### Vibration control of washing machine using MR damper

The washing machine object of this work is a prototype based on the LG F1402FDS washer manufactured by LG Electronics (Seoul, South Korea). A three-dimensional (3D) schematic diagram of the washer is shown in Figure 1. It is characterized by a suspended tub (basket) to store the water for washing linked to the cabinet by two springs and two dampers. The rotor is directly connected with the drum which rotates against the tub while the stator is fixed on the back of the tub. When the drum is rotating, the unbalanced mass due to the eccentricity of laundry causes the vibration of the tub assembly. The vibration of the tub assembly is then transmitted to the cabinet and the bottom through the springs and the dampers. In Figure 2, a two-dimensional (2D) simplified schematic of the machine is depicted. From the figure, the following governing equation of the washing machine can be derived:

\mathit{m}\mathit{\xfc}+\mathit{c}\dot{\mathit{u}}\left[{sin}^{2}\left(\mathit{\phi}+{\mathit{\beta}}_{2}\right)+{sin}^{2}\left(\mathit{\phi}-{\mathit{\beta}}_{1}\right)\right]+\mathit{ku}\left[{sin}^{2}\left(\mathit{\phi}+{\mathit{\alpha}}_{1}\right)+{sin}^{2}\left(\mathit{\phi}-{\mathit{\alpha}}_{2}\right)\right]={\mathit{F}}_{\mathit{u}}\left(\mathit{t}\right)

(1)

where *m* is the mass of the suspended tub assembly including the drum, laundry, shaft, counter weight, rotor, and stator. For the prototype washing machine, *m* is roughly estimated about 40 kg. *c* is the damping coefficient of each damper, and *k* is the stiffness of each spring which is assumed to be 8 kN/m in this study. *φ* is the angle of an arbitrary direction (*u* direction) in which the vibration is considered. *F*_{
u
} is the excitation force due to unbalanced mass in the *u* direction, *F*_{
u
} *= F*_{0}cos*ωt = m*_{
u
}*ω*^{2}*R*_{
u
}cos*ωt*, in which *m*_{
u
} and *R*_{
u
} are the mass and radius from the rotation axis of the unbalanced mass. From Equation 1, the damped frequency of the suspended tub assembly is calculated by

{\mathit{\omega}}_{\mathit{d}}={\mathit{\omega}}_{\mathit{n}}\sqrt{1-{\mathit{\xi}}^{2}}

(2)

where {\mathit{\omega}}_{\mathit{n}}=\sqrt{\frac{\mathit{k}\left[{sin}^{2}\left(\mathit{\phi}+{\mathit{\alpha}}_{1}\right)+{sin}^{2}\left(\mathit{\phi}-{\mathit{\alpha}}_{2}\right)\right]}{\mathit{m}}} and \mathit{\xi}=\frac{\mathit{c}\left[{sin}^{2}\left(\mathit{\phi}+{\mathit{\beta}}_{2}\right)+{sin}^{2}\left(\mathit{\phi}-{\mathit{\beta}}_{1}\right)\right]}{2\sqrt{\mathit{mk}\left[{sin}^{2}\left(\mathit{\phi}+{\mathit{\alpha}}_{1}\right)+{sin}^{2}\left(\mathit{\phi}-{\mathit{\alpha}}_{2}\right)\right]}}.

It is seen that the damped frequency and natural frequency of the tub assembly in the *u* direction are a function of *φ*. Therefore, in general, in a different direction of vibration, the tub assembly exhibits different resonant frequency. This causes the vibration to become more severe and hard to control. In the design of the suspension system for the tub assembly, the frequency range of the resonance in all directions should be as small as possible. From the above, it is easy to show that, by choosing *α*_{1} + *α*_{2} = 90° and *β*_{2} *+ β*_{2} = 90°, Equation 1 can be simplified to yield

\mathit{m}\mathit{\xfc}+\mathit{c}\dot{\mathit{u}}+\mathit{ku}={\mathit{F}}_{\mathit{u}}

(3)

In this case, the damped frequency and natural frequency of the tub assembly do not depend on the direction of the vibration. We have

{\mathit{\omega}}_{\mathit{d}}={\mathit{\omega}}_{\mathit{n}}\sqrt{1-{\mathit{\xi}}^{2}}

(4)

where {\mathit{\omega}}_{\mathit{n}}=\sqrt{\frac{\mathit{k}}{\mathit{m}}} and \mathit{\xi}=\frac{\mathit{c}}{2\sqrt{\mathit{mk}}}.

An inherent drawback of the conventional damper is its high transmissibility of vibration at high excitation frequency. In order to solve this issue, semi-active suspension systems such as ER and MR dampers are potential candidates. In this study, two MR dampers are employed to control the vibration of the tub assembly.

Figure 3 shows the schematic configuration of a flow-mode MR damper proposed for the prototype washing machine. From the figure, it is observed that an MR valve structure is incorporated in the MR damper. The outer and inner pistons are combined to form the MR valve structure which divides the MR damper into two chambers: the upper and lower chambers. These chambers are fully filled with the MR fluid. As the piston moves, the MR fluid flows from one chamber to the other through the annular duct (orifice). The floating piston incorporated with a gas chamber functions as an accumulator to accommodate the piston shaft volume as it enters and leaves the fluid chamber.

By neglecting the frictional force and assuming quasi-static behavior of the damper, the damping force can be calculated by [7]

{\mathit{F}}_{\mathit{d}}={\mathit{P}}_{\mathit{a}}{\mathit{A}}_{\mathit{s}}+{\mathit{C}}_{\mathrm{vis}}{\dot{\mathit{x}}}_{\mathit{p}}+{\mathit{F}}_{\mathrm{MR}}sgn\left({\dot{\mathit{x}}}_{\mathit{p}}\right)

(5)

where *P*_{
a
}, *c*_{vis}, and *F*_{MR}, respectively, are the pressure in the gas chamber, the viscous coefficient, and the yield stress force of the MR damper which are determined as follows:

{\mathit{P}}_{\mathit{a}}={\mathit{P}}_{0}{\left(\frac{{\mathit{V}}_{0}}{{\mathit{V}}_{0}+{\mathit{A}}_{\mathit{s}}{\mathit{x}}_{\mathit{p}}}\right)}^{\mathit{\gamma}}

(6)

{\mathit{C}}_{\mathrm{vis}}=\frac{12\mathit{\eta L}}{\mathit{\pi}{\mathit{R}}_{\mathit{d}}{\mathit{t}}_{\mathit{d}}^{3}}{\left({\mathit{A}}_{\mathit{p}}-{\mathit{A}}_{\mathit{s}}\right)}^{2}

(7)

{\mathit{F}}_{\mathrm{MR}}=2\left({\mathit{A}}_{\mathit{p}}-{\mathit{A}}_{\mathit{s}}\right)\frac{2.85{\mathit{L}}_{\mathit{p}}}{{\mathit{t}}_{\mathit{d}}}{\mathit{\tau}}_{\mathit{y}}

(8)

In the above, *τ*_{
y
} is the induced yield stress of the MR fluid which is an unknown and can be estimated from magnetic analysis of the damper and *η* is the post-yield viscosity of the MR fluid which is assumed to be field independent. *R*_{
d
} is the average radius of the annular duct given by *R*_{
d
} = *R* − *t*_{
h
} − 0.5*t*_{
d
}. *L* and *R* are the overall length and outside radius of the MR valve, respectively. *t*_{
h
} is the valve housing thickness, *t*_{
d
} is the annular duct gap, and *L*_{
p
} is the magnetic pole length.

In this work, the commercial MR fluid (MRF132-DG) made by Lord Corporation (Cary, NC, USA) is used. The post-yield viscosity of the MR fluid is assumed to be independent on the applied magnetic field, *η* = 0.1 Pas. The induced yield stress of the MR fluid as a function of the applied magnetic field intensity (*H*) is shown in Figure 4. By applying the least squares curve fitting method, the yield stress of the MR fluid can be approximately expressed by

{\mathit{\tau}}_{\mathit{y}}={\mathit{C}}_{0}+{\mathit{C}}_{1}\mathit{H}+{\mathit{C}}_{2}{\mathit{H}}^{2}+{\mathit{C}}_{3}{\mathit{H}}^{3}

(9)

In Equation 9, the unit of the yield stress is kilopascal while that of the magnetic field intensity is kA/m. The coefficients *C*_{0}, *C*_{1}, *C*_{2}, and *C*_{3} are respectively identified as 0.044, 0.4769, −0.0016, and 1.8007E-6. In order to estimate the induced yield stress using Equation 9, first the magnetic field intensity across the active MRF duct must be calculated. In this study, the commercial FEM software, ANSYS, is used to analyze the magnetic problem of the proposed MR damper.

### Optimal design of the MR damper

In this study, the optimal design of the proposed MR damper is considered based on the quasi-static model of the MR damper and dynamic equation of the tub assembly developed in the ‘Vibration control of washing machine using MR damper’ section. From Figure 2 and Equation 3, the force transmissibility of the tub assembly to the cabinet can be obtained as follows:

\mathrm{TR}=\sqrt{\frac{1+{\left(2\mathit{\xi r}\right)}^{2}}{{\left(1-{\mathit{r}}^{2}\right)}^{2}+{\left(2\mathit{\xi r}\right)}^{2}}}

(10)

where *r* is the frequency ratio, *r = ω/ω*_{
n
}. The dependence of the force transmissibility on excitation frequency is presented in Figure 5. As shown from the figure, at low damping, the resonant transmissibility is relatively large, while the transmissibility at higher frequencies is quite low. As the damping is increased, the resonant peaks are attenuated, but vibration isolation is lost at high frequency. This illustrates the inherent trade-off between resonance control and high-frequency isolation associated with the design of passive suspension systems. It is also observed from the figure that when the damping ratio is 0.7 or greater, the resonant peaks are almost attenuated. Thus, the higher value of damping ratio is not necessary. It is noted that in Equation 5, the third term *F*_{MR} is much greater than the other and the behavior of the MR damper can is similar to that of a dry friction damper. By introducing an equivalent damping coefficient *C*_{eq} such that the work per cycle due to this equivalent damping coefficient equals that due to the yield stress damping force of the MR damper, the following equation holds [8]:

{\mathit{C}}_{\mathrm{eq}}=\frac{4\left|{\mathit{F}}_{\mathrm{MR}}\right|}{\mathit{X\omega \pi}}

(11)

or

\left|{\mathit{F}}_{\mathrm{MR}}\right|=\frac{\mathit{X\omega \pi}{\mathit{C}}_{\mathrm{eq}}}{4}=\frac{\mathit{X\omega \pi \xi}\sqrt{\mathit{km}}}{2}=\frac{\mathit{kX\pi \xi r}}{2}

(12)

In the above, *X* is the magnitude of the tub vibration which is determined by

\mathit{X}=\frac{{\mathit{F}}_{0}}{\mathit{k}}\sqrt{\frac{1}{{\left(1-{\mathit{r}}^{2}\right)}^{2}+{\left(2\mathit{\xi r}\right)}^{2}}}=\frac{{\mathit{m}}_{\mathit{u}}{\mathit{r}}^{2}{\mathit{R}}_{\mathit{u}}}{\mathit{m}}\sqrt{\frac{1}{{\left(1-{\mathit{r}}^{2}\right)}^{2}+{\left(2\mathit{\xi r}\right)}^{2}}}

(13)

From Equations 12 and 13, the required value of *F*_{MR} can be determined from a required value of the damping ratio *ξ* as follows:

\left|{\mathit{F}}_{\mathrm{MR}}\right|=\frac{\mathit{k\pi \xi}{\mathit{m}}_{\mathit{u}}{\mathit{r}}^{3}{\mathit{R}}_{\mathit{u}}}{2\mathit{m}}\sqrt{\frac{1}{{\left(1-{\mathit{r}}^{2}\right)}^{2}+{\left(2\mathit{\xi r}\right)}^{2}}}

(14)

In this study, it is assumed that the spring stiffness is *k* = 10 kN/m, the mass of the suspended tub assembly is *m* = 40 kg, and the equivalent unbalanced mass is *m*_{
u
} = 10 kg located at the radius *R*_{
u
} = 0.15 m. With the required damping ratio *ξ* = 0.7, at the resonance \mathit{r}=\sqrt{1-{\mathit{\xi}}^{2}}, the required value of *F*_{MR} can be calculated from Equation 14 which is around 150 N in this study.

Taking all above into consideration, the optimal design of the MR brake for the washing machine can be summarized as follows: Find the optimal value of significant dimensions of the MR damper such as the pole length *L*_{
p
}, the housing thickness *t*_{
h
}, the core radius *R*_{
c
}, the width of the MR duct *t*_{
d
}, the width of the coil *W*_{
c
}, and the overall length of the valve structure *L* that

\mathrm{Minimize}\phantom{\rule{0.25em}{0ex}}\mathrm{the}\phantom{\rule{0.25em}{0ex}}\mathrm{viscous}\phantom{\rule{0.25em}{0ex}}\mathrm{coefficient}\phantom{\rule{0.25em}{0ex}}\left(\mathrm{objective}\phantom{\rule{0.25em}{0ex}}\mathrm{function}\right),\mathrm{OBJ}={\mathit{c}}_{\mathrm{vis}}=\frac{12\mathit{\eta L}}{\mathit{\pi}{\mathit{R}}_{\mathit{d}}{\mathit{t}}_{\mathit{d}}^{3}}{\left({\mathit{A}}_{\mathit{p}}-{\mathit{A}}_{\mathit{s}}\right)}^{2}

\mathrm{Subjected}\phantom{\rule{0.25em}{0ex}}\mathrm{to}:{\mathit{F}}_{\mathrm{MR}}=2\left({\mathit{A}}_{\mathit{p}}-{\mathit{A}}_{\mathit{s}}\right)\frac{2.85{\mathit{L}}_{\mathit{p}}}{{\mathit{t}}_{\mathit{d}}}{\mathit{\tau}}_{\mathit{y}}\ge 150\phantom{\rule{0.25em}{0ex}}\mathrm{N}

In order to obtain the optimal solution, a finite element analysis code integrated with an optimization tool is employed. In this study, the first-order method with the golden section algorithm of the ANSYS optimization tool is used. Figure 6 shows the flow chart to achieve optimal design parameters of the MR damper. Firstly, an analysis ANSYS file for solving the magnetic circuit of the damper and calculating the objective function is built using ANSYS parametric design language (APDL). In the analysis file, the design variables (DVs) such as the pole length *L*_{
p
}, the housing thickness *t*_{
h
}, the core radius *R*_{
c
}, the width of the MR duct *t*_{
d
}, the width of the coil *W*_{
c
}, and the overall length of the valve structure *L* must be coded as variables and initial values are assigned to them. The geometric dimensions of the valve structure are varied during the optimization process; the meshing size therefore should be specified by the number of elements per line rather than the element size. Because the magnetic field intensity is not constant along the pole length, it is necessary to define paths along the MR active volume where magnetic flux passes. The average magnetic field intensity across the MR ducts (*H*_{mr}) is calculated by integrating the field intensity along the defined path then divided by the path length. Thus, the magnetic field intensity is determined as follows:

\phantom{\rule{0.25em}{0ex}}{\mathit{H}}_{\mathrm{mr}}=\frac{1}{{\mathit{L}}_{\mathit{p}}}{\displaystyle \underset{\phantom{\rule{0.25em}{0ex}}0}{\overset{\phantom{\rule{0.5em}{0ex}}{\mathit{L}}_{\mathit{p}}}{\int}}\mathit{H}\left(\mathit{s}\right)\mathit{ds}}

(15)

where *H*(*s*) is the magnetic flux density and magnetic field intensity at each nodal point on the defined path.

From the figure, it is observed that the optimization is started with the initial value of DVs. By executing the analysis file, first the magnetic field intensity is derived. Then the yield stress, yield stress damping force, and objective function are respectively calculated from Equations 9, 7, and 8. The ANSYS optimization tool then transforms the optimization problem with constrained design variables to an unconstrained one via penalty functions. The dimensionless, unconstrained objective function *f* is formulated as follows:

\mathit{f}\left(\mathit{x}\right)=\frac{\mathrm{OBJ}}{{\mathrm{OBJ}}_{0}}+{\displaystyle \sum _{\mathit{i}=1}^{\mathit{n}}{\mathit{P}}_{{\mathit{x}}_{\mathit{i}}}\left({\mathit{x}}_{\mathit{i}}\right)}

(16)

where OBJ_{0} is the reference objective function value that is selected from the current group of design sets. {\mathit{P}}_{{\mathit{x}}_{\mathit{i}}} is the exterior penalty function for the design variable *x*_{
i
}. For the initial iteration (*j* = 0), the search direction of DVs is assumed to be the negative of the gradient of the unconstrained objective function. Thus, the direction vector is calculated by

{\mathit{d}}^{\left(0\right)}=-\nabla \mathit{f}\left({\mathit{x}}^{\left(0\right)}\right)

(17)

The values of DVs in the next iteration (*j + 1*) is obtained from the following equation:

{\mathit{x}}^{\left(\mathit{j}+1\right)}={\mathit{x}}^{\left(\mathit{j}\right)}+{\mathit{s}}_{\mathit{j}}{\mathit{d}}^{\left(\mathit{j}\right)}

(18)

where the line search parameter *s*_{
j
} is calculated by using a combination of the golden section algorithm and a local quadratic fitting technique. The analysis file is then executed with the new values of DVs, and the convergence of the objective function is checked. If the convergence occurs, the values of DVs at this iteration are the optimum. If not, the subsequent iterations will be performed. In the subsequent iterations, the procedures are similar to those of the initial iteration except for that the direction vectors are calculated according to the Polak-Ribiere recursion formula as follows:

{\mathit{d}}^{\left(\mathit{j}\right)}=-\nabla \mathit{f}\left({\mathit{x}}^{\left(\mathit{j}\right)}\right)+{\mathit{r}}_{\mathit{j}-1}{\mathit{d}}^{\left(\mathit{j}-1\right)}

(19)

\mathrm{where}\phantom{\rule{0.25em}{0ex}}{\mathit{r}}_{\mathit{j}-1}=\frac{{\left[\nabla \mathit{f}\left({\mathit{x}}^{\left(\mathit{j}\right)}\right)-\nabla \mathit{f}\left({\mathit{x}}^{\left(\mathit{j}-1\right)}\right)\right]}^{\mathit{T}}\nabla \mathit{f}\left({\mathit{x}}^{\left(\mathit{j}\right)}\right)}{{\left|\nabla \mathit{f}\left({\mathit{x}}^{\left(\mathit{j}-1\right)}\right)\right|}^{2}}.

(20)