# Vibration Analysis of Piezoelectric Cantilever Beams with Bimodular Functionally-Graded Properties
## 4 Numerical Simulation
### Simulation for Free Damping Vibration  
As shown in Table 3, we have finished the frequency extraction of 15 eigenvalues for the case \(\alpha_{1}=-2,\ \alpha_{2}=-3\), in which the maximum frequency reads 28,006 Hz; thus, the corresponding period is \(1/28006\ \mathrm{s}=0.00003571\ \mathrm{s}\). Since the time increment in the analysis step of transient modal should be less than this period value, the time increment is determined as \(3\times 10^{-5}\ \mathrm{s}\). The details are shown below.  
1. Establishment of dynamic analysis step of transient modal  
Although the material properties considered in this study are bimodular, functionally-graded and piezoelectric--that is, it is not linear--the nonlinear behavior of the material itself has little impact on the dynamic response of the structure. Besides, the motion attributes to the small deflection bending vibration thus there is no geometrical nonlinearity here. Furthermore, the damping of the structural system is relatively small, and the frequency involved in the analysis is low. Therefore, the system may be regarded as linear, which is suitable for linear transient dynamic analysis.  
Frequency extraction analysis step is followed by this step. To observe the attenuation process of the vibration, the analytical step time is set to be 0.5 s and the time increment is determined as \(3\times 10^{-5}\ \mathrm{s}\), as indicated above.  
1. Setting of the damping  
Direct modal damping is used here and the value of the damping ratio is 0.03. The starting mode order is 1 and the terminating mode order is 15.
2. Setting of the output of historical variable  
In the output of historical variable, the displacement component is selected.
3. Definition of load  
In order to make the model produce the initial displacement of 2 mm, a short-term load is applied on the model, in which the time of duration of the load is determined by the variation of load amplitudes with time and the amplitude is given in tabulate. In the time period 0-0.005 s, the amplitude is 1; and in the later time period, the amplitude is set as 0. The smoothness parameter is set as 0.25. Lastly, along the negative direction of \(z\)-axis, the load is applied and the magnitude of the load is 25 N.  
1. Submission of analysis and post-processing  
After a job is established, we may submit the job and calculate and output the results. Figures 5 and 6 show cloud diagrams of stress and displacement of 15 th order mode at the end of the transient modal analysis, in which Figure 5 is for the case \(\alpha_{1}=-2,\ \alpha_{2}=-3\) and Figure 6 is for the case \(\alpha_{1}=2,\ \alpha_{2}=3\).  
Note that although the numerical implementation in ABAQUS is three-dimensional in the form, we still put out the final results in the form of two-dimensional case, including three stresses, \(\sigma_{x}\), \(\sigma_{z}\) and \(\tau_{xz}\), as well as two displacements, \(u\) and \(w\), which are typical in two-dimensional plane problem. In fact, the studied problem is a two-dimensional plane problem, even in some cases (for example, a slim beam), the problem may be further simplified as a one-dimensional problem, like our analytical solution based on Euler-Bernoulli beam theory.  
Figure 5: Cloud diagram of stress and displacement for the case \(\alpha_{1}=-2\), \(\alpha_{2}=-3\): (**a**) \(\sigma_{x^{\prime}}\)(**b**) \(\sigma_{z^{\prime}}\)(**c**) \(\tau_{xz}\); (**d**) \(u\); (**e**) \(w\).