# Effects of Nonlinear Damping on Vibrations of Microbeam
## 2 Methods  
In the modified couple stress theory, the symmetric couple stress characterizes the small-scale effect, and the strain energy density can be written as [13; 23]:  
\[U=\frac{1}{2}\big{(}\sigma_{ij}\varepsilon_{ij}+m_{ij}\chi_{ij}\big{)},\ (i,j=x,y,z) \tag{1}\]  
where \(\varepsilon_{ij}\) and \(\sigma_{ij}\) are the stress and strain, \(m_{ij}=2G\xi^{2}\chi_{ij}\) is the couple stress tensor, \(\chi_{ij}=\frac{1}{2}\Big{[}\
abla\omega+(\
abla\omega)^{T}\Big{]}\) is the symmetrical curvature tensor, \(\omega=(\
abla\times u)/2\) with \(u\) being the displacement vector [13], and \(G=E/[2(1+\
u)]\). Here, \(E\) and \(\
u\) are the [PERSON]'s modulus and [PERSON]'s ratio, \(\zeta\) is the material length scale parameter obtained from the experimentations. Based on the modified couple stress theory, [PERSON] established an integral differential equation to model the mechanical properties of microbeams, as shown in Figure 1. This theory takes into account the small-scale effect and nonlinear damping, which are induced by the elongation of the microbeam [16]:  
\[m\frac{\partial^{2}w}{\partial t^{2}}+\Big{(}EI+GS\xi^{2}\Big{)}\frac{\partial ^{4}w}{\partial x^{4}}+\left[P_{0}-\frac{ES}{2L}\int\limits_{0}^{L}\left( \frac{\partial w}{\partial x}\right)^{2}dx\right]\frac{\partial^{2}w}{ \partial x^{2}}=F(x,t) \tag{2}\]  
where \(w\) is the vertical deflection, \(P_{0}\) and \(m\) are the initial axial load and mass per unit length, respectively, \(S\) and \(I\) are the cross-sectional area and moment of inertia. From Equation (2), it is found that the small-scale effect significantly affects the static bending and vibration frequencies of microbeams [16]. However, Equation (2) is an integral differential equation, and it is not easy to find solutions to the equation. As shown below, we will develop a new differential equation to replace Equation (2). Figure 1: Portrait of the structure: (**a**) The undeformed coordinate system, (**b**) cross-section, (**c**) the displacement and rotation angle. Suppose that the average axial force, which is induced by the axial elongation of the beam, is \(\overline{P}_{1}\), thus we obtain:  
\[\overline{P}_{1}=\frac{ES\Big{(}\widetilde{L}-L\Big{)}}{L}\approx\frac{ES}{2L} \int\limits_{0}^{L}{(\partial w/\partial x)^{2}dx} \tag{3}\]  
where \(\widetilde{L}=\int\limits_{0}^{L}{\Big{[}1+(\partial w/\partial x)^{2}/2\Big{]}dx}\) is the deformed length of the beam axle. On the beam element, the \(z\) component of the axial force increment is \((dP_{1})\cdot\sin\theta\), and \(\theta\) is the rotation angle of the cross-section, as shown in Figure 1c. Considering \(\overline{P}_{1}\approx P_{1}\), \(\sin\theta\approx\theta\), \(\theta\approx\partial w/\partial x\), and using Equation (3), we obtain:  
\[\frac{\partial}{\partial x}(P_{1}\sin\theta)dx\approx\frac{\partial}{\partial x }\bigg{(}\overline{P}_{1}\frac{\partial w}{\partial x}\bigg{)}dx\approx\frac{ ES}{2L}\left\{\left[\frac{\partial^{2}w}{\partial x^{2}}\int\limits_{0}^{L}{ \left(\frac{\partial w}{\partial x}\right)^{2}dx}\right]\right\}dx \tag{4}\]  
The right part of Equation (4) is also the nonlinear term of Equation (2). Equation (3) indicates that the accurate axial force \(P_{1}\) is replaced by the average axial force \(\overline{P}_{1}\) in Equation (2). The accurate force, which is induced by the axial elongation, can be written as \(P_{1}=ES(\Delta L/dx)\approx(ES/2)(\partial w/\partial x)^{2}\), where \(\Delta\)\(L\) is the elongation of the element. Considering \(d(P_{1}\sin\theta)\approx(3 ES/2)(\partial w/\partial x)^{2}\big{(}\partial^{2} w/\partial x)^{2}dx\), and Equation (4), we obtain:  
\[\left[\frac{ES}{2L}\int\limits_{0}^{L}{\left(\frac{\partial w}{\partial x} \right)^{2}dx}\right]\frac{\partial^{2}w}{\partial x^{2}}\approx\frac{3 ES}{2} \bigg{(}\frac{\partial w}{\partial x}\bigg{)}^{2}\frac{\partial^{2}w}{ \partial x^{2}} \tag{5}\]  
Equation (5) indicates that the nonlinear differential term can replace the nonlinear term of Equation (2).