# Vortex-Induced Micro-Cantilever Vibrations with Small and Large Amplitudes in Rarefied Gas Flow
## 3 The Mathematical Model of the Elastic Obstacle
The beam was discretized by one hundred 20-node brick SOLID186 finite elements and 963 nodes. As one can see, the differences between the frequencies obtained by the three different approaches are much less than 1%. Thus, we decided to continue with the geometrically nonlinear version of the Euler-Bernoulli mode. The last check was to compare the time history diagram obtained by the 3D finite element model and the reduced three degrees of freedom model. The beams modelled by both models were loaded with step loading equal to 12,000 Pa, which is approximately the value of the loading caused by fluid flow on the beam. The comparison can be seen in Figure 3.  
As can be seen, the agreement between the results is excellent, and the differences are negligible. Similar results were obtained using the Timoshenko beam theory instead of the Euler-Bernoulli beam theory. Due to all these reasons and the fact that the equations of motion for the Euler-Bernoulli beam are a little simpler than the ones for the Timoshenko beam, we decided to continue the computations using the geometrically nonlinear version of the Euler-Bernoulli beam theory. The equations of the beam motion of the beam, according to this theory, are [5; 25]  
\[EI\frac{\partial^{4}w}{\partial x^{4}}+c_{d}\frac{\partial w}{\partial t}+ \rho A\frac{\partial^{2}w}{\partial t^{2}}=p(y,t)+G_{l}(y,t) \tag{5}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{\Re}\) & **Euler–Bernoulli** & **[PERSON]** & **ANSYS** \\ \hline
1 & 21.452 MHz & 21.225 MHz & 21.48 MHz \\
2 & 134.44 MHz & 134.15 MHz & 134.41 MHz \\
3 & 376.42 MHz & 374.5 MHz & 375.4 MHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Natural frequencies of the beam, MHz. Here, the longitudinal inertia is neglected, and by \(G_{L}\), the nonlinear term due to large deflection is denoted as  
\[G_{l}(y,t)=\frac{EI}{L}\left(\frac{6}{h^{2}}\int\limits_{0}^{L}\left(\frac{ \partial w}{\partial y}\right)^{2}dy\right)\frac{\partial^{2}w}{\partial y^{2}} \tag{6}\]  
and \(c_{\rm d}\) is a term connected with internal damping. In (5) and (6), \(I\) is the geometrical moment of inertia of the beam cross-section. In the case of a rectangular cross-section of the beam, it is \(I=blt^{3}/12\)  
Figure 3: Response of the tip of the beam subjected to a step load \(p\) = 12,000 Pa according to the reduced model (large deflection beam theory) and ANSYS. Figure 2: Finite element discretization of the beam. The boundary conditions for the elastic obstacle are for a cantilever (clamped at one end and free at the tip) [22]  
\[ \begin{split}\text{For }y&=0:w(0,t)=0,\ \frac{ \partial w(0,t)}{\partial y}=0,\\ \text{For }y&=L\ :\frac{\partial^{2}w(0,t)}{ \partial y^{2}}=0,\ \frac{\partial^{2}w(0,t)}{\partial y^{3}}=0\end{split}  \tag{7}\]  
Dimensionless variables:  
As far as the considered case is concerned, the micro-objects and the dimensions of the channel and the beam are very small; therefore, it is necessary to use dimensionless variables in the numerical computations. The following dimensionless variables are introduced  
\[\overline{y}=y/L,\ \overline{w}=w/L,\ \overline{t}=tc/L,\ c=\sqrt{E/\rho} \tag{8}\]  
Then, Equations (5) and (6) are transformed into:  
\[ \begin{split}&\frac{\partial^{4}\overline{w}}{\partial\overline{ y}^{4}}+c_{v}\frac{\partial\overline{w}}{\partial\overline{t}}+\alpha^{-1} \frac{\partial^{2}\overline{w}}{\partial\overline{t}^{2}}=p_{1}+\overline{ G}_{l}(y,t),\\ & p_{1}=\frac{p(\overline{y},t)L^{3}}{EI},\ \overline{G}_{L}=\frac{1}{2 \alpha}\left(\int\limits_{0}^{1}\left(\frac{\partial\overline{w}}{\partial \overline{\xi}}\right)^{2}d\xi\right)\frac{\partial^{2}\overline{w}}{\partial \overline{y}^{2}},\ c_{v}=\frac{c_{d}}{EI}\end{split}  \tag{9}\]  
In the following equations, for convenience, the bars over the dimensionless variables will be omitted. Equation (9) has to be coupled with the gas flow solution. For this purpose, a reduced model of the beam was developed here. According to the Galerkin procedure, the solution is sought by series Equation (3).