# Vortex-Induced Micro-Cantilever Vibrations with Small and Large Amplitudes in Rarefied Gas Flow
## 5 Results and Discussion  
We examined the vibration of an elastic beam modelled by the linear and nonlinear model in its resonance zone. The resonance arises due to the frequency of VIV, caused by the Karman vortex street in the rarefied gas, being close to the beam's first natural frequency. The selection of rarefied gas flow regimes and cantilever material is essential. While numerical simulations give very accurate results, parameter analysis allows the possibility of finding relations between the parameters of two interacting media: a rarefied gas flow and an elastic beam. We follow the dimensional analysis in [6], where we selected to vary cantilever thickness (\(h\)). Here is the briefly presented analysis. Oscillating flows are described by the Strouhal number, which is defined as:  
\[St^{g}=\frac{f_{\mathcal{C}_{L}}^{g}L}{u_{intel}^{g}} \tag{23}\]  
where \(f_{\mathcal{C}_{L}}^{g}\) is the vortex detachment frequency. \(f_{\mathcal{C}_{L}}^{g}\) was determined with continuous 1D wavelet transform (CWT) with the bump wavelet of the lift force (force along the \(y\)-axis) of the cantilever at the established unsteady regime. We focused on Mach and Knudsen numbers as the primary governing parameters because the Strouhal number can be expressed as a function of both. In the micro-gas flow regime, the Knudsen number is especially important; note that Mach, Reynolds, and Knudsen numbers are related (see Equation (21)). A detailed, dimensionless analysis of gas parameters is presented in [6], and a linear model was used to represent the beam, enabling us to derive a straightforward expression for the cantilever's resonance zone. The derived expression (24) made it possible to identify which gas and beam parameters influence the cantilever's resonance. To investigate the resonance between the flow and the beam, we equated the VIV frequency to the beam's first natural frequency. By fixing the gas properties (Mach number, Knudsen number, surface temperature, and the type of gas), the material properties, and the length of the beam, the only parameter that one can vary is the dimensionless beam thickness (\(c_{h}=h/L\)):  
\[St^{g}\,u_{inlet}^{g}=\frac{1.01498}{2\pi}\ c_{h}\sqrt{\frac{E}{\rho}} \tag{24}\]  
where the left-hand side represents the flow-induced frequency (a function of Mach and Knudsen numbers), and the right-hand side depends on the beam's elasticity (Young's modulus \(E\)) and density \(\rho\). In order to investigate the oscillations of the beam in the resonance zone, 14 cases with varying beam thicknesses were studied. Each case individually requires substantial computational resources, and solving this series of problems demands significant computational effort and efficiency. All simulations were performed using our DSMC code on the supercomputer HEMUS at the Centre of Excellence in Informatics and ICT, financed by the OP SESG and co-financed by the EU through the ESIF. At time t = 0, the flow with an initial velocity equal to the free-stream velocity is initiated. As it interacts with the elastic beam, the beam undergoes bending. Figure 5 presents the time history of the load on the elastic micro-beam at a node near the tip from the start of the simulation. During the interval from 0 to 6 us, the system exhibits a transitional regime. Subsequently, vortex shedding begins to develop. According to Figure 6, which shows the displacement of the beam tip, the established unsteady regime is reached after approximately 15 us. Periodic fluctuations in the load are observed because of the vortex shedding detachment period, i.e., the VIV period. The resulting unsteady loading and vortex dynamics induce sustained vibrations in the beam. The filtered load clearly shows the period of VIV (see Figure 5). The data were filtered using a one-dimensional median filter with a window size of 500 points. Figure 5: Load at a node near the tip in time; beam thickness is \(c_{h}=0.0170\), nonlinear elastic beam model.