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Vibration characteristics analysis of anisotropic metal rubber medium-thick cylindrical shells

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Abstract

This paper is to analyze the vibration characteristics of a metal rubber medium-thick cylindrical shell (MR-MTCS) influenced by material anisotropy. The parameters of the metal rubber (MR) are determined by the inclined helix-pyramid model (IHPM). The control equations for MR-MTCS are determined by first-order shear deformation theory (FSDT) and Rayleigh–Ritz method. The results show that the anisotropy of MR has a significant effect on the vibration characteristics of MR-MTCS. Among them, the anisotropy of Young's modulus has the most significant effect on the vibration characteristics. In addition, the effect of MR anisotropy on the vibration characteristics of MR-MTCS increases with the increase of axial half-wave number m and decreases with the increase of axial wave number n.

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Data availability

All data generated during this study are included in this published paper.

Abbreviations

h :

Cylindrical shell thickness (m)

R :

The radius of the middle surface of the cylindrical shell (m)

L :

Cylindrical shell length (m)

u, v, w :

Displacement of the shell along x, θ, z direction (m)

ψ x, ψ θ :

Rotation angles normal to the direction of x- and θ-axis (rad)

a x , a θ :

Lame coefficient in the x, θ direction in the middle surface (−)

R x, R θ :

The radius of curvature in the x, θ direction in the middle surface (m)

A x, A θ :

Lamé coefficient along x, θ direction in cylindrical shell (−)

n m :

Number of screws of equivalent spring unit (−)

E i :

Young's modulus of MR (MPa)

ε x, ε θ :

Normal strain of shell along x, θ direction (−)

γ xθ, γ θz, γxz :

Shear strain of the shell normal to the z-axis, x-axis and θ-axis (−)

k u ,k v ,k w :

Stiffness coefficient along x, θ, z direction

κ x, κ θ :

Slope of strain εx, εθ along the z-direction (−)

E, E m :

Young's modulus of cylindrical shell and metal wire (GPa)

μ :

Poisson's ratio of MR (−)

μ m :

Poisson's ratio of metal wire

ρ :

Density of the shell (kg/m3)

N, Q s :

Membrane stresses (Pa)

M :

Membrane moments (Pa.m)

K s :

Transverse shear correction factor (−)

U,U spr :

Strain energy of the shell and elastic support (N.m)

T :

Shell kinetic energy (N.m)

δ :

Variable score symbols (−)

t,T 0,T 1 :

Times, starting moment, ending moment (s)

ω :

Round frequency(rad)

n :

Circumferential wave number (−)

m :

Axial half wave number (−)

δ kl :

Kronecker constant (−)

f :

Natural frequency of MR-MTCS (Hz)

η :

Modal loss factor of MR-MTCS

β :

Half cone angle of equivalent spring unit (rad)

\({\upeta }_{{\text{ m,n}}}^{{\text{ N}}}\) :

Modal loss factor at N (%)

\({\text{f}} _{{\text{m,n}}}^{{\text{ N}}}\) :

Natural frequency at N (Hz)

W L ,W U :

Mechanical work during loading and unloading (N.m)

ρm, \({\overline{\rho }} \) :

Relative density of MR samples and blanks (%)

\(\mu_{c} { }\) :

Friction coefficient of 304 stainless steel (%)

a, b :

Load correction factor (−)

α :

Guidance angle (rad)

d, D :

Wire diameter and helix diameter (mm)

N :

Polynomial truncation number (−)

C :

Spring index of the helix unit (−)

φ , \({\overline{{\varphi }}}\) :

Inclination angle of the helix unit and its mean value (rad)

r :

Molding ratio of MR(−)

K T :

Axial stiffness of the helix unit (N/m)

K V :

Tangential stiffness of the helix unit (N/m)

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Acknowledgements

The authors thank Haiyang Liu for helpful discussion.

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Yan Bai & He Li

Authors
  1. Yan Bai
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YB contributed to conceptualization, methodology, software, and writing—original draft. HL contributed to supervision, project administration, funding acquisition, and writing—review and editing.

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Correspondence to He Li.

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Appendices

Appendix A

Equation (19) for M, K and Kspr has the following expressions:

$$ {\varvec{M}} = \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{11}^{sub} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{M}}_{14}^{sub} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{22}^{sub} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{M}}_{25}^{sub} } \\ {\mathbf{0}} & {\mathbf{0}} & {{\varvec{M}}_{33}^{sub} } & {\mathbf{0}} & {\mathbf{0}} \\ {{\varvec{M}}_{41}^{sub} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{M}}_{44}^{sub} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{52}^{sub} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{M}}_{55}^{sub} } \\ \end{array} } \right] $$
(A1)
$$ {\varvec{M}}_{11}^{sub} = {\varvec{M}}_{22}^{sub} = {\varvec{M}}_{33}^{sub} = \frac{1}{2}hL\pi R^{2} \rho \omega^{2} \int_{0}^{1} {{\varvec{P}}_{u} {\varvec{P}}_{u} dx} $$
(A2)
$$ {\varvec{M}}_{41}^{sub} = {\varvec{M}}_{14}^{sub} = {\varvec{M}}_{52}^{sub} = {\varvec{M}}_{25}^{sub} = \frac{1}{24}h^{3} L\pi R\rho \omega^{2} \int_{0}^{1} {{\varvec{P}}_{u} {\varvec{P}}_{u} dx} $$
(A3)
$$ {\varvec{M}}_{44}^{sub} = {\varvec{M}}_{55}^{sub} = \frac{1}{24}h^{3} L\pi R^{2} \rho \omega^{2} \int_{0}^{1} {{\varvec{P}}_{u} {\varvec{P}}_{u} dx} $$
(A4)
$$ {\varvec{K}} = {\varvec{K}}^{(1)} + {\varvec{K}}^{(2)} + {\varvec{K}}^{(3)} $$
(A5)
$$ {\varvec{K}}_{spr} \user2{ = K}_{spr}^{\left( 0 \right)} + {\varvec{K}}_{spr}^{\left( 1 \right)} $$
(A6)
$$ {\varvec{K}}^{(1)} = \left[ {\begin{array}{*{20}c} {{\varvec{K}}_{11}^{\left( 1 \right)} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{14}^{\left( 1 \right)} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{K}}_{22}^{\left( 1 \right)} } & {{\varvec{K}}_{23}^{\left( 1 \right)} } & {\mathbf{0}} & {{\varvec{K}}_{25}^{\left( 1 \right)} } \\ {\mathbf{0}} & {{\varvec{K}}_{32}^{\left( 1 \right)} } & {{\varvec{K}}_{33}^{\left( 1 \right)} } & {\mathbf{0}} & {{\varvec{K}}_{35}^{\left( 1 \right)} } \\ {{\varvec{K}}_{41}^{\left( 1 \right)} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{44}^{\left( 1 \right)} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{K}}_{52}^{\left( 1 \right)} } & {{\varvec{K}}_{53}^{\left( 1 \right)} } & {\mathbf{0}} & {{\varvec{K}}_{55}^{\left( 1 \right)} } \\ \end{array} } \right] $$
(A7)
$$ {\varvec{K}}^{(2)} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{\varvec{K}}_{12}^{\left( 2 \right)} } & {{\varvec{K}}_{13}^{\left( 2 \right)} } & {\mathbf{0}} & {{\varvec{K}}_{15}^{\left( 2 \right)} } \\ {{\varvec{K}}_{21}^{\left( 2 \right)} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{24}^{\left( 2 \right)} } & {\mathbf{0}} \\ {{\varvec{K}}_{31}^{\left( 2 \right)} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{24}^{\left( 2 \right)} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{K}}_{42}^{\left( 2 \right)} } & {{\varvec{K}}_{43}^{\left( 2 \right)} } & {\mathbf{0}} & {{\varvec{K}}_{45}^{\left( 2 \right)} } \\ {{\varvec{K}}_{51}^{\left( 2 \right)} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{54}^{\left( 2 \right)} } & {\mathbf{0}} \\ \end{array} } \right] $$
(A8)
$$ {\varvec{K}}^{(3)} = \left[ {\begin{array}{*{20}c} {{\varvec{K}}_{11}^{\left( 3 \right)} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{14}^{\left( 3 \right)} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{K}}_{22}^{\left( 3 \right)} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{25}^{\left( 3 \right)} } \\ {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{33}^{\left( 3 \right)} } & {\mathbf{0}} & {\mathbf{0}} \\ {{\varvec{K}}_{41}^{\left( 2 \right)} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{44}^{\left( 2 \right)} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{K}}_{52}^{\left( 2 \right)} } & {\mathbf{0}} & {\mathbf{0}} & {{\varvec{K}}_{53}^{\left( 2 \right)} } \\ \end{array} } \right] $$
(A9)
$$ {\varvec{K}}_{11}^{(1)} = \frac{{G_{12} Ln^{2} \pi R}}{{2\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left[ \begin{gathered} - L_{9} + L_{5} \nu_{12} \nu_{21} + L_{6} \left( { - 1 + \nu_{12} \nu_{21} } \right) + \hfill \\ L_{7} \left( { - 1 + \nu_{12} \nu_{21} } \right) \hfill \\ \end{gathered} \right]\int_{0}^{1} {{\varvec{P}}_{u} {\varvec{P}}_{u} dx} $$
(A10)
$$ {\varvec{K}}_{14}^{(1)} = \frac{L\pi R}{{648\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left( \begin{gathered} - 324G_{12} hn^{2} + 648G_{12} L_{2} n^{2} R - \hfill \\ 648G_{12} L_{3} n^{2} R + 648G_{12} L_{4} n^{2} R + \hfill \\ 324G_{12} hn^{2} \nu_{12} \nu_{21} - 648G_{12} L_{2} n^{2} R\nu_{12} \nu_{21} + \hfill \\ 648G_{12} L_{3} n^{2} R\nu_{12} \nu_{21} - 648G_{12} L_{4} n^{2} R\nu_{12} \nu_{21} \hfill \\ \end{gathered} \right)\int_{0}^{1} {{\varvec{P}}_{u} {\varvec{P}}_{{\varphi_{x} }} dx} $$
(A11)
$$ {\varvec{K}}_{22}^{(1)} = \frac{L\pi R}{{2\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left\{ \begin{gathered} L_{14} \left[ { - E_{1} n^{2} + G_{13} K_{s} \left( { - 1 + \nu_{12} \nu_{21} } \right)} \right] + \hfill \\ L_{12} \left[ { - E_{2} n^{2} + G_{23} K_{s} \left( { - 1 + \nu_{12} \nu_{21} } \right)} \right] + \hfill \\ L_{13} \left[ { - E_{2} n^{2} + G_{23} K_{s} \left( { - 1 + \nu_{12} \nu_{21} } \right)} \right] \hfill \\ \end{gathered} \right\}\int_{0}^{1} {{\varvec{P}}_{v} {\varvec{P}}_{v} dx} $$
(A12)
$$ K_{23}^{(1)} = \frac{L\pi R}{{648\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left( \begin{gathered} 324E_{2} L_{5} n + 324G_{23} K_{s} L_{5} n + 324E_{2} L_{6} n + \hfill \\ 324G_{23} K_{s} L_{6} n + 324E_{1} L_{7} n + 324G_{13} K_{s} L_{7} n - \hfill \\ 324G_{23} K_{s} L_{5} n\nu_{12} \nu_{21} - 324G_{23} K_{s} L_{6} n\nu_{12} \nu_{21} - \hfill \\ 324G_{13} K_{s} L_{7} n\nu_{12} \nu_{21} \hfill \\ \end{gathered} \right)\int_{0}^{1} {{\varvec{P}}_{v} {\varvec{P}}_{w} d} x $$
(A13)
$$ {\varvec{K}}_{25}^{(1)} = \frac{L\pi R}{{648\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left( \begin{gathered} - 108E_{1} hn^{2} - 216E_{2} hn^{2} + \hfill \\ 324G_{23} K_{s} L_{5} R + 324G_{23} K_{s} L_{6} R + \hfill \\ 324G_{13} K_{s} L_{7} R + 648E_{1} L_{2} n^{2} R - \hfill \\ 648E_{2} L_{3} n^{2} R + 648E_{2} L_{4} n^{2} R - \hfill \\ 324G_{23} K_{s} L_{5} R\nu_{12} \nu_{21} - 324G_{23} K_{s} L_{6} R\nu_{12} \nu_{21} - \hfill \\ 324G_{13} K_{s} L_{7} R\nu_{12} \nu_{21} \hfill \\ \end{gathered} \right)\int_{0}^{1} {{\varvec{P}}_{v} {\varvec{P}}_{{\varphi_{\theta } }} dx} $$
(A14)
$$ {\varvec{K}}_{32}^{(1)} = \frac{Ln\pi R}{{2\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left\{ \begin{gathered} L_{14} \left[ {E_{1} + G_{13} K_{s} \left( {1 - \nu_{12} \nu_{21} } \right)} \right] + \hfill \\ L_{12} \left[ {E_{2} + G_{23} K_{s} \left( {1 - \nu_{12} \nu_{21} } \right)} \right] + \hfill \\ L_{13} \left[ {E_{2} + G_{23} K_{s} \left( {1 - \nu_{12} \nu_{21} } \right)} \right] \hfill \\ \end{gathered} \right\}\int_{0}^{1} {{\varvec{P}}_{w} {\varvec{P}}_{v} dx} $$
(A15)
$$ {\varvec{K}}_{33}^{(1)} = \frac{L\pi R}{{648\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left( \begin{gathered} - 324E_{2} L_{5} - 324E_{2} L_{6} - 324E_{1} L_{7} - \hfill \\ 324G_{23} K_{s} L_{6} n^{2} - 324G_{13} K_{s} L_{7} n^{2} - \hfill \\ 324G_{23} K_{s} L_{9} n^{2} + 324G_{23} K_{s} L_{5} n^{2} \nu_{12} \nu_{21} + \hfill \\ 324G_{23} K_{s} L_{6} n^{2} \nu_{12} \nu_{21} + 324G_{13} K_{s} L_{7} n^{2} \nu_{12} \nu_{21} \hfill \\ \end{gathered} \right)\int_{0}^{1} {{\varvec{P}}_{w} {\varvec{P}}_{w} dx} $$
(A16)
$$ {\varvec{K}}_{35}^{(1)} = \frac{L\pi R}{{648\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left( \begin{gathered} 108E_{1} hn + 216E_{2} hn - 648E_{1} L_{2} nR + \hfill \\ 648E_{2} L_{3} nR - 648E_{2} L_{4} nR - \hfill \\ 324G_{23} K_{s} L5nR - 324G_{23} K_{s} L_{6} nR - \hfill \\ 324G_{13} K_{s} L_{7} nR + 324G_{23} K_{s} L_{5} nR\nu_{12} \nu_{21} + \hfill \\ 324G_{23} K_{s} L_{6} nR\nu_{12} \nu_{21} + 324G_{13} K_{s} L_{7} nR\nu_{12} \nu_{21} \hfill \\ \end{gathered} \right)\int_{0}^{1} {{\varvec{P}}_{w} {\varvec{P}}_{{\varphi_{\theta } }} dx} $$
(A17)
$$ {\varvec{K}}_{41}^{(1)} = \frac{1}{2}G_{12} Ln^{2} \pi R\left( {h - 2L_{2} R + 2L_{3} R - 2L_{4} R} \right)\int_{0}^{1} {{\varvec{P}}_{{\varphi_{x} }} {\varvec{P}}_{u} dx} $$
(A18)
$$ {\varvec{K}}_{44}^{(1)} = \frac{{L\pi R^{2} }}{{ - 6 + 6\nu_{12} \nu_{21} }}\left\{ \begin{gathered} \left( { - 1 + \nu_{12} \nu_{21} } \right)\left[ {G_{23} K_{s} h + 3G_{23} K_{s} \left( { - 2L_{2} + L_{7} } \right)R - } \right. \hfill \\ \left. {3G_{12} n^{2} h - 3G_{12} n^{2} \left( {2L_{2} - 2L_{8} + L_{9} } \right)R} \right] + \hfill \\ G_{13} K_{s} \left[ {2h\left( { - 1 + \nu_{12} \nu_{21} } \right) - 3R\left( { - 4L_{4} + L_{5} + } \right.} \right. \hfill \\ L_{6} - 2L_{8} + L_{9} + 4L_{4} \nu_{12} \nu_{21} - L_{6} \nu_{12} \nu_{21} + \hfill \\ \left. {\left. {2L_{8} \nu_{12} \nu_{21} - 2L_{9} \nu_{12} \nu_{21} + 4L_{3} - 4L_{3} \nu_{12} \nu_{21} } \right)} \right] \hfill \\ \end{gathered} \right\}\int_{0}^{1} {{\varvec{P}}_{{\varphi_{x} }} {\varvec{P}}_{{\varphi_{x} }} } dx $$
(A19)
$$ {\varvec{K}}_{52}^{(1)} = \frac{L\pi R}{{ - 6 + 6\nu_{12} \nu_{21} }}\left\{ \begin{gathered} - E_{1} n^{2} \left( {h - 6L_{2} R} \right) - 2E_{2} n^{2} \left[ {h + 3\left( {L_{3} - L_{4} } \right)R} \right] - \hfill \\ 3K_{s} R\left[ {G_{23} \left( {L_{12} + L_{13} } \right) + G_{13} L_{14} } \right]\left( { - 1 + \nu_{12} \nu_{21} } \right) \hfill \\ \end{gathered} \right\}\int_{0}^{1} {{\varvec{P}}_{{\varphi_{\theta } }} {\varvec{P}}_{v} dx} $$
(A20)
$$ K_{53}^{(1)} = \frac{Ln\pi R}{{ - 6 + 6\nu_{12} \nu_{21} }}\left\{ \begin{gathered} E_{1} \left( {h - 6L_{2} R} \right) + 2E_{2} \left[ {h + 3\left( {L_{3} - L_{4} } \right)R} \right] + \hfill \\ 3K_{s} R\left[ {G_{23} \left( {L_{5} + L_{6} } \right) + G_{13} L_{7} } \right]\left( { - 1 + \nu_{12} \nu_{21} } \right) \hfill \\ \end{gathered} \right\}\int_{0}^{1} {{\varvec{P}}_{{\varphi_{\theta } }} {\varvec{P}}_{w} dx} $$
(A21)
$$ K_{55}^{(1)} = \frac{{L\pi R^{2} }}{{ - 6 + 6\nu_{12} \nu_{21} }}\left\{ \begin{gathered} E_{1} n^{2} \left( {h - 6L_{2} R} \right) + E_{2} n^{2} \left( {2h + 6L_{8} R - 3L_{9} R} \right) + \hfill \\ 3K_{s} \left( {G_{23} \left( {L_{5} + L_{6} } \right) + G_{13} L_{7} } \right)R\left( { - 1 + \nu_{12} \nu_{21} } \right) \hfill \\ \end{gathered} \right\}\int_{0}^{1} {{\varvec{P}}_{{\varphi_{\theta } }} {\varvec{P}}_{{\varphi_{\theta } }} dx} $$
(A22)
$$ {\varvec{K}}_{11}^{(2)} = \frac{{Ln\pi R\left[ { - E_{1} \nu_{12} + G_{12} \left( { - 1 + \nu_{12} \nu_{21} } \right)} \right]}}{{ - 2 + 2\nu_{12} \nu_{21} }}\left[ \begin{gathered} h - 2L_{2} R + 2L_{3} R - \hfill \\ 2L_{4} R + L_{5} R + L_{6} R + L_{7} R \hfill \\ \end{gathered} \right]\int_{0}^{1} {{\varvec{P}}_{u} \user2{P^{\prime}}_{u} dx} $$
(A23)
$$ {\varvec{K}}_{13}^{(2)} = \frac{{E_{1} L\pi R\nu_{12} }}{{ - 2 + 2\nu_{12} \nu_{21} }}\left[ \begin{gathered} h - 2L_{2} R + 2L_{3} R - \hfill \\ 2L_{4} R + L_{6} R + L_{7} R + \hfill \\ L_{9} R \hfill \\ \end{gathered} \right]\int_{0}^{1} {{\varvec{P}}_{u} \user2{P^{\prime}}_{w} dx} $$
(A24)
$$ {\varvec{K}}_{15}^{(2)} = \frac{{L\left( {2L_{3} - 2L_{4} - 2L_{8} + L_{9} } \right)n\pi R^{3} \left( {G_{12} + E_{1} \nu_{12} - G_{12} \nu_{12} \nu_{21} } \right)}}{{2 - 2\nu_{12} \nu_{21} }}\int_{0}^{1} {{\varvec{P}}_{u} {\varvec{P}}_{{\varphi_{\theta } }} dx} $$
(A25)
$$ {\varvec{K}}_{21}^{(2)} = - \frac{{Ln\pi R\left( { - E_{1} \nu_{12} + G_{12} \left( { - 1 + \nu_{12} \nu_{21} } \right)} \right)}}{{ - 2 + 2\nu_{12} \nu_{21} }}\left[ \begin{gathered} h - 2L_{2} R + 2L_{3} R - \hfill \\ 2L_{4} R + L_{5} R + L_{6} R + \hfill \\ L_{7} R \hfill \\ \end{gathered} \right]\int_{0}^{1} {{\varvec{P}}_{v} {\varvec{P}}_{u} dx} $$
(A26)
$$ {\varvec{K}}_{24}^{(2)} = \frac{{Ln\pi R^{3} \left( {2L_{3} - 2L_{4} - 2L_{8} + L_{9} } \right)\left( {G_{12} + E_{1} \nu_{12} - G_{12} \nu_{12} \nu_{21} } \right)}}{{ - 2 + 2\nu_{12} \nu_{21} }}\int_{0}^{1} {{\varvec{P}}_{v} {\varvec{P}}_{{\varphi_{x} }} dx} $$
(A16)
$$ {\varvec{K}}_{31}^{(2)} = - \frac{{E_{1} L\pi R\nu_{12} }}{{ - 2 + 2\nu_{12} \nu_{21} }}\left( \begin{gathered} h - 2L_{2} R + \hfill \\ 2L_{3} R - 2L_{4} R + \hfill \\ L_{5} R + L_{6} R + L_{7} R \hfill \\ \end{gathered} \right)\int_{0}^{1} {{\varvec{P}}_{w} {\varvec{P}}_{u} dx} $$
(A27)
$$ {\varvec{K}}_{34}^{(2)} = - \frac{L\pi R}{{ - 6 + 6\nu_{12} \nu_{21} }}\left\{ \begin{gathered} 3E_{1} R\nu_{12} \left( {2L_{3} - 2L_{4} - 2L_{8} + L_{9} } \right) + \hfill \\ G_{23} K_{s} \left( {h - 6L_{2} R + 3L_{7} R} \right)\left( { - 1 + \nu_{12} \nu_{21} } \right) + \hfill \\ G_{13} K_{s} \left[ {2h\left( { - 1 + \nu_{12} \nu_{21} } \right) + 3R\left( { - 4L_{3} + 4L_{4} - } \right.} \right. \hfill \\ L_{6} + 2L_{8} - 2L_{9} + 4L_{3} \nu_{12} \nu 21 - 4L_{4} \nu_{12} \nu_{21} + \hfill \\ \left. {\left. {L_{5} \nu_{12} \nu_{21} + L_{6} \nu_{12} \nu_{21} - 2L_{8} \nu_{12} \nu_{21} + L_{9} \nu_{12} \nu_{21} } \right)} \right] \hfill \\ \end{gathered} \right\}\int_{0}^{1} {{\varvec{P}}_{w} {\varvec{P}}_{{\varphi_{x} }} dx} $$
(A28)
$$ {\varvec{K}}_{42}^{(2)} = \frac{{Ln\pi R^{3} \left( {2L_{3} - 2L_{4} - 2L_{8} + L_{9} } \right)\left( {G_{12} + E_{1} \nu_{12} - G_{12} \nu_{12} \nu_{21} } \right)}}{{2 - 2\nu_{12} \nu_{21} }}\int_{0}^{1} {{\varvec{P}}_{{\varphi_{x} }} {\varvec{P}}_{v} dx} $$
(A29)
$$ {\varvec{K}}_{43}^{(2)} = \frac{{L\pi R^{2} }}{{ - 6 + 6\nu_{12} \nu_{21} }}\left\{ \begin{gathered} 3E_{1} \left( {2L_{3} - 2L_{4} - 2L_{8} + L_{9} } \right)R\nu_{12} + \hfill \\ G_{23} K_{s} \left( {h - 6L_{2} R + 3L7R} \right)\left( { - 1 + \nu_{12} \nu_{21} } \right) + \hfill \\ G_{13} K_{s} \left[ {2h\left( { - 1 + \nu_{12} \nu_{21} } \right) - 3R\left( { - 4L_{4} + L_{5} + } \right.} \right. \hfill \\ L_{6} - 2L_{8} + L_{9} + 4L_{4} \nu_{12} \nu_{21} - L_{6} \nu_{12} \nu_{21} + \hfill \\ 2L_{8} \nu_{12} \nu_{21} - 2L_{9} \nu_{12} \nu_{21} + L_{3} \left( {4 - 4\nu_{12} \nu_{21} } \right) \hfill \\ \end{gathered} \right\}\int_{0}^{1} {{\varvec{P}}_{{\varphi_{x} }} {\varvec{P}}_{w} dx} $$
(A30)
$$ {\varvec{K}}_{45}^{(2)} = \frac{{Ln\pi R\left[ {h^{3} + 12\left( {L_{10} + 2L_{3} - 2L_{8} + L_{9} } \right)R^{3} } \right]\left[ { - E_{1} \nu_{12} + G_{12} \left( { - 1 + \nu_{12} \nu_{21} } \right)} \right]}}{{24\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\int_{0}^{1} {{\varvec{P}}_{{\varphi_{x} }} {\varvec{P}}_{w} dx} $$
(A31)
$$ {\varvec{K}}_{51}^{(2)} = \frac{{Ln\pi R^{3} \left( {2L_{3} - 2L_{4} - 2L_{8} + L_{9} } \right)\left( {G_{12} + E_{1} \nu_{12} - G_{12} \nu_{12} \nu_{21} } \right)}}{{ - 2 + 2\nu_{12} \nu_{21} }}\int_{0}^{1} {{\varvec{P}}_{{\varphi_{\theta } }} {\varvec{P}}_{u} dx} $$
(A32)
$$ {\varvec{K}}_{54}^{(2)} = - \frac{{Ln\pi R\left[ {G_{12} \left( { - 1 + \nu_{12} \nu_{21} } \right) - E_{1} \nu_{12} } \right]}}{{24\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left( \begin{gathered} 12R^{3} L_{10} + 24R^{3} L_{3} - \hfill \\ 24R^{3} L_{8} + 12R^{3} L_{9} \hfill \\ \end{gathered} \right)\int_{0}^{1} {{\varvec{P}}_{{\varphi_{\theta } }} {\varvec{P}}_{{\varphi_{x} }} dx} $$
(A33)
$$ {\varvec{K}}_{11}^{\left( 3 \right)} = \frac{{L\pi R^{2} }}{{ - 6 + 6\nu_{12} \nu_{21} }}\left[ \begin{gathered} 2E_{1} h + E_{2} h + 3E_{2} \left( { - 2L_{2} + L_{7} } \right)R + \hfill \\ 3E_{1} \left( {4L_{3} - 4L_{4} + L_{6} - 2L_{8} + 2L_{9} } \right)R \hfill \\ \end{gathered} \right]\int_{0}^{1} {{\varvec{P}}_{u} {\varvec{P}}_{u} dx} $$
(A34)
$$ {\varvec{K}}_{14}^{\left( 3 \right)} = \frac{L\pi R}{{648\left( { - 1 + \nu 12\nu 21} \right)}}\left[ \begin{gathered} 26E1h^{3} + E2h^{3} + \hfill \\ 324E1\left( {L10R^{3} + } \right. \hfill \\ 4L3R^{3} - 2L4R^{3} - \hfill \\ \left. {4L8R^{3} + 2L9R^{3} } \right) \hfill \\ \end{gathered} \right]\int_{0}^{1} {{\varvec{P}}_{u} {\varvec{P}}_{{\varphi_{x} }} dx} $$
(A35)
$$ {\varvec{K}}_{22}^{\left( 3 \right)} = - \frac{1}{2}G_{12} L\pi R^{2} \left( \begin{gathered} h - 2L_{2} R + 4L_{3} R - \hfill \\ 4L_{4} R + L_{5} R + L_{6} R + \hfill \\ L_{7} R - 2L_{8} R + L_{9} R \hfill \\ \end{gathered} \right)\int_{0}^{1} {{\varvec{P}}_{v} {\varvec{P}}_{v} dx} $$
(A36)
$$ {\varvec{K}}_{25}^{\left( 3 \right)} = - \frac{1}{24}G_{12} L\pi R\left( \begin{gathered} h^{3} + 12L_{10} R^{3} + 48L_{3} R^{3} - \hfill \\ 24L_{4} R^{3} - 48L_{8} R^{3} + 24L_{9} R^{3} \hfill \\ \end{gathered} \right)\int_{0}^{1} {{\varvec{P}}_{v} {\varvec{P}}_{{\varphi_{\theta } }} dx} $$
(A37)
$$ {\varvec{K}}_{33}^{\left( 3 \right)} = - \frac{{K_{s} L\pi R^{2} }}{{ - 6 + 6\nu_{12} \nu_{21} }}\left\{ \begin{gathered} G_{23} \left( {h + 3\left( { - 2L_{2} + L_{7} } \right)R} \right)\left( { - 1 + \nu_{12} \nu_{21} } \right) + \hfill \\ G_{13} \left[ {2h\left( { - 1 + \nu_{12} \nu_{21} } \right) + 3R\left( { - 4L_{3} + 4L_{4} - } \right.} \right. \hfill \\ L_{6} + 2L_{8} - 2L_{9} + 4L3\nu_{12} \nu_{21} - 4L_{4} \nu_{12} \nu_{21} + \hfill \\ L_{5} \nu_{12} \nu_{21} + L_{6} \nu_{12} \nu_{21} - 2L_{8} \nu_{12} \nu_{21} + L_{9} \nu_{12} \nu_{21} \hfill \\ \end{gathered} \right\}\int_{0}^{1} {{\varvec{P}}_{v} {\varvec{P}}_{{\varphi_{\theta } }} dx} $$
(A38)
$$ {\varvec{K}}_{41}^{\left( 3 \right)} = - \frac{L\pi R}{{648\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left[ \begin{gathered} 26E_{1} h^{3} + E_{2} h^{3} + \hfill \\ 324E_{1} R^{3} \left( {L_{10} + } \right. \hfill \\ \left. {4L_{3} - 2L_{4} - 4L_{8} + 2L_{9} } \right) \hfill \\ \end{gathered} \right]\int_{0}^{1} {{\varvec{P}}_{{\varphi_{x} }} {\varvec{P}}_{u} dx} $$
(A39)
$$ {\varvec{K}}_{44}^{\left( 3 \right)} = \frac{{L\pi R^{2} }}{{648\left( { - 1 + \nu_{12} \nu_{21} } \right)}}\left[ \begin{gathered} 26E_{1} h^{3} + E_{2} h^{3} + \hfill \\ 648E_{1} R^{3} \left( {L_{10} + } \right. \hfill \\ \left. {2L_{3} - 2L_{8} + L_{9} } \right) \hfill \\ \end{gathered} \right]\int_{0}^{1} {{\varvec{P}}_{{\varphi_{x} }} {\varvec{P}}_{{\varphi_{x} }} dx} $$
(A40)
$$ {\varvec{K}}_{52}^{\left( 3 \right)} = - \frac{1}{24}G_{12} L\pi R\left[ {h^{3} + 12\left( {L_{10} + 4L_{3} - 2L_{4} - 4L_{8} + 2L_{9} } \right)R^{3} } \right]\int_{0}^{1} {{\varvec{P}}_{{\varphi_{\theta } }} {\varvec{P}}_{{\varphi_{x} }} dx} $$
(A41)
$$ {\varvec{K}}_{55}^{\left( 3 \right)} = - \frac{1}{24}G_{12} L\pi R^{2} \left[ {h^{3} + 24\left( {L_{10} + 2L_{3} - 2L_{8} + L_{9} } \right)R^{3} } \right]\int_{0}^{1} {{\varvec{P}}_{{\varphi_{\theta } }} {\varvec{P}}_{{\varphi_{\theta } }} dx} $$
(A42)
$$ {\varvec{K}}_{spr}^{\left( 0 \right)} = \frac{1}{2}\pi R^{2} \left[ {\begin{array}{*{20}c} {K_{{u_{0} }} {\varvec{P}}_{u} {\varvec{P}}_{u} } & 0 & 0 & 0 & 0 \\ 0 & {K_{{v_{0} }} {\varvec{P}}_{v} {\varvec{P}}_{v} } & 0 & 0 & 0 \\ 0 & 0 & {K_{{w_{0} }} {\varvec{P}}_{w} {\varvec{P}}_{w} } & 0 & 0 \\ 0 & 0 & 0 & {K_{{\varphi_{x0} }} {\varvec{P}}_{{\varphi_{x} }} {\varvec{P}}_{{\varphi_{x} }} } & 0 \\ 0 & 0 & 0 & 0 & {K_{{\varphi_{\theta 0} }} {\varvec{P}}_{{\varphi_{\theta } }} {\varvec{P}}_{{\varphi_{\theta } }} } \\ \end{array} } \right]_{x = 0} $$
(A43)
$$ {\varvec{K}}_{spr}^{\left( 1 \right)} = \frac{1}{2}\pi R^{2} \left[ {\begin{array}{*{20}c} {K_{{u_{1} }} {\varvec{P}}_{u} {\varvec{P}}_{u} } & 0 & 0 & 0 & 0 \\ 0 & {K_{{v_{1} }} {\varvec{P}}_{v} {\varvec{P}}_{v} } & 0 & 0 & 0 \\ 0 & 0 & {K_{{w_{1} }} {\varvec{P}}_{w} {\varvec{P}}_{w} } & 0 & 0 \\ 0 & 0 & 0 & {K_{{\varphi_{x1} }} {\varvec{P}}_{{\varphi_{x} }} {\varvec{P}}_{{\varphi_{x} }} } & 0 \\ 0 & 0 & 0 & 0 & {K_{{\varphi_{\theta 1} }} {\varvec{P}}_{{\varphi_{\theta } }} {\varvec{P}}_{{\varphi_{\theta } }} } \\ \end{array} } \right]_{x = 1} $$
(A44)

Appendix B

Expressions exist in Appendix A as shown below:

$$ L_{1} = \ln \left( { - \frac{h + 2R}{{h - 2R}}} \right) $$
(B1)
$$ L_{2} = {\varvec{arccoth}}\left( \frac{6R}{h} \right) $$
(B2)
$$ L_{3} = {\varvec{arccoth}}\left( {\frac{h}{2h - 6R}} \right) $$
(B3)
$$ L_{4} = {\varvec{arccoth}}\left( {\frac{h}{2h + 6R}} \right) $$
(B4)
$$ L_{5} = \ln \left[ {\frac{{3\left( {h + 2R} \right)}}{h + 6R}} \right] $$
(B5)
$$ L_{6} = \ln \left( {\frac{h - 6R}{{3h - 6R}}} \right) $$
(B6)
$$ L_{7} = \ln \left( {1 - \frac{2h}{{h - 6R}}} \right) $$
(B7)
$$ L_{8} = {\varvec{arccoth}}\left( {2 - \frac{6R}{h}} \right) $$
(B8)
$$ L_{9} = \ln \left( {1 + \frac{2h}{{h + 6R}}} \right) $$
(B9)
$$ L_{10} = \ln \left( {\frac{h + 6R}{{3h + 6R}}} \right) $$
(B10)
$$ L_{11} = \ln \left[ {\frac{{\left( {h - 6R} \right)\left( {h + 2R} \right)}}{{\left( {h - 2R} \right)\left( {h + 6R} \right)}}} \right] $$
(B11)
$$ L_{12} = \ln \left( {\frac{h - 6R}{{h - 2R}}} \right) $$
(B12)
$$ L_{13} = \ln \left( {\frac{h + 2R}{{h + 6R}}} \right) $$
(B13)
$$ L_{14} = \ln \left( { - \frac{h + 6R}{{h - 6R}}} \right) $$
(B14)

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Bai, Y., Li, H. Vibration characteristics analysis of anisotropic metal rubber medium-thick cylindrical shells. Arch Appl Mech 93, 3553–3579 (2023). https://doi.org/10.1007/s00419-023-02453-w

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