## 4 Numerical Results
Figure 5: Dependence of the calibrated central deflection on the porosity parameter \((a/h=20,n=0.3)\) (**a**) for different boundary conditions, (**b**) for different cosine-type porosity distributions. Figure 7a compares the variations of calibrated fundamental frequency \(\overline{\omega}_{(1,1)}\) to the porosity parameter \(e\) between the even and uneven porosity distributions, where the fundamental frequency is calibrated by \(\overline{\omega}_{(1,1)}=\omega_{(1,1)}a^{2}\sqrt{\rho_{m}/E_{m}}/h\). Similarly to the calibrated central deflection shown in Figure 4a, the \(\overline{\omega}_{(1,1)}\) increases in proportion to \(e\), except for the even distribution with \(n=0.6\). This is because the decrease in density with increasing \(e\) gives a higher effect on the free vibration than the decrease inelastic modulus proportional to \(e\) and vice versa for the exceptional case. Meanwhile, the uneven distribution provides the higher values of \(\overline{\omega}_{(1,1)}\) than the even distribution when the ceramic index \(n\) is \(\geq\)0.3 since the porosity in the uneven distribution is biased near the mid-surface, while the ceramic with the higher elastic modulus becomes more concentrated at the bottom proportional to \(n\). Figure 7b represents the variation in \(\overline{\omega}_{(1,1)}\) to the porosity for different \(w/t\) ratios \(a/h\), where the \(\overline{\omega}_{(1,1)}\) uniformly increases in proportion to \(a/h\) owing to the calibration. According to this calibration, it is seen that the relative difference in \(\overline{\omega}_{(1,1)}\) between even and uneven porosity distributions uniformly increases in proportion to \(a/h\). The uneven porosity distribution provides higher fundamental frequencies than the even porosity distribution when the \(a/h\) equals to and larger than 5.0. The relative difference in \(\overline{\omega}_{(1,1)}\) between even and uneven porosity distributions uniformly increases with the ceramic index and the \(w/t\) ratio. This is because the difference in thickness-wise material composition distributions between even and uneven porosity distributions becomes more apparent in proportion to these two factors. Figure 8a compares the variations in \(\overline{\omega}_{(1,1)}\) to the porosity parameter for different boundary conditions, where the highest frequency appears at CCCC and the second and third and lowest ones are shown at SCSC, SSSS and SFSF, respectively. This order of the magnitude of \(\overline{\omega}_{(1,1)}\) is completely contrary to that of \(\overline{w}\) shown in Figure 5a, which is consistent with the fact that the stronger the constraint is, the higher the fundamental frequency is. Figure 8b shows the variation in \(\overline{\omega}_{(1,1)}\) to the porosity parameter for three different porosity distributions, where the order of the magnitude of \(\overline{\omega}_{(1,1)}\) is completely contrary to that of \(\overline{w}\) shown in Figure 5b. This is because the natural frequency and the central deflection show opposite variations to each other in their magnitudes to the structure stiffness. The unsym 1 distribution shows the lowest level and a decreasing trend for the porosity parameter, because both the porosity and the ceramic with the higher elastic modulus are simultaneously biased towards the bottom. And, the decrease in the elastic modulus with increasing the value of \(e\) gives a higher effect on the free vibration than the decrease in the density along with the value of \(e\).