# Vibration Analysis of Piezoelectric Cantilever Beams with Bimodular Functionally-Graded Properties
## 2 The Problem Description  
An FGPM orthotropic cantilever beam with different tensile and compressive properties is considered here, and the corresponding Cartesian coordinates system \((x,y,z)\) is established, as depicted in Figure 2, where the right end of the beam is free and the left end is fully fixed; \(h\) is the rectangular section height of the beam, \(b\) is the section width and \(l\) is the beam length (\(h<<l\)). By exerting a concentrated load at the right end of the beam, an initial displacement \(v_{0}\) is generated at this end of the beam, as depicted in Figure 2. Then, by removing the load suddenly, the beam will freely vibrate until it ceases due to the existence of damping. During the bending vibration, the beam will continuously present downward bending and upwards bending up to the last cease. In the downward (or upward) bending, the tensile and compressive areas, bound by the neutral layer, will take turns to generate; this physical phenomenon is different from the corresponding static problem, which has an unchanged tensile area and compressive area [27; 28]. Therefore, unlike a bending beam in static analysis, in the vibration problem here, it seems that there is no definite tensile or compressive area in the beam. For the convenience of the following analysis, however, we assume in this study that the tensile or compressive area is defined in the present of initial displacement, i.e., the upper part of the beam is in compression and the lower part is in tension, as shown in Figure 2.  
Note that the physical parameters of materials of the beam are also functions of coordinates due to the functionally-graded property. In present study, we assume physical parameters vary only along  
Figure 2: Scheme of the functionally-graded piezoelectric materials (FGPM) bimodular cantilever beam. the thickness direction. Thus, all the material parameters are assumed to change with \(z\), according to the following relations. \[ \begin{array}{l}s_{ij}^{+}=s_{ij}^{0}F^{+}(z),d_{ij}^{+}=d_{ij}^{0}F^{+}(z), \lambda_{ij}^{+}=\lambda_{ij}^{0}F^{+}(z),\\ s_{ij}^{-}=s_{ij}^{0}F^{-}(z),d_{ij}^{-}=d_{ij}^{0}F^{-}(z),\lambda_{ij}^{-}= \lambda_{ij}^{0}F^{-}(z)\end{array} \, \tag{1}\]  
in which, \(F^{+}(z)=e^{\alpha_{12}z/h}\), \(F^{-}(z)=e^{\alpha_{2}z/h}\) are the tensile and compressive gradient functions, respectively; superscript \"+\" represents tension and \"\(-\)\" compression; \(d_{ij}^{+/-},\tau_{ij}^{+/-},s_{ij}^{+/-}\) are piezoelectric coefficient, dielectric coefficient and elastic coefficient, respectively; \(d_{ij}^{0},\lambda_{ij}^{0},s_{ij}^{0}\) are values at the neutral layer of the corresponding material parameters. It should be noted here that the neutral layer is defined at \(z=0\), whose determination has been reported in our previous study on static problem [27; 28]. The proposal of the neutral layer stems from the static problem but is still adopted in dynamic counterpart; otherwise, the so-called subarea in tension and compression cannot be realized. Furthermore, note that a set of very small electrodes are adhered discontinuously to the lower and upper surfaces of the beam and the beam is then poled along the direction of \(z\). Suppose that, in a two-dimensional problem, the stress and strain components is denoted by \(\sigma_{x}^{+/-},\sigma_{z}^{+/-},\tau_{zx}^{+/-}\) and \(\varepsilon_{x}^{+/-},\varepsilon_{z}^{+/-},\gamma_{zx}^{+/-}\), respectively; the electrical displacement and the electrical field components by \(D_{x}^{+/-},D_{z}^{+/-}\) and \(E_{x}^{+/-},E_{z}^{+/-}\), respectively. Therefore, the physical equations are  
\[\left\{ \begin{array}{l}\varepsilon_{x}^{+/-}\\ \varepsilon_{x}^{+/-}\\ \gamma_{zx}^{+/-}\end{array} \right\}=\left[ \begin{array}{ccc}s_{11}^{+/-}&s _{13}^{+/-}&0\\ s_{13}^{+/-}&s_{33}^{+/-}&0\\ 0&0&s_{44}^{+/-}\end{array} \right]\left\{ \begin{array}{l}\sigma_{x}^{+/-}\\ \sigma_{z}^{+/-}\\ \tau_{zx}^{+/-}\end{array} \right\}+\left[ \begin{array}{l}0&d_{31}^{+/-}\\ 0&d_{33}^{+/-}\\ d_{15}^{+/-}&0\end{array} \right]\left\{