Analysis on nonlinear dynamics of a deploying composite laminated cantilever plate

Abstract

This paper presents the analysis on the nonlinear dynamics of a deploying orthotropic composite laminated cantilever rectangular plate subjected to the aerodynamic pressures and the in-plane harmonic excitation. The third-order nonlinear piston theory is employed to model the transverse air pressures. Based on Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the nonlinear governing equations of motion are derived for the deploying composite laminated cantilever rectangular plate. The Galerkin method is utilized to discretize the partial differential governing equations to a two-degree-of-freedom nonlinear system. The two-degree-of-freedom nonlinear system is numerically studied to analyze the stability and nonlinear vibrations of the deploying composite laminated cantilever rectangular plate with the change of the realistic parameters. The influences of different parameters on the stability of the deploying composite laminated cantilever rectangular plate are analyzed. The numerical results show that the deploying velocity and damping coefficient have great effects on the amplitudes of the nonlinear vibrations, which may lead to the jumping phenomenon of the amplitudes for first-order and second-order modes. The increase of the damping coefficient can suppress the increase of the amplitudes of the nonlinear vibration.

Appendices

Appendix A

Letting $\alpha = \pi^{2}( \frac{E}{l_{0}b\rho} )^{1 / 2}$, the coefficients of the non-dimensional equation (15) are given as follows:

$$\begin{aligned} &{a_{10} = 1,\quad a_{11} = \frac{\bar{A}_{66}}{\bar{A}_{11}} \frac{l_{0}^{2}}{b^{2}},} \\ &{a_{12} = \frac{\bar{A}_{12} + \bar{A}_{66}}{\bar{A}_{11}},\quad a_{13} = \frac{\bar{A}_{12} + \bar{A}_{66}}{\bar{A}_{11}}\frac{h^{2}}{b^{2}},} \\ &{a_{14} = \frac{h^{2}}{l_{0}^{2}},\quad a_{15} = \frac{\bar{A}_{66}}{\bar{A}_{11}}\frac{h^{2}}{b^{2}},} \\ &{a_{16} = - \frac{\bar{I}_{0}}{\bar{A}_{11}}l_{0}^{2} \alpha^{2}\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}},\qquad a_{17} = \frac{\bar{I}_{0}}{\bar{A}_{11}}l_{0}^{2}\alpha^{2},} \\ &{a_{18} = \frac{\bar{I}_{1} - \bar{c}_{1}\bar{I}_{3}}{\bar{A}_{11}}l_{0} \alpha^{2},\quad a_{19} = - \frac{\bar{c}_{1}\bar{I}_{3}}{\bar{A}_{11}}h \alpha^{2}.} \\ &{b_{10} = 1,\quad b_{11} = \frac{\bar{A}_{66}}{\bar{A}_{22}} \frac{b^{2}}{l_{0}^{2}},\quad b_{12} = \frac{\bar{A}_{12} + \bar{A}_{66}}{\bar{A}_{22}},} \\ &{ b_{13} = \frac{\bar{A}_{12} + \bar{A}_{66}}{\bar{A}_{22}}\frac{h^{2}}{l_{0}^{2}},\quad b_{14} = \frac{h^{2}}{b^{2}},} \\ &{b_{15} = \frac{\bar{A}_{66}}{\bar{A}_{22}}\frac{h^{2}}{l_{0}^{2}},\quad b_{16} = \frac{\bar{I}_{0}}{\bar{A}_{22}}b^{2}\alpha^{2},} \\ &{b_{17} = \frac{\bar{I}_{1} - \bar{c}_{1}\bar{I}_{3}}{\bar{A}_{22}}b\alpha^{2},\quad b_{18} = - \frac{\bar{c}_{1}\bar{I}_{3}}{\bar{A}_{22}}h\alpha^{2}.} \end{aligned}$$

(21)

Letting $\lambda_{1} = \bar{A}_{55} - \bar{c}_{2}\bar{D}_{55} + \bar{c}_{2}^{2}\bar{F}_{55}$, we have

$$\begin{aligned} &{c_{10} = 1,\quad c_{11} = \frac{\bar{A}_{44} - 2\bar{c}_{2}\bar{D}_{44} + \bar{c}_{2}^{2}\bar{F}_{44}}{\lambda_{1}} \frac{l_{0}^{2}}{b^{2}} - \frac{f_{y}}{\lambda_{1}}\frac{l_{0}^{2}}{b^{2}},} \\ &{c_{12} = \frac{\bar{A}_{11}}{\lambda_{1}},\quad c_{13} = \frac{\bar{A}_{21}}{\lambda_{1}},\quad c_{14} = \frac{2\bar{A}_{66}}{\lambda_{1}} \frac{l_{0}^{2}}{b^{2}},} \\ &{c_{15} = \frac{\bar{A}_{11}}{\lambda_{1}},\quad c_{16} = \frac{\bar{A}_{66}}{\lambda_{1}},\quad c_{17} = \frac{\bar{A}_{21} + \bar{A}_{66}}{\lambda_{1}} \frac{l_{0}^{2}}{b^{2}},} \\ &{c_{18} = \frac{\bar{A}_{12} + \bar{A}_{66}}{\lambda_{1}},\quad c_{19} = \frac{\bar{A}_{22}}{\lambda_{1}}\frac{l_{0}^{2}}{b^{2}},\quad c_{20} = \frac{\bar{A}_{12}}{\lambda_{1}},} \\ &{c_{21} = \frac{\bar{A}_{22}}{\lambda_{1}}\frac{l_{0}^{2}}{b^{2}},\quad c_{22} = \frac{2\bar{A}_{66}}{\lambda_{1}},\quad c_{23} = \frac{\bar{A}_{66}}{\lambda_{1}}\frac{l_{0}^{2}}{b^{2}},} \\ &{c_{24} = \frac{3}{2}\frac{\bar{A}_{11}}{\lambda_{1}} \frac{h^{2}}{l_{0}^{2}},\quad c_{25} = \frac{\frac{1}{2}\bar{A}_{21} + \bar{A}_{66}}{\lambda_{1}} \frac{h^{2}}{b^{2}},} \\ &{c_{26} = \frac{\frac{1}{2}\bar{A}_{12} + \bar{A}_{66}}{\lambda_{1}}\frac{h^{2}}{b^{2}},\quad c_{27} = \frac{3\bar{A}_{22}}{2\lambda_{1}}\frac{l_{0}^{2}h^{2}}{b^{4}},} \\ &{c_{28} = \frac{\bar{A}_{12} + \bar{A}_{21} + 4\bar{A}_{66}}{\lambda_{1}}\frac{h^{2}}{b^{2}},} \\ &{c_{29} = \frac{\bar{A}_{55} - 2\bar{c}_{2}\bar{D}_{55} + \bar{c}_{2}^{2}\bar{F}_{55}}{\lambda_{1}}\frac{l_{0}}{h},} \\ &{c_{30} = \frac{\bar{A}_{44} - 2\bar{c}_{2}\bar{D}_{44} + \bar{c}_{2}^{2}\bar{F}_{44}}{\lambda_{1}}\frac{l_{0}^{2}}{bh},} \end{aligned}$$

(22)

$$\begin{aligned} &{c_{31} = \frac{\bar{F}_{11}\bar{c}_{1} - \bar{H}_{11}\bar{c}_{1}^{2}}{\lambda_{1}}\frac{1}{l_{0}h},} \\ &{c_{32} = \frac{\bar{F}_{22}\bar{c}_{1} - \bar{H}_{22}\bar{c}_{1}^{2}}{\lambda_{1}}\frac{l_{0}^{2}}{hb^{3}},} \\ &{c_{33} = - \frac{\bar{c}_{1}^{2}\bar{H}_{11}}{\lambda_{1}}\frac{1}{l_{0}^{2}},\quad c_{34} = - \frac{\bar{c}_{1}^{2}\bar{H}_{22}}{\lambda_{1}}\frac{l_{0}^{2}}{b^{4}},} \\ &{c_{35} = \frac{\bar{c}_{1}\bar{F}_{12} + 2\bar{c}_{1}\bar{F}_{66} - \bar{H}_{12}\bar{c}_{1}^{2} - 2\bar{H}_{66}\bar{c}_{1}^{2}}{\lambda_{1}}\frac{1}{hb},} \\ &{c_{36} = \frac{\bar{c}_{1}\bar{F}_{21} + 2\bar{c}_{1}\bar{F}_{66} - \bar{H}_{21}\bar{c}_{1}^{2} - 2\bar{H}_{66}\bar{c}_{1}^{2}}{\lambda_{1}}\frac{l_{0}}{hb^{2}},} \\ &{c_{37} = - \frac{\bar{c}_{1}^{2}(\bar{H}_{12} + \bar{H}_{21} + 4\bar{H}_{66})}{\lambda_{1}}\frac{1}{b^{2}},} \\ &{c_{38} = - \frac{4q_{d}\gamma l_{0}^{2}\alpha}{\lambda_{1}M_{\infty} v}\frac{\mathrm{d}x}{\mathrm{d}t},} \\ &{c_{39} = - \frac{4q_{d}\gamma l_{0}^{2}}{\lambda_{1}M_{\infty} b},\quad c_{40} = - \frac{4q_{d}\gamma l_{0}^{2}\alpha}{\lambda_{1}M_{\infty} v} - \frac{\delta}{ \lambda_{1}}l_{0}^{2}\alpha,} \\ &{c_{41} = - \frac{q_{d}( \kappa + 1 )\gamma^{3}M_{\infty}}{3\lambda_{1}}\frac{l_{0}^{2}}{h},\quad c_{42} = \frac{h\alpha}{v}\frac{\mathrm{d}x}{\mathrm{d}t},} \\ &{c_{43} = \frac{h}{b},\quad c_{44} = \frac{h\alpha}{v},\quad c_{45} = \frac{\bar{I}_{0}l_{0}^{2}}{\lambda_{1}} \alpha^{2},} \\ &{ c_{46} = - \frac{\bar{c}_{1}^{2}\bar{I}_{6}}{\lambda_{1}}\alpha^{2},\quad c_{47} = - \frac{\bar{c}_{1}^{2}\bar{I}_{6}}{\lambda_{1}}\frac{l_{0}^{2}}{b^{2}}\alpha^{2},} \\ &{c_{48} = \frac{\bar{c}_{1}\bar{I}_{3}}{\lambda_{1}}\frac{l_{0}^{2}}{h} \alpha^{2},\quad c_{49} = \frac{\bar{c}_{1}\bar{I}_{4} - \bar{c}_{1}^{2}\bar{I}_{6}}{\lambda_{1}} \frac{l_{0}}{h}\alpha^{2},} \\ &{c_{50} = \frac{\bar{c}_{1}\bar{I}_{4} - \bar{c}_{1}^{2}\bar{I}_{6}}{\lambda_{1}}\frac{l_{0}^{2}}{bh} \alpha^{2}.} \end{aligned}$$

Letting $\lambda_{2} = \bar{D}_{11} - 2\bar{r}_{1}\bar{F}_{11} + \bar{r}_{1}^{2}\bar{H}_{11}$, we have

$$\begin{aligned} &{d_{10} = 1,\quad d_{11} = \frac{\bar{D}_{66} - 2\bar{c}_{1}\bar{F}_{66} + \bar{c}_{1}^{2}\bar{H}_{66}}{\lambda_{2}} \frac{l_{0}^{2}}{b^{2}},} \\ &{d_{12} = \frac{\bar{c}_{1}^{2}\bar{H}_{11} - \bar{c}_{1}\bar{F}_{11}}{\lambda_{2}}\frac{h}{l},} \\ &{ d_{13} = \frac{\bar{D}_{12} - 2\bar{c}_{1}\bar{F}_{12} + \bar{D}_{66} - 2\bar{c}_{1}\bar{F}_{66} + \bar{c}_{1}^{2}\bar{H}_{12} + \bar{c}_{1}^{2}\bar{H}_{66}}{\lambda_{2}}\frac{l_{0}}{b},} \\ &{d_{14} = \frac{2\bar{c}_{2}\bar{D}_{55} - \bar{A}_{55} - \bar{c}_{2}^{2}\bar{F}_{55}}{\lambda_{2}}l_{0}h,} \\ &{ d_{15} = \frac{ - \bar{c}_{1}\bar{F}_{12} - 2\bar{c}_{1}\bar{F}_{66} + \bar{c}_{1}^{2}\bar{H}_{12} + 2\bar{c}_{1}^{2}\bar{H}_{66}}{\lambda_{2}}\frac{l_{0}h}{b^{2}},} \\ &{d_{16} = \frac{2\bar{c}_{2}\bar{D}_{55} - \bar{A}_{55} - \bar{c}_{2}^{2}\bar{F}_{55}}{\lambda_{2}}l_{0}^{2},} \\ &{d_{17} = \frac{\bar{I}_{1} - \bar{c}_{1}\bar{I}_{3}}{\lambda_{2}}l_{0}^{3} \alpha^{2},} \\ &{d_{18} = \frac{\bar{I}_{2} - 2\bar{c}_{1}\bar{I}_{4} - \bar{c}_{1}^{2}\bar{I}_{6}}{\lambda_{2}}l_{0}^{2} \alpha^{2},} \\ &{d_{19} = \frac{- \bar{c}_{1}\bar{I}_{4} + \bar{c}_{1}^{2}\bar{I}_{6}}{\lambda_{2}}hl_{0} \alpha^{2}.} \end{aligned}$$

(23)

Letting $\lambda_{3} = \bar{D}_{22} - 2\bar{r}_{1}\bar{F}_{22} + 2\bar{r}_{1}^{2}\bar{H}_{22}$, we have

$$\begin{aligned} &{e_{10} = 1,\quad e_{11} = \frac{\bar{D}_{66} - 2\bar{c}_{1}\bar{F}_{66} + \bar{c}_{1}^{2}\bar{H}_{66}}{\lambda_{3}} \frac{b^{2}}{l_{0}^{2}},} \\ &{e_{12} = \frac{\bar{c}_{1}^{2}\bar{H}_{22} - \bar{c}_{1}\bar{F}_{22}}{\lambda_{3}}\frac{h}{b},} \\ &{e_{13} = \frac{\bar{D}_{21} - 2\bar{c}_{1}\bar{F}_{21} + \bar{D}_{66} - 2\bar{c}_{1}\bar{F}_{66} + \bar{c}_{1}^{2}\bar{H}_{21} + \bar{c}_{1}^{2}\bar{H}_{66}}{\lambda_{3}}\frac{b}{l},} \\ &{e_{14} = \frac{2\bar{c}_{2}\bar{D}_{44} - \bar{A}_{44} - \bar{c}_{2}^{2}\bar{F}_{44}}{\lambda_{3}}bh,} \\ &{e_{15} = \frac{ - \bar{c}_{1}\bar{F}_{21} - 2\bar{c}_{1}\bar{F}_{66} + \bar{c}_{1}^{2}\bar{H}_{21} + 2\bar{c}_{1}^{2}\bar{H}_{66}}{\lambda_{3}}\frac{bh}{l_{0}^{2}},} \\ &{e_{16} = \frac{2\bar{c}_{2}\bar{D}_{44} - \bar{A}_{44} - \bar{c}_{2}^{2}\bar{F}_{44}}{\lambda_{3}}\frac{bh}{l_{0}^{2}},} \\ &{e_{17} = \frac{\bar{I}_{1} - \bar{c}_{1}\bar{I}_{3}}{\lambda_{3}}b^{3} \alpha^{2},\quad e_{18} = \frac{\bar{I}_{2} - 2\bar{c}_{1}\bar{I}_{4} - \bar{c}_{1}^{2}\bar{I}_{6}}{\lambda_{3}}b^{2} \alpha^{2},} \\ &{e_{19} = \frac{ - \bar{c}_{1}\bar{I}_{4} + \bar{c}_{1}^{2}\bar{I}_{6}}{\lambda_{3}}hb\alpha^{2}.} \end{aligned}$$

(24)

Appendix B

Using the Galerkin method, we obtain the coefficients of the first two modes for the displacement variables u ₀, v ₀, w ₀, ϕ _x , ϕ _y . The coefficients of u ₁(t) are obtained as

$$\begin{aligned} &{k_{11} = - 0.0314\frac{\pi^{2}}{4l^{2}}a_{10},\quad k_{12} = - 0.0314\frac{\pi^{2}}{b^{2}}a_{11},} \\ &{k_{13} = 0.1390\frac{\pi^{2}}{2lb}a_{12},\quad k_{14} = 0.0872\frac{3\pi^{2}}{lb}a_{12},} \\ &{k_{15} = 0.6180\frac{k_{1}k_{2}^{2}}{lb^{2}}a_{13},} \\ &{ k_{16} = 0.2403\frac{k_{2}k_{4}( k_{1} + k_{3} )}{lb^{2}}a_{13},} \\ &{k_{17} = - 0.0029\frac{k_{2}k_{3}k_{4}}{lb^{2}}a_{13},} \\ &{k_{18} = 0.6180\frac{k_{1}^{3}}{l^{3}}a_{14},} \\ &{k_{19} = - 0.0953\frac{k_{1}k_{3}( k_{1} + k_{3} )}{l^{3}}a_{14},} \\ &{k_{20} = 0.0426\frac{k_{3}^{3}}{l^{3}}a_{14},\quad k_{21} = - 0.4319\frac{k_{1}k_{2}^{2}}{lb^{2}}a_{15},} \\ &{k_{22} = - 0.2047\frac{( k_{1}k_{4}^{2} + k_{3}k_{2}^{2} )}{lb^{2}}a_{15},} \\ &{k_{23} = 0.0548\frac{k_{3}k_{4}^{2}}{lb^{2}}a_{15},\quad k_{24} = 0.1534a_{16}.} \end{aligned}$$

(25)

The coefficients of u ₂(t) are obtained as

$$\begin{aligned} &{l_{11} = - 0.1599\frac{9\pi^{2}}{4l^{2}}a_{10},\quad l_{12} = - 0.1599\frac{4\pi^{2}}{b^{2}}a_{11},} \\ &{l_{13} = 0.3094\frac{\pi^{2}}{2lb}a_{12},\quad l_{14} = 0.2073\frac{3\pi^{2}}{lb}a_{12},} \\ &{l_{15} = 0.8094\frac{k_{1}k_{2}^{2}}{lb^{2}}a_{13},} \\ &{l_{16} = 0.7573\frac{k_{2}k_{4}( k_{1} + k_{3} )}{lb^{2}}a_{13},} \\ &{l_{17} = - 0.0194\frac{k_{2}k_{3}k_{4}}{lb^{2}}a_{13},} \\ &{l_{18} = 0.8094\frac{k_{1}^{3}}{l^{3}}a_{14},} \\ &{l_{19} = - 0.2368\frac{k_{1}k_{3}( k_{1} + k_{3} )}{l^{3}}a_{14},} \\ &{l_{20} = 0.0927\frac{k_{3}^{3}}{l^{3}}a_{14},\quad l_{21} = - 0.1343\frac{k_{1}k_{2}^{2}}{lb^{2}}a_{15},} \\ &{l_{22} = - 0.1248\frac{( k_{1}k_{4}^{2} + k_{3}k_{2}^{2} )}{lb^{2}}a_{15},} \\ &{l_{23} = 0.1051\frac{k_{3}k_{4}^{2}}{lb^{2}}a_{15},\quad l_{24} = 0.3354a_{16}.} \end{aligned}$$

(26)

The coefficients of v ₁(t) are obtained as

$$\begin{aligned} &{m_{11} = - 0.0049\frac{\pi^{2}}{b^{2}}b_{10},\quad m_{12} = - 0.0049\frac{\pi^{2}}{4l^{2}}b_{11},} \\ &{m_{13} = 0.0219\frac{\pi^{2}}{2lb}b_{12},\quad m_{14} = 0.0300\frac{\pi^{2}}{lb}b_{12},} \\ &{m_{15} = - 0.0176\frac{k_{1}^{2}k_{2}}{bl^{2}}b_{13},} \\ &{m_{16} = 0.00008\frac{k_{1}k_{3}( k_{4} + k_{2} )}{bl^{2}}b_{13},} \\ &{m_{17} = 0.2206\frac{k_{3}^{2}k_{4}}{bl^{2}}b_{13},} \\ &{m_{18} = - 0.0176\frac{k_{2}^{3}}{b^{3}}b_{14},} \\ &{m_{19} = - 0.0522\frac{k_{2}k_{4}( k_{4} + k_{2} )}{b^{3}}b_{14},} \\ &{m_{20} = 0.0822\frac{k_{4}^{3}}{b^{3}}b_{14},} \\ &{m_{21} = - 0.0010\frac{k_{1}^{2}k_{2}}{l^{2}b}b_{15},} \\ &{m_{22} = 0.5992\frac{( k_{4}k_{1}^{2} + k_{2}k_{3}^{2} )}{l^{2}b}b_{15},} \\ &{m_{23} = - 0.0012\frac{k_{3}^{2}k_{4}}{l^{2}b}b_{15}.} \end{aligned}$$

(27)

The coefficients of v ₂(t) are obtained as

$$\begin{aligned} &{n_{11} = - 0.1135\frac{4\pi^{2}}{b^{2}}b_{10},\quad n_{12} = - 0.1135\frac{9\pi^{2}}{4l^{2}}b_{11},} \\ &{n_{13} = 0.1062\frac{\pi^{2}}{2lb}b_{12},\quad n_{14} = 0.1472\frac{3\pi^{2}}{lb}b_{12},} \\ &{n_{15} = - 0.1214\frac{k_{1}^{2}k_{2}}{bl^{2}}b_{13},} \\ &{n_{16} = - 0.0048\frac{k_{1}k_{3}( k_{4} + k_{2} )}{bl^{2}}b_{13},} \\ &{n_{17} = 1.1086\frac{k_{3}^{2}k_{4}}{bl^{2}}b_{13},} \\ &{n_{18} = - 0.1218\frac{k_{2}^{3}}{b^{3}}b_{14},} \\ &{n_{19} = - 0.1878\frac{k_{2}k_{4}( k_{4} + k_{2} )}{b^{3}}b_{14},} \\ &{n_{20} = 0.3605\frac{k_{4}^{3}}{b^{3}}b_{14},} \\ &{n_{21} = - 0.0074\frac{k_{1}^{2}k_{2}}{l^{2}b}b_{15},} \\ &{n_{22} = 3.0182\frac{( k_{4}k_{1}^{2} + k_{2}k_{3}^{2} )}{l^{2}b}b_{15},} \\ &{n_{23} = - 0.0060\frac{k_{3}^{2}k_{4}}{l^{2}b}b_{15}.} \end{aligned}$$

(28)

The coefficients of ϕ _x1(t) are obtained as

$$\begin{aligned} &{k_{31} = - 0.0314\frac{\pi^{2}}{4l^{2}}d_{10},\quad k_{32} = - 0.0314\frac{\pi^{2}}{b^{2}}d_{11},} \\ &{k_{33} = - 0.2696\frac{k_{1}^{3}}{l^{3}}d_{12},\quad k_{34} = - 0.0126\frac{k_{3}^{3}}{l^{3}}d_{12},} \\ &{k_{35} = 0.0104\frac{\pi^{2}}{2lb}d_{13},\quad k_{36} = 0.0362\frac{2\pi^{2}}{lb}d_{13},} \\ &{k_{37} = 0.1291\frac{k_{1}}{l}d_{14},\quad k_{38} = 0.0025\frac{k_{3}}{l}d_{14},} \\ &{k_{39} = - 0.3729\frac{k_{1}k_{2}^{2}}{lb^{2}}d_{15},\quad k_{40} = 0.0077\frac{k_{3}k_{4}^{2}}{lb^{2}}d_{15},} \\ &{ k_{41} = d_{16}.} \end{aligned}$$

(29)

The coefficients of ϕ _x2(t) are obtained as

$$\begin{aligned} &{l_{31} = - 0.0485\frac{\pi^{2}}{l^{2}}d_{10},\quad l_{32} = - 0.0485\frac{4\pi^{2}}{b^{2}}d_{11},} \\ &{ l_{33} = - 0.5242\frac{k_{1}^{3}}{l^{3}}d_{12},\quad l_{34} = - 0.0453\frac{k_{3}^{3}}{l^{3}}d_{12},} \\ &{l_{35} = 0.0136\frac{\pi^{2}}{2lb}d_{13},\quad l_{36} = 0.0482\frac{2\pi^{2}}{lb}d_{13},} \\ &{ l_{37} = 0.2500\frac{k_{1}}{l}d_{14},\quad l_{38} = 0.0010\frac{k_{3}}{l}d_{14},} \\ &{k_{39} = - 0.7307\frac{k_{1}k_{2}^{2}}{lb^{2}}d_{15},} \\ &{l_{40} = - 0.0025\frac{k_{3}k_{4}^{2}}{lb^{2}}d_{15},\quad l_{41} = d_{16}.} \end{aligned}$$

(30)

The coefficients of ϕ _y1(t) are obtained as

$$\begin{aligned} &{m_{31} = 0.0005\frac{\pi^{2}}{b^{2}}e_{10},\quad m_{32} = 0.0154\frac{\pi^{2}}{4l^{2}}e_{11},} \\ &{ m_{33} = - 0.0365\frac{k_{2}^{3}}{b^{3}}e_{12},\quad m_{34} = 0.0110\frac{k_{4}^{3}}{b^{3}}e_{12},} \\ &{m_{35} = 0.0051\frac{\pi^{2}}{2lb}e_{13},\quad m_{36} = 0.0083\frac{2\pi^{2}}{lb}e_{13},} \\ &{m_{37} = - 0.0257\frac{k_{2}}{b}e_{14},\quad m_{38} = - 0.0176\frac{k_{4}}{b}e_{14},} \\ &{m_{39} = - 0.0067\frac{k_{1}^{2}k_{2}}{l^{2}b}e_{15},\quad m_{40} = 0.0140\frac{k_{3}^{2}k_{4}}{l^{2}b}e_{15},} \\ &{ m_{41} = e_{16}.} \end{aligned}$$

(31)

The coefficients of ϕ _y2(t) are obtained as

$$\begin{aligned} &{n_{31} = 0.0017\frac{4\pi^{2}}{b^{2}}e_{10},\quad n_{32} = 0.0485\frac{\pi^{2}}{l^{2}}e_{11},} \\ &{ n_{33} = - 0.0095\frac{k_{2}^{3}}{b^{3}}e_{12},\quad n_{34} = 0.0525\frac{k_{4}^{3}}{b^{3}}e_{12},} \\ &{n_{35} = 0.0136\frac{\pi^{2}}{2lb}e_{13},\quad n_{36} = 0.0224\frac{2\pi^{2}}{lb}e_{13},} \\ &{ n_{37} = - 0.0698\frac{k_{2}}{b}e_{14},\quad n_{38} = - 0.0724\frac{k_{4}}{b}e_{14},} \\ &{n_{39} = - 0.0186\frac{k_{1}^{2}k_{2}}{l^{2}b}e_{15},\quad n_{40} = 0.0576\frac{k_{3}^{2}k_{4}}{l^{2}b}e_{15},} \\ &{ n_{41} = e_{16}.} \end{aligned}$$

(32)

The coefficients of w ₁(t) are obtained as

$$\begin{aligned} &{r_{11} = 0.1386\frac{k_{1}^{2}}{l^{2}}\biggl( c_{10} + c_{45}\biggl( \frac{\mathrm{d}l}{\mathrm{d}t} \biggr)^{2} \biggr),} \\ &{ r_{12} = - 0.4359\frac{k_{3}^{2}}{l^{2}}\biggl( c_{10} + c_{45}\biggl( \frac{\mathrm{d}l}{\mathrm{d}t} \biggr)^{2} \biggr),} \\ &{r_{13} = - 0.1813\frac{k_{2}^{2}}{b^{2}}c_{11},\quad r_{14} = - 0.2417\frac{k_{4}^{2}}{b^{2}}c_{11},} \\ &{ r_{15} = 0.0673\frac{\pi k_{1}^{2}}{2l^{3}}c_{12},\quad r_{16} = - 0.2065\frac{\pi k_{3}^{2}}{2l^{3}}c_{12},} \\ &{ r_{17} = 0.1229\frac{3\pi k_{1}^{2}}{2l^{3}}c_{12},\quad r_{18} = - 0.2843\frac{3\pi k_{3}^{2}}{2l^{3}}c_{12},} \\ &{ r_{19} = - 0.0835\frac{\pi k_{2}^{2}}{2lb^{2}}c_{13},\quad r_{20} = - 0.1127\frac{\pi k_{4}}{2lb^{2}}c_{13},} \\ &{ r_{21} = - 0.1035\frac{3\pi k_{2}^{2}}{2lb^{2}}c_{13},\quad r_{22} = - 0.1517\frac{3\pi k_{4}^{2}}{2lb^{2}}c_{13},} \\ &{ r_{23} = 0.0043\frac{\pi k_{1}k_{2}}{lb^{2}}c_{14},\quad r_{24} = - 0.0064\frac{\pi k_{3}k_{4}}{lb^{2}}c_{14},} \\ &{r_{25} = 0.0289\frac{2\pi k_{1}k_{2}}{lb^{2}}c_{14},} \\ &{r_{26} = - 0.0289\frac{2\pi k_{31}k_{4}}{lb^{2}}c_{14},} \\ &{ r_{27} = 0.1456\frac{\pi^{2}k_{1}}{4l^{3}}c_{15},\quad r_{28} = - 0.0243\frac{\pi^{2}k_{3}}{4l^{3}}c_{15},} \\ &{ r_{29} = 0.3616\frac{9\pi^{2}k_{1}}{4l^{3}}c_{15},\quad r_{30} = - 0.0109\frac{9\pi^{2}k_{3}}{4l^{3}}c_{15},} \\ &{ r_{31} = - 0.0029\frac{\pi^{2}k_{2}}{4l^{2}b}c_{16},\quad r_{32} = - 0.0353\frac{\pi^{2}k_{4}}{4l^{2}b}c_{16},} \\ &{r_{33} = - 0.0195\frac{9\pi^{2}k_{2}}{4l^{2}b}c_{16},} \\ &{r_{34} = - 0.1750\frac{9\pi^{2}k_{4}}{4l^{2}b}c_{16},} \end{aligned}$$

(33)

$$\begin{aligned} &{ r_{35} = - 0.0105\frac{\pi^{2}k_{2}}{2lb^{2}}c_{17},\quad r_{36} = - 0.1384\frac{\pi^{2}k_{4}}{2lb^{2}}c_{17},} \\ &{ r_{37} = - 0.0186\frac{3\pi^{2}k_{2}}{lb^{2}}c_{17},} \\ &{r_{38} = - 0.0186\frac{3\pi^{2}k_{4}}{lb^{2}}c_{17},} \\ &{ r_{39} = 0.5712\frac{\pi^{2}k_{1}}{2l^{2}b}c_{18},\quad r_{40} = - 0.0246\frac{\pi^{2}k_{3}}{2l^{2}b}c_{18},} \\ &{r_{41} = 0.3884\frac{3\pi^{2}k_{1}}{l^{2}b}c_{18},\quad r_{42} = 0.0109\frac{3\pi^{2}k_{3}}{l^{2}b}c_{18},} \\ &{r_{43} = 0.0029\frac{\pi^{2}k_{2}}{b^{3}}c_{19},\quad r_{44} = 0.0353\frac{\pi^{2}k_{4}}{b^{3}}c_{19},} \\ &{ r_{45} = 0.0195\frac{4\pi^{2}k_{2}}{b^{3}}c_{19},\quad r_{46} = 0.1750\frac{4\pi^{2}k_{4}}{b^{3}}c_{19},} \\ &{r_{47} = 0.0125\frac{\pi k_{1}}{l^{2}b}c_{20},\quad r_{48} = - 0.1116\frac{\pi k_{3}}{l^{2}b}c_{20},} \\ &{r_{49} = 0.0366\frac{2\pi k_{1}}{l^{2}b}c_{20},\quad r_{50} = - 0.2873\frac{2\pi k_{3}}{l^{2}b}c_{20},} \\ &{r_{51} = - 0.0461\frac{\pi k_{2}}{b^{3}}c_{21},\quad r_{52} = - 0.0572\frac{\pi k_{4}}{b^{3}}c_{21},} \\ &{ r_{53} = - 0.1084\frac{2\pi k_{2}}{b^{3}}c_{21},\quad r_{54} = - 0.1450\frac{2\pi k_{4}}{b^{3}}c_{21},} \\ &{ r_{55} = - 0.0167\frac{\pi k_{1}k_{2}}{2l^{2}b}c_{22},\quad r_{56} = 0.0065\frac{\pi k_{3}k_{4}}{2l^{2}b}c_{22},} \\ &{ r_{57} = - 0.0311\frac{3\pi k_{1}k_{2}}{2l^{2}b}c_{22},} \\ &{r_{58} = - 0.0055\frac{3\pi k_{3}k_{4}}{2l^{2}b}c_{22},} \end{aligned}$$

$$\begin{aligned} &{ r_{59} = 0.1456\frac{\pi^{2}k_{1}^{2}}{lb^{2}}c_{23},\quad r_{60} = - 0.0243\frac{\pi^{2}k_{3}^{2}}{lb^{2}}c_{23},} \\ &{ r_{61} = 0.3616\frac{4\pi^{2}k_{1}^{2}}{lb^{2}}c_{23},\quad r_{62} = - 0.0571\frac{4\pi^{2}k_{3}^{2}}{lb^{2}}c_{23},} \\ &{ r_{63} = 0.2217\frac{k_{1}^{4}}{l^{4}}c_{24},} \\ &{r_{64} = - 1.3458\frac{( k_{1}^{2}k_{3}^{2} + 2k_{1}^{3}k_{3} )}{l^{4}}c_{24},} \\ &{ r_{65} = 0.3432\frac{( k_{1}^{2}k_{3}^{2} + 2k_{1}k_{3}^{3} )}{l^{4}}c_{24},} \\ &{r_{66} = - 0.4387\frac{k_{3}^{4}}{l^{4}}c_{24},\quad r_{67} = - 0.4455\frac{k_{1}^{2}k_{2}^{2}}{l^{2}b^{2}}c_{25},} \\ &{ r_{68} = - 0.7402\frac{( k_{1}^{2}k_{4}^{2} + 2k_{2}^{2}k_{1}k_{3} )}{l^{2}b^{2}}c_{25},} \\ &{r_{69} = - 0.7008\frac{( k_{1}^{2}k_{3}^{2} + 2k_{4}^{2}k_{1}k_{3} )}{l^{2}b^{2}}c_{25},} \\ &{r_{70} = - 0.2480\frac{k_{3}^{2}k_{4}^{2}}{l^{2}b^{2}}c_{25},\quad r_{71} = - 0.0099\frac{k_{1}^{2}k_{2}^{2}}{l^{2}b^{2}}c_{26},} \\ &{ r_{72} = - 0.8399\frac{( k_{2}^{2}k_{3}^{2} + 2k_{1}^{2}k_{2}k_{4} )}{l^{2}b^{2}}c_{26},} \\ &{r_{73} = 0.1511\frac{( k_{1}^{2}k_{4}^{2} + 2k_{3}^{2}k_{2}k_{4} )}{l^{2}b^{2}}c_{26},} \end{aligned}$$

$$\begin{aligned} &{r_{74} = 3.8143\frac{k_{3}^{2}k_{4}^{2}}{l^{2}b^{2}}c_{26},\quad r_{75} = - 0.0693\frac{k_{2}^{4}}{b^{4}}c_{27},} \\ &{r_{76} = - 0.0084\frac{( k_{2}^{2}k_{4}^{2} + 2k_{2}^{3}k_{4} )}{b^{4}}c_{27},} \\ &{r_{77} = - 0.3677\frac{( k_{2}^{2}k_{4}^{2} + 2k_{2}k_{4}^{3} )}{b^{4}}c_{27},} \\ &{r_{78} = - 0.2997\frac{k_{4}^{4}}{b^{4}}c_{27},\quad r_{79} = 0.1909\frac{k_{1}^{2}k_{2}^{2}}{l^{2}b^{2}}c_{28},} \\ &{r_{80} = - 0.0051\frac{( k_{1}^{2}k_{2}k_{4} + k_{1}k_{2}^{2}k_{3} + k_{1}k_{2}k_{3}k_{4} )}{l^{2}b^{2}}c_{28},} \\ &{r_{81} = - 0.2473\frac{( k_{4}^{2}k_{1}k_{3} + k_{2}k_{3}^{2}k_{4} + k_{1}k_{2}k_{3}k_{4} )}{l^{2}b^{2}}c_{28},} \\ &{ r_{82} = 0.1107\frac{k_{3}^{2}k_{4}^{2}}{l^{2}b^{2}}c_{28},\quad r_{83} = 0.3059\frac{\pi}{2l}c_{29},} \\ &{r_{84} = 0.2763\frac{\pi}{l}c_{29},\quad r_{85} = - 0.0012\frac{\pi}{ b}c_{30},} \\ &{r_{86} = - 0.0100\frac{2\pi}{b}c_{30},\quad r_{87} = 0.3059\frac{\pi^{3}}{8l^{3}}c_{31},} \\ &{ r_{88} = 0.2763\frac{\pi^{3}}{l^{3}}c_{31},\quad r_{89} = 0.0012\frac{\pi^{3}}{b^{3}}c_{32},} \\ &{r_{90} = 0.0100\frac{8\pi^{3}}{b^{3}}c_{32},\quad r_{91} = 0.3836\frac{k_{1}^{4}}{l^{4}}c_{33},} \\ &{r_{92} = 0.5509\frac{k_{3}^{4}}{l^{4}}c_{33},\quad r_{93} = 0.3836\frac{k_{2}^{4}}{b^{4}}c_{34},} \\ &{ r_{94} = 0.5509\frac{k_{4}^{4}}{b^{4}}c_{34},\quad r_{95} = 0.0364\frac{\pi^{3}}{4l^{2}b}c_{35},} \\ &{r_{96} = 0.0710\frac{2\pi^{3}}{l^{2}b}c_{35},\quad r_{97} = 0.3059\frac{\pi^{3}}{2lb^{2}}c_{36},} \\ &{r_{98} = 0.2763\frac{4\pi^{3}}{lb^{2}}c_{36},\quad r_{99} = - 0.0655\frac{k_{1}^{2}k_{2}^{2}}{l^{2}b^{2}}c_{37},} \end{aligned}$$

$$\begin{aligned} &{r_{100} = 0.1913\frac{k_{3}^{2}k_{4}^{2}}{l^{2}b^{2}}c_{37},} \\ &{r_{101} = 0.6087\frac{k_{1}}{l}\biggl( c_{38} + c_{40}\frac{\mathrm{d}l}{\mathrm{d}t} + c_{45}\frac{\mathrm{d}^{2}l}{\mathrm{d}t^{2}} \biggr),} \\ &{r_{102} = - 0.0301\frac{k_{3}}{l}\biggl( c_{38} + c_{40}\frac{\mathrm{d}l}{\mathrm{d}t} + c_{45}\frac{\mathrm{d}^{2}l}{\mathrm{d}t^{2}} \biggr),} \\ &{r_{103} = - 0.0862\frac{k_{2}}{b}c_{39},} \\ &{r_{104} = - 0.4821\frac{k_{4}}{b}c_{39},\quad r_{105} = 0.5912c_{40},} \end{aligned}$$

$$\begin{aligned} &{r_{106} = c_{41}\biggl( 1.663\frac{k_{1}^{3}}{l^{3}}c_{42}^{3} - 0.216\frac{3k_{1}^{2}k_{2}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{r_{106} =} {}+ 0.135\frac{3k_{1}k_{2}^{2}}{lb^{2}}c_{42}c_{43}^{2} - 0.0302\frac{k_{2}^{3}}{b^{3}}c_{43}^{3} \biggr),} \\ &{r_{107} = c_{41}\biggl( - 0.226 \frac{k_{1}^{2}k_{3}}{l^{3}}c_{42}^{3} + 0.025\frac{2k_{1}k_{2}k_{3}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{r_{107} =} {}- 0.012\frac{k_{2}^{2}k_{3}}{lb^{2}}c_{42}^{2}c_{43} - 0.966\frac{k_{1}^{2}k_{4}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{r_{107} =} {} + 0.215\frac{2k_{1}k_{2}k_{4}}{lb^{2}}c_{42}^{2}c_{43} - 0.026\frac{k_{2}^{2}k_{4}}{b^{3}}c_{43}^{3} \biggr),} \\ &{r_{108} = c_{41}\biggl( 0.542\frac{k_{1}k_{3}^{2}}{l^{3}}c_{42}^{3} + 0.078\frac{2k_{1}k_{3}k_{4}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{r_{108} =} {} + 1.01\frac{k_{1}k_{4}^{2}}{lb^{2}}c_{42}c_{43}^{2} - 0.039\frac{k_{2}k_{3}^{2}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{r_{108} =} {}+ 0.0003\frac{2k_{2}k_{3}k_{4}}{lb^{2}}c_{42}c_{43}^{2} - 0.25\frac{k_{2}k_{4}^{2}}{b^{3}}c_{43}^{3} \biggr),} \\ &{r_{109} = c_{41}c_{44}\biggl( 1.09 \frac{k_{1}^{2}}{l^{2}}c_{42}^{2} - 0.15\frac{2k_{1}k_{2}}{lb}c_{42}c_{43}} \\ &{\phantom{r_{109} =} {} + 0.10\frac{k_{2}^{2}}{b^{2}}c_{43}^{2} \biggr),} \\ &{r_{110} = c_{41}\biggl( 0.77\frac{k_{1}}{l}c_{42} - 0.114\frac{k_{2}}{b}c_{43} \biggr)c_{44}^{2},} \end{aligned}$$

$$\begin{aligned} &{r_{111} = c_{41}c_{44}\biggl( 0.35 \frac{k_{3}^{2}}{l^{2}}c_{42}^{2} + 0.11\frac{2k_{3}k_{4}}{lb}c_{42}c_{43}} \\ &{\phantom{r_{111} =} {} + 0.65\frac{k_{4}^{2}}{b^{2}}c_{43}^{2} \biggr),} \\ &{r_{112} = c_{41}\biggl( - 0.29\frac{k_{3}}{l}c_{42} - 0.43\frac{k_{4}}{b}c_{43} \biggr)c_{44}^{2},} \\ &{r_{113} = c_{41}c_{44}\biggl( - 0.30 \frac{k_{1}k_{3}}{l^{2}}c_{42}^{2} - 0.62\frac{k_{1}k_{4}}{lb}c_{42}c_{43}} \\ &{\phantom{r_{113} =} {}+ 0.02\frac{k_{2}k_{3}}{lb}c_{42}c_{43} + 0.15\frac{k_{2}k_{4}}{b^{2}}c_{43}^{2} \biggr),} \\ &{r_{114} = c_{41}\biggl( - 0.04\frac{k_{3}^{3}}{l^{3}}c_{42}^{3} - 0.20\frac{3k_{3}^{2}k_{4}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{r_{114} =} {}- 0.05\frac{3k_{3}k_{4}^{2}}{lb^{2}}c_{42}c_{43}^{2} - 1.32\frac{k_{4}^{3}}{b^{3}}c_{43}^{3} \biggr),} \\ &{r_{115} = 0.298c_{41}c_{44}^{3}, \quad r_{116} = 0.3836c_{45},} \\ &{r_{117} = 0.6087c_{45}\frac{2k_{1}}{l} \frac{\mathrm{d}l}{\mathrm{d}t},} \\ &{r_{118} = - 0.03c_{45}\frac{2k_{3}}{l} \frac{\mathrm{d}l}{\mathrm{d}t}.} \end{aligned}$$

The coefficients of w ₂(t) are obtained as

$$\begin{aligned} &{s_{11} = 0.4601\frac{k_{1}^{2}}{l^{2}}\biggl( c_{10} + c_{45}\biggl( \frac{\mathrm{d}l}{\mathrm{d}t} \biggr)^{2} \biggr),} \\ &{ s_{12} = 0.00001\frac{k_{3}^{2}}{l^{2}}\biggl( c_{10} + c_{45}\biggl( \frac{\mathrm{d}l}{\mathrm{d}t} \biggr)^{2} \biggr),} \\ &{s_{13} = - 0.2014\frac{k_{2}^{2}}{b^{2}}c_{11},\quad s_{14} = - 1.2199\frac{k_{4}^{2}}{b^{2}}c_{11},} \\ &{s_{15} = 0.2242\frac{\pi k_{1}^{2}}{2l^{3}}c_{12},\quad s_{16} = - 0.6139\frac{\pi k_{3}^{2}}{2l^{3}}c_{12},} \\ &{s_{17} = 0.4000\frac{3\pi k_{1}^{2}}{2l^{3}}c_{12},\quad s_{18} = - 0.7953\frac{3\pi k_{3}^{2}}{2l^{3}}c_{12},} \\ &{s_{19} = - 0.0944\frac{\pi k_{2}^{2}}{2lb^{2}}c_{13},\quad s_{20} = - 0.5487\frac{\pi k_{4}}{2lb^{2}}c_{13},} \\ &{s_{21} = - 0.1286\frac{3\pi k_{2}^{2}}{2lb^{2}}c_{13},\quad s_{22} = - 0.6692\frac{3\pi k_{4}^{2}}{2lb^{2}}c_{13},} \\ &{s_{23} = 0.0609\frac{\pi k_{1}k_{2}}{lb^{2}}c_{14},\quad s_{24} = - 0.0001\frac{\pi k_{3}k_{4}}{lb^{2}}c_{14},} \\ &{ s_{25} = - 0.2882\frac{2\pi k_{1}k_{2}}{lb^{2}}c_{14},} \\ &{s_{26} = 0.0016\frac{2\pi k_{31}k_{4}}{lb^{2}}c_{14},} \\ &{ s_{27} = 0.2192\frac{\pi^{2}k_{1}}{4l^{3}}c_{15},\quad s_{28} = - 0.0018\frac{\pi^{2}k_{3}}{4l^{3}}c_{15},} \\ &{ s_{29} = 0.5740\frac{9\pi^{2}k_{1}}{4l^{3}}c_{15},\quad s_{30} = - 0.2790\frac{9\pi^{2}k_{3}}{4l^{3}}c_{15},} \\ &{ s_{31} = 0.0371\frac{\pi^{2}k_{2}}{4l^{2}b}c_{16},\quad s_{32} = - 0.0244\frac{\pi^{2}k_{4}}{4l^{2}b}c_{16},} \\ &{ s_{33} = 0.1733\frac{9\pi^{2}k_{2}}{4l^{2}b}c_{16},\quad s_{34} = - 0.1166\frac{9\pi^{2}k_{4}}{4l^{2}b}c_{16},} \end{aligned}$$

(34)

$$\begin{aligned} &{ s_{35} = 0.1455\frac{\pi^{2}k_{2}}{2lb^{2}}c_{17},\quad s_{36} = - 0.1050\frac{\pi^{2}k_{4}}{2lb^{2}}c_{17},} \\ &{s_{37} = 0.1864\frac{3\pi^{2}k_{2}}{lb^{2}}c_{17},\quad s_{38} = - 0.1492\frac{3\pi^{2}k_{4}}{lb^{2}}c_{17},} \\ &{s_{39} = 0.9565\frac{\pi^{2}k_{1}}{2l^{2}b}c_{18},\quad s_{40} = 0.2794\frac{\pi^{2}k_{3}}{2l^{2}b}c_{18},} \\ &{s_{41} = 0.7154\frac{3\pi^{2}k_{1}}{l^{2}b}c_{18},\quad s_{42} = 0.2790\frac{3\pi^{2}k_{3}}{l^{2}b}c_{18},} \\ &{s_{43} = - 0.0371\frac{\pi^{2}k_{2}}{b^{3}}c_{19},\quad s_{44} = 0.0244\frac{\pi^{2}k_{4}}{b^{3}}c_{19},} \\ &{s_{45} = - 0.1733\frac{4\pi^{2}k_{2}}{b^{3}}c_{19},\quad s_{46} = 0.1166\frac{4\pi^{2}k_{4}}{b^{3}}c_{19},} \\ &{ s_{47} = 0.0565\frac{\pi k_{1}}{l^{2}b}c_{20},\quad s_{48} = - 0.3162\frac{\pi k_{3}}{l^{2}b}c_{20},} \\ &{s_{49} = 0.1589\frac{2\pi k_{1}}{l^{2}b}c_{20},\quad s_{50} = - 0.7486\frac{2\pi k_{3}}{l^{2}b}c_{20},} \\ &{s_{51} = - 0.0481\frac{\pi k_{2}}{b^{3}}c_{21},\quad s_{52} = - 0.2547\frac{\pi k_{4}}{b^{3}}c_{21},} \\ &{s_{53} = - 0.1195\frac{2\pi k_{2}}{b^{3}}c_{21},\quad s_{54} = - 0.5461\frac{2\pi k_{4}}{b^{3}}c_{21},} \\ &{ s_{55} = 0.2659\frac{\pi k_{1}k_{2}}{2l^{2}b}c_{22},\quad s_{56} = - 0.0171\frac{\pi k_{3}k_{4}}{2l^{2}b}c_{22},} \end{aligned}$$

$$\begin{aligned} &{s_{57} = 0.3592\frac{3\pi k_{1}k_{2}}{2l^{2}b}c_{22},} \\ &{s_{58} = - 0.0343\frac{3\pi k_{3}k_{4}}{2l^{2}b}c_{22},} \\ &{s_{59} = 0.2192\frac{\pi^{2}k_{1}^{2}}{lb^{2}}c_{23},\quad s_{60} = - 0.0018\frac{\pi^{2}k_{3}^{2}}{lb^{2}}c_{23},} \\ &{ s_{61} = 0.5740\frac{4\pi^{2}k_{1}^{2}}{lb^{2}}c_{23},\quad s_{62} = 0.0133\frac{4\pi^{2}k_{3}^{2}}{lb^{2}}c_{23},} \\ &{ s_{63} = 0.9333\frac{k_{1}^{4}}{l^{4}}c_{24},} \\ &{s_{64} = - 2.7017\frac{( k_{1}^{2}k_{3}^{2} + 2k_{1}^{3}k_{3} )}{l^{4}}c_{24},} \\ &{ s_{65} = 1.0896\frac{( k_{1}^{2}k_{3}^{2} + 2k_{1}k_{3}^{3} )}{l^{4}}c_{24},} \\ &{s_{66} = - 1.0052\frac{k_{3}^{4}}{l^{4}}c_{24},\quad s_{67} = - 0.6388\frac{k_{1}^{2}k_{2}^{2}}{l^{2}b^{2}}c_{25},} \\ &{s_{68} = - 1.7862\frac{( k_{1}^{2}k_{4}^{2} + 2k_{2}^{2}k_{1}k_{3} )}{l^{2}b^{2}}c_{25},} \\ &{s_{69} = - 1.2414\frac{( k_{1}^{2}k_{3}^{2} + 2k_{4}^{2}k_{1}k_{3} )}{l^{2}b^{2}}c_{25},} \\ &{s_{70} = - 0.9225\frac{k_{3}^{2}k_{4}^{2}}{l^{2}b^{2}}c_{25},} \\ &{s_{71} = 0.0013\frac{k_{1}^{2}k_{2}^{2}}{l^{2}b^{2}}c_{26},} \end{aligned}$$

$$\begin{aligned} &{s_{72} = - 0.4524\frac{( k_{2}^{2}k_{3}^{2} + 2k_{1}^{2}k_{2}k_{4} )}{l^{2}b^{2}}c_{26},} \\ &{s_{73} = 0.1850\frac{( k_{1}^{2}k_{4}^{2} + 2k_{3}^{2}k_{2}k_{4} )}{l^{2}b^{2}}c_{26},} \\ &{s_{74} = 9.8455\frac{k_{3}^{2}k_{4}^{2}}{l^{2}b^{2}}c_{26},\quad s_{75} = - 0.0189\frac{k_{2}^{4}}{b^{4}}c_{27},} \\ &{s_{76} = - 0.5655\frac{( k_{2}^{2}k_{4}^{2} + 2k_{2}^{3}k_{4} )}{b^{4}}c_{27},} \\ &{s_{77} = - 0.3610\frac{( k_{2}^{2}k_{4}^{2} + 2k_{2}k_{4}^{3} )}{b^{4}}c_{27},} \\ &{s_{78} = - 1.2131\frac{k_{4}^{4}}{b^{4}}c_{27},\quad s_{79} = 0.0729\frac{k_{1}^{2}k_{2}^{2}}{l^{2}b^{2}}c_{28},} \\ &{s_{80} = - 0.0911\frac{( k_{1}^{2}k_{2}k_{4} + k_{1}k_{2}^{2}k_{3} + k_{1}k_{2}k_{3}k_{4} )}{l^{2}b^{2}}c_{28},} \\ &{s_{81} = 1.2927\frac{( k_{4}^{2}k_{1}k_{3} + k_{2}k_{3}^{2}k_{4} + k_{1}k_{2}k_{3}k_{4} )}{l^{2}b^{2}}c_{28},} \end{aligned}$$

$$\begin{aligned} &{s_{82} = 0.3068\frac{k_{3}^{2}k_{4}^{2}}{l^{2}b^{2}}c_{28},\quad s_{83} = 0.1761\frac{\pi}{2l}c_{29},} \\ &{s_{84} = 0.2909\frac{\pi}{l}c_{29},\quad s_{85} = 0.0039\frac{\pi}{ b}c_{30},} \\ &{s_{86} = 0.0253\frac{2\pi}{b}c_{30},\quad s_{87} = 0.1761\frac{\pi^{3}}{8l^{3}}c_{31},} \\ &{s_{88} = 0.2909\frac{\pi^{3}}{l^{3}}c_{31}, \quad s_{89} = - 0.0039\frac{\pi^{3}}{b^{3}}c_{32},} \\ &{s_{90} = - 0.0253\frac{8\pi^{3}}{b^{3}}c_{32},\quad s_{91} = 0.5509\frac{k_{1}^{4}}{l^{4}}c_{33},} \\ &{s_{92} = 1.8503\frac{k_{3}^{4}}{l^{4}}c_{33},\quad s_{93} = 0.5509\frac{k_{2}^{4}}{b^{4}}c_{34},} \\ &{ s_{94} = 1.8503\frac{k_{4}^{4}}{b^{4}}c_{34},\quad s_{95} = - 0.1506\frac{\pi^{3}}{4l^{2}b}c_{35},} \\ &{s_{96} = - 0.2276\frac{2\pi^{3}}{l^{2}b}c_{35},\quad s_{97} = 0.1761\frac{\pi^{3}}{2lb^{2}}c_{36},} \\ &{s_{98} = 0.6084\frac{4\pi^{3}}{lb^{2}}c_{36},\quad s_{99} = - 0.1382\frac{k_{1}^{2}k_{2}^{2}}{l^{2}b^{2}}c_{37},} \\ &{s_{100} = 0.8774\frac{k_{3}^{2}k_{4}^{2}}{l^{2}b^{2}}c_{37},} \end{aligned}$$

$$\begin{aligned} &{s_{101} = 1.0012\frac{k_{1}}{l}\biggl( c_{38} + c_{40}\frac{\mathrm{d}l}{\mathrm{d}t} + c_{45}\frac{\mathrm{d}^{2}l}{\mathrm{d}t^{2}} \biggr),} \\ &{s_{102} = 0.2868\frac{k_{3}}{l}\biggl( c_{38} + c_{40}\frac{\mathrm{d}l}{\mathrm{d}t} + c_{45}\frac{\mathrm{d}^{2}l}{\mathrm{d}t^{2}} \biggr),} \\ &{s_{103} = 0.2608\frac{k_{2}}{b}c_{39},\quad s_{104} = - 0.4385\frac{k_{4}}{b}c_{39},} \\ &{s_{105} = 0.3627c_{40},} \\ &{s_{106} = c_{41}\biggl( 3.12\frac{k_{1}^{3}}{l^{3}}c_{42}^{3} - 0.15\frac{3k_{1}^{2}k_{2}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{s_{106} =} {}+ 0.05\frac{3k_{1}k_{2}^{2}}{lb^{2}}c_{42}c_{43}^{2} + 0.32\frac{k_{2}^{3}}{b^{3}}c_{43}^{3} \biggr),} \\ &{s_{107} = c_{41}\biggl( 0.19\frac{k_{1}^{2}k_{3}}{l^{3}}c_{42}^{3} + 0.03\frac{2k_{1}k_{2}k_{3}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{s_{107} =} {} - 0.12\frac{k_{2}^{2}k_{3}}{lb^{2}}c_{42}^{2}c_{43} - 1.26\frac{k_{1}^{2}k_{4}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{s_{107} =} {} - 0.002\frac{2k_{1}k_{2}k_{4}}{lb^{2}}c_{42}^{2}c_{43} - 0.05\frac{k_{2}^{2}k_{4}}{b^{3}}c_{43}^{3} \biggr),} \\ &{s_{108} = c_{41}\biggl( 1.22\frac{k_{1}k_{3}^{2}}{l^{3}}c_{42}^{3} - 0.08\frac{2k_{1}k_{3}k_{4}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{s_{108} =} {}+ 0.94\frac{k_{1}k_{4}^{2}}{lb^{2}}c_{42}c_{43}^{2} + 0.11\frac{k_{2}k_{3}^{2}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{s_{108} =} {} - 0.02\frac{2k_{2}k_{3}k_{4}}{lb^{2}}c_{42}c_{43}^{2} + 0.21\frac{k_{2}k_{4}^{2}}{b^{3}}c_{43}^{3} \biggr),} \end{aligned}$$

$$\begin{aligned} &{s_{109} = c_{41}c_{44}\biggl( 1.83 \frac{k_{1}^{2}}{l^{2}}c_{42}^{2} - 0.10\frac{2k_{1}k_{2}}{lb}c_{42}c_{43}} \\ &{\phantom{s_{109} =} {} + 0.03\frac{k_{2}^{2}}{b^{2}}c_{43}^{2} \biggr),} \\ &{s_{110} = c_{41}c_{44}^{2} \biggl( 1.18\frac{k_{1}}{l}c_{42} - 0.07\frac{k_{2}}{b}c_{43} \biggr),} \\ &{s_{111} = c_{41}c_{44}\biggl( 0.61 \frac{k_{3}^{2}}{l^{2}}c_{42}^{2} + 0.09\frac{2k_{3}k_{4}}{lb}c_{42}c_{43}} \\ &{\phantom{s_{111} =} {}+ 0.55\frac{k_{4}^{2}}{b^{2}}c_{43}^{2} \biggr),} \\ &{s_{112} = c_{41}c_{44}^{2} \biggl( - 0.35\frac{k_{3}}{l}c_{42} - 0.47\frac{k_{4}}{b}c_{43} \biggr),} \\ &{s_{113} = c_{41}c_{44}\biggl( - 0.26 \frac{k_{1}k_{3}}{l^{2}}c_{42}^{2} - 0.74\frac{k_{1}k_{4}}{lb}c_{42}c_{43}} \\ &{\phantom{s_{113} =} {}+ 0.04\frac{k_{2}k_{3}}{lb}c_{42}c_{43} - 0.001\frac{k_{2}k_{4}}{b^{2}}c_{43}^{2} \biggr),} \\ &{s_{114} = c_{41}\biggl( 0.63\frac{k_{3}^{3}}{l^{3}}c_{42}^{3} - 0.33\frac{3k_{3}^{2}k_{4}}{l^{2}b}c_{42}^{2}c_{43}} \\ &{\phantom{s_{114} =} {}+ 0.07\frac{3k_{3}k_{4}^{2}}{lb^{2}}c_{42}c_{43}^{2} - 0.87\frac{k_{4}^{3}}{b^{3}}c_{43}^{3} \biggr),} \\ &{s_{115} = 0.3967c_{41}c_{44}^{3}, \quad s_{116} = 0.5509c_{45},} \\ &{s_{117} = 1.0012c_{45}\frac{2k_{1}}{l} \frac{\mathrm{d}l}{\mathrm{d}t},} \\ &{ s_{118} = 0.2868c_{45}\frac{2k_{3}}{l} \frac{\mathrm{d}l}{\mathrm{d}t}.} \end{aligned}$$

The coefficients of α _i in Eq. (20a) are obtained as

$$\begin{aligned} &{\alpha_{17}' = r_{15} + r_{19} + r_{23} + r_{27} + r_{35} + r_{59},} \\ &{\alpha_{18}' = r_{16} + r_{20} + r_{24} + r_{28} + r_{36} + r_{60},} \\ &{\alpha_{19}' = r_{17} + r_{21} + r_{25} + r_{29} + r_{37} + r_{61},} \\ &{\alpha_{20}' = r_{18} + r_{22} + r_{26} + r_{30} + r_{38} + r_{62},} \\ &{\alpha_{21}' = r_{31} + r_{39} + r_{43} + r_{47} + r_{51} + r_{55},} \\ &{\alpha_{22}' = r_{32} + r_{40} + r_{44} + r_{48} + r_{52} + r_{56},} \\ &{\alpha_{23}' = r_{33} + r_{41} + r_{45} + r_{49} + r_{53} + r_{57},} \\ &{\alpha_{24}' = r_{34} + r_{42} + r_{46} + r_{50} + r_{54} + r_{58},} \\ &{\alpha_{25}' = r_{83} + r_{87} + r_{97},\quad \alpha_{26}' = r_{84} + r_{88} + r_{98},} \\ &{\alpha_{27}' = r_{85} + r_{89} + r_{95},\quad \alpha_{28}' = r_{86} + r_{90} + r_{96}.} \\ &{\alpha_{1} = r_{116},\quad \alpha_{2} = r_{105} + r_{117},\quad \alpha_{3} = r_{118},} \\ &{\alpha_{4} = r_{11} + r_{91} + r_{93} + r_{99} + r_{101} + r_{103}} \\ &{\phantom{\alpha_{4} =} {}- \frac{\alpha_{17}'}{( k_{11} + k_{12} )}k_{24}a_{16} - \frac{\alpha_{19}'}{( l_{11} + l_{12} )}l_{24}a_{16}} \\ &{\phantom{\alpha_{4} =} {}- \frac{\alpha_{25}'}{( k_{31} + k_{32} + k_{41} )}( k_{33} + k_{37} + k_{39} )} \\ &{\phantom{\alpha_{4} =} {} - \frac{\alpha_{26}'}{( l_{31} + l_{32} + l_{41} )}( l_{33} + l_{37} + l_{39} )} \\ &{\phantom{\alpha_{4} =} {}- \frac{\alpha_{27}'}{( m_{31} + m_{32} + m_{41} )}( m_{33} + m_{37} + m_{39} )} \\ &{\phantom{\alpha_{4} =} {} - \frac{\alpha_{28}'}{( n_{31} + n_{32} + n_{41} )}( n_{33} + n_{37} + n_{39} ),} \\ &{\alpha_{5} = r_{13},} \\ &{\alpha_{6} = r_{12} + r_{14} + r_{92} + r_{94} + r_{100} + r_{102} + r_{104}} \\ &{\phantom{\alpha_{6} =} {} - \frac{\alpha_{18}'}{( k_{11} + k_{12} )}k_{24}a_{16} - \frac{\alpha_{20}'}{( l_{11} + l_{12} )}l_{24}a_{16}} \\ &{\phantom{\alpha_{6} =} {}- \frac{\alpha_{25}'}{( k_{31} + k_{32} + k_{41} )}( k_{31} + k_{32} + k_{41} )} \\ &{\phantom{\alpha_{6} =} {} - \frac{\alpha_{26}'}{( l_{31} + l_{32} + l_{41} )}( l_{34} + l_{38} + l_{40} )} \\ &{\phantom{\alpha_{6} =} {}- \frac{\alpha_{27}'}{( m_{31} + m_{32} + m_{41} )}( m_{34} + m_{38} + m_{40} )} \\ &{\phantom{\alpha_{6} =} {} - \frac{\alpha_{28}'}{( n_{31} + n_{32} + n_{41} )}( n_{34} + n_{38} + n_{40} ),} \end{aligned}$$

(35)

$$\begin{aligned} &{\alpha_{7} = r_{63} + r_{67} + r_{71} + r_{75} + r_{79} + r_{106}} \\ &{\phantom{\alpha_{7} =} {}- \frac{\alpha_{17}'}{( k_{11} + k_{12} )}( k_{15} + k_{18} + k_{21} ) } \\ &{\phantom{\alpha_{7} =} {}- \frac{\alpha_{19}'}{( l_{11} + l_{12} )}( l_{15} + l_{18} + l_{21} )} \\ &{\phantom{\alpha_{7} =} {}- \frac{\alpha_{21}'}{( m_{11} + m_{12} )}( m_{15} + m_{18} + m_{21} )} \\ &{\phantom{\alpha_{7} =} {} - \frac{\alpha_{23}'}{( n_{11} + n_{12} )}( n_{15} + n_{18} + n_{21} ),} \\ &{\alpha_{8} = r_{64} + r_{68} + r_{72} + r_{76} + r_{80} + 3r_{107}} \\ &{\phantom{\alpha_{8} =} {}- \frac{\alpha_{17}'}{( k_{11} + k_{12} )}( k_{16} + k_{19} + k_{22} )} \\ &{\phantom{\alpha_{8} =} {}- \frac{\alpha_{18}'}{( k_{11} + k_{12} )}( k_{15} + k_{18} + k_{21} )} \\ &{\phantom{\alpha_{8} =} {}- \frac{\alpha_{19}'}{( l_{11} + l_{12} )}( l_{16} + l_{19} + l_{22} )} \\ &{\phantom{\alpha_{8} =} {} - \frac{\alpha_{20}'}{( l_{11} + l_{12} )}( l_{15} + l_{18} + l_{21} )} \\ &{\phantom{\alpha_{8} =} {}- \frac{\alpha_{21}'}{( m_{11} + m_{12} )}( m_{16} + m_{19} + m_{22} )} \\ &{\phantom{\alpha_{8} =} {} - \frac{\alpha_{22}'}{( m_{11} + m_{12} )}( m_{15} + m_{18} + m_{21} )} \\ &{\phantom{\alpha_{8} =} {}- \frac{\alpha_{23}'}{( n_{11} + n_{12} )}( n_{16} + n_{19} + n_{22} )} \\ &{\phantom{\alpha_{8} =} {} - \frac{\alpha_{24}'}{( n_{11} + n_{12} )}( n_{15} + n_{18} + n_{21} ),} \end{aligned}$$

$$\begin{aligned} &{\alpha_{9} = r_{65} + r_{69} + r_{73} + r_{77} + r_{81} + 3r_{108}} \\ &{\phantom{\alpha_{9} =} {} - \frac{\alpha_{17}'}{( k_{11} + k_{12} )}( k_{17} + k_{20} + k_{23} )} \\ &{\phantom{\alpha_{9} =} {} - \frac{\alpha_{18}'}{( k_{11} + k_{12} )}( k_{16} + k_{19} + k_{22} )} \\ &{\phantom{\alpha_{9} =} {}- \frac{\alpha_{19}'}{( l_{11} + l_{12} )}( l_{17} + l_{20} + l_{23} )} \\ &{\phantom{\alpha_{9} =} {} - \frac{\alpha_{20}'}{( l_{11} + l_{12} )}( l_{16} + l_{19} + l_{22} )} \\ &{\phantom{\alpha_{9} =} {}- \frac{\alpha_{21}'}{( m_{11} + m_{12} )}( m_{17} + m_{20} + m_{23} )} \\ &{\phantom{\alpha_{9} =} {} - \frac{\alpha_{22}'}{( m_{11} + m_{12} )}( m_{16} + m_{19} + m_{22} )} \\ &{\phantom{\alpha_{9} =} {}- \frac{\alpha_{23}'}{( n_{11} + n_{12} )}( n_{17} + n_{20} + n_{23} )} \\ &{\phantom{\alpha_{9} =} {} - \frac{\alpha_{24}'}{( n_{11} + n_{12} )}( n_{16} + n_{19} + n_{22} ),} \\ &{\alpha_{10} = r_{66} + r_{70} + r_{74} + r_{78} + r_{82} + r_{114}} \\ &{\phantom{\alpha_{10} =} {} - \frac{\alpha_{18}'}{( k_{11} + k_{12} )}( k_{17} + k_{20} + k_{23} )} \\ &{\phantom{\alpha_{10} =} {} - \frac{\alpha_{20}'}{( l_{11} + l_{12} )}( l_{17} + l_{20} + l_{23} )} \\ &{\phantom{\alpha_{10} =} {}- \frac{\alpha_{22}'}{( m_{11} + m_{12} )}( m_{17} + m_{20} + m_{23} )} \\ &{\phantom{\alpha_{10} =} {} - \frac{\alpha_{24}'}{( n_{11} + n_{12} )}( n_{17} + n_{20} + n_{23} ),} \\ &{\alpha_{11} = 3r_{109},\quad \alpha_{12} = 3r_{110},\quad \alpha_{13} = 3r_{111},} \\ &{ \alpha_{14} = 3r_{112},\quad \alpha_{15} = 6r_{113},\quad \alpha_{16} = r_{115}.} \end{aligned}$$

The coefficients of β _i in Eq. (20b) are obtained as

$$\begin{aligned} &{\beta_{17}' = s_{15} + s_{19} + s_{23} + s_{27} + s_{35} + s_{59},} \\ &{\beta_{18}' = s_{16} + s_{20} + s_{24} + s_{28} + s_{36} + s_{60},} \\ &{\beta_{19}' = s_{17} + s_{21} + s_{25} + s_{29} + s_{37} + s_{61},} \\ &{\beta_{20}' = s_{18} + s_{22} + s_{26} + s_{30} + s_{38} + s_{62},} \\ &{\beta_{21}' = s_{31} + s_{39} + s_{43} + s_{47} + s_{51} + s_{55},} \\ &{\beta_{22}' = s_{32} + s_{40} + s_{44} + s_{48} + s_{52} + s_{56},} \\ &{\beta_{23}' = s_{33} + s_{41} + s_{45} + s_{49} + s_{53} + s_{57},} \\ &{\beta_{24}' = s_{34} + s_{42} + s_{46} + s_{50} + s_{54} + s_{58},} \\ &{\beta_{25}' = s_{83} + s_{87} + s_{97},\quad \beta_{26}' = s_{84} + s_{88} + s_{98},} \\ &{\beta_{27}' = s_{85} + s_{89} + s_{95},\quad \beta_{28}' = s_{86} + s_{90} + s_{96}.} \\ &{\beta_{1} = s_{116},\quad \beta_{2} = s_{105} + s_{118},\quad \beta_{3} = s_{117},} \\ &{\beta_{4} = s_{12} + s_{92} + s_{94} + s_{100} + s_{102} + s_{104}} \\ &{\phantom{\beta_{4} =} {}- \frac{\beta_{18}'}{( k_{11} + k_{12} )}k_{24}a_{16} - \frac{\beta_{20}'}{( l_{11} + l_{12} )}l_{24}a_{16}} \\ &{\phantom{\beta_{4} =} {}- \frac{\beta_{25}'}{( k_{31} + k_{32} + k_{41} )}( k_{34} + k_{38} + k_{40} )} \\ &{\phantom{\beta_{4} =} {} - \frac{\beta_{26}'}{( l_{31} + l_{32} + l_{41} )}( l_{34} + l_{38} + l_{40} )} \\ &{\phantom{\beta_{4} =} {}- \frac{\beta_{27}'}{( m_{31} + m_{32} + m_{41} )}( m_{34} + m_{38} + m_{40} )} \\ &{\phantom{\beta_{4} =} {} - \frac{\beta_{28}'}{( n_{31} + n_{32} + n_{41} )}( n_{34} + n_{38} + n_{40} ),} \\ &{\beta_{5} = s_{14},} \\ &{\beta_{6} = s_{11} + s_{91} + s_{93} + s_{99} + s_{101} + s_{103}} \\ &{\phantom{\beta_{6} =} {} - \frac{\beta_{17}'}{( k_{11} + k_{12} )}k_{24}a_{16} - \frac{\beta_{19}'}{( l_{11} + l_{12} )}l_{24}a_{16}} \\ &{\phantom{\beta_{6} =} {}- \frac{\beta_{25}'}{( k_{31} + k_{32} + k_{41} )}( k_{33} + k_{37} + k_{39} )} \\ &{\phantom{\beta_{6} =} {} - \frac{\beta_{26}'}{( l_{31} + l_{32} + l_{41} )}( l_{33} + l_{37} + l_{39} )} \\ &{\phantom{\beta_{6} =} {}- \frac{\beta_{27}'}{( m_{31} + m_{32} + m_{41} )}( m_{33} + m_{37} + m_{39} )} \\ &{\phantom{\beta_{6} =} {} - \frac{\beta_{28}'}{( n_{31} + n_{32} + n_{41} )}( n_{33} + n_{37} + n_{39} ),} \end{aligned}$$

(36)

$$\begin{aligned} &{\beta_{7} = s_{63} + s_{67} + s_{71} + s_{75} + s_{79} + s_{106}} \\ &{\phantom{\beta_{7} =} {} - \frac{\beta_{17}'}{( k_{11} + k_{12} )}( k_{15} + k_{18} + k_{21} )} \\ &{\phantom{\beta_{7} =} {} - \frac{\beta_{19}'}{( l_{11} + l_{12} )}( l_{15} + l_{18} + l_{21} )} \\ &{\phantom{\beta_{7} =} {}- \frac{\beta_{21}'}{( m_{11} + m_{12} )}( m_{15} + m_{18} + m_{21} )} \\ &{\phantom{\beta_{7} =} {} - \frac{\beta_{23}'}{( n_{11} + n_{12} )}( n_{15} + n_{18} + n_{21} ),} \\ &{\beta_{8} = s_{64} + s_{68} + s_{72} + s_{76} + s_{80} + 3s_{107}} \\ &{\phantom{\beta_{8} =} {}- \frac{\beta_{17}'}{( k_{11} + k_{12} )}( k_{16} + k_{19} + k_{22} )} \\ &{\phantom{\beta_{8} =} {} - \frac{\beta_{18}'}{( k_{11} + k_{12} )}( k_{15} + k_{18} + k_{21} )} \\ &{\phantom{\beta_{8} =} {}- \frac{\beta_{19}'}{( l_{11} + l_{12} )}( l_{16} + l_{19} + l_{22} )} \\ &{\phantom{\beta_{8} =} {} - \frac{\beta_{20}'}{( l_{11} + l_{12} )}( l_{15} + l_{18} + l_{21} )} \\ &{\phantom{\beta_{8} =} {}- \frac{\beta_{21}'}{( m_{11} + m_{12} )}( m_{16} + m_{19} + m_{22} )} \\ &{\phantom{\beta_{8} =} {} - \frac{\beta_{22}'}{( m_{11} + m_{12} )}( m_{15} + m_{18} + m_{21} )} \\ &{\phantom{\beta_{8} =} {}- \frac{\beta_{23}'}{( n_{11} + n_{12} )}( n_{16} + n_{19} + n_{22} )} \\ &{\phantom{\beta_{8} =} {} - \frac{\beta_{24}'}{( n_{11} + n_{12} )}( n_{15} + n_{18} + n_{21} ),} \end{aligned}$$

$$\begin{aligned} &{\beta_{9} = s_{65} + s_{69} + s_{73} + s_{77} + s_{81} + 3s_{108}} \\ &{\phantom{\beta_{9} =} {} - \frac{\beta_{17}'}{( k_{11} + k_{12} )}( k_{17} + k_{20} + k_{23} )} \\ &{\phantom{\beta_{9} =} {} - \frac{\beta_{18}'}{( k_{11} + k_{12} )}( k_{16} + k_{19} + k_{22} )} \\ &{\phantom{\beta_{9} =} {}- \frac{\beta_{19}'}{( l_{11} + l_{12} )}( l_{17} + l_{20} + l_{23} )} \\ &{\phantom{\beta_{9} =} {} - \frac{\beta_{20}'}{( l_{11} + l_{12} )}( l_{16} + l_{19} + l_{22} )} \\ &{\phantom{\beta_{9} =} {}- \frac{\beta_{21}'}{( m_{11} + m_{12} )}( m_{17} + m_{20} + m_{23} )} \\ &{\phantom{\beta_{9} =} {} - \frac{\beta_{22}'}{( m_{11} + m_{12} )}( m_{16} + m_{19} + m_{22} )} \\ &{\phantom{\beta_{9} =} {}- \frac{\beta_{23}'}{( n_{11} + n_{12} )}( n_{17} + n_{20} + n_{23} )} \\ &{\phantom{\beta_{9} =} {} - \frac{\beta_{24}'}{( n_{11} + n_{12} )}( n_{16} + n_{19} + n_{22} ),} \\ &{\beta_{10} = s_{66} + s_{70} + s_{74} + s_{78} + s_{82} + s_{114}} \\ &{\phantom{\beta_{9} =} {} - \frac{\beta_{18}'}{( k_{11} + k_{12} )}( k_{17} + k_{20} + k_{23} )} \\ &{\phantom{\beta_{9} =} {} - \frac{\beta_{20}'}{( l_{11} + l_{12} )}( l_{17} + l_{20} + l_{23} )} \\ &{\phantom{\beta_{9} =} {}- \frac{\beta_{22}'}{( m_{11} + m_{12} )}( m_{17} + m_{20} + m_{23} )} \\ &{\phantom{\beta_{9} =} {} - \frac{\beta_{24}'}{( n_{11} + n_{12} )}( n_{17} + n_{20} + n_{23} ),} \\ &{\beta_{11} = 3s_{109},\quad \beta_{12} = 3s_{110},\quad \beta_{13} = 3s_{111},} \\ &{\beta_{14} = 3s_{112},\quad \beta_{15} = 6s_{113},\quad \beta_{16} = s_{115}.} \end{aligned}$$