## 3 Free Vibration and Buckling Analysis
### Galerkin Solution  
In order to analyze free vibration of 3D-GrF microshells with other boundary conditions, a Galerkin-based solution technique is utilized. Therein, the spatial displacement field of the microshell can be expressed as [58]  
\[u(x,\theta,t)=U_{mn}\frac{\partial\phi(x)}{\partial x}\cos(n\theta)\mathrm{e}^{ \mathrm{i}\omega t} \tag{61}\]  
\[v(x,\theta,t)=V_{mn}\phi(x)\sin(n\theta)\mathrm{e}^{\mathrm{i}\omega t} \tag{62}\]  
\[w(x,\theta,t)=W_{mn}\phi(x)\cos(n\theta)\mathrm{e}^{\mathrm{i}\omega t}. \tag{63}\]  
Thereinto, the axial modal function \(\phi(x)\) is:  
\[\phi(x)=c_{1}\cosh\left(\frac{\lambda_{i}x}{L}\right)+c_{2}\cos\left(\frac{ \lambda_{i}x}{L}\right)-\zeta_{i}\bigg{[}c_{3}\mathrm{sinh}\left(\frac{ \lambda_{i}x}{L}\right)+c_{4}\sin\left(\frac{\lambda_{i}x}{L}\right)\bigg{]}. \tag{64}\]  
The simply supported boundary condition is expressed as  
\[\frac{\partial^{2}\phi(x)}{\partial x^{2}}=\phi(x)=0\qquad(x=0,L). \tag{65}\]  
The clamped boundary condition is expressed as  
\[\frac{\partial\phi(x)}{\partial x}=\phi(x)=0\qquad(x=0,L). \tag{66}\]  
The free boundary condition is  
\[\frac{\partial^{2}\phi(x)}{\partial x^{2}}=\frac{\partial^{3}\phi(x)}{ \partial x^{3}}=0\qquad(x=0,L). \tag{67}\]  
Herein, free-simply supported (F-SS), clamped-free (C-F), clamped-simply supported (C-SS), free-free (F-F), and clamped-clamped (C-C) boundary conditions are taken into account. Based on Equations (65)-(67), the constants \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\), \(\lambda_{i}\), and \(\zeta_{i}\) (\(i=1,2,3,4\)... ) are given in Table 2.  
Substituting Equations (61)-(63) into Equations (42)-(44) and neglecting axial load \(N_{xx}^{0}\), then applying the [PERSON] method [59; 60; 61; 62], yields  
\[\left(\mathbf{K}_{1}-\omega^{2}\mathbf{M}_{1}\right)\left\{\begin{array}{l}U _{mn}\\ V_{mn}\\ W_{mn}\end{array}\right\}=\left\{\begin{array}{l}0\\ 0\\ 0\end{array}\right\} \tag{68}\]  
where \(\mathbf{M}_{1}\) and \(\mathbf{K}_{1}\) represent the mass matrix and stiffness matrix, respectively. By solving the eigenvalue problem, the natural frequencies and eigenvectors are obtained.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Boundary Condition** & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) & \(\mathbf{c_{3}}\) & \(\mathbf{c_{4}}\) & \(\mathbf{\zeta_{i}}\) & \(\mathbf{\lambda_{i}}\) \\ \hline C–C & \(1\) & \(-1\) & \(1\) & \(-1\) & \(\frac{\cosh(\lambda_{i})-\cos(\lambda_{i})}{\sinh(\lambda_{i})-\sin(\lambda_{i})}\) & \(\cosh(\lambda_{i})\cdot\cos(\lambda_{i})=1\) \\ C–SS & \(1\) & \(-1\) & \(1\) & \(-1\) & \(\frac{\cosh(\lambda_{i})-\cos(\lambda_{i})}{\sinh(\lambda_{i})-\sin(\lambda_{i})}\) & \(\tan(\lambda_{i})=\tanh(\lambda_{i})\) \\ F–F & \(1\) & \(1\) & \(1\) & \(1\) & \(\frac{\cosh(\lambda_{i})-\cos(\lambda_{i})}{\sinh(\lambda_{i})-\sin(\lambda_{i})}\) & \(\cosh(\lambda_{i})\cdot\cos(\lambda_{i})=1\) \\ C–F & \(1\) & \(-1\) & \(1\) & \(-1\) & \(\frac{\sinh(\lambda_{i})-\sin(\lambda_{i})}{\cosh(\lambda_{i})-\cos(\lambda_{i} )}\) & \(\cosh(\lambda_{i})\cdot\cos(\lambda_{i})=-1\) \\ F–SS & \(1\) & \(1\) & \(1\) & \(1\) & \(\frac{\cosh(\lambda_{i})-\cos(\lambda_{i})}{\sinh(\lambda_{i})-\sin(\lambda_{i} )}\) & \(\tan(\lambda_{i})=\tanh(\lambda_{i})\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values of \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\), \(\zeta_{i}\), and \(\lambda_{i}\) for different boundary conditions.