## 4 Results and Discussion
Consequently, there is a coupling between the effects of the nonlocal parameter, viscoelastic damping, and detuning parameter on the forced amplitude. The expression of detuning parameter can be obtained from Equation (25), given by  
\[\upsilon=\pm\sqrt{\left(\frac{2\Lambda_{\rm m}}{\rho\sqrt{\Lambda}\mathrm{Ib} \omega_{0}^{4}\sqrt{1-\pi^{2}\tau^{2}}}\right)^{2}-\left(\frac{\kappa}{2 \omega_{0}^{2}}\right)^{2}}-\frac{3\mathrm{b}^{2}}{8\omega_{0}^{2}\sqrt{1- \pi^{2}\tau^{2}}}. \tag{26}\]  
Figure 4: Forced amplitude versus viscoelastic damping coefficient. Figure 3: Frequency response curves of the micro-beam. In Equation (26), assuming \(\kappa=\rho\sqrt{\Lambda\mathrm{I}}=\omega_{0}=0.1\), we obtain the relations between the excitation amplitude and forced amplitude for different detuning parameters and nonlocal parameters, shown in Figure 5. Unlike the softening effect shown before, a hardening effect is observed. The results show that for the case of \(\upsilon=-20\) with a small excitation amplitude, there are multiple forced amplitudes corresponding to the same excitation amplitude, which is caused by the aforementioned leap phenomenon. On the other hand, while the excitation amplitude becomes larger, the forced amplitude increases with the increase in the excitation amplitude, but the rate of increase is smaller and smaller. In addition, the forced amplitude decreases with an increase in the detuning parameter. For the case of \(\upsilon=-10\) and \(\upsilon=-20\), the forced amplitude with a nonlocal effect (\(\tau=0.15\)) is smaller than that without a nonlocal effect (\(\tau=0\)) or the classical counterpart. For the case of \(\upsilon=20\), the nonlocal result (\(\tau=0.15\)) is larger than the classical result (\(\tau=0\)). This is because the main vibration function is deflected as a result of the existence of the nonlocal effect. Predictably, the larger the nonlocal parameter is, the larger the numerical result is. Assuming \(\omega_{0}=0.1\) and \(\upsilon=10\), Figure 6 shows the variation of forced amplitude with excitation amplitude under different dimensionless viscoelastic damping parameters and nonlocal parameters. In addition to a significant nonlocal effect, where the nonlocality reduces the forced amplitude with the same other conditions, it is also observed that, for different viscoelastic damping parameters, the forced amplitude increases with an increase in the excitation amplitude. Hence, the hardening phenomenon occurs again. Therefore, the present work proves that both the softening and hardening effects can be found in a micro/nano-structure or device, and this is similar to the nonlinear system with macrostructures. It is worth mentioning that, in early years, there was controversy about whether micro/nano-structures showed nonlinear softening or hardening mechanisms [22]. [PERSON] et al. [24] and [PERSON] et al. [39] confirmed that both exist at the micro/nanoscale. The present results support this point again. In addition, the smaller the viscoelastic damping parameter, the larger the forced amplitude. For a completely elastic microbeam, the forced amplitude will reach the maximum. Figure 5: Forced amplitude versus excitation amplitude with different detuning parameters. Additionally, we gain the expression of complex function B as  
\[\text{B}(\text{T}_{1})=\frac{1}{2}\text{b}_{0}\text{e}^{\text{i}\left[\left( \sqrt{\left(\frac{\text{a}_{0}}{2\text{b}_{0}w_{0}^{2}\sqrt{1-\pi^{2}\tau^{2}}} \right)^{2}-\left(\frac{\kappa}{2w_{0}^{2}}\right)^{2}}-\frac{3\text{b}_{0}^{2} }{8\omega_{0}^{2}\sqrt{1-\pi^{2}\tau^{2}}}\right)\text{T}_{1}+\beta_{0}\right] \tag{27}\]  
where \(\text{b}_{0}\) and \(\beta_{0}\) are integral constants and they can be determined by certain initial conditions.