Appendix
The complete form of the equations, referred by their equation number in the main text:
$$\begin{aligned}&(cv_1 )\ddot{V}+(cv_2 )\dot{V}+(cv_3 )V+(cv_4 )E=\varepsilon \left\{ (cv_{n1} )r\ddot{V}\right. \\&\quad +(cv_{n2} )EW +(cv_{n3} )VW+(cv_{n4} )\ddot{E}W\\&\quad +(cv_{n5} )\dot{E}\dot{W}+(cv_{n6} )\ddot{r}V-(cv_{n7} k_s )rV \\&\quad +(cv_{n8} )\dot{r}\dot{V} +(cv_{W^{2}V} )W^{2}V+(cv_{E^{2}V} )E^{2}V\\&\quad +(cv_{V^{3}} )V^{3} +(cv_{E^{3}} )E^{3}+(cv_{EV^{2}} )EV^{2} \\&\quad +(cv_{EW^{2}} )EW^{2}+(cv_{\ddot{V}W^{2}} )\ddot{V}W^{2}\\&\quad +2(cv_{2\dot{V}\dot{W}W} )\dot{V}\dot{W}W+(cv_{r^{2}\ddot{V}} )r^{2}\ddot{V}\\&\quad +(cv_{r\dot{r}\dot{V}} )r\dot{r}\dot{V}+(cv_{r\ddot{r}V} )r\ddot{r}V+(cv_{r^{2}V} )r^{2}V\\&\quad +(cv_{\dot{r}^{2}V} )\dot{r}^{2}V+2(cv_{LV} )(V^{2}\ddot{V}+V\dot{V}^{2})\\&\quad +2(cv_{LW} )(VW\ddot{W}+V\dot{W}^{2}) \\&\quad \left. +\frac{1}{2}\left( \int \limits _0^1 \phi _v (s)ds\right) (f_v \Omega _v^2 ) (e^{i\Omega _v T_0 } +e^{-i\Omega _v T_0 })\right\} ,\end{aligned}$$
(4.4)
$$\begin{aligned}&(cw_1 )\ddot{W}+(cw_2 )\dot{W}+(cw_3 )W=\varepsilon \left\{ (cw_{n1} )r\ddot{W}\right. \\&\quad +(cw_{n2} )EV+(cw_{n3} )\dot{E}\dot{V}+(cw_{n4} )E^{2} \\&\quad +(cw_{n5} )V^{2}+(cw_{n6} )\ddot{r}W-(cw_{n7} k_s )rW \\&\quad +(cw_{r^{2}\ddot{W}} )r^{2}\ddot{W} +(cw_{r^{2}W} )r^{2}W+(cv_{r\ddot{r}W} )r\ddot{r}W \\&\quad + (cv_{\dot{r}^{2}W} )\dot{r}^{2}W+(cw_{WV^{2}} )WV^{2}+(cw_{W^{3}} )W^{3}\\&\quad +(cw_{W\dot{V}^{2}} )W\dot{V}^{2}+(cw_{E^{2}W} )E^{2}W\\&\quad +(cw_{EVW} )EVW +2(cw_{LV} )(VW\ddot{V}\\&\quad +W\dot{V}^{2})+2(cw_{LW} )(W^{2}\ddot{W}+W\dot{W}^{2})\\&\quad \left. +\frac{1}{2}\left( \int \limits _0^1 \phi _w (s)ds\right) (f_w \Omega _w^2 ) (e^{i\Omega _w T_0 }+e^{-i\Omega _w T_0 })\right\} , \end{aligned}$$
(4.5)
$$\begin{aligned}&(c\phi _1 )\ddot{E}+(c\phi _2 )\dot{E}+(c\phi _3 )E+(c\phi _4 )V\\&\quad =\varepsilon \left\{ (c\phi _{n1} )VW +(c\phi _{n2} )EW+(c\phi _{n3} )\ddot{V}W\right. \\&\qquad +(c\phi _{n4} )\dot{V}\dot{W} +(c\phi _{E^{2}V} )E^{2}V+(c\phi _{V^{3}} )V^{3}\\&\qquad +(c\phi _{EV^{2}} )EV^{2} +(c\phi _{EW^{2}} )EW^{2}\\&\qquad \left. +\frac{1}{2}\left( \int \limits _0^1 \phi _\phi (s)ds\right) (f_\phi \Omega _\phi ^2 ) (e^{i\Omega _\phi T_0 }+e^{-i\Omega _\phi T_0 })\right\} , \end{aligned}$$
(4.6)
$$\begin{aligned}&m_s \ddot{r}+k_s r-k_t \dot{q}=\varepsilon \{(cr_{n1} )V\ddot{V} +(cr_{n2} )\dot{V}^{2}\\&\quad +(cr_{n3} )W\ddot{W}+(cr_{n4} )\dot{W}^{2}+(cr_{rW^{2}} )rW^{2} \\&\quad + cr_{r\dot{V}^{2}} (r\dot{V}^{2})+cr_{r\dot{W}^{2}} (r\dot{W}^{2}) \\&\quad +cr_{rV\ddot{V}} (rV\ddot{V})+cr_{rW\ddot{W}} (rW\ddot{W})\}. \end{aligned}$$
(4.7)
$$\begin{aligned}&k_q \ddot{q}+\dot{q}+\varepsilon k_e \dot{r}=0, \end{aligned}$$
(4.8)
$$\begin{aligned}&(cv_1 )D_0^2 V_1 +(cv_3 )V_1 +(cv_4 )E_1 =-2(cv_1 )D_0 D_1 V_0 \\&\quad -(cv_2 )D_0 V_0 +(cv_{n1} )r_1 D_0^2 V_0 \\&\quad +(cv_{n2} )E_0 W_0 + (cv_{n3} )V_0 W_0 +(cv_{n4} )D_0^2 E_0 W_0 \\&\quad +(cv_{n5} )D_0 E_0 D_0 W_0+(cv_{n6} )D_0^2 r_0 V_0 \\&\quad -(cv_{n7} k_s )r_0 V_0 +(cv_{n8} )D_0 r_0 D_0 V_0 \\&\quad + (cv_{w^{2}v} )W_0 ^{2}V_0 +(cv_{E^{2}v} )E_0 ^{2}V_0 +(cv_{V^{3}} )V_0 ^{3}\\&\quad +(cv_{E^{3}} )E_0 ^{3}+(cv_{EV^{2}} )E_0 V_0 ^{2}+(cv_{EW^{2}} )E_0 W_0 ^{2} \\&\quad + (cv_{\ddot{V}W^{2}} )D_0^2 V_0 W_0 ^{2}+2(cv_{2\dot{V}\dot{W}W} )D_0 V_0 D_0 W_0 W_0\\&\quad +(cv_{r^{2}\ddot{V}} )r_0^2 D_0^2 V+(cv_{r\dot{r}\dot{V}} )r_0 D_0 r_0 D_0 V_0 \\&\quad + (cv_{r\ddot{r}V} )r_0 D_0^2 r_0 V_0 +(cv_{r^{2}V} )r_0 ^{2}V_0 \\&\quad +(cv_{\dot{r}^{2}V} )(D_0 r_0 )^{2}V_0+2(cv_{LV} ) \\&\quad + (V_0 ^{2}D_0^2 V_0 +V_0 (D_0 V_0 )^{2})2(cv_{LW} )\\&\quad \times (V_0 W_0 D_0^2 W_0 +V_0 (D_0 W_0 )^{2})\\&\quad +\frac{1}{2}\left( \int \limits _0^1 \phi _v (s)ds\right) (f_v \Omega _v^2 ) (e^{i\Omega _v T_0 }+e^{-i\Omega _v T_0 }), \end{aligned}$$
(5.11)
$$\begin{aligned}&(cw_1 )D_0^2 W_1 +(cw_3 )W_1 =-2D_0 D_1 W_0 -(cw_2 )D_0 W_0 \\&\quad +(cw_{n1} )r_0 D_0^2 W_0 +(cw_{n2} )E_0 V_0 \\&\quad + (cw_{n3} )D_0 E_0 D_0 V_0 +(cw_{n4} )E_0 ^{2}+(cw_{n5} )V_0 ^{2} \\&\quad +(cw_{n6} )D_0^2 r_0 W_0 -(cw_{n7} k_s )r_0 W_0 \\&\quad + (cw_{r^{2}\ddot{W}} )r_0 ^{2}D_0^2 W_0 +(cw_{r^{2}W} )r_0 ^{2}W_0 \\&\quad +(cv_{r\ddot{r}W} )r_0 D_0^2 r_0 W_0 +(cv_{\dot{r}^{2}W} )(D_0 r_0 )^{2}W_0 \\&\quad + (cw_{WV^{2}} )W_0 V_0 ^{2}+(cw_{W^{3}} )W_0 ^{3}\\&\quad +(cw_{W\dot{V}^{2}} )W_0 (D_0 V_0 )^{2}+(cw_{E^{2}W} )E_0 ^{2}W_0 \\&\quad + (cw_{EVW} )E_0 V_0 W_0 +2(cw_{LV} )(V_0 W_0 D_0^2 V_0 \\&\quad +W_0 (D_0 V_0 )^{2}) + 2(cw_{LW} )(W_0 ^{2}D_0^2 W_0 +W_0 (D_0 W_0)^{2}) \\&\quad +\frac{1}{2}\left( \int \limits _0^1 \phi _w (s)ds\right) (f_w \Omega _w^2 ) (e^{i\Omega _w T_0 }+e^{-i\Omega _w T_0 }), \end{aligned}$$
(5.12)
$$\begin{aligned}&(c\phi _1 )D_0^2 E_1 +(c\phi _3 )E_1 +(c\phi _4 )V_1 =-2D_0 D_1 E_0 \\&\quad -(c\phi _2 )D_0 E_0 +(c\phi _{n1} )V_0 W_0 +(c\phi _{n2} )E_0 W_0 \\&\quad + (c\phi _{n3} )D_0^2 V_0 W_0 +(c\phi _{n4} )D_0 V_0 D_0 W_0 \\&\quad +(c\phi _{E^{2}V} )E_0^2 V_0 +(c\phi _{V^{3}} )V_0 ^{3}+(c\phi _{EV^{2}} )E_0 V_0 ^{2} \\&\quad + (c\phi _{EW^{2}} )E_0 W_0 ^{2} \\&\quad +\frac{1}{2}\left( \int \limits _0^1 \phi _\phi (s)ds\right) (f_\phi \Omega _\phi ^2 ) (e^{i\Omega _\phi T_0 }+e^{-i\Omega _\phi T_0 }), \end{aligned}$$
(5.13)
$$\begin{aligned}&m_s D_0^2 r_1 +k_s r_1 -k_t D_0 q_1 =(cr_{n1} )V_0 D_0^2 V_0 \\&\quad +(cr_{n2} )(D_0 V_0 )^{2}+(cr_{n3} )W_0 D_0^2 W_0 \\&\quad +(cr_{n4} )(D_0 W_0 )^{2} +(cr_{rW^{2}} )r_0 W_0 ^{2}\\&\quad +(cr_{r\dot{V}^{2}} )(r_0 (D_0 V_0 )^{2}) +(cr_{r\dot{W}^{2}} )(r_0 (D_0 W_0 )^{2})\\&\quad -2m_s D_0 D_1 r_0 -k_t D_1 q_0 +(cr_{rW\ddot{W}} )(r_0 W_0 D_0^2 W_0 )\\&\quad +(cr_{rV\ddot{V}} )(r_0 V_0 D_0^2 V_0 ). \end{aligned}$$
(5.14)
$$\begin{aligned}&k_q D_0^2 q_1 +D_0 q_1 =-2k_q D_0 D_1 q_0 -D_1 q_0 -k_e D_0 r_0 , \end{aligned}$$
(5.15)
$$\begin{aligned}&\!\!\!H_{vj} (s,t)={v}''(s,t)\left\{ m_s \ddot{u}+m_s \ddot{r}\left( 1-\frac{{v}'^{2}}{2}+\frac{{w}'^{2}}{2}\right) \right. \nonumber \\&\qquad \left. -\,\,2m_s \dot{r}{\dot{v}}'{v}'+k_s r\left( 1-\frac{{v}'^{2}}{2}+\frac{{w}'^{2}}{2}\right) \right\} _{s=r(t)} \nonumber \\&\qquad -\,\,\left\{ m_s ({\dot{v}}'\dot{r})+m_s \ddot{r}{v}'+m_s \dot{r}(\dot{r}{v}''+{\dot{v}}') \right. \nonumber \\&\qquad \left. +k_s r({v}'+{v}'{w}'^{2}) \right\} \delta (s-r(t)) \end{aligned}$$
(8.1)
$$\begin{aligned}&\!\!\!H_{wj} (s,t)\!=\!{w}''(s,t)\!\left\{ m_s \ddot{u}+m_s \ddot{r}\left( \!1-\frac{{v}'^{2}}{2}+\frac{{w}'^{2}}{2}\right) \right. \nonumber \\&\qquad \left. -\,\,2m_s \dot{r}{\dot{v}}'{v}'+k_s r\left( 1-\frac{{v}'^{2}}{2}+\frac{{w}'^{2}}{2}\right) \right\} _{s=r(t)} \nonumber \\&\qquad -\,\,\left\{ m_s \dot{r}{\dot{w}}'-m_s \ddot{r}{w}'-m_s \dot{r}(\dot{r}{w}''+{\dot{w}}')\right. \nonumber \\&\qquad \left. -\,\,k_s r\left( {w}'+\frac{{v}'^{2}}{2}{w}'\right) \right\} \delta (s-r(t)) \end{aligned}$$
(8.2)
$$\begin{aligned}&H_r (t)=\left\{ -m_s \ddot{u}-2m_s \dot{u}{\dot{w}}'{w}'-m_s \ddot{v}{v}' \right. \nonumber \\&\qquad \left. -\,\,m_s \ddot{w}{w}'-k_s r{w}'^{2} \right\} _{s=r(t)} \end{aligned}$$
(8.3)
$$\begin{aligned}&\!\!\!cv_1 =\int \limits _0^1 {\phi _v^2 } (s)ds+J_\zeta \int \limits _0^1 {{\phi '}_v^2 } (s)ds +m_s \phi _v (r_e ),cv_2 \nonumber \\&\quad =c_v \int \limits _0^1 {\phi _v^2 } (s)ds,cv_3 =\beta _{33} \int \limits _0^1 {{\phi ''}_v^2 } (s)ds, \nonumber \\&\!\!\!cv_4 =\beta _{13} \int \limits _0^1 {\phi _\phi ^{\prime }} (s){\phi }''_v (s)ds,cv_{n1} \nonumber \\&\quad =-2m_s \phi _v (r_e ){\phi }'_v (r_e ), \nonumber \\&\!\!\!cv_{n2} =\{\phi _v \beta _{11} ({\phi }'_\phi {\phi }'_w )' -\phi _v (\beta _{22} -\beta _{33} )(\phi _\phi {\phi }''_w )' \nonumber \\&\quad -{\phi }'_v \beta _{11} ({\phi }'_\phi {\phi }'_w )+{\phi }'_v (\beta _{22} -\beta _{33} )(\phi _\phi {\phi }''_w )\}_{s=1} \nonumber \\&\!\!\!-\beta _{11} \int \limits _0^1 {\phi _v } ({\phi }'_\phi {\phi }'_w )''ds\nonumber \\&\quad +(\beta _{22} -\beta _{33} )\int \limits _0^1 {\phi _v } (\phi _\phi {\phi }''_w )''ds,cv_{n3} \nonumber \\&\quad =\{\phi _v \beta _{13} (2{\phi }''_v {\phi }'_w )\}_{s=1} -\beta _{13} \int \limits _0^1 {\phi _v } (2{\phi }''_v {\phi }'_w )''ds, \nonumber \\&\!\!\!cv_{n4} =\{-\phi _v J_\xi (\phi _\phi {\phi }'_w )\}_{s=1} \nonumber \\&\quad +J_\xi \int \limits _0^1 {\phi _v } (\phi _\phi {\phi }'_w )'ds,cv_{n5} =cv_{n4} , \nonumber \\&\!\!\!cv_{n6} =m_s \int \limits _0^1 {\phi _v } {\phi }''_v ds-m_s \phi _v (r_e ){\phi }'_v (r_e ),cv_{n7} \nonumber \\&\quad =-\int \limits _0^1 {\phi _v } {\phi }''_v ds+\phi _v (r_e ){\phi }'_v (r_e ),cv_{n8} \nonumber \\&\quad =-2m_s \phi _v (r_e ){\phi }'_v (r_e ), \end{aligned}$$
(8.4)
$$\begin{aligned}&\!\!\!cw_1 =\int \limits _0^1 {\phi _w^2 } (s)ds+J_\eta \int \limits _0^1 {{\phi '}_w^2 } (s)ds+m_s \phi _w (r_e ),cw_2 \nonumber \\&\quad =c_w \int \limits _0^1 {\phi _w^2 } (s)ds,cw_3 =\beta _{22} \int \limits _0^1 {{\phi ''}_w^2 } (s)ds, \nonumber \\&\!\!\!cw_{n1} =-2m_s \phi _w (r_e ){\phi }'_w (r_e ), \nonumber \\&\!\!\!cw_{n2} =\{\phi _w \beta _{11} ({\phi }'_\phi {\phi }''_v )-\phi _w (\beta _{22} -\beta _{33} )(\phi _\phi {\phi }''_v )'\nonumber \\&\quad +{\phi }'_w (\beta _{22} -\beta _{33} )(\phi _\phi {\phi }''_v )\}_{s=1} \nonumber \\&\!\!\!+\beta _{11} \int \limits _0^1 {\phi _w } ({\phi }'_\phi {\phi }''_v )'ds\nonumber \\&\quad +(\beta _{22} -\beta _{33} )\int \limits _0^1 {\phi _w } (\phi _\phi {\phi }''_v )''ds,cw_{n3} \nonumber \\&\quad =\{\phi _w J_\xi (\phi _\phi {\phi }'_v )\}_{s=1} -J_\xi \int \limits _0^1 {\phi _w } (\phi _\phi {\phi }'_v )'ds, \nonumber \\&\!\!\!cw_{n4} =\{\phi _w \beta _{13} (\phi _\phi {\phi }'_\phi )'-{\phi }'_w \beta _{13} \phi _\phi {\phi }'_\phi \}_{s=1} \nonumber \\&\quad -\beta _{13} \int \limits _0^1 {\phi _w } (\phi _\phi {\phi }'_\phi )''ds, \nonumber \\&\!\!\!cw_{n5} =\{-\phi _w \beta _{13} ({\phi }''_v )^{2}\}_{s=1}\nonumber \\&\quad +\beta _{13} \int \limits _0^1 {\phi _w ({\phi }''_v )'^{2}} ds,cw_{n6} \nonumber \\&\quad =m_s \int \limits _0^1 {\phi _w } {\phi }''_w ds+m_s \phi _w (r_e ){\phi }'_w (r_e ), \nonumber \\&\!\!\! cw_{n7} =-\int \limits _0^1 {\phi _w } {\phi }''_w ds-\phi _w (r_e ){\phi }'_w (r_e ),cv_{n8} \nonumber \\&\quad =-2m_s \phi _v (r_e ){\phi }'_v (r_e ), \end{aligned}$$
(8.5)
$$\begin{aligned}&\!\!\!c\phi _1 =J_\xi \int \limits _0^1 {\phi _\phi ^2 } (s)ds,c\phi _2 =c_\phi \int \limits _0^1 {\phi _\phi ^2 } (s)ds,c\phi _3 \nonumber \\&\quad =\beta _{11} \int \limits _0^1 {{\phi '}_\phi ^2 } (s)ds, \nonumber \\&\!\!\!c\phi _4 =\beta _{13} \int \limits _0^1 {{\phi '}_v^2 } (s)ds,c\phi _{n1} =\{-\phi _\phi \beta _{11} {\phi ''}_v {\phi '}_w \}_{s=1} \nonumber \\&\quad +\beta _{11} \int \limits _0^1 {\phi _\phi ({\phi ''}_v {\phi '}_w )'} ds+(\beta _{22} -\beta _{33} )\nonumber \\&\quad \int \limits _0^1 {\phi _\phi ({\phi ''}_v {\phi ''}_w )} ds, \nonumber \\&\!\!\!c\phi _{n2} =\{-\phi _\phi \beta _{13} (\phi _\phi {\phi }''_w )\}_{s=1} +\beta _{13} \int \limits _0^1 {\phi _\phi ^2 } {\phi }'''_w ds,c\phi _{n3} \nonumber \\&\quad =-J_\xi \int \limits _0^1 {\phi _\phi } {\phi }'_v {\phi }'_w ds,c\phi _{n4} =c\phi _{n3} , \end{aligned}$$
(8.6)
$$\begin{aligned}&\!\!\!cr_{n1} =m_s \left( \int \limits _0^s {{\phi '}_v^2 } (s)ds\right) _{s=r_e }\nonumber \\&\quad -m_s (\phi _v {\phi '}_v )_{s=r_e } ,cr_{n2} \nonumber \\&\quad =m_s \left( \int \limits _0^s {{\phi '}_v^2 } (s)ds\right) _{s=r_e } , \nonumber \\&\!\!\!cr_{n3} =m_s \left( \int \limits _0^s {{\phi '}_w^2 } (s)ds\right) _{s=r_e }\nonumber \\&\quad -m_s (\phi _w {\phi '}_w )_{s=r_e } ,cr_{n4} \nonumber \\&\quad =m_s \left( \int \limits _0^s {{\phi '}_w^2 } (s)ds\right) _{s=r_e } , \end{aligned}$$
(8.7)
Coefficients of nonlinear cubic terms:
$$\begin{aligned}&\!\!\!cv_{W^{2}V} =\{\phi _v \beta _{11} ({\phi '}_w^2 {\phi ''}_v )'+\phi _v \beta _{33} ({\phi '}_v ({\phi '}_w {\phi ''}_w )')\nonumber \\&\quad -{\phi '}_v \beta _{11} ({\phi ''}_v {\phi '}_w^2 )+{\phi '}_v (\beta _{22} -\beta _{33} ){\phi '}_v {\phi '}_w {\phi ''}_w \}_{s=1} \nonumber \\&\quad - \beta _{11} \int _0^1 {\phi _v } ({\phi ''}_v {\phi '}_w^2 )''ds\nonumber \\&\quad -\beta _{33} \int _0^1 {\phi _v } ({\phi '}_v ({\phi '}_w {\phi ''}_w )')'ds, \nonumber \\&\!\!\!cv_{E^{2}V} =\{\phi _v (\beta _{22} -\beta _{33} )(\phi _\phi ^2 {\phi }''_v )'\nonumber \\&\quad -{\phi }'_v (\beta _{22} -\beta _{33} )(\phi _\phi ^2 {\phi }''_v )\}_{s=1} \nonumber \\&\quad -(\beta _{22} -\beta _{33} )\int _0^1 {\phi _v } (\phi _\phi ^2 {\phi }''_v )''ds, \nonumber \\&\!\!\!cv_{V^{3}} =\{\phi _v \beta _{33} ({\phi }'_v ({\phi }'_v {\phi }''_v )')\}_{s=1}\nonumber \\&\quad -\beta _{33} \int _0^1 {\phi _v } ({\phi }'_v ({\phi }'_v {\phi }''_v )')'ds, \nonumber \\&\!\!\!cv_{E^{3}} =\{-\phi _v \beta _{13} (1/2)(\phi _\phi ^2 {\phi }'_\phi )'\nonumber \\&\quad +{\phi }'_v \beta _{13} (1/2)(\phi _\phi ^2 {\phi }'_\phi )\}_{s=1} \nonumber \\&\quad +\beta _{13} (1/2)\int _0^1 {\phi _v (\phi _\phi ^2 {\phi }'_\phi )''ds,} \nonumber \\&\!\!\!cv_{EV^{2}} =\{\phi _v \beta _{13} (1/2)({\phi '}_v^2 {\phi ''}_\phi ) \nonumber \\&\quad -{\phi '}_v \beta _{13} (1/2)({\phi '}_v^2 {\phi '}_\phi )\}_{s=1} \nonumber \\&\quad -\beta _{13} (1/2)\int _0^1 {\phi _v ({\phi ''}_\phi {\phi '}^{2}_v )'ds,} \nonumber \\&\!\!\!cv_{EW^{2}} =\{\phi _v \beta _{13} \phi _\phi ({\phi }'_w {\phi }''_w )'-{\phi }'_v \beta _{13} \phi _\phi {\phi }'_w {\phi }''_w \}_{s=1} \nonumber \\&\quad -\beta _{13} \int _0^1 {\phi _v (\phi _\phi ({\phi }'_w {\phi }''_w )')'ds} , \nonumber \\&\!\!\!cv_{\ddot{V}W^{2}} =\{-\phi _v J_\xi ({\phi '}_w^2 {\phi '}_v )\}_{s=1} \nonumber \\&\quad +J_\xi \int _0^1 {\phi _v ({\phi '}_w^2 {\phi '}_v )'ds,} cv_{2\dot{V}W\dot{W}} =cv_{\ddot{V}W^{2}} , \nonumber \\&\!\!\!cv_{r^{2}\ddot{V}} =-m_s ({\phi '}_v^2 )_{s=r_e } , cv_{r\dot{r}\dot{V}} \nonumber \\&\quad =-2m_s (\phi _v {\phi ''}_v +{\phi '}_v^2 )_{s=r_e } , cv_{r\ddot{r}V} =( cv_{r\dot{r}\dot{V}} /2), \nonumber \\&\!\!\!cv_{r^{2}V} =-k_s (\phi _v {\phi ''}_v +{\phi '}_v^2 )_{s=r_e } , cv_{\dot{r}^{2}V} \nonumber \\&\quad =-m_s (\phi _v {\phi ''}_v )_{s=r_e } , \nonumber \\&\!\!\!cv_{Lv} =\int _0^1 {\phi _v {\phi ''}_v } ds\left\{ -\frac{m_s }{2}\left( \int _0^s {{\phi '}_v^2 ds} \right) _{s=r_e } \right\} \nonumber \\&\quad -\frac{1}{2}\int _0^1 {\phi _v \left[ {\phi '}_v \int _1^s {\int _0^s {{\phi '}_v^2 dsds} } \right] 'ds,} \nonumber \\&\!\!\!cv_{Lw} =\int _0^1 {\phi _v {\phi ''}_v } ds\left\{ -\frac{m_s }{2}\left( \int _0^s {{\phi '}_w^2 ds} \right) _{s=r_e } \right\} \nonumber \\&\quad -\frac{1}{2}\int _0^1 {\phi _v \left[ {\phi '}_v \int _1^s {\int _0^s {{\phi '}_w^2 dsds} } \right] 'ds,} \end{aligned}$$
(8.8)
$$\begin{aligned}&\!\!\!cw_{r^{2}\ddot{W}} =-m_s ({\phi '}_w^2 )_{s=r_e } ,cw_{r^{2}W} \nonumber \\&\quad =k_s (\phi _w {\phi ''}_w +{\phi '}_w^2 )_{s=r_e } , cw_{r\ddot{r}W} =(\frac{m_s }{k_s })cw_{r^{2}W} , \nonumber \\&\!\!\!cw_{\dot{r}^{2}W} =m_s (\phi _w {\phi }''_w )_{s=r_e } , cw_{WV^{2}} \nonumber \\&\quad =\{-\phi _w \beta _{11} {\phi '}_w {\phi ''}_v^2 +\phi _w \beta _{33} {\phi '}_w ({\phi '}_v {\phi ''}_v )'\}_{s=1} \nonumber \\&\quad +\beta _{11} \int _0^1 \phi _w ({\phi '}_w {\phi ''}_v^2 )'ds\nonumber \\&\quad -\,\,\beta _{33} \int _0^1 \phi _w [ {\phi '}_w ({\phi }'_v {\phi }''_v )']'ds, \nonumber \\&\!\!\!cw_{W^{3}} =\{\phi _w \beta _{22} {\phi '}_w ({\phi '}_w {\phi ''}_w )'\}_{s=1}\nonumber \\&\quad -\beta _{22} \int _0^1 \phi _w [{\phi '}_w ({\phi '}_w {\phi ''}_w )']'ds, \nonumber \\&\!\!\!cw_{W\dot{V}^{2}} =\{\phi _w J_\xi ({\phi '}_w {\phi '}_v^2 )\}_{s=1} \!-\!J_\xi \!\!\int _0^1 {\phi _w ({\phi '}_w {\phi '}_v^2 )'ds,} \nonumber \\&\!\!\!cw_{E^{2}W} =\{-\phi _w (\beta _{22} -\beta _{33} )(\phi _\phi ^2 {\phi }''_w )'\nonumber \\&\quad +{\phi }'_w (\beta _{22} -\beta _{33} )(\phi _\phi ^2 {\phi }''_w )\}_{s=1} \nonumber \\&\quad +(\beta _{22} -\beta _{33} )\int _0^1 {\phi _w (\phi _\phi ^2 {\phi }''_w )''ds} , \nonumber \\&\!\!\!cw_{EVW} \!=\!\{\phi _w \beta _{13} {\phi }'_w (\phi _\phi {\phi }'_v )''\!-\!{\phi }'_w \beta _{13} {\phi }'_w (\phi _\phi {\phi }'_v )'\}_{s=1} \nonumber \\&\quad -\beta _{13} \int _0^1 {\phi _w [{\phi }'_w (\phi _\phi {\phi }'_v )''} ]'ds, \nonumber \\&\!\!\!cw_{Lv} =\int _0^1 {\phi _w {\phi ''}_w } ds\left\{ -\frac{m_s }{2}\left( \int _0^s {{\phi '}_v^2 ds} \right) _{s=r_e } \right\} \nonumber \\&\quad -\frac{1}{2}\int _0^1 {\phi _w \left[ {\phi '}_w \int _1^s {\int _0^s {{\phi '}_v^2 dsds} } \right] 'ds,} \nonumber \\&\!\!\!cw_{Lw} =\int _0^1 {\phi _w {\phi }''_w } ds\left\{ -\frac{m_s }{2}\left( \int _0^s {{\phi '}_w^2 ds} \right) _{s=r_e } \right\} \nonumber \\&\quad -\frac{1}{2}\int _0^1 {\phi _w \left[ {\phi '}_w \int _1^s {\int _0^s {{\phi '}_w^2 dsds} } \right] 'ds,} \end{aligned}$$
(8.9)
$$\begin{aligned}&\!\!\!c\phi _{E^{2}V} =\{\phi _\phi \frac{1}{2}\beta _{13} (\phi _\phi ^2 {\phi }''_v )\}_{s=1} \nonumber \\&\quad -\frac{1}{2}\beta _{13} \int _0^1 {\phi _\phi } (\phi _\phi ^2 {\phi }'''_v )ds, \nonumber \\&\!\!\!c\phi _{V^{3}} =\{-\phi _\phi \frac{1}{2}\beta _{13} ({\phi '}_v^2 {\phi ''}_v )\}_{s=1}\nonumber \\&\quad +\frac{1}{2}\beta _{13} \int _0^1 {\phi _\phi } ({\phi '}_v^2 {\phi ''}_v )'ds, \nonumber \\&\!\!\!c\phi _{VW^{2}} =\left\{ -\phi _\phi \frac{1}{2}\beta _{13} ({\phi }'_v {\phi }'_w {\phi }''_w )\right\} _{s=1} \nonumber \\&\quad +\beta _{13} \int _0^1 {\phi _\phi } {\phi }'_v ({\phi }'_w {\phi }''_w )'ds, \nonumber \\&\!\!\!c\phi _{EV^{2}} =-(\beta _{22} -\beta _{33} )\int _0^1 {\phi _\phi ^2 {\phi ''}_v^2 ds, } c\phi _{EW^{2}} \nonumber \\&\quad =(\beta _{22} -\beta _{33} )\int _0^1 {\phi _\phi ^2 {\phi ''}_w^2 ds, } \end{aligned}$$
(8.10)
$$\begin{aligned}&\!\!\!cr_{rW^{2}} =-k_s ({\phi '}_w^2 )_{s=r_e } , cr_{r\dot{V}^{2}} =m_s ({\phi '}_v^2 )_{s=r_e } ,\nonumber \\&\quad cr_{r\dot{W}^{2}} =m_s ({\phi '}_v^2 )_{s=r_e } , \nonumber \\&\!\!\!cr_{rV\ddot{V}} =-m_s (\phi _v {\phi ''}_v )_{s=r_e } ,cr_{rW\ddot{W}} \nonumber \\&\quad =-m_s (\phi _w {\phi ''}_w )_{s=r_e } , \end{aligned}$$
(8.11)
Coefficients of modulation equations:
$$\begin{aligned}&\!\!\!\Gamma _1 =-cv_1 -\alpha c\phi _1 , \nonumber \\&\!\!\!\Gamma _2 =\frac{(cv_2 +\alpha c\phi _2 )}{2}, \nonumber \\&\!\!\!\Gamma _3 =-\omega ^{2}cv_{r^{2}\ddot{V}} -\omega _r^2 cv_{r\ddot{r}V} +cv_{r^{2}V} +\omega _r^2 cv_{\dot{r}^{2}V} , \nonumber \\&\!\!\!\Gamma _4 =cv_{W^{2}V} +\alpha cv_{EW^{2}} -\omega ^{2}cv_{\ddot{V}W^{2}} \nonumber \\&\quad +c\phi _{VW^{2}} +\alpha c\phi _{EW^{2}} , \nonumber \\&\!\!\!\Gamma _5 =\alpha ^{2}cv_{E^{2}V} +cv_{V^{3}} +\alpha ^{3}cv_{E^{3}} +\alpha cv_{EV^{2}}\nonumber \\&\quad -(4/3)\omega ^{2}cv_{LV} +\alpha ^{2}c\phi _{E^{2}V} +c\phi _{V^{3}} +\alpha c\phi _{EV^{2}} , \nonumber \\&\!\!\!\Gamma _6 =-\omega ^{2}cv_{r^{2}\ddot{V}} +\omega \omega _r cv_{r\dot{r}\dot{V}} \nonumber \\&\quad -\omega _r^2 cv_{r\ddot{r}V} +cv_{r^{2}V} -\omega _r^2 cv_{\dot{r}^{2}V} , \nonumber \\&\!\!\!\Gamma _7 =-\omega ^{2}cv_{n1} -\omega _r^2 cv_{n6} -k_s cv_{n7} +\omega \omega _r cv_{n8} , \nonumber \\&\!\!\!\Gamma _8 =(1/2)f_v \Omega _v^2 \int \limits _0^1 {\phi _v (s)ds,} \end{aligned}$$
(8.12)
$$\begin{aligned}&\!\!\!\Lambda _1 =-cw_1 , \Lambda _2 =\frac{(cw_2 )}{2}, \Lambda _3 =-\rho ^{2}cw_{r^{2}\ddot{W}} \nonumber \\&\quad +cw_{r^{2}W} -\omega _r^2 cw_{r\ddot{r}W} +\omega _r^2 cw_{\dot{r}^{2}W} , \nonumber \\&\!\!\!\Lambda _4 =cw_{WV^{2}} +\omega ^{2}cw_{W\dot{V}^{2}} +\alpha ^{2}cw_{E^{2}W} \nonumber \\&\quad +\alpha cw_{EVW} , \Lambda _5 =cw_{W^{3}} -(4/3)\rho ^{2}cw_{LW} , \nonumber \\&\!\!\!\Lambda _6 =-\rho ^{2}cw_{r^{2}\ddot{W}} +cw_{r^{2}W} -\omega _r^2 cw_{r\ddot{r}W} \nonumber \\&\quad -\omega _r^2 cw_{\dot{r}^{2}W} , \Lambda _7 =-\rho ^{2}cw_{n1} -\omega _r^2 cw_{n6} -k_s cw_{n7} , \nonumber \\&\!\!\!\Lambda _8 =\frac{f_w \Omega _w^2 }{2}\int \limits _0^1 {\phi _w (s)ds,} \end{aligned}$$
(8.13)
$$\begin{aligned}&\!\!\!\mathrm{X}_1 =-m_s , \mathrm{X}_2 =\omega ^{2}cr_{r\dot{V}^{2}} -\omega ^{2}cr_{rV\ddot{V}} , \mathrm{X}_3 \nonumber \\&\quad =\rho ^{2}cr_{r\dot{W}^{2}} +cr_{rW^{2}} -\rho ^{2}cr_{rW\ddot{W}} , \nonumber \\&\!\!\!\mathrm{X}_4 =-\omega ^{2}cr_{r\dot{V}^{2}} -\omega ^{2}cr_{rV\ddot{V}} , \mathrm{X}_5 \nonumber \\&\quad =-\omega ^{2}cr_{n1} -\omega ^{2}cr_{n2} , \nonumber \\&\!\!\!\mathrm{X}_6 =-\rho ^{2}cr_{r\dot{W}^{2}} +cr_{rW^{2}} -\rho ^{2}cr_{rW\ddot{W}} , \mathrm{X}_7 \nonumber \\&\quad =-\rho ^{2}cr_{n3} -\rho ^{2}cr_{n4} , \end{aligned}$$
(8.14)
