Skip to main content
SpringerLink
Log in

Free vibration analysis of combined composite laminated conical–cylindrical shells with varying thickness using the Haar wavelet method

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

This paper presents the free vibration analysis of combined composite laminated conical–cylindrical shells with varying thickness using the Haar wavelet method (HWM). The displacement field of the combined shell is set based on the first-order shear deformation theory (FSDT), the displacement components, and rotation of individual shells including boundary conditions that are expanded by the Haar wavelet and Fourier series in the meridional and the circumferential direction. By solving the vibration characteristic equation discretized by the Haar wavelet, the vibrational results of combined shells are obtained. Then, the results of the proposed method are compared with those of published literature and finite element analysis (FEA). The results show that HWM has high convergence and high accuracy for the free vibration analysis of the combined composite laminated conical–cylindrical shells with varying thickness. Also, the effects of the parameters such as thickness variation parameters, material properties, geometrical dimensions, and different boundary conditions, on the vibrational behavior of the combined shells are investigated. Finally, new numerical results are provided to illustrate the free vibration behavior of the combined composite laminated conical–cylindrical shells with varying thickness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data Availability

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

References

  1. Irie, T., Yamada, G., Muramoto, Y.: Free vibration of joined conical-cylindrical shells. J. Sound Vib. 95(1), 31–39 (1984)

    Article  Google Scholar 

  2. Hu, W., Raney, J.P.: Experimental and analytical study of vibrations of joined shells. AIAA J. 5(5), 976–980 (2012)

    Article  Google Scholar 

  3. Benjeddou, A.: Vibrations of complex shells of revolution using B-spline finite elements. Comput. Struct. 74(4), 429–440 (2000)

    Article  Google Scholar 

  4. Caresta, M., Kessissoglou, N.J.: Free vibrational characteristics of isotropic coupled cylindrical-conical shells. J. Sound Vib. 329(6), 733–751 (2010)

    Article  Google Scholar 

  5. Damatty, A., Saafan, M.S., Sweedan, A.: Dynamic characteristics of combined conical-cylindrical shells. Thin Walled Struct. 43(9), 1380–1397 (2005)

    Article  Google Scholar 

  6. Qu, Y., Yong, C., Long, X., et al.: A modified variational approach for vibration analysis of ring-stiffened conical-cylindrical shell combinations. Eur. J. Mech. A Solids 37, 200–215 (2013)

    Article  MathSciNet  Google Scholar 

  7. Qu, Y., Wu, S., Chen, Y., et al.: Vibration analysis of ring-stiffened conical-cylindrical-spherical shells based on a modified variational approach. Int. J. Mech. Sci. 69, 72–84 (2013)

    Article  Google Scholar 

  8. Qu, Y., Yong, C., Long, X., et al.: A new method for vibration analysis of joined cylindrical-conical shells. J. Vib. Control 19(16), 2319–2334 (2012)

    Article  Google Scholar 

  9. Wu, S., Qu, Y., Hua, H.: Vibration characteristics of a spherical-cylindrical-spherical shell by a domain decomposition method. Mech. Res. Commun. 49, 17–26 (2013)

    Article  Google Scholar 

  10. Wu, S., Qu, Y., Hua, H.: Vibrations characteristics of joined cylindrical-spherical shell with elastic-support boundary conditions. J. Mech. Sci. Technol. 27(5), 1265–1272 (2013)

    Article  Google Scholar 

  11. Ma, X., Jin, G., Xiong, Y., et al.: Free and forced vibration analysis of coupled conical-cylindrical shells with arbitrary boundary conditions. Int. J. Mech. Sci. 88, 122–137 (2014)

    Article  Google Scholar 

  12. Ma, X., Jin, G., Shi, S., Ye, T., Liu, Z.: An analytical method for vibration analysis of cylindrical shells coupled with annular plate under general elastic boundary and coupling conditions. J. Vib. Control 2, 691–693 (2015)

    MATH  Google Scholar 

  13. Bagheri, H., Kiani, Y., Eslami, M.R.: Free vibration of joined conical-cylindrical-conical shells. Acta Mech. 229, 2751–2764 (2018)

    Article  MathSciNet  Google Scholar 

  14. Bagheri, H., Kiani, Y., Eslami, M.R.: Free vibration of joined conical-conical shells. Thin Walled Struct. 120, 446–457 (2017)

    Article  Google Scholar 

  15. Su, Z., Jin, G.: Vibration analysis of coupled conical-cylindrical-spherical shells using a Fourier spectral element method. J. Acoust. Soc. Am. 140(5), 3925–3940 (2016)

    Article  Google Scholar 

  16. Cheng, L., Nicolas, J.: Free vibration analysis of a cylindrical shell-circular plate system with general coupling and various boundary conditions. J. Sound Vib. 155(2), 231–247 (1992)

    Article  Google Scholar 

  17. Chen, M., Xie, K., Jia, W., et al.: Free and forced vibration of ring-stiffened conical–cylindrical shells with arbitrary boundary conditions. Ocean Eng. 108, 241–256 (2015)

    Article  Google Scholar 

  18. Efraim, E., Eisenberger, M.: Exact vibration frequencies of segmented axisymmetric shells. Thin Walled Struct. 44(3), 281–289 (2006)

    Article  Google Scholar 

  19. Kang, J.H.: Three-dimensional vibration analysis of joined thick conical-cylindrical shells of revolution with variable thickness. J. Sound Vib. 331(18), 4187–4198 (2012)

    Article  Google Scholar 

  20. Carrera, E., Antona, E.: A class of two-dimensional theories for anisotropic multilayered plates analysis. Accademia delle Scienze (1995)

  21. Carrera, E., Giunta, G., Petrolo, M.: Beam structures: classical and advanced theories. John Wiley & Sons (2011)

    Book  Google Scholar 

  22. Carrera, E., Filippi, M., Zappino, E.: Free vibration analysis of rotating composite blades via Carrera Unified Formulation. Compos. Struct. 106, 317–325 (2013)

    Article  Google Scholar 

  23. Pagani, A., Carrera, E., Boscolo, M., Banerjee, J.R.: Refined dynamic stiffness elements applied to free vibration analysis of generally laminated composite beams with arbitrary boundary conditions. Compos. Struct. 110, 305–316 (2014)

    Article  Google Scholar 

  24. Carrera, E.: Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch. Comput. Methods Eng. 9(2), 87–140 (2002)

    Article  MathSciNet  Google Scholar 

  25. Carrera, E.: Historical review of Zig-Zag theories for multilayered plates and shells. Appl. Mech. Rev. 56(3), 287–309 (2003)

    Article  MathSciNet  Google Scholar 

  26. Xie, X., Jin, G., Liu, Z.: Free vibration analysis of cylindrical shells using the Haar wavelet method. Int. J. Mech. Sci. 77, 47–56 (2013)

    Article  Google Scholar 

  27. Xie, X., Jin, G., Yan, Y., et al.: Free vibration analysis of composite laminated cylindrical shells using the Haar wavelet method. Compos. Struct. 109(1), 169–177 (2014)

    Article  Google Scholar 

  28. Xie, X., Jin, G., Li, W., et al.: A numerical solution for vibration analysis of composite laminated conical, cylindrical shell and annular plate structures. Compos. Struct. 111, 20–30 (2014)

    Article  Google Scholar 

  29. Jin, G., Xie, X., Liu, Z.: The Haar wavelet method for free vibration analysis of functionally graded cylindrical shells based on the shear deformation theory. Compos. Struct. 108, 435–448 (2014)

    Article  Google Scholar 

  30. Dai, Q., Cao, Q.: Parametric instability analysis of truncated conical shells using the Haar wavelet method. Mech. Syst. Signal Process. 105, 200–213 (2018)

    Article  Google Scholar 

  31. Xie, X., Jin, G., Ye, T., et al.: Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method. Appl. Acoust. 85, 130–142 (2014)

    Article  Google Scholar 

  32. Talebitooti, R., Anbardan, V.S.: Haar wavelet discretization approach for frequency analysis of the functionally graded generally doubly-curved shells of revolution. Appl. Math. Model. 67, 645–675 (2019)

    Article  MathSciNet  Google Scholar 

  33. Kim, K., Kwak, S., Choe, K., et al.: Application of Haar wavelet method for free vibration of laminated composite conical-cylindrical coupled shells with elastic boundary condition. Phys. Scr. 96(3), 035223 (2021). https://doi.org/10.1088/1402-4896/abd9f7

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for carefully reading the paper and their very valuable comments. The authors also 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 all those who made contributions to the completion of this article.

Author information

Authors and Affiliations

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

    Kwanghun Kim

  2. College of Mechanical Science and Technology, Kim Chaek University of Technology, Pyongyang, 999093, Democratic People’s Republic of Korea

    Songhun Kwak

  3. Department of Material Science and Technology, Kim Chaek University of Technology, Pyongyang, 999093, Democratic People’s Republic of Korea

    Cholho Pang

  4. Department of Organic Chemistry, Hamhung University of Chemical Engineering, Hamhung, Democratic People’s Republic of Korea

    Kyongjin Pang

  5. Department of Information Engineering, Pyongyang University of Mechanical Engineering, Pyongyang, 999093, Democratic People’s Republic of Korea

    Kwangil Choe

Authors
  1. Kwanghun Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Songhun Kwak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cholho Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kyongjin Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kwangil Choe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kwanghun Kim.

Ethics declarations

Conflict of interest

No potential conflict of interest was reported by the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix 1

For the conical shell:

$$\begin{aligned} & L_{{11,co}} = A_{{11,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{2A_{{16,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{A_{{66,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + \frac{{A_{{11,co}} \sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - A_{{22,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{{\partial A_{{11,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{1}{{R_{{co}} }}\frac{{\partial A_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{\partial A_{{12,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }} - I_{{0,co}} \frac{{\partial ^{2} }}{{\partial t^{2} }}, \\ \end{aligned}$$
$$\begin{aligned} & L_{{12,co}} = A_{{16,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{A_{{12,co}} + A_{{66,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{A_{{26,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} - \frac{{A_{{26,co}} \sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \left( {A_{{22,co}} + A_{{66_{{co}} }} } \right)\frac{{\sin \varphi }}{{R_{{co}}^{2} }}\frac{\partial }{{\partial \theta _{{co}} }} \\ & \quad\quad\quad\,\,\, + A_{{26,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }}\, + \frac{{\partial A_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{1}{{R_{{co}} }}\frac{{\partial A_{{12,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} - \frac{{\partial A_{{16,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }}, \\ \end{aligned}$$
$$L_{13,co} = A_{12,co} \frac{\cos \varphi }{{R_{co} }}\frac{\partial }{{\partial x_{co} }} + A_{26,co} \frac{\cos \varphi }{{R_{co}^{2} }}\frac{\partial }{{\partial \theta_{co} }} - A_{22,co} \frac{\sin \varphi \cos \varphi }{{R_{co}^{2} }} + \frac{{\partial A_{12,co} }}{{\partial x_{co} }}\frac{\cos \varphi }{{R_{co} }},$$
$$\begin{aligned} & L_{{14,co}} = B_{{11,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{2B_{{16}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{B_{{66}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + B_{{11,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - B_{{22,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} + \frac{{\partial B_{{11,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} \\ & \quad\quad\quad\,\,\, + \frac{1}{{R_{{co}} }}\frac{{\partial B_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{\partial B_{{12,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }} - I_{{1,co}} \frac{{\partial ^{2} }}{{\partial t^{2} }}, \\ \end{aligned}$$
$$\begin{aligned} & L_{{15,co}} = B_{{16,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{B_{{12,co}} + B_{{66,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{B_{{26,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} - B_{{26,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \left( {B_{{22,co}} + B_{{66,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}}^{2} }}\frac{\partial }{{\partial \theta _{{co}} }} \\ & \quad\quad\quad\,\,\, + B_{{26,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }}\, + \frac{{\partial B_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{1}{{R_{{co}} }}\frac{{\partial B_{{12,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} - \frac{{\partial B_{{16,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }}, \\ \end{aligned}$$
$$\begin{aligned} & L_{{21,co}} = A_{{16,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{A_{{12,co}} + A_{{66,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{A_{{26,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + \left( {A_{{26,co}} + 2A_{{16,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} \\ & \quad\quad\quad\,\,\, + \left( {A_{{22,co}} + A_{{66,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}}^{2} }}\frac{\partial }{{\partial \theta _{{co}} }}\, + A_{{26,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }}\, + \frac{{\partial A_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{1}{{R_{{co}} }}\frac{{\partial A_{{66,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{\partial A_{{26,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }} \\ \end{aligned}$$
$$\begin{aligned} & L_{{22,co}} = A_{{66,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{2A_{{26,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + A_{{22,co}} \frac{1}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + A_{{66,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} \\ & \quad\quad\quad\,\,\, - \left( {A_{{66,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} - A_{{44,co}} \frac{{\cos ^{2} \varphi }}{{R_{{co}}^{2} }}} \right) \\ & \quad\quad\quad\,\,\, + \frac{{\partial A_{{26,co}} }}{{\partial x_{{co}} }}\frac{1}{{R_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{\partial A_{{66,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \frac{{\partial A_{{66,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }} - I_{{0,co}} \frac{{\partial ^{2} }}{{\partial t^{2} }}, \\ \end{aligned}$$
$$L_{23,co} = \left( {A_{26,co} + A_{45,co} } \right)\frac{\cos \varphi }{{R_{co} }}\frac{\partial }{{\partial x_{co} }} + \left( {A_{22,co} + A_{44,co} } \right)\frac{\cos \varphi }{{R_{co}^{2} }}\frac{\partial }{{\partial \theta_{co} }} + A_{26,co} \frac{\sin \varphi \cos \varphi }{{R_{co}^{2} }} + \frac{{\partial A_{26,co} }}{{\partial x_{co} }}\frac{\cos \varphi }{{R_{co} }},$$
$$\begin{gathered} L_{{24,co}} = B_{{16,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{B_{{12,co}} + B_{{66,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{B_{{26,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + \left( {B_{{26,co}} + 2B_{{16,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} \hfill \\ \quad\quad \,\,\,\,\,\,\,\,\,\, + \left( {B_{{22,co}} + B_{{66,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}}^{2} }}\frac{\partial }{{\partial \theta _{{co}} }}\, + \left( {B_{{26,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} + A_{{45,co}} \frac{{\cos \varphi }}{{R_{{co}} }}} \right)\, \hfill \\ \quad\quad \,\,\,\,\,\,\,\,\,\, + \frac{{\partial B_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{{\partial B_{{66,co}} }}{{\partial x_{{co}} }}\frac{1}{{R_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{B_{{26,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }}, \hfill \\ \end{gathered}$$
$$\begin{aligned} & L_{{25,co}} = B_{{66,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{2B_{{26,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{B_{{22,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + B_{{66,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \left( {B_{{66,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} - A_{{44,co}} \frac{{\cos \varphi }}{{R_{{co}} }}} \right) \\ &\quad\quad\quad\,\,\, \, + \frac{{B_{{66,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{{\partial B_{{26,co}} }}{{\partial x_{{co}} }}\frac{1}{{R_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} - \frac{{\partial B_{{66,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }} - I_{{1,co}} \frac{{\partial ^{2} }}{{\partial t^{2} }} \\ \end{aligned}$$
$$L_{31,co} = - A_{12,co} \frac{\cos \varphi }{{R_{co} }}\frac{\partial }{{\partial x_{co} }} - A_{26,co} \frac{\cos \varphi }{{R_{co}^{2} }}\frac{\partial }{{\partial \theta_{co} }} - A_{22,co} \frac{\sin \varphi \cos \varphi }{{R_{co}^{2} }},$$
$$L_{32,co} = - \left( {A_{26,co} + A_{45,co} } \right)\frac{\cos \varphi }{{R_{co} }}\frac{\partial }{{\partial x_{co} }} - \left( {A_{22,co} + A_{44,co} } \right)\frac{\cos \varphi }{{R_{co}^{2} }}\frac{\partial }{{\partial \theta_{co} }} + A_{26,co} \frac{\sin \varphi \cos \varphi }{{R_{co}^{2} }} - \frac{{\partial A_{45,co} }}{{\partial x_{co} }}\frac{\cos \varphi }{{R_{co} }},$$
$$\begin{aligned} & L_{{33,co}} = A_{{55,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{2A_{{45,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{A_{{44,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + A_{{55,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} \\ & \quad\quad\quad\,\, \, - A_{{22,co}} \frac{{\cos ^{2} \varphi }}{{R_{{co}}^{2} }} + \frac{{\partial A_{{55,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{1}{{R_{{co}} }}\frac{{\partial A_{{45,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} - I_{{0,co}} \frac{{\partial ^{2} }}{{\partial t^{2} }}, \\ \end{aligned}$$
$$\begin{aligned} & L_{{34,co}} = \left( {A_{{55,co}} - B_{{12,co}} \frac{{\cos \varphi }}{{R_{{co}} }}} \right)\frac{\partial }{{\partial x_{{co}} }} + \left( {\frac{{A_{{45,co}} }}{{R_{{co}} }} - B_{{26,co}} \frac{{\cos \varphi }}{{R_{{co}}^{2} }}} \right)\frac{\partial }{{\partial \theta _{{co}} }} \\ & \quad\quad\quad\,\,\, + \left( {A_{{55,co}} \frac{{\sin \varphi }}{{R_{{co}} }} - B_{{22,co}} \frac{{\sin \varphi \cos \varphi }}{{R_{{co}}^{2} }}} \right) + \frac{{\partial A_{{55,co}} }}{{\partial x_{{co}} }}, \\ \end{aligned}$$
$$\begin{aligned} & L_{{35,co}} = \left( {A_{{45,co}} - B_{{26,co}} \frac{{\cos \varphi }}{{R_{{co}} }}} \right)\frac{\partial }{{\partial x_{{co}} }} + \left( {\frac{{A_{{44,co}} }}{{R_{{co}} }} - B_{{22,co}} \frac{{\cos \varphi }}{{R_{{co}}^{2} }}} \right)\frac{\partial }{{\partial \theta _{{co}} }} \\ & \quad\quad\quad\,\,\, + \left( {A_{{45,co}} \frac{{\sin \varphi }}{{R_{{co}} }} + B_{{26,co}} \frac{{\sin \varphi \cos \varphi }}{{R_{{co}}^{2} }}} \right) + \frac{{\partial A_{{45,co}} }}{{\partial x_{{co}} }}, \\ \end{aligned}$$
$$\begin{gathered} L_{41,co} = B_{11,co} \frac{{\partial^{2} }}{{\partial x_{co}^{2} }} + \frac{{2B_{16,co} }}{{R_{co} }}\frac{{\partial^{2} }}{{\partial x_{co} \partial \theta_{co} }} + B_{66,co} \frac{1}{{R_{co}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{co}^{2} }} + B_{11,co} \frac{\sin \varphi }{{R_{co} }}\frac{\partial }{{\partial x_{co} }} - B_{22,co} \frac{{\sin^{2} \varphi }}{{R_{co}^{2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{{B_{11,co} }}{{\partial x_{co} }}\frac{\partial }{{\partial x_{co} }} + \frac{1}{{R_{co} }}\frac{{B_{16,co} }}{{\partial x_{co} }}\frac{\partial }{{\partial \theta_{co} }} + \frac{{B_{12,co} }}{{\partial x_{co} }}\frac{\sin \varphi }{{R_{co} }} - I_{1,co} \frac{{\partial^{2} }}{{\partial t^{2} }}, \hfill \\ \end{gathered}$$
$$\begin{aligned} & L_{{42,co}} = B_{{16,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{B_{{12,co}} + B_{{66,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{B_{{26,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} v}}{{\partial \theta _{{co}}^{2} }} \\ &\quad\quad\quad\,\,\, - B_{{26,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \left( {B_{{22,co}} + B_{{66,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}}^{2} }}\frac{\partial }{{\partial \theta _{{co}} }} \\ &\quad\quad\quad\,\,\, \, + \left( {B_{{26,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} + A_{{45,co}} \frac{{\cos \varphi }}{{R_{{co}} }}} \right)\, + \frac{1}{{R_{{co}} }}\frac{{B_{{12,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{B_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \frac{{B_{{16,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }}, \\ \end{aligned}$$
$$L_{43,co} = \left( {B_{12,co} \frac{\cos \varphi }{{R_{co} }} - A_{55,co} } \right)\frac{\partial }{{\partial x_{co} }} + \left( {B_{26,co} \frac{\cos \varphi }{{R_{co}^{2} }} - A_{45,co} \frac{1}{{R_{co} }}} \right)\frac{\partial }{{\partial \theta_{co} }} - B_{22,co} \frac{\sin \varphi \cos \varphi }{{R_{co}^{2} }} + \frac{{B_{12,co} }}{{\partial x_{co} }}\frac{\cos \varphi }{{R_{co} }},$$
$$\begin{aligned} & L_{{44,co}} = D_{{11,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{2D_{{16,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x\partial \theta }} + \frac{{D_{{66,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + D_{{11,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \left( {D_{{22,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} + A_{{55,co}} } \right) \\ &\quad\quad\quad\,\,\, + \frac{{D_{{11,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{{D_{{12,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }} + \frac{1}{{R_{{co}} }}\frac{{D_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} - I_{{2,co}} \frac{{\partial ^{2} }}{{\partial t^{2} }}, \\ \end{aligned}$$
$$\begin{aligned} & L_{{45,co}} = D_{{16,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{D_{{12,co}} + D_{{66,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{D_{{26,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} - D_{{26,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} \\ &\quad\quad\quad\,\,\, - \left( {D_{{22,co}} + D_{{66,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}}^{2} }}\frac{\partial }{{\partial \theta _{{co}} }} + \left( {D_{{26,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} - A_{{45,co}} } \right) \\ & \quad\quad\quad\,\,\, + \frac{{D_{{12,co}} }}{{\partial x_{{co}} }}\frac{1}{{R_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{D_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \frac{{D_{{16,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }}, \\ \end{aligned}$$
$$\begin{aligned} & L_{{51,co}} = B_{{16,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{B_{{12,co}} + B_{{66,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{B_{{26,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + \left( {2B_{{16,co}} + B_{{26,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} \\ & \quad\quad\quad\,\,\, + \left( {B_{{22,co}} + B_{{66,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}}^{2} }}\frac{\partial }{{\partial \theta _{{co}} }} + B_{{26,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }}\, + \frac{{B_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{{B_{{66,co}} }}{{\partial x_{{co}} }}\frac{1}{{R_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{B_{{26,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }} ,\\ \end{aligned}$$
$$\begin{aligned} & L_{{52,co}} = B_{{66,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{2B_{{26,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{B_{{22,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + B_{{66,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \left( {A_{{44,co}} \frac{{\cos \varphi }}{{R_{{co}} }} - B_{{66,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }}} \right) \\ & \quad\quad\quad\,\,\, + \frac{{B_{{66,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{{B_{{26,co}} }}{{\partial x_{{co}} }}\frac{1}{{R_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} - \frac{{B_{{66,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }} - I_{{1,co}} \frac{{\partial ^{2} }}{{\partial t^{2} }}, \\ \end{aligned}$$
$$L_{53,co} = \left( {B_{26,co} \frac{\cos \varphi }{{R_{co} }} - A_{45,co} } \right)\frac{\partial }{{\partial x_{co} }} + \left( {B_{22,co} \frac{\cos \varphi }{{R_{co}^{2} }} - A_{44,co} \frac{1}{{R_{co} }}} \right)\frac{\partial }{{\partial \theta_{co} }} + B_{26,co} \frac{\sin \varphi \cos \varphi }{{R_{co}^{2} }} + \frac{{B_{26,co} }}{{\partial x_{co} }}\frac{\cos \varphi }{{R_{co} }},$$
$$\begin{aligned} & L_{{54,co}} = D_{{16,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{D_{{12,co}} + D_{{66,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x_{{co}} \partial \theta _{{co}} }} + \frac{{D_{{26,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + \left( {2D_{{16,co}} + D_{{26,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} \\ & \quad\quad\quad\,\,\, + \left( {D_{{22,co}} + D_{{66,co}} } \right)\frac{{\sin \varphi }}{{R_{{co}}^{2} }}\frac{\partial }{{\partial \theta _{{co}} }} + \left( {D_{{26,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} - A_{{45,co}} } \right) \\ & \quad\quad\quad\,\,\, + \frac{{D_{{16,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} + \frac{1}{{R_{{co}} }}\frac{{D_{{66,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{D_{{26,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }}, \\ \end{aligned}$$
$$\begin{aligned} & L_{{55,co}} = D_{{66,co}} \frac{{\partial ^{2} }}{{\partial x_{{co}}^{2} }} + \frac{{2D_{{26,co}} }}{{R_{{co}} }}\frac{{\partial ^{2} }}{{\partial x\partial \theta }} + \frac{{D_{{22,co}} }}{{R_{{co}}^{2} }}\frac{{\partial ^{2} }}{{\partial \theta _{{co}}^{2} }} + D_{{66,co}} \frac{{\sin \varphi }}{{R_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \left( {D_{{66,co}} \frac{{\sin ^{2} \varphi }}{{R_{{co}}^{2} }} - A_{{44,co}} } \right) \\ & \quad\quad\quad\,\,\, + \frac{{D_{{26,co}} }}{{\partial x_{{co}} }}\frac{1}{{R_{{co}} }}\frac{\partial }{{\partial \theta _{{co}} }} + \frac{{D_{{66,co}} }}{{\partial x_{{co}} }}\frac{\partial }{{\partial x_{{co}} }} - \frac{{D_{{66,co}} }}{{\partial x_{{co}} }}\frac{{\sin \varphi }}{{R_{{co}} }}, \\ \end{aligned}$$

For the cylindrical shell:

$$\begin{gathered} L_{11,cy} = A_{11,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \frac{{2A_{16,cy} }}{{R_{cy} }}\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{A_{66,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }} - I_{0,cy} \frac{{\partial^{2} }}{{\partial t^{2} }}\;,\; \hfill \\ L_{12,cy} = A_{16,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \left( {\frac{{A_{12,cy} }}{{R_{cy} }} + \frac{{A_{66,cy} }}{{R_{cy} }}} \right)\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{A_{26,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }},L_{13,cy} = \left( {\frac{{A_{12,cy} }}{{R_{cy} }}\frac{\partial }{{\partial x_{cy} }} + \frac{{A_{26,cy} }}{{R_{cy}^{2} }}\frac{\partial }{{\partial \theta_{cy} }}} \right), \hfill \\ L_{14,cy} = B_{11,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \frac{{2B_{16,cy} }}{{R_{cy} }}\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{B_{66,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }} - I_{1,cy} \frac{{\partial^{2} }}{{\partial t^{2} }}, \hfill \\ L_{15,cy} = B_{16,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \left( {\frac{{B_{12,cy} }}{{R_{cy} }} + \frac{{B_{66,cy} }}{{R_{cy} }}} \right)\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{B_{26,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }}, \hfill \\ \end{gathered}$$
$$\begin{gathered} L_{21,cy} = L_{12,cy} ,\;\;\;\;L_{22,cy} = A_{66,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \frac{{2A_{26,cy} }}{{R_{cy} }}\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{A_{22,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }} - \frac{{\kappa A_{44,cy} }}{{R_{cy}^{2} }} - I_{0,cy} \frac{{\partial^{2} }}{{\partial t^{2} }},\;\; \hfill \\ L_{23,cy} = \left( {\frac{{A_{26,cy} }}{{R_{cy} }} + \frac{{\kappa A_{45,cy} }}{{R_{cy} }}} \right)\frac{\partial }{{\partial x_{cy} }} + \left( {\frac{{A_{22} }}{{R_{cy}^{2} }} + \frac{{\kappa A_{44,cy} }}{{R_{cy}^{2} }}} \right)\frac{\partial }{{\partial \theta_{cy} }}, \hfill \\ L_{24,cy} = B_{16} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \left( {\frac{{B_{12,cy} }}{{R_{cy} }} + \frac{{B_{66,cy} }}{{R_{cy} }}} \right)\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{B_{26,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }} + \frac{{\kappa A_{45} }}{{R_{cy} }},\; \hfill \\ L_{25,cy} = B_{66,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \frac{{2B_{26,cy} }}{R}\frac{{\partial^{2} }}{\partial x\partial \theta } + \frac{{B_{22,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }} + \frac{{\kappa A_{44,cy} }}{R} - I_{1,cy} \frac{{\partial^{2} }}{{\partial t^{2} }}, \hfill \\ \end{gathered}$$
$$\begin{gathered} L_{31,cy} = L_{13,cy} ,\;\;L_{32,cy} = L_{23,cy} , \hfill \\ L_{33,cy} = \frac{{A_{22,cy} }}{{R_{cy}^{2} }} - \left( {\kappa A_{55,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \frac{{2\kappa A_{45,cy} }}{{R_{cy} }}\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{\kappa A_{44,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }}} \right) - I_{0,cy} \frac{{\partial^{2} }}{{\partial t^{2} }}, \hfill \\ L_{34,cy} = \left( {\frac{{B_{12,cy} }}{{R_{cy} }} - \kappa A_{55,cy} } \right)\frac{\partial }{{\partial x_{cy} }} + \left( {\frac{{B_{26,cy} }}{{R_{cy}^{2} }} - \frac{{\kappa A_{45,cy} }}{{R_{cy} }}} \right)\frac{\partial }{{\partial \theta_{cy} }},\;\; \hfill \\ L_{35,cy} = \left( {\frac{{B_{26,cy} }}{{R_{cy} }} - \kappa A_{45,cy} } \right)\frac{\partial }{{\partial x_{cy} }} + \left( {\frac{{B_{22,cy} }}{{R_{cy}^{2} }} - \frac{{\kappa A_{44,cy} }}{R}} \right)\frac{\partial }{{\partial \theta_{cy} }}, \hfill \\ \end{gathered}$$
$$\begin{gathered} L_{41,cy} = L_{14,cy} ,\;\;\;L_{42,cy} = L_{24,cy} ,\;\;\;L_{43,cy} = L_{34,cy} ,\;\,\, \hfill \\ L_{44,cy} = D_{11,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \frac{{2D_{16,cy} }}{{R_{cy} }}\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{D_{66,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }} - \kappa A_{55,cy} - I_{2,cy} \frac{{\partial^{2} }}{{\partial t^{2} }}, \hfill \\ L_{45,cy} = D_{16,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \left( {\frac{{D_{12,cy} }}{{R_{cy} }} + \frac{{D_{66,cy} }}{{R_{cy} }}} \right)\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{D_{26,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }} - \kappa A_{45,cy}, \hfill \\ \end{gathered}$$
$$\begin{gathered} L_{51,cy} = L_{15,cy} ,\;\;\,\,\,\,L_{52,cy} = L_{25,cy} ,\;\;\,\,\,\,L_{53,cy} = L_{35,cy} ,\;\;\,\,\,L_{54,cy} = L_{45,cy} ,\;\;\;\; \hfill \\ L_{55,cy} = D_{66,cy} \frac{{\partial^{2} }}{{\partial x_{cy}^{2} }} + \frac{{2D_{26,cy} }}{{R_{cy} }}\frac{{\partial^{2} }}{{\partial x_{cy} \partial \theta_{cy} }} + \frac{{D_{22,cy} }}{{R_{cy}^{2} }}\frac{{\partial^{2} }}{{\partial \theta_{cy}^{2} }} - \kappa A_{44,cy} - I_{2,cy} \frac{{\partial^{2} }}{{\partial t^{2} }}. \hfill \\ \end{gathered}$$

Appendix 2

Conical shell:

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

Cylindrical shell:

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, K., Kwak, S., Pang, C. et al. Free vibration analysis of combined composite laminated conical–cylindrical shells with varying thickness using the Haar wavelet method. Acta Mech 233, 1567–1597 (2022). https://doi.org/10.1007/s00707-022-03173-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-022-03173-y

Search

Navigation