# Modeling the Response of Magnetorheological Fluid Dampers under Seismic Conditions
## 2 Mathematical Model
### Magneto-Hydrodynamic Model  
The magneto-hydrodynamic (MHD) model simulates the interaction between an electrically conducting fluid flow and the applied electromagnetic field. There are two main aspects of the coupling between the fluid flow field and the magnetic field: electromagnetic induction and Lorentz force. Electromagnetic induction is the effect of electric current induction as a result of the movement of conductive material in a magnetic field. Lorentz force is generated as a result between the interaction of the electric current and the magnetic field. In MR fluid dampers, the macroscopic physics can be understood as occurring due to a Lorentz force. As illustrated in Figure 3, when the electromagnetic coil is in the on-state, a magnetic field \(B\) is generated in the horizontal direction. When the damper is actuated, the MR fluid experiences a changing magnetic field. According to [PERSON]'s law of induction, an electromotive force (EMF) is induced and a current will flow around the annular orifice with current density \(j\). The interaction between the current density and magnetic field creates a force that opposes the motion of the fluid [45]. This force is called a Lorentz Force and is equal to the cross product of the current density and magnetic field \(j\times B\). It is this interaction of the fluid and magnetic field that creates a damping force and will result in a higher pressure differential across the annular orifice. Figure 3: Schematic of the magnetohydrodynamics model for a magnetorheological fluid damper. In the MHD model, [PERSON]'s equations in the following form are used to describe the electromagnetic fields:  
\[\
abla\cdot\mathbf{B}=0 \tag{7}\]  
\[\
abla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t} \tag{8}\]  
where \(\mathbf{B}\) is the magnetic field in T, and \(\mathbf{E}\) is the electric field in V/m. In addition:  
\[\
abla\times\mathbf{H}=\mathbf{j}+\frac{\partial\mathbf{j}}{\partial t} \tag{9}\]  
\[\
abla\cdot\mathbf{D}=q \tag{10}\]  
where \(\mathbf{H}\) and \(\mathbf{D}\) are the induction fields for the magnetic and electric fields, respectively. \(q\) is the electric charge density in C/m\({}^{3}\), and \(\mathbf{j}\) is the electric current density vector in A/m\({}^{2}\). For media with sufficiently high conductivity, the electric charge density \(q\) and the displacement current \(\frac{\partial\mathbf{D}}{\partial t}\) are considered negligible. The induction fields \(\mathbf{H}\) and \(\mathbf{D}\) can be determined by:  
\[\mathbf{H}=\frac{1}{\mu}\mathbf{B} \tag{11}\]  
\[\mathbf{D}=\varepsilon\mathbf{E} \tag{12}\]  
where \(\mu\) is the magnetic permeability and \(\varepsilon\) is the electric permittivity. The electric current density vector \(\mathbf{j}\) may be determined using two different approaches, namely the magnetic induction method and the electric potential method. The magnetic induction method is derived from [PERSON]'s law and the [PERSON]'s equation, whereas the electric potential method solves the electric potential equation to determine the current density using [PERSON]'s law. In this study, the electric potential method is used. The current density is defined by [PERSON]'s law using:  
\[\mathbf{j}=\sigma\mathbf{E} \tag{13}\]  
where \(\sigma\) is the electrical conductivity of the medium. In a flow under an external magnetic field \(\mathbf{B}\), [PERSON]'s law can be expressed as:  
\[\mathbf{j}=\sigma(\mathbf{E}+\mathbf{U}\times\mathbf{B}) \tag{14}\]  
where \(\mathbf{U}\) is the velocity of the fluid flow. In the electric potential approach, the electric field \(\mathbf{E}\) is given as:  
\[\mathbf{E}=-\
abla\varphi-\frac{\partial\mathbf{A}}{\partial t} \tag{15}\]  
where \(\varphi\) is the scalar potential and \(\mathbf{A}\) is the vector potential.