# Vibration Analysis of Piezoelectric Cantilever Beams with Bimodular Functionally-Graded Properties
## 5 Comparisons and Discussion  
### Comparison of Theoretical and Numerical Results  
For the purpose of the full comparison between the theoretical results and numerical ones, both under the two cases of different modulus--that is, the tensile elastic modulus is greater than the compressive one \(E^{+}(z)>E^{-}(z)\), or reversely, \(E^{+}(z)<E^{-}(z)\)--we should select an object of study from the beam, for example, the upper surface of the free end, as our study object. In order to investigate the displacement variation with time under two cases of different modulus, we take the following two cases of functionally-grade index: (a) \(\alpha_{1}=-2,\ \alpha_{2}=-3\), which corresponds to \(E^{+}(z)>E^{-}(z)\), and the case (b) \(\alpha_{1}=2,\ \alpha_{2}=3\), which corresponds to \(E^{+}(z)<E^{-}(z)\). The numerical results are taken from the 15 th order mode. For the theoretical results, we should compute the equivalent modulus of elasticity first. For this purpose, substituting Equation (1) into Equation (8), we have  
\[E^{*}=\frac{\lambda_{33}^{+/-}}{s_{11}^{+/-}\lambda_{33}^{+/-}-\left(d_{31}^{+ /-}\right)^{2}}=\frac{\lambda_{33}^{0}}{s_{11}^{0}\lambda_{33}^{0}-\left(d_{31 }^{0}\right)^{2}}e^{-\alpha_{i}z/h} \tag{26}\]  
Substituting the values of \(s_{11}^{0}\), \(d_{31}^{0}\) and \(\lambda_{33}^{0}\) from Table 1 into the above equation and also noting that the upper surface is in compression; thus, \(\alpha_{i}=\alpha_{2}\), the values of the equivalent modulus of elasticity, may be determined and is listed in Table 6. Note that for the first case, \(E^{+}(z)>E^{-}(z)\), we take \(\alpha_{i}=\alpha_{2}=-3\) and \(z=-0.0014\) m, and for the second case, \(E^{+}(z)<E^{-}(z)\), we have \(\alpha_{i}=\alpha_{2}=3\) and \(z=-0.0006\) m, in which the values of \(z\) may refer to the cases (a) and (b) in Figure 4. Besides, the theoretical value of vibration frequency may be obtained via the expression \(\omega^{2}=a^{4}E^{*}I^{y}/\overline{m}\) and the numerical result of frequency is also from the 15 th order modes, which are also listed in Table 6.  
Figures 7 and 8 show the two time-displacement curves from theoretical and numerical results, in which Figure 7 is for \(E^{+}(z)>E^{-}(z)\) and Figure 8 is for \(E^{+}(z)<E^{-}(z)\). It is easy to see that despite some differences, the theoretical curve basically agrees with the curve from numerical simulation, this validates the theoretical solution to some extent. From Figures 7 and 8, we may also see that under two cases \(E^{+}(z)>E^{-}(z)\) and \(E^{+}(z)<E^{-}(z)\), the attenuation speed of the vibration from the numerical result is faster than that from theoretical solution, this is because the natural frequency from numerical simulation is greater than the one from theoretical solution, which may be easily seen from Table 6, in which for \(E^{+}(z)>E^{-}(z)\), the value from numerical simulation is 28,006 Hz while the counterpart from theoretical solution is 17,506 Hz; for \(E^{+}(z)<E^{-}(z)\), the two values from numerical simulation and theoretical solution are 34,599 Hz and 28,019 Hz, respectively. In addition, for \(E^{+}(z)<E^{-}(z)\), the attenuation speed is obviously faster than that in the case \(E^{+}(z)>E^{-}(z)\); this also may be easily seen from Table 6, in which the natural frequency from \(E^{+}(z)<E^{-}(z)\) is greater than the one from \(E^{+}(z)>E^{-}(z)\), for example, for theoretical solutions 28,019 Hz is greater than 17,506 Hz, while the numerical simulation 34,599 Hz is also greater than 28,006 Hz. It may be concluded that the relative magnitudes of the tensile and compressive moduli have influence on the attenuation speed of the vibration.