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A solution method for free vibrration analysis of the elastically joined functionally graded shells

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Abstract

In this paper, for the first time, the free vibration characteristics of a joined shell structure in which three shells of revolution made of functionally graded material are elastically bonded are reported. The structure of the joined shell is formed by elastically bonding double-curved shells to both sides of the cylindrical shell in the middle, and the hyperbolic shells have elliptic, parabolic, and hyperbolic shapes. The Haar wavelet discretization method (HWDM), one of the effective, convenient and accurate numerical solution methods, is applied to investigate the free vibration behavior of various elastically joined functionally graded shells (EJFGSs). The theoretical model of the EJFGSs is formulated based on first-order shear deformation theory (FSDT). The displacement and rotation of arbitrary point of the EJFGS are expended as Haar wavelet series in the meridian direction and as a Fourier series in the circumferential direction. The boundary condition at both ends of the EJFGS and the continuous condition between its shells are modeled by the spring stiffness technique. By solving the governing equation of the whole system, the vibration characteristics of the EJFGS such as the natural frequency and the corresponding mode shape can be obtained. Through the comparisons with the previous literature and finite element method (FEM) results, the accuracy and reliability of our method are verified. Finally, new free vibration analysis results of various EJFGSs, which can be used as benchmark data for researchers in this field, are reported with parameter study.

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Data Availability

The data that supports the findings of this study are available within the article.

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Acknowledgements

The authors would like to thank the anonymous reviewers for carefully reading the paper and their very valuable comments. The authors gratefully acknowledge the supports from Pyongyang University of Mechanical Engineering of DPRK. In addition, the authors would like to take the opportunity to express my hearted gratitude to Dr. Paeksan Jang who made contribution to the completion of my article.

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Authors and Affiliations

  1. Information Center, Kim Chaek University of Technology, Pyongyang, 950003, Democratic People’s Republic of Korea

    Kwangil An & Yongsu Jon

  2. Department of Engineering Machine, Pyongyang University of Mechanical Engineering, Pyongyang, 999093, Democratic People’s Republic of Korea

    Kwanghun Kim & Cholsik Kim

  3. Department of Information Engineering, Chongjin Mine & Metal University, Chongjin, 999091, Democratic People’s Republic of Korea

    Sok Kim

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  1. Kwangil An
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  2. Yongsu Jon
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  3. Kwanghun Kim
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  4. Sok Kim
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Correspondence to Kwanghun Kim.

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Appendices

Appendix 1: Detailed expressions of the constant coefficients \(L_{ij}^{\psi } ,C_{ij}^{\psi } ,R_{ij}^{\psi }\)

  • For left double-curved shell

    $$ \begin{gathered} L_{11}^{0} = - A_{11} R_{{\varphi_{l} }}^{2} \cos^{2} \varphi_{l} - A_{12} R_{{\varphi_{l} }} R_{{0_{l} }} \sin \varphi_{l} - A_{66} n^{2} R_{{\varphi_{l} }}^{2} - \kappa A_{66} R_{{0_{l} }}^{2} \hfill \\ L_{11}^{1} = A_{11} \cos \varphi_{l} R_{{\varphi_{l} }} R_{{0_{l} }} - A_{11} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }},\,\,\,\,\,\,L_{11}^{2} = A_{11} R_{{0_{l} }}^{2} ,\,\,\,\,\,L_{12}^{0} = - \left( {A_{11} + A_{66} } \right)n\cos \varphi_{l} R_{{\varphi_{l} }}^{2} , \hfill \\ L_{12}^{1} = \left( {A_{12} + A_{66} } \right)nR_{{\varphi_{l} }} R_{{0_{l} }} ,\,\,\,\,\,L_{13}^{0} = - A_{11} \sin \varphi_{l} \cos \varphi_{l} R_{{\varphi_{l} }}^{2} + A_{11} \cos \varphi_{l} R_{{\varphi_{l} }} R_{{0_{l} }} - A_{11} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }} \hfill \\ L_{13}^{1} = A_{11} R_{{0_{l} }}^{2} + A_{12} \sin \varphi_{l} R_{{\varphi_{l} }} R_{{0_{l} }} + \kappa A_{66} R_{{0_{l} }}^{2} \hfill \\ L_{14}^{0} = - B_{11} R_{{\varphi_{l} }}^{2} \cos^{2} \varphi_{l} - B_{12} R_{{\varphi_{l} }} R_{{0_{l} }} \sin^{2} \varphi_{l} - B_{66} n^{2} R_{{\varphi_{l} }}^{2} + \kappa A_{66} R_{{0_{l} }}^{2} R_{{\varphi_{l} }} \hfill \\ L_{14}^{1} = B_{11} \cos \varphi_{l} R_{{\varphi_{l} }} R_{{0_{l} }} - B_{11} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }},\,\,\,\,\,\,L_{14}^{2} = B_{11} R_{{0_{l} }}^{2} \hfill \\ L_{15}^{0} = - \left( {B_{11} + B_{66} } \right)n\cos \varphi_{l} R_{{\varphi_{l} }}^{2} ,\,\,\,\,\,L_{15}^{1} = \left( {B_{12} + B_{66} } \right)nR_{{\varphi_{l} }} R_{{0_{l} }} \hfill \\ \end{gathered} $$
    $$ \begin{gathered} L_{21}^{0} = - \left( {A_{11} + A_{66} } \right)n\cos \varphi_{l} R_{{\varphi_{l} }}^{2} ,\,\,\,\,\,\,L_{21}^{1} = - \left( {A_{12} + A_{66} } \right)nR_{{0_{l} }} R_{{\varphi_{l} }} \hfill \\ L_{22}^{0} = - A_{66} \cos^{2} \varphi_{l} R_{{\varphi_{l} }}^{2} + A_{66} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - A_{11} n^{2} R_{{\varphi_{l} }}^{2} - \kappa A_{66} \sin^{2} \varphi_{l} R_{{\varphi_{l} }}^{2} \hfill \\ L_{22}^{1} = A_{66} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - A_{66} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }},\,\,\,\,\,\,L_{22}^{2} = A_{66} R_{{0_{l} }}^{2} , \hfill \\ L_{23}^{0} = - A_{11} n\sin \varphi_{l} R_{{\varphi_{l} }}^{2} - \kappa A_{66} n\sin \varphi_{l} R_{{\varphi_{l} }}^{2} - A_{12} nR_{{0_{l} }} R_{{\varphi_{l} }} ,\,\,\,\,\,L_{24}^{0} = - \left( {B_{11} + B_{66} } \right)n\cos \varphi_{l} R_{{\varphi_{l} }}^{2} , \hfill \\ L_{24}^{1} = - \left( {B_{12} + B_{66} } \right)nR_{{0_{l} }} R_{{\varphi_{l} }} ,\,\,\,\,\,L_{25}^{0} = - B_{66} \cos^{2} \varphi_{l} R_{{\varphi_{l} }}^{2} + B_{66} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - B_{11} n^{2} R_{{\varphi_{l} }}^{2} + \kappa A_{66} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }}^{2} \hfill \\ L_{25}^{1} = B_{66} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - B_{66} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }},\,\,\,\,\,\,L_{25}^{2} = B_{66} R_{{0_{l} }}^{2} \hfill \\ \end{gathered} $$
    $$ \begin{gathered} L_{31}^{0} = - A_{12} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - A_{11} \sin \varphi_{l} \cos \varphi_{l} R_{{\varphi_{l} }}^{2} + \kappa A_{66} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} + \kappa A_{66} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }} \hfill \\ L_{31}^{1} = - A_{11} R_{{0_{l} }}^{2} - A_{12} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - \kappa A_{66} R_{{0_{l} }}^{2} ,\,\,\,\,\,L_{32}^{0} = - \left( {A_{11} + \kappa A_{66} } \right)n\sin \varphi_{l} R_{{\varphi_{l} }}^{2} - A_{12} nR_{{0_{l} }} R_{{\varphi_{l} }} \hfill \\ L_{33}^{0} = - A_{11} R_{{0_{l} }}^{2} - A_{12} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - A_{11} \sin \varphi_{l} R_{{\varphi_{l} }}^{2} - \kappa A_{66} n^{2} R_{{\varphi_{l} }}^{2} \hfill \\ L_{33}^{1} = \kappa A_{66} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - \kappa A_{66} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }},\,\,\,\,\,L_{33}^{2} = \kappa A_{66} R_{{0_{l} }}^{2} \hfill \\ L_{34}^{0} = \kappa A_{66} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }}^{2} - B_{12} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - B_{11} \sin \varphi_{l} \cos \varphi_{l} R_{{\varphi_{l} }}^{2} \hfill \\ L_{34}^{1} = \kappa A_{66} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }}^{2} - B_{11} R_{{0_{l} }}^{2} - B_{12} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} \hfill \\ L_{35}^{0} = \kappa A_{66} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }}^{2} - B_{12} nR_{{0_{l} }} R_{{\varphi_{l} }} - B_{11} n\sin \varphi_{l} R_{{\varphi_{l} }}^{2} \hfill \\ \end{gathered} $$
    $$ \begin{gathered} L_{41}^{0} = - B_{12} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - B_{11} \cos^{2} \varphi_{l} R_{{\varphi_{l} }}^{2} - B_{66} n^{2} R_{{\varphi_{l} }}^{2} + \kappa A_{66} R_{{\varphi_{l} }} R_{{0_{l} }}^{2} \hfill \\ L_{41}^{1} = B_{11} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - B_{11} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }},\,\,\,\,\,L_{41}^{2} = B_{11} R_{{0_{l} }}^{2} ,\,\,\,\,\,L_{42}^{0} = - \left( {B_{11} + B_{66} } \right)n\cos \varphi_{l} R_{{\varphi_{l} }}^{2} , \hfill \\ L_{42}^{1} = \left( {B_{12} + B_{66} } \right)nR_{{0_{l} }} R_{{\varphi_{l} }} ,\,\,\,\,\,L_{43}^{0} = B_{11} \sin \varphi_{l} \cos \varphi_{l} R_{{\varphi_{l} }}^{2} + B_{11} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - B_{11} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }} \hfill \\ L_{43}^{1} = B_{11} R_{{0_{l} }}^{2} + B_{12} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} + \kappa A_{66} R_{{\varphi_{l} }} R_{{0_{l} }}^{2} \hfill \\ L_{44}^{0} = - D_{12} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - D_{11} \cos^{2} \varphi_{l} R_{{\varphi_{l} }}^{2} - D_{66} n^{2} R_{{\varphi_{l} }}^{2} - \kappa A_{66} R_{{\varphi_{l} }}^{2} R_{{0_{l} }}^{2} \hfill \\ L_{44}^{1} = D_{11} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - D_{11} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }},\,\,\,\,\,L_{44}^{2} = D_{11} R_{{0_{l} }}^{2} \hfill \\ L_{45}^{0} = - \left( {D_{11} + D_{66} } \right)n\cos \varphi_{l} R_{{\varphi_{l} }}^{2} ,\,\,\,\,\,L_{45}^{1} = \left( {D_{12} + D_{66} } \right)nR_{{0_{l} }} R_{{\varphi_{l} }} \hfill \\ \end{gathered} $$
    $$ \begin{gathered} L_{51}^{0}\, { = } - \left( {B_{11} + B_{66} } \right)n\cos \varphi_{l} R_{{\varphi_{l} }}^{2} ,\,\,\,\,\,L_{51}^{1} = - \left( {B_{12} + B_{66} } \right)nR_{{0_{l} }} R_{{\varphi_{l} }} \hfill \\ L_{52}^{0} = \kappa A_{66} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }}^{2} - B_{11} n^{2} R_{{\varphi_{l} }}^{2} - B_{66} \cos^{2} \varphi_{l} R_{{\varphi_{l} }}^{2} + B_{66} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} \hfill \\ L_{52}^{1} = B_{66} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - B_{66} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }},\,\,\,\,\,\,L_{52}^{2} = B_{66} R_{{0_{l} }}^{2} \hfill \\ L_{53}^{0} = - B_{11} n\sin \varphi_{l} R_{{\varphi_{l} }}^{2} - B_{12} nR_{{0_{l} }} R_{{\varphi_{l} }} + \kappa A_{66} nR_{{0_{l} }} R_{{\varphi_{l} }}^{2} ,\,\,\,\,\,L_{54}^{0} = - \left( {D_{11} + D_{66} } \right)n\cos \varphi_{l} R_{{\varphi_{l} }}^{2} , \hfill \\ L_{54}^{1} = - \left( {D_{12} + D_{66} } \right)nR_{{0_{l} }} R_{{\varphi_{l} }} ,\,\,\,\,\,L_{55}^{0} = - D_{11} n^{2} R_{{\varphi_{l} }}^{2} - D_{66} \cos^{2} \varphi_{l} R_{{\varphi_{l} }}^{2} + D_{66} \sin \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - \kappa A_{66} \sin \varphi_{l} R_{{0_{l} }}^{2} R_{{\varphi_{l} }}^{2} \hfill \\ L_{55}^{1} = D_{66} \cos \varphi_{l} R_{{0_{l} }} R_{{\varphi_{l} }} - D_{66} R_{{0_{l} }}^{2} \frac{1}{{R_{{\varphi_{l} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{l} }},\,\,\,\,\,\,L_{52}^{2} = D_{66} R_{{0_{l} }}^{2} \hfill \\ \end{gathered} $$

  • For cylindrical shell

    $$ \begin{gathered} C_{{11}}^{1} = - A_{{66}} n^{2} ,\;\;\;C_{{11}}^{3} = A_{{11}} R^{2} ,\;\;\;C_{{12}}^{2} = A_{{12}} Rn{\text{ + }}A_{{66}} Rn,\;\;\;\;C_{{13}}^{2} = A_{{12}} R,\; \hfill \\ C_{{14}}^{1} = - B_{{66}} n^{2} ,\;\;C_{{14}}^{3} = B_{{11}} R^{2} ,\;\;\;\;C_{{15}}^{2} = B_{{12}} Rn{\text{ + }}B_{{66}} Rn \hfill \\ C_{{21}}^{2} = - A_{{12}} Rn - A_{{66}} Rn,\;\;C_{{22}}^{1} = - A_{{22}} n^{2} - \kappa A_{{44}} ,\;\;C_{{22}}^{3} = A_{{66}} R^{2} ,\;C_{{23}}^{1} = - A_{{22}} n - \kappa A_{{44}} n, \hfill \\ C_{{24}}^{2} = - B_{{12}} Rn - B_{{66}} Rn,\,\,\,C_{{25}}^{1} = - B_{{22}} n^{2} + \kappa A_{{44}} R,\;\;\;C_{{25}}^{3} = B_{{66}} R^{2} \hfill \\ C_{{31}}^{2} = A_{{12}} R,\;\;\;C_{{32}}^{1} = A_{{22}} n{\text{ + }}\kappa A_{{44}} n,\;\;C_{{33}}^{1} = \kappa A_{{44}} n^{2} + A_{{22}} ,\;\;\;C_{{33}}^{3} = - \kappa A_{{55}} R^{2} ,\;\;\; \hfill \\ C_{{34}}^{2} = B_{{12}} R - \kappa A_{{55}} R^{2} ,\;C_{{35}}^{1} = B_{{22}} n{\text{ - }}\kappa A_{{44}} Rn,\; \hfill \\ C_{{41}}^{1} = - B_{{66}} n^{2} ,\;\;\;C_{{41}}^{3} = B_{{11}} R^{2} ,\;\;C_{{42}}^{2} = B_{{12}} Rn{\text{ + }}B_{{66}} Rn,\;\;\;\;C_{{43}}^{2} = B_{{12}} R - \kappa A_{{55}} R^{2} ,\;\; \hfill \\ C_{{44}}^{1} = - D_{{66}} n^{2} - \kappa A_{{55}} R^{2} ,\;\;C_{{44}}^{3} = D_{{11}} R^{2} ,\;\;\;\;\;C_{{45}}^{2} = D_{{12}} Rn{\text{ + }}D_{{66}} Rn,\; \hfill \\ C_{{51}}^{2} = - B_{{12}} Rn - B_{{66}} Rn,\;\;C_{{52}}^{1} = \kappa A_{{44}} R - B_{{22}} n^{2} ,\;\;\;C_{{52}}^{3} = B_{{66}} R^{2} ,\;C_{{53}}^{1} = \kappa A_{{44}} Rn - B_{{22}} n,\; \hfill \\ C_{{54}}^{2} = - D_{{12}} Rn - D_{{66}} Rn,\;\;C_{{55}}^{1} = - D_{{22}} n^{2} - \kappa A_{{44}} R^{2} ,\;\;\;\;\;C_{{55}}^{3} = D_{{66}} R^{2} \hfill \\ \end{gathered} $$

  • For right double-curved shell

    $$ \begin{gathered} R_{11}^{0} = - A_{11} R_{{\varphi_{r} }}^{2} \cos^{2} \varphi_{r} - A_{12} R_{{\varphi_{r} }} R_{{0_{r} }} \sin \varphi_{r} - A_{66} n^{2} R_{{\varphi_{r} }}^{2} - \kappa A_{66} R_{{0_{r} }}^{2} \hfill \\ R_{11}^{1} = A_{11} \cos \varphi_{r} R_{{\varphi_{r} }} R_{{0_{r} }} - A_{11} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }},\,\,\,\,\,\,R_{11}^{2} = A_{11} R_{{0_{r} }}^{2} ,\,\,\,\,\,R_{12}^{0} = - \left( {A_{11} + A_{66} } \right)n\cos \varphi_{r} R_{{\varphi_{r} }}^{2} , \hfill \\ R_{12}^{1} = \left( {A_{12} + A_{66} } \right)nR_{{\varphi_{r} }} R_{{0_{r} }} ,\,\,\,\,\,R_{13}^{0} = - A_{11} \sin \varphi_{r} \cos \varphi_{r} R_{{\varphi_{r} }}^{2} + A_{11} \cos \varphi_{r} R_{{\varphi_{r} }} R_{{0_{r} }} - A_{11} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }} \hfill \\ R_{13}^{1} = A_{11} R_{{0_{r} }}^{2} + A_{12} \sin \varphi_{r} R_{{\varphi_{r} }} R_{{0_{r} }} + \kappa A_{66} R_{{0_{r} }}^{2} \hfill \\ R_{14}^{0} = - B_{11} R_{{\varphi_{r} }}^{2} \cos^{2} \varphi_{r} - B_{12} R_{{\varphi_{r} }} R_{{0_{r} }} \sin^{2} \varphi_{r} - B_{66} n^{2} R_{{\varphi_{r} }}^{2} + \kappa A_{66} R_{{0_{r} }}^{2} R_{{\varphi_{r} }} \hfill \\ R_{14}^{1} = B_{11} \cos \varphi_{l} R_{{\varphi_{r} }} R_{{0_{r} }} - B_{11} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }},\,\,\,\,\,\,R_{14}^{2} = B_{11} R_{{0_{r} }}^{2} \hfill \\ R_{15}^{0} = - \left( {B_{11} + B_{66} } \right)n\cos \varphi_{r} R_{{\varphi_{r} }}^{2} ,\,\,\,\,\,R_{15}^{1} = \left( {B_{12} + B_{66} } \right)nR_{{\varphi_{r} }} R_{{0_{r} }} \hfill \\ \end{gathered} $$
    $$ \begin{gathered} R_{21}^{0} = - \left( {A_{11} + A_{66} } \right)n\cos \varphi_{r} R_{{\varphi_{r} }}^{2} ,\,\,\,\,\,\,R_{21}^{1} = - \left( {A_{12} + A_{66} } \right)nR_{{0_{r} }} R_{{\varphi_{r} }} \hfill \\ R_{22}^{0} = - A_{66} \cos^{2} \varphi_{r} R_{{\varphi_{r} }}^{2} + A_{66} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - A_{11} n^{2} R_{{\varphi_{r} }}^{2} - \kappa A_{66} \sin^{2} \varphi_{r} R_{{\varphi_{r} }}^{2} \hfill \\ R_{22}^{1} = A_{66} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - A_{66} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }},\,\,\,\,\,\,R_{22}^{2} = A_{66} R_{{0_{r} }}^{2} , \hfill \\ R_{23}^{0} = - A_{11} n\sin \varphi_{r} R_{{\varphi_{r} }}^{2} - \kappa A_{66} n\sin \varphi_{r} R_{{\varphi_{r} }}^{2} - A_{12} nR_{{0_{r} }} R_{{\varphi_{r} }} ,\,\,\,\,\,R_{24}^{0} = - \left( {B_{11} + B_{66} } \right)n\cos \varphi_{r} R_{{\varphi_{r} }}^{2} , \hfill \\ R_{24}^{1} = - \left( {B_{12} + B_{66} } \right)nR_{{0_{r} }} R_{{\varphi_{r} }} ,\,\,\,\,\,R_{25}^{0} = - B_{66} \cos^{2} \varphi_{r} R_{{\varphi_{r} }}^{2} + B_{66} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - B_{11} n^{2} R_{{\varphi_{r} }}^{2} + \kappa A_{66} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }}^{2} \hfill \\ R_{25}^{1} = B_{66} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - B_{66} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }},\,\,\,\,\,\,R_{25}^{2} = B_{66} R_{{0_{r} }}^{2} \hfill \\ \end{gathered} $$
    $$ \begin{gathered} R_{31}^{0} = - A_{12} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - A_{11} \sin \varphi_{r} \cos \varphi_{r} R_{{\varphi_{r} }}^{2} + \kappa A_{66} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} + \kappa A_{66} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }} \hfill \\ R_{31}^{1} = - A_{11} R_{{0_{l} }}^{2} - A_{12} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - \kappa A_{66} R_{{0_{r} }}^{2} ,\,\,\,\,\,R_{32}^{0} = - \left( {A_{11} + \kappa A_{66} } \right)n\sin \varphi_{r} R_{{\varphi_{r} }}^{2} - A_{12} nR_{{0_{r} }} R_{{\varphi_{r} }} \hfill \\ R_{33}^{0} = - A_{11} R_{{0_{r} }}^{2} - A_{12} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - A_{11} \sin \varphi_{r} R_{{\varphi_{r} }}^{2} - \kappa A_{66} n^{2} R_{{\varphi_{r} }}^{2} \hfill \\ R_{33}^{1} = \kappa A_{66} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - \kappa A_{66} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{l} }} }}{{d\varphi_{r} }},\,\,\,\,\,R_{33}^{2} = \kappa A_{66} R_{{0_{r} }}^{2} \hfill \\ R_{34}^{0} = \kappa A_{66} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }}^{2} - B_{12} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - B_{11} \sin \varphi_{r} \cos \varphi_{r} R_{{\varphi_{r} }}^{2} \hfill \\ R_{34}^{1} = \kappa A_{66} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }}^{2} - B_{11} R_{{0_{r} }}^{2} - B_{12} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} \hfill \\ R_{35}^{0} = \kappa A_{66} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }}^{2} - B_{12} nR_{{0_{r} }} R_{{\varphi_{r} }} - B_{11} n\sin \varphi_{r} R_{{\varphi_{r} }}^{2} \hfill \\ \end{gathered} $$
    $$ \begin{gathered} R_{41}^{0} = - B_{12} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - B_{11} \cos^{2} \varphi_{r} R_{{\varphi_{r} }}^{2} - B_{66} n^{2} R_{{\varphi_{r} }}^{2} + \kappa A_{66} R_{{\varphi_{r} }} R_{{0_{r} }}^{2} \hfill \\ R_{41}^{1} = B_{11} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - B_{11} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }},\,\,\,\,\,R_{41}^{2} = B_{11} R_{{0_{r} }}^{2} ,\,\,\,\,\,R_{42}^{0} = - \left( {B_{11} + B_{66} } \right)n\cos \varphi_{r} R_{{\varphi_{r} }}^{2} , \hfill \\ R_{42}^{1} = \left( {B_{12} + B_{66} } \right)nR_{{0_{r} }} R_{{\varphi_{r} }} ,\,\,\,\,\,R_{43}^{0} = B_{11} \sin \varphi_{r} \cos \varphi_{r} R_{{\varphi_{r} }}^{2} + B_{11} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - B_{11} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }} \hfill \\ R_{43}^{1} = B_{11} R_{{0_{r} }}^{2} + B_{12} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} + \kappa A_{66} R_{{\varphi_{r} }} R_{{0_{r} }}^{2} \hfill \\ R_{44}^{0} = - D_{12} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - D_{11} \cos^{2} \varphi_{r} R_{{\varphi_{r} }}^{2} - D_{66} n^{2} R_{{\varphi_{r} }}^{2} - \kappa A_{66} R_{{\varphi_{r} }}^{2} R_{{0_{r} }}^{2} \hfill \\ R_{44}^{1} = D_{11} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - D_{11} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }},\,\,\,\,\,R_{44}^{2} = D_{11} R_{{0_{r} }}^{2} \hfill \\ R_{45}^{0} = - \left( {D_{11} + D_{66} } \right)n\cos \varphi_{l} R_{{\varphi_{l} }}^{2} ,\,\,\,\,\,R_{45}^{1} = \left( {D_{12} + D_{66} } \right)nR_{{0_{l} }} R_{{\varphi_{l} }} \hfill \\ \end{gathered} $$
    $$ \begin{gathered} R_{51}^{0} { = } - \left( {B_{11} + B_{66} } \right)n\cos \varphi_{r} R_{{\varphi_{r} }}^{2} ,\,\,\,\,\,R_{51}^{1} = - \left( {B_{12} + B_{66} } \right)nR_{{0_{r} }} R_{{\varphi_{r} }} \hfill \\ R_{52}^{0} = \kappa A_{66} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }}^{2} - B_{11} n^{2} R_{{\varphi_{r} }}^{2} - B_{66} \cos^{2} \varphi_{r} R_{{\varphi_{r} }}^{2} + B_{66} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} \hfill \\ R_{52}^{1} = B_{66} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - B_{66} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }},\,\,\,\,\,\,R_{52}^{2} = B_{66} R_{{0_{r} }}^{2} \hfill \\ R_{53}^{0} = - B_{11} n\sin \varphi_{r} R_{{\varphi_{r} }}^{2} - B_{12} nR_{{0_{r} }} R_{{\varphi_{r} }} + \kappa A_{66} nR_{{0_{r} }} R_{{\varphi_{r} }}^{2} ,\,\,\,\,\,R_{54}^{0} = - \left( {D_{11} + D_{66} } \right)n\cos \varphi_{r} R_{{\varphi_{r} }}^{2} , \hfill \\ R_{54}^{1} = - \left( {D_{12} + D_{66} } \right)nR_{{0_{r} }} R_{{\varphi_{r} }} ,\,\,\,\,\,R_{55}^{0} = - D_{11} n^{2} R_{{\varphi_{r} }}^{2} - D_{66} \cos^{2} \varphi_{r} R_{{\varphi_{r} }}^{2} + D_{66} \sin \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - \kappa A_{66} \sin \varphi_{r} R_{{0_{r} }}^{2} R_{{\varphi_{r} }}^{2} \hfill \\ R_{55}^{1} = D_{66} \cos \varphi_{r} R_{{0_{r} }} R_{{\varphi_{r} }} - D_{66} R_{{0_{r} }}^{2} \frac{1}{{R_{{\varphi_{r} }} }}\frac{{dR_{{\varphi_{r} }} }}{{d\varphi_{r} }},\,\,\,\,\,\,R_{52}^{2} = D_{66} R_{{0_{r} }}^{2} \hfill \\ \end{gathered} $$

Appendix 2. Detailed expressions of the mass and stiffness matrix

$$ \begin{gathered} {\mathbf{K}}_{{dd}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{dd,l}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{K}}_{{dd,c}} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{{dd,r}} } \\ \end{array} } \right],\quad {\mathbf{K}}_{{bd}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{bd,l}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{K}}_{{bd,c}} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{{bd,r}} } \\ \end{array} } \right] \hfill \\ {\mathbf{K}}_{{bb}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{bb,l}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{{bb,r}} } \\ {{\mathbf{K}}_{{bb,l1}} } & {{\mathbf{K}}_{{bb,lc}} } & {\mathbf{0}} \\ {{\mathbf{K}}_{{bb,cl}} } & {{\mathbf{K}}_{{bb,c0}} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{K}}_{{bb,c1}} } & {{\mathbf{K}}_{{bb,cr}} } \\ {\mathbf{0}} & {{\mathbf{K}}_{{bb,rc}} } & {{\mathbf{K}}_{{bb,r1}} } \\ \end{array} } \right],\quad {\mathbf{K}}_{{bd}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{bd,l}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{{bd,r}} } \\ {{\mathbf{K}}_{{bd,l1}} } & {{\mathbf{K}}_{{bd,lc}} } & {\mathbf{0}} \\ {{\mathbf{K}}_{{bd,cl}} } & {{\mathbf{K}}_{{bd,c0}} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{K}}_{{bd,c1}} } & {{\mathbf{K}}_{{bd,cr}} } \\ {\mathbf{0}} & {{\mathbf{K}}_{{bd,rc}} } & {{\mathbf{K}}_{{bd,r1}} } \\ \end{array} } \right] \hfill \\ {\mathbf{A}}_{d} \;= \left[ {{\mathbf{A}}_{{dl}} ,{\mathbf{A}}_{{dc}} ,{\mathbf{A}}_{{dr}} } \right]^{T} ,{\mathbf{A}}_{b} = \left[ {{\mathbf{A}}_{{bl}} ,{\mathbf{A}}_{{bc}} ,{\mathbf{A}}_{{br}} } \right]^{T} \hfill \\ {\mathbf{A}}_{{d\varsigma }} = \left[ {\varvec{a}_{\varsigma } ,\;\varvec{b}_{\varsigma } ,\;\varvec{c}_{\varsigma } ,\;\varvec{d}_{\varsigma } ,\;\varvec{e}_{\varsigma } } \right]^{T} ,{\mathbf{A}}_{{b\varsigma }} = \left[ {\varvec{f}_{\varsigma } ,\;\varvec{g}_{\varsigma } ,\;\varvec{h}_{\varsigma } ,\;\varvec{k}_{\varsigma } ,\;\varvec{l}_{\varsigma } } \right]^{T} \hfill \\ \end{gathered} $$

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An, K., Jon, Y., Kim, K. et al. A solution method for free vibrration analysis of the elastically joined functionally graded shells. Eur. Phys. J. Plus 136, 767 (2021). https://doi.org/10.1140/epjp/s13360-021-01748-7

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