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Parametric study and stability analysis on nonlinear traveling wave vibrations of rotating thin cylindrical shells

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Abstract

Parametric study and stability analysis on nonlinear traveling wave vibrations of rotating thin cylindrical shells with simply supported boundary conditions are carried out in the paper. Considering the Coriolis forces as well as the initial hoop tension due to rotation, an infinite-dimensional gyro system model with nonlinearity is established by using Lagrange equations. Based on this model, convergence analysis is performed and the most significant modes dominating the nonlinear behavior are recognized to discretize it to a finite multi-degree system. Then, the periodic solutions of the system are tracked by using harmonic balance method combined with arc length continuation technique. Furthermore, parametric studies are performed and the effects of rotating speed, damping ratio and the amplitude of excitation on the nonlinear dynamic behavior of the shell are investigated. Meanwhile, the Floquet theory is employed to carry out stability analysis of the periodic solutions. The results shown in this paper illustrate the nonlinear dynamic evolution of the traveling wave vibration for rotating thin cylindrical shells.

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Acknowledgements

The authors are grateful to the National Natural Science Foundation of China (Grant Nos. 11802129 and 11902184), Shandong Provincial Natural Science Foundation of China (Grant No. ZR2020QA039), and the Fundamental Research Funds of Shandong University for financial support in this study.

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Authors and Affiliations

  1. School of Civil Engineering, Shandong University, Jinan, 250061, People’s Republic of China

    Shupeng Sun

  2. Institute of Dynamics and Control Science, Shandong Normal University, Jinan, 250014, People’s Republic of China

    Lun Liu

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Appendices

Appendix A

\({\mathbf{c}}_{\lambda } \left( {\lambda = u,v,w} \right)\), and \({\mathbf{H}}_{i} \left( {i = 1,2, \ldots ,18} \right)\) are given by the following:

$$\begin{aligned} \begin{array}{*{20}l} {\mathbf{c}_{\lambda } = \left[ {\begin{array}{*{20}l} \ddots \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {c_{{c,mn}}^{\lambda } } \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {c_{{s,mn}}^{\lambda } } \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & \ddots \hfill \\ \end{array} } \right] \triangleq \left[ {\begin{array}{*{20}l} \ddots \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {c_{{0,mn}}^{\lambda } } \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {c_{{0,mn}}^{\lambda } } \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & \ddots \hfill \\ \end{array} } \right]\quad \left( {\lambda = u,v,w} \right).} \hfill & {} \hfill \\ {{\mathbf{H}}_{1} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {{\mathbf{U}}^{T} \left( {\xi ,\theta } \right){\mathbf{U}}\left( {\xi ,\theta } \right){\text{d}}\xi {\text{d}}\theta } } } \hfill & {{\mathbf{H}}_{2} = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]} } ^{T} \frac{{\partial U\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\text{d}}\xi {\text{d}}\theta } \hfill \\ {{\mathbf{H}}_{3} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]} } ^{T} \frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\text{d}}\xi {\text{d}}\theta } \hfill & {{\mathbf{H}}_{4} = \int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]} } ^{T} \frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\text{d}}\xi {\text{d}}\theta } \hfill \\ {{\mathbf{H}}_{5} = \int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]} } ^{T} \frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\text{d}}\xi {\text{d}}\theta } \hfill & {{\mathbf{H}}_{6} = \int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]} } ^{T} {\mathbf{W}}\left( {\xi ,\theta } \right){\text{d}}\xi {\text{d}}\theta } \hfill \\ {{\mathbf{H}}_{7} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {{\mathbf{V}}^{T} \left( {\xi ,\theta } \right){\mathbf{V}}\left( {\xi ,\theta } \right){\text{d}}\xi {\text{d}}\theta } } } \hfill & {{\mathbf{H}}_{8} = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} \frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\text{d}}\xi {\text{d}}\theta } } } \hfill \\ {{\mathbf{H}}_{9} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {{\mathbf{V}}^{T} \left( {\xi ,\theta } \right){\mathbf{W}}\left( {\xi ,\theta } \right){\text{d}}\xi {\text{d}}\theta } } } \hfill & {{\mathbf{H}}_{{10}} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} \frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\text{d}}\xi {\text{d}}\theta } } } \hfill \\ {{\mathbf{H}}_{{11}} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} {\mathbf{W}}\left( {\xi ,\theta } \right){\text{d}}\xi {\text{d}}\theta } } } \hfill & {{\mathbf{H}}_{{12}} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {{\mathbf{V}}^{T} \left( {\xi ,\theta } \right)\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\text{d}}\xi {\text{d}}\theta } } } \hfill \\ {{\mathbf{H}}_{{13}} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {W^{T} \left( {\xi ,\theta } \right){\mathbf{W}}\left( {\xi ,\theta } \right){\text{d}}\xi {\text{d}}\theta } } } \hfill & {{\mathbf{H}}_{{14}} = \frac{1}{{L^{3} }}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi ^{2} }}} \right]^{T} \frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi ^{2} }}{\text{d}}\xi {\text{d}}\theta } } } \hfill \\ {{\mathbf{H}}_{{15}} = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left\{ {\left[ {\frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi ^{2} }}} \right]^{T} \frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta ^{2} }} + \left[ {\frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta ^{2} }}} \right]^{T} \frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi ^{2} }}} \right\}{\text{d}}\xi {\text{d}}\theta } } } \hfill & {} \hfill \\ {{\mathbf{H}}_{{16}} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta ^{2} }}} \right]^{T} \frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta ^{2} }}{\text{d}}\xi {\text{d}}\theta } } } \hfill & {{\mathbf{H}}_{{17}} = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi \partial \theta }}} \right]^{T} \frac{{\partial ^{2} {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi \partial \theta }}{\text{d}}\xi {\text{d}}\theta } } } \hfill \\ {{\mathbf{H}}_{{18}} = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} \frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\text{d}}\xi {\text{d}}\theta } } } \hfill & {} \hfill \\ \end{array} \hfill \\ \hfill \\ \end{aligned}$$

Appendix B

\({\mathbf{N}}_{u}\),\({\mathbf{N}}_{v}\) and \({\mathbf{N}}_{w}\) are given by the following:

$$\begin{aligned} {\mathbf{N}}_{u} & = \frac{RH}{2}Q_{11} {\mathbf{N}}_{1} + \frac{H}{2R}Q_{12} {\mathbf{N}}_{2} + \frac{H}{R}G{\mathbf{N}}_{3} , \\ {\mathbf{N}}_{v} & = \frac{H}{2}Q_{12} {\mathbf{N}}_{4} + \frac{H}{{2R^{2} }}Q_{22} {\mathbf{N}}_{5} + HG{\mathbf{N}}_{6} , \\ {\mathbf{N}}_{w} & = \frac{{RHQ_{11} }}{2}{\mathbf{N}}_{7} + \frac{{HQ_{22} }}{{2R^{3} }}{\mathbf{N}}_{8} + \left( {\frac{{HQ_{12} }}{2R} + \frac{HG}{R}} \right){\mathbf{N}}_{9} + RHQ_{11} {\mathbf{N}}_{10} + HQ_{12} {\mathbf{N}}_{11} + \frac{{HQ_{12} }}{2}{\mathbf{N}}_{12} + HQ_{12} {\mathbf{N}}_{13} + HQ_{12} \frac{1}{R}{\mathbf{N}}_{14} \\ & \quad + \frac{H}{{R^{2} }}Q_{22} {\mathbf{N}}_{15} + \frac{H}{{2R^{2} }}Q_{22} {\mathbf{N}}_{16} + \frac{H}{{R^{2} }}Q_{22} {\mathbf{N}}_{17} + \frac{H}{R}G{\mathbf{N}}_{18} + HG{\mathbf{N}}_{19} , \\ \end{aligned}$$

, where

$$\begin{aligned} {\mathbf{N}}_{1} & = \frac{1}{{L^{2} }}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\mathbf{w}}\left( t \right)} \right]} } ^{2} \left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{2} & = \int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\mathbf{w}}\left( t \right)} \right]} } ^{2} \left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{3} & = \int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\mathbf{w}}\left( t \right)} \right]\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\mathbf{w}}\left( t \right)} \right]\left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} } } {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{4} & = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\mathbf{w}}\left( t \right)} \right]} } ^{2} \left[ {\frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{5} & = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\mathbf{w}}\left( t \right)} \right]} } ^{2} \left[ {\frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{6} & = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\mathbf{w}}\left( t \right)} \right]} } \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\mathbf{w}}\left( t \right)} \right]\left[ {\frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{7} & = \frac{1}{{L^{3} }}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\mathbf{w}}\left( t \right)} \right]^{3} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} {\text{d}}\xi {\text{d}}\theta } } \\ {\mathbf{N}}_{8} & = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\mathbf{w}}\left( t \right)} \right]^{3} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} {\text{d}}\xi {\text{d}}\theta } } \\ {\mathbf{N}}_{9} & = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left\{ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\mathbf{w}}\left( t \right)\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\mathbf{w}}\left( t \right)} \right]^{2} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} + \frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}w\left( t \right)\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\mathbf{w}}\left( t \right)} \right]^{2} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} } \right\}{\text{d}}\xi {\text{d}}\theta } } \\ {\mathbf{N}}_{{10}} & = \frac{1}{{L^{2} }}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\mathbf{u}}\left( t \right)} \right]} } \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\mathbf{w}}\left( t \right)} \right]\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{{11}} & = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial V\left( {\xi ,\theta } \right)}}{{\partial \theta }}v\left( t \right)} \right]} } \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]{\mathbf{w}}\left( t \right){\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{{12}} & = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}{\mathbf{w}}\left( t \right)} \right]^{2} {\mathbf{W}}^{T} \left( {\xi ,\theta } \right)} } {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{{13}} & = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {{\mathbf{W}}\left( {\xi ,\theta } \right){\mathbf{w}}\left( t \right)} \right]} } \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]{\mathbf{w}}\left( t \right){\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{{14}} & = \int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}u\left( t \right)} \right]} } \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]{\mathbf{w}}\left( t \right){\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{{15}} & = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}v\left( t \right)} \right]} } \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]{\mathbf{w}}\left( t \right){\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{{16}} & = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}{\mathbf{w}}\left( t \right)} \right]^{2} {\mathbf{W}}^{T} \left( {\xi ,\theta } \right){\text{d}}\xi {\text{d}}\theta } } \\ {\mathbf{N}}_{{17}} & = L\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {{\mathbf{W}}\left( {\xi ,\theta } \right){\mathbf{w}}\left( t \right)} \right]\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]{\mathbf{w}}\left( t \right)} } {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{{18}} & = \int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{U}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}u\left( t \right)} \right]\left\{ {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right] + \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]} \right\}{\mathbf{w}}\left( t \right)} } {\text{d}}\xi {\text{d}}\theta \\ {\mathbf{N}}_{{19}} & = \frac{1}{L}\int_{0}^{{2\pi }} {\int_{0}^{1} {\left[ {\frac{{\partial {\mathbf{V}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}v\left( t \right)} \right]\left\{ {\left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]^{T} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right] + \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \xi }}} \right]^{T} \left[ {\frac{{\partial {\mathbf{W}}\left( {\xi ,\theta } \right)}}{{\partial \theta }}} \right]} \right\}{\mathbf{w}}\left( t \right)} } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned}$$

Appendix C

\({\overline{\mathbf{H}}}_{i} \;(i = 1,2, \ldots ,18)\) are given by the following:

$$\begin{aligned} & {\overline{\mathbf{H}}}_{1} = H_{1} /L,\quad {\overline{\mathbf{H}}}_{2} = H_{2} L,\quad {\overline{\mathbf{H}}}_{3} = H_{3} /L,\quad {\overline{\mathbf{H}}}_{4} = H_{4} ,\quad {\overline{\mathbf{H}}}_{5} = H_{5} ,\quad {\overline{\mathbf{H}}}_{6} = H_{6} ,\quad {\overline{\mathbf{H}}}_{7} = H_{7} /L, \\ & {\overline{\mathbf{H}}}_{8} = H_{8} L,\quad {\overline{\mathbf{H}}}_{9} = H_{9} /L,\quad {\overline{\mathbf{H}}}_{10} = H_{10} /L,\quad {\overline{\mathbf{H}}}_{11} = H_{11} /L,\quad {\overline{\mathbf{H}}}_{12} = H_{12} /L,\quad {\overline{\mathbf{H}}}_{13} = H_{13} /L, \\ & {\overline{\mathbf{H}}}_{14} = H_{14} L^{3} ,\quad {\overline{\mathbf{H}}}_{15} = H_{15} L,\quad {\overline{\mathbf{H}}}_{16} = H_{16} /L,\quad {\overline{\mathbf{H}}}_{17} = H_{17} L,\quad {\overline{\mathbf{H}}}_{18} = H_{18} /L. \\ \end{aligned}$$

\({\overline{\mathbf{N}}}_{\lambda }^{{}} \;\left( {\lambda = u,v,w} \right)\) are given by the following:

$$\begin{aligned} {\overline{\mathbf{N}}}_{u} & = \frac{{H_{r} }}{{2L_{r}^{3} }}{\overline{\mathbf{N}}}_{1} { + }\frac{{\mu H_{r} }}{{2L_{r} }}{\overline{\mathbf{N}}}_{2} { + }\frac{{\left( {1 - \mu } \right)H_{r} }}{{2L_{r} }}{\overline{\mathbf{N}}}_{3} , \\ {\overline{\mathbf{N}}}_{v} & = \frac{{\mu H_{r} }}{{2L_{r}^{2} }}{\overline{\mathbf{N}}}_{4} + \frac{{H_{r} }}{2}{\overline{\mathbf{N}}}_{5} + \frac{{\left( {1 - \mu } \right)H_{r} }}{{2L_{r}^{2} }}{\overline{\mathbf{N}}}_{6} , \\ {\overline{\mathbf{N}}}_{w} & = \frac{{H_{r}^{2} }}{{2L_{r}^{4} }}{\overline{\mathbf{N}}}_{7} + \frac{{H_{r}^{2} }}{2}{\overline{\mathbf{N}}}_{8} \, + \frac{{H_{r}^{2} }}{{2L_{r}^{2} }}{\overline{\mathbf{N}}}_{9} + \frac{{H_{r} }}{{L_{r}^{3} }}{\overline{\mathbf{N}}}_{10} + \frac{{\mu H_{r} }}{{L_{r}^{2} }}{\overline{\mathbf{N}}}_{11} + \frac{{\mu H_{r} }}{{2L_{r}^{2} }}{\overline{\mathbf{N}}}_{12} + \frac{{\mu H_{r} }}{{L_{r}^{2} }}{\overline{\mathbf{N}}}_{13} + \frac{{\mu H_{r} }}{{L_{r} }}{\overline{\mathbf{N}}}_{14} \\ & \quad + H_{r} {\overline{\mathbf{N}}}_{15} + \frac{{H_{r} }}{2}{\overline{\mathbf{N}}}_{16} + H_{r} {\overline{\mathbf{N}}}_{17} + \frac{{\left( {1 - \mu } \right)H_{r} }}{{2L_{r} }}{\overline{\mathbf{N}}}_{18} + \frac{{\left( {1 - \mu } \right)H_{r} }}{{2L_{r}^{2} }}{\overline{\mathbf{N}}}_{19} , \\ \end{aligned}$$

, where

$$\begin{aligned} & {\overline{\mathbf{N}}}_{1} = {\mathbf{N}}_{1} L^{2} /H^{2} ,\quad {\overline{\mathbf{N}}}_{2} = {\mathbf{N}}_{2} /H^{2} ,\quad {\overline{\mathbf{N}}}_{3} = {\mathbf{N}}_{3} /H^{2} ,\quad {\overline{\mathbf{N}}}_{4} = {\mathbf{N}}_{4} L/H^{2} ,\quad {\overline{\mathbf{N}}}_{5} = {\mathbf{N}}_{5} /LH^{2} , \\ & {\overline{\mathbf{N}}}_{6} = {\mathbf{N}}_{6} L/H^{2} ,\quad {\overline{\mathbf{N}}}_{7} = {\mathbf{N}}_{7} L^{3} /H^{3} ,\quad {\overline{\mathbf{N}}}_{8} = {\mathbf{N}}_{8} /LH^{3} ,\quad {\overline{\mathbf{N}}}_{9} = {\mathbf{N}}_{9} L/H^{3} ,\quad {\overline{\mathbf{N}}}_{10} = {\mathbf{N}}_{10} L^{2} /H^{2} , \\ & {\overline{\mathbf{N}}}_{11} = {\mathbf{N}}_{11} L/H^{2} ,\quad {\overline{\mathbf{N}}}_{12} = {\mathbf{N}}_{12} L/H^{2} ,\quad {\overline{\mathbf{N}}}_{13} = {\mathbf{N}}_{13} L/H^{2} ,\quad {\overline{\mathbf{N}}}_{14} = {\mathbf{N}}_{14} /H^{2} ,\quad {\overline{\mathbf{N}}}_{15} = {\mathbf{N}}_{15} /LH^{2} , \\ & {\overline{\mathbf{N}}}_{16} = {\mathbf{N}}_{16} /LH^{2} ,\quad {\overline{\mathbf{N}}}_{17} = {\mathbf{N}}_{17} /LH^{2} ,\quad {\overline{\mathbf{N}}}_{18} = {\mathbf{N}}_{18} /H^{2} ,\quad {\overline{\mathbf{N}}}_{19} = {\mathbf{N}}_{19} L/H^{2} . \\ \end{aligned}$$

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Sun, S., Liu, L. Parametric study and stability analysis on nonlinear traveling wave vibrations of rotating thin cylindrical shells. Arch Appl Mech 91, 2833–2851 (2021). https://doi.org/10.1007/s00419-021-01934-0

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