## 2 Theoretical Modeling and Formulations  
Considering a Euler-Bernoulli microbeam clamped at both ends with the length L, density \(\rho\), uniform cross sectional area A, we investigate its excited vibration that usually appears in the MEMS. The X-axis is measured from the left side along the axial direction, while the Y-axis is measured from the bottom along the transverse direction, shown in Figure 1. The dynamic equilibrium chart for a microbeam element straying from the left side X, with length dX of the microbeam is illustrated in Figure 2, where \(\rm p(T)\cos(\pi X/L)\) is the external harmonic excitation at the time T and position X of unit length [35], M is the bending moment, N is the internal axial tension, Q is the shear force, and \(\theta\) is the rotation angle with respect to X-axis of the microbeam. Figure 1: Sketch of the microbeam. Figure 2: Dynamic equilibrium chart for a microbeam element. It is noticed that the shear deformation is ignored, according to the Euler-Bernoulli beam theory. Based on the d'Alembert principle, the equilibrium equation of the element provided with respect to the Y-axis can be obtained as  
\[ \begin{split}& Q-\left(Q+\frac{\partial Q}{\partial X}dX\right)+N \theta-N\left(\theta+\frac{\partial\theta}{\partial X}dX\right)+\\ & p(T)\cos\frac{\pi\chi}{L}dX-\rho A\frac{\partial^{2}U}{ \partial T^{2}}dX-\mu\frac{\partial U}{\partial T}dX=0\end{split}  \tag{1}\]  
where U is the lateral displacement, the viscoelastic damping of the microstructures is considered, and \(\mu\) is the viscoelastic damping coefficient. Based on the moment balance condition and the relation between displacement and rotation angle, the above equation can be changed into the following one  
\[\frac{\partial^{2}M_{\text{ef}}}{\partial X^{2}}-N\frac{\partial^{2}U}{ \partial X^{2}}+p(T)\cos\frac{\pi\chi}{L}-\rho A\frac{\partial^{2}U}{\partial T ^{2}}-\mu\frac{\partial U}{\partial T}=0 \tag{2}\]  
where \(M_{\text{ef}}\) is the effective nonlocal bending moment defined in the nonlocal stress theory [22]. The axial tension N shown in Equation (2) caused by the lateral deformation is given by  
\[\frac{N}{A}=E\frac{\int_{0}^{L}\delta_{\text{dX}}dX}{L}=\frac{E}{L}\int_{0}^{ L}\left[\sqrt{1+\left(\frac{\partial U}{\partial X}\right)^{2}}-1\right]dX \tag{3}\]  
where \(\delta_{\text{dX}}\) is the axial elongation of a unit length d\(X\), and E is the Young's modulus of the microbeam. Making Taylor expansion series for Equation (3) and dropping the higher-order terms yield  
\[N=\frac{EA}{2L}\int_{0}^{L}\left(\frac{\partial U}{\partial X}\right)^{2}dX \tag{4}\]  
Substituting Equation (4) into (2), one obtains  
\[\frac{\partial^{2}M_{\text{ef}}}{\partial X^{2}}-\left[\frac{EA}{2L}\int_{0}^{ L}\left(\frac{\partial U}{\partial X}\right)^{2}dX\right]\frac{\partial^{2}U}{ \partial X^{2}}+p(T)\cos\frac{\pi\chi}{L}-\rho A\frac{\partial^{2}U}{\partial T ^{2}}-\mu\frac{\partial U}{\partial T}=0 \tag{5}\]  
It is known that the relationship between the effective nonlocal bending moment and lateral displacement based on the nonlocal stress theory is [22; 23]  
\[M_{\text{ef}}=-EI\sum_{n=1}^{\infty}(e_{0}a)^{2(n-1)}\frac{\partial^{2n}U}{ \partial X^{2n}}, \tag{6}\]  
where \(I=\iint_{A}Y^{2}dA\) is the usual area moment of inertia of the microstructural cross section, a is an internal characteristic length scale and \(e_{0}\) is a constant dependent on material properties [19]. The magnitude of \(e_{0}\) is determined experimentally or approximated by matching the theoretical predictions with those of molecular dynamics simulations. The combination of \(e_{0}a\) is defined as a nonlocal modulus, which approaches the Dirac delta function in the limit that the nonlocal modulus is much smaller than the external characteristic length of the microbeam. In that case, the classical continuum mechanics will be recovered. Moreover, it is interesting that the effective nonlocal bending moment turns into the classical bending moment when \(n=1\) in Equation (6).