## 4 Results and Discussion
Using the combination of Equations (15), (19), (22) and (27), one obtains the non-dimensional time-dependent displacement as  
\[\text{s}=\frac{1}{2}\text{b}_{0}\text{e}^{\text{i}\Lambda}-\frac{\text{b}_{0} ^{3}\varepsilon}{64\sqrt{1-\pi^{2}\tau^{2}}}\omega_{0}^{2}\text{e}^{3\text{i} \Lambda}. \tag{28}\]  
where \(\Lambda=\left(\sqrt{\left(\frac{\text{a}_{0}}{2\text{b}_{0}\omega_{0}^{2} \sqrt{1-\pi^{2}\tau^{2}}}\right)^{2}-\left(\frac{\kappa}{2\omega_{0}^{2}} \right)^{2}}-\frac{3\text{b}_{0}^{2}}{8\omega_{0}^{2}\sqrt{1-\pi^{2}\tau^{2}} }\right)\varepsilon t+\sqrt{1-\pi^{2}\tau^{2}}\text{t}+\beta_{0}\). Relations between the real-part of the time-dependent displacement and dimensionless time coordinate given in Equation (28) can also be illustrated in Figure 7, where \(\text{b}_{0}=\beta_{0}=\varepsilon=1\), \(\omega_{0}=\kappa=0.1\) and \(\text{a}_{\text{m}}=0.1\) are assumed; it indicates that the nonlocal parameter \(\tau\) causes the displacement to draft along the time coordinate axis. Consequently, the nonlocal hysteresis effect is found in the lateral excited vibration of a beam-like microstructure, which is mainly caused by the small-scale effect and viscoelastic effect. It is noted that the hysteresis effect has not been reported in the nonlocal free vibration of micro/nano-structures [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. Resultingly, it is a special phenomenon in excited vibration. With a longer time, the nonlocal hysteresis effect seems more obvious. From Equations (8) and (28) we can deduce the dimensionless lateral displacement as:  
\[\text{u}(\text{x},\text{t})=\left(\frac{1}{2}\text{b}_{0}\text{e}^{\text{i} \Lambda}-\frac{\text{b}_{0}^{3}\varepsilon}{64\sqrt{1-\pi^{2}\tau^{2}}\omega_ {0}^{2}}\text{e}^{3\text{i}\Lambda}\right)\cos(\tau\text{x}). \tag{29}\]  
where \(\text{u}=\text{U}\sqrt{\frac{\Lambda}{\Upsilon}}\), \(\text{x}=\frac{\chi}{\Gamma}\) and the cubic nonlinearity is reflected again here. The evolvement of lateral displacement with respect to axial and time coordinates is given in Figure 8, where \(\tau=0.1\) is adopted besides the assumptions in Figure 7. It clearly reveals the changes of lateral excited displacement with time and spatial variables, which are different from the results in the classical beam excited vibration, because of the nonlocal effect. For example, the displacement of the left and right halves of the microbeam is not symmetrical. Addition of the lateral displacement with respect to axial and time coordinates is given in Figure 8.  
Figure 4:0 Forced amplitude versus excitation amplitude with a different viscoelastic damping coefficient. Finally, further research into this topic can be undertaken: (i) The one-dimensional viscoelastic microbeam can be extended to microplate and microshell structures, and their lateral excited vibration responses need to be analyzed; (ii) beyond the nonlocal effect, the strain gradient effect, surface effect, and quantum effect of the excited vibration of viscoelastic microstructures can be considered, which are helpful to explain the key dynamics mechanisms at the microscale; (iii) based on the above two points, it is necessary to develop new perturbation methods in order to solve complex dynamic equations with higher dimensions and more variables.