A control system for MR damper-based suspension of front-loaded washing machines featuring magnetic induction coils and phase-lead compensator

Abstract

This paper studies a semi-active control system for magneto-rheological (MR) damper-based suspension of front-loaded washing machines. In this system, a permanent magnet is fastened to the shaft end of a shear-mode MR damper and an induction coil is wound directly on the slots of damper housing. Vibration of the washing machine tube leads to the relative reciprocal movement between the magnet and induction coil, which results in an induction voltage. The induced voltage is then served as an input signal to control the damping force of the MR damper. Because of the MR fluid hysteresis, a phase-lead compensator and an amplifier are employed in the controller of the MR damper. After the controller design, simulations and experiments are performed to evaluate the control system effectiveness. From the experimental results, performance characteristics of the proposed control system are discussed and compared with those of constant current and uncontrolled states.

Keywords

MR damper semi-active suspension washing machine induction coil phase-lead compensator

1. Introduction

MR fluid belongs to a family of smart materials that possesses a fast, adjustable and reversible transition from free-moving to semi-solid states when subjected to an applied magnetic field, thereby experiences increases in its rheological characteristics such as yield stress and viscosity. Thus, MR fluid is increasingly investigated in semi-active application systems to take its great intrinsic advantage of continuous controllability in a wide dynamic range. MR fluid-based devices can be found in various industrial fields such as structural protectors (Abdeddaim et al., 2022; Bagherkhani and Baghlani, 2021; Rayegani and Nouri, 2022; Weber, 2014), actuators (Diep et al., 2021; Kaluvan et al., 2016; Kikuchi et al., 2021), vehicle suspensions (Bai et al., 2013; Devikiran et al., 2022; Du et al., 2020; Oh et al., 2021), and trains (Bhardawaj et al., 2020; Guo et al., 2015; Hua et al., 2022). For washing machines, several scholars have performed researches on semi-active suspension systems using MR dampers (Bui et al., 2018; Carlson, 1999; Nguyen et al., 2014; Spelta et al., 2009; Ulasyar and Lazoglu, 2018). It is broadly known that vibration of washing machines mainly results from the unbalanced laundry mass in the drum. Especially in front-loaded washing machines, the vibration is more formidable because of the gravity influence on the laundry. This causes force transmitted from the tube assembly to the cabinet and ground through the suspension, which results in noises, unpleasantness and engine gradual failure. In conventional suspension systems of washing machines, passive dampers and springs are employed which results in inevitable problems of vibration isolation and noises at high spinning speeds, especially at 1000 rpm or higher. Therefore, the application of a smart semi-active suspension system such as MR damper-based one is timely and desirable to washing machines. Compared with active control systems, the semi-active ones are simpler, safer and more cost-effective while performing equivalent vibration isolation characteristics.

A key factor for MR damper-based semi-active suspension systems to achieve successful performance is to employ a proper control scheme. There have been many controllers proposed for MR fluid devices that can be categorized into three approaches: classic, advanced, and hybrid controllers. The classic control strategies such as sky-hook controller (Kwak et al., 2014; Seong et al., 2011; Talib et al., 2021) and proportional–integral–derivative (PID) controller (Erol et al., 2012; Kavyashree et al., 2022; Oh et al., 2014) can provide favorable output performance, but is short of robustness under exterior disturbance sources and parameter uncertainties. In spite of this shortcoming, the classic methods are commonly adopted in semi-active control systems featuring MR fluid due to their simplicity and low computational cost. The advanced control techniques overcome the disadvantage of the classic ones as they can guarantee robust stability as well as deal with perturbations and nonlinearities of the system; however, they are more sophisticated and expensive. Several controllers included in this approach are sliding mode controller (Chae and Choi, 2015; Mata et al., 2021; Yao et al., 2013), fuzzy controller (Ali and Ramaswamy, 2009; Bitaraf et al., 2010; Lin et al., 2021) and adaptive controller (Behboodi et al., 2021; Chen and Liao, 2010; Dong et al., 2011). The hybrid control methods, including fuzzy neural network controller (Eslaminasab et al., 2007; Yan et al., 2020; Yu et al., 2009), fuzzy PID controller (Ding et al., 2021; Rashid et al., 2011) and sky-hook sliding mode controller (Chen, 2009; Chen et al., 2021), incorporate two or more conventional controllers to improve the system overall performance. In the above controllers, sensors were employed to measure the motion of the tube assembly. This results in high cost impeding the application of MR dampers in washing machines.

Consequently, this study focuses on development of a low cost control system for MR damper-based suspension of front-loaded washing machines. The research is the succession of the work in (Bui et al., 2018), in which a new shear-mode MR damper was developed. The proposed control system includes two components: system and damper controllers. The system controller employs the sky-hook algorithm to generate desired damping force from plant responses. The damper one then tunes the command current fed to the MR damper coils to track this desired force. For MR fluid hysteresis compensation, a phase-lead compensator is integrated into the damper controller. In addition, a permanent magnet and an induction coil are attached at the damper end to enable the self-sensing ability. With this configuration, the proposed control system is expected to be effective, compact and economical. Following dynamic modeling of the front-loaded washing machine installed with MR damper-based suspension, the semi-active control system is configured, designed and fabricated. A test rig is established to verify the control system efficiency and the output performance is then analyzed via both simulation and experiment.

2. Self-sensing MR damper for suspension of front-loaded washing machines

In this paper, the objective suspension of front-loaded washing machine utilizes the shear-mode MR damper developed in Bui et al. (2018). A self-sensing part is supplemented to the MR damper configuration, as shown in Figure 1. The damping part comprises two exciting coils wound directly on the housing and a MR fluid layer positioned between the shearing shaft and inner surface of housing. When the exciting coils are electrically powered, magnetic fields are generated around the coils. By designing a thin wall between the MR fluid gap and coil grooves, the magnetic flux passing through it swiftly reaches to saturation and is driven across the gap. The MR fluid thereby becomes semi-solid and produces damping force via friction against the moving shaft. To increase the active length of the MR fluid in the gap, chamfer geometry is added to the coil grooves. The self-sensing part basically follows the damping part. As shown in the figure, a permanent magnet is installed on the shaft end and an induction coil is wound directly on the outer stator core. When the washing machine tube oscillates, an induced voltage is generated in the induction coil from the relative movement between it and the magnet. The voltage intensity increases with the vibration power. This varying induced voltage is then served as an input signal for the control system.

Figure 2 shows experimental step responses of the self-sensing MR damper when its exciting coils are supplied with different constant currents by an external power unit. As shown from the figure, the energy dissipation of the damper significantly depends on the applied current intensity. It is also seen that the MRF in the gap exhibits a hysteresis phenomenon as transforming from the off- to active-states (at zero time point when the damper coils start to be excited). This will be treated suitably in the next design phase of the control system.

3. Control system for MR damper-based suspension of front-loaded washing machines

3.1. Dynamic modeling of front-loaded washing machines

Prior to designing control system, it is firstly necessary to analyze the dynamic model of front-loaded washing machines. Figure 3 shows the 2D mechanical model of the washing machine. From the figure, the governing equation of motion can be written as follows

f_{I} + f_{D} + f_{S} = f_{u} (t)

(1)

where f_I, f_D, and f_S are respectively the inertia force, damping force and elastic resisting force, f_u(t) is the exciting force causing the displacement u(t) in an arbitrary vibratory u-direction for the tube center. The inertia force f_I caused by the tube assembly mass m is defined

f_{I} = m \overset{\cdot\cdot}{u} (t)

(2)

The damping force f_D is yielded from the vectorial sum of the damping forces of each damper

f_{D} = \sqrt{f_{D 1}^{2} + f_{D 2}^{2} + 2 f_{D 1} f_{D 2} \cos (β_{1} + β_{2})}

(3a)

f_{D 1} = c \overset{\cdot}{u} (t) \cos [π - (φ + \frac{π}{2} + β_{2})] = c \overset{\cdot}{u} (t) \sin (φ + β_{2})

(3b)

f_{D 2} = c \overset{\cdot}{u} (t) \cos (φ + \frac{π}{2} - β_{1}) = - c \overset{\cdot}{u} (t) \sin (φ - β_{1})

(3c)

Similarly, the elastic resisting force f_S is determined by

f_{S} = \sqrt{f_{S 1}^{2} + f_{S 2}^{2} + 2 f_{S 1} f_{S 2} \cos (α_{1} + α_{2})}

(4a)

f_{S 1} = ku (t) \cos [\frac{π}{2} - (φ + α_{1})] = c \overset{\cdot}{u} (t) \sin (φ + α_{1})

(4b)

f_{S 2} = ku (t) \cos (\frac{π}{2} - φ + α_{2}) = c \overset{\cdot}{u} (t) \sin (φ - α_{2})

(4c)

In the above equations, c is the damping coefficient of each damper, k is the stiffness of each spring, α_i and β_i (i = 1, 2) are the angles referred to the vertical y-axis of the springs and dampers, respectively. It is explained later that in case of α₁ + α₂ = 90° and β₁ + β₂ = 90°, equations (3a) and (4a) become

f_{D} = c \overset{\cdot}{u} (t) {[\sin^{2} (φ + β_{2}) + \sin^{2} (φ - β_{1})]}^{1 / 2}

(5)

f_{S} = ku (t) {[\sin^{2} (φ + α_{1}) + \sin^{2} (φ - α_{2})]}^{1 / 2}

(6)

and the governing equation (1) is rewritten as follows

\begin{matrix} m \overset{\cdot\cdot}{u} (t) + c \overset{\cdot}{u} (t) {[\sin^{2} (φ + β_{2}) + \sin^{2} (φ - β_{1})]}^{1 / 2} \\ + ku (t) {[\sin^{2} (φ + α_{1}) + \sin^{2} (φ - α_{2})]}^{1 / 2} = f_{u} (t) \end{matrix}

(7)

The damping ratio ξ, natural free-vibration frequency ω_n and damped frequency ω_d and are calculated by

ξ = \frac{c}{2 \sqrt{mk}} \frac{{[\sin^{2} (φ + β_{2}) + \sin^{2} (φ - β_{1})]}^{1 / 2}}{{[\sin^{2} (φ + α_{1}) + \sin^{2} (φ - α_{2})]}^{1 / 4}}

(8)

ω_{n} = \sqrt{\frac{k}{m}} {[\sin^{2} (φ + α_{1}) + \sin^{2} (φ - α_{2})]}^{1 / 4}

(9)

ω_{d} = ω_{n} \sqrt{1 - ξ^{2}}

(10)

From the above equations, it can be observed that the dynamic parameters involve the indefinite angle φ. As a result, the system vibration is harder to be controlled since the resonance occurs in various vibratory directions. Because the resonance should be attenuated as much as possible in any direction, the initial constraints of the suspension structure are complemented. By setting α₁ + α₂ = 90° and β₁ + β₂ = 90°, we can easily simplify (7) to obtain the following

m \overset{\cdot\cdot}{u} (t) + c \overset{\cdot}{u} (t) + ku (t) = f_{u} (t)

(11)

Then the damping ratio and natural free-vibration frequency become independent of the vibratory direction

ξ = \frac{c}{2 \sqrt{mk}}, ω_{n} = \sqrt{\frac{k}{m}}

(12)

Actually, the parameters α_i and β_i vary with the vibration of the washing machine. Although the above treatment of these parameters may results in an error, it significantly simplifies the mathematical model, which is a greatly important criterion for practical applications. In addition, the largest error only occurs in the spin-drying phase when the speed of the washing drum increases/decreases to the resonant point, but it is transient and does not affect much the whole process. Furthermore, an effective control system is also of help to timely suppress the vibration source and reduce the error. Considering the manufacturing, assembly and operation aspects, the springs and dampers of the suspension are often designed symmetrically in pair, or α₁ = α₂ = β₁ = β₂ = 45°.

During operation of washing machines, the exciting force can be expressed as a harmonic loading with exciting frequency ω and amplitude F_u

f_{u} (t) = F_{u} \cos (ω t)

(13)

From equations (11) and (13), the steady-state displacement response of the tube assembly can be derived by

u (t) = \frac{F_{u}}{k} D \sin (ω t - θ)

(14)

where θ is the phase angle and D is the dynamic magnification factor

θ = \arctan (\frac{2 ξ r}{1 - r^{2}}); D = \frac{1}{\sqrt{{(1 - r^{2})}^{2} + {(2 ξ r)}^{2}}}

(15)

in which r is the ratio of the exciting frequency to the natural frequency, r = ω/ω_n. By using equation (14) and its first-derivative, the forces transmitted from the tube assembly to the ground via the springs and dampers become

f_{k} (t) = ku (t) = F_{u} D \sin (ω t - θ)

(16)

f_{d} (t) = c \overset{\cdot}{u} (t) = 2 ξ r F_{u} D \cos (ω t - θ)

(17)

Since the phase angle between these two forces is π/2, the amplitude of the total transmitted force is evidently obtained by

F_{t} (t) = \sqrt{f_{k . max} {(t)}^{2} + f_{d . max} {(t)}^{2}} = F_{u} D \sqrt{1 + {(2 ξ r)}^{2}}

(18)

Let m_u be the unbalanced mass and R_u be its radius from the turning axis, F_u can be determined by F_u = m_uω²R_u. Then the total transmitted force is rewritten to yield the following

F_{t} (t) = m_{u} ω^{2} R_{u} D \sqrt{1 + {(2 ξ r)}^{2}}

(19)

3.2. Design of semi-active vibration control system

Based on the abovementioned dynamic modeling of front-loaded washing machines, a close-loop semi-active vibration control system is developed for the suspension featuring the MR dampers. In MR damper-based vibration control systems, the inverse model provides desired values of command current/voltage and thus plays an important role in achieving good vibration suppression performance. The structure of the inverse model should be also compact to facilitate practical application. For these, the forward model of the MR damper needs to be established first. Figure 4 shows the simplified 2D model of the washing machine installed with the MR dampers. From the figure, equation (11) is rewritten

m \overset{\cdot\cdot}{u} (t) + F_{d} (t) + ku (t) = f_{u} (t)

(20)

The damping force F_d(t) of the system can be decomposed into two components: the passive damping force F_pass(t) and the controllable active damping force F_MR(t).

F_{d} (t) = F_{pass} (t) + F_{MR} (t)

(21)

The passive damping force F_pass(t) is caused by the viscous damping of the MR fluid and Coulomb friction element between the shaft and O-rings, which is expressed by

F_{pass} (t) = c_{pass} \overset{\cdot}{u} (t)

(22)

where c_pass is the equivalent passive damping coefficient of each damper. Previous literature has shown that the viscous damping coefficient of MR dampers varies according to excitation conditions. Flow-mode MR dampers usually possess high force capacity; therefore, the change of the viscous damping coefficient should be taken into account as a controllable dynamic parameter. With respect to the shear-mode configuration of the MR dampers in this work (maximum damping force is only up to 80 N), the change is however so small that it can be neglected and hence the viscous damping force is considered as a part of uncontrolled passive damping force. This dealing is of help to simplify the computational cost, which is greatly significant in real applications.

The active damping force F_MR(t) due to the MR fluid yield stress is the decisive factor of the damping force. This force component increases with the current intensity applied to the exciting coils of the MR dampers, as presented in Section 2. Based on the experimental responses in Figure 2, an exponential progression with the current intensity is proposed for the active damping force. It is also observed from the figure that the MRF undergoes a time hysteresis behavior at the transition from off- to active-states (for about the first 0.2 s); therefore, a time hysteresis factor should be complemented to the determination of the active damping force. Because the relation between this force and time is also an exponential one, the product of these above two exponential functions yields the final formulation of the active damping force

F_{M R} (t) = f_{1} [1 - e^{(- f_{2} I)}] [1 - e^{(- f_{3} t)}]

(23)

In this equation, the coefficient f₁ represents the peak force value at saturation. The remaining coefficients f₂ and f₃ are left to adjust the slope at origin and thereby control the response time of the MRF. These coefficients are determined using the curve fitting combined with least square method in MATLAB. The objective function is to minimize the mean square error between the estimated and experimental active damping forces, which is defined by

OBJ = min \sum_{j = 1}^{n} {(F_{m . j} - F_{\exp . j})}^{2}

(24)

Where n is the number of data points, F_m.j and F_exp.j are respectively the jth estimated and experimental active damping forces. The tuning results are shown in Figure 5 and the three coefficients f₁, f₂, and f₃ respectively take values of 64.28 N, 2.65 A^–1, and 12.5 s^–1.

From the forward model in equation (23), the inverse model can be analytically obtained as the following expressions

I = - \frac{1}{f_{2}} \ln {1 - \frac{F_{c}}{f_{1} [1 - e^{(- f_{3} t)}]}}

(25)

I_{c} = {\begin{matrix} I for I \leq I_{max} \\ I_{max} for I > I_{max} \end{matrix}

(26)

where F_c is the desired output damping force of the MR dampers to mitigate the vibration, I_c is the required input command current and I_max is the maximum current intensity designed for the exciting coils. With the given desired damping force F_c, response time t and maximum current I_max, the required command current I_c can be easily to find out then.

Figure 6 shows the block diagram of the semi-active vibration control system for the suspension featuring the MR dampers. The proposed control system consists of two controllers: the system controller and the damper one. The system controller performs the task of generating desired damping force F_c based on the input signal (induced voltage) from the self-sensing part. The active damping force F_MR of the MR dampers cannot be controlled directly, but only the current I applied to their exciting coils can be adjusted immediately. Therefore, the damper controller is employed to provide the command current I_c for the power supply, whereby the corresponding active damping force F_MR can be produced to track the desired damping force F_c.

The induced voltage E_emf(t) results from the reciprocal movement of the magnet inside the induction coil of the self-sensing part is derived by (Choi and Wereley, 2009)

E_{emf} (t) = - N \frac{d Φ}{dt} = - 2 π λ N B_{rem} r_{o} \overset{\cdot}{u} (t)

(27)

In this equation, Φ is the magnetic flux, λ is the empirical magnetic efficiency, B_rem and r_o is respectively the remanent flux density and outer radius of the magnet and N is the number of effective turns of the induction coil, which can be approximated by (Chen and Liao, 2012) $N = 2 A_{c} / (\sqrt{3} d_{w}^{2})$ where d_w is the wire diameter and A_c is the effective cross-section area of the coil.

Based on this input signal, the desired damping force F_c is generated by the system controller using sky-hook control algorithm as follows

F_{c} = C_{s} E_{emf}

(28)

where C_s is the control gain. In this research, it is studied later that 0.1265 is sufficient for a good output of vibration depression. Then the damper controller calculates the required command current I_c for the power supply to electrically excite the MR damper via equations (25) and (26).

In order to compensate for the MR fluid hysteresis, a phase-lead compensator is employed for the damper controller, which is given as follows

G_{c} (s) = K_{c} \frac{s + {T_{c}}^{- 1}}{s + {(α_{c} T_{c})}^{- 1}}, (α_{c} < 1)

(29)

where s is the Laplace variable, and K_c, T_c, and α_c are the control factors that respectively take values of 6.425, 0.0258, and 0.4181. As compared with the traditional controllers (without compensator), the phase-lead compensator improves transient response and provides faster response time, which is appropriate to solve the hysteresis phenomenon of the MR fluid.

4. Results and discussions

In this section, the responses of the front-loaded washing machine using the proposed semi-active control system are simulated in MATLAB Simulink. The plant subjects to a harmonic excitation with frequency varying linearly for 100 s, which is corresponding to spindle speed 0–1200 rpm of the washing tube. The objective front-loaded washing machine of this research is the WF8690NGW one produced by Samsung Electronics Co., Ltd. The parameters for the control system are determined on the basis of the washing machine prototype as follows: m = 40 kg, c_pass = 180 N.s/m, k = 10 kN/m, m_u = 7 kg, R_u = 0.125 m, λ = 0.85, A_c = 40 mm², d_w = 0.34 mm, B_rem = 1.17 T, r_o = 14 mm. To evaluate performance characteristics and illustrate the advantages of the added phase-lead compensator, the uncontrolled state, 1 A constant current controller and the proposed control system without compensator are also included in the simulation model.

Figure 7(a) shows the simulated responses of the washing machine in relations to time and spindle speed. It can be observed from the figure that at low resonance frequencies (spindle speed of about 150 rpm), the forces transmitted from the tube assembly to the ground of the proposed and constant current control systems are considerably attenuated as compared with that of the uncontrolled state. However, the resonance peaks of the proposed controller without compensator are not suppressed as good as those with compensator. This mainly comes from the opportune and adequate response characteristic of the added phase-lead compensator. Conversely at higher excitation frequencies, the proposed control system and uncontrolled state express vibration isolation efficiency better than the constant current controller. Against the two others, the proposed control system therefore has more impressive vibration control performance during the operation of the washing machine. By using the Spectrum Analyzer tool of MATLAB Simulink, the washing machine responses are simulated in excitation frequency domain, as shown in Figure 7(b). The same remarks can also be drawn from the figure.

For the reasonable chosen of control gain C_s in equation (28), the dependence of vibration control performance on the control gain parameter is studied in this part. Figure 8 shows the influence of the control gain on the vibration reduction at the low resonance frequencies. It is noted that the higher the value of control gain is, the more the resonance peaks are attenuated. Thus, the smaller-than-0.122 values of control gain are not included in this figure. Nevertheless, a high value of control gain leads to a high operational cost of the system. A convergence of vibration reduction performance is expected when the control gain increases to a certain value. As shown in Figure 8(b), the reduction of resonance peaks almost reaches to convergence at the control gain value of 0.1265. The error is around 0.1% when the value increases from 0.1265 to 0.1275. Therefore, the control gain parameter reasonably takes this value.

In order to verify the operating efficiency of the proposed semi-active control system, an experimental setup on the Samsung washing machine prototype assembled with the MR dampers is established, as shown in Figure 9. A mass of 7 kg is fixed in the washing drum for excitation. An accelerometer is mounted on the cabinet to archive the transmitted vibration indices. The tests are carried out during the spin-drying phase of the washing machine, which is illustrated in Figure 10. The data are collected for 3 min when the spindle speed of the drum increases from 0 to 900 rpm.

Figure 11 shows the experimental vibratory acceleration indices in x-, y-, and z-directions of the washing machine prototype installed with the MR dampers under the three control states. As shown in the figure, the constant current controller remarkably reduces the washing machine vibration at low frequencies (about first 77 s). However, its vibration isolation characteristics at higher excitation frequencies are almost unachievable. The uncontrolled state is on the contrary. Against these two control states, the proposed control system provides an appropriate and timely controllability, thereby improves the vibration suppression performance throughout the operating process of the washing machine. This is clearer indicated by the means of average acceleration indices in Figure 12. It can be realized that the proposed control system inherits advantages of the two others in contribution to feasibility of washing machine suspension in particular and vibration control system based on MR dampers in general.

5. Conclusions

To take full advantages of MR dampers in application to suspension of front-loaded washing machines, a semi-active control system was proposed and experimentally evaluated in this research work. By employing the sky-hook algorithm and phase-lead compensator for the MR fluid hysteresis compensation, the proposed control system is expected to be simple, effective and economical. The integration of the self-sensing part into the MR damper also makes the system structure more compact. Following the introduction of the MR damper configuration, the semi-active control system was designed based on the dynamic modeling of front-loaded washing machines. In order to assess performance characteristics, the responses of the washing machine under different control states, including the uncontrolled, constant current and proposed controllers, were first analyzed via simulations in MATLAB Simulink. The results showed that there was an inherent compromise between the resonance controllability at low frequencies and high-frequency vibration isolation with respect to the uncontrolled state and constant current controller. However, the proposed control system was more desirable as it could reduce the transmitted force in a wide frequency range. The simulation also illustrated the advantages of the added phase-lead compensator since it could better attenuate the resonance peak than the controller without compensator. The performance of the proposed control system was clearer represented when it was tested on the front-loaded washing machine prototype installed with the MR dampers. The acceleration indices of the washing machine featuring the proposed control system were significantly improved throughout the spin-drying operating phase compared with those of the two others. Possessing advantages of the uncontrolled state and constant current controller, the proposed control system will be of help in efforts to further develop MR damper-based application systems.

Footnotes

Appendix

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Quoc-Duy Bui

Quoc Hung Nguyen

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