## 3 Reduced-order model
### Governing equations of the reduced-order model  
The reduced-order model is obtained by premultiplying both sides of equation (10a) with \(\bar{w}\), integrating the resulting equation over \(\Omega\), and substituting into the resulting equation the approximations given by equations (14b), (15), and (22). We thus obtain  
\[\int_{\Omega}\bar{w}^{2}\ddot{\zeta}\mathrm{d}\Omega+\int_{\Omega} \bar{w}\frac{\partial^{2}}{\partial x_{k}\partial x_{k}}\left(\frac{\partial^{ 2}\bar{w}}{\partial x_{j}\partial x_{j}}\right)\zeta\mathrm{d}\Omega\] \[\quad-12\alpha\int_{\Omega}\bar{w}\frac{\partial}{\partial x_{j} }\left(\bar{\varepsilon}_{jk}\frac{\partial\bar{w}}{\partial x_{k}}+\frac{1}{ 2}\frac{\partial\bar{w}}{\partial x_{k}}\frac{\partial\bar{w}}{\partial x_{k} }\frac{\partial\bar{w}}{\partial x_{j}}\right)\zeta^{3}\mathrm{d}\Omega\] \[\quad+\beta\int_{\Omega}\bar{w}\frac{\partial^{2}\bar{w}}{ \partial x_{k}\partial x_{k}}\zeta\mathrm{d}\Omega+\lambda\int_{\Omega}\frac{ \bar{w}}{(1+\bar{w}\zeta)^{2}}\mathrm{d}\Omega+\mu\int_{\Omega}\frac{\bar{w}} {(1+\bar{w}\zeta)^{3}}\mathrm{d}\Omega=0, \tag{24}\]  
where  
\[\bar{\varepsilon}_{ij}=\sum_{p=1}^{P}\Xi_{(p)}\left(\frac{1-\
u}{2}\left( \frac{\partial\bar{u}_{(p)i}}{\partial x_{j}}+\frac{\partial\bar{u}_{(p)j}}{ \partial x_{i}}\right)+\
u\left(\frac{\partial\bar{u}_{(p)k}}{\partial x_{k} }\right)\delta_{ij}\right). \tag{25}\]  
From equations (23) and (25), we note that the normalization constants \(A_{mn}\) introduced in equation (21) do not affect \(\bar{\varepsilon}_{ij}\). By using the divergence theorem and by imposing the boundary conditions (13a), we obtain the following equation for the reduced-order system  
\[m\ddot{\zeta}+\left(k_{1}+\beta k_{2}\right)\zeta+\alpha k_{3}\zeta^{3}+ \lambda f_{\mathrm{e}}(\zeta)+\mu f_{\mathrm{vdW}}(\zeta)=0, \tag{26}\]  
where  
\[m =\int_{\Omega}\bar{w}^{2}\mathrm{d}\Omega, \tag{27a}\] \[k_{1} =\int_{\Omega}\left(\frac{\partial^{2}\bar{w}}{\partial x_{k} \partial x_{k}}\right)^{2}\mathrm{d}\Omega,\qquad k_{2}=-\int_{\Omega}\frac{ \partial\bar{w}}{\partial x_{k}}\frac{\partial\bar{w}}{\partial x_{k}}\mathrm{ d}\Omega,\] (27b) \[k_{3} =12\int_{\Omega}\left(\bar{\varepsilon}_{jk}\frac{\partial\bar{ w}}{\partial x_{j}}\frac{\partial\bar{w}}{\partial x_{k}}+\frac{1}{2}\left( \frac{\partial\bar{w}}{\partial x_{k}}\frac{\partial\bar{w}}{\partial x_{k} }\right)^{2}\right)\mathrm{d}\Omega,\] (27c) \[f_{\mathrm{e}}(\zeta)=\int_{\Omega}\frac{\bar{w}}{(1+\bar{w} \zeta)^{2}}\mathrm{d}\Omega,\qquad f_{\mathrm{vdW}}(\zeta)=\int_{\Omega}\frac{ \bar{w}}{(1+\bar{w}\zeta)^{3}}\mathrm{d}\Omega. \tag{27d}\]  
Equation (26) describes a nonlinear mass-spring single degree-of-freedom system, where the spring force is the combination of a number of concurring phenomena: linear bending (\(k_{1}\zeta\)), thermal stress (\(k_{2}\zeta\)), nonlinear membrane stretching (\(k_{3}\zeta^{3}\)), the Coulomb force (\(\lambda f_{\mathrm{e}}(\zeta)\)), and the van der Waals force (\(\mu f_{\mathrm{vdW}}(\zeta)\)). From equation (26), we note that an increase in the device temperature, that is, decreasing the parameter \(\beta\), results in mechanical softening effect, whereas a decrease in the device temperature yields a mechanical stiffening effect. As the parameter \(\beta\) decreases mechanical buckling due to thermal stress can be reached.  
In what follows, we discuss two equivalent methods to extract static pull-in parameters. The first method is based on the solution of the static problem, while the second one is based on the study of small vibrations of the system around its static equilibrium configurations. The method based on the vibration analysis can be used to experimentally determine the pull-in voltage without potentially damaging the MEMS [51].