On the symplectic superposition method for analytic free vibration solutions of right triangular plates

Abstract

The analytic free vibration solutions of triangular plates are important for both rapid analyses and preliminary designs of similar structures. Due to the difficulty in solving the complex boundary value problems of the governing high-order partial differential equations, the current knowledge about the analytic solutions is limited. This study presents a first attempt to explore an up-to-date symplectic superposition method for analytic free vibration solutions of right triangular plates. Specifically, an original problem is regarded as the superposition of three fundamental subproblems of the corresponding rectangular plates that are solved by the symplectic eigenexpansion within the Hamiltonian-system framework, involving the coordinate transformation. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition. By comparison with the numerical results for the right triangular plates under six different combinations of clamped and simply supported boundary constraints, the fast convergence and high accuracy of the present approach are well confirmed. Within the current solution framework, the extension to the problems of more polygonal plates is possible.

Appendix

The expressions for ${f_1}\left( {m,p,\phi ,{\bar{\omega }} ,{{\bar{K}}_p},{{{\bar{L}}}_p}} \right)$, ${f_2}\left( {m,p,\phi ,{\bar{\omega }} ,{{{\bar{K}}}_p},{{{\bar{L}}}_p}} \right)$, ${f_3}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{K}}}_p},{{{\bar{L}}}_p}} \right)$, ${f_4}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{K}}}_p},{{{\bar{L}}}_p}} \right)$, ${f_5}\left( {m,p,\phi ,{\bar{\omega }} ,{{{\bar{E}}}_m},{{{\bar{F}}}_m}} \right)$, ${f_{\text {6}}}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{\bar{G}}_n},{{{\bar{H}}}_n}} \right)$, ${f_{\text {7}}}\left( {m,p,\phi ,\bar{\omega },{{{\bar{E}}}_m},{{{\bar{F}}}_m}} \right)$, ${f_{\text {8}}}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{G}}}_n},{{{\bar{H}}}_n}} \right)$, ${f_9}\left( {m,n,\phi ,{\bar{\omega }} ,{{{\bar{G}}}_n},{{{\bar{H}}}_n}} \right)$, ${f_{10}}\left( {m,p,\phi ,{\bar{\omega }} ,{{\bar{K}}_p},{{{\bar{L}}}_p}} \right)$, ${f_{11}}\left( {m,n,{\bar{\phi }} ,\hat{\omega },{{{\bar{E}}}_m},{{{\bar{F}}}_m}} \right)$, ${f_{12}}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{K}}}_p},{{{\bar{L}}}_p}} \right)$, ${f_{13}}\left( {m,p,\phi ,{\bar{\omega }} ,{{{\bar{E}}}_m},{{{\bar{F}}}_m}} \right)$, and ${f_{14}}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{\bar{G}}_n},{{{\bar{H}}}_n}} \right)$ are as follows:

$$\begin{aligned}&{f_1}\left( {m,p,\phi ,{\bar{\omega }} ,{{{\bar{K}}}_p},{{{\bar{L}}}_p}} \right) \nonumber \\&\quad = \frac{{2mp{\pi ^2}\phi }}{{{\bar{\omega }} {\xi _{m5}}{\xi _{m6}}{\xi _{m7}}{\xi _{m8}}\sqrt{1 + {\phi ^2}} }} \nonumber \\&\qquad \times \, \left\{ {{\varsigma _p}{\xi _{m5}}{\xi _{m7}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{\xi _{p1}}} \right) \left[ {{\text {cth}} \left( {\frac{{\phi {\varsigma _p}}}{{1 + {\phi ^2}}}} \right) - \cos \left( {m\pi } \right) \cos \left( {\frac{{p\pi }}{{1 + {\phi ^2}}}} \right) {\text {csch}} \left( {\frac{{\phi {\varsigma _p}}}{{1 + {\phi ^2}}}} \right) } \right] } \right. \nonumber \\&\qquad \left. { -\, {\varepsilon _p}{\xi _{m6}}{\xi _{m8}}\left( {{{\bar{L}}_p} + {{{\bar{K}}}_p}{\xi _{p2}}} \right) \left[ {{\text {cth}} \left( {\frac{{\phi {\varepsilon _p}}}{{1 + {\phi ^2}}}} \right) - \cos \left( {m\pi } \right) \cos \left( {\frac{{p\pi }}{{1 + {\phi ^2}}}} \right) {\text {csch}} \left( {\frac{{\phi {\varepsilon _p}}}{{1 + {\phi ^2}}}} \right) } \right] } \right\} \end{aligned}$$

(A1)

where ${\xi _{m5}} = {\pi ^2}\left[ {{{\left( {m + p} \right) }^2} + {m^2}{\phi ^2}} \right] + {\phi ^2}{\bar{\omega }}$, ${\xi _{m6}} = {\pi ^2}\left[ {{{\left( {m + p} \right) }^2} + {m^2}{\phi ^2}} \right] - {\phi ^2}{\bar{\omega }}$, ${\xi _{m7}} = {\pi ^2}\left[ {{{\left( {m - p} \right) }^2} + {m^2}{\phi ^2}} \right] + {\phi ^2}{\bar{\omega }}$, and ${\xi _{m8}} = {\pi ^2}\left[ {{{\left( {m - p} \right) }^2} + {m^2}{\phi ^2}} \right] - {\phi ^2}{\bar{\omega }}$.

$$\begin{aligned}&{f_2}\left( {m,p,\phi ,{\bar{\omega }} ,{{{\bar{K}}}_p},{{{\bar{L}}}_p}} \right) \nonumber \\&\quad = \frac{{2mp{\pi ^2}\phi }}{{{\bar{\omega }} {\xi _{m5}}{\xi _{m6}}{\xi _{m7}}{\xi _{m8}}\sqrt{1 + {\phi ^2}} }} \nonumber \\&\qquad \times \, \left\{ {{\varepsilon _p}{\xi _{m1}}{\xi _{m6}}{\xi _{m8}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{\xi _{p2}}} \right) \left[ {{\text {cth}} \left( {\frac{{\phi {\varepsilon _p}}}{{1 + {\phi ^2}}}} \right) - \cos \left( {m\pi } \right) \cos \left( {\frac{{p\pi }}{{1 + {\phi ^2}}}} \right) {\text {csch}} \left( {\frac{{\phi {\varepsilon _p}}}{{1 + {\phi ^2}}}} \right) } \right] } \right. \nonumber \\&\qquad \left. { -\, {\varsigma _p}{\xi _{m2}}{\xi _{m5}}{\xi _{m7}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{\xi _{p1}}} \right) \left[ {{\text {cth}} \left( {\frac{{\phi {\varsigma _p}}}{{1 + {\phi ^2}}}} \right) - \cos \left( {m\pi } \right) \cos \left( {\frac{{p\pi }}{{1 + {\phi ^2}}}} \right) {\text {csch}} \left( {\frac{{\phi {\varsigma _p}}}{{1 + {\phi ^2}}}} \right) } \right] } \right\} \end{aligned}$$

(A2)

$$\begin{aligned}&{f_3}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{K}}}_p},{{{\bar{L}}}_p}} \right) \nonumber \\&\quad = \frac{{2np{\pi ^2}{\bar{\phi }} \cos \left( {p\pi } \right) }}{{{\hat{\omega }} {{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n6}}{{\bar{\xi }}_{n7}}{{{\bar{\xi }} }_{n8}}\sqrt{1 + {{{\bar{\phi }} }^2}} }} \nonumber \\&\qquad \times \, \left\{ {{{{\bar{\varsigma }} }_p}{{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n7}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{{\bar{\xi }}_{p1}}} \right) \left[ {{\text {cth}} \left( {\frac{{\bar{\phi }{{{\bar{\varsigma }} }_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) - \cos \left( {n\pi } \right) \cos \left( {\frac{{p\pi }}{{1 + {{{\bar{\phi }} }^2}}}} \right) {\text {csch}} \left( {\frac{{{\bar{\phi }} {{\bar{\varsigma }}_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) } \right] } \right. \nonumber \\&\qquad \left. { -\, {{{\bar{\varepsilon }} }_p}{{{\bar{\xi }} }_{n6}}{{{\bar{\xi }} }_{n8}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{{{\bar{\xi }} }_{p2}}} \right) \left[ {{\text {cth}} \left( {\frac{{{\bar{\phi }} {{\bar{\varepsilon }}_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) - \cos \left( {n\pi } \right) \cos \left( {\frac{{p\pi }}{{1 + {{{\bar{\phi }} }^2}}}} \right) {\text {csch}} \left( {\frac{{{\bar{\phi }} {{\bar{\varepsilon }}_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) } \right] } \right\} \end{aligned}$$

(A3)

where $\ {{\bar{\xi }} _{n5}} = {\pi ^2}\left[ {{{\left( {n + p} \right) }^2} + {n^2}{{{\bar{\phi }} }^2}} \right] + {{\bar{\phi }} ^2}\hat{\omega }$, $\ {{\bar{\xi }} _{n6}} = {\pi ^2}\left[ {{{\left( {n + p} \right) }^2} + {n^2}{{{\bar{\phi }} }^2}} \right] - {{\bar{\phi }} ^2}\hat{\omega }$, $\ {{\bar{\xi }} _{n7}} = {\pi ^2}\left[ {{{\left( {n - p} \right) }^2} + {n^2}{{{\bar{\phi }} }^2}} \right] + {{\bar{\phi }} ^2}\hat{\omega }$, ${{\bar{\xi }} _{n8}} = {\pi ^2}\left[ {{{\left( {n - p} \right) }^2} + {n^2}{{{\bar{\phi }} }^2}} \right] - {{\bar{\phi }} ^2}\hat{\omega }$, ${{\bar{\varepsilon }} _p} = \sqrt{{p^2}{\pi ^2} + \hat{\omega }\left( {1 + {{{\bar{\phi }} }^2}} \right) }$, ${{\bar{\varsigma }} _p} = \sqrt{{p^2}{\pi ^2} - {\hat{\omega }} \left( {1 + {{{\bar{\phi }} }^2}} \right) }$, ${{\bar{\xi }} _{p1}} = {p^2}{\pi ^2}\left( {1 - \nu } \right) + {\hat{\omega }} \left( {1 + {{{\bar{\phi }} }^2}} \right)$, and ${{\bar{\xi }} _{p2}} = {p^2}{\pi ^2}\left( {1 - \nu } \right) - \hat{\omega }\left( {1 + {{{\bar{\phi }} }^2}} \right)$.

$$\begin{aligned}&{f_4}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{K}}}_p},{{{\bar{L}}}_p}} \right) \nonumber \\&\quad = \frac{{2np{\pi ^2}{\bar{\phi }} \cos \left( {p\pi } \right) }}{{{\hat{\omega }} {{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n6}}{{\bar{\xi }}_{n7}}{{{\bar{\xi }} }_{n8}}\sqrt{1 + {{{\bar{\phi }} }^2}} }} \nonumber \\&\qquad \times \, \left\{ {{{{\bar{\varepsilon }} }_p}{{{\bar{\xi }} }_{n1}}{{{\bar{\xi }} }_{n6}}{{{\bar{\xi }} }_{n8}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{{{\bar{\xi }} }_{p2}}} \right) \left[ {{\text {cth}} \left( {\frac{{{\bar{\phi }} {{{\bar{\varepsilon }} }_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) - \cos \left( {n\pi } \right) \cos \left( {\frac{{p\pi }}{{1 + {{{\bar{\phi }} }^2}}}} \right) {\text {csch}} \left( {\frac{{{\bar{\phi }} {{{\bar{\varepsilon }} }_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) } \right] } \right. \nonumber \\&\qquad -\, \left. {{{\bar{\varsigma }}_p}{{{\bar{\xi }} }_{n2}}{{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n7}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{{{\bar{\xi }} }_{p1}}} \right) \left[ {{\text {cth}} \left( {\frac{{{\bar{\phi }} {{\bar{\varsigma }}_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) - \cos \left( {n\pi } \right) \cos \left( {\frac{{p\pi }}{{1 + {{{\bar{\phi }} }^2}}}} \right) {\text {csch}} \left( {\frac{{{\bar{\phi }} {{{\bar{\varsigma }} }_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) } \right] } \right\} \end{aligned}$$

(A4)

$$\begin{aligned}&{f_5}\left( {m,p,\phi ,{\bar{\omega }} ,{{{\bar{E}}}_m},{{{\bar{F}}}_m}} \right) \nonumber \\&\quad = \frac{{2mp{\pi ^2}\phi }}{{{\bar{\omega }} {\xi _{m5}}{\xi _{m6}}{\xi _{m7}}{\xi _{m8}}\sqrt{1 + {\phi ^2}} }} \nonumber \\&\qquad \times \, \left\{ {{\zeta _m}{\xi _{m5}}{\xi _{m7}}\left( {{{{\bar{F}}}_m} + {{{\bar{E}}}_m}{\xi _{m1}}} \right) \left[ {{\text {cth}} \left( {\phi {\zeta _m}} \right) - \cos \left( {m\pi } \right) \cos \left( {p\pi } \right) {\text {csch}} \left( {\phi {\zeta _m}} \right) } \right] } \right. \nonumber \\&\qquad \left. { -\, {\eta _m}{\xi _{m6}}{\xi _{m8}}\left( {{{{\bar{F}}}_m} + {{{\bar{E}}}_m}{\xi _{m2}}} \right) \left[ {{\text {cth}} \left( {\phi {\eta _m}} \right) - \cos \left( {m\pi } \right) \cos \left( {p\pi } \right) {\text {csch}} \left( {\phi {\eta _m}} \right) } \right] } \right\} \end{aligned}$$

(A5)

$$\begin{aligned}&{f_6}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{G}}}_n},{{{\bar{H}}}_n}} \right) \nonumber \\&\quad = \frac{{2np{\pi ^2}{\bar{\phi }} \cos \left( {p\pi } \right) }}{{{\hat{\omega }} {{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n6}}{{\bar{\xi }}_{n7}}{{{\bar{\xi }} }_{n8}}\sqrt{1 + {{{\bar{\phi }} }^2}} }} \nonumber \\&\qquad \times \, \left\{ {{{{\bar{\zeta }} }_n}{{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n7}}\left( {{{{\bar{H}}}_n} + {{{\bar{G}}}_n}{{\bar{\xi }}_{n1}}} \right) \left[ {{\text {cth}} \left( {{\bar{\phi }} {{{\bar{\zeta }} }_n}} \right) - \cos \left( {n\pi } \right) \cos \left( {p\pi } \right) {\text {csch}} \left( {{\bar{\phi }} {{{\bar{\zeta }} }_n}} \right) } \right] } \right. \nonumber \\&\qquad \left. { -\, {{{\bar{\eta }} }_n}{{{\bar{\xi }} }_{n6}}{{{\bar{\xi }} }_{n8}}\left( {{{{\bar{H}}}_n} + {{{\bar{G}}}_n}{{{\bar{\xi }} }_{n2}}} \right) \left[ {{\text {cth}} \left( {\phi {{{\bar{\eta }} }_n}} \right) - \cos \left( {n\pi } \right) \cos \left( {p\pi } \right) {\text {csch}} \left( {\bar{\phi }{{{\bar{\eta }} }_n}} \right) } \right] } \right\} \end{aligned}$$

(A6)

$$\begin{aligned}&{f_7}\left( {m,p,\phi ,{\bar{\omega }} ,{{{\bar{E}}}_m},{{{\bar{F}}}_m}} \right) \nonumber \\&\quad = \frac{{2mp{\pi ^2}\phi }}{{{\bar{\omega }} {\xi _{m5}}{\xi _{m6}}{\xi _{m7}}{\xi _{m8}}\sqrt{1 + {\phi ^2}} }} \nonumber \\&\qquad \times \, \left\{ {{\eta _m}{\xi _{p1}}{\xi _{m6}}{\xi _{m8}}\left( {{{{\bar{F}}}_m} + {{{\bar{E}}}_m}{\xi _{m2}}} \right) \left[ {{\text {cth}} \left( {\phi {\eta _m}} \right) - \cos \left( {m\pi } \right) \cos \left( {p\pi } \right) {\text {csch}} \left( {\phi {\eta _m}} \right) } \right] } \right. \nonumber \\&\qquad \left. { -\, {\zeta _m}{\xi _{p2}}{\xi _{m5}}{\xi _{m7}}\left( {{{{\bar{F}}}_m} + {{{\bar{E}}}_m}{\xi _{m1}}} \right) \left[ {{\text {cth}} \left( {\phi {\zeta _m}} \right) - \cos \left( {m\pi } \right) \cos \left( {p\pi } \right) {\text {csch}} \left( {\phi {\zeta _m}} \right) } \right] } \right\} \end{aligned}$$

(A7)

$$\begin{aligned}&{f_8}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{\bar{G}}_n},{{{\bar{H}}}_n}} \right) \nonumber \\&\quad = \frac{{2np{\pi ^2}{\bar{\phi }} \cos \left( {p\pi } \right) }}{{{\hat{\omega }} {{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n6}}{{{\bar{\xi }} }_{n7}}{{{\bar{\xi }} }_{n8}}\sqrt{1 + {{{\bar{\phi }} }^2}} }} \nonumber \\&\qquad \times \, \left\{ {{{{\bar{\eta }} }_n}{{{\bar{\xi }} }_{p1}}{{{\bar{\xi }} }_{n6}}{{{\bar{\xi }} }_{n8}}\left( {{{{\bar{H}}}_n} + {{{\bar{G}}}_n}{{{\bar{\xi }} }_{n2}}} \right) \left[ {{\text {cth}} \left( {\phi {{{\bar{\eta }} }_n}} \right) - \cos \left( {n\pi } \right) \cos \left( {p\pi } \right) {\text {csch}} \left( {\bar{\phi }{{{\bar{\eta }} }_n}} \right) } \right] } \right. \nonumber \\&\qquad \left. -\, {{{{\bar{\zeta }} }_n}{{{\bar{\xi }} }_{p2}}{{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n7}}\left( {{{{\bar{H}}}_n} + {{{\bar{G}}}_n}{{{\bar{\xi }} }_{n1}}} \right) \left[ {{\text {cth}} \left( {{\bar{\phi }} {{{\bar{\zeta }} }_n}} \right) - \cos \left( {n\pi } \right) \cos \left( {p\pi } \right) {\text {csch}} \left( {{\bar{\phi }} {{{\bar{\zeta }} }_n}} \right) } \right] } \right\} \end{aligned}$$

(A8)

$$\begin{aligned}&{f_9}\left( {m,n,\phi ,{\bar{\omega }} ,{{{\bar{G}}}_n},{{{\bar{H}}}_n}} \right) \nonumber \\&\quad = \frac{{2mn{\pi ^2}{\phi ^2}\cos \left( {m\pi } \right) \cos \left( {n\pi } \right) \left\{ {{{{\bar{H}}}_n} + {{{\bar{G}}}_n}{\pi ^2}\left[ {{m^2}{\phi ^2} + {n^2}\left( {2 - \nu } \right) } \right] } \right\} }}{{{\phi ^4}{{{\bar{\omega }} }^2} - {\pi ^4}{{\left( {{n^2} + {m^2}{\phi ^2}} \right) }^2}}} \end{aligned}$$

(A9)

$$\begin{aligned}&{f_{10}}\left( {m,p,\phi ,{\bar{\omega }} ,{{\bar{K}}_p},{{{\bar{L}}}_p}} \right) \nonumber \\&\quad = \frac{\pi }{{2{\bar{\omega }} {\xi _{m5}}{\xi _{m6}}{\xi _{m7}}{\xi _{m8}}{{\left( {1 + {\phi ^2}} \right) }^{3/2}}}} \nonumber \\&\qquad \times \, \left\{ {{\xi _{m6}}{\xi _{m8}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{\xi _{p2}}} \right) \left[ {p\pi \phi \left( {4m\varepsilon _p^2 + {\xi _{m4}}{\xi _{m5}} + {\xi _{m3}}{\xi _{m7}}} \right) } \right. } \right. \nonumber \\&\qquad \left. { +\, {\varepsilon _p}{\xi _{p4}}\left( {{\xi _{m3}} + {\xi _{m4}}} \right) \cos \left( {m\pi } \right) \sin \left( {\frac{{p\pi }}{{1 + {\phi ^2}}}} \right) {\text {csch}} \left( {\frac{{\phi {\varepsilon _p}}}{{1 + {\phi ^2}}}} \right) } \right] \nonumber \\&\qquad -\, {\xi _{m5}}{\xi _{m7}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{\xi _{p1}}} \right) \left[ {p\pi \phi \left( {4m\varsigma _p^2 + {\xi _{m4}}{\xi _{m6}} + {\xi _{m3}}{\xi _{m8}}} \right) } \right. \nonumber \\&\qquad \left. {\left. { +\, {\varsigma _p}{\xi _{p3}}\left( {{\xi _{m3}} + {\xi _{m4}}} \right) \cos \left( {m\pi } \right) \sin \left( {\frac{{p\pi }}{{1 + {\phi ^2}}}} \right) {\text {csch}} \left( {\frac{{\phi {\varsigma _p}}}{{1 + {\phi ^2}}}} \right) } \right] } \right\} \end{aligned}$$

(A10)

where ${\xi _{m3}} = m + p + m{\phi ^2}$, ${\xi _{m4}} = m - p + m{\phi ^2}$, ${\xi _{p3}} = {\pi ^2}\left[ {{p^2} - {m^2}\left( {1 + {\phi ^2}} \right) } \right] + {\phi ^2}{\bar{\omega }}$, and ${\xi _{p4}} = {\pi ^2}\left[ {{p^2} - {m^2}\left( {1 + {\phi ^2}} \right) } \right] - {\phi ^2}{\bar{\omega }}$.

$$\begin{aligned} {f_{11}}\left( {m,n,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{E}}}_m},{{{\bar{F}}}_m}} \right)= & {} \frac{{2mn{\pi ^2}{{{\bar{\phi }} }^2}\cos \left( {m\pi } \right) \cos \left( {n\pi } \right) \left\{ {{{{\bar{F}}}_m} + {{{\bar{E}}}_m}{\pi ^2}\left[ {{n^2}{{{\bar{\phi }} }^2} + {m^2}\left( {2 - \nu } \right) } \right] } \right\} }}{{{{{\bar{\phi }} }^4}{{{\hat{\omega }} }^2} - {\pi ^4}{{\left( {{m^2} + {n^2}{{{\bar{\phi }} }^2}} \right) }^2}}} \end{aligned}$$

(A11)

$$\begin{aligned} {f_{12}}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{K}}}_p},{{\bar{L}}_p}} \right)= & {} \frac{{\cos \left( {p\pi } \right) \pi }}{{2\hat{\omega }{{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n6}}{{{\bar{\xi }} }_{n7}}{{\bar{\xi }}_{n8}}{{\left( {1 + {{{\bar{\phi }} }^2}} \right) }^{3/2}}}} \nonumber \\&\times \, \left\{ {{{{\bar{\xi }} }_{n6}}{{{\bar{\xi }} }_{n8}}\left( {{{\bar{L}}_p} + {{{\bar{K}}}_p}{{{\bar{\xi }} }_{p2}}} \right) \left[ {p\pi \bar{\phi }\left( {4n{\bar{\varepsilon }} _p^2 + {{{\bar{\xi }} }_{n4}}{{{\bar{\xi }} }_{n5}} + {{{\bar{\xi }} }_{n3}}{{{\bar{\xi }} }_{n7}}} \right) } \right. } \right. \nonumber \\&\left. { +\,{{{\bar{\varepsilon }} }_p}{{{\bar{\xi }} }_{p4}}\left( {{{\bar{\xi }}_{n3}} + {{{\bar{\xi }} }_{n4}}} \right) \cos \left( {n\pi } \right) \sin \left( {\frac{{p\pi }}{{1 + {{{\bar{\phi }} }^2}}}} \right) {\text {csch}} \left( {\frac{{{\bar{\phi }} {{\bar{\varepsilon }}_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) } \right] \nonumber \\&-\, {{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n7}}\left( {{{{\bar{L}}}_p} + {{{\bar{K}}}_p}{{{\bar{\xi }} }_{p1}}} \right) \left[ {p\pi {\bar{\phi }} \left( {4n{\bar{\varsigma }} _p^2 + {{{\bar{\xi }} }_{n4}}{{{\bar{\xi }} }_{n6}} + {{{\bar{\xi }} }_{n3}}{{{\bar{\xi }} }_{n8}}} \right) } \right. \nonumber \\&\left. +\,{{{{\bar{\varsigma }} }_p}{{{\bar{\xi }} }_{p3}}\left( {{{{\bar{\xi }} }_{n3}} + {{{\bar{\xi }} }_{n4}}} \right) \cos \left( {n\pi } \right) \sin \left( {\frac{{p\pi }}{{1 + {{{\bar{\phi }} }^2}}}} \right) {\text {csch}} \left( {\frac{{{\bar{\phi }} {{{\bar{\varsigma }} }_p}}}{{1 + {{{\bar{\phi }} }^2}}}} \right) } \right\} \end{aligned}$$

(A12)

where ${{\bar{\xi }} _{n3}} = n + p + n{{\bar{\phi }} ^2}$, ${{\bar{\xi }} _{n4}} = n - p + n{{\bar{\phi }} ^2}$, ${{\bar{\xi }} _{p3}} = {\pi ^2}\left[ {{p^2} - {n^2}\left( {1 + {{{\bar{\phi }} }^2}} \right) } \right] + {{\bar{\phi }} ^2}{\hat{\omega }}$, and ${{\bar{\xi }} _{p4}} = {\pi ^2}\left[ {{p^2} - {n^2}\left( {1 + {{{\bar{\phi }} }^2}} \right) } \right] - {{\bar{\phi }} ^2}{\hat{\omega }}$.

$$\begin{aligned} {f_{13}}\left( {m,p,\phi ,{\bar{\omega }} ,{{{\bar{E}}}_m},{{{\bar{F}}}_m}} \right)= & {} \frac{{mp{\pi ^2}\phi }}{{{\bar{\omega }} {\xi _{m5}}{\xi _{m6}}{\xi _{m7}}{\xi _{m8}}\sqrt{1 + {\phi ^2}} }} \nonumber \\&\times \, \left\{ {{\xi _{m5}}{\xi _{m7}}\left( {{{{\bar{F}}}_m} + {{{\bar{E}}}_m}{\xi _{m1}}} \right) \left[ {{\xi _{p3}} + 2\zeta _m^2\left( {1 + {\phi ^2}} \right) } \right] } \right. \nonumber \\&\left. { -\, {\xi _{m6}}{\xi _{m8}}\left( {{{{\bar{F}}}_m} + {{\bar{E}}_m}{\xi _{m2}}} \right) \left[ {{\xi _{p4}} + 2\eta _m^2\left( {1 + {\phi ^2}} \right) } \right] } \right\} \end{aligned}$$

(A13)

$$\begin{aligned} {f_{14}}\left( {n,p,{\bar{\phi }} ,{\hat{\omega }} ,{{{\bar{G}}}_n},{{\bar{H}}_n}} \right)= & {} \frac{{np{\pi ^2}{\bar{\phi }} \cos \left( {p\pi } \right) }}{{{\hat{\omega }} {{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n6}}{{\bar{\xi }}_{n7}}{{{\bar{\xi }} }_{n8}}\sqrt{1 + {{{\bar{\phi }} }^2}} }} \nonumber \\&\times \, \left\{ {{{{\bar{\xi }} }_{n5}}{{{\bar{\xi }} }_{n7}}\left( {{{{\bar{H}}}_n} + {{{\bar{G}}}_n}{{{\bar{\xi }} }_{n1}}} \right) \left[ {{{{\bar{\xi }} }_{p3}} + 2{\bar{\zeta }} _n^2\left( {1 + {{{\bar{\phi }} }^2}} \right) } \right] } \right. \nonumber \\&\left. { -\, {{{\bar{\xi }} }_{n6}}{{{\bar{\xi }} }_{n8}}\left( {{{{\bar{H}}}_n} + {{\bar{G}}_n}{{{\bar{\xi }} }_{n2}}} \right) \left[ {{{{\bar{\xi }} }_{p4}} + 2\bar{\eta }_n^2\left( {1 + {{{\bar{\phi }} }^2}} \right) } \right] } \right\} \end{aligned}$$

(A14)