# Development of a Magnetorheological Damper with Self-Powered Ability for Washing Machines
## 4 MR Damping Part  
### Modeling of the MR Damping Part  
The MR damping part is coaxial with and in front of the EH part. Figure 6 shows the schematic diagram of the shear-mode MR damping part. The shaft and the housing are made of C45 steel. The O-rings made of 70-durometer NBR rubber are used to seal the MR fluid. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter (mm)** & **Value** & **Parameter (mm)** & **Value** \\ \hline
**Coil height \(h_{cm}\)** & 4.4 & Air gap thickness \(t_{gm}\) & 0.8 \\
**Coil width \(w_{cm}\)** & 4.56 & Thin wall thickness \(t_{mm}\) & 0.8 \\
**Coil pitch \(p_{c}\)** & 6.74 & Housing thickness \(t_{gm}\) & 2 \\
**Magnet length \(I_{m}\)** & 7 & Outer radius \(R\) & 22 \\
**Pole pair pitch \(p_{m}\)** & 13.48 & Generated power \(P\) (W) & 19.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal parameters of the energy-harvesting part. Figure 4: Finite element model of the energy-harvesting part. Figure 5: Finite element analysis of the optimized energy-harvesting part: (**a**) magnetic flux lines and (**b**) magnetic flux density. In the design of MR damper-based devices, the damping forces should be first established as functions of the applied magnetic field and geometric dimensions. For the proposed MR damping part illustrated in Figure 6, it is supposed that if the MR fluid velocity in the gap between the housing and the shaft is linear, then the generated damping force \(F_{d}\) and zero-field friction force \(F_{0}\) are defined by  
\[F_{d}=2\pi r_{s}L_{e}(\tau_{y}+\eta_{t}\frac{v}{t_{g}})+2F_{or} \tag{11}\]  
\[F_{0}=2\pi r_{s}L(\tau_{y0}+\eta_{0}\frac{v}{t_{g}})+2F_{or} \tag{12}\]  
where \(R_{s}\) is the shaft radius; \(t_{g}\) is the MR fluid gap thickness; \(v\) is the velocity between the housing and the shaft; \(\tau_{y}\) and \(\eta\) are the yield stress and post-yield viscosity of the active MR fluid in the gap, respectively; \(L\) is the MR fluid gap length; and \(L_{e}\) is the effective length of the active MR fluid in the gap. In this configuration, \(L_{e}\approx L\). The Coulomb friction force \(F_{or}\) between the O-ring and the shaft can be determined by [23]  
\[F_{or}=f_{c}L_{r}+f_{h}A_{r} \tag{13}\]  
where \(L_{r}\) is the seal surface length and in this case, equals to the circumference of the shaft;\(f_{c}\) is the friction per unit length caused by the compression of the O-ring; \(A_{r}\) is the projected area of the seal; and \(f_{h}\) is the friction force caused by the fluid pressure on a unit projected area of the O-ring. It is noted that in the configuration of the shear-mode, the pressure on the O-rings could be ignored since it is almost negligible, \(f_{h}\approx 0\). Moreover, the O-rings should be compressed moderately so that the zero-field friction force is not so high while the sealing ability is still guaranteed during the working process of washing machines. Accordingly, a compression of 15% is set for the O-rings and \(f_{c}\) is found to be 175.1 N/m. For our research, the 132-DG MR fluid produced by Lord Corporation is employed, and the Bingham model is used to represent the general behavior of the MR fluid. Despite the inconsistency with experimental responses at low shear rate, the Bingham model is beneficial to the design of MR damper-based devices due to its simplicity and quick modeling. Based on this model, the dependence of MR fluid rheological properties on the applied magnetic field are given by [24]  
\[Y=Y_{\infty}+(Y_{0}-Y_{\infty})(2e^{-Ba_{SY}}-e^{-2 Ba_{SY}}) \tag{14}\]  
where \(B\) is the applied magnetic density, \(Y\) expresses one of the MR fluid rheological properties such as the postyield viscosity \(\eta\) and the yield stress \(\tau_{y}\), and \(\alpha_{SY}\) is the saturation moment of the \(Y\) property. The \(Y\) value varies from the value of off-state \(Y_{0}\) to the value of saturation \(Y_{\infty}\).