## 3 Governing Equations of the Problem  
The governing equations of the problem are determined by applying the Hamiltonian principle within the time interval \([t_{0},t_{1}]\) as follows:  
\[\int_{t_{0}}^{t_{1}}[\delta U-\delta K-\delta W]\ =\ 0, \tag{26}\]  
where \(\delta U\), \(\delta W\) and \(\delta K\) are the variations in the strain energy, external work and kinetic energy, respectively. More specifically, the strain energy of the piezoelectric plate is defined as follows:  
\[ \begin{array}{rl}U=&\frac{1}{2}\bigg{(}\int_{r_{i}}^{r_{o}}\int_{0}^{2\pi} \int_{\frac{h_{i}}{2}}^{\frac{h_{i}}{2}}\left(\sigma_{r}^{t}\varepsilon_{r}^{ t}+\sigma_{\theta}^{t}\varepsilon_{\theta}^{t}+2\tau_{rz}^{t}\varepsilon_{rz} ^{t}-D_{r}^{t}\varepsilon_{r}^{t}-D_{z}^{t}E_{z}^{t}-B_{r}^{t}H_{r}^{t}-B_{z}^ {t}H_{z}^{t}\right)rd\theta drd\xi^{t}\\ &+\int_{r_{i}}^{r_{o}}\int_{0}^{2\pi}\int_{\frac{h_{i}}{2}}^{\frac{h_{i}}{2}} \left(\sigma_{r}^{t}\varepsilon_{r}^{t}+\sigma_{\theta}^{t}\varepsilon_{ \theta}^{t}+2\tau_{rz}^{t}\varepsilon_{rz}^{t}\right)rd\theta drd\xi^{c}\\ &\int_{r_{i}}^{r_{o}}\int_{0}^{2\pi}\int_{\frac{h_{i}}{2}}^{\frac{h_{i}}{2}} \left(\sigma_{r}^{t}\varepsilon_{r}^{t}+\sigma_{\theta}^{t}\varepsilon_{ \theta}^{t}+2\tau_{rz}^{t}\varepsilon_{rz}^{t}-D_{r}^{t}E_{z}^{t}-B_{z}^{t}H_ {r}^{b}-B_{z}^{b}H_{r}^{b}-B_{z}^{b}H_{z}^{b}\right)rd\theta drd\xi^{b}\bigg{)} \end{array}  \tag{27}\]  
The total kinetic energy of the annular sandwich plate is expressed as follows:  
\[K\ =\ \frac{1}{2}\bigg{(}\int_{r_{i}}^{r_{o}}\int_{0}^{2\pi}\int_{-\frac{h}{2}} ^{\frac{h}{2}}p^{i}\bigg{\{}\left(\left(\dot{U}^{i}\right)^{2}+\left(\dot{W}^ {i}\right)^{2}\right)\bigg{\}}rdrd\theta d\xi^{i} \tag{28}\]  
where \(h\) is the total thickness of structure. Moreover, the work conducted by the external forces is defined as follows:  
\[ \begin{array}{rl}W\ =\ \frac{1}{2}\int_{r_{i}}^{r_{o}}\int_{0}^{2\pi} \bigg{(}(N_{t}+N_{e}+N_{m})\bigg{(}\frac{\partial^{2}w}{\partial\tau^{2}}+ \frac{1}{r}\frac{\partial w}{\partial r}\bigg{)}+k_{w}w-k_{g}\bigg{(}\frac{ \partial^{2}w}{\partial\tau^{2}}+\frac{1}{r}\frac{\partial w}{\partial\tau} \bigg{)}\bigg{)}rdrd\theta,\end{array}  \tag{29}\]  
where \(k_{w}\) and \(k_{g}\) define the Winkler and Pasternak stiffnesses, respectively; \(N_{t}\), \(N_{c}\), \(N_{e}\) and \(N_{m}\) refer to the normal forces induced by the temperature variation, \(\Delta T\), moisture change, \(\Delta C\), and external electric and magnetic potentials, which are, in turn, defined as follows:  
\[ \begin{array}{rl}N_{e}\ =\ 2\varepsilon_{31}V_{0},\ N_{t}\ =\ \lambda h_{i}\Delta T,\ N_{c}\ =\ \varsigma h_{i}\Delta C\ \ N_{m}\ =\ 2d_{31}\lambda_{0}.\end{array}  \tag{30}\]  
By substituting Equations (27)-(29) into Equation (26), we obtain the following governing equations of motion:  
\[ \begin{array}{rl}\frac{N_{r}^{b}-N_{r}^{b}}{r}+\frac{\partial N_{r}^{b}}{ \partial r}+\frac{N_{r}^{c}-N_{r}^{c}}{r}+\frac{\partial N_{r}^{c}}{r}+\frac {N_{r}^{i}-N_{r}^{d}}{r}+\frac{\partial N_{r}^{i}}{\partial r}\ =\ \Big{(}l_{0}^{b}+l_{0}^{b}+l_{0}^{t} \Big{)}\ddot{u}_{0}\\ +\Big{(}l_{0}^{t}\frac{h_{i}}{2}+l_{1}^{t}\Big{)}\ddot{\theta}_{r}^{t}+l_{0}^{b }\ddot{\theta}_{r}^{c}+\Big{(}l_{1}^{b}-l_{0}^{b}\frac{h_{i}}{2}\Big{)}\ddot{ \theta}_{r}^{b},\end{array}  \tag{31}\]\[