## 4 Results and Discussion
Shown in Figure 13 are the dimensionless natural frequency (\((\omega_{n}L^{2}/h_{t})\sqrt{12\rho_{\max}/C_{\max}}\))-porosity coefficient (\(\kappa_{0}\)) curves for different distributions of the core layer (\(m_{s}=0.5I_{0}L\), \(W_{0}=0.01I_{t}\), \(R=1k\Omega\), \(P=0\)). The influence of the porosity coefficient of the sandwich layer on the vibration frequency of the composite beam is also significant. In the figure, for the same value of the porosity coefficient (\(\kappa_{0}\)), the mass of the three types of distribution is equal. It can be found that the vibration frequencies of distribution 1 and distribution 3 decrease rapidly with the increase in the porosity coefficient, while the decrease in the vibration frequency of distribution 2 is not obvious. In addition, for the same porosity coefficient, the vibration frequency of distribution 2 is the largest, the vibration frequency of distribution 3 is the second largest, and the vibration frequency of distribution 1 is the smallest. Figure 12: Dimensionless optimal root mean square voltage output \((\frac{a_{\text{vis}}}{\varepsilon_{311}}\frac{V_{\text{max}}}{W_{\text{max}}})\)-axial load \((\frac{12L^{2}P}{\pi^{2}\text{D}\text{C}_{\text{max}}L_{\text{s}}^{2}})\) curves \((m_{s}=0.5l_{0}L\), \(W_{0}=0.01h_{t}\), \(\kappa_{0}=0.5\), \(R=1k\Omega)\). The dimensionless optimal root mean square voltage output (\(\frac{\mu_{31}}{c_{331}}\frac{V_{rms}}{W_{\max}}\)) - porosity coefficients (\(\kappa_{0}\)) curves for different distributions of the core layer are shown in Figure 14 (\(m_{s}=0.5I_{0}L\), \(W_{0}=0.01h_{t}\), \(R=1k\Omega\), \(P=0\)). The influence of the porosity coefficient of the sandwich layer on the root mean square voltage output of the composite beam is also significant. In the figure, the mass of three types of distribution is assumed to be equal. It can be seen from the figure that the optimal RMS voltage output for the three type distributions decreases rapidly with the increasing porosity coefficient. Besides, for the same porosity coefficient, distribution 2 has the largest optimal RMS voltage output, distribution 3 has the second largest optimal RMS voltage output, and distribution 1 has the smallest optimal RMS voltage output among the three types of distribution. Shown in Figure 15 are the dimensionless amplitude (\(W_{\max}/W_{0}\)) - excitation amplitude (\(W_{0}\)) curves for linear/nonlinear vibration and different porosity coefficients (\(\kappa_{0}\)) (\(m_{s}=0\), \(R=1k\Omega\), \(P=0\) for distribution 1). It can be found that the dimensionless amplitude of nonlinear vibration decreases as the excitation amplitude increases, while the dimensionless amplitude of the linear vibration does not change with the excitation amplitude. This is because the stiffness of the structure increases when considering nonlinear vibrations. In addition, for the same excitation amplitude, the larger the porosity coefficient, the larger the dimensionless amplitude. Shown in Figure 16 are the dimensionless RMS voltage output (\(\frac{\mu_{31}}{c_{331}}\frac{V_{rms}}{W_{\max}}\)) - excitation amplitude (\(W_{0}\)) curves for linear/non-linear vibration and different porosity coefficients (\(\kappa_{0}\)) (\(m_{s}=0\), \(R=1k\Omega\), \(P=0\) for distribution 1). It can be seen that the RMS voltage output of nonlinear vibration increases with the increase in the excitation amplitude, while the RMS voltage output of linear vibration does not vary with the excitation amplitude. This indicates that considering nonlinear vibration can enhance the mechanical electrical conversion efficiency. Figure 16: Dimensionless root mean square voltage output (\(\frac{\delta_{0}}{\varepsilon_{\text{max}}}\), \(\frac{V_{\text{max}}}{V_{\text{max}}}\)) – excitation amplitude (\(W_{0}\)) curves for linear/nonlinear vibration and different porosity coefficients (\(\kappa_{0}\)) (\(m_{s}=0\), \(R=1k\Omega\), \(P=0\) for distribution 1).