# Effects of Nonlinear Damping on Vibrations of Microbeam
## 4 Discussion  
In this section, we use the following parameters as an example to study the effects of nonlinear damping on microbeam vibrations [16]: \(E=1.44\) GPa, \(G=0.5127\) GPa, and mass density \(\rho=1220\) kg/m\({}^{3}\). The beam's geometrical parameters are \(L=100\) um, \(b=10\) um, and \(h=20\) um, thus we have \(S=2\times 10^{2}\)\(\left(\upmu\right)^{2}\), \(m=\rho S=2.44\times 10^{-7}\) kg/m, \(I=2\times 10^{4}/3\)\(\left(\upmu\right)^{4}\). Let \(\xi=20\) um and \(P_{0}=0.188\) N, thus we obtain \(\omega_{0}=1.42\times 10^{7}\), \(\omega=0.901\), \(d=21.3\), and \(C_{2}=450C_{1}\). Suppose that \(\xi_{1}=0.01\), thus we obtain \(c_{1}=6.75\).  
In Section 3.1, it is theoretically demonstrated that nonlinear damping does not affect the vibration amplitudes under the non-resonant condition. Here, the numerical calculations of Equation (11) with \((f_{1},\Omega_{1})=(0.5,1.5)\) confirm this prediction, as shown in Figure 2. However, when the exciting amplitude increases, for example \((f_{1},\Omega_{1})=(10,1.5)\), multi-frequency vibrations will appear in the system without nonlinear damping, as shown in Figure 3. In this case, the perturbation methods are no longer applicable, since the strong excitations allow for Equation (11) to appear nonlinearly strong. When the large-amplitude vibrations of non-resonant excitations appear, nonlinear damping significantly reduces the vibration amplitudes, and suppresses the vibration components of non-excited frequencies, as shown in Figure 3.  
The influence of nonlinear damping on primary resonance is mainly shown in the following two aspects: First, when the exciting frequency is greater than the modal frequency of the microbeam, nonlinear damping can significantly change the bifurcation points of the load and strongly affects the dynamic behavior of the microbeam, as shown in Figures 4 and 5. Second, when the exciting frequency is less than the modal frequency of the microbeam, the impact of nonlinear damping is slight, as shown in Figures 5 and 6. The numerical calculations of Equation (11) confirm the aforementioned results, as shown in Figures 7 and 8. For example, when \(f_{2}=3\) and \(\sigma=10\), the vibration amplitude with \(c_{2}=0\) is about six times larger than the amplitude with \(c_{2}=6.75\), as shown in Figure 7. This difference is essential in practice.  
Figure 4: Amplitude of the response as a function of the excitation amplitude of primary resonance with \(\sigma=10\) for several damping parameters.  
Figure 5: Frequency-response curves of primary resonance for \(f_{2}=6\).  
Figure 3: The phase portraits of non-resonant excitations for \(\Omega_{1}=1.5\) and \(f_{1}=10\).  
Nonlinear elastic effects play an essential role in the dynamics of MEMS [1,2]. Nonlinear dissipation effects in micromechanical oscillators are often overlooked. However, our work shows that arbitrarily overlooking nonlinear damping may lead to qualitative mistakes. In this paper, we especially focus on the system's behavior near the bifurcation points of primary resonance. The results show that nonlinear damping has a significant impact on the dynamics of microbeams, as shown in Figures 4 and 6. Moreover, it is necessary to research particular vibrations, such as the super-resonance and sub-resonance. Furthermore, the experimental results suggest that linear Kelvin-Voigt damping cannot be used to study the dynamics of the microbeam.  
Figure 8: Time evolution of the microbeam’s midpoint displacement of primary resonance for \(f_{2}=30\) and \(\sigma=-10\).  
Figure 6: Amplitude of the response as a function of the excitation amplitude of primary resonance with \(\sigma=-10\) for several damping parameters.  
Figure 7: Time evolution of the microbeam’s midpoint displacement of primary resonance for \(f_{2}=3\) and \(\sigma=10\).  
fully account for all of the experimentally observed nonlinear dissipation [22]. We believe that the effects of nonlinear damping in microbeams necessitate further studies, both experimentally and theoretically.