## 3 Reduced-order model
### Extraction of pull-in parameters from the static problem  
At the onset of instability the system's tangent stiffness  
\[K(\beta,\lambda,\mu,\zeta)=k_{1}+\beta k_{2}+3\alpha k_{3}\zeta^{2}+\lambda g_{ \mathrm{e}}(\zeta)+\mu g_{\mathrm{vdW}}(\zeta) \tag{28}\]  
vanishes. The quantities in equation (28) are defined by  
\[g_{\mathrm{e}}(\zeta)=\frac{\mathrm{d}f_{\mathrm{e}}}{\mathrm{d} \zeta}(\zeta) =-\int_{\Omega}\frac{2\bar{w}^{2}}{(1+\bar{w}\zeta)^{3}}\mathrm{d}\Omega, \tag{29a}\] \[g_{\mathrm{vdW}}(\zeta)=\frac{\mathrm{d}f_{\mathrm{vdW}}}{ \mathrm{d}\zeta}(\zeta) =-\int_{\Omega}\frac{3\bar{w}^{2}}{(1+\bar{w}\zeta)^{4}}\mathrm{d}\Omega. \tag{29b}\]  
Therefore, at the pull-in instability, the reduced-order model satisfies equation (26) with \(\ddot{\zeta}=0\) and the additional condition  
\[K(\beta,\lambda,\mu,\zeta)=0. \tag{30}\]  
We solve the problem for \(\lambda=0\) to compute the critical value, say \(\mu_{\mathrm{cr}}\), of the van der Waals force parameter. When \(\mu=\mu_{\mathrm{cr}}\) the system collapses spontaneously without applying any voltage, that is, for \(V=0\). We note that the critical value of the van der Waals force parameter depends on the parameter \(\beta\) that directly affects the mechanical stiffness. By eliminating \(\mu\) from equations (26) and (30) with \(\ddot{\zeta}=0\) and \(\lambda=0\), we obtain the nonlinear equation  
\[\left(\left(k_{1}+\beta k_{2}\right)\zeta_{\mathrm{cr}}+\alpha k_{3}\zeta_{ \mathrm{cr}}^{3}\right)g_{\mathrm{vdW}}(\zeta_{\mathrm{cr}})=\left(k_{1}+\beta k _{2}+3\alpha k_{3}\zeta_{\mathrm{cr}}^{2}\right)f_{\mathrm{vdW}}\left(\zeta_{ \mathrm{cr}}\right), \tag{31}\]  
whose positive first root gives \(\zeta_{\mathrm{cr}}\). The corresponding value of \(\mu_{\mathrm{cr}}\) is given by  
\[\mu_{\mathrm{cr}}=-\frac{\left(k_{1}+\beta k_{2}\right)\zeta_{\mathrm{cr}}+ \alpha k_{3}\zeta_{\mathrm{cr}}^{3}}{f_{\mathrm{vdW}}\left(\zeta_{\mathrm{cr} }\right)}=-\frac{k_{1}+\beta k_{2}+3\alpha k_{3}\zeta_{\mathrm{cr}}^{2}}{g_{ \mathrm{vdW}}\left(\zeta_{\mathrm{cr}}\right)}. \tag{32}\]  
The effect of the van der Waals force on pull-in parameters \(\lambda_{\mathrm{PI}}\) and \(\|w_{\mathrm{PI}}\|_{\infty}\) is investigated by solving equations (26) and (30) with \(\ddot{\zeta}=0\) and variable \(\lambda\) for different values of \(\mu\) in the range \([0,\mu_{\mathrm{cr}}]\). Here, \(\|\cdot\|_{\infty}\) is defined as \(\max_{(x_{1},x_{2})\in\Omega}|\cdot|\). By solving equation (30) for \(\lambda\) and by substituting in equation (26), we obtain the following nonlinear equation in \(\zeta_{\mathrm{PI}}\):  
\[\left(\left(k_{1}+\beta k_{2}\right)\zeta_{\mathrm{PI}}+\alpha k_{3}\zeta_{ \mathrm{PI}}^{3}+\mu f_{\mathrm{vdW}}\left(\zeta_{\mathrm{PI}}\right)\right)g _{\mathrm{vdW}}\left(\zeta_{\mathrm{PI}}\right)=\left(k_{1}+\beta k_{2}+3 \alpha k_{3}\zeta_{\mathrm{PI}}^{2}+\mu g_{\mathrm{vdW}}\left(\zeta_{\mathrm{ PI}}\right)\right)f_{\mathrm{vdW}}\left(\zeta_{\mathrm{PI}}\right). \tag{33}\]  
The lowest positive root of equation (33) gives the non-dimensional pull-in deflection \(\zeta_{\mathrm{PI}}\). The corresponding non-dimensional pull-in voltage is thereby determined from equation (26) with \(\ddot{\zeta}=0\) as  
\[\lambda_{\mathrm{PI}}=-\frac{\left(k_{1}+\beta k_{2}\right)\zeta_{\mathrm{PI}} +\alpha k_{3}\zeta_{\mathrm{PI}}^{3}+\mu f_{\mathrm{vdW}}\left(\zeta_{\mathrm{ PI}}\right)}{f_{\mathrm{e}}\left(\zeta_{\mathrm{PI}}\right)}=-\frac{k_{1}+\beta k _{2}+3\alpha k_{3}\zeta_{\mathrm{PI}}^{2}+\mu g_{\mathrm{vdW}}\left(\zeta_{ \mathrm{PI}}\right)}{g_{\mathrm{e}}\left(\zeta_{\mathrm{PI}}\right)}. \tag{34}\]