# Vibration Analysis of Piezoelectric Cantilever Beams with Bimodular Functionally-Graded Properties
## 5 Comparisons and Discussion
### Comparison of Theoretical and Experimental Results  
At present, the relevant vibration experiment for the functionally-graded piezoelectric cantilever beam has not been found, not to mention the consideration for bimodular effect of the materials,  
Figure 8: Comparison of two time-displacement curves for \(E^{+}(z)<E^{-}(z)\)  
Figure 7: Comparison of two time-displacement curves for \(E^{+}(z)>E^{-}(z)\)  
The comparison should therefore be based on the experimental data available. Aiming at purely piezoelectric materials (which means there is no bimodular effect and also no functionally-graded characteristic), [PERSON] et al. [33] performed an experiment of free damping vibration of a cantilever beam. In this section, the theoretical solution derived in this study with the experimental results from [33] will be compared. Before comparison, it is necessary for us to reduce our theoretical solution to a purely piezoelectric case, to agree with the material model adopted in [33]. For this purpose, we need to presume \(\alpha_{1}=\alpha_{2}\)\(=0\) in Equation (8); thus, the equivalent modulus of elasticity \(E^{\star}\) may be simplified as  
\[E^{\prime}=\frac{\lambda_{33}}{s_{11}\lambda_{33}-\left(d_{31}\right)^{2}} \tag{27}\]  
in which \(E^{\prime}\) stands for the equivalent modulus of elasticity without bimodular and functionally-graded characteristic. Accordingly, Equation (18) may be changed as  
\[\omega_{D}^{\prime}=\omega^{\prime}\sqrt{1-\xi^{2}} \tag{28}\]  
in which \(\omega_{D}^{\prime}\) is the natural damped frequency in the same case of materials. In the comparison we should take the same parameters in accordance with the experimental model in [33], including the shape dimension of the beam, \(h=0.0001\) m, \(b=0.02\) m, \(l=0.05\) m, the damping ration \(\xi=0.03\) and the uniformly-distributed mass \(\overline{m}=0.015\) Kg/m, as well as the material parameters of PZT-5 (shown in Table 7). Besides, three different initial displacements in [33] are also adopted in our theoretical results; thus, Table 8 lists the vibration frequencies from experimental measurement and theoretical solution. It is easily seen that the differences between experimental results and theoretical results are small, which indicates the theoretical vibration frequency is reliable. Figures 9-11 show time-displacement curves from theoretical and experimental results under the three different initial displacements. It is readily found that the theoretical results agree with the experimental ones, although there are some differences between them, which may be caused by some uncontrollable factors in the experimental operation. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Initial Displacements**} & \multicolumn{6}{c}{**Vibration Frequency (Hz)**} \\ \cline{2-9}  & \multicolumn{2}{c}{**Experimental Results**} & \multicolumn{2}{c}{**Theoretical Results**} & \multicolumn{2}{c}{**Relative Errors**} \\  & \multicolumn{2}{c}{**Reference [33]**} & \multicolumn{2}{c}{**(This Paper)**} & \multicolumn{2}{c}{**(\%)**} \\ \hline
0.475 & 123.25 & 123.220 & 0.02 \\
0.750 & 123.25 & 123.220 & 0.02 \\
1.342 & 123.25 & 123.220 & 0.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Frequencies of experimental and theoretical results. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Initial Displacements**} & \multicolumn{6}{c}{**Vibration Frequency (Hz)**} \\ \cline{2-9}  & \multicolumn{2}{c}{**Experimental Results**} & \multicolumn{2}{c}{**Theoretical Results**} & \multicolumn{2}{c}{**Relative Errors**} \\  & \multicolumn{2}{c}{**Reference [33]**} & \multicolumn{2}{c}{**(This Paper)**} & \multicolumn{2}{c}{**(\%)**} \\ \hline
0.475 & 123.25 & 123.220 & 0.02 \\
0.750 & 123.25 & 123.220 & 0.02 \\
1.342 & 123.25 & 123.220 & 0.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Frequencies of experimental and theoretical results. Figure 11: Comparison of two time-displacement curves under initial displacement 1.342 mm.