# Vortex-Induced Micro-Cantilever Vibrations with Small and Large Amplitudes in Rarefied Gas Flow
## 6 Conclusions  
In this work, theoretical and numerical modelling of the interaction between micro-cantilevers and rarefied gas flows was presented. The theoretical framework includes reduced-order modelling of the geometrically nonlinear vibrations of the beam. The rarefied gas flow is modelled using the Direct Simulation Monte Carlo (DSMC) method. A hybrid numerical approach ensures a full coupling between the mechanical and fluid dynamics, enabling comprehensive fluid-structure interaction (FSI) simulation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\omega_{1}\)/\(\omega_{vortex}\)} & **Cantilever** & **Amplitude of** & **Amplitude of** & \\  & **Thickness,** & **Displacement,** & **Displacement,** & **Displacement,** & \(\Delta\)_uv_/_w\({}_{LDBT}\)_ \\  & \(c_{h}\) = _h/L_ & **SDBT (Linear), m** & **LDBT (Nonlinear), m** & \\ \hline
0.74728 & 0.015 & 1.60 \(\times\) 10\({}^{-8}\) & 2.73 \(\times\) 10\({}^{-8}\) & 0.415 \\
0.84693 & 0.017 & 2.06 \(\times\) 10\({}^{-8}\) & 2.80 \(  
Extensive computations were carried out using parallel codes executed on a supercomputer, examining both linear and geometrically nonlinear Euler-Bernoulli beam theories. A comparative analysis of the beam's responses under these two theories was performed, with variations in beam thickness thus affecting the natural frequencies. Notably, the Karman vortex street generated behind the elastic beam closely matches the first natural frequency of the specific beams under consideration.  
The numerical studies confirmed that vibrations induced by the vortex could produce resonance-like phenomena, leading to increased vibration amplitudes in beams whose natural frequencies aligned with the vortex frequency. The results emphasize the necessity of employing geometrically nonlinear beam theory, particularly in the resonance zone, where the SDBT may yield inaccurate predictions. This insight is crucial for applications requiring resonance suppression or maximal amplitude generation, such as energy harvesting. This detailed analysis is expected to benefit designers of sensors and other microelectromechanical systems (MEMS) where the short time scales and small cantilever dimensions are important.  
[PERSON] conceptualized the work, created the mathematical model of the elastic beam, computer code for the beam vibration, and some visualization related to the beam. [PERSON] developed the DSMC code, numerical calculations, and visualizations related to the rarefied gas flows. [PERSON] performed validation of the elastic beam model and some visualizations related to the beam. All authors have read and agreed to the published version of the manuscript.  
This work was accomplished with financial support from the Bulgarian research fund, grant KP-06-N72/7, 2023. 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. The work was partially supported by the Centre of Excellence in Informatics and ICT established under Grant No BG05M2 OP001-1.001-0003, financed by the Science and Education for Smart Growth Operational Program and co-financed by the European Union through the European Structural and Investment funds.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.