Research on nonlinear vibration control of laminated cylindrical shells with discontinuous piezoelectric layer

Abstract

A model of laminated cylindrical shells with discontinuous piezoelectric layer is proposed. Based on the first-order shear nonlinear shell theory, the nonlinear vibration control of the piezoelectric laminated cylindrical shell model with point-supported elastic boundary condition is analyzed. In this model, a series of artificial springs are introduced to simulate the arbitrary boundary conditions. And the elastic-electrically coupled differential equations of piezoelectric laminated cylindrical shells are obtained by the Chebyshev polynomials and Lagrange equations and decoupled by using the negative velocity feedback adjustment. Then, the frequency–amplitude responses of the piezoelectric laminated cylindrical shells are obtained by the incremental harmonic balance method. Finally, the influence of the constant gain, size, and position of the piezoelectric layer on the nonlinear amplitude–frequency response is investigated. The results show that the constant gain and the position and size of the piezoelectric layer have a significant influence on the amplitude of the nonlinear amplitude–frequency and time–frequency response.

Appendices

Appendix A: Expressions for the generalized coordinates

$$ \begin{aligned} {\varvec{q}}_{u} & =\left[ {a_{11} \, a_{12} \, \cdot \cdot \cdot a_{1n} \, a_{21} \, a_{22} \, \cdot \cdot \cdot a_{2n} \, \cdot \cdot \cdot a_{mn} \, {\mathbf{0}}} \right]^{{\text{T}}} e^{ - j\omega t} \\ {\varvec{q}}_{v} & =\left[ {b_{11} \, b_{12} \, \cdot \cdot \cdot b_{1n} \, b_{21} \, b_{22} \, \cdot \cdot \cdot b_{2n} \, \cdot \cdot \cdot b_{mn} \, {\mathbf{0}}} \right]^{{\text{T}}} e^{ - j\omega t} \\ {\varvec{q}}_{w} & =\left[ {c_{11} \, c_{12} \, \cdot \cdot \cdot c_{1n} \, c_{21} \, c_{22} \, \cdot \cdot \cdot c_{2n} \, \cdot \cdot \cdot c_{mn} \, c_{1} \, c_{2} \, \cdot \cdot \cdot c_{m} } \right]^{{\text{T}}} e^{ - j\omega t} \\ {\varvec{q}}_{{\phi_{x} }} & =\left[ {d_{11} \, d_{12} \, \cdot \cdot \cdot d_{1n} \, d_{21} \, d_{22} \, \cdot \cdot \cdot d_{2n} \, \cdot \cdot \cdot d_{mn} \, {\mathbf{0}}} \right]^{{\text{T}}} e^{ - j\omega t} \\ {\varvec{q}}_{{\phi_{\theta } }} & =\left[ {e_{11} \, e_{12} \, \cdot \cdot \cdot e_{1n} \, e_{21} \, e_{22} \, \cdot \cdot \cdot e_{2n} \, \cdot \cdot \cdot e_{mn} \, {\mathbf{0}}} \right]^{{\text{T}}} e^{ - j\omega t} \\ {\varvec{q}}_{{\psi_{\text{s}} }} & =\left[ {f_{11} \, f_{12} \, \cdot \cdot \cdot f_{1n} \, f_{21} \, f_{22} \, \cdot \cdot \cdot f_{2n} \, \cdot \cdot \cdot f_{mn} \, {\mathbf{0}}} \right]^{{\text{T}}} e^{ - j\omega t} \\ {\varvec{q}}_{{\psi_{\text{a}} }} & =\left[ {g_{11} \, g_{12} \, \cdot \cdot \cdot g_{1n} \, g_{21} \, g_{22} \, \cdot \cdot \cdot g_{2n} \, \cdot \cdot \cdot g_{mn} \, {\mathbf{0}}} \right]^{{\text{T}}} e^{ - j\omega t} \\ \end{aligned} $$

(33)

where 0 is a 1 × m zero vector, so that all the vectors ${\varvec{q}}_{u} ,{\varvec{q}}_{v} \cdots {\varvec{q}}_{{\psi_{{\text{a}}} }}$ have the same dimension.

Appendix B: Expressions for the mass matrix M

The generalized mass matrix of the piezoelectric laminated shell is expressed as

$$ \user2{M}_{{qq}} = \left[ {\begin{array}{*{20}l} {\user2{M}^{{uu}} } \hfill & 0 \hfill & 0 \hfill & {\frac{1}{2}\user2{M}^{{u\phi _{x} }} } \hfill & 0 \hfill \\ 0 \hfill & {\user2{M}^{{vv}} } \hfill & 0 \hfill & 0 \hfill & {\frac{1}{2}\user2{M}^{{v\phi _{\theta } }} } \hfill \\ 0 \hfill & 0 \hfill & {\user2{M}^{{ww}} } \hfill & 0 \hfill & 0 \hfill \\ {\frac{1}{2}\user2{M}^{{u\phi _{x} \text{T}}} } \hfill & 0 \hfill & 0 \hfill & {\user2{M}^{{\phi _{x} \phi _{x} }} } \hfill & 0 \hfill \\ 0 \hfill & {\frac{1}{2}\user2{M}^{{v\phi _{\theta } \text{T}}} } \hfill & 0 \hfill & 0 \hfill & {\user2{M}^{{\phi _{\theta } \phi _{\theta } }} } \hfill \\ \end{array} } \right] $$

(34)

where

$$ \begin{aligned} {\varvec{M}}^{uu} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\user2{\overline{U}\overline{U}}^{\text{T}} } } {\text{d}}\xi {\text{d}}\theta } \cdot I_{0} {\kern 1pt} {\kern 1pt} + LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta^{\prime}}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\user2{\overline{U}\overline{U}}^{\text{T}} } \text{d}\xi \text{d}\theta } \cdot \widetilde{I}_{0} {\kern 1pt} } \\ {\varvec{M}}^{uu} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\user2{\overline{U}\overline{U}}^{\text{T}} } } {\text{d}}\xi {\text{d}}\theta } \cdot I_{0} {\kern 1pt} {\kern 1pt} + LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta^{\prime}}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\user2{\overline{U}\overline{U}}^{\text{T}} } \text{d}\xi \text{d}\theta } \cdot \widetilde{I}_{0} {\kern 1pt} } \\ {\varvec{M}}^{{u\phi_{x} }} & = {2}LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\user2{\overline{U}\overline{\Phi }}_{x}^{\text{T}} } } {\text{d}}\xi {\text{d}}\theta } \cdot I_{1} {\kern 1pt} {\kern 1pt} + 2LR{\kern 1pt} \sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta^{\prime}}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\user2{\overline{U}\overline{\Phi }}_{x}^{\text{T}} } \text{d}\xi \text{d}\theta } \cdot \widetilde{I}_{1} {\kern 1pt} } \\ {\varvec{M}}^{vv} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\user2{\overline{V}\overline{V}}^{\text{T}} } } {\text{d}}\xi {\text{d}}\theta } \cdot I_{0} {\kern 1pt} {\kern 1pt} + LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta^{\prime}}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\user2{\overline{V}\overline{V}}^{\text{T}} } \text{d}\xi \text{d}\theta } \cdot \widetilde{I}_{0} {\kern 1pt} } \\ {\varvec{M}}^{{v\phi_{\theta } }} & = {2}LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\user2{\overline{U}\overline{\Phi }}_{\theta }^{\text{T}} } } {\text{d}}\xi {\text{d}}\theta } \cdot I_{1} {\kern 1pt} {\kern 1pt} + 2LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta^{\prime}}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\user2{\overline{U}\overline{\Phi }}_{\theta }^{\text{T}} } \text{d}\xi \text{d}\theta } \cdot \widetilde{I}_{1} {\kern 1pt} } \\ {\varvec{M}}^{ww} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\user2{\overline{W}\overline{W}}^{\text{T}} } } {\text{d}}\xi {\text{d}}\theta } \cdot I_{0} {\kern 1pt} {\kern 1pt} + LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta^{\prime}}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\user2{\overline{W}\overline{W}}^{\text{T}} } \text{d}\xi \text{d}\theta } \cdot \widetilde{I}_{0} {\kern 1pt} } \\ {\varvec{M}}^{{\phi_{x} \phi_{x} }} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\overline{\user2{\Phi }}_{x} \overline{\user2{\Phi }}_{x}^{\text{T}} } } {\text{d}}\xi {\text{d}}\theta } \cdot I_{2} {\kern 1pt} {\kern 1pt} + LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta^{\prime}}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\overline{\user2{\Phi }}_{x} \overline{\user2{\Phi }}_{x}^{\text{T}} } \text{d}\xi \text{d}\theta } \cdot \widetilde{I}_{2} {\kern 1pt} } \\ {\varvec{M}}^{{\phi_{\theta } \phi_{\theta } }} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\overline{\user2{\Phi }}_{\theta } \overline{\user2{\Phi }}_{\theta }^{\text{T}} } } {\text{d}}\xi {\text{d}}\theta } \cdot I_{2} {\kern 1pt} {\kern 1pt} + LR{\kern 1pt} \sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta^{\prime}}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\overline{\user2{\Phi }}_{\theta } \overline{\user2{\Phi }}_{\theta }^{\text{T}} } \text{d}\xi \text{d}\theta } \cdot \widetilde{I}_{2} {\kern 1pt} } \\ \end{aligned} $$

Appendix C: Expressions of the stiffness matrix K

The generalized stiffness matrix of the piezoelectric laminated shell is expressed as

$$ \user2{K}_{{qq}} {\text{ = }}\left[ {\begin{array}{*{20}l} {\user2{K}^{{uu}} } \hfill & {\frac{1}{2}\user2{K}^{{uv}} } \hfill & {\frac{1}{2}\user2{K}^{{uw}} } \hfill & {\frac{1}{2}\user2{K}^{{u\phi _{x} }} } \hfill & {\frac{1}{2}\user2{K}^{{u\phi _{\theta } }} } \hfill \\ {\frac{1}{2}\user2{K}^{{uv{T}}} } \hfill & {\user2{K}^{{vv}} } \hfill & {\frac{1}{2}\user2{K}^{{vw}} } \hfill & {\frac{1}{2}\user2{K}^{{v\phi _{x} }} } \hfill & {\frac{1}{2}\user2{K}^{{v\phi _{\theta } }} } \hfill \\ {\frac{1}{2}\user2{K}^{{uw{T}}} } \hfill & {\frac{1}{2}\user2{K}^{{vw{T}}} } \hfill & {\user2{K}^{{ww}} } \hfill & {\frac{1}{2}\user2{K}^{{w\phi _{x} }} } \hfill & {\frac{1}{2}\user2{K}^{{w\phi _{\theta } }} } \hfill \\ {\frac{1}{2}\user2{K}^{{u\phi _{x} {T}}} } \hfill & {\frac{1}{2}\user2{K}^{{v\phi _{x} {T}}} } \hfill & {\frac{1}{2}\user2{K}^{{w\phi _{x} {T}}} } \hfill & {\user2{K}^{{\phi _{x} \phi _{x} }} } \hfill & {\frac{1}{2}\user2{K}^{{\phi _{x} \phi _{\theta } }} } \hfill \\ {\frac{1}{2}\user2{K}^{{u\phi _{\theta } {T}}} } \hfill & {\frac{1}{2}\user2{K}^{{v\phi _{\theta } {T}}} } \hfill & {\frac{1}{2}\user2{K}^{{w\phi _{\theta } {T}}} } \hfill & {\frac{1}{2}\user2{K}^{{\phi _{x} \phi _{\theta } {T}}} } \hfill & {\user2{K}^{{\phi _{\theta } \phi _{\theta } }} } \hfill \\ \end{array} } \right]$$

(35)

where

$$ \begin{aligned} {\varvec{K}}^{uu} & =LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{A_{11} }}{{L^{2} }}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\frac{{\partial \overline{\user2{U}}^{\text{T}} }}{\partial \xi } + \frac{{A_{{{66}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{U}}}}{\partial \theta }\frac{{\partial \overline{\user2{U}}^{\text{T}} }}{\partial \theta }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{\tilde{A}_{11} }}{{L^{2} }}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\frac{{\partial \overline{\user2{U}}^{\text{T}} }}{\partial \xi } + \frac{{\tilde{A}_{{{66}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{U}}}}{\partial \theta }\frac{{\partial \overline{\user2{U}}^{\text{T}} }}{\partial \theta }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{uv} & =LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{2{\mkern 1mu} A_{12} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\frac{{\partial \overline{\user2{V}}^{\text{T}} }}{\partial \theta } + \frac{{2{\mkern 1mu} A_{{{66}}} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{U}}}}{\partial \theta }\frac{{\partial \overline{\user2{V}}^{\text{T}} }}{\partial \xi }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{2{\mkern 1mu} \tilde{A}_{12} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\frac{{\partial \overline{\user2{V}}^{\text{T}} }}{\partial \theta } + \frac{{2{\mkern 1mu} \tilde{A}_{{{66}}} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{U}}}}{\partial \theta }\frac{{\partial \overline{\user2{V}}^{\text{T}} }}{\partial \xi }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ {\varvec{K}}^{uw} =LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{2{\mkern 1mu} A_{12} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\overline{\user2{W}}^{\text{T}} } \right)} } {\text{d}}\xi {\text{d}}\theta } + LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{2{\mkern 1mu} \tilde{A}_{12} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\overline{\user2{W}}^{\text{T}} } \right)} } } {\text{d}}\xi {\text{d}}\theta $$

$$ \begin{aligned} {\varvec{K}}^{{u\phi_{x} }} & =LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{2B_{11} }}{{L^{2} }}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \xi } + {\mkern 1mu} \frac{{2B_{66} }}{{R^{2} }}\frac{{\partial \overline{\user2{U}}}}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \theta }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{2\tilde{B}_{11} }}{{L^{2} }}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \xi } + {\mkern 1mu} \frac{{2\tilde{B}_{66} }}{{R^{2} }}\frac{{\partial \overline{\user2{U}}}}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \theta }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{{u\phi_{\theta } }} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{2B_{12} }}{LR}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta } + {\mkern 1mu} \frac{{2B_{66} }}{LR}\frac{{\partial \overline{\user2{U}}}}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \xi }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( { + \frac{{2\tilde{B}_{12} }}{LR}\frac{{\partial \overline{\user2{U}}}}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta } + {\mkern 1mu} \frac{{2\tilde{B}_{66} }}{LR}\frac{{\partial \overline{\user2{U}}}}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \xi }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{vv} & =LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{A_{{{22}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \theta }\frac{{\partial \overline{\user2{V}}^{\text{T}} }}{\partial \theta }{\mkern 1mu} + \frac{{k_{c} A_{{{44}}} }}{{R^{2} }}\user2{\overline{V}\overline{V}}^{\text{T}} + \frac{{A_{{{66}}} }}{{L^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \xi }\frac{{\partial \overline{\user2{V}}^{\text{T}} }}{\partial \xi }{\mkern 1mu} {\mkern 1mu} } \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{\tilde{A}_{{{22}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \theta }\frac{{\partial \overline{\user2{V}}^{\text{T}} }}{\partial \theta }{\mkern 1mu} + \frac{{k_{c} \tilde{A}_{{{44}}} }}{{R^{2} }}\user2{\overline{V}\overline{V}}^{\text{T}} + \frac{{\tilde{A}_{{{66}}} }}{{L^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \xi }\frac{{\partial \overline{\user2{V}}^{\text{T}} }}{\partial \xi }{\mkern 1mu} } \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{gathered} {\varvec{K}}^{vw} =LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{2A_{{{22}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \theta }\overline{\user2{W}}^{\text{T}} - \frac{{2k_{c} A_{{{44}}} }}{{R^{2} }}\overline{\user2{V}}\frac{{\partial \overline{\user2{W}}^{\text{T}} }}{\partial \theta }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ + LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{2\tilde{A}_{{{22}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \theta }\overline{\user2{W}}^{\text{T}} - \frac{{2k_{c} \tilde{A}_{{{44}}} }}{{R^{2} }}\overline{\user2{V}}\frac{{\partial \overline{\user2{W}}^{\text{T}} }}{\partial \theta }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{gathered} $$

$$ \begin{aligned} {\varvec{K}}^{{v\phi_{x} }} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {{\mkern 1mu} \frac{{2B_{12} }}{LR}\frac{{\partial \overline{\user2{V}}}}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \xi } + {\mkern 1mu} \frac{{2B_{66} }}{LR}\frac{{\partial \overline{\user2{V}}}}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \theta }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {{\mkern 1mu} \frac{{2\tilde{B}_{12} }}{LR}\frac{{\partial \overline{\user2{V}}}}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \xi } + {\mkern 1mu} \frac{{2\tilde{B}_{66} }}{LR}\frac{{\partial \overline{\user2{V}}}}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \theta }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{{v\phi_{\theta } }} & =LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {{\mkern 1mu} \frac{{2B_{22} }}{{R^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta } + {\mkern 1mu} \frac{{2B_{66} }}{{L^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \xi } - \frac{{2k_{c} A_{{{44}}} }}{R}\user2{\overline{V}\overline{\Phi }}_{\theta }^{\text{T}} } \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {{\mkern 1mu} \frac{{2\tilde{B}_{22} }}{{R^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta } + {\mkern 1mu} \frac{{2\tilde{B}_{66} }}{{L^{2} }}\frac{{\partial \overline{\user2{V}}}}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \xi } - \frac{{2k_{c} \tilde{A}_{{{44}}} }}{R}\user2{\overline{V}\overline{\Phi }}_{\theta }^{\text{T}} } \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{ww} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{k_{c} A_{{{44}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{W}}}}{\partial \theta }\frac{{\partial \overline{\user2{W}}^{\text{T}} }}{\partial \theta } + \frac{{k_{c} A_{55} }}{{L^{2} }}\frac{{\partial \overline{\user2{W}}}}{\partial \xi }\frac{{\partial \overline{\user2{W}}^{\text{T}} }}{\partial \xi } + \frac{{A_{{{22}}} }}{{R^{2} }}\user2{\overline{W}\overline{W}}^{\text{T}} } \right)} } {\text{d}}\xi {\text{d}}\theta {\kern 1pt} } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{k_{c} \tilde{A}_{{{44}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{W}}}}{\partial \theta }\frac{{\partial \overline{\user2{W}}^{\text{T}} }}{\partial \theta } + \frac{{k_{c} \tilde{A}_{55} }}{{L^{2} }}\frac{{\partial \overline{\user2{W}}}}{\partial \xi }\frac{{\partial \overline{\user2{W}}^{\text{T}} }}{\partial \xi } + \frac{{\tilde{A}_{{{22}}} }}{{R^{2} }}\user2{\overline{W}\overline{W}}^{\text{T}} } \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{{w\phi_{x} }} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{2k_{c} A_{{{55}}} }}{L}\frac{{\partial \overline{\user2{W}}}}{\partial \xi }\overline{\user2{\Phi }}_{x}^{\text{T}} + \frac{{2B_{12} }}{LR}\overline{\user2{W}}\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \xi }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{2k_{c} \tilde{A}_{{{55}}} }}{L}\frac{{\partial \overline{\user2{W}}}}{\partial \xi }\overline{\user2{\Phi }}_{x}^{\text{T}} + \frac{{2\tilde{B}_{12} }}{LR}\overline{\user2{W}}\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \xi }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{{w\phi_{\theta } }} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{2k_{c} A_{{{44}}} }}{R}\frac{{\partial \overline{\user2{W}}}}{\partial \theta }\overline{\user2{\Phi }}_{\theta }^{\text{T}} + \frac{{2B_{22} }}{{R^{2} }}\overline{\user2{W}}\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{2k_{c} \tilde{A}_{{{44}}} }}{R}\frac{{\partial \overline{\user2{W}}}}{\partial \theta }\overline{\user2{\Phi }}_{\theta }^{\text{T}} + \frac{{2\tilde{B}_{22} }}{{R^{2} }}\overline{\user2{W}}\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{{\phi_{x} \phi_{x} }} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{D_{11} }}{{L^{2} }}\frac{{\partial \overline{\user2{\Phi }}_{x} }}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \xi }{\mkern 1mu} + \frac{{D_{{{66}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{\Phi }}_{x} }}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \theta } + k_{c} A_{55} \overline{\user2{\Phi }}_{x} \overline{\user2{\Phi }}_{x}^{\text{T}} } \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{\tilde{D}_{11} }}{{L^{2} }}\frac{{\partial \overline{\user2{\Phi }}_{x} }}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \xi }{\mkern 1mu} + \frac{{\tilde{D}_{{{66}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{\Phi }}_{x} }}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{x}^{\text{T}} }}{\partial \theta } + k_{c} \tilde{A}_{55} \overline{\user2{\Phi }}_{x} \overline{\user2{\Phi }}_{x}^{\text{T}} } \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{{\phi_{x} \phi_{\theta } }} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{2D_{12} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{\Phi }}_{x} }}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta } + \frac{{2D_{{{66}}} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{\Phi }}_{x} }}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \xi }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{2\tilde{D}_{12} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{\Phi }}_{x} }}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta } + \frac{{2\tilde{D}_{{{66}}} }}{{L{\mkern 1mu} R}}\frac{{\partial \overline{\user2{\Phi }}_{x} }}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \xi }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

$$ \begin{aligned} {\varvec{K}}^{{\phi_{\theta } \phi_{\theta } }} & = LR\sum\limits_{s = 1}^{NP} {\int_{{\theta_{s} }}^{{\theta^{\prime}_{s} }} {\int_{{\xi_{s} }}^{{\xi^{\prime}_{s} }} {\left( {\frac{{D_{{{22}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{\Phi }}_{\theta } }}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta } + k_{c} A_{44} \overline{\user2{\Phi }}_{\theta } \overline{\user2{\Phi }}_{\theta }^{\text{T}} + \frac{{D_{{{66}}} }}{{L^{2} }}\frac{{\partial \overline{\user2{\Phi }}_{\theta } }}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \xi }} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ & \quad + \,LR\sum\limits_{r = 1}^{{\overline{NP} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( {\frac{{\tilde{D}_{{{22}}} }}{{R^{2} }}\frac{{\partial \overline{\user2{\Phi }}_{\theta } }}{\partial \theta }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \theta } + k_{c} \tilde{A}_{44} \overline{\user2{\Phi }}_{\theta } \overline{\user2{\Phi }}_{\theta }^{\text{T}} + \frac{{\tilde{D}_{{{66}}} }}{{L^{2} }}\frac{{\partial \overline{\user2{\Phi }}_{\theta } }}{\partial \xi }\frac{{\partial \overline{\user2{\Phi }}_{\theta }^{\text{T}} }}{\partial \xi }} \right)} } } {\text{d}}\xi {\text{d}}\theta \\ \end{aligned} $$

The electromechanical coupling stiffness and the electrical stiffness matrix of the piezoelectric laminated shell are expressed as

$$ {\varvec{K}}_{q\psi } \user2=\left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill \\ {\frac{1}{2}{\varvec{K}}^{{v{\kern 1pt} \psi_{a} }} } \hfill & {\frac{1}{2}{\varvec{K}}^{{v{\kern 1pt} \psi_{s} }} } \hfill \\ {\frac{1}{2}{\varvec{K}}^{{w{\kern 1pt} \psi_{a} }} } \hfill & {\frac{1}{2}{\varvec{K}}^{{w{\kern 1pt} \psi_{s} }} } \hfill \\ {\frac{1}{2}{\varvec{K}}^{{\phi_{x} {\kern 1pt} \psi_{a} }} } \hfill & {\frac{1}{2}{\varvec{K}}^{{\phi_{x} {\kern 1pt} \psi_{s} }} } \hfill \\ {\frac{1}{2}{\varvec{K}}^{{\phi_{\theta } {\kern 1pt} \psi_{a} }} } \hfill & {\frac{1}{2}{\varvec{K}}^{{\phi_{\theta } {\kern 1pt} \psi_{s} }} } \hfill \\ \end{array} } \right] $$

(36)

$$ \user2{K}_{{\psi \psi }} \user2{ = }\left[ {\begin{array}{*{20}l} {\user2{K}^{{{\kern 1pt} \psi _{a} \psi _{a} }} } \hfill & 0 \hfill \\ 0 \hfill & {\user2{K}^{{\psi _{s} \psi _{s} }} } \hfill \\ \end{array} } \right] $$

(37)

where

$$ \begin{aligned} \user2{K}^{{v\psi _{a} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left( {{\mkern 1mu} \frac{{e_{{24{\mkern 1mu} e}} ^{a} h_{a} ^{3} }}{{3R^{2} }}\bar{V}\frac{{\partial \user2{\bar{\Psi }}_{a} ^{\text{T}} }}{{\partial \theta }}} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ \user2{K}^{{v\psi _{s} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left( {\frac{{e_{{24{\mkern 1mu} e}} ^{s} h_{s} ^{3} }}{{3R^{2} }}\bar{V}\frac{{\partial \user2{\bar{\Psi }}_{s} ^{\text{T}} }}{{\partial \theta }}} \right)} } {\text{d}}\xi {\text{d}}\theta } \\ \user2{K}^{{w\psi _{a} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left\{ { - \frac{{e_{{15{\mkern 1mu} e}} ^{a} h_{a} ^{3} }}{{3L^{2} }}\frac{{\partial \user2{\bar{W}}}}{{\partial \xi }}\frac{{\partial \user2{\bar{\Psi }}_{a} ^{\text{T}} }}{{\partial \xi }} - \frac{{e_{{24{\mkern 1mu} e}} ^{a} h_{a} ^{3} }}{{3R^{2} }}\frac{{\partial \user2{\bar{W}}}}{{\partial \theta }}\frac{{\partial \user2{\bar{\Psi }}_{a} ^{\text{T}} }}{{\partial \theta }}} \right\}} } {\text{d}}\xi {\text{d}}\theta } \\ \user2{K}^{{w\psi _{s} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left\{ { - \frac{{e_{{15{\mkern 1mu} e}} ^{s} h_{s} ^{3} }}{{3L^{2} }}\frac{{\partial \user2{\bar{W}}}}{{\partial \xi }}\frac{{\partial \user2{\bar{\Psi }}_{s} ^{\text{T}} }}{{\partial \xi }} - \frac{{e_{{24{\mkern 1mu} e}} ^{s} h_{s} ^{3} }}{{3R^{2} }}\frac{{\partial \user2{\bar{W}}}}{{\partial \theta }}\frac{{\partial \user2{\bar{\Psi }}_{s} ^{\text{T}} }}{{\partial \theta }}} \right\}} } {\text{d}}\xi {\text{d}}\theta } \\ \user2{K}^{{\phi _{x} \psi _{a} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left\{ { - \frac{{e_{{15{\mkern 1mu} e}} ^{a} h_{a} ^{3} }}{{3L}}\user2{\bar{\Phi }}_{x} \frac{{\partial \user2{\bar{\Psi }}_{a} ^{\text{T}} }}{{\partial \xi }} + \frac{{e_{{31{\mkern 1mu} e}} ^{a} h_{a} ^{3} }}{{3L}}\frac{{\partial \user2{\bar{\Phi }}_{x} }}{{\partial \xi }}\user2{\bar{\Psi }}_{a} ^{\text{T}} } \right\}} } {\text{d}}\xi {\text{d}}\theta } \\ \user2{K}^{{\phi _{x} \psi _{s} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left\{ {{\mkern 1mu} - \frac{{e_{{15{\mkern 1mu} e}} ^{s} h_{s} ^{3} }}{{3L}}\user2{\bar{\Phi }}_{x} \frac{{\partial \user2{\bar{\Psi }}_{s} ^{\text{T}} }}{{\partial \xi }} + \frac{{e_{{31{\mkern 1mu} e}} ^{s} h_{s} ^{3} }}{{3L}}\frac{{\partial \user2{\bar{\Phi }}_{x} }}{{\partial \xi }}\user2{\bar{\Psi }}_{s} ^{\text{T}} {\mkern 1mu} } \right\}} } {\text{d}}\xi {\text{d}}\theta } \\ \user2{K}^{{\phi _{\theta } \psi _{a} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left\{ {{\mkern 1mu} \frac{{e_{{32{\mkern 1mu} e}} ^{a} h_{a} ^{3} }}{{3R}}\frac{{\partial \user2{\bar{\Phi }}_{\theta } }}{{\partial \theta }}\user2{\bar{\Psi }}_{a} ^{\text{T}} - \frac{{e_{{24{\mkern 1mu} e}} ^{a} h_{a} ^{3} }}{{3R}}\user2{\bar{\Phi }}_{\theta } \frac{{\partial \user2{\bar{\Psi }}_{a} ^{\text{T}} }}{{\partial \theta }}} \right\}} } {\text{d}}\xi {\text{d}}\theta } \\ \user2{K}^{{\phi _{\theta } \psi _{s} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left\{ {\frac{{e_{{32{\mkern 1mu} e}} ^{s} h_{s} ^{3} }}{{3R}}\frac{{\partial \user2{\bar{\Phi }}_{\theta } }}{{\partial \theta }}\user2{\bar{\Psi }}_{s} ^{\text{T}} - {\mkern 1mu} \frac{{e_{{24{\mkern 1mu} e}} ^{s} h_{s} ^{3} }}{{3R}}\user2{\bar{\Phi }}_{\theta } \frac{{\partial \user2{\bar{\Psi }}_{s} ^{\text{T}} }}{{\partial \theta }}} \right\}} } {\text{d}}\xi {\text{d}}\theta } \\ \user2{K}^{{\psi _{a} \psi _{a} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left\{ { - \frac{{\zeta _{{11{\mkern 1mu} e}} ^{a} h_{a} ^{5} }}{{30{\mkern 1mu} L^{2} }}\frac{{\partial \user2{\bar{\Psi }}_{a} }}{{\partial \xi }}\frac{{\partial \user2{\bar{\Psi }}_{a} ^{\text{T}} }}{{\partial \xi }} - {\mkern 1mu} \frac{{\zeta _{{22{\mkern 1mu} e}} ^{a} h_{a} ^{5} }}{{30R^{2} }}\frac{{\partial \user2{\bar{\Psi }}_{a} }}{{\partial \theta }}\frac{{\partial \user2{\bar{\Psi }}_{a} ^{\text{T}} }}{{\partial \theta }} - \frac{{\zeta _{{33{\mkern 1mu} e}} ^{a} h_{a} ^{3} }}{3}\user2{\bar{\Psi }}_{a} \user2{\bar{\Psi }}_{a} ^{\text{T}} } \right\}} } {\text{d}}\xi {\text{d}}\theta } \\ \user2{K}^{{\psi _{s} \psi _{s} }} & =LR\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left\{ { - \frac{{\zeta _{{11{\mkern 1mu} e}} ^{s} h_{s} ^{5} }}{{30{\mkern 1mu} L^{2} }}\frac{{\partial \user2{\bar{\Psi }}_{s} }}{{\partial \xi }}\frac{{\partial \user2{\bar{\Psi }}_{s} ^{\text{T}} }}{{\partial \xi }} - \frac{{\zeta _{{22{\mkern 1mu} e}} ^{s} h_{s} ^{5} }}{{30R^{2} }}\frac{{\partial \user2{\bar{\Psi }}_{s} }}{{\partial \theta }}\frac{{\partial \user2{\bar{\Psi }}_{s} ^{\text{T}} }}{{\partial \theta }} - {\mkern 1mu} \frac{{\zeta _{{33{\mkern 1mu} e}} ^{s} h_{s} ^{3} }}{3}\user2{\bar{\Psi }}_{s} \user2{\bar{\Psi }}_{s} ^{\text{T}} } \right\}} } {\text{d}}\xi {\text{d}}\theta } \\ \end{aligned} $$

Appendix D: Expressions of the spring stiffness matrix K _spr

The spring stiffness matrix of the piezoelectric laminated shell is shown as

$$ {\varvec{K}}_{{{\text{spr}}}} = \left[ {\begin{array}{*{20}l} {{\varvec{K}}_{{{\text{spr}}}}^{uu} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {{\varvec{K}}_{{{\text{spr}}}}^{vv} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {{\varvec{K}}_{{{\text{spr}}}}^{ww} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {{\varvec{K}}_{{{\text{spr}}}}^{{\phi_{x} \phi_{x} }} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {{\varvec{K}}_{{{\text{spr}}}}^{{\phi_{\theta } \phi_{\theta } }} } \hfill \\ \end{array} } \right] $$

(38)

where

$$ \begin{gathered} K_{{\text{spr}}}^{{uu}} = \sum\limits_{{p = 1}}^{{\text{NA}}} {\left( {k_{{u,p}}^{o} \user2{\bar{U}}\left( {0,\theta _{p} } \right)\user2{\bar{U}}^{\text{T}} \left( {0,\theta _{p} } \right) + k_{{u,p}}^{1} \user2{\bar{U}}\left( {0,\theta _{p} } \right)\user2{\bar{U}}^{\text{T}} \left( {0,\theta _{p} } \right)} \right)} \hfill \\ K_{{\text{spr}}}^{{vv}} = \sum\limits_{{p = 1}}^{{\text{NA}}} {\left( {k_{{v,p}}^{0} \user2{\bar{V}}\left( {0,\theta _{p} } \right)\user2{\bar{V}}^{\text{T}} \left( {0,\theta _{p} } \right) + k_{{v,p}}^{1} \user2{\bar{V}}\left( {1,\theta _{p} } \right)\user2{\bar{V}}^{\text{T}} \left( {1,\theta _{p} } \right)} \right)} \hfill \\ K_{{\text{spr}}}^{{ww}} = \sum\limits_{{p = 1}}^{{\text{NA}}} {\left( {k_{{w,p}}^{0} \user2{\bar{W}}\left( {0,\theta _{p} } \right)\user2{\bar{W}}^{\text{T}} \left( {0,\theta _{p} } \right)} \right.} + \left. {k_{{w,p}}^{\prime1} \user2{\bar{W}}\left( {1,\theta _{p} } \right)\user2{\bar{W}}^{\text{T}} \left( {1,\theta _{p} } \right)} \right)~ \hfill \\ K_{{\text{spr}}}^{{\phi _{x} \phi _{x} }} = \sum\limits_{{p = 1}}^{{\text{NA}}} {\left( {k_{{x,p}}^{0} \user2{\bar{\Phi }}_{x} \left( {0,\theta _{p} } \right)\user2{\bar{\Phi }}_{x} ^{\text{T}} \left( {0,\theta _{p} } \right)} \right.} + \left. {k_{{x,p}}^{1} \user2{\bar{\Phi }}_{x} \left( {1,\theta _{p} } \right)\user2{\bar{\Phi }}_{x} ^{\text{T}} \left( {1,\theta _{p} } \right)} \right)~ \hfill \\ K_{{\text{spr}}}^{{\phi _{\theta } \phi _{\theta } }} = \sum\limits_{{p = 1}}^{{\text{NA}}} {\left( {k_{{\theta ,p}}^{0} \user2{\bar{\Phi }}_{\theta } \left( {0,\theta _{p} } \right)\user2{\bar{\Phi }}_{\theta } ^{\text{T}} \left( {0,\theta _{p} } \right)} \right.} + \left. {k_{{\theta ,p}}^{1} \user2{\bar{\Phi }}_{\theta } \left( {1,\theta _{p} } \right)\user2{\bar{\Phi }}_{\theta } ^{\text{T}} \left( {1,\theta _{p} } \right)} \right)~ \hfill \\ \end{gathered} $$

Appendix E: Expressions of the nonlinear item Q _N

$$ \user2{Q}_{N} = \frac{{LR}}{2}\sum\limits_{{s = 1}}^{{NP}} {\int_{{\theta _{s} }}^{{\theta ^{\prime}_{s} }} {\int_{{\xi _{s} }}^{{\xi ^{\prime}_{s} }} {\left( \begin{gathered} \frac{{A_{{12}} }}{{LR^{2} }}{\user2{q}}_{u}^{T} \frac{{\partial \overline{\user2{U}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} + \frac{{A_{{11}} }}{{L^{3} }}{\user2{q}}_{u}^{T} \frac{{\partial \overline{\user2{U}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }} \hfill \\ + \frac{{2A_{{66}} }}{{LR^{2} }}{\user2{q}}_{u}^{T} \frac{{\partial \overline{\user2{U}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} + \frac{{2A_{{66}} }}{{L^{2} R}}{\user2{q}}_{v}^{T} \frac{{\partial \overline{\user2{V}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}\user2{q}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} \hfill \\ + \frac{{A_{{12}} }}{{L^{2} R}}{\user2{q}}_{v}^{T} \frac{{\partial \overline{\user2{V}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} q_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }} + \frac{{A_{{22}} }}{{R^{3} }}{\user2{q}}_{v}^{T} \frac{{\partial \overline{\user2{V}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} \hfill \\ + \frac{{A_{{12}} }}{{L^{2} R}}{\user2{q}}_{w}^{T} \overline{\user2{W}} \frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }} + \frac{{A_{{22}} }}{{R^{3} }}{\user2{q}}_{w}^{T} \overline{\user2{W}} \frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} \hfill \\ + \frac{{B_{{12}} }}{{R^{2} L}}{\user2{q}}_{{\phi _{x} }}^{T} \frac{{\partial \user2{\bar{\Phi }}_{x} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} + \frac{{B_{{11}} }}{{L^{3} }}{\user2{q}}_{{\phi _{x} }}^{T} \frac{{\partial \user2{\bar{\Phi }}_{x} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} \hfill \\ + \frac{{2B_{{66}} }}{{R^{2} L}}{\user2{q}}_{{\phi _{x} }}^{T} \frac{{\partial \user2{\bar{\Phi }}_{x} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} + \frac{{2{\kern 1pt} B_{{66}} }}{{L^{2} R}}{\user2{q}}_{{\phi _{\theta } }}^{T} \frac{{\partial \user2{\bar{\Phi }}_{\theta } }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}q_{w} \hfill \\ + \frac{{B_{{12}} }}{{L^{2} R}}{\user2{q}}_{{\phi _{\theta } }}^{T} \frac{{\partial \user2{\bar{\Phi }}_{\theta } }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} + \frac{{B_{{22}} }}{{R^{3} }}{\user2{q}}_{{\phi _{\theta } }}^{T} \frac{{\partial \user2{\bar{\Phi }}_{\theta } }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}q_{w} \hfill \\ + \frac{{A_{{11}} }}{{4L^{4} }}{\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{{\user2{W}^{T} }} }}{{\partial \xi }}{\user2{q}}_{w} \hfill \\ + {\kern 1pt} \frac{{A_{{22}} }}{{4R^{4} }}{\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} \hfill \\ + \frac{{A_{{12}} }}{{4R^{2} L^{2} }}{\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}\frac{{\partial \overline{{\user2{W}^{T} }} }}{{\partial \theta }}{\user2{q}}_{w} \hfill \\ + \frac{{A_{{12}} }}{{4R^{2} L^{2} }}{\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \xi }}{\user2{q}}_{w} \hfill \\ + \frac{{A_{{66}} }}{{R^{2} L^{2} }}{\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{T} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{T} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}q_{w} \hfill \\ \end{gathered} \right)} } {\text{d}}\xi {\text{d}}\theta } $$

$$ + \frac{{LR}}{2}\sum\limits_{{r = 1}}^{{\overline{{NP}} }} {\int_{{\tilde{\theta }_{r} }}^{{\tilde{\theta }^{\prime}_{r} }} {\int_{{\tilde{\xi }_{r} }}^{{\tilde{\xi }^{\prime}_{r} }} {\left( \begin{gathered} \frac{{\tilde{A}_{{12}} }}{{LR^{2} }}{\user2{q}}_{u}^{{\text{T}}} \frac{{\partial \overline{\user2{U}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}}^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} + \frac{{\tilde{A}_{{11}} }}{{L^{3} }}{\user2{q}}_{u}^{{\text{T}}} \frac{{\partial \overline{\user2{U}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }} \hfill \\ + \frac{{2\tilde{A}_{{66}} }}{{LR^{2} }}{\user2{q}}_{u}^{{\text{T}}} \frac{{\partial \overline{\user2{U}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} + \frac{{2\tilde{A}_{{66}} }}{{L^{2} R}}{\user2{q}}_{v}^{{\text{T}}} \frac{{\partial \overline{V} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} \hfill \\ + \frac{{\tilde{A}_{{12}} }}{{L^{2} R}}{\user2{q}}_{v}^{{\text{T}}} \frac{{\partial \overline{\user2{V}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }} + \frac{{\tilde{A}_{{22}} }}{{R^{3} }}{\user2{q}}_{v}^{{\text{T}}} \frac{{\partial \overline{\user2{V}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} \hfill \\ + \frac{{\tilde{A}_{{12}} }}{{L^{2} R}}{\user2{q}}_{w}^{{\text{T}}} \overline{\user2{W}} \frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }} + \frac{{\tilde{A}_{{22}} }}{{R^{3} }}{\user2{q}}_{w}^{{\text{T}}} \overline{\user2{W}} \frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }} \hfill \\ + \frac{{\tilde{B}_{{12}} }}{{R^{2} L}}{\user2{q}}_{{\phi _{x} }}^{{\text{T}}} \frac{{\partial \user2{\bar{\Phi }}_{x} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} + \frac{{\tilde{B}_{{11}} }}{{L^{3} }}{\user2{q}}_{{\phi _{x} }}^{{\text{T}}} \frac{{\partial \user2{\bar{\Phi }}_{x} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} \hfill \\ + \frac{{2\tilde{B}_{{66}} }}{{R^{2} L}}{\user2{q}}_{{\phi _{x} }}^{{\text{T}}} \frac{{\partial \user2{\bar{\Phi }}_{x} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} + \frac{{2{\kern 1pt} \tilde{B}_{{66}} }}{{L^{2} R}}{\user2{q}}_{{\phi _{\theta } }}^{{\text{T}}} \frac{{\partial \user2{\bar{\Phi }}_{\theta } }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} \hfill \\ + \frac{{\tilde{B}_{{12}} }}{{L^{2} R}}{\user2{q}}_{{\phi _{\theta } }}^{{\text{T}}} \frac{{\partial \user2{\bar{\Phi }}_{\theta } }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} + \frac{{\tilde{B}_{{22}} }}{{R^{3} }}{\user2{q}}_{{\phi _{\theta } }}^{{\text{T}}} \frac{{\partial \user2{\bar{\Phi }}_{\theta } }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} \frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} \hfill \\ + \frac{{\tilde{A}_{{11}} }}{{4L^{4} }}{\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{{\user2{W}^{{\text{T}}} }} }}{{\partial \xi }}{\user2{q}}_{w} \hfill \\ + {\kern 1pt} \frac{{\tilde{A}_{{22}} }}{{4R^{4} }}{\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} \hfill \\ + \frac{{\tilde{A}_{{12}} }}{{4R^{2} L^{2} }}{\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}\frac{{\partial \overline{{\user2{W}^{{\text{T}}} }} }}{{\partial \theta }}{\user2{q}}_{w} \hfill \\ + \frac{{\tilde{A}_{{12}} }}{{4R^{2} L^{2} }}{\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \xi }}q_{w} \hfill \\ + \frac{{\tilde{A}_{{66}} }}{{R^{2} L^{2} }}{\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} ^{{\text{T}}} }}{{\partial \theta }}{\user2{q}}_{w} {\user2{q}}_{w}^{{\text{T}}} \frac{{\partial \overline{\user2{W}} }}{{\partial \xi }}\frac{{\partial \overline{\user2{W}} }}{{\partial \theta }}q_{w} \hfill \\ \end{gathered} \right)} } } d\xi d\theta $$

Appendix F: Incremental Harmonic Balance Method

By presenting the new non-dimensional time variable τ, we write the nonlinear vibration differential equation to a modified form

$$ \begin{aligned} & \omega^{2} {\varvec{M}}_{qq} \user2{X^{\prime\prime}} + \omega \left( {{\varvec{C}}_{A} + {\varvec{C}}_{R} } \right)\user2{X^{\prime}} + \left( {{\varvec{K}}_{qq} + {\varvec{K}}_{spr} + {\varvec{K}}_{{q{\kern 1pt} \psi }}^{S} {\varvec{K}}_{\psi \psi }^{S - 1} {\varvec{K}}_{{q{\kern 1pt} \psi }}^{ST} + {\varvec{K}}_{{\text{N}}}^{\left( 2 \right)} + {\varvec{K}}_{{\text{N}}}^{\left( 3 \right)} } \right){\varvec{X}} = {\varvec{F}}{\text{cos}}\tau \\ & \tau = \omega_{{{\text{EX}}}} t,\,{\varvec{X}}=\left[ {X_{1} ,X_{2} ,X_{3} ,\ldots,X_{n} } \right]^{\text{T}} ,\,{\varvec{F}} = \left[ {\begin{array}{*{20}c} {{\varvec{f}}_{u} } & {{\varvec{f}}_{v} } & {{\varvec{f}}_{w} } & {{\varvec{f}}_{{\phi_{x} }} } & {{\varvec{f}}_{{\phi_{\theta } }} } \\ \end{array} } \right]^{\text{T}} \\ \end{aligned} $$

(39)

Firstly, the differential equations are linearized by using the Newton–Raphson procedure. The corresponding adding increments are used to represent the neighboring state as

$$ \begin{aligned} X_{j} & = X_{j0} + \Delta X,\quad j = 1,2, \ldots ,n \\ \omega & = \omega_{0} + \Delta \omega \\ \end{aligned} $$

(40)

Taking Eqs. (40) to (39), the higher-order terms are neglected, and Eq. (39) is written as

$$ \begin{aligned} & \omega_{0}^{2} {\varvec{M}}_{qq} \user2{X^{\prime\prime}} + \omega_{0} \left( {{\varvec{C}}_{A} + {\varvec{C}}_{R} } \right)\user2{X^{\prime}} + \left( {{\varvec{K}}_{qq} + {\varvec{K}}_{spr} + {\varvec{K}}_{{q{\kern 1pt} \psi }}^{S} {\varvec{K}}_{\psi \psi }^{S- 1} {\varvec{K}}_{{q{\kern 1pt} \psi }}^{ST} + 2{\varvec{K}}_{{\text{N}}}^{\left( 2 \right)} + 3{\varvec{K}}_{{\text{N}}}^{\left( 3 \right)} } \right){\varvec{X}} \\ & = {\mathbf{Re}} - \left( {2\omega_{0} {\varvec{M}}_{qq} \user2{X^{\prime\prime}}_{0} + \left( {{\varvec{C}}_{A} + {\varvec{C}}_{R} } \right)\user2{X^{\prime}}_{0} } \right)\Delta \omega \\ \end{aligned} $$

(41)

$$ {\mathbf{Re}} = {\varvec{F}}{\text{cos}}\tau - \omega_{0}^{2} {\varvec{M}}_{qq} \user2{X^{\prime\prime}}_{0} - \omega_{0}^{{}} \left( {{\varvec{C}}_{A} + {\varvec{C}}_{R} } \right)\user2{X^{\prime}}_{0} - \left( {{\varvec{K}}_{qq} + {\varvec{K}}_{{q{\kern 1pt} \psi }}^{S} {\varvec{K}}_{\psi \psi }^{S- 1} {\varvec{K}}_{{q{\kern 1pt} \psi }}^{ST} \user2{ + }2{\varvec{K}}_{{\text{N}}}^{\left( 2 \right)} + 3{\varvec{K}}_{{\text{N}}}^{\left( 3 \right)} } \right){\varvec{X}}_{0} $$

(42)

where ${\varvec{X}}_{{\varvec{0}}} =\left[ {X_{10} ,X_{20} ,X_{30} ,\ldots,X_{n0} } \right]^{\text{T}} ,\Delta {\varvec{X}}=\left[ {\Delta X_{1} ,\Delta X_{2} ,\Delta X_{3} ,\ldots,\Delta X_{n} } \right]^{\text{T}}$. When the solution is the exact value, the residue “Re” becomes zero.

Next, by using the Galerkin’s technique, the steady-state response with a truncated Fourier series is written as

$$ \begin{aligned} X_{{j0}} & = A_{0} + \sum\limits_{{k = 1}}^{r} {\left( {a_{{jk}} \cos k\tau + b_{{jk}} \sin k\tau } \right)} = \user2{T}_{c} \user2{A}_{j} \\ \Delta X_{j} & = \Delta A + \sum\limits_{{k = 1}}^{r} {\left( {a_{{jk}} \cos k\tau + b_{{jk}} \sin k\tau } \right)} = \user2{T}_{c} \Delta \user2{A}_{j} \\ \end{aligned} $$

(43)

where

$$ \begin{aligned} T_{c} & = \left[ {1,\;\cos \tau ,\;\sin \tau ,\;\cos 2\tau ,\;\sin 2\tau , \ldots ,\cos r\tau ,\;\sin r\tau } \right] \\ A_{j} & = \left\{ {a_{{j1}} ,a_{{j2}} ,...,a_{{jn}} ,b_{{j1}} ,b_{{j2}} ,...,b_{{jn}} } \right\}^{{\text{T}}} \\ \Delta A_{j} & = \left\{ {\Delta a_{{j1}} ,\Delta a_{{j2}} ,...,\Delta a_{{jn}} ,\Delta b_{{j1}} ,\Delta b_{{j2}} ,...,\Delta b_{{jn}} } \right\}^{{\text{T}}} \\ \end{aligned} $$

(44)

where $a_{jk}$,$b_{jk}$ are the Fourier coefficients, and n are the numbers of cosine and sine terms. Using these vectors of Fourier coefficients A and its increments $\Delta {\varvec{A}}$, the vectors of the unknown parameters are written as

$$ {X}_{0} {\text{ = }}{SA}\quad \Delta {X}{\text{ = }}{S}\Delta {A} $$

(45)

in which

$$ {S} = \left[ {\begin{array}{*{20}l} {{T}_{c} } \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {{T}_{c} } \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & \ddots \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {{T}_{c} } \hfill \\ \end{array} } \right],{A} = \left\{ {{A}_{1} ,{A}_{2} ,\ldots,{A}_{n} } \right\}^{\text{T}} ,\quad \Delta {A} = \left\{ {\Delta {A}_{1} ,\Delta {A}_{2} ,\ldots,\Delta {A}_{n} } \right\}^{\text{T}} $$

(46)

Taking Eqs. (45) to (41), Galerkin’s technique is used

$$ \begin{aligned} & \int_{0}^{{2{\varvec{\pi}}}} {\delta \left( {\Delta {X}} \right)}^{{\varvec{T}}} \left[ {\omega_{0}^{2} {\varvec{M}}_{qq} \user2{X^{\prime\prime}} + \omega_{0} \left( {{\varvec{C}}_{A} + {\varvec{C}}_{R} } \right)\user2{X^{\prime}} + \left( {{K}_{qq} + {K}_{spr} + {\varvec{K}}_{{q{\kern 1pt} \psi }}^{S} {\varvec{K}}_{\psi \psi }^{S- 1} {\varvec{K}}_{{q{\kern 1pt} \psi }}^{ST} + 2{K}_{{\text{N}}}^{\left( 2 \right)} + 3{K}_{{\text{N}}}^{\left( 3 \right)} } \right)\Delta {X}} \right]{\text{d}}\tau \\ & = \int_{0}^{{2{\varvec{\pi}}}} {\delta \left( {\Delta {X}} \right)}^{{\varvec{T}}} \left[ {{\mathbf{Re}} - \left( {2\omega_{0} {\varvec{M}}_{qq} \user2{X^{\prime\prime}}_{0} } \right)\Delta \omega - \left( {{\varvec{C}}_{A} + {\varvec{C}}_{R} } \right)\user2{X^{\prime}}_{0} } \right]{\text{d}}\tau \\ \end{aligned} $$

(47)

The linear equation including $\Delta {\varvec{A}}$ and $\Delta \omega$ is written as

$$ {\varvec{K}}_{{{\text{mc}}}} \Delta {\varvec{A}} + {\varvec{R}}_{{{\text{mc}}}} {\varvec{A}}\Delta \omega = {\varvec{R}}_{{{\text{m}}1}} {\varvec{A}} + {\varvec{R}}_{{{\text{m}}2}} $$

(48)

where

$$ \begin{aligned} {\varvec{K}}_{{{\text{mc}}}} & = \int_{0}^{{2{\uppi }}} {{\varvec{S}}^{{\text{T}}} \left[ {\omega^{2} {\varvec{M}}_{qq} \user2{\ddot{S}} + \omega \left( {{\varvec{C}}_{A} + {\varvec{C}}_{R} } \right)\dot{\user2{S}} + \left( {{\varvec{K}}_{qq} + {\varvec{K}}_{spr} + {\varvec{K}}_{{q{\kern 1pt} \psi }}^{S} {\varvec{K}}_{\psi \psi }^{S - 1} {\varvec{K}}_{{q{\kern 1pt} \psi }}^{ST} + 2{\varvec{K}}_{{\text{N}}}^{\left( 2 \right)} + 3{\varvec{K}}_{{\text{N}}}^{\left( 3 \right)} } \right){\varvec{S}}} \right]{\text{d}}\tau } \\ {\varvec{R}}_{{{\text{mc}}}} & = \int_{0}^{{2{\uppi }}} {{\varvec{S}}^{{\text{T}}} \left( {2\omega {\varvec{M}}_{qq} \user2{\ddot{S}} + \left( {{\varvec{C}}_{A} + {\varvec{C}}_{R} } \right)\dot{\user2{S}}} \right){\text{d}}\tau } \\ {\varvec{R}}_{{{\text{m}}1}} & = - \int_{0}^{{2{\uppi }}} {{\varvec{S}}^{{\text{T}}} \left[ {\omega^{2} {\varvec{M}}_{qq} \user2{\ddot{S}} + \omega \left( {{\varvec{C}}_{A} + {\varvec{C}}_{R} } \right)\dot{\user2{S}} + \left( {{\varvec{K}}_{qq} + {\varvec{K}}_{spr} + {\varvec{K}}_{{q{\kern 1pt} \psi }}^{S} {\varvec{K}}_{\psi \psi }^{S - 1} {\varvec{K}}_{{q{\kern 1pt} \psi }}^{ST} + {\varvec{K}}_{{\text{N}}}^{\left( 2 \right)} + {\varvec{K}}_{{\text{N}}}^{\left( 3 \right)} } \right){\varvec{S}}} \right]{\text{d}}\tau } \\ {\varvec{R}}_{{{\text{m2}}}} & = \int_{0}^{{2{\uppi }}} {{\varvec{S}}^{{\text{T}}} {\varvec{F}}\cos \tau {\text{d}}\tau } \\ \end{aligned} $$

(49)

Finally, we use the arc-length method to solve Eq. (48) for obtaining the frequency–amplitude response.