Modal coupled vibration behavior of piezoelectric L-shaped resonator induced by added mass

Abstract

This paper studies the mode coupling behavior and complex nonlinear dynamics of a piezoelectric driven L-shaped beam considering the added mass. To realize natural frequency adjustment and energy exchange among different modes, the added mass is delicately designed. The nonlinear governing equations, representing the first and second modes, are obtained by the Hamilton principle and Galerkin method. Perturbation and bifurcation analyses show that mode coupled vibration can lead to complex dynamic phenomena such as amplitude jump, amplitude saturation, double Hopf bifurcation and amplitude persistence. The physical conditions of amplitude jump, the critical voltage of amplitude saturation and the discriminant formula of double Hopf bifurcation are deduced theoretically and verified numerically. To be more convincing, an experimental test system is set up to observe the nonlinear dynamic behaviors. It is found that when the driving frequency is less than 26.18 Hz or more than 26.52 Hz, the first-order mode vibration jumps under the bifurcation driving voltage, which is qualitatively consistent with the theoretical results. Through mechanism investigation on subcritical Hopf bifurcation induced structure jumping phenomenon, the linear relationship between bifurcation voltage and perturbation mass is deduced. Both theoretical and experimental results demonstrate that small disturbance of added mass can significantly affect the bifurcation voltage of modal coupled vibration, which can realize the detection of micro-mass. The research results of this paper provide theoretical basis and experimental support for the development of micro-resonance device.

Appendix 1

The solutions of Eqs. (18) and (19) can be expressed in the following form

$$ \begin{gathered} w_{1} = \sum\limits_{i = 1}^{M} {\phi_{1i} (x_{1} )\eta_{i} (t)} \hfill \\ \overline{w}_{2} = \sum\limits_{i = 1}^{M} {\phi_{2i} (x_{2} )\eta_{i} (t)} \hfill \\ \end{gathered} $$

(39)

where $\phi_{1i} (x_{1} )$ and $\phi_{2i} (x_{2} )$ represent the linear undamped mode shapes of the horizontal and vertical beams; $\eta_{i} (t)$ represent the deformation coefficient of the i-th mode. Since horizontal and vertical beams can produce coupled vibration behavior, their frequencies should be consistent, and we assume that each mode function satisfies the following expressions

$$ \begin{gathered} { - }\beta_{i}^{2} \phi_{1i} (x_{1} ) + \phi_{1i}^{{{\text{iv}}}} (x_{1} ){ = }0 \hfill \\ { - }\beta_{i}^{2} \phi_{2i} (x_{2} ) + \frac{{h_{b2}^{2} L_{1}^{4} }}{{h_{b1}^{2} L_{2}^{4} }}\phi_{2i}^{{{\text{iv}}}} (x_{2} ){ = }0 \hfill \\ \end{gathered} $$

(40)

Through Eq. (20), the boundary conditions of the mode are as follows

$$ \begin{gathered} \phi^{\prime}_{1i} (0) = \phi_{1i} (0) = 0 \hfill \\ \phi_{2i} (0) = \phi^{\prime\prime}_{2i} (1) = \phi^{\prime\prime\prime}_{2i} (1) = 0 \hfill \\ \phi^{\prime}_{2i} (0){ = }\phi^{\prime}_{1i} (1) \hfill \\ \phi^{\prime\prime\prime}_{1i} (1){ = } - \frac{{h_{b2} L_{2} }}{{h_{b1} L_{1} }}\beta_{i}^{2} \phi_{1i} (1) \hfill \\ \phi^{\prime\prime}_{1i} (1){ = }\frac{{h_{b2}^{3} L_{1} }}{{h_{b1}^{3} L_{2} }}\phi^{\prime\prime}_{2i} (0) \hfill \\ \end{gathered} $$

Introduce expressions for modal functions

$$ \begin{gathered} \phi_{1i} (x_{1} ){ = }A_{1i} \sin \lambda_{1i} x_{1} + B_{1i} \cos \lambda_{1i} x_{1} + C_{1i} sh\lambda_{1i} x_{1} + D_{1i} ch\lambda_{1i} x_{1} \hfill \\ \phi_{2i} (x_{2} ){ = }A_{2i} \sin \lambda_{2i} x_{2} + B_{2i} \cos \lambda_{2i} x_{2} + C_{2i} sh\lambda_{2i} x_{2} + D_{2i} ch\lambda_{2i} x_{2} \hfill \\ \end{gathered} $$

(41)

From Eqs. (40–41), we derive the following expression

$$ \lambda_{2i} L_{1} \sqrt {h_{b2} } = \lambda_{1i} L_{2} \sqrt {h_{b1} } $$

(42)

Here, Fig. 

Fig. 13

First and second mode shapes obtained by the Eq. (41). a: Transverse deformation of the horizontal beam; b: Transverse deformation of the vertical beam)

13 shows the first and second mode shapes obtained by the mathematical model. It is found that when the first mode vibrates, the vertical beam and the horizontal beam are in phase. When the second mode vibrates, the vertical beam and the horizontal beam are out of phase, which is consistent with COMSOL simulation results (Fig. 4).

Considering the first two modes, Eq. (39) can be rewritten in the following form

$$ \begin{gathered} w_{1} = \phi_{11} (x_{1} )\eta_{1} (t){ + }\phi_{12} (x_{1} )\eta_{2} (t) \hfill \\ \overline{w}_{2} = \phi_{21} (x_{2} )\eta_{1} (t){ + }\phi_{22} (x_{2} )\eta_{2} (t) \hfill \\ \end{gathered} $$

(43)

Substituting Eq. (43) into the resulting Eqs. (18) and (19), multiplying by $\phi_{1i} (x_{1} )$, $\phi_{2i} (x_{2} )$, and integrating the outcome from 0 to 1, yield

$$ \begin{gathered} \ddot{\eta }_{1} [\phi_{11}^{2} (x_{1} - \frac{{l_{1} }}{{L_{1} }})m + \int_{0}^{1} {\phi_{11} \phi_{11} dx_{1} } {] + }\beta_{1}^{2} \eta_{1} \int_{0}^{1} {\phi_{11} \phi_{11} dx_{1} } + c_{1n} \eta_{1} \int_{0}^{1} {\phi_{11} \phi_{11} dx_{1} } \hfill \\ - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}(\ddot{\eta }_{1} \eta_{1} \int_{0}^{1} {\phi_{21} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{11} \phi_{11} dx_{1} } + \ddot{\eta }_{1} \eta_{2} \int_{0}^{1} {\phi_{21} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{12} \phi_{11} dx_{1} } \hfill \\ + \ddot{\eta }_{2} \eta_{1} \int_{0}^{1} {\phi_{22} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{11} \phi_{11} dx_{1} } + \ddot{\eta }_{2} \eta_{2} \int_{0}^{1} {\phi_{22} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{12} \phi_{11} dx_{1} } ) \hfill \\ = \alpha V\cos (\Omega t)[\phi^{\prime}_{11} (x_{b} ) - \phi^{\prime}_{11} (x_{a} )] \hfill \\ \end{gathered} $$

(44)

$$ \begin{gathered} \ddot{\eta }_{1} \int_{0}^{1} {\phi_{21} \phi_{21} dx_{1} } { + }\beta_{1}^{2} \eta_{1} \int_{0}^{1} {\phi_{21} \phi_{21} dx_{1} } + c_{2n} \eta_{1} \int_{0}^{1} {\phi_{21} \phi_{21} dx_{1} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}[\ddot{\eta }_{1} \eta_{1} \int_{0}^{1} {(1 - x_{2} )\phi_{11} (1)\phi^{\prime\prime}_{21} \phi_{21} dx_{2} } + \ddot{\eta }_{1} \eta_{2} \int_{0}^{1} {(1 - x_{2} )\phi_{11} (1)\phi^{\prime\prime}_{22} \phi_{21} dx_{2} } \hfill \\ + \ddot{\eta }_{2} \eta_{1} \int_{0}^{1} {(1 - x_{2} )\phi_{12} (1)\phi^{\prime\prime}_{21} \phi_{21} dx_{2} } + \ddot{\eta }_{2} \eta_{2} \int_{0}^{1} {(1 - x_{2} )\phi_{12} (1)\phi^{\prime\prime}_{22} \phi_{21} dx_{2} } ] \hfill \\ - \frac{{L_{1} }}{{L_{2} }}[\ddot{\eta }_{1} \eta_{1} \int_{0}^{1} {\phi_{11} (1)\phi^{\prime}_{21} \phi_{21} dx_{2} } + \ddot{\eta }_{1} \eta_{2} \int_{0}^{1} {\phi_{11} (1)\phi^{\prime}_{22} \phi_{21} dx_{2} } \hfill \\ + \ddot{\eta }_{2} \eta_{1} \int_{0}^{1} {\phi_{12} (1)\phi^{\prime}_{21} \phi_{21} dx_{2} } + \ddot{\eta }_{2} \eta_{2} \int_{0}^{1} {\phi_{12} (1)\phi^{\prime}_{22} \phi_{21} dx_{2} } ] \hfill \\ { + }\frac{{L_{1} }}{{L_{2} }}(\dot{\eta }_{1}^{2} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{11} dx_{1} } + \dot{\eta }_{2}^{2} \int_{0}^{1} {\phi^{\prime}_{12} \phi^{\prime}_{12} dx_{1} } + 2\dot{\eta }_{1} \dot{\eta }_{2} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } )\int_{0}^{1} {\phi_{21} dx_{2} } \hfill \\ { + }\frac{{L_{1} }}{{L_{2} }}(\ddot{\eta }_{1} \eta_{1} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{11} dx_{1} } + \ddot{\eta }_{2} \eta_{2} \int_{0}^{1} {\phi^{\prime}_{12} \phi^{\prime}_{12} dx_{1} } + \ddot{\eta }_{1} \eta_{2} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } \hfill \\ + \eta_{1} \ddot{\eta }_{2} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } )\int_{0}^{1} {\phi_{21} dx_{2} } = 0 \hfill \\ \end{gathered} $$

(45)

$$ \begin{gathered} \ddot{\eta }_{2} [\phi_{12}^{2} (x_{1} - \frac{{l_{1} }}{{L_{1} }})m + \int_{0}^{1} {\phi_{12} \phi_{12} dx_{1} } {] + }\beta_{2}^{2} \eta_{2} \int_{0}^{1} {\phi_{12} \phi_{12} dx_{1} } + c_{1n} \eta_{2} \int_{0}^{1} {\phi_{12} \phi_{12} dx_{1} } \hfill \\ - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}(\ddot{\eta }_{1} \eta_{1} \int_{0}^{1} {\phi_{21} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{11} \phi_{12} dx_{1} } + \ddot{\eta }_{1} \eta_{2} \int_{0}^{1} {\phi_{21} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{12} \phi_{12} dx_{1} } \hfill \\ + \ddot{\eta }_{2} \eta_{1} \int_{0}^{1} {\phi_{22} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{11} \phi_{12} dx_{1} } + \ddot{\eta }_{2} \eta_{2} \int_{0}^{1} {\phi_{22} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{12} \phi_{12} dx_{1} } ) \hfill \\ = \alpha V\cos (\Omega t)[\phi^{\prime}_{12} (x_{b} ) - \phi^{\prime}_{12} (x_{a} )] \hfill \\ \end{gathered} $$

(46)

$$ \begin{gathered} \ddot{\eta }_{2} \int_{0}^{1} {\phi_{22} \phi_{22} dx_{1} } { + }\beta_{2}^{2} \eta_{2} \int_{0}^{1} {\phi_{22} \phi_{22} dx_{1} } + c_{2n} \eta_{2} \int_{0}^{1} {\phi_{22} \phi_{22} dx_{1} } + \frac{{L_{1} }}{{L_{2} }}[\ddot{\eta }_{1} \eta_{1} \int_{0}^{1} {(1 - x_{2} )\phi_{11} (1)\phi^{\prime\prime}_{21} \phi_{22} dx_{2} } + \hfill \\ \ddot{\eta }_{1} \eta_{2} \int_{0}^{1} {(1 - x_{2} )\phi_{11} (1)\phi^{\prime\prime}_{22} \phi_{22} dx_{2} } + \ddot{\eta }_{2} \eta_{1} \int_{0}^{1} {(1 - x_{2} )\phi_{12} (1)\phi^{\prime\prime}_{21} \phi_{22} dx_{2} } + \ddot{\eta }_{2} \eta_{2} \int_{0}^{1} {(1 - x_{2} )\phi_{12} (1)\phi^{\prime\prime}_{22} \phi_{22} dx_{2} } ] \hfill \\ - \frac{{L_{1} }}{{L_{2} }}[\ddot{\eta }_{1} \eta_{1} \int_{0}^{1} {\phi_{11} (1)\phi^{\prime}_{21} \phi_{22} dx_{2} } + \ddot{\eta }_{1} \eta_{2} \int_{0}^{1} {\phi_{11} (1)\phi^{\prime}_{22} \phi_{22} dx_{2} } + \ddot{\eta }_{2} \eta_{1} \int_{0}^{1} {\phi_{12} (1)\phi^{\prime}_{21} \phi_{22} dx_{2} } \hfill \\ + \ddot{\eta }_{2} \eta_{2} \int_{0}^{1} {\phi_{12} (1)\phi^{\prime}_{22} \phi_{22} dx_{2} } ] + \frac{{L_{1} }}{{L_{2} }}(\dot{\eta }_{1}^{2} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{11} dx_{1} } + \dot{\eta }_{2}^{2} \int_{0}^{1} {\phi^{\prime}_{12} \phi^{\prime}_{12} dx_{1} } + 2\dot{\eta }_{1} \dot{\eta }_{2} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } )\int_{0}^{1} {\phi_{22} dx_{2} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}(\ddot{\eta }_{1} \eta_{1} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{11} dx_{1} } + \ddot{\eta }_{2} \eta_{2} \int_{0}^{1} {\phi^{\prime}_{12} \phi^{\prime}_{12} dx_{1} } + \ddot{\eta }_{1} \eta_{2} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } + \eta_{1} \ddot{\eta }_{2} \int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } )\int_{0}^{1} {\phi_{22} dx_{2} } = 0 \hfill \\ \end{gathered} $$

(47)

According to the principle of minimum energy, the first two modes are superimposed to obtain the two degrees of freedom equations of motion, which represent the first-order vibration behavior and the second-order vibration behavior of L-shaped beam, respectively.

$$ \begin{gathered} \ddot{\eta }_{1} { + }\omega_{1}^{2} \eta_{1} + c\dot{\eta }_{1} { + }g_{11} \dot{\eta }_{1}^{2} + g_{12} \dot{\eta }_{1} \dot{\eta }_{2} + g_{22} \dot{\eta }_{2}^{2} + s_{11} \ddot{\eta }_{1} \eta_{1} + s_{12} \ddot{\eta }_{1} \eta_{2} \hfill \\ + s_{21} \ddot{\eta }_{2} \eta_{1} + s_{22} \ddot{\eta }_{2} \eta_{2} = f\cos \Omega t \hfill \\ \end{gathered} $$

(48)

$$ \begin{gathered} \ddot{\eta }_{2} { + }\omega_{2}^{2} \eta_{2} + c\dot{\eta }_{2} { + }\overline{g}_{11} \dot{\eta }_{1}^{2} + \overline{g}_{12} \dot{\eta }_{1} \dot{\eta }_{2} + \overline{g}_{22} \dot{\eta }_{2}^{2} + \overline{s}_{11} \ddot{\eta }_{1} \eta_{1} + \overline{s}_{12} \ddot{\eta }_{1} \eta_{2} \hfill \\ + \overline{s}_{21} \ddot{\eta }_{2} \eta_{1} + \overline{s}_{22} \ddot{\eta }_{2} \eta_{2} = \overline{f}\cos \Omega t \hfill \\ \end{gathered} $$

(49)

where $c_{1n} = c_{2n} = c$.

In this article, we express the motion state of the resonator by the lateral displacement $\overline{w}_{2} (1,t)$ of the free end of the vertical beam. The first modal displacement and the second modal displacement can be expressed as $\phi_{21} (1)\eta_{1} (t)$ and $\phi_{22} (1)\eta_{2} (t)$, respectively. The specific coefficients are as follows:

$$ M_{1} = \phi_{11}^{2} (x_{1} - \frac{{l_{1} }}{{L_{1} }})m + \int_{0}^{1} {\phi_{11} \phi_{11} dx_{1} } + \int_{0}^{1} {\phi_{21} \phi_{21} dx_{1} } $$

(50)

$$ \omega_{1}^{2} = \frac{{\beta_{1}^{2} \int_{0}^{1} {\phi_{21} \phi_{21} dx_{1} } + \beta_{1}^{2} \int_{0}^{1} {\phi_{11} \phi_{11} dx_{1} } }}{{M_{1} }}\; $$

(51)

$$ g_{11} = \frac{{\frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{11} dx_{1} } \int_{0}^{1} {\phi_{21} dx_{2} } }}{{M_{1} }}\; $$

(52)

$$ g_{12} = \frac{{2\frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{21} dx_{2} } }}{{M_{1} }} $$

(53)

$$ g_{22} = \frac{{\frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{12} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{21} dx_{2} } }}{{M_{1} }} $$

(54)

$$ s_{11} = \frac{\begin{gathered} \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {(1 - x_{2} )\phi_{11} (1)\phi^{\prime\prime}_{21} \phi_{21} dx_{2} } - \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi_{11} (1)\phi^{\prime}_{21} \phi_{21} dx_{2} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{11} dx_{1} } \int_{0}^{1} {\phi_{21} dx_{2} } - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}\int_{0}^{1} {\phi_{21} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{11} \phi_{11} dx_{1} } \hfill \\ \end{gathered} }{{M_{1} }} $$

(55)

$$ s_{12} = \frac{\begin{gathered} \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {(1 - x_{2} )\phi_{11} (1)\phi^{\prime\prime}_{22} \phi_{21} dx_{2} } - \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi_{11} (1)\phi^{\prime}_{22} \phi_{21} dx_{2} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{21} dx_{2} } - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}\int_{0}^{1} {\phi_{21} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{12} \phi_{11} dx_{1} } \hfill \\ \end{gathered} }{{M_{1} }} $$

(56)

$$ s_{21} = \frac{\begin{gathered} \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {(1 - x_{2} )\phi_{12} (1)\phi^{\prime\prime}_{21} \phi_{21} dx_{2} } - \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi_{12} (1)\phi^{\prime}_{21} \phi_{21} dx_{2} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{21} dx_{2} } - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}\int_{0}^{1} {\phi_{22} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{11} \phi_{11} dx_{1} } \hfill \\ \end{gathered} }{{M_{1} }} $$

(57)

$$ s_{22} = \frac{\begin{gathered} \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {(1 - x_{2} )\phi_{12} (1)\phi^{\prime\prime}_{22} \phi_{21} dx_{2} } - \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi_{12} (1)\phi^{\prime}_{22} \phi_{21} dx_{2} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{12} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{21} dx_{2} } - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}\int_{0}^{1} {\phi_{22} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{12} \phi_{11} dx_{1} } \hfill \\ \end{gathered} }{{M_{1} }} $$

(58)

$$ f = \frac{{\alpha V[\phi^{\prime}_{11} (x_{b} ) - \phi^{\prime}_{11} (x_{a} )]}}{{M_{1} }} $$

(59)

$$ M_{2} = \phi_{12}^{2} (x_{1} - \frac{{l_{1} }}{{L_{1} }})m + \int_{0}^{1} {\phi_{12} \phi_{12} dx_{1} } + \int_{0}^{1} {\phi_{22} \phi_{22} dx_{1} } $$

(60)

$$ \omega_{2}^{2} = \frac{{\beta_{2}^{2} \int_{0}^{1} {\phi_{22} \phi_{22} dx_{1} } + \beta_{2}^{2} \int_{0}^{1} {\phi_{12} \phi_{12} dx_{1} } }}{{M_{2} }}\; $$

(61)

$$ \overline{g}_{11} = \frac{{\frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{11} dx_{1} } \int_{0}^{1} {\phi_{22} dx_{2} } }}{{M_{2} }}\; $$

(62)

$$ \overline{g}_{12} = \frac{{2\frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{22} dx_{2} } }}{{M_{2} }} $$

(63)

$$ \overline{g}_{22} = \frac{{\frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{12} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{22} dx_{2} } }}{{M_{2} }} $$

(64)

$$ \overline{s}_{11} = \frac{\begin{gathered} \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {(1 - x_{2} )\phi_{11} (1)\phi^{\prime\prime}_{21} \phi_{22} dx_{2} } - \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi_{11} (1)\phi^{\prime}_{21} \phi_{22} dx_{2} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{11} dx_{1} } \int_{0}^{1} {\phi_{22} dx_{2} } - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}\int_{0}^{1} {\phi_{21} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{11} \phi_{12} dx_{1} } \hfill \\ \end{gathered} }{{M_{2} }} $$

(65)

$$ \overline{s}_{12} = \frac{\begin{gathered} \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {(1 - x_{2} )\phi_{11} (1)\phi^{\prime\prime}_{22} \phi_{22} dx_{2} } - \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi_{11} (1)\phi^{\prime}_{22} \phi_{22} dx_{2} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{22} dx_{2} } - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}\int_{0}^{1} {\phi_{21} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{12} \phi_{12} dx_{1} } \hfill \\ \end{gathered} }{{M_{2} }} $$

(66)

$$ \overline{s}_{21} = \frac{\begin{gathered} \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {(1 - x_{2} )\phi_{12} (1)\phi^{\prime\prime}_{21} \phi_{22} dx_{2} } - \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi_{12} (1)\phi^{\prime}_{21} \phi_{22} dx_{2} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{11} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{22} dx_{2} } - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}\int_{0}^{1} {\phi_{22} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{11} \phi_{12} dx_{1} } \hfill \\ \end{gathered} }{{M_{2} }} $$

(67)

$$ \overline{s}_{22} = \frac{\begin{gathered} \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {(1 - x_{2} )\phi_{12} (1)\phi^{\prime\prime}_{22} \phi_{22} dx_{2} } - \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi_{12} (1)\phi^{\prime}_{22} \phi_{22} dx_{2} } \hfill \\ + \frac{{L_{1} }}{{L_{2} }}\int_{0}^{1} {\phi^{\prime}_{12} \phi^{\prime}_{12} dx_{1} } \int_{0}^{1} {\phi_{22} dx_{2} } - \frac{{L_{2}^{2} h_{b2} }}{{L_{1}^{2} h_{b1} }}\int_{0}^{1} {\phi_{22} dx_{2} } \int_{0}^{1} {\phi^{\prime\prime}_{12} \phi_{12} dx_{1} } \hfill \\ \end{gathered} }{{M_{2} }} $$

(68)

$$ \overline{f} = \frac{{\alpha V[\phi^{\prime}_{12} (x_{b} ) - \phi^{\prime}_{12} (x_{a} )]}}{{M_{2} }} $$

(69)