# Effects of Nonlinear Damping on Vibrations of Microbeam
## 2 Methods
Furthermore, Equation (2) does not consider the structural damping. Here, we suppose that Kelvin-Voigt damping is in the beams, thus \(E\) in Equation (2) will be replaced by \(E+\widetilde{E}\partial/\partial t\)[21; 24], where \(\widetilde{E}\) is the viscous damping coefficient. Substituting Equation (5) into Equation (2), the nonlinear equation, which includes the small-scale effect and nonlinear damping term, can be obtained as:  
\[ \begin{array}{c}m\frac{\partial^{2}w}{\partial t^{2}}+\widetilde{E}\frac{ \partial}{\partial t}\left[\frac{\partial^{4}w}{\partial x^{4}}-\frac{3S}{2} \Big{(}\frac{\partial w}{\partial x}\Big{)}^{2}\frac{\partial^{2}w}{\partial x ^{2}}\right]+P_{0}\frac{\partial^{2}w}{\partial x^{2}}\\ +\left(EI+GS\delta^{2}\right)\frac{\partial^{4}w}{\partial x^{4}}-\frac{3 ES}{2} \left(\frac{\partial w}{\partial x}\right)^{2}\frac{\partial^{2}w}{\partial x ^{2}}=F\end{array}  \tag{6}\]  
[PERSON] et al. studied the primary resonance, super-resonance, and sub-resonance of microbeams using Equation (2) with linear damping \(\widetilde{E}\partial w/\partial t\)[11]. In the present paper, we will focus on the effects of nonlinear damping. Let \(\widetilde{x}=x/L\), \(\widetilde{w}=w/L\), \(\widetilde{t}=\omega_{0}t\) normalize the variables in Equation (6), thus we obtain:  
\[ \begin{array}{c}\frac{\partial^{2}\overline{w}}{\partial t^{2}}+\pi^{-4} \frac{4^{4}\overline{w}}{\partial\overline{x}^{4}}+P\frac{\partial^{2} \overline{w}}{\partial\overline{x}^{2}}+C_{1}\frac{\partial^{5}\overline{w}} {\partial\overline{x}^{2}\partial\overline{x}^{4}}\\ -D\Big{(}\frac{\partial\overline{w}}{\partial\overline{x}}\Big{)}^{2}\frac{ \partial^{2}\overline{w}}{\partial\overline{x}^{2}}-C_{2}\frac{\partial}{ \partial t}\left[\left(\frac{\partial\overline{w}}{\partial\overline{x}} \right)^{2}\frac{\partial^{2}\overline{w}}{\partial\overline{x}^{2}}\right]= \overline{f}\end{array}  \tag{7}\]  
The parameters in Equation (7) can be written as:  
\[P=\frac{P_{0}}{mL^{2}\omega_{0}^{2}},\ C_{1}=\frac{\widetilde{E}I}{mL^{4}\omega _{0}},\ C_{2}=\frac{3\widetilde{E}S}{2 mL^{2}\omega_{0}},\ D=\frac{3 ES}{2 mL^{2} \omega_{0}^{2}},\ \overline{f}=\frac{F}{Lm\omega_{0}^{2}} \tag{8}\]  
where \(\omega_{0}=\sqrt{\pi^{4}(EI+GS\delta^{2})\left(mL^{4}\right)^{-1}}\) is the natural frequency of the microbeam. For the same \(I/L\)[4], the microbeam's natural frequency \(\omega_{0}\) is more than the macrobeam, since the macrobeam's mass per unit length is less than the microbeam. Therefore, the nonlinear damping term of the microbeam is larger than the correspondent term of the macrobeam due to \(D/C_{2}=E/\left(\widetilde{E}\omega_{0}\right)\). The boundary conditions for hinged-hinged microbeams can be written as:  
\[\overline{w}\left(0,\overline{t}\right)=\overline{w}\left(1,\overline{t} \right)=0,\ \frac{\partial^{2}\overline{w}}{\partial\overline{x}^{2}}\left(0, \overline{t}\right)=\frac{\partial^{2}\overline{w}}{\partial\overline{x}^{2}} \left(1,\overline{t}\right)=0 \tag{9}\]  
Moreover, it is not easy to obtain an accurate analytical solution, since Equation (7) is the nonlinear differential equation. Therefore, we will use the [PERSON] method to obtain an approximate analytical solution [24; 25; 26]. Suppose that the solution of Equation (7) can be written as:  
\[\overline{w}=\sum_{j=1}^{\infty}\sin(j\pi\overline{x})\ \eta_{j}\left( \overline{t}\right) \tag{10}\]  
Substituting Equation (10) into Equation (7), then multiplying \(\sin\pi\overline{x}\) by both sides of the equations, and integrating it in \([0,1]\), the first-mode truncation of [PERSON] method [24], which is an ordinary differential equation in time, can be obtained as:  
\[\ddot{\eta}+2\widetilde{c}_{1}\dot{\eta}+\omega^{2}\eta+d\eta^{3}+c_{2}\dot{ \eta}\eta^{2}=f\left(\overline{t}\right). \tag{11}\]  
Here, we let \(\eta=\eta_{1}\) and \(\omega^{2}=1-N\pi^{2}\), \(\widetilde{c}_{1}=\pi^{4}C_{1}/2\), \(d=\pi^{4}D/4\), \(c_{2}=3\pi^{4}C_{2}/4\).