# Development of a Magnetorheological Damper with Self-Powered Ability for Washing Machines
## 3 Energy-Harvesting Part
### Modeling of the Energy-Harvesting Part  
Figure 3 shows the schematic of the EH part. The pole spacers and stator core are made of commercial C45 steel, which is a common material with high magnetic permeability. The NdFeB grade N35 ring-shaped magnets are fitted on an aluminum shaft end alternately with the pole spacers. The magnet has an outer diameter of 28 mm, an inner diameter of 6 mm, and a length of 7 mm. In our research, the prototype front-loaded washing machine is the WF8690 NGW washing machine produced by Samsung Electronics Co., Ltd., Seoul, Korea. Considering the available assembly space of dampers in the washing machine, the number of magnets and coils on the stator core are chosen to be 2 and 7, respectively. In this way, the maximum number of coils in the electromagnetic working state will be 4 out of 7 coils.  
By neglecting the reluctance of pole spacers and stator core, the magnetic flux of the cylindrical air gap between the magnets and the stator core \(\varPhi_{g}\) is given as [17; 18; 19; 20]  
\[\varPhi_{g}=\lambda\frac{B_{rem}l_{m}l_{0}H_{core}A_{gm}}{2t_{gm}B_{rem}+l_{m}l_{ 0}H_{core}\big{(}A_{gm}/A_{m}\big{)}} \tag{1}\]  
where \(\lambda\) expresses the efficiency of the magnetic flux considering leakage effect, \(B_{rem}\) characterizes the magnet remanent flux density, \(H_{core}\) denotes the magnetic coercivity of the magnet, \(\mu_{0}\) represents the relative magnetic permeability and is equal to \(4\pi\times 10^{-7}\) N/A\({}^{2}\), \(l_{m}\) is the length of the magnet, \(t_{gm}\) is the thickness of the air gap, \(A_{gm}\) is the cylindrical surface area of the air gap, and \(A_{m}\) is the sectional area of the magnet. \(A_{gm}\) and \(A_{m}\) can be determined by  
\[A_{gm}=2\pi\bigg{(}r_{o}+\frac{t_{gm}}{2}\bigg{)}\bigg{(}\frac{p_{m}-l_{m}}{2} \bigg{)} \tag{2}\]  
\[A_{m}=\pi\Big{(}r_{o}^{2}-r_{i}^{2}\Big{)} \tag{3}\]  
where \(r_{i}\) and \(r_{o}\) are the inner and outer radii of magnet, respectively, and \(p_{m}\) is the pitch of the pole pair. The induced voltage \(E\) in the inducing coil is defined as  
\[E=-N\varPhi_{g}\frac{\pi}{p_{m}}\sin\bigg{(}\frac{\pi}{p_{m}}x+\varphi_{0} \bigg{)}\frac{dx}{dt} \tag{4}\]  
where \(x\) and \(\frac{dx}{dt}\) are the displacement and velocity of the shaft, respectively, and \(\varphi_{0}\) is the initial phase angle of the inducing coil. The number of winding turns of each inducing coil \(N\) is calculated by [17]  
\[N=\frac{2A_{c}}{\sqrt{3}d_{w}^{2}} \tag{5}\]  
where \(d_{w}\) is the diameter of the copper wire and \(A_{c}\) is the cross-sectional area of the coil determined by the product of the coil height \(h_{cm}\) and the coil width \(w_{cm}\). Since the pitch of the pole pair \(p_{m}\) is designed to double the pitch of the coil slot \(p_{cm}\), the phase angle is \(\pi\)/2 between each nearby coil. Accordingly, the induced voltages in four active coils are  
\[E_{1}=-N\varPhi_{g}\frac{\pi}{p_{m}}\sin\bigg{(}\frac{\pi}{p_{m}}x\bigg{)} \frac{dx}{dt} \tag{6}\]  
Figure 3: Schematic of the energy-harvesting part.  
\[E_{3} =-E_{1} \tag{8}\] \[E_{4} =-E_{2} \tag{9}\]  
In case of low excitation frequencies, by neglecting the coil inductance, the coil is supposed to work as a resistor. The power of four active coils \(P\) is obtained by  
\[P=2(P_{1}+P_{2})=2\left(\frac{E_{1}^{2}}{R_{c}}+\frac{E_{2}^{2}}{R_{c}}\right)= 2\frac{\left(N\Phi_{g}\frac{\pi}{p_{m}}\right)^{2}}{R_{c}}v^{2} \tag{10}\]  
where \(P_{1}\) and \(P_{2}\) are the powers of coils 1 and 2, respectively; \(R_{c}\) is the resistance of each coil; and \(v\) is the velocity of the shaft. Equation (10) describes the generated power of four active coils as a quadratic function of the excitation velocity. From the equation, it is also observed that geometric dimensions of the stator core and the pole spacers considerably affect the power magnitude. Consequently, to achieve the best performance of power generation, an optimization procedure for the design of the EH part should be implemented.