## 2 Theoretical Modeling and Formulations
That is, the effective nonlocal bending moment is an infinite higher-order variable based on the classical counterpart. However, with the increase of \(n\), the influence of higher-order terms will be smaller and smaller because \(e_{0}a\) is a small quantity. Consequently, only the first two terms in the effective nonlocal bending moment are considered and further substitute it into Equation (5), one can obtain the following nonlinear partial differential equation that governs the lateral vibration as  
\[ \begin{split}&\text{EI}\Big{[}\frac{\partial^{2}\text{U}}{ \partial\text{X}^{4}}+(\text{e}_{0}\text{a})^{2}\frac{\partial^{2}\text{U}}{ \partial\text{X}^{6}}\Big{]}+\left[\frac{\text{EA}}{\text{L}}\int_{0}^{\text{ L}}\left(\frac{\partial\text{U}}{\partial\text{X}}\right)^{2}\text{dX}\right]\frac{ \partial^{2}\text{U}}{\partial\text{X}^{2}}-\\ &\text{p}(\text{T})\cos\frac{\text{\pi X}}{\text{L}}+\rho\text{A} \frac{\partial^{2}\text{U}}{\partial\text{T}^{2}}+\mu\frac{\partial\text{U}}{ \partial\text{T}}=0\end{split}  \tag{7}\]  
It is shown that, unlike the classical vibration equation of beams with a fourth-order differential equation [36], the governing equation based on the nonlocal theory is a higher-order one. For the above partial differential Equation (7), it is difficult to determine the exact solutions directly. In this paper, we attempt to observe effects of the nonlocal parameter on the amplitude-frequency characteristics of the microbeam, in an effort to separate the spatial and temporal variables. According to the form of the external force, the approximate solution for the lateral vibration of the microbeam can be assumed as  
\[\text{U}(\text{X},\text{T})=\text{S}(\text{T})\cos\frac{\text{\pi X}}{\text{L}} \tag{8}\]  
where S is the time-dependent displacement and \(\cos(\text{\pi X}/\text{L})\) is the approximate resonance vibration form of the microbeam clamped at both ends. By use of Equation (8), the governing Equation (7) then becomes  
\[\text{EI}\left[\frac{\text{\pi}^{4}}{\text{L}^{4}}\text{S}-(\text{e}_{0}\text {a})^{2}\frac{\text{\pi}^{6}}{\text{L}^{6}}\text{S}\right]-\frac{\text{EA} \text{\pi}^{4}}{4\text{L}^{4}}\text{S}^{3}-\text{p}(\text{T})+\rho\text{A} \frac{\text{d}^{2}\text{S}}{\text{dT}^{2}}+\mu\frac{\text{dS}}{\text{dT}}=0 \tag{9}\]  
Introduce the following non-dimensional variables and parameters:  
\[\text{s}=\text{S}\sqrt{\frac{\text{A}}{\text{T}}},\,\omega_{0}=\sqrt{\frac{ \text{EI}\text{\pi}^{4}}{\rho\text{AL}^{4}}},\,\text{t}=\omega_{0}\text{T}, \,\xi=\frac{\mu}{\rho\text{A}},\,\tau=\frac{\text{e}_{0}\text{a}}{\text{L}}, \,\,\varepsilon=\frac{\omega_{0}^{2}}{4} \tag{10}\]  
where \(\varepsilon\) is a small variable and \(\tau\) is the non-dimensional nonlocal parameter which describes the nanoscale effects. For a microbeam with certain material and length, the non-dimensional nonlocal parameter is a constant. Using the above variables and parameters, Equation (9) can be cast into the dimensionless form as follows  
\[\omega_{0}^{2}\frac{\text{d}^{2}\text{s}}{\text{dt}^{2}}-\varepsilon\text{s}^ {3}+\omega_{0}^{2}\text{s}-\omega_{0}^{2}\text{\pi}^{2}\text{\tau}^{2}\text{s }+\varepsilon\omega_{0}\frac{\text{ds}}{\text{dt}}=\frac{\text{A}_{\text{m}} \sin(\text{\Omega T})}{\rho\sqrt{\text{AI}}} \tag{11}\]  
where \(\text{p}(\text{T})=\text{A}_{\text{m}}\sin(\text{\Omega T})\) is assumed and \(\text{A}_{\text{m}}\) is the amplitude of the external excitation, \(\Omega\) is the excitation frequency. Further, the following equation can be determined and shown as  
\[\omega_{0}^{2}\frac{\text{d}^{2}\text{s}}{\text{dt}^{2}}-\varepsilon\text{s}^ {3}+\omega_{0}^{2}\text{s}-\omega_{0}^{2}\text{\pi}^{2}\text{\tau}^{2}\text{s }+\varepsilon\kappa\frac{\text{ds}}{\text{dt}}=\varepsilon\text{a}_{\text{m}} \sin(\text{\omega t}) \tag{12}\]  
where \(\text{a}_{\text{m}}=\frac{\text{A}_{\text{m}}}{\rho\varepsilon\sqrt{\text{AI}}}\), \(\omega=\frac{\Omega}{\omega_{0}}\approx 1\) and \(\kappa=\frac{\varepsilon\omega_{0}}{\varepsilon}\).