Analysis of a reduced-order nonlinear model of a multi-physics beam

Abstract

This article studies nonlinear behaviors of a two-dimensional beam with piezoelectric patches on it. Governing electromechanical equations of the beam taking into account several coupled piezoelectric patches, at different positions, are derived. Then, spatiotemporal variables of the system are separated. A methodology is proposed, for the detection of different mode functions and corresponding frequencies of the multi-physics beam, via using space equations of the system. As a representative example, a homogeneous beam with a single piezoelectric patch is considered. The paper is followed by consideration of two particular cases: (i) a single mode of the system is in resonance with the direct lateral-base excitation and (ii) two modes of the system present an internal resonance, while the first one is in resonance with the lateral-base excitation. For both cases, the electromechanical system equations are projected on its targeted mode(s). The temporal equations are treated via a multiple scale method leading to detections of its fixed points. The effects of one of the nonlinear coefficients of the piezoelectric patch, on the overall responses of the multi-physics beam, in terms of changing its behavior from hardening to softening (or vice versa), are discussed and commented upon. Moreover, it is shown that the piezoelectric patch is able to control the targeted mode(s) of the system.

Appendices

Appendix 1

We suppose that the neutral axis of the beam undergoes deformation from an inertial coordinate system $(\mathbf {e}_x,\mathbf {e}_y,\mathbf {e}_z)$ to a local curvilinear coordinate system $(\mathbf {e}_\xi ,\mathbf {e}_\eta ,\mathbf {e}_\zeta )$. The beam presents three Euler-angle rotations, namely $\chi (s,t)$, $\theta (s,t)$ and $\beta (s,t)$, where s stands for the distance along the deformed beam axis measured from the origin and t for the time, see Fig. 22. The absolute angular velocity $\varPi (s,t)$ for the curvilinear axis system is defined as [21, 22]:

$$\begin{aligned} \varPi (s,t)= & {} (\dot{\beta }-\dot{\xi }\sin (\theta ))\mathbf {e}_\xi +(\dot{\xi }\cos (\theta )\sin (\beta )\nonumber \\&+\,\dot{\theta }\cos (\beta ))\mathbf {e}_\eta \nonumber \\&+\,\big (\dot{\xi }\cos (\theta )\cos (\beta )-\dot{\theta }\sin (\beta )\big )\mathbf {e}_\zeta \nonumber \\= & {} \varPi _\xi \mathbf {e}_\xi +\varPi _\eta \mathbf {e}_\eta +\varPi _\zeta \mathbf {e}_\zeta \end{aligned}$$

(117)

where “$\ \dot{} \$” stands for the time derivative of the argument, i.e., $\displaystyle {\frac{\partial }{\partial t}}$. With the Kirchhoff’s kinetic analogy [32], the curvature vector $\rho (s,t)$ can be defined by replacing the time derivatives by the spatial derivatives in Eq. 117; it reads [21, 22]:

$$\begin{aligned} \rho (s,t)= & {} (\beta '-\xi '\sin (\theta ))\mathbf {e}_\xi +(\xi '\cos (\theta )\sin (\beta )\nonumber \\&+\,\theta '\cos (\beta ))\mathbf {e}_\eta \nonumber \\&+\,\big (\xi '\cos (\theta )\cos (\beta )-\theta '\sin (\beta )\big )\mathbf {e}_\zeta \nonumber \\= & {} \rho _\xi \mathbf {e}_\xi +\rho _\eta \mathbf {e}_\eta +\rho _\zeta \mathbf {e}_\zeta \end{aligned}$$

(118)

where “$\ ' \$” stands for the space derivative of the argument, i.e., $\displaystyle {\frac{\partial }{\partial s}}$.

It is assumed that the cross section remains straight. Let us study the kinetic of an elementary length ds on the neutral axis due to the deformation. After a transformation $\digamma (t)$, it becomes $ds^*$, see Fig. 23. This transformation $\digamma (t)$ corresponds to change of displacement components in the inertial coordinate to the curvilinear coordinate, i.e., change of (u, v, w) to $(u+du,v+dv,w+dw)$. The components du, dv and dw are defined via the Euler angles, as shown in Fig. 23:

$$\begin{aligned} \tan (\chi )= & {} \frac{v'}{1+u'} \nonumber \\ \tan (\theta )= & {} \displaystyle {\frac{w'}{\sqrt{(1+u')^2+v'^2}}} \end{aligned}$$

(119)

The strain e is defined as:

$$\begin{aligned} \displaystyle {e=\frac{ds^*-ds}{ds}=\sqrt{(1+u')^2+v'^2+w'^2}-1} \end{aligned}$$

(120)

The beam’s neutral axis is supposed to be inextensional, i.e., $e=0$. Equation 120 reads:

$$\begin{aligned} (1+u')^2+v'^2+w'^2=1 \end{aligned}$$

(121)

Let us consider a particle P on an arbitrary coordinate of the cross section and a particle C on its neutral axis in the initial configuration of the beam. The local coordinates of the particle P read as $(\eta ,\zeta )$. The particle C is at a distance s from the origin O, as depicted in Fig. 24. C and P are transferred to $C^*$ and $P^*$ due to the transformation $\digamma (t)$ of the beam. Thus, we can write:

Fig. 22

The Euler-angle rotations protocol: $\chi (s,t)$, $\theta (s,t)$ and $\beta (s,t)$

Fig. 23

Transformation $\digamma (t)$ of an elementary length ds to the element $ds^*$ due to the deformation

Fig. 24

Transformation $\digamma (t)$ of a particle P on the cross section of the beam due to the deformation ($P\rightarrow P^*$)

$$\begin{aligned}&\overrightarrow{OC}+\overrightarrow{CP}=s\mathbf {e}_x+\eta \mathbf {e}_y+\zeta \mathbf {e}_z\nonumber \\&\overrightarrow{OC^*}+\overrightarrow{C^*P^*}=(s+u)\mathbf {e}_x+v\mathbf {e}_y\nonumber \\&\quad +\,w\mathbf {e}_z+\eta \mathbf {e}_\eta +\zeta \mathbf {e}_\zeta \end{aligned}$$

(122)

or,

$$\begin{aligned} d(\overrightarrow{OP})= & {} ds\mathbf {e}_x+d\eta \mathbf {e}_y+d\zeta \mathbf {e}_z\nonumber \\ d(\overrightarrow{OP^*})= & {} (1+u')ds\mathbf {e}_x+v'ds\mathbf {e}_y+w'ds\mathbf {e}_z\nonumber \\&+\,d\eta \mathbf {e}_\eta +\eta d\mathbf {e}_\eta +d\zeta \mathbf {e}_\zeta +\zeta d\mathbf {e}_\zeta \end{aligned}$$

(123)

For a fixed s, it is also known that:

$$\begin{aligned} \displaystyle {\frac{d\mathbf {e}_\wp }{dt}=\varPi \times \mathbf {e}_\wp } \ \ \ \text {with} \ \ \wp \rightarrow \xi ,\eta ,\zeta \end{aligned}$$

(124)

when $\times$ stands for the vector product of two vectors. With Kirchhoff’s analogy [32]:

$$\begin{aligned} \displaystyle {\frac{d\mathbf {e}_\wp }{ds}=\rho \times \mathbf {e}_\wp } \ \ \ \text {with} \ \ \wp \rightarrow \xi ,\eta ,\zeta \end{aligned}$$

(125)

Thus, the deformations of the vectors become:

$$\begin{aligned} d(\overrightarrow{OP})= & {} ds\mathbf {e}_x+d\eta \mathbf {e}_y+d\zeta \mathbf {e}_z\nonumber \\ d(\overrightarrow{OP^*})= & {} (1+\eta \zeta \rho _\eta -\eta \zeta \rho _\eta )ds\mathbf {e}_\xi \nonumber \\&+\,(d\eta -\zeta \xi \rho _\eta ds)\mathbf {e}_\eta \nonumber \\&+\,(d\zeta +\eta \xi \rho _\eta ds)\mathbf {e}_\zeta \end{aligned}$$

(126)

From the definition of the Green’s strain tensor:

$$\begin{aligned}&d(\overrightarrow{OP^*})\cdot d(\overrightarrow{OP^*})-d(\overrightarrow{OP})\cdot d(\overrightarrow{OP})\nonumber \\&=2[ds \ \ d\eta \ \ d\zeta ]\varvec{\epsilon _{b}}[ds \ \ d\eta \ \ d\zeta ]^T \end{aligned}$$

(127)

where $\varvec{\epsilon _{b}}$ is the Green’s strain tensor of the homogeneous beam. From Eq. 126, it is obtained:

$$\begin{aligned}&d(\overrightarrow{OP^*})\cdot d(\overrightarrow{OP^*})-d(\overrightarrow{OP})\cdot d(\overrightarrow{OP})\nonumber \\&\quad =2(\zeta \rho _\eta -\eta \rho _\zeta -2\zeta \eta \rho _\zeta \rho _\eta )ds^2\nonumber \\&\qquad -\,2\zeta \rho _\xi d\eta ds +2\eta \rho _\xi dsd\zeta \end{aligned}$$

(128)

neglecting higher-order terms of $\rho _\wp$ ($\wp \mapsto \xi ,\zeta ,\eta$), Eq. 128 becomes:

$$\begin{aligned}&d(\overrightarrow{OP^*})\cdot d(\overrightarrow{OP^*})\nonumber \\&\quad -\,d(\overrightarrow{OP})\cdot d(\overrightarrow{OP})=2(\zeta \rho _\eta -\eta \rho _\zeta )ds^2\nonumber \\&\quad -\,2\zeta \rho _\xi d\eta ds+2\eta \rho _\xi dsd\zeta \end{aligned}$$

(129)

Appendix 2

The matrix $\varvec{G}$ of Eq. 38 reads as:

$$\begin{aligned} \varvec{G}= & {} \begin{bmatrix} \varvec{C1}&\varvec{O}&...&\varvec{O}&\varvec{C2}\\ \varvec{CL}(K_1,x_1)&-\varvec{CL}(K_2,x_1)&...&\varvec{O}&\varvec{O}\\ \vdots&\vdots&\ddots&\vdots&\vdots \\ \varvec{O}&\varvec{O}&...&\varvec{CL}(K_2,x_{2n})&-\varvec{CL}(K_1,x_{2n}) \end{bmatrix}\nonumber \\ \end{aligned}$$

(130)

where $\varvec{O}$ is a $4\times 4$ zero matrix, and:

$$\begin{aligned} \varvec{C1}= & {} \begin{bmatrix} 1&0&1&0 \\ 0&1&0&1 \\ 0&0&0&0 \\ 0&0&0&0 \end{bmatrix} \end{aligned}$$

(131)

$$\begin{aligned}&\varvec{C2}= \begin{bmatrix} 0&0&0&0 \\ 0&0&0&0 \\ -K_1^2\cos (K_1L)&-K_1^2\sin (K_1L)&K_1^2\cosh (K_1L)&K_1^2\sinh (K_1L) \\ K_1^3\sin (K_1L)&-K_1^3\cos (K_1L)&K_1^3\sinh (K_1L)&K_1^3\cosh (K_1L) \end{bmatrix} \end{aligned}$$

(132)

$$\begin{aligned}&\varvec{CL}(K,x)= \begin{bmatrix} \cos (Kx)&\sin (Kx)&\cosh (Kx)&\sinh (Kx) \\ -K\sin (Kx)&K\cos (Kx)&K\sin (Kx)&\cosh (Kx)\\ -K^2EI\cos (Kx)&-K^2EI\sin (Kx)&K^2EI\cosh (Kx)&K^2EI\sinh (Kx)\\ K^3EI\sin (Kx)&-K^3EI\cos (Kx)&K^3EI\sinh (Kx)&K^3EI\cosh (Kx) \end{bmatrix}\nonumber \\ \end{aligned}$$

(133)

with general EI and K, from Eq. 35, related as:

$$\begin{aligned} K^4=\frac{\omega _{v}^2\mu }{EI} \end{aligned}$$

(134)

Appendix 3

$$\begin{aligned} \varvec{\tilde{G}}= \begin{bmatrix} \varvec{\tilde{C1}}&\varvec{\tilde{0}}&...&\varvec{\tilde{0}}&\varvec{\tilde{C2}}\\ \varvec{\tilde{CL}}(K_1,x_1)&-\varvec{CL}(K_2,x_1)&...&\varvec{O}&\varvec{O}\\ \vdots&\vdots&\ddots&\vdots&\vdots \\ \varvec{\tilde{O}}&\varvec{O}&...&\varvec{CL}(K_2,x_{2n})&-\varvec{CL}(K_1,x_{2n}) \end{bmatrix}\nonumber \\ \end{aligned}$$

(135)

when $\varvec{\tilde{O}}$ and $\varvec{\tilde{0}}$ are $3\times 3$ and $3\times 4$ zero matrix, respectively, and:

$$\begin{aligned} \varvec{\tilde{C1}}= & {} \begin{bmatrix} 1&0&1\\ 0&0&0\\ 0&0&0 \end{bmatrix} \end{aligned}$$

(136)

$$\begin{aligned}&\varvec{\tilde{C2}}= \begin{bmatrix} 0&\qquad 0&\qquad 0&\qquad 0\\ -K^2\cos (K_1L)&\qquad -K^2\sin (K_1L)&\qquad K^2\cosh (K_1L)&\qquad K^2\sinh (K_1L) \\ K^3\sin (K_1L)&\qquad -K^3\cos (K_1L)&\qquad K^3\sinh (K_1L)&\qquad K^3\cosh (K_1L) \end{bmatrix}\end{aligned}$$

(137)

$$\begin{aligned}&\varvec{\tilde{CL}(K_1,x_1)}= \begin{bmatrix} K_1\cos (K_1x_1)&\qquad K_1\sin (K_1x_1)&\qquad \cosh (K_1x_1)\\ -K_1^2EI_1\sin (K_1x_1)&\qquad K_1^2EI_1\cosh (K_1x_1)&\qquad K_1^2EI_1\sinh (K_1x_1)\\ -K_1^3EI_1\cos (K_1x_1)&\qquad K_1^3EI_1\sinh (K_1x_1)&\qquad K_1^3EI_1\cosh (K_1x_1) \end{bmatrix} \end{aligned}$$

(138)

$$\begin{aligned} \varvec{\tilde{E}}= & {} \begin{bmatrix} 0\\ 0\\ 0\\ -A_1\cos (K_1x_1)\\ A_1K_1\sin (K_1x_1)\\ A_1K_1^2EI_1\cos (K_1x_1)\\ -A_1K_1^3EI_1\sin (K_1x_1)\\ 0\\ \vdots \\ 0 \end{bmatrix} \end{aligned}$$

(139)

Appendix 4

1.1 Stability analysis of equilibrium points

Let us perturb $|a_\tau |$ and $\alpha _\tau$ linearly in Eq. 81 as:

$$\begin{aligned} \begin{array}{lll} |a_\tau |&{}\rightarrow &{}|a_\tau |+\varDelta |a_\tau | \\ \alpha _\tau &{}\rightarrow &{}\alpha _\tau +\varDelta \alpha _\tau \end{array} \end{aligned}$$

(140)

Separation of the real and imaginary parts leads to:

$$\begin{aligned} \begin{bmatrix} D_2|a_\tau | \\ D_2\alpha _\tau \end{bmatrix} =\varvec{Inst} \times \begin{bmatrix} \varDelta |a_\tau | \\ \varDelta \alpha _\tau \end{bmatrix} \end{aligned}$$

(141)

with:

$$\begin{aligned} \begin{array}{lcc} \varvec{Inst}=\frac{1}{2\omega _{v\tau }G_0} \begin{bmatrix} Inst_{11} &{} Inst_{12}\\ Inst_{21}&{} Inst_{22} \end{bmatrix} \\ Inst_{11}=-\omega _{v\tau }G_1+\mathcal {Z}_i+3\mathcal {P}_i|a_\tau |^2 \\ Inst_{12}=(2\omega _{v\tau }G_0\sigma +\mathcal {Z}_r)|a_\tau |\\ \quad +\,(3G_4-4\omega _{v\tau }^2G_5+\mathcal {P}_r)|a_\tau |^3 \\ Inst_{21}=\frac{2\omega _{v\tau }G_0+\mathcal {Z}_r+3\mathcal {P}_r|a_\tau |^2}{a_\tau } \\ Inst_{22}= -\omega _{v\tau }G_1+\mathcal {Z}_i+\mathcal {P}_i|a_\tau |^2 \end{array} \end{aligned}$$

(142)

Then, if the real part of at least one of the eigenvalues of $\varvec{Inst}$ is positive, the point $(|a_\tau |,\sigma )$ is unstable.

Appendix 5

$$\begin{aligned} \begin{array}{l c l} G_0=\int _{0}^{L}\mu (s)\phi _\iota ^2(s)ds \\ G_1=c_v\int _{0}^{L}\phi _\iota ^2(s)ds \\ G_2=F\int _{0}^{L}\mu (s)\phi _\iota (s)ds \\ G_3=\int _{0}^{L}(-EI(s))\phi _\iota ^{(iv)}(s)\phi _\iota (s)ds \\ G_4=\int _{0}^{L}(-EI(s))[\phi _\iota '(s)\big (\phi _\iota '(s)\phi _\iota ''(s)\big )']'ds \\ G_5=\int _{0}^{L}[\phi _\iota '(s)\int _{s}^{L}\frac{-\mu (s)}{2}\int _{s}^{0}\phi _\iota '(s)^2ds \ ds]'\phi _\iota (s) ds \\ G_6=\int _{x_1}^{x_2}\int _{y_1}^{y_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}s_{111}\eta ^3d\zeta d\eta \phi _\iota ''(s)^3ds \\ G_7=\int _{x_1}^{x_2}\int _{y_1}^{y_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\frac{d_{311}}{h_p}\eta ^2d\zeta d\eta \phi _\iota ''(s)^2ds \\ \displaystyle {G_8=\int _{x_1}^{x_2}\int _{y_1}^{y_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\eta \frac{-d_{31}}{h_p}d\zeta d\eta \phi _\iota ''(s) ds} \\ \displaystyle {G_9=\int _{x_1}^{x_2}\int _{y_1}^{y_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\eta \frac{r_{331}}{h_p^2}d\zeta d\eta \phi _\iota ''(s) ds } \end{array} \end{aligned}$$

(143)

$$\begin{aligned} \begin{array}{lll} F_1=\frac{1}{R} \\ F_2=\int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}d_{31}(y_1-y_2)\phi _\iota ''(s)d\zeta ds \\ F_3=\int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\frac{2(y_2-y_1)}{h_p}r_{331}\phi _\iota ''(s)d\zeta ds \\ F_4=\int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}2d_{311}(y_2^2-y_1^2)\phi _\iota ''(s)^2d\zeta ds \\ F_5=\int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\frac{\xi _{33}}{h_p}d\zeta ds \\ F_6=\int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\frac{-2\xi _{333}}{h_p^2}d\zeta ds \\ \end{array} \end{aligned}$$

(144)

Appendix 6

Supposing only $G_2=G_1=0$ imposes that coefficients in Eq. 85 are simplified as:

$$\begin{aligned} \gimel _0'= & {} (3G_4-4\omega _{v\tau }^2G_5+\mathcal {P}_r)^2+\mathcal {P}_i^2\nonumber \\ \gimel _1'= & {} 4G_0\omega _{v\tau }(3G_4-4\omega _{v\tau }^2G_5+\mathcal {P}_r)\sigma \nonumber \\ \gimel _2'= & {} (2G_0\omega _{v\tau })^2\sigma ^2\nonumber \\ \gimel _3'= & {} 0 \end{aligned}$$

(145)

Equation 84, if $|a_\tau |\ne 0$, can be written as:

$$\begin{aligned} \mathcal {X}(\gimel _0'\mathcal {X}^2+\gimel _1'\sigma \mathcal {X}+\gimel _2'\sigma ^2)=0 \end{aligned}$$

(146)

with $\mathcal {X}=|a_\tau |^2$, $\gimel _0'>0$ and $\gimel _2'>0$.

Thus, the discriminant of Eq. 146 is defined as:

$$\begin{aligned} \varDelta _d=\gimel _1'^2-4\gimel _0'\gimel _2' \end{aligned}$$

(147)

From the expression of the constants $\gimel$ in Eqs. 85 and 145, the discriminant is reduced as:

$$\begin{aligned} \displaystyle {\varDelta _d=-16w_1^2G_0^2\mathcal {P}_i^2} \end{aligned}$$

(148)

This way, $\varDelta _d$ is negative and there is no real solution for $\mathcal {X}$ except for the special case $\mathcal {P}_i=0$. This case corresponds to the case when there is no current in the system (see Eqs. 67–69).

Appendix 7

$$\begin{aligned} N_0= & {} \int _{0}^{L}\mu (s)\phi _n^2(s)ds \nonumber \\ N_1= & {} c_v\int _{0}^{L}\phi _n^2(s)ds \nonumber \\ N_2= & {} F\int _{0}^{L}\mu (s)\phi _n(s)ds \nonumber \\ N_3= & {} \int _{0}^{L}(-EI(s))\phi _n^{(iv)}(s)\phi _n(s)ds \nonumber \\ N_4= & {} \int _{0}^{L}(-EI(s))[\phi _m'(s)\big (\phi _m'(s)\phi _m''(s)\big )']'\phi _n(s)ds \nonumber \\ N_5= & {} \int _{0}^{L}(-EI(s))[\phi _m'(s)\big (\phi _n'(s)\phi _n''(s)\big )'\nonumber \\&+\,\phi _n'(s)\big (\phi _m'(s)\phi _n''(s)\big )'\nonumber \\&+\,\phi _n'(s)\big (\phi _n'(s)\phi _m''(s)\big )']'\phi _n(s)ds \nonumber \\ N_6= & {} \int _{0}^{L}(-EI(s))[\phi _n'(s)\big (\phi _m'(s)\phi _m''(s)\big )'\nonumber \\&+\,\phi _m'(s)\big (\phi _m'(s)\phi _n''(s)\big )'\nonumber \\&+\,\phi _m'(s)\big (\phi _n'(s)\phi _m''(s)\big )']'\phi _n(s)ds \nonumber \\ N_7= & {} \int _{0}^{L}(-EI(s))[\phi _n'(s)\big (\phi _n'(s)\phi _n''(s)\big )']'\phi _n(s)ds \nonumber \\ N_8= & {} \int _{0}^{L}[\phi _m'(s)\int _{s}^{L}\frac{-\mu (s)}{2}\int _{s}^{0}\phi _m'(s)^2ds \ ds]\phi _n(s)ds \nonumber \\ N_9= & {} \int _{0}^{L}[\phi _m'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _m'(s)\phi _n'(s)ds \ ds]\phi _n(s)ds \nonumber \\ N_{10}= & {} \int _{0}^{L}[\phi _n'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _m'(s)\phi _n'(s)ds \ ds]\phi _n(s)ds \nonumber \\ N_{11}= & {} \int _{0}^{L}[\phi _n'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _n'(s)^2ds \ ds]\phi _n(s)ds \nonumber \\ N_{12}= & {} \int _{0}^{L}[\phi _m'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _n'(s)^2ds \ ds]\phi _n(s)ds \nonumber \\ N_{13}= & {} \int _{0}^{L}[\phi _n'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _m'(s)^2ds \ ds]\phi _n(s)ds \nonumber \\ N_{14}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}s_{111}\eta ^3 \ d\eta d\zeta \phi _m''(s)^2\phi _n''(s) \ ds \nonumber \\ N_{15}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}2s_{111}\eta ^3 \ d\eta d\zeta \phi _m''(s)\phi _n''(s)^2 \ ds \nonumber \\ N_{16}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}s_{111}\eta ^3 \ d\eta d\zeta \phi _n''(s)^3 \ ds \nonumber \\ N_{17}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}2\frac{d_{311}}{h_p}\eta ^2 \ d\eta d\zeta \phi _n''(s)^2 \ ds\end{aligned}$$

(149)

$$\begin{aligned} N_{18}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}2\frac{d_{311}}{h_p}\eta ^2 \ d\eta d\zeta \phi _n''(s)\phi _m''(s) \ ds \nonumber \\ N_{19}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}\frac{-d_{31}}{h_p}\eta \ d\eta d\zeta \phi _n''(s) \ ds \nonumber \\ N_{20}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}\frac{r_{331}}{h_p^2}\eta \ d\eta d\zeta \phi _n''(s) \ ds \nonumber \\ M_0= & {} \int _{0}^{L}\mu (s)\phi _m^2(s)ds \nonumber \\ M_1= & {} c_w\int _{0}^{L}\phi _m^2(s)ds \nonumber \\ M_2= & {} F\int _{0}^{L}\mu (s)\phi _m(s)ds \nonumber \\ M_3= & {} \int _{0}^{L}(-EI(s))\phi _n^{(iv)}(s)\phi _m(s)ds \nonumber \\ M_4= & {} \int _{0}^{L}(-EI(s))[\phi _m'(s)\big (\phi _m'(s)\phi _m''(s)\big )']'\phi _m(s)ds \nonumber \\ M_5= & {} \int _{0}^{L}(-EI(s))[\phi _m'(s)\big (\phi _n'(s)\phi _n''(s)\big )'\nonumber \\&+\,\phi _n'(s)\big (\phi _m'(s)\phi _n''(s)\big )'\nonumber \\&+\,\phi _n'(s)\big (\phi _n'(s)\phi _m''(s)\big )']'\phi _m(s)ds \nonumber \\ M_6= & {} \int _{0}^{L}(-EI(s))[\phi _n'(s)\big (\phi _m'(s)\phi _m''(s)\big )'\nonumber \\&+\,\phi _m'(s)\big (\phi _m'(s)\phi _n''(s)\big )'\nonumber \\&+\,\phi _m'(s)\big (\phi _n'(s)\phi _m''(s)\big )']'\phi _m(s)ds \nonumber \\ M_7= & {} \int _{0}^{L}(-EI(s))[\phi _n'(s)\big (\phi _n'(s)\phi _n''(s)\big )']'\phi _m(s)ds \nonumber \\ M_8= & {} \int _{0}^{L}[\phi _m'(s)\int _{s}^{L}\frac{-\mu (s)}{2}\int _{s}^{0}\phi _m'(s)^2ds \ ds]\phi _m(s)ds \nonumber \\ M_9= & {} \int _{0}^{L}[\phi _m'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _m'(s)\phi _n'(s)ds \ ds]\phi _m(s)ds \nonumber \\ M_{10}= & {} \int _{0}^{L}[\phi _n'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _m'(s)\phi _n'(s)ds \ ds]\phi _m(s)ds \nonumber \\ M_{11}= & {} \int _{0}^{L}[\phi _n'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _n'(s)^2ds \ ds]\phi _m(s)ds \nonumber \\ M_{12}= & {} \int _{0}^{L}[\phi _m'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _n'(s)^2ds \ ds]\phi _m(s)ds \nonumber \\ M_{13}= & {} \int _{0}^{L}[\phi _n'(s)\int _{s}^{L}(-\mu (s))\int _{s}^{0}\phi _m'(s)^2ds \ ds]\phi _m(s)ds \nonumber \\ M_{14}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}s_{111}\eta ^3 \ d\eta d\zeta \phi _m''(s)^3 \ ds\end{aligned}$$

(150)

$$\begin{aligned} M_{15}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}2s_{111}\eta ^3 \ d\eta d\zeta \phi _m''(s)^2\phi _n''(s) \ ds \nonumber \\ M_{16}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}s_{111}\eta ^3 \ d\eta d\zeta \phi _n''(s)^2\phi _m''(s) \ ds \nonumber \\ M_{17}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}2\frac{d_{311}}{h_p}\eta ^2 \ d\eta d\zeta \phi _n''(s)\phi _m''(s) \ ds \nonumber \\ M_{18}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}2\frac{d_{311}}{h_p}\eta ^2 \ d\eta d\zeta \phi _m''(s)^2 \ ds \nonumber \\ M_{19}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}\frac{-d_{31}}{h_p}\eta \ d\eta d\zeta \phi _m''(s) \ ds \nonumber \\ M_{20}= & {} \int _{x_1}^{x_2}\int _{\frac{-b}{2}}^{\frac{b}{2}}\int _{y_1}^{y_2}\frac{r_{331}}{h_p^2}\eta \ d\eta d\zeta \phi _m''(s) \ ds \end{aligned}$$

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To simplify, we suppose $c_v= c_w$.

$$\begin{aligned} S_1= & {} \frac{1}{R} \nonumber \\ S_2= & {} \int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}(y_1-y_2)d_{31}\phi _m''(s) d\zeta \ ds \nonumber \\ S_3= & {} \int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}(y_1-y_2)d_{31}\phi _n''(s) d\zeta \ ds \nonumber \\ S_4= & {} \int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\frac{2}{h_p}r_{331}\phi _m''(s) d\zeta \ ds \nonumber \\ S_5= & {} \int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\frac{2}{h_p}r_{331}\phi _n''(s) d\zeta \ ds \nonumber \\ S_6= & {} \int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}d_{311}(y_2^2-y_1^2)\phi _m''(s)^2d\zeta \ ds \nonumber \\ S_7= & {} \int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}d_{311}(y_2^2-y_1^2)\phi _n''(s)^2d\zeta \ ds \nonumber \\ S_8= & {} \int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\frac{\xi _{33}}{h_p}d\zeta \ ds \nonumber \\ S_9= & {} \int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}\frac{-2\xi _{333}}{h_p^2}d\zeta \ ds \nonumber \\ S_{10}= & {} \int _{x_1}^{x_2}\int _{-\frac{b}{2}}^{\frac{b}{2}}d_{311}(y_2^2-y_1^2)\phi _n''(s)\phi _m''(s)d\zeta \ ds\nonumber \\ \end{aligned}$$

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Appendix 8

$$\begin{aligned} \varGamma _{2n}= & {} \frac{N_{16}+\varLambda _nN_{17}+\varLambda _n^2N_{20}}{-(2\omega _{vn})^2N_0-N_3} \nonumber \\ \varGamma _{2m}= & {} \frac{N_{14}+\varLambda _mN_{18}+\varLambda _m^2N_{20}}{-(2\omega _{vm})^2N_0-N_3} \nonumber \\ \varGamma _{mn}= & {} \frac{\varLambda _mN_{17}+\varLambda _nN_{18}+2\varLambda _n\varLambda _mN_{20}}{-(\omega _{vm}+\omega _{vn})^2N_0-N_3} \nonumber \\ \varGamma _{m\overline{n}}= & {} \frac{\varLambda _mN_{17}+\overline{\varLambda }_nN_{18}+2\varLambda _m\overline{\varLambda }_nN_{18}}{-(\omega _{vm}-\omega _{vn})^2R_0-R_3} \nonumber \\ \varGamma _{cn}= & {} \frac{2N_{16}+\varLambda _nN_{17}+\overline{\varLambda }_nN_{17}+2\varLambda _n\overline{\varLambda }_nN_{20}}{-2N_3} \nonumber \\ \varGamma _{cm}= & {} \frac{2N_{14}+\varLambda _mN_{18}+\overline{\varLambda }_mN_{18}+2\varLambda _m\overline{\varLambda }_mN_{20}}{-2N_3} \nonumber \\ \varUpsilon _{2n}= & {} \frac{M_{16}+\varLambda _nM{17}+\varLambda _n^2N_{20}}{-(2\omega _{vn})^2M_0-M_3} \nonumber \\ \varUpsilon _{2m}= & {} \frac{M_{14}+\varLambda _mM_{18}+\varLambda _m^2M_{20}}{-(2\omega _{vm})^2M_0-M_3} \nonumber \\ \varUpsilon _{mn}= & {} \frac{M_{15}+\varLambda _mM_{17}+\varLambda _nM_{18}+\varLambda _m\varLambda _nM_{20}}{-(\omega _{vm}-\omega _{vn}))^2M_0-M_3} \nonumber \\ \varUpsilon _{m\overline{n}}= & {} \frac{M_{15}+\varLambda _mM_{17}+\overline{\varLambda }_nM_{18}+2\varLambda _m\overline{\varLambda }_nM_{20}}{-(\omega _{vm}-\omega _{vn}))^2M_0-M_3} \nonumber \\ \varUpsilon _{cm}= & {} \frac{2M_{16}+\varLambda _nM_{17}+\overline{\varLambda }_mM_{17}+2\varLambda _n\overline{\varLambda }_nM_{20}}{-2M_3} \nonumber \\ \varUpsilon _{cn}= & {} \frac{2M_{14}+\varLambda _mM_{18}+\overline{\varLambda }_mM_{18}+2\varLambda _m\overline{\varLambda }_mM_{20}}{-2M_3}\nonumber \\ \end{aligned}$$

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Appendix 9

$$\begin{aligned} \varLambda _{2m}= & {} \frac{(i\omega _{vm})(2\varUpsilon _{2m}S_2+2\varGamma _{2m}S_3+2\varLambda _mS_4+2\varLambda _mS_5+S_6+\varLambda _m^2S_9)}{S_1-2i\omega _{vm}S_8} \nonumber \\ \varLambda _{2n}= & {} \frac{(i\omega _{vn})(2\varUpsilon _{2n}S_2+2\varGamma _{2n}S_3+S_7+\varLambda _n^2S_9)}{S_1-2i\omega _{vn}S_8} \nonumber \\ \varLambda _{mn}= & {} \frac{(i\omega _{vn}+i\omega _{vm})(S_{10}+\varUpsilon _{mn}S_2+\varGamma _{mn}S_3+\varLambda _nS_4+\varLambda _nS_5+\varLambda _n\varLambda _mS_9)}{S_1-i(\omega _{vn}+\omega _{vm})S_8} \nonumber \\ \varLambda _{m\overline{n}}= & {} \frac{(-i\omega _{vn}+i\omega _{vm})(S_{10}+\varUpsilon _{m\overline{n}}S_2+\varGamma _{m\overline{n}}S_3+\overline{\varLambda }_nS_4+\overline{\varLambda }_nS_5+\overline{\varLambda }_n\varLambda _mS_9)}{S_1-i(-\omega _{vn}+\omega _{vm})S_8} \end{aligned}$$

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Appendix 10

$$\begin{aligned} \varXi _0= & {} 2\omega _{vn}N_0 \nonumber \\ \varXi _1= & {} \frac{N_2\omega _{vn}^2}{2} \nonumber \\ \varXi _2= & {} \varUpsilon _{2n}N_{15}+(\varUpsilon _{cn}+\overline{\varUpsilon }_{cn})N_{15}+\varLambda _{2n}N_{17}\nonumber \\&+\overline{\varLambda }_n\varGamma _{2n}N_{17}+\varLambda _n(\varGamma _{cn}+\overline{\varGamma }_{cn})N_{17}-4\omega _{vn}^2N_{11} \nonumber \\&+3N_7+2\varLambda _{2n}\overline{\varLambda }_nN_{20}+\overline{\varLambda }_n\varUpsilon _{2n}N_{18}+\varLambda _n(\varUpsilon _{cn}\nonumber \\&+\overline{\varUpsilon }_{cn})N_{18}+2\varUpsilon _{2n}N_{16}+2(\varUpsilon _{cn}+\overline{\varUpsilon }_{cn})N_{16}\nonumber \\ \varXi _3= & {} 2(\varUpsilon _{mn}+\varUpsilon _{\overline{m}n})N_{14}+(\varLambda _{mb}+\varLambda _{\overline{m}n})N_{18}\nonumber \\&+\varLambda _n(\varUpsilon _{cm}+\overline{\varUpsilon }_{cm})N_{18}+\varLambda _m\varUpsilon _{\overline{m}n}N_{18}\nonumber \\&+\overline{\varLambda }_m\varUpsilon _{mn}N_{18}\nonumber \\&+2\varLambda _m\varLambda _{\overline{m}n}N_{20}+2\overline{\varLambda }_m\varLambda _{mn}N_{20}+2N_6+2(\varGamma _{cm}\nonumber \\&+\overline{\varGamma }_{cm})N_{16}+(\varUpsilon _{cm}+\overline{\varUpsilon }_{cm})N_{15}\nonumber \\&+(\varGamma _{mn}+\varGamma _{\overline{m}n})N_{15}+\varLambda _n(\varGamma _{cm}+\overline{\varGamma }_{cm})N_{17}\nonumber \\&+\varLambda _m\varGamma _{\overline{m}n}N_{17}+\overline{\varLambda }_m\varGamma _{mn}N_{17}-2(\omega _{vn}^2+\omega _{vm}^2)N_9 \nonumber \\ \varXi _4= & {} 2\overline{\varUpsilon }_{2n}N_{14}+\overline{\varLambda }_{2n}N_{18}+\varLambda _m\overline{\varUpsilon }_{2n}N_{18}\nonumber \\&+\overline{\varLambda }_n\varUpsilon {m\overline{n}}N_{18}+2\overline{\varLambda }_{2n}\varLambda _m\nonumber \\&+2\varLambda _{m\overline{n}}\overline{\varLambda }_nN_{20}+\varUpsilon _{m\overline{n}}N_{15} \nonumber \\&+\overline{\varGamma }_{2n}N_{15}+N_5+2N_{16}\varGamma _{m\overline{n}}+\varLambda _{m\overline{n}}N_{17}\nonumber \\&+\varLambda _m\overline{\varGamma }_{2n}N_{17}+\overline{\varLambda }_n\varGamma _{m\overline{n}}N_{17}\nonumber \\&-8\omega _{vn}^2N_{12}-(\omega _{vn}-\omega _{vm})^2N_{10} \nonumber \\ \varXi _5= & {} N_1\omega _{vn}\nonumber \\ \varTheta _1= & {} 2M_4-4\omega _{vm}^2M_8+2\varUpsilon _{2m}M_{14}+2(\varUpsilon _{cm}\nonumber \\&+\overline{\varUpsilon }_{cm})M_{14} +\varLambda _{2m}M_{18}\nonumber \\&+\overline{\varLambda }_mM_{18}+\varLambda _m(\varUpsilon _{cm}+\overline{\varUpsilon }_{cm})M_{18}+\varGamma _{2m}M_{15}\nonumber \\&+(\varGamma _{cm}+\overline{\varGamma }_{cm})M_{18}+\varGamma _{2m}M_{15}+(\varGamma _{cm}+\overline{\varGamma }_{cm})M_{15}\nonumber \\&+\overline{\varLambda }_m\varGamma _{2m}M_{17}+\varLambda _m(\varGamma _{cm}+\overline{\varGamma }_{cm})M_{17}\nonumber \\ \varTheta _2= & {} M_7-4\omega _{vn}^2M_{11}+2\varGamma _{2n}M_{16}+\varUpsilon _{2n}M_{15}+\varLambda _{2n}M_{17}\nonumber \\&+\varLambda _n\varGamma _{2n}M_{17}+\varLambda _n\varUpsilon _{2n}M_{18}\nonumber \\ \varTheta _3= & {} 2M_5-2M_{10}(\omega _{vn}^2+\omega _{vm}^2)+2M_{14}(\varUpsilon _{cn}+\overline{\varUpsilon }_{cn})\nonumber \\&+\varLambda _m(\varUpsilon _{cn}+\overline{\varUpsilon }_{cn})M_{18} \nonumber \\&+\varLambda _n\varUpsilon _{m\overline{n}}M_{18}+\overline{\varLambda }_n\varUpsilon _{mn}M_{18}+2(\varGamma _{mn}\nonumber \\&+\varGamma _{m\overline{n}})M_{16}+\varUpsilon _{mn}M_{15}+\varUpsilon _{m\overline{n}}M_{15}+(\varGamma _{cn}\nonumber \\&+\overline{\varGamma }_{cn})M_{15} \nonumber \\&+\varLambda _{mn}M_{17}+\varLambda _{m\overline{n}}M_{17}+\varLambda _m(\varGamma _{cn}+\overline{\varGamma }_{cn})M_{17}\nonumber \\&+\overline{\varLambda }_n\varGamma _{mn}M_{17}+\varLambda _n\varGamma {m\overline{n}}M_{17} \nonumber \\ \varTheta _4= & {} 2\omega _{vm}M_0 \nonumber \\ \varTheta _5= & {} \omega _{vm}M_1 \end{aligned}$$

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