frac{\partial }{\partial \theta }} &{} {\frac{\kappa _{1} }{\beta _{2} }\frac{\partial }{\partial z}} \\ {\frac{C_{44} }{r}\frac{\partial }{\partial \theta }} &{} {C_{44} \frac{\partial }{\partial z}} &{} 0 &{} 0 &{} 0 &{} {\frac{e_{24} C_{44} }{r\alpha }} &{} {\frac{e_{15} e_{24} }{r\alpha }} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {K_{4} \frac{\partial }{\partial z}} &{} 0 \\ {\frac{\beta _{4} C_{13} }{\beta _{2} }\frac{\partial }{\partial z}} &{} {\frac{\vartheta _{2} }{r}\frac{\partial }{\partial \theta }} &{} {\frac{\vartheta _{5} }{r}\frac{\partial }{\partial \theta }} &{} {-\frac{\beta _{4} }{\beta _{2} }} &{} {\frac{\beta _{6} }{\beta _{2} }} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{\vartheta _{2} }{r}} &{} {\frac{\vartheta _{5} }{r}\frac{\partial }{\partial \theta }} &{} {\frac{\beta _{4} e_{13} }{\beta _{2} }\frac{\partial }{\partial z}} \\ 0 &{} 0 &{} {K_{4} \frac{\partial }{\partial z}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ {\frac{\kappa _{2} }{\beta _{2} }\frac{\partial }{\partial z}} &{} {\frac{\kappa _{1} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {\frac{\beta _{4} e_{13} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {-\frac{K_{1} e_{13} }{\beta _{2} }} &{} {\frac{R_{1} e_{13} }{\beta _{2} }} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{\kappa _{1} }{r\beta _{2} }} &{} {\frac{\beta _{4} e_{13} }{r\beta _{2} }\frac{\partial }{\partial \theta }} &{} {-\frac{\kappa _{3} }{\beta _{2} }\frac{\partial }{\partial z}} \\ {\frac{e_{24} }{r}\frac{\partial }{\partial \theta }} &{} {e_{24} \frac{\partial }{\partial z}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{\chi _{22} }{r}\frac{\partial }{\partial \theta }} \\ \end{array} }} \right] , \end{aligned}$$

where

$$\begin{aligned} \alpha= & {} C_{55} \chi _{11} +e_{15}^{2} , \beta _{1} =R_{1}^{2} +C_{11} K_{2} , \beta _{2} =R_{1}^{2} -C_{11} K_{1} , \beta _{3} =C_{12} K_{1} +R_{1}^{2} , \beta _{4} =R_{1} (K_{1} -K_{2} ),\\ \beta _{5}= & {} R_{1} (C_{11} +C_{12} ), \beta _{6} =R_{1}^{2} -C_{11} K_{2} , \beta _{7} =C_{12}^{2} -C_{11}^{2} , \beta _{8} =K_{1} (C_{12} -C_{11} ), \beta _{9} =K_{2}^{2} -K_{1}^{2} , \\ \beta _{10}= & {} R_{1} /[C_{66} (K_{1} -K_{2} )],\kappa _{1} =e_{32} \beta _{2} +e_{31} \beta _{3} , \kappa _{2} =e_{33} \beta _{2} +e_{31} C_{13} K_{1} , \kappa _{3} =\chi _{33} \beta _{2} -e_{31}^{2} K_{1} ,\\ \kappa _{4}= & {} C_{33} \beta _{2} +C_{13}^{2} K_{1} , \kappa _{5} =C_{13} (\beta _{2} +\beta _{3} ), \kappa _{6} =C_{23} +C_{44} ,\vartheta _{1} =\kappa _{5} /\beta _{2} ,\vartheta _{2} =\beta _{4} \beta _{5} /(R_{1} \beta _{2} ),\\ \vartheta _{3}= & {} (2R_{1} \beta _{5} +K_{1} \beta _{7} )/\beta _{2} , \vartheta _{4} =-(2R_{1}^{2} +\beta _{8} )/\beta _{2} , \vartheta _{5} =(2R_{1} \beta _{4} +C_{11} \beta _{9} )/\beta _{2} , \lambda \text{= }m\pi R/L. \end{aligned}$$

The operational matrixes C\(^{i}\), C\(^{c}\), and D\(^{i}\) in Eqs. (7)–(9) can also be presented in similar form.

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Huang, Y., Li, Y., Zhang, L. et al. Dynamic analysis of a multilayered piezoelectric two-dimensional quasicrystal cylindrical shell filled with compressible fluid using the state-space approach. Acta Mech 231, 2351–2368 (2020). https://doi.org/10.1007/s00707-020-02641-7

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