# Vortex-Induced Micro-Cantilever Vibrations with Small and Large Amplitudes in Rarefied Gas Flow
## 3 The Mathematical Model of the Elastic Obstacle  
The elastic element inserted in the channel was modelled as an elastic cantilever beam. The geometrically nonlinear version of the Euler-Bernoulli beam theory and the Timoshenko beam theory were checked and compared to analyse the beam vibration. The equations of motion of the beam, according to the Euler beam theory, which considers large displacements, are [25]:  
\[ \begin{array}{l}\frac{\partial N}{\partial y}+c_{\mathrm{d}}\frac{\partial u} {\partial t}+\rho A\frac{\partial^{2}u}{\partial t^{2}}=0\\ \frac{\partial^{2}M}{\partial y^{2}}+c_{\mathrm{d}}\frac{\partial w}{\partial t }+N\frac{\partial^{2}w}{\partial y^{2}}+\rho A\frac{\partial^{2}w}{\partial t ^{2}}=p(y,t)\hskip 28.452756 pt0<y<L\end{array}  \tag{1}\]  
and the equations  
\[ \begin{array}{l}\frac{\partial N}{\partial y}-c_{\mathrm{d}}\frac{\partial u }{\partial t}-\rho\,F\frac{\partial^{2}u}{\partial t^{2}}=0,\hskip 56.905512 pt-h/2 \;\leq\;z\;\leq\;h/2\\ -\frac{\partial M}{\partial y}+Q-c_{\mathrm{d}}\frac{\partial t}{\partial t}- \rho I\frac{\partial^{2}u}{\partial t^{2}}=0\hskip 56.905512 pt0\;<\;y\;<\;L,\;t\;> \;0\\ \frac{\partial Q}{\partial y}+N\left(\frac{\partial^{2}w}{\partial y^{2}} \right)+\frac{\partial N}{\partial y}\frac{\partial w}{\partial y}-c_{ \mathrm{d}}\frac{\partial w}{\partial y}-\rho A\frac{\partial^{2}w}{\partial t ^{2}}=-p(y,t)\end{array}  \tag{2}\]  
according to the Timoshenko beam theory. In (1) and (2), \(N\), \(M\) and \(Q\) are the generalized stresses  
\[N=b\int\limits_{-h/2}^{h/2}\sigma_{y}dz,\;M=b\int\limits_{-h/2}^{h/2}\sigma_{ y}dz,\;Q=kb\int\limits_{-h/2}^{h/2}\tau_{yz}dz\]  
\(u\) is the displacement along \(y\)-axis (longitudinal displacement), \(w\) is the displacement along \(x\)-axis (transverse beam displacement), \(p(y,t)\) is the transverse loading of the beam, \(A=bh\) is the area of the beam cross-section, and \(b\) is the width of the beam, \(t\) is the time and by c\({}_{\mathrm{1}}\), c\({}_{\mathrm{2}}\) and c\({}_{\mathrm{d}}\) the material damping coefficient are denoted. The shear force \(Q\) is considered only in the case of Timoshenko beam theory, introducing the so-called shear correction factor \(k\). A reduced model of the beam was used to describe the beam motion. By using the [PERSON] procedure, the solutions of the equations of motion were sought by series  
\[w(y,t)=\sum\limits_{n=1}^{N_{f}}w_{n}(y)q_{n}(t) \tag{3}\]  
Figure 1: Geometry of flow past elastic cantilever beam placed normal to free stream. for the Euler-Bernoulli beam and  
\[ \begin{split} w(y,t)&=\sum\limits_{i=1}^{N_{f}}W_{i}(y )q_{i}(t),\\ \psi(y,t)&=\sum\limits_{i=1}^{N_{f}}\Psi_{i}(y)q_{i}(t) \end{split}  \tag{4}\]  
for Timoshenko beam, where \(w_{\text{n}}(y)\) are space functions representing vibration modes according to the Euler-Bernoulli beam theory, \(W_{\text{n}}\) are the normal forms of vibrations according to the Timoshenko beam theory and \(\Psi_{i}(y)\) are angular rotations according to the Timoshenko beam theory. These functions satisfy the geometrical boundary conditions, \(q_{n}(t)\) are time functions, and \(N_{f}\) is the number of modes in the expansion. The Timoshenko beam theory usually gives closer results to the ones obtained by 3D beam theory than the Euler-Bernoulli beam theory, especially in the case of thicker beams and when high frequencies are involved in the response. The case considered by us is with fixed beam length and parameter of the rarefied gas flow. For the selected geometrical parameters of the beam, we calculated the natural frequencies and some forced responses of the beam obtained by both theories. The verification of the cantilever's models was performed for a beam with the same material and geometrical properties, which are used in all computations (see section Statement of the problem), with thickness \(h=L/20\). The results are written in Table 1.  
For the same beam, a 3D finite element model was created with ANSYS 2022R2 software (see Figure 2).