## 4 Results and Discussion  
Figure 3 shows the relationship between the detuning parameter and the forced amplitude, in which \(a_{m}=\omega_{0}=\kappa=0.1\) is assumed. Note that the condition of \(\tau=0\) in Figure 3 means zero dimensionless nonlocal parameter. Considering that the dimensionless nonlocal parameter is defined as the ratio of internal and external characteristic scale parameters, \(\tau=0\) indicates that the external scale parameter (e.g., length of a beam) is much larger than the internal scale parameter (e.g., a lattice parameter), or the internal characteristic scale can be ignored in comparison with the corresponding external counterpart. On this condition, the size of the beam deviates from the micro-scale, and, as a result, a classical macrobeam and related results of classical vibration theory recover. However, the condition of \(\tau=0.15\) represents that the values of internal and external characteristic scale parameters are comparable, as, notably, the magnitudes of internal and external characteristic scales are almost at the same level, namely, the size of the beam is at a micro-scale. For a microbeam, the external characteristic scale parameter is also in the order of microns/nanometers, the nonlocal parameter that measures the significant nonlocal effect should not be zero; that is the essence of the nonlocal theory. It is found that, when the absolute value of the detuning parameter becomes larger, the forced amplitude becomes smaller. Moreover, it is noted that, in the ranges of \(\upsilon_{1}\)-\(\upsilon_{2}\) and \(\upsilon^{\prime}_{1}\)-\(\upsilon^{\prime}_{2}\), for the same detuning parameter, there are multiple stable-steady excited vibrations with different amplitudes. Therefore, a leap phenomenon exists in the two steady vibrations. It is observed that such a phenomenon also exists in the case of \(\tau=0.15\), which indicates that some mechanisms at the micro-scale are identical to those at a macro-scale. For example, the leap phenomenon and related characteristics (e.g., a softening effect) exist in both the classical and nonlocal vibration models. However, some mechanisms are unique only at small-scales. Comparing the case of \(\tau=0\) with \(\tau=0.15\), that is, without and with the nonlocal effect, we find that the nonlocal effect deflects the curve in Figure 3 where the relationship between forced amplitude and detuning parameter changes. In the range of \(\upsilon^{\prime}_{3}\)-\(\upsilon^{\prime}_{4}\), the forced amplitude decreases when the nonlocal effect is taken into account, while beyond the range, the amplitude increases, which indicates that the nonlocal effect significantly affects the forced amplitude. Therefore, the nonlocal effect plays an indispensable role in the analysis of excited vibrations at the micro-scale. Figure 4 shows the relationship between the forced amplitude and the viscoelastic damping coefficient, in which \(a_{m}=\omega_{0}=0.1\) is assumed. The softening effect is observed again where the forced amplitude gradually decreases with the increase of the viscoelastic damping coefficient, and the decreasing speed is slow at first, and then fast, and finally the curves tend to be flat. In addition, it has been observed that the dimensionless viscoelastic damping parameter has a significant influence on the nonlocal effect in the forced amplitude corresponding to the same detuning parameter. For the case of \(\upsilon=10\), the nonlocal effect increases the forced amplitude, and the difference becomes a little larger as \(\kappa\) increases and  
then gradually decreases when \(\kappa\) is more than 0.6. Corresponding to \(\upsilon=-10\), the forced amplitude with a nonlocal effect (\(\tau=0.15\)) is smaller than that without the nonlocal effect (\(\tau=0\)) when \(\kappa\) is small, but with the increase of \(\kappa\), the situation reverses, namely, the forced amplitude with nonlocal effect exceeds that without the nonlocal effect.