# Development of a Magnetorheological Damper with Self-Powered Ability for Washing Machines
## 4 MR Damping Part
### Optimal Design of the MR Damping Part  
In order to improve the performance of the MR damping part during the operating process of washing machines, the maximum damping force (the damping force when a magnetic field is applied) determined by Equation (11) reaches to a required value at low resonance frequency whereas the zero-field friction force in Equation (12) is minimized at high excitation frequencies. Therefore, the dynamic model of washing machines should be first analyzed. The 2D mechanical modeling of the washing machine Samsung WF8690 NGW is shown in Figure 7.  
The vibration equation of the washing machine shown in Figure 7 can be represented [10] as  
\[m\ddot{u}+ci\dot{u}\big{[}\sin^{2}(\varphi+\beta_{2})+\sin^{2}(\varphi-\beta_{1 })\big{]}+ku\big{[}\sin^{2}(\varphi+\alpha_{1})+\sin^{2}(\varphi-\alpha_{2}) \big{]}=F_{u}(t) \tag{15}\]  
where \(k\) is the spring stiffness, \(c\) is the damping coefficient, \(m\) is the total mass of the tub assembly, \(\varphi\) is the angle from the \(x\) axis of an arbitrary direction \(u\) of the vibration analysis, and \(\Delta u\) is the displacement of the center of the tube in the \(u\) direction. The exciting force \(F_{u}\) due to an imbalanced mass in the \(u\) direction is defined by  
\[F_{u}=F_{u0}\cos(\omega t)=m_{u}\alpha^{2}R_{u}\cos(\omega t) \tag{16}\]  
where \(F_{u0}\) is the magnitude of the exciting force and \(R_{u}\) and \(m_{u}\) are the radius and mass from the rotary axis of the imbalanced mass, respectively. From Equation (15), the damped frequency \(\omega_{d}\), natural frequency \(\omega_{n}\), and damping ratio \(\xi\) of the suspended tub assembly are given by  
\[\omega_{d}=\omega_{n}\sqrt{1-\xi^{2}} \tag{17}\]  
\[\omega_{n}=\sqrt{\frac{k\big{[}\sin^{2}(\varphi+\alpha_{1})+\sin^{2}(\varphi- \alpha_{2})\big{]}}{m}} \tag{18}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Parameters** & \\ \hline \(\eta_{0}=0.1\) Pa-s & \(\tau_{y0}=15\) Pa & \(\alpha_{S\eta}=4.5\) T\({}^{-1}\) \\ \hline \(\eta_{\infty}=3.8\) Pa-s & \(\tau_{yoo}=40,000\) Pa & \(\alpha_{Sty}=2.9\) T\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Rheological properties of the magnetorheological (MR) fluid 132-DG. Figure 7: A 2D mechanical modeling of the washing machine. \[\xi=\frac{c\Big{[}\sin^{2}(\varphi+\beta_{2})+\sin^{2}(\varphi-\beta_{1})\Big{]}}{ 2\sqrt{mk\Big{[}\sin^{2}(\varphi+\alpha_{1})+\sin^{2}(\varphi-\alpha_{2})\Big{]}}} \tag{19}\]  
The resonance in all directions of vibration should be attenuated as much as possible. For this, by designing \(\alpha_{1}+\alpha_{2}=90^{\circ}\) and \(\beta_{1}+\beta_{2}=90^{\circ}\), Equations (15), (18) and (19) can be simplified to be independent of the angle \(\varphi\) as follows  
\[m\ddot{u}+c\dot{u}+ku=F_{u}(t) \tag{20}\]  
\[\omega_{n}=\sqrt{\frac{k}{m}};\ \xi=\frac{c}{2\sqrt{mk}} \tag{21}\]  
In this case, the force transmissibility \(T_{f}\) from the tub assembly to the cabinet is derived by  
\[T_{f}=\sqrt{\frac{1+\left(2\xi r_{\omega}\right)^{2}}{\left(1-r_{\omega}^{2} \right)^{2}+\left(2\xi r_{\omega}\right)^{2}}} \tag{22}\]  
where \(r_{\omega}=\omega/\omega_{n}\) is the frequency ratio. Figure 8 presents the relation between the force transmissibility and the excitation frequency under different damping ratios. From the figure, it is observed that the resonant peaks are nearly eliminated as the damping ratio \(\xi\) reaches to 0.7. Thus, it is not necessary to get the greater value of \(\xi\).