# Vibration Analysis of Piezoelectric Cantilever Beams with Bimodular Functionally-Graded Properties
## 3 Equivalent Modulus of Elasticity and Analytical Solution  
For a relatively shallow beam, the stress and strain along \(x\) direction, \(\sigma_{x}^{+/-}\) and \(\varepsilon_{x}^{+/-}\), are dominant, while other stresses and strains along \(z\) direction, \(\sigma_{z}^{+/-}\) and \(\tau_{zx}^{+/-}\) as well as \(\varepsilon_{z}^{+/-}\) and \(\gamma_{zx}^{+/-}\), are less important. Thus, the constitutive relation of FGPM with bimodular effect, that is, Equations (2) and (3), may be further simplified as,  
\[\left\{ \begin{array}{l}\varepsilon_{x}^{+/-}=s_{11}^{+/-}\sigma_{x}^{+/-}+d_ {31}^{+/-}E_{z}^{+/-}\\ \varepsilon_{z}^{+/-}=0\\ \gamma_{zx}^{+/-}=0\end{array} \right. \tag{4}\]  
and  
\[\left\{ \begin{array}{l}D_{x}^{+/-}=\lambda_{11}^{+/-}E_{x}^{+/-}\\ D_{z}^{+/-}=d_{31}^{+/-}\sigma_{x}^{+/-}+\lambda_{33}^{+/-}E_{z}^{+/-}\end{array} \.\right. \tag{5}\]  
In existing studies for the two-dimensional problem [27; 29], \(D_{x}>>D_{z}\) may be found; thus, it may be assumed that \(D_{z}\approx 0\) in a one-dimensional problem, especially if a long and shallow beam is considered here. From the second one of Equation (5), we have  
\[E_{z}^{+/-}=-\frac{d_{31}^{+/-}}{\lambda_{33}^{+/-}}\sigma_{x}^{+/-} \tag{6}\]Plugging Equation (6) into the first one of Equation (4) yields  
\[e_{x}^{+/-}=\left[\frac{s_{11}^{+/-}\lambda_{33}^{+/-}-\left(d_{31}^{+/-}\right)^ {2}}{\lambda_{33}^{+/-}}\right]\sigma_{x}^{+/-}=\frac{\sigma_{x}^{+/-}}{E^{*}}, \tag{7}\]  
in which \(E^{*}\) represents the equivalent modulus of elasticity, that is  
\[E^{*}=\frac{\lambda_{33}^{+/-}}{s_{11}^{+/-}\lambda_{33}^{+/-}-\left(d_{31}^{+ /-}\right)^{2}}. \tag{8}\]  
We note that Equation (8) may be rewritten as the form \(E^{*}=\left[s_{11}^{+/-}-\left(d_{31}^{+/-}\right)^{2}/\lambda_{33}^{+/-} \right]^{-1}\), clearly showing the piezoelectric effect on the elastic modulus. Then, \(E^{*}=1/s_{11}^{+/-}\) is obtained when \(s_{11}^{+/-}>>\left(d_{31}^{+/-}\right)^{2}/\lambda_{33}^{+/-}\); this, exactly, stands for the reciprocal relation of stiffness coefficient and flexibility coefficient. Meanwhile, the existence of the term \(\left(d_{31}^{+/-}\right)^{2}/\lambda_{33}^{+/-}\) also reveals the well-known piezoelectric stiffening effect. It is this term that the resulting equivalent modulus becomes larger than \(1/s_{11}^{+/-}\), thus stiffening the mechanical performance of piezoelectric materials and structures under external loading. Now, the vibration equation of free damping of the bimodular FGPM cantilever beam may be easily obtained, by only replacing \(E\) in a classical equation [30] by \(E^{*}\),  
\[E^{*}I_{y}\frac{\partial^{4}v(x,t)}{\partial x^{4}}+\overline{m}\frac{ \partial^{2}v(x,t)}{\partial t^{2}}+c\frac{\partial v(x,t)}{\partial t}=0. \tag{9}\]  
in which \(\overline{m}\) is the uniformly-distributed mass, \(v(x,t)\) is the displacement along \(z\) direction, \(t\) is the time variable, \(E^{*}I_{y}\) is the equivalent bending stiffness of the beam, \(I_{y}\) is the moment of inertia with respect to \(y\) axis, \(c\) is the damping parameter, and \(c=2\xi\overline{m}\omega\), in which, \(\xi\) is the damping ratio and \(\omega\) is the undamped frequency. Equation (9) may be solved under the following boundary conditions:  
\[v(x,t)=0\text{ and }\frac{\partial v(x,t)}{\partial x}=0,\text{ at }x=0 \tag{10}\]  
\[EI_{y}\frac{\partial^{2}v(x,t)}{\partial x^{2}}=0\text{ and }EI_{y}\frac{ \partial^{3}v(x,t)}{\partial x^{3}}=0,\text{ at }x=l. \tag{11}\]  
and the conditions of initial values  
\[v(x,t)=v_{0}\text{ and }\frac{\partial v(x,t)}{\partial t}=0,\text{ at }x=l,t=0 \tag{12}\]  
The variable separation is first needed to solve the Equation (9); suppose that  
\[v(x,t)=\phi(x)Y(t) \tag{13}\]  
Plugging Equation (13) into Equation (9), we may obtain the following two equations:  
\[\frac{d^{4}\phi(x)}{dt^{4}}-a^{4}\phi(x)=0 \tag{14}\]  
and  
\[\frac{d^{2}Y(t)}{dt^{2}}+\frac{c}{\overline{m}}\frac{dY(t)}{dt}+\omega^{2}Y(t)=0 \tag{15}\]  
in which \(a\) is an unknown constant: \(\omega^{2}=a^{4}E^{*}I_{y}/\overline{m}\).