Appendix 1: The Partial Differential Operators \(L_{jL}\) and \(L_{jR}\) of Eq. (15)
$$\begin{aligned} L_{1L} = & \, A_{11} \frac{{\partial^{2} u_{0} }}{{\partial x^{2} }} + \frac{1}{{R^{2} }}A_{66} \frac{{\partial^{2} u_{0} }}{{\partial \theta^{2} }} + \frac{1}{R}A_{11} \frac{{\partial u_{0} }}{\partial x}\sin \beta - \frac{1}{{R^{2} }}A_{22} u_{0} \cos^{2} \beta + \frac{1}{R}(A_{12} + A_{66} )\frac{{\partial^{2} v_{0} }}{\partial x\partial \theta } \\ & \quad - \frac{1}{{R^{2} }}(A_{22} + A_{66} )\frac{{\partial v_{0} }}{\partial \theta }\sin \beta + \frac{1}{{R^{2} }}A_{66} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\frac{{\partial w_{0} }}{\partial x} + \frac{1}{{R^{2} }}(A_{12} + A_{66} )\frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\frac{{\partial w_{0} }}{\partial \theta } + B_{11} \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} \\ & \quad + A_{11} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\frac{{\partial w_{0} }}{\partial x} + \frac{1}{R}(B_{12} + B_{66} )\frac{{\partial^{2} \phi_{\theta } }}{\partial x\partial \theta } + \frac{1}{2R}(A_{11} - A_{12} )\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\sin \beta + \frac{1}{R}A_{12} \frac{{\partial w_{0} }}{\partial x}\cos \beta \\ & \quad - \frac{1}{{2R^{3} }}(A_{12} - A_{22} )\frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\sin \beta - \frac{1}{{R^{2} }}(B_{22} + B_{66} )\frac{{\partial \phi_{\theta } }}{\partial \theta }\sin \beta - \frac{1}{{R^{2} }}A_{22} w_{0} \sin \beta \cos \beta \\ & \quad + \frac{1}{{R^{2} }}B_{66} \frac{{\partial^{2} \phi_{x} }}{{\partial \theta^{2} }} + \frac{1}{R}B_{11} \frac{{\partial \phi_{x} }}{\partial x}\sin \beta - \frac{1}{{R^{2} }}B_{22} \phi_{x} \sin^{2} \beta + \frac{1}{R}(N_{xx}^{T} - N_{\theta \theta }^{T} )\sin \beta , \\ \end{aligned}$$
(20)
$$L_{1R} = I_{0} \ddot{u}_{0} + I_{1} \ddot{\phi }_{x} ,$$
(21)
$$\begin{aligned} L_{2L} = & \frac{1}{R}(A_{12} + A_{66} )\frac{{\partial^{2} u_{0} }}{\partial x\partial \theta } + \frac{1}{{R^{2} }}(A_{22} + A_{66} )\frac{{\partial u_{0} }}{\partial \theta }\sin \beta + A_{66} \frac{{\partial^{2} v_{0} }}{{\partial x^{2} }} + \frac{1}{{R^{2} }}KA_{44} \frac{{\partial w_{0} }}{\partial \theta }\cos \beta \\ & \quad - \frac{1}{{R^{2} }}\left( {KA_{44} \cos^{2} \beta + A_{66} \sin^{2} \beta } \right)v_{0} + \frac{1}{R}A_{66} \frac{{\partial w_{0} }}{\partial \theta }\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }} + \frac{1}{{R^{3} }}A_{22} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\frac{{\partial w_{0} }}{\partial \theta } \\ & \quad + \frac{1}{{R^{2} }}A_{22} \frac{{\partial^{2} v_{0} }}{{\partial \theta^{2} }} + \frac{1}{R}A_{66} \frac{{\partial v_{0} }}{\partial x}\sin \beta + \frac{1}{{R^{2} }}A_{66} \frac{{\partial w_{0} }}{\partial \theta }\frac{{\partial w_{0} }}{\partial x}\sin \beta + \frac{1}{{R^{2} }}A_{22} \frac{{\partial w_{0} }}{\partial \theta }\cos \beta \\ & \quad + \frac{1}{R}\left( {A_{12} + A_{66} } \right)\frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\frac{{\partial w_{0} }}{\partial x} + \frac{1}{R}\left( {B_{12} + B_{66} } \right)\frac{{\partial^{2} \phi_{x} }}{\partial x\partial \theta } + \frac{1}{{R^{2} }}\left( {B_{22} + B_{66} } \right)\frac{{\partial \phi_{x} }}{\partial \theta }\sin \beta \\ & \quad + \frac{1}{R}B_{66} \frac{{\partial \phi_{\theta } }}{\partial x}\sin \beta + \frac{1}{{R^{2} }}B_{22} \frac{{\partial^{2} \phi_{\theta } }}{{\partial \theta^{2} }} + \left( {\frac{1}{R}KA_{44} \cos \beta - \frac{1}{{R^{2} }}B_{66} \sin^{2} \beta } \right)\phi_{\theta } , \\ \end{aligned}$$
(22)
$$L_{2R} = I_{0} \ddot{v}_{0} + I_{1} \ddot{\phi }_{\theta },$$
(23)
$$\begin{aligned} L_{3L} = & \frac{2}{{R^{2} }}B_{66} \frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\frac{{\partial \phi_{x} }}{\partial \theta } + \frac{1}{2R}A_{11} \left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{3} \sin \beta - \frac{1}{R}N_{\theta \theta }^{T} \cos \beta + \frac{1}{R}\left( {B_{11} + B_{22} } \right)\frac{{\partial \phi_{x} }}{\partial x}\frac{{\partial w_{0} }}{\partial x} \\ & \quad + A_{11} \frac{{\partial^{2} u_{0} }}{{\partial x^{2} }}\frac{{\partial w_{0} }}{\partial x} - \frac{1}{{R^{2} }}A_{66} \frac{{\partial v_{0} }}{\partial \theta }\frac{{\partial w_{0} }}{\partial x}\sin \beta - \frac{1}{{R^{3} }}(A_{12} + A_{66} )\frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\frac{{\partial w_{0} }}{\partial x} + B_{11} \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }}\frac{{\partial w_{0} }}{\partial x} \\ & \quad + \frac{2}{{R^{2} }}(A_{12} + 2A_{66} )\frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\frac{{\partial w_{0} }}{\partial \theta }\frac{{\partial w_{0} }}{\partial x} + \frac{1}{R}(B_{12} + B_{66} )\frac{{\partial^{2} \phi_{\theta } }}{\partial x\partial \theta }\frac{{\partial w_{0} }}{\partial x} + \frac{1}{{R^{2} }}A_{66} \frac{{\partial^{2} u_{0} }}{{\partial \theta^{2} }}\frac{{\partial w_{0} }}{\partial x} \\ & \quad + \frac{1}{{2R^{2} }}(A_{12} + 2A_{66} )\frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} + \frac{1}{R}\left( {KA_{55} + N_{xx}^{T} } \right)\frac{{\partial w_{0} }}{\partial x}\sin \beta + \frac{3}{2}A_{11} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} \\ & \quad - \frac{1}{{R^{2} }}B_{66} \frac{{\partial \phi_{\theta } }}{\partial x}\frac{{\partial w_{0} }}{\partial x}\sin \beta + \frac{1}{2R}A_{12} \left( {\frac{{\partial w_{0} }}{\partial x}} \right)^{2} \cos \beta + \frac{1}{R}(A_{11} + A_{12} )\frac{{\partial w_{0} }}{\partial x}\frac{{\partial u_{0} }}{\partial x}\sin \beta \\ & \quad + \frac{1}{R}A_{12} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}w_{0} \cos \beta + \frac{1}{{R^{2} }}A_{12} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\frac{{\partial u_{0} }}{\partial x} + A_{11} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\frac{{\partial u_{0} }}{\partial x} + \frac{1}{R}B_{12} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\phi_{x} \sin \beta \\ & \quad + \frac{2}{R}A_{66} \frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\frac{{\partial v_{0} }}{\partial x} - \frac{1}{{R^{2} }}A_{66} \frac{{\partial w_{0} }}{\partial \theta }\frac{{\partial v_{0} }}{\partial x} + \frac{1}{{R^{3} }}A_{22} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}w_{0} \cos \beta - A_{12} \frac{{\partial u_{0} }}{\partial x}\cos \beta \\ & \quad - \frac{1}{{R^{2} }}A_{22} w_{0} \cos \beta + \frac{1}{{R^{3} }}B_{22} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\phi_{x} \sin \beta + \frac{1}{R}(A_{12} + A_{66} )\frac{{\partial^{2} v_{0} }}{\partial x\partial \theta }\frac{{\partial w_{0} }}{\partial x} \\ & \quad + \left( {\frac{1}{R}KA_{55} - \frac{1}{{R^{2} }}B_{22} \cos \beta } \right)\varphi_{x} \sin \beta + \frac{1}{{R^{3} }}B_{66} \frac{{\partial w_{0} }}{\partial \theta }\varphi_{\theta } \sin^{2} \beta - \frac{2}{{R^{2} }}B_{66} \frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\varphi_{\theta } \sin \beta \\ & \quad + \frac{1}{{R^{3} }}A_{22} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}u_{0} \sin \beta - \frac{1}{{R^{2} }}A_{22} u_{0} \sin \beta \cos \beta + \frac{1}{{R^{3} }}\left( {B_{22} - B_{66} } \right)\frac{{\partial \phi_{x} }}{\partial \theta }\frac{{\partial w_{0} }}{\partial \theta } - \gamma \dot{w}_{0} \\ & \quad - \frac{2}{{R^{2} }}A_{66} \frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }v_{0} \sin \beta + \left( {KA_{55} + N_{xx}^{T} - (p_{0} + p_{1} \cos \varOmega_{2} t)} \right)\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }} + \frac{1}{R}A_{66} \frac{{\partial^{2} v_{0} }}{{\partial x^{2} }}\frac{{\partial w_{0} }}{\partial \theta } \\ & \quad + \frac{1}{{R^{3} }}A_{66} \frac{{\partial^{2} v_{0} }}{{\partial \theta^{2} }} + \frac{1}{R}A_{12} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}u_{0} \sin \beta + \frac{1}{{R^{3} }}A_{66} \frac{{\partial w_{0} }}{\partial \theta }v_{0} \sin^{2} \beta + \frac{1}{R}B_{66} \frac{{\partial^{2} \phi_{\theta } }}{{\partial x^{2} }}\frac{{\partial w_{0} }}{\partial \theta } \\ & \quad + \frac{1}{{R^{2} }}\left( {KA_{44} + N_{\theta \theta }^{T} } \right)\frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }} + \frac{1}{R}A_{12} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\frac{{\partial v_{0} }}{\partial \theta } + \frac{1}{{R^{3} }}A_{22} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\frac{{\partial v_{0} }}{\partial \theta } + KA_{55} \frac{{\partial \phi_{x} }}{\partial x}\frac{{\partial w_{0} }}{\partial \theta } \\ & \quad - \frac{1}{{R^{2} }}(KA_{44} + A_{22} )\frac{{\partial v_{0} }}{\partial \theta }\cos \beta + B_{11} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\frac{{\partial \varphi_{x} }}{\partial x} - \frac{1}{R}B_{12} \frac{{\partial \varphi_{x} }}{\partial x}\cos \beta + \frac{1}{{R^{2} }}B_{12} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\frac{{\partial \varphi_{x} }}{\partial x} \\ & \quad + \frac{3}{{2R^{4} }}A_{22} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\left( {\frac{{\partial w_{0} }}{\partial \theta }} \right)^{2} + \frac{2}{{R^{2} }}A_{66} \frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\frac{{\partial u_{0} }}{\partial \theta } + \frac{1}{{R^{2} }}\left( {B_{12} + B_{66} } \right)\frac{{\partial^{2} \phi_{x} }}{\partial x\partial \theta }\frac{{\partial w_{0} }}{\partial \theta } \\ & \quad + \frac{1}{{R^{2} }}B_{66} \frac{{\partial^{2} \phi_{x} }}{{\partial \theta^{2} }}\frac{{\partial w_{0} }}{\partial x} + \frac{1}{{2R^{3} }}A_{22} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\cos \beta + \frac{2}{R}B_{66} \frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\frac{{\partial \phi_{\theta } }}{\partial x} + \frac{1}{{R^{3} }}B_{22} \frac{{\partial^{2} \phi_{\theta } }}{{\partial \theta^{2} }}\frac{{\partial w_{0} }}{\partial \theta } \\ & \quad + \frac{1}{{R^{2} }}(A_{12} + A_{66} )\frac{{\partial^{2} u_{0} }}{\partial x\partial \theta }\frac{{\partial w_{0} }}{\partial \theta } - \frac{1}{{R^{2} }}B_{66} \frac{{\partial w_{0} }}{\partial \theta }\frac{{\partial \varphi_{\theta } }}{\partial x}\sin \beta + \frac{1}{{R^{3} }}(A_{22} - A_{66} )\frac{{\partial w_{0} }}{\partial \theta }\frac{{\partial u_{0} }}{\partial \theta }\sin \beta \\ & \quad + \frac{1}{2R}(A_{12} + 2A_{66} )\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\left( {\frac{{\partial w_{0} }}{\partial \theta }} \right)^{2} + F\cos \varOmega_{1} t, \\ \end{aligned}$$
(24)
$$L_{3R} = I_{0} \ddot{w}_{0} ,$$
(25)
$$\begin{aligned} L_{4L} = & \, B_{11} \frac{{\partial^{2} u_{0} }}{{\partial x^{2} }} + \frac{1}{{R^{2} }}B_{66} \frac{{\partial^{2} u_{0} }}{{\partial \theta^{2} }} + \frac{1}{R}B_{11} \frac{{\partial u_{0} }}{\partial x}\sin \beta - \frac{1}{{R^{2} }}B_{22} u_{0} \sin^{2} \beta + \frac{1}{R}(D_{12} + D_{66} )\frac{{\partial^{2} \phi_{x} }}{\partial x\partial \theta } \\ & \quad + \frac{1}{R}(B_{12} + B_{66} )\frac{{\partial^{2} v_{0} }}{\partial x\partial \theta } - \frac{1}{{R^{2} }}(B_{22} + B_{66} )\frac{{\partial v_{0} }}{\partial \theta }\sin \beta + \frac{1}{2R}(B_{11} - B_{12} )\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\sin \beta \\ & \quad + \frac{1}{R}B_{12} \frac{{\partial w_{0} }}{\partial x}\cos \beta + B_{11} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\frac{{\partial w_{0} }}{\partial x} + \frac{1}{{R^{2} }}B_{66} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\frac{{\partial w_{0} }}{\partial x} - KA_{55} \frac{{\partial w_{0} }}{\partial x} + \frac{1}{{R^{2} }}D_{66} \frac{{\partial^{2} \phi_{x} }}{{\partial \theta^{2} }} \\ & \quad - \frac{1}{{R^{2} }}B_{22} w_{0} \sin \beta \cos \beta + \frac{1}{R}D_{11} \frac{{\partial \phi_{x} }}{\partial x}\sin \beta - \frac{1}{{2R^{2} }}(B_{12} + B_{22} )\frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\sin \beta + D_{11} \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} \\ & \quad + \frac{1}{{R^{2} }}(B_{12} + B_{66} )\frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\frac{{\partial w_{0} }}{\partial \theta } - \left( {KA_{55} + \frac{1}{{R^{2} }}D_{22} } \right)\phi_{x} - \frac{1}{{R^{2} }}(D_{22} + D_{66} )\frac{{\partial \phi_{\theta } }}{\partial \theta }\sin \beta \\ & \quad + \frac{1}{R}M_{xx}^{T} \sin \beta - \frac{1}{R}M_{\theta \theta }^{T} \sin \beta , \\ \end{aligned}$$
(26)
$$L_{4R} = I_{1} \ddot{u}_{0} + I_{2} \ddot{\phi }_{x},$$
(27)
$$\begin{aligned} L_{5L} = & \frac{1}{R}(B_{12} + B_{66} )\frac{{\partial^{2} u_{0} }}{\partial x\partial \theta } + \frac{1}{{R^{2} }}(B_{22} + B_{66} )\frac{{\partial u_{0} }}{\partial \theta }\sin \beta + B_{66} \frac{{\partial^{2} v_{0} }}{{\partial x^{2} }} + \frac{1}{{R^{2} }}B_{22} \frac{{\partial^{2} v_{0} }}{{\partial \theta^{2} }} \\ & \quad + \left( {\frac{1}{R}KA_{44} \cos \beta - \frac{1}{{R^{2} }}B_{66} \sin^{2} \beta } \right)v_{0} + \frac{1}{{R^{2} }}\left( {D_{22} + D_{66} } \right)\frac{{\partial \phi_{\theta } }}{\partial \theta } + \frac{1}{{R^{2} }}B_{66} \frac{{\partial w_{0} }}{\partial x}\frac{{\partial w_{0} }}{\partial \theta }\sin \beta \\ & \quad + \left( {\frac{1}{{R^{2} }}B_{22} \cos \beta - \frac{1}{R}KA_{44} } \right)\frac{{\partial w_{0} }}{\partial \theta } + \frac{1}{{R^{3} }}B_{22} \frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}\frac{{\partial w_{0} }}{\partial \theta } + \frac{1}{R}B_{66} \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}\frac{{\partial w_{0} }}{\partial \theta } + \frac{1}{{R^{2} }}D_{22} \frac{{\partial^{2} \varphi_{\theta } }}{{\partial \theta^{2} }} \\ & \quad + \frac{1}{R}B_{66} \frac{{\partial v_{0} }}{\partial x}\sin \beta - \left( {KA_{44} + \frac{1}{{R^{2} }}D_{66} \sin^{2} \beta } \right)\phi_{\theta } + \frac{1}{R}\left( {B_{12} + B_{66} } \right)\frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }\frac{{\partial w_{0} }}{\partial x} \\ & \quad + \frac{1}{R}(D_{12} + D_{66} )\frac{{\partial^{2} \phi_{x} }}{\partial x\partial \theta } + D_{66} \frac{{\partial^{2} \phi_{\theta } }}{{\partial x^{2} }} + \frac{1}{R}D_{66} \frac{{\partial \phi_{\theta } }}{\partial x}, \\ \end{aligned}$$
(28)
$$L_{5R} = I_{1} \ddot{v}_{0} + I_{2} \ddot{\phi }_{\theta }.$$
(29)
Appendix 2: Matrixes and Vectors of Eq. (19)
$$\varvec{W} = \left[ {\begin{array}{*{20}c} {w_{1} } & {w_{2} } \\ \end{array} } \right]^{T} ,\;\varvec{\mu}= \left[ {\begin{array}{*{20}c} {\mu_{1} } & 0 \\ 0 & {\mu_{2} } \\ \end{array} } \right],$$
(30)
$$\varvec{M} = \left[ {\begin{array}{*{20}c} {\xi_{10} } & 0 \\ 0 & {\xi_{20} } \\ \end{array} } \right],\,\varvec{P} = \left[ {\begin{array}{*{20}c} {\xi_{18} } & 0 \\ 0 & {\xi_{28} } \\ \end{array} } \right](p_{0} + p_{1} \cos \varOmega_{2} t),$$
(31)
$$\varvec{NL} = \xi \left[ {\begin{array}{*{20}c} {w_{1}^{3} } & {w_{1}^{2} w_{2} } & {w_{1}^{2} } & {w_{1} w_{2} } & {w_{2}^{2} } & {w_{1} w_{2}^{2} } & {w_{2}^{3} } \\ \end{array} } \right]^{\text{T}} ,$$
(32)
where
$$\xi = \left[ {\begin{array}{*{20}c} {\xi_{11} } & {\xi_{12} } & {\xi_{13} } & {\xi_{14} } & {\xi_{15} } & {\xi_{16} } & {\xi_{17} } \\ {\xi_{21} } & {\xi_{22} } & {\xi_{23} } & {\xi_{24} } & {\xi_{25} } & {\xi_{26} } & {\xi_{27} } \\ \end{array} } \right],$$
(33)
$$\varvec{C} = \left[ {\begin{array}{*{20}c} {\xi_{1} } & {\xi_{2} } \\ \end{array} } \right]^{\text{T}} ,\;\varvec{F} = \left[ {\begin{array}{*{20}c} {\xi_{19} F_{1} } & 0 \\ 0 & {\xi_{29} F_{2} } \\ \end{array} } \right].$$
(34)
