Appendix
Recall that the kinetic energy and potential energy of the system are:
$$\begin{aligned} U & = \frac{D}{2}\mathop {\iint }\nolimits \left\{ {\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{\partial^{2} w}}{{\partial y^{2} }}} \right)^{2} - 2(1 - v)\left[ {\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }}} \right) - \left( {\frac{{\partial^{2} w}}{\partial x\partial y}} \right)^{2} } \right]} \right\}dA \\ & \quad + \frac{1}{2}\mathop {\iint }\nolimits D_{1} (x,y)\left\{ {\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{\partial^{2} w}}{{\partial y^{2} }}} \right)^{2} - 2(1 - v)\left[ {\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }}} \right) - \left( {\frac{{\partial^{2} w}}{\partial x\partial y}} \right)^{2} } \right]} \right\}dA \\ & \quad + \frac{1}{2}\mathop {\iint }\nolimits D_{2} (x,y)\left\{ {\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{\partial^{2} w}}{{\partial y^{2} }}} \right)^{2} - 2(1 - v_{pzt} )\left[ {\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }}} \right) - \left( {\frac{{\partial^{2} w}}{\partial x\partial y}} \right)^{2} } \right]} \right\}dA \\ & \quad + B\left( t \right)\mathop {\iint }\nolimits S\left( {x,y} \right)\left[ {\left( {d_{31} + v_{pzt} d_{32} } \right)\frac{{\partial^{2} w}}{{\partial x^{2} }} + \left( {d_{32} + v_{pzt} d_{31} } \right)\frac{{\partial^{2} w}}{{\partial y^{2} }}} \right]dA \\ T & = \frac{1}{2}\rho h\mathop {\iint }\nolimits \left( {\frac{\partial w}{\partial t}} \right)^{2} {\text{d}}A + \frac{1}{2}\rho_{pzt} t_{p} \mathop {\iint }\nolimits \left( {\frac{\partial w}{\partial t}} \right)^{2} S\left( {x,y} \right){\text{d}}A + \frac{1}{2}\mathop {\iint }\nolimits \mathop \sum \limits_{r} m_{r} \delta (x - x_{r} ,y - y_{r} )\left( {\frac{\partial w}{\partial t}} \right)^{2} {\text{d}}A \\ \end{aligned}$$
The assumed solution given in double series as in (8), is substituted into these energy equations. The product of multiplication between two double series or its derivatives will have four conditions, i.e. (i) m = p and n = q, (ii) m = p and n ≠ q, (iii) m ≠ p and n = q and (iv) m ≠ p and n ≠ q, and there are nine integrals involved which can be solved as follows:
$$\begin{aligned} I_{1} & = {\iint }\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{\partial^{2} w}}{{\partial y^{2} }}} \right)^{2} dA \\ & = \sum\limits_{m} {\sum\limits_{n} {\sum\limits_{p} {\sum\limits_{q} {W_{mn} } } } } W_{pq} \left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left( {\frac{q\pi }{b}} \right)^{2} } \right]\int\limits_{0}^{a} {\sin {\mkern 1mu} \frac{m\pi x}{a}{\mkern 1mu} \sin {\mkern 1mu} \frac{p\pi x}{a}dx} \int\limits_{0}^{b} {\sin {\mkern 1mu} \frac{n\pi y}{b}{\mkern 1mu} \sin {\mkern 1mu} \frac{q\pi y}{b}dy} \\ & = \frac{ab}{4}\sum\limits_{m} {\sum\limits_{n} {W_{{mn}}^{2} } } \left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right]^{2} \\ \end{aligned}$$
$$\begin{aligned} I_{2} & = \mathop {\iint }\nolimits \frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }}dA \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p} \mathop \sum \limits_{q} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}\mathop \smallint \limits_{0}^{a} \,\sin \,\frac{m\pi x}{a}\,\sin \,\frac{p\pi x}{a}dx\mathop \smallint \limits_{0}^{b} \,\sin \,\frac{n\pi y}{b}\,\sin \,\frac{q\pi y}{b}dy \\ & = \frac{ab}{4}\mathop \sum \limits_{m} \mathop \sum \limits_{n} W_{{mn}}^{2} \frac{{m^{2} n^{2} \pi^{4} }}{{a^{2} b^{2} }} \\ \end{aligned}$$
$$\begin{aligned} I_{3} & = \mathop {\iint }\nolimits \left( {\frac{{\partial^{2} w}}{\partial x\partial y}} \right)^{2} dA \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p} \mathop \sum \limits_{q} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}\mathop \smallint \limits_{0}^{a} cos\frac{m\pi x}{a}cos\frac{p\pi x}{a}dx\mathop \smallint \limits_{0}^{b} cos\frac{n\pi y}{b}cos\frac{q\pi y}{b}dy \\ & = \frac{ab}{4}\mathop \sum \limits_{m} \mathop \sum \limits_{n} W_{{mn}}^{2} \frac{{m^{2} n^{2} \pi^{4} }}{{a^{2} b^{2} }} \\ \end{aligned}$$
$$\begin{aligned} I_{4} & = \mathop {\iint }\nolimits S\left( {x,y} \right)\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{\partial^{2} w}}{{\partial y^{2} }}} \right)^{2} dA \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p} \mathop \sum \limits_{q} W_{mn} W_{pq} \left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left( {\frac{q\pi }{b}} \right)^{2} } \right]\mathop \smallint \limits_{x1}^{x2} \sin \frac{m\pi x}{a}\sin \frac{p\pi x}{a}dx\mathop \smallint \limits_{y1}^{y2} \sin \frac{n\pi y}{b}\sin \frac{q\pi y}{b}dy \\ & { = }\mathop \sum \limits_{m} \mathop \sum \limits_{n} W_{{mn}}^{2} \left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right]^{2} C_{1x} C_{1y} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q \ne n} W_{mn} W_{pq} \left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left( {\frac{q\pi }{b}} \right)^{2} } \right]C_{2x}\cdot C_{2y} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p = m} \mathop \sum \limits_{q \ne n} W_{mn} W_{pq} \left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left( {\frac{q\pi }{b}} \right)^{2} } \right]C_{1x}\cdot C_{2y} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q = n} W_{mn} W_{pq} \left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left( {\frac{q\pi }{b}} \right)^{2} } \right]C_{2x}\cdot C_{1y} \\ \end{aligned}$$
$$\begin{aligned} I_{5} & = \mathop {\iint }\nolimits S\left( {x,y} \right)\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }}dA \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p} \mathop \sum \limits_{q} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}\mathop \smallint \limits_{x1}^{x2} \sin \frac{m\pi x}{a}\sin \frac{p\pi x}{a}dx\mathop \smallint \limits_{y1}^{y2} \sin \frac{n\pi y}{b}\sin \frac{q\pi y}{b}dy \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} W_{mn}^{2} \frac{{m^{2} n^{2} \pi^{4} }}{{a^{2} b^{2} }}C_{1x} C_{1y} + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q \ne n} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}C_{2x} C_{2y} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p = m} \mathop \sum \limits_{q \ne n} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}C_{1x} C_{2y} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q = n} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}C_{2x} C_{1y} \\ \end{aligned}$$
$$\begin{aligned} I_{6} & = \mathop {\iint }\nolimits S\left( {x,y} \right)\left( {\frac{{\partial^{2} w}}{\partial x\partial y}} \right)^{2} dA \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p} \mathop \sum \limits_{q} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}\mathop \smallint \limits_{x1}^{x2} cos\frac{m\pi x}{a}cos\frac{p\pi x}{a}dx\mathop \smallint \limits_{y1}^{y2} cos\frac{n\pi y}{b}cos\frac{q\pi y}{b}dy \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} W_{mn}^{2} \frac{{m^{2} n^{2} \pi^{4} }}{{a^{2} b^{2} }}C_{3x} C_{3y} + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q \ne n} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}C_{4x} C_{4y} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p = m} \mathop \sum \limits_{q \ne n} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}C_{3x} C_{4y} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q = n} W_{mn} W_{pq} \frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}C_{4x} C_{3y} \\ \end{aligned}$$
$$\begin{aligned} I_{7} & = \mathop {\iint }\nolimits S\left( {x,y} \right)\frac{{\partial^{2} w}}{{\partial x^{2} }}dA = - \mathop \sum \limits_{m} \mathop \sum \limits_{n} W_{mn} \left( {\frac{m\pi }{a}} \right)^{2} \mathop \smallint \limits_{x1}^{x2} \sin \frac{m\pi x}{a}dx\mathop \smallint \limits_{y1}^{y2} \sin \frac{n\pi y}{b}dy \\ & = - \mathop \sum \limits_{m} \mathop \sum \limits_{n} W_{mn} \left( {\frac{m\pi }{a}} \right)^{2} C_{5x} C_{5y} \\ \end{aligned}$$
$$\begin{aligned} I_{8} & = \mathop {\iint }\nolimits S\left( {x,y} \right)\frac{{\partial^{2} w}}{{\partial y^{2} }}dA = - \mathop \sum \limits_{m} \mathop \sum \limits_{n} W_{mn} \left( {\frac{n\pi }{b}} \right)^{2} \mathop \smallint \limits_{x1}^{x2} \sin \frac{m\pi x}{a}dx\mathop \smallint \limits_{y1}^{y2} \sin \frac{n\pi y}{b}dy \\ & = - \mathop \sum \limits_{m} \mathop \sum \limits_{n} W_{mn} \left( {\frac{n\pi }{b}} \right)^{2} C_{5x} C_{5y} \\ \end{aligned}$$
$$\begin{aligned} I_{9} & = \mathop {\iint }\nolimits \left( {\frac{\partial w}{\partial t}} \right)^{2} {\text{d}}A = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p} \mathop \sum \limits_{q} \dot{W}_{mn} \dot{W}_{pq} \mathop \smallint \limits_{x1}^{x2} \sin\frac{m\pi x}{a}\sin\frac{p\pi x}{a}dx\mathop \smallint \limits_{y1}^{y2} \sin\frac{n\pi y}{b}\sin\frac{q\pi y}{b}dy \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \dot{W}_{mn}^{2} \frac{ab}{4} \\ \end{aligned}$$
$$\begin{aligned} I_{10} & = \mathop {\iint }\nolimits S\left( {x,y} \right)\left( {\frac{\partial w}{\partial t}} \right)^{2} {\text{d}}A \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p} \mathop \sum \limits_{q} \dot{W}_{mn} \dot{W}_{pq} \mathop \smallint \limits_{x1}^{x2} \sin\frac{m\pi x}{a}\sin\frac{p\pi x}{a}dx\mathop \smallint \limits_{y1}^{y2} \sin\frac{n\pi y}{b}\sin\frac{q\pi y}{b}dy \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \dot{W}_{mn}^{2} \left\{ {\frac{\rho hab}{8} + \frac{{\rho_{piezo} t_{p} }}{2}C_{1x} C_{1y} } \right\} + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q \ne n} \dot{W}_{mn} \dot{W}_{pq} \frac{{\rho_{piezo} t_{p} }}{2}C_{2x} C_{2y} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p = m} \mathop \sum \limits_{q \ne n} \dot{W}_{mn} \dot{W}_{pq} \frac{{\rho_{piezo} t_{p} }}{2}C_{1x} C_{2y} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q = n} \dot{W}_{mn} \dot{W}_{pq} \frac{{\rho_{piezo} t_{p} }}{2}C_{2x} C_{1y} \\ \end{aligned}$$
$$\begin{aligned} I_{11} & = \mathop {\iint }\nolimits \delta (x - x_{r} ,y - y_{r} )\left( {\frac{\partial w}{\partial t}} \right)^{2} {\text{d}}A \\ & = \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p} \mathop \sum \limits_{q} \dot{W}_{mn} \dot{W}_{pq} \sin\frac{{m\pi x_{r} }}{a}\sin\frac{{p\pi x_{r} }}{a}\sin\frac{{n\pi y_{r} }}{b}\sin\frac{{q\pi y_{r} }}{b} \\ \end{aligned}$$
where the coefficients C are
-
\(C_{1x} = \mathop \smallint \limits_{x1}^{x2} \sin^{2} \frac{m\pi x}{a}dx = \frac{1}{2}\left[ {\left( {x_{2} - x_{1} } \right) - \frac{a}{2m\pi }(\sin \frac{{2m\pi x_{2} }}{a} - \sin \frac{{2m\pi x_{1} }}{a})} \right]\)
-
\(C_{1y} = \mathop \smallint \limits_{y1}^{y2} \sin^{2} \frac{n\pi y}{b}dy = \frac{1}{2}\left[ {\left( {y_{2} - y_{1} } \right) - \frac{b}{2n\pi }(\sin \frac{{2n\pi y_{2} }}{b} - \sin \frac{{2n\pi y_{1} }}{b})} \right]\)
-
\(\begin{aligned} C_{2x} = \mathop \smallint \limits_{x1}^{x2} \sin \frac{m\pi x}{a}\sin \frac{p\pi x}{a}dx\, =&\, \frac{a}{{2\left( {m - p} \right)\pi }}\left[ {\sin \frac{{\left( {m - p} \right)\pi x_{2} }}{a} - \sin \frac{{\left( {m - p} \right)\pi x_{1} }}{a}} \right]\\ &\,- \frac{a}{{2\left( {m + p} \right)\pi }}\left[ {\sin \frac{{\left( {m + p} \right)\pi x_{2} }}{a} - \sin \frac{{\left( {m + p} \right)\pi x_{1} }}{a}} \right]\end{aligned}\)
-
\(\begin{aligned} C_{2y} = \mathop \smallint \limits_{y1}^{y2} \sin \frac{n\pi y}{b}\sin \frac{q\pi y}{b}dy \,=&\, \frac{b}{{2\left( {n - q} \right)\pi }}\left[ {\sin \frac{{\left( {n - q} \right)\pi y_{2} }}{b} - \sin \frac{{\left( {n - q} \right)\pi y_{1} }}{b}} \right]\\ &\,- \frac{b}{2(n + q)\pi }\left[ {\sin \frac{{(n + q)\pi y_{2} }}{b} - \sin \frac{{(n + q)\pi y_{1} }}{b}} \right] \end{aligned}\)
-
\(C_{3x} = \mathop \smallint \limits_{x1}^{x2} cos^{2} \frac{m\pi x}{a}dx = \frac{1}{2}\left[ {\left( {x_{2} - x_{1} } \right) + \frac{a}{2m\pi }(\sin \frac{{2m\pi x_{2} }}{a} - \sin \frac{{2m\pi x_{1} }}{a})} \right]\)
-
\(C_{3y} = \mathop \smallint \limits_{y1}^{y2} cos^{2} \frac{n\pi y}{b}dy = \frac{1}{2}\left[ {\left( {y_{2} - y_{1} } \right) + \frac{b}{2n\pi }(\sin \frac{{2n\pi y_{2} }}{b} - \sin \frac{{2n\pi y_{1} }}{b})} \right]\)
-
\(\begin{aligned} C_{4x} = \mathop \smallint \limits_{x1}^{x2} cos\frac{m\pi x}{a}cos\frac{p\pi x}{a}dx\,=&\, \frac{a}{{2\left( {m - p} \right)\pi }}\left[ {\sin \frac{{\left( {m - p} \right)\pi x_{2} }}{a} - \sin \frac{{\left( {m - p} \right)\pi x_{1} }}{a}} \right]\\&\, + \frac{a}{2(m + p)\pi }\left[ {\sin \frac{{(m + p)\pi x_{2} }}{a} - \sin \frac{{(m + p)\pi x_{1} }}{a}} \right]\end{aligned}\)
-
\(\begin{aligned} C_{4y} = \mathop \smallint \limits_{y1}^{y2} cos\frac{n\pi y}{b}cos\frac{q\pi y}{b}dy\, =&\, \frac{b}{{2\left( {n - q} \right)\pi }}\left[ {\sin \frac{{\left( {n - q} \right)\pi y_{2} }}{b} - \sin \frac{{\left( {n - q} \right)\pi y_{1} }}{b}} \right]\\&\, + \frac{b}{2(n + q)\pi }\left[ {\sin \frac{{(n + q)\pi y_{2} }}{b} - \sin \frac{{(n + q)\pi y_{1} }}{b}} \right]\end{aligned}\)
-
\(C_{5x} = \mathop \smallint \limits_{x1}^{x2} \sin^{2} \frac{m\pi x}{a}dx = \frac{a}{m\pi }(\cos \frac{{m\pi x_{1} }}{a} - \cos \frac{{m\pi x_{2} }}{a})\)
-
\(C_{5y} = \mathop \smallint \limits_{y1}^{y2} \sin^{2} \frac{n\pi y}{b}dy = \frac{b}{n\pi }(\cos \frac{{n\pi y_{1} }}{b} - \cos \frac{{n\pi y_{2} }}{b})\)
Thus, the potential and kinetic energies of the system are:
$$\begin{aligned} U_{total} & = \mathop \sum \limits_{m}\mathop\sum \limits_{n} W_{mn}^{2} \left\{{\frac{abD}{8}\left[{\left( {\frac{m\pi }{a}} \right)^{2} + \left({\frac{n\pi }{b}}\right)^{2} } \right]^{2} } \right. \\ & {\quad\qquad\qquad\qquad\qquad+ \frac{{D_{1} }}{2}\left\{ {\left[{\left({\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi}{b}}\right)^{2} } \right]^{2} C_{1x} C_{1y} - 2\left( {1 -v}\right)\left( {\frac{{mn\pi^{2} }}{ab}} \right)^{2} \left({C_{1x}C_{1y} - C_{3x} C_{3y} } \right)} \right\}} \\ &\quad\qquad\qquad\qquad\qquad + \frac{{D_{2} }}{2}\left\{{\left[{\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi}{b}}\right)^{2} } \right]^{2} C_{1x} C_{1y} } \right. \\ &\left.{\left. { \quad\qquad\qquad\qquad \qquad\qquad\qquad - 2\left({1 -v_{pzt} } \right)\left( {\frac{{mn\pi^{2} }}{ab}}\right)^{2}\left({C_{1x} C_{1y} - C_{3x} C_{3y} } \right)} \right\}}\right\} \\&\quad + \mathop \sum \limits_{m} \mathop \sum\limits_{n} \mathop\sum \limits_{p \ne m} \mathop \sum\limits_{q \ne n} W_{mn} W_{pq}\left\{ {\frac{{D_{1}}}{2}\left\{ {\left[ {\left( {\frac{m\pi }{a}}\right)^{2} +\left( {\frac{n\pi }{b}} \right)^{2} } \right]\left[{\left({\frac{p\pi }{a}} \right)^{2} + \left( {\frac{q\pi}{b}}\right)^{2} } \right]C_{2x} C_{2y} } \right.} \right. \\&\left. { \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad - 2\left( {1 -v}\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}}\right)\left({C_{2x} C_{2y} - C_{4x} C_{4y} } \right)} \right\} \\&\qquad\qquad \qquad\qquad\qquad\qquad\qquad\quad\quad \quad+\frac{{D_{2} }}{2}\left\{ {\left[{\left( {\frac{m\pi }{a}}\right)^{2} + \left( {\frac{n\pi }{b}}\right)^{2} } \right]\left[{\left( {\frac{p\pi }{a}} \right)^{2}+ \left( {\frac{q\pi }{b}}\right)^{2} } \right]C_{2x} C_{2y}} \right.\\ & \left. {\left. {\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad-2\left( {1 - v_{pzt} } \right)\left({\frac{{mnpq\pi^{4} }}{{a^{2}b^{2} }}} \right)\left({C_{2x}C_{2y} - C_{4x} C_{4y} } \right)} \right\}} \right\}\\&\quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n}\mathop\sum \limits_{p = m} \mathop \sum \limits_{q \ne n}W_{mn}W_{pq} \left\{ {\frac{{D_{1} }}{2}\left\{ {\left[{\left({\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi}{b}}\right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}}\right)^{2}+ \left( {\frac{q\pi }{b}} \right)^{2} } \right]C_{1x}C_{2y}} \right.} \right. \\ & \left. { \qquad\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \quad-2\left( {1 - v}\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2}}}} \right)\left({C_{1x} C_{2y} - C_{3x} C_{4y} } \right)}\right\} \\ & \quad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \quad+\frac{{D_{2}}}{2}\left\{ {\left[ {\left( {\frac{m\pi }{a}}\right)^{2} +\left( {\frac{n\pi }{b}} \right)^{2} } \right]\left[{\left({\frac{p\pi }{a}} \right)^{2} + \left({\frac{q\pi }{b}}\right)^{2} } \right]C_{1x} C_{2y} } \right.\\& \left. {\left. {\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad -2\left( {1 - v_{pzt} }\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2}b^{2} }}} \right)\left({C_{1x} C_{2y} - C_{3x} C_{4y} } \right)}\right\}} \right\}\\ & \quad+ \mathop \sum \limits_{m} \mathop \sum\limits_{n} \mathop\sum \limits_{p \ne m} \mathop \sum \limits_{q =n} W_{mn}W_{pq} \left\{ {\frac{{D_{1} }}{2}\left\{ {\left[{\left({\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi}{b}}\right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}}\right)^{2}+ \left( {\frac{q\pi }{b}} \right)^{2} } \right]C_{2x}C_{1y}} \right.} \right. \\ & \left. { \qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad-2\left( {1 - v}\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2}}}} \right)\left({C_{2x} C_{1y} - C_{4x} C_{3y} } \right)}\right\} \\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad + \frac{{D_{2}}}{2}\left\{{\left[ {\left( {\frac{m\pi }{a}} \right)^{2} +\left( {\frac{n\pi}{b}} \right)^{2} } \right]\left[ {\left({\frac{p\pi }{a}}\right)^{2} + \left({\frac{q\pi }{b}}\right)^{2} } \right]C_{2x} C_{1y} } \right.\\& \left. {\left. { \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad- 2\left({1 - v_{pzt} }\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}}\right)\left({C_{2x} C_{1y} - C_{4x} C_{3y} } \right)} \right\}}\right\}\\ &\quad + \mathop \sum \limits_{m} \mathop \sum\limits_{n} W_{mn}B\left( t \right)\left[ {\left( {d_{31} + v_{pzt}d_{32} }\right)\left( {\frac{m\pi }{a}} \right)^{2} + \left( {d_{32}+v_{pzt} d_{31} } \right)\left( {\frac{n\pi }{b}} \right)^{2}}\right]C_{5x} C_{5y} \\ \end{aligned}$$
$$\begin{aligned} T_{total} & = \mathop \sum \limits_{m} \mathop\sum \limits_{n} \dot{W}_{mn}^{2} \left\{ {\frac{\rho hab}{8} +\frac{{\rho_{piezo} t_{p} }}{2}C_{1x} C_{1y} } \right\} +\mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum\limits_{p \ne m} \mathop \sum \limits_{q \ne n} \dot{W}_{mn}\dot{W}_{pq} \frac{{\rho_{piezo} t_{p} }}{2}C_{2x} C_{2y}\\ &\quad+ \mathop \sum \limits_{m} \mathop \sum \limits_{n}\mathop \sum \limits_{p = m} \mathop \sum \limits_{q \ne n}\dot{W}_{mn} \dot{W}_{pq} \frac{{\rho_{piezo} t_{p}}}{2}C_{1x} C_{2y} \\ &\quad + \mathop \sum\limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q = n} \dot{W}_{mn} \dot{W}_{pq}\frac{{\rho_{piezo} t_{p} }}{2}C_{2x} C_{1y} \\ &\quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n}\mathop \sum \limits_{p} \mathop \sum \limits_{q} \dot{W}_{mn}\dot{W}_{pq} \sin\frac{{m\pi x_{r} }}{a}\sin\frac{{p\pi x_{r}}}{a}\sin\frac{{n\pi y_{r} }}{b}\sin\frac{{q\pi y_{r} }}{b} \\ \end{aligned}$$
Lagrange’s method is used to derive the EOM of the system. Since there is no external force acting on the system, (that is, force is generated internally by the piezoelectric patch), the Lagrange’s equation becomes
$$\frac{\partial }{\partial t}\left( {\frac{\partial L}{{\partial\dot{W}_{mn} }}} \right) - \frac{\partial L}{{\partial W_{mn}}} = 0\quad where\quad L = T - U$$
the final EOM is as follows:
$$\begin{aligned} & - \omega^{2} \left\{ {W_{mn} \left\{{\frac{\rho hab}{4} + \rho_{pzt} t_{p} C_{1x} C_{1y} }\right\}} \right. \\ & \quad\quad + \frac{{\rho_{pzt}t_{p} }}{2}\left\{ {\mathop \sum \limits_{p \ne m} \mathop \sum\limits_{q \ne n} W_{pq} C_{2x} C_{2y} + \mathop \sum\limits_{p = m} \mathop \sum \limits_{q \ne n} W_{pq} C_{1x}C_{2y} + \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q= n} W_{pq} C_{2x} C_{1y} } \right\}\\& \left. { \quad\quad+ \mathop \sum \limits_{r} m_{r} \mathop \sum\limits_{p} \mathop \sum \limits_{q} W_{pq} \sin\frac{{m\pi x_{r}}}{a}\sin\frac{{p\pi x_{r} }}{a}\sin\frac{{n\pi y_{r}}}{b}\sin\frac{{q\pi y_{r} }}{b}} \right\} \\ & + 2W_{mn}\left\{ {\frac{abD}{8}\left[ {\left( {\frac{m\pi }{a}}\right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right]^{2} }\right. \\ & \quad\quad\quad + \frac{{D_{1} }}{2}\left\{ {\left[{\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}}\right)^{2} } \right]^{2} C_{1x} C_{1y} - 2\left( {1 - v}\right)\left( {\frac{{mn\pi^{2} }}{ab}} \right)^{2} \left({C_{1x} C_{1y} - C_{3x} C_{3y} } \right)} \right\} \\&\quad\quad\quad + \frac{{D_{2} }}{2}\left\{ {\left[ {\left({\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}}\right)^{2} } \right]^{2} C_{1x} C_{1y} } \right. \\ & \left.{\left. { \quad\quad\quad- 2\left( {1 - v_{pzt} } \right)\left({\frac{{mn\pi^{2} }}{ab}} \right)^{2} \left( {C_{1x} C_{1y} -C_{3x} C_{3y} } \right)} \right\}} \right\}\\ & + \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q\ne n} W_{pq} \left\{ {\frac{{D_{1} }}{2}\left\{ {\left[ {\left({\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}}\right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2}+ \left( {\frac{q\pi }{b}} \right)^{2} } \right]C_{2x} C_{2y}} \right.} \right. \\ & \left. { \quad\quad\quad\quad\quad\quad - 2\left({1 - v} \right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}}\right)\left( {C_{2x} C_{2y} - C_{4x} C_{4y} } \right)}\right\} \\ & \quad\quad\quad\quad\quad\quad + \frac{{D_{2} }}{2}\left\{{\left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left({\frac{n\pi }{b}} \right)^{2} } \right]\left[ {\left( {\frac{p\pi}{a}} \right)^{2} + \left( {\frac{q\pi }{b}} \right)^{2} }\right]C_{2x} C_{2y} } \right.\\ & \left. {\left. {\quad\quad\quad\quad\quad\quad - 2\left( {1 - v_{pzt} }\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}} \right)\left({C_{2x} C_{2y} - C_{4x} C_{4y} } \right)} \right\}} \right\}\\ & + \mathop \sum \limits_{p = m} \mathop \sum \limits_{q \ne n} W_{pq} \left\{ {\frac{{D_{1} }}{2}\left\{ {\left[ {\left({\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}}\right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2}+ \left( {\frac{q\pi }{b}} \right)^{2} } \right]C_{1x} C_{2y}} \right.} \right. \\ & \left. {\quad\quad\quad\quad\quad\quad- 2\left( {1- v} \right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}}\right)\left( {C_{1x} C_{2y} - C_{3x} C_{4y} } \right)}\right\} \\ &\quad\quad\quad\quad\quad\quad + \frac{{D_{2} }}{2}\left\{{\left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left({\frac{n\pi }{b}} \right)^{2} } \right]\left[ {\left( {\frac{p\pi}{a}} \right)^{2} + \left( {\frac{q\pi }{b}} \right)^{2} }\right]C_{1x} C_{2y} } \right.\\ & \left. {\left. {\quad\quad\quad\quad\quad\quad - 2\left( {1 - v_{pzt} }\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}} \right)\left({C_{1x} C_{2y} - C_{3x} C_{4y} } \right)} \right\}} \right\}\\ & + \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q =n} W_{pq} \left\{ {\frac{{D_{1} }}{2}\left\{ {\left[ {\left({\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}}\right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2}+ \left( {\frac{q\pi }{b}} \right)^{2} } \right]C_{2x} C_{1y}} \right.} \right. \\ & \left. {\quad\quad\quad\quad\quad\quad - 2\left( {1- v} \right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}}\right)\left( {C_{2x} C_{1y} - C_{4x} C_{3y} } \right)}\right\} \\ &\quad\quad\quad\quad\quad\quad+ \frac{{D_{2} }}{2}\left\{{\left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left({\frac{n\pi }{b}} \right)^{2} } \right]\left[ {\left( {\frac{p\pi}{a}} \right)^{2} + \left( {\frac{q\pi }{b}} \right)^{2} }\right]C_{2x} C_{1y} } \right.\\ & \left. {\left. {\quad\quad\quad\quad\quad\quad - 2\left( {1 - v_{pzt} }\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}} \right)\left({C_{2x} C_{1y} - C_{4x} C_{3y} } \right)} \right\}} \right\}\\ & = - B\left( t \right)\left[ {\left( {d_{31} + v_{pzt}d_{32} } \right)\left( {\frac{m\pi }{a}} \right)^{2} + \left({d_{32} + v_{pzt} d_{31} } \right)\left( {\frac{n\pi }{b}}\right)^{2} } \right]C_{5x} C_{5y} \\ \end{aligned}$$
Terms in mass matrix (denoted by M
mnpq
), stiffness matrix (denoted by K
mnpq
) and force vector (denoted by F
mn
) are:
$$\begin{aligned} M_{mnpq} & = \frac{\rho hab}{4} + \rho_{pzt} t_{p} C_{1x} C_{1y} + \frac{{\rho_{pzt} t_{p} }}{2}\left\{ {\mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q \ne n} C_{2x} C_{2y} + \mathop \sum \limits_{p = m} \mathop \sum \limits_{q \ne n} C_{1x} C_{2y} + \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q = n} C_{2x} C_{1y} } \right\} \\ & \quad + \mathop \sum \limits_{r} m_{r} \mathop \sum \limits_{p} \mathop \sum \limits_{q} \sin\frac{{m\pi x_{r} }}{a}\sin\frac{{p\pi x_{r} }}{a}\sin\frac{{n\pi y_{r} }}{b}\sin\frac{{q\pi y_{r} }}{b} \\ \end{aligned}$$
$$\begin{aligned} K_{mnpq} & = 2\left\{ {\frac{abD}{8}\left[{\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}}\right)^{2} } \right]^{2} } \right. \\ &\quad \quad + \frac{{D_{1}}}{2}\left\{ {\left[ {\left( {\frac{m\pi }{a}} \right)^{2} +\left( {\frac{n\pi }{b}} \right)^{2} } \right]^{2} C_{1x}C_{1y} - 2\left( {1 - v} \right)\left( {\frac{{mn\pi^{2}}}{ab}} \right)^{2} \left( {C_{1x} C_{1y} - C_{3x} C_{3y}} \right)} \right\} \\ &\quad \quad + \frac{{D_{2} }}{2}\left\{{\left[ {\left( {\frac{m\pi }{a}} \right)^{2} + \left({\frac{n\pi }{b}} \right)^{2} } \right]^{2} C_{1x} C_{1y} }\right. \\ & \left. {\quad \quad \left. { - 2\left( {1 - v_{pzt}} \right)\left( {\frac{{mn\pi^{2} }}{ab}} \right)^{2} \left({C_{1x} C_{1y} - C_{3x} C_{3y} } \right)} \right\}} \right\}\\ & \quad+ \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q\ne n} \left\{ {\frac{{D_{1} }}{2}\left\{ {\left[ {\left({\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}}\right)^{2} } \right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2}+ \left( {\frac{q\pi }{b}} \right)^{2} } \right]C_{2x} C_{2y}} \right.} \right. \\ & \left. { \quad\quad - 2\left( {1 - v}\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}} \right)\left({C_{2x} C_{2y} - C_{4x} C_{4y} } \right)} \right\} \\ &\quad\quad + \frac{{D_{2} }}{2}\left\{ {\left[ {\left( {\frac{m\pi}{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} }\right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left({\frac{q\pi }{b}} \right)^{2} } \right]C_{2x} C_{2y} } \right.\\ & \left. {\left. {\quad \quad - 2\left( {1 - v_{pzt} }\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}} \right)\left({C_{2x} C_{2y} - C_{4x} C_{4y} } \right)} \right\}} \right\}\\ &\quad + \mathop \sum \limits_{p = m} \mathop \sum \limits_{q \ne n} \left\{ {\frac{{D_{1} }}{2}\left\{ {\left[ {\left( {\frac{m\pi}{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} }\right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left({\frac{q\pi }{b}} \right)^{2} } \right]C_{1x} C_{2y} }\right.} \right. \\ & \left. {\quad \quad - 2\left( {1 - v}\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}} \right)\left({C_{1x} C_{2y} - C_{3x} C_{4y} } \right)} \right\} \\ &\quad\quad + \frac{{D_{2} }}{2}\left\{ {\left[ {\left( {\frac{m\pi}{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} }\right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left({\frac{q\pi }{b}} \right)^{2} } \right]C_{1x} C_{2y} } \right.\\ & \left. {\left. { \quad\quad - 2\left( {1 - v_{pzt} }\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}} \right)\left({C_{1x} C_{2y} - C_{3x} C_{4y} } \right)} \right\}} \right\}\\ &\quad + \mathop \sum \limits_{p \ne m} \mathop \sum \limits_{q =n} \left\{ {\frac{{D_{1} }}{2}\left\{ {\left[ {\left( {\frac{m\pi}{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} }\right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left({\frac{q\pi }{b}} \right)^{2} } \right]C_{2x} C_{1y} }\right.} \right. \\ & \left. {\quad \quad - 2\left( {1 - v}\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}} \right)\left({C_{2x} C_{1y} - C_{4x} C_{3y} } \right)} \right\} \\ &\quad\quad + \frac{{D_{2} }}{2}\left\{ {\left[ {\left( {\frac{m\pi}{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} }\right]\left[ {\left( {\frac{p\pi }{a}} \right)^{2} + \left({\frac{q\pi }{b}} \right)^{2} } \right]C_{2x} C_{1y} } \right.\\ & \left. {\left. {\quad \quad - 2\left( {1 - v_{pzt} }\right)\left( {\frac{{mnpq\pi^{4} }}{{a^{2} b^{2} }}} \right)\left({C_{2x} C_{1y} - C_{4x} C_{3y} } \right)} \right\}} \right\}\\ F_{mn} & = - B\left( t \right)\left[ {\left( {d_{31} +v_{pzt} d_{32} } \right)\left( {\frac{m\pi }{a}} \right)^{2} +\left( {d_{32} + v_{pzt} d_{31} } \right)\left( {\frac{n\pi}{b}} \right)^{2} } \right]C_{5x} C_{5y} \\ \end{aligned}$$
where
$$\left[ {\mathbf{M}} \right] = \left[ {\begin{array}{*{20}c} {M_{1111} } & {M_{1112} } & \ldots & {M_{11pq} } \\ {M_{1211} } & {M_{1212} } & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots \\ {M_{mn11} } & \ldots & \ldots & {M_{mnpq} } \\ \end{array} } \right] \quad \left[ {\mathbf{K}} \right]\left[ { = \begin{array}{*{20}c} {K_{1111} } & {K_{1112} } & \ldots & {K_{11pq} } \\ {K_{1211} } & {K_{1212} } & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots \\ {K_{mn11} } & \ldots & \ldots & {K_{mnpq} } \\ \end{array} } \right] \quad \left[ {\mathbf{F}} \right] = \left[ { \begin{array}{*{20}c} {F_{11} } \\ {F_{12} } \\ \ldots \\ {F_{mn} } \\ \end{array}} \right]$$
