## 4 Results and Discussion
It is because the nonlocal strain gradient theory can explain both the strengthening and weakening of the size-effect of the structure. Whether the overall effect of the nonlocal strain gradient strengthening or weakening is related to the relative quantity between the nonlocal parameter \(e_{0}a\) and the strain gradient constant \(l\). On the other hand, the nonlocal strain gradient shows the opposite effect on the natural frequency at the prebuckling and postbuckling state. The nonlocal parameter \(e_{0}a\) decreases the natural frequency at the prebuckling state, while it increases the natural frequency at the postbuckling state. The strain gradient constant \(l\) increases the natural frequency at the prebuckling state, while it decreases the natural frequency at the postbuckling state. Shown in Figure 7 are the dimensionless amplitude \((W_{\max}/W_{0})\)-excitation frequency \((\left(\omega_{n}L^{2}/h_{t}\right)\sqrt{12\rho_{\max}/C_{\max}})\) curves for different axial loads at the prebuckling state (\(P\leq P_{cr}\), \(m_{s}=0\), \(R=1k\Omega\), \(W_{0}=0.01h_{t}\), \(\kappa_{0}=0\) for distribution 1). It seems that at the prebuckling state, the dimensionless amplitude increases to a peak value and then decreases with the increase in dimensionless excitation frequency. Moreover, when \(P=0.2P_{cr}\), \(P=0.4P_{cr}\), \(P=0.6P_{cr}\), \(P=0.8P_{cr}\) and \(P=P_{cr}\), the maximum dimensionless amplitude decreases by 10.4%, 22.4%, 36.9%, 55.0% and 91.1%, respectively, compared with that of without axial load. The resonance amplitude also decreases as the axial load increases. When the axial load equals the buckling load \(P_{cr}\), the resonance phenomenon cannot be observed because the natural frequency of the structure decreases to zero as the structure is critically buckled. In such the case, the dimensionless amplitude increases from 0 to 1 as the excitation frequency increases. Figure 6: Effects of the nonlocal parameter and strain gradient constant on the natural frequency \(((\omega_{n}L^{2}/h_{t})\sqrt{12\rho_{\max}/C_{\max}})\) and buckling load for distribution 1 (\(\kappa_{0}=0\), \(m_{s}=0\)). Shown in Figure 8 are the dimensionless root mean square voltage (\(\frac{\sigma_{21}}{\varepsilon_{311}}\frac{V_{\text{max}}}{W_{\text{max}}}\))-excitation frequency (\(\omega_{n}L^{2}/h_{t}\sqrt{12\rho_{\text{max}}/C_{\text{max}}}\)) curves for different axial loads at the prebuckling state (\(P\)\(\leq P_{cr}\), \(m_{s}=0\), \(R=1k\Omega\), \(W_{0}=0.01h_{t}\), \(\kappa_{0}=0\) for distribution 1). It seems that the root mean square voltage output increases to a peak and then decreases as the excitation frequency increases in the prebuckling procedure. The peak value is acquired when the excitation frequency is close to the natural frequency of the structure. Further, as the axial load increases, the peak value decreases. When the axial load is equal to the buckling load, the RMS voltage output increases as the excitation frequency increases, and the value of the RMS voltage remains at a low level. Shown in Figure 9 are the dimensionless vibrational amplitude (\((W-W_{s})_{\text{max}}/W_{0}\))-excitation frequency (\((\omega_{n}L^{2}/h_{t})\sqrt{12\rho_{\text{max}}/C_{\text{max}}}\)) curves for different axial loads at the postbuckling state (\(P\geq P_{cr}\), \(m_{s}=0\), \(R=1k\Omega\), \(W_{0}=0.01h_{t}\), \(\kappa_{0}=0\) for distribution 1). It is worth noting that the vibrational amplitude only calculates the vibrational component of the beam, and displacement due to the axial load (\(W_{s}\)) is taken off in Figure 9, because it would not produce an electric field from this part of the displacement. In contrast with the curves at the prebuckling state (shown in Figure 7), the resonance frequency increases as the axial load increases due to the decreases in the natural frequency as the axial load increases at postbuckling state. Further, the resonance amplitude increases as the axial load increases, which is different from the results at the prebuckling state.