# Shunt Damping Vibration Control Technology: A Review
## 2 Fundamentals of the Piezoelectric and Electromagnetic Transducers
### Piezoelectric Transducer  
Since [PERSON] [PERSON] found the piezoelectric effect in 1880, piezoelectric ceramics has been widely applied to engineering structure. The phenomenon where a volume material undergoes shape transformation when exposed to an electric field is known as the piezoelectric effect. Conversely, the piezoelectric materials generate force or displacement output as the external charge is applied, which is the inverse piezoelectric effect. Therefore, piezoelectric ceramics is used as a sensor and an actuator in adaptive structures [42].  
The electro-mechanical coupling of a piezoelectric transducer is [43,44],  
\[\begin{split}\varepsilon^{i}&=S_{ij}^{E}\sigma_{j }+d_{mi}E_{m}\\ D_{m}&=d_{mi}\sigma_{i}+\varsigma_{mk}E_{k}\end{split} \tag{1}\]  
where the indices \(i,j=1,2,\ldots,6\) and \(m,k=1,2,3\) refer to different directions within the material coordinate system. \(\varepsilon,\sigma,D\) and \(E\) are the strain, stress, electrical displacement (charge per unit area) and the electric field, respectively. \(S^{E},d\) and \(\varsigma\) denote the elastic compliance, piezoelectric strain constant, and material permittivity, respectively.  
The piezoelectric strain constant \(d_{mi}\) is defined as the free strain developed in the \(i^{th}\) direction per unit (going from 1 to 6) of applied electric field in the \(m^{th}\) direction (going from 1 to 3). Assuming that a piezoelectric transducer is polarized along the z direction, the z directional strain is [45],  
\[\varepsilon^{3}=d_{33}\frac{V}{h} \tag{2}\]  
where \(V\) and \(h\) are the external voltage and the thickness of the piezoelectric ceramics. By definition, a voltage of the same polarity as the poling orientation leads to a positive strain or elongation. Elongation in the z direction implies a contraction in the \(x\) and \(y\) directions, thus \(d_{31}\) and \(d_{32}\) are opposite in sign to \(d_{33}\). The developed \(x\) and \(y\) directional strain are as \[\begin{array}{c}\varepsilon^{1}=d_{31}\frac{V}{\hbar}\\ \varepsilon^{2}=d_{32}\frac{V}{\hbar}\end{array} \tag{3}\]  
In electrical field, we have \(D_{m}=e_{mi}S_{i}+\zeta_{mk}E_{k}\). Solving it for the electric field (the index subscripts have been dropped because this is now a scalar equation) results in \(E=D/\varsigma-eS/\varsigma\). Multiplying by the thickness \(t\) of the piezoelectric results in an expression for the voltage across the piezoelectric material  
\[v=Et=\frac{Dt}{\varsigma}-\frac{eSt}{\varsigma} \tag{4}\]  
\(v\) is the voltage across the piezoelectric element. Define \(q^{c}/A=D\) as the free charge area on the piezoelectric element, and \(q_{p}/A=eS\) the piezoelectric polarization charge per electrode area. The voltage can be written as [46]  
\[v=\frac{q_{c}}{C^{s}}-\frac{q_{p}}{C^{s}} \tag{5}\]  
where \(C^{s}=\zeta A/t\). The charge \(q_{p}\) is proportional to the strain of the piezoelectric transducer and the charge \(q_{c}\) is the applied or free charge. There is no charge applied on the piezoelectric transducer as to the shunt damping technology. Hence, the electrical circuit model is shown in Figure 2. The piezoelectric transducers is equivalent to a capacitor \(C_{p}\) with a voltage source \(v_{p}\) as the structure vibrates [47].