# Shunt Damping Vibration Control Technology: A Review
## 2 Fundamentals of the Piezoelectric and Electromagnetic Transducers
### Governing Equations of the Coupled System  
For a single degree of freedom system, a discrete multimode degree of freedom system and a continuous system, it is easy to obtain the corresponding governing equation according to vibration theories [62; 63; 64]. The dynamic equations of the discrete systems are easy to analyze. However, methods for solving continuous structures are complicated. Some simple structures, such as an Euler-Bernoulli beam, a Kirchhoff plate and a shell, the governing equations can be derived if the boundary conditions are simple. In many cases, engineering structures are complicated and it is hard to obtain the theoretical model. Finite element method (FEM) can provide a feasible solution to obtain their modal characteristics. Either with the theoretical model or the finite element model, the system  
Figure 3: Equivalent electrical model of the electromagnetic transducer. can be reduced to several single mode vibrations. Then the response of the system can be approximated using the modal superposition method. The process can be found in many professional books. Because the shunted electromagnetic transducer adds a control force to the host structure, the governing equation can be obtained by combining the electrical equation of the circuit with the mechanical equation [28]. The governing equation presented by [PERSON] et al. [65] and [PERSON] et al. [66] can be used,  
\[\left[ \begin{array}{cc}\mathbf{M}+\mathbf{M}_{p}&\mathbf{0}\\ \mathbf{0}&\mathbf{L}\end{array} \right]\left[ \begin{array}{c}\ddot{\mathbf{q }}\\ \ddot{\mathbf{Q}}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{C}+ \mathbf{C}_{s}&\mathbf{0}\\ \mathbf{0}&\mathbf{R}\end{array} \right]\left[ \begin{array}{c}\dot{\mathbf{q }}\\ \dot{\mathbf{Q}}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{K}+\mathbf{ K}_{p}&\mathbf{\Theta}\\ \mathbf{0}&\mathbf{C}_{p}^{-1}\end{array} \right]\left[ \begin{array}{c}\mathbf{ q}\\ \mathbf{Q}\end{array} \right]=\left[ \begin{array}{c}\mathbf{F}\\ \mathbf{v}_{P}\end{array} \right] \tag{11}\]  
where the \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) denote the mass, damping and stiffness matrix of the host structure, respectively. \(\mathbf{M}_{p}\), \(\mathbf{C}_{s}\) and \(\mathbf{K}_{p}\) are the mass, damping and stiffness of piezoelectric transducers. \(\mathbf{F}\) is the external force. \(\mathbf{\Theta}\) is the electromechanical coupling matrix. The \(\mathbf{q}\) and \(\mathbf{Q}\) are the displacement vector of the host structure and the electrical charge vector. \(\mathbf{C}_{p}\) is the inherent capacitance vector. \(\mathbf{L}\) and \(\mathbf{R}\) are the total inductance and resistance vector of the electrical circuit. The governing equation of EMSD can be obtained by combing the structural and electrical equation together [34],  
\[\left[ \begin{array}{cc}\mathbf{M}+\mathbf{M}_{E}&\mathbf{0}\\ \mathbf{0}&\mathbf{L}\end{array} \right]\left[ \begin{array}{c}\ddot{\mathbf{ q}}\\ \ddot{\mathbf{Q}}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{C}+ \mathbf{C}_{E}&\mathbf{0}\\ \mathbf{0}&\mathbf{R}\end{array} \right]\left[ \begin{array}{c}\dot{\mathbf{ q}}\\ \dot{\mathbf{Q}}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{K}&\mathbf{C}_{m }\\ \mathbf{0}&\mathbf{C}_{p}^{-1}\end{array} \right]\left[ \begin{array}{c}\mathbf{ q}\\ \mathbf{Q}\end{array} \right]=\left[ \begin{array}{c}\mathbf{F}\\ \mathbf{V}_{e}\end{array} \right] \tag{12}\]  
where \(\mathbf{M}_{E}\) and \(\mathbf{C}_{E}\) are the additional mass of the electromagnetic transducer and the electromagnetic shunt damping coefficient. \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are same as that in (11). Then (11) and (12) can be unified as  
\[\left[ \begin{array}{cc}\mathbf{M}+\mathbf{M}_{E}&\mathbf{0}\\ \mathbf{0}&\mathbf{L}\end{array} \right]\left[ \begin{array}{c}\ddot{\mathbf{ q}}\\ \ddot{\mathbf{q}}_{E}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{C}+ \mathbf{C}_{E}&\mathbf{0}\\ \mathbf{0}&\mathbf{R}\end{array} \right]\left[ \begin{array}{c}\dot{\mathbf{ q}}\\ \dot{\mathbf{q}}_{E}\end{array}