# Vortex-Induced Micro-Cantilever Vibrations with Small and Large Amplitudes in Rarefied Gas Flow
## 3 The Mathematical Model of the Elastic Obstacle
The natural modes of vibrations, \(w_{n}\) which satisfy the boundary conditions, have the expression  
\[w_{n}(y)=A_{n}[\sin\beta_{n}y-\sinh\beta_{n}y-B_{n}(\cos\beta_{n}y+\cosh\beta_{ n}y)] \tag{10}\]  
where the following notations are used  
\[\beta_{n}^{4}=\omega_{n}^{2}\alpha,\ \ \ \ \ \alpha=12\left(\frac{L}{\hbar} \right)^{2}\ \ \ \ B_{n}=\frac{\sin\beta_{n}+\sinh\beta_{n}}{\cos\beta_{n}+\cosh\beta_{n}} \tag{11}\]  
and \(\omega_{n}\) are the dimensionless natural frequencies of the linear Euler-Bernoulli beam obtained from the frequency equation  
\[\mathbf{cos}\beta\mathbf{cosh}\beta=-1 \tag{12}\]  
Substituting Equation (3) into a partial differential Equation (5), multiplying it by the modal functions, integrating through the beam length, invoking the orthogonality condition  
\[\int\limits_{0}^{1}(w_{n}w_{m})dy= \begin{cases}1\ \ \text{for}\ \ \ n=m\\ 0\ \ \text{for}\ \ \ n\
eq m\end{cases}  \tag{13}\]  
and assuming the proportional damping, the equation is transformed into a set of coupled ordinary differential equations of motion  
\[\ddot{q}_{n}(t)+2\dot{\xi}_{n}\omega_{n}\dot{q}_{n}+\omega_{n}^{2}\dot{q}_{n}( t)=F_{n}(t), \tag{14}\]where \(\xi_{n}\) are modal damping parameters and  
\[F_{n}(t)=\int\limits_{0}^{1}w_{n}(y)\left[P(y,t)+G_{l}(y,t)\right]dy \tag{15}\]  
In the present work, we chose the three-mode reduction (\(N_{f}=3\)). The preliminary studies of the vibration behaviour of the beam subjected to step loading showed that the higher modes do not influence the response. On the other hand, using three-mode reduction leads to manageable calculations to obtain the system of ordinary differential equations, which could be solved quickly and easily by the available software. Using the three-modes reduction, the geometrically nonlinear term has the form  
\[ \begin{split} G^{L}(x,t)=-\frac{1}{2}\big{(}I_{11}q_{1}^{2}+I_{2 2}q_{2}^{2}+I_{33}q_{3}^{2}+2I_{12}q_{1}q_{2}+2I_{23}q_{2}q_{3}+2I_{13}q_{1}q_{ 3}\big{)}.\\ (w^{\prime\prime}{}_{1}q_{1}+w^{\prime\prime}{}_{2}q_{2}+w^{\prime \prime}{}_{3}q_{3}),\end{split}  \tag{16}\]  
where  
\[I_{ij}=\int\limits_{0}^{1}w^{\prime};w^{\prime}jdy \tag{17}\]  
So, the mathematical model based on the three modes reduction takes the form:  
\[ \begin{split}\ddot{q}_{n}(t)+2\xi_{n}\omega_{n}\dot{q}_{n}+ \omega_{n}^{2}q_{n}(t)+D_{111}^{n}q_{1}^{3}+D_{222}^{n}q_{2}^{3}+\\ D_{333}^{n}q_{3}^{3}+D_{112}^{n}q_{1}^{2}q_{2}+D_{113}^{n}q_{1}^{2 }q_{3}+D_{221}^{n}q_{1}^{2}q_{2}+\\ D_{223}^{n}q_{2}^{2}q_{3}+D_{331}^{n}q_{1}q_{3}^{2}+D_{332}^{n}q_{2}q_{ 3}^{2}+D_{123}^{n}q_{1}q_{2}q_{3}=F_{n}^{p}(t),\quad n=1,2,3\end{split}  \tag{18}\]  
Here  
\[D_{ijk}^{n}=0.5\int\limits_{0}^{1}w_{n}C_{ijk}dy\quad i,j,k=1,2,3 \tag{19}\]  
and  
\[ \begin{split} C_{111}=I_{11}w^{\prime\prime}{}_{1}\,\ C_{222}=I_{22}w^{\prime\prime}{}_{2}\,\ C_{333}=I_{33}w^{\prime\prime}{}_{3}\\ C_{112}=I_{11}w^{\prime\prime}{}_{2}\ +2I_{12}w^{\prime\prime}{}_{1} \ C_{113}=I_{11}w^{\prime\prime}{}_{3}\ +2I_{13}w^{\prime\prime}{}_{1}\\ C_{221}=I_{22}w^{\prime\prime}{}_{1}\ +2I_{12}w^{\prime\prime}{}_{2} \,\ C_{223}=I_{22}w^{\prime\prime}{}_{3}\ +2I_{23}w^{\prime\prime}{}_{2}\\ C_{331}=I_{33}w^{\prime\prime}{}_{1}\ +2I_{13}w^{\prime\prime}{}_{3}\,\ C_{332}=I_{33}w^{\prime\prime}{}_{2}\ +2I_{23}w^{\prime\prime}{}_{3}\end{split}  \tag{20}\]  
The nonlinear system of Equation (18) was solved using [PERSON]'s method [26] from the IMSL library. [PERSON]'s numerical method is generated by validated for solving complicated nonlinear differential equations. This implicit method is appropriate for stiff problems that require long integration times. The integrals in Equations (15), (17), and (19) are computed using [PERSON]'s rule.