# Natural Frequencies and Modes of Electrostatically Actuated Curved Bell-Shaped Microplates
## 2 Materials and Methods
### Reduced-Order Model
#### 2.1.2 Free Vibrations  
For the spectral response, the representation of Equations (8) and (9) with the base functions given by Equation (10) and the initial shape given by Equation (12) were substituted into Equations (6) and (7). The [PERSON] procedure results in the following linear algebraic eigenvalue problem in terms of the generalized vibrational amplitudes \(q_{k},p_{k}\)  
\[\alpha^{2}(1-\
u)\sum_{i=1}^{N}q_{i}\int_{0}^{1}\[PERSON]{(}\sum_{j =1}^{N}q_{j}^{s}\frac{d\[PERSON]_{j}}{dr}\frac{d\[PERSON]_{i}}{dr}\[PERSON]{)}r^{2}H_{k}rdr +\alpha^{2}\sum_{i=1}^{N}q_{i}\int_{0}^{1}\[PERSON]{(}\sum_{j=1}^{N}q_{j}^{s} \frac{d\[PERSON]_{j}}{dr}\frac{d^{2}\[PERSON]_{i}}{dr^{2}}\[PERSON]{)}r^{3}H_{k}rdr\\ +\alpha^{2}\sum_{i=1}^{N}q_{i}\int_{0}^{1}\[PERSON]{(}\sum_{j=1}^{N} q_{j}^{s}\frac{d^{2}\[PERSON]_{j}}{dr^{2}}\frac{d\[PERSON]_{i}}{dr}\[PERSON]{)}r^{2}H_{k}rdr+ \sum_{i=1}^{N}p_{i}\int_{0}^{1}\biggl{(}\frac{d^{2}H_{i}}{dr^{2}}r^{3}-H_{i}r+ \frac{dH_{i}}{dr}r^{2}\biggr{)}H_{k}rdr=0 \tag{16}\]\[-\omega^{2}q_{k}\int_{0}^{1} \[PERSON]_{k}\[PERSON]_{k}rdr\] \[+\sum_{i=1}^{N}q_{i}\int_{0}^{1}\left(\frac{d^{4}(\[PERSON]_{i})}{dr^{4}}+ \frac{2}{r}\frac{d^{3}(\[PERSON]_{i})}{dr^{3}}-\frac{1}{r^{2}}\frac{d^{2}(\[PERSON]_{i})} {dr^{2}}+\frac{1}{r^{3}}\frac{d(\[PERSON]_{i})}{dr}\right)\[PERSON]_{k}rdr\] \[-\gamma\alpha^{2}\sum_{i=1}^{N}q_{i}\int_{0}^{1}\frac{1}{r}\frac{ d}{dr}\left(r\left(\sum_{i=1}^{N}q_{j}^{5}\frac{d\[PERSON]_{j}}{dr}\right)^{2} \frac{d\[PERSON]_{i}}{dr}\right)\[PERSON]_{k}rdr\] \[-\gamma\sum_{i=1}^{N}p_{i}\int_{0}^{1}\frac{1}{r}\frac{d}{dr} \left(\sum_{j=1}^{N}q_{j}^{5}\frac{rdH_{i}}{dr}\frac{d\[PERSON]_{j}}{dr}+vH_{i} \sum_{j=1}^{N}q_{j}^{5}\frac{d\[PERSON]_{j}}{dr}\right)\[PERSON]_{k}rdr\] \[-\gamma\frac{\alpha^{2}}{2}\sum_{i=1}^{N}q_{i}\int_{0}^{1}\frac{ 1}{r}\frac{d}{dr}\left(r\left(\sum_{j=1}^{N}q_{j}^{5}\frac{d\[PERSON]_{j}}{dr} \right)^{2}\frac{d\[PERSON]_{i}}{dr}\right)\[PERSON]_{k}rdr\] \[+\gamma\frac{\alpha^{2}}{2}\frac{2}{\beta_{0}^{2}}\sum_{i=1}^{N}q _{i}\int_{0}^{1}\frac{1}{r}\frac{d}{dr}\left(r\left(\frac{d\[PERSON]_{1}}{r}\right) ^{2}\frac{d\[PERSON]_{i}}{dr}\right)\[PERSON]_{k}rdr\] \[-\gamma\sum_{i=1}^{N}q_{i}\int_{0}^{1}\frac{1}{r}\frac{d}{dr} \left(r\left(\sum_{j=1}^{N}p_{j}^{5}\frac{dH_{j}}{dr}+v\frac{H_{j}}{r}\right) \frac{d\[PERSON]_{i}}{dr}\right)\[PERSON]_{k}rdr\] \[-\sum_{i=1}^{N}q_{i}\beta\int_{0}^{1}\frac{2\[PERSON]_{i}}{\left(1+ \sum_{j=1}^{N}q_{j}^{5}\[PERSON]_{j}\right)^{3}}\[PERSON]_{k}rdr=0 \tag{17}\]  
Equations (16) and (17) allow finding the approximation the frequencies and the modes of the free undamped vibrations of the plate around the stable equilibria. Note that while the eigenvalue problem itself, Equations (16) and (17), is linear, the associated static problem, which should be solved to find the equilibrium configuration, is nonlinear. Namely, the eigenvalues and eigenvectors of Equations (16) and (17) depend on the (parameterized by the voltage parameter \(\beta\)) equilibrium values of the generalized coordinates \(q_{k}^{s},\ p_{k}^{s}\), which are found numerically using Maple software, as the solutions of the system of nonlinear algebraic Equations (13) and (14). To track unstable branches of the equilibrium curves, arc-length continuation (the Ricks method) was implemented, as detailed in study [5]. The obtained static deflections \(q_{k}^{s},\ p_{k}^{s}\) are then used to calculate the coefficients (the stiffness and mass matrices) of the dynamic equations Equations (16) and (17). The eigenvalues and the eigenvectors of Equations (16) and (17) are then found numerically using an eigenvalue problem solver in Matlab.