# Vortex-Induced Micro-Cantilever Vibrations with Small and Large Amplitudes in Rarefied Gas Flow
## 5 Results and Discussion
On the other hand, the original load of the flow on the beam is very irregular and looks stochastic (see Figure 5). This loading leads to vibration as the ones shown in Figures 6 and 7. In Figure 6, the dimensionless beam thickness \(c_{h}\) = 0.017 and in Figure 7 \(c_{h}\) = 0.020073. \(c_{h}\) was varied within the interval [0.015, 0.0275]. As can be seen, at the beginning, the beam bends in the direction of the flow and oscillates with large amplitudes. After the transition period, which is usually around 12 us, the vibrations continue in a steady-state regime around a new bent position of the beam. Figure 6: Time history diagram of the vibration of the tip of the beam in case \(c_{h}\) = 0.0170. Figure 7: Time history diagram of the vibration of the tip of the beam in case \(c_{h}\) = 0.0190. In these figures are plotted the vibrations of the tip of the beam obtained by the small deflection beam theory (SDBT) in blue colour and by the large deflection beam theory (LDBT) in red colour. In the cases shown, the amplitudes of vibrations obtained by LDBT are larger than the ones obtained by SDBT, and the differences are bigger for \(c_{h}\) = 0.019. The time history diagrams for the other cases of \(c_{h}\) are similar to those shown in Figures 6 and 7. For the selected interval of \(c_{h}\), the first natural frequencies of the beams are placed in the interval \(\omega/\omega_{\text{vortex}}\subset[0.75,\,0.125]\)  
Note the short time scales and small cantilever dimensions considered in this study. This suggests that such a sensor--or an array of sensors--could be capable of detecting rapid gas-phase processes in micro-electromechanical systems (MEMS), as well as those occurring in the boundary layer of high-speed vehicles such as racing cars or airplanes, under real-world conditions rather than in controlled laboratory settings. Assuming that the transition period ends at 15 us for all cases, the vibrations after that point are considered fully established (steady-state) (see Figures 6 and 7). For the study-state vibration, the difference between the maximal and minimal value of the vibration amplitude, i.e., the range of oscillation, is plotted in Figure 8a,b. It is seen that the difference between the range of the amplitudes obtained by SDBT and LDBT is larger for the thinner beams. This is because for thinner beams, the influence of the geometrically nonlinear terms, caused by large deflections, influences the response. More interesting is the fact that increasing the thickness of the beam the range of the amplitudes increases. The explanation is that for \(c_{h}\) = 0.020073 the first natural frequency of the beam is almost equal to the frequency of the vortex shedding and a resonance-like phenomenon occurs. The maximal range of the amplitudes occurs for computations by LDBT at \(c_{h}\) = 0.019. This means that the resonance of the beam described by LDBT occurs at \(\omega/\omega_{\text{vortex}}\approx 0.92\). It is confirmed that LDBT, which considers the existence of longitudinal force, leads to increasing the frequency of the beam. For the linear beam theory, the maximal range of the oscillations is \(\omega/\omega_{\text{vortex}}\)-1. In the beams with natural frequencies higher than the frequency of the flow, the range of oscillations quickly decreases, and the difference between the results obtained by SDBT and LDBT becomes negligible. The significance of the nonlinearity on the cantilever's response can be more clearly seen in Table 2 and Figure 9. Table 2 shows the values of the maximal amplitudes of vibration of the cantilever tip for different beam thicknesses obtained by SDBT and LDBT. The relative differences \(\Delta w/w_{\text{LDBT}}\) = \((w_{\text{LDBT}}-w_{\text{SDBT}})/w_{\text{LDBT}}\) in dependence of the thickness (first natural frequency) is shown in Figure 9a,b.