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Vibration Analysis of a Horizontal Washing

Machine, Part I: Vibration Displacement and

Velocity

Galal Ali Hassaan

Department of Mechanical Design & Production, Faculty of Engineering.

Cairo University, Giza, Egypt

Abstract— Horizontal washing machine are an important class of automatic washing machines. During spinning

extreme unbalance occurs producing forced vibrations excited by the rotating unbalance. The purpose of this paper

is to investigate the effect of the various parameters of the washing machine on the vibration amplitude and velocity

of the machine drum in terms of the spinning speed. This covers the effect of the isolators stiffness, damping

coefficient and the drum mass for specific laundry capacity.

Keywords— Horizontal washing machines, Spinning speed, vibration amplitude and velocity, parametric

analysis.

I. INTRODUCTION

Horizontal washing machines are domestic automatic type subjected to severe vibration

during the spinning process. Parametric analysis is important for the proper design of the

washing machine, proper selection of its isolators, proper selection of its spinning speed and

finally for the optimal design of the machine.

Papadopoulos and Papadimitrious (2001) presented a simplified three-dimensional

dynamic model of a horizontal washing machine. They presented a control-based method to

eliminate instability and vibrations associated with active balancing [1]. Kiozumi et. al.

(2005) presented the modeling of a vibration analysis model for a drum type washing

machine. They conducted parameter tests to reduce the vibration of the washing machine [2].

Sichani and Mahjoob (2007) measured vibration responses of a horizontal washing machine

during run-up and run-down. Their results supported the operational modal analysis of the

machine [3]. Spelta, Savaresi, Fraternale and Gandiano (2008) analysed and designed a

control system for the reduction of vibration and perceived acoustic noise in a washing

machine. They outlined the design procedure and described the semiactive MR damper

located in the suspension between drum and cabinet [4].

Tyan, Chao and Tu (2009) developed a multibody dynamic model for front loading

washing machine. The suspension system was composed of two springs and two MR

dampers between the case and basket [5]. Yorukoglu and Altug (2009) proposed an approach

to evaluate the angular position and mass of the unbalanced mass of the washing machine.

They developed a simulation model and performed various experiments [6]. Nygards and

Berbyuk (2010) focused on several aspects of vibration dynamics in washing machines. They

built a computational model of a washing machine in Adams/View based on production

drawing. They parameterized the models of the functional components and used them for

suspension optimization [7]. Nygards and Berbyuk (2011) presented a computational model

of a horizontal washing machine. They presented several results of numerical studies of the

vibration dynamics of washing machines including sensitivity study of system dynamics with

respect to suspension parameters [8].

Kolhar and Patel (2013) formulated a mathematical model for reducing drum vibration and

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an improved drum design was proposed to further reduce the vibrations. They could reduce

the watder consumption, drum vibration and power consumption [9]. Mukherje and Bhardwaj

(2014) studied vibration control methods of front loaded washing machine. They used

vibration isolation / dampening pads [10]. Zhang, Wu and Shan (2015) used a test to analyze

the vibration signal comparing with Lissajou’s figure providing the reference for the optimal

design of the washing machine [11]. Ma, Hu and Liu (2015) built a rigid- flexible coupling

model of drum washing machine with parameterized characteristics using ANSYS. They

used ADAMS / View and ADAMS / Vibration platform to achieve the simulation of the

system in time and frequency domains [12]. Yalcin and Erol (2015) developed a semiactive

control method for dynamic stability of a horizontal washing machine based on adjusting the

maximum force produced by the semiactive suspension elements using vibration data of the

machine. They tested their proposed method during the spinning cycle from 200 to 900

rev/min. They showed the effectiveness of the proposed control through experimental results

[13].

II. PHYSICAL MODEL OF THE HORIZONTAL WASHING MACHINE

Fig.1 shows a physical model of the horizontal washing machine [6].

Fig.1 Physical model of a horizontal washing machine [6].

The isolators consist of two springs each of a stiffness k and two dampers each of a

damping coefficient c. The springs inclination with the vertical direction is Ɵ and the

inclination of the dampers is β. The drum total mass is M and the laundry mass is m. The

rotor speed is N and the laundry eccentricity during spinning is e.

III. VIBRATION ANALYSIS DURING SPINNING

The internal rotor holding the laundry rotates with a speed N (rev/min) and an angular

speed ω (rad/s). The inertia force due to a laundry of mass m is proportional to the laundry

acceleration. For a laundry of eccentricity e during spinning, the inertia force acting on the

rotor is:

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Fi = mω2e (1)

This force will have an orientation ωt with the x-axis (Fig.1). The inertia (exciting) force

will have a x-component:

Fx = mω2e cos ωt (2)

The dynamic model of the system can be derived in terms of the dynamic motion x and y

in the two directions (x and y). However, it is possible to perform simpler analysis by

considering only the motion in one direction (say x) [9].

This approach is followed as follows:

The drum dynamic motion in the horizontal direction is x. The deflection in the spring δs

depends on tits inclination angle Ɵ. That is:

δs = xcos(90 – Ɵ) = xsinƟ (3)

In the same way, the deflection of the damper piston δd is:

δd = xcos(90 – β) = xsinβ (4)

Using the two elastic forces and two damping forces acting on the drum, the differential

equation of the drum is obtained using the second law of motion of Newton as:

Mx’’ + 2cx’(sinβ)2 + 2kx(sinƟ)2 = mω2e cos ωt (5)

Where x’ is the vibration velocity and x’’ is the vibration acceleration.

With ke = 2k(sinƟ)2

And ce = 2c(sinβ)2

Eq.5 becomes:

Mx’’ + cex’ + kex = mω2e cos ωt (6)

Using Eq.6, the natural frequency and damping ratio of the vibrating system are given by:

ωn = √(ke / M) rad/s

or fn = ωn / (2π) Hz

and: ζ = ce / (2M ωn )

The steady-state solution of Eq.6 reveals the steady state vibration motion of the drum as:

x = Xsin(ωt – φ) (7)

where X is the peak amplitude of the drum vibration and φ is the phase angle between the

output motion and the exciting force.

The exciting frequency is ω related to the motor speed N through:

ω = 2 N / 60 rad/s (8)

The frequency ratio r is the ratio between the exciting and natural frequencies of

the system. That is:

r = ω / ωn

The vibration peak amplitude X is obtained by combining Eqs.6 and 7. This gives:

X = (me/M)r2 / √{(1 – r2)2 + (2ζr)2} (9)

The vibration peak amplitude of Eq.9 is function of the suspension parameters, k, c, θ and β,

drum mass M, unbalance me and exciting frequency ω. The effect of the vibrating system

parameters is investigated as follows:

- The inclination angles θ and β are kept constant at 45 degrees.

- The drum mass M is changed in the range: 6.68 ≤ M ≤ 30 kg.

- The isolator stiffness k is changed in the range: 5 ≤ k ≤ 15 kN/m.

- The isolator damping coefficient c is changed in the range: 50 ≤ c ≤ 550 Ns/m.

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IV. PARAMETRIC EFFECT

1. Effect on the natural frequency of the vibrating system:

The effect of the drum mass and isolators stiffness on the natural frequency of the

vibrating system is shown in Fig.2.

Fig.2 Effect of M and k on the system natural frequency.

Decreasing the isolator stiffness and increasing the drum mass will decrease the natural

frequency of the vibrating system giving a change to avoid the resonance condition away

from the spinning speed.

2. Effect on the damping ratio of the vibrating system:

The effect of the drum mass and isolators stiffness and damping coefficient on the

damping ratio of the vibrating system is shown in Fig.3.

Fig.3 Effect of M, k and c on the damping ratio of the system.

The diagram of Fig.3 helps the vibration engineer to set easily any desired damping ratio for

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the vibrating system through the three parameters M, k and c.

3. Effect of M, k and c on the peak vibration amplitude and vibration velocity in mm/s

RMS:

- Effect of M = 6.68 kg and k = 5 kN/m: Fig.4.

Fig.4 Vibration displacement and velocity for M=6.68 kg and k=5 kN/m.

- Effect of M = 6.68 kg and k = 10 kN/m: Fig.5.

Fig.5 Vibration displacement and velocity for M=6.68 kg and k=10 kN/m.

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Effect of M = 6.68 kg and k = 15 kN/m: Fig.6.

Fig.6 Vibration displacement and velocity for M=6.68 kg and k=15 kN/m.

- Effect of M = 20 kg and k = 5 kN/m: Fig.7.

Fig.7 Vibration displacement and velocity for M=20 kg and k=5 kN/m.

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Effect of M = 20 kg and k = 10 kN/m: Fig.8.

Fig.8 Vibration displacement and velocity for M=20 kg and k=10 kN/m.

- Effect of M = 20 kg and k = 15 kN/m: Fig.9.

Fig.9 Vibration displacement and velocity for M=20 kg and k=15 kN/m.

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Effect of M = 30 kg and k = 5 kN/m: Fig.10.

Fig.10 Vibration displacement and velocity for M=30 kg and k=5 kN/m.

Effect of M = 30 kg and k = 10 kN/m: Fig.11.

Fig.11 Vibration displacement and velocity for M=30 kg and k=10 kN/m.

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Effect of M = 30 kg and k = 15 kN/m: Fig.12.

Fig.12 Vibration displacement and velocity for M=30 kg and k=15 kN/m.

V. BEST PARAMETRIC COMBINATION

Going Through Fig3.4 through 12 we come easily to a solid conclusion that the best

combination of the isolators and drum mass parameters are:

M = 30 kg , k = 5 kN/s and c = 390 Ns/m

As the spinning speed has to be > 300 rev/min [1], the effect of the spinning speed from

400 to 1200 rev/min on the drum vibration displacement amplitude and velocity is shown in

Table 1.

TABLE 1

VIBRATION DISPLACEMENT AND VELOCITY FOR M = 30, K= 5 kN/m and c = 390 Ns/m.

Spinning speed

(rev/min)

Vibration displacement

(mm peak)

Vibration velocity

(mm/s RMS)

400 1.0452 30.9536

500 1.0296 38.1056

600 1.0205 45.3244

700 1.0151 52.6108

800 1.0116 59.9180

900 1.0092 67.2462

1000 1.0075 74.5892

1100 1.0062 81.9430

1200 1.0052 89.3051

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VI. CONCLUSION

- A horizontal washing machine was dynamically modeled for sake of vibration

analysis during the spinning cycle.

- The drum vibration was excited by the wet laundry mass during spinning.

- The drum vibration displacement and velocity were simulated assuming a SDOF

vibrating system.

- The vibration amplitude was affected by the spinning speed, laundry mass and

eccentricity, drum mass and suspension parameters.

- The orientation of the isolators centerlines with the vertical direction was considered

constant at 45 degrees.

- The effect of isolators stiffness and drum mass on system natural frequency (critical

speed) was investigated. This revealed a critical speed in the range from 175 to 640

rev/min.

- The effect of isolators stiffness and drum mass on system damping ratio was

investigated. This revealed a damping ratio in the range from 0.025 to 2.2.

- The effect of the drum mass, isolators stiffness and damping coefficient on the

vibration displacement and velocity was investigated in details. Those three

parameters are used to investigate the drum vibration for a range of spinning speed

from 200 to 1200 rev/min.

- It was possible by the proper selection of the drum mass and isolators parameters to

go down with the vibration peak amplitude to about one mm.

- With a drum mass of 30 kg, isolator stiffness of 5 kN/m and a damping coefficient of

390 Ns/m, the drum vibration amplitude decreased from 1.0452 to 1.0052 mm for a

spinning speed increase from 400 to 1200 rev/min.

- The vibration velocity increased from 30.9536 to 89.3051 mm/s RMS for a spinning

speed increase from 400 to 1200 rev/min.

- These high levels of vibration velocity have to be checked against ISO standard for

vibration severity of washing machines. The author is in contact with the ISO

organization representative and member of its vibration committee to clarify this

matter.

REFERENCES

[1] E. Papadopoulos and I. Papadimitriou, ”Modeling, design and control of a portable washing machine during

the spinning cycle”, Proceedings of the IEEE/ASME International Conference on Advanced Intelligent

Mechatronics Systems, Como, Italy, 8-11 July, pp.899-904, 2001.

[2] T. Koizumi, N. Tsujiuchi, Y. Nishimura and N. Yamaroka, “Modeling and vibration analysis of a drum

type washing machine”, Proceedings of the 12th International Congress on Sound and Vibration, Lisbon

Portugal, 11-14 July, 8 pages, 2005.

[3] M. Sichani and M. Mahjoob, “Operational model analysis applied to a horizontal washing machine: A

comparative approach”, International Operational Modal Analysis Conference, Copenhagen, Denmark,

30April – 2 May , 8 pages, 2007.

[4] C. Spelta, S. Savaresi, G. Fraternate and N. Gandiano, “Vibration reduction in a washing machine via

damping control”, Proceedings of the 17th World Congress IFAC, Seoul, Korea, 6-11 July, pp.11835-

11840, 2008.

[5] F. Tyan, C. Chao and S. Tu, “Modeling and vibration control of a drum type washing machine via MR fluid

dampers”, Proceedings of CACS International Automatic Control Cionference, Taipei, Taiwan, 27-29

November, 5 pages, 2009.

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[6] A. Yorukoglu and E. Altug, “Determining the mass and angular position of the unbalanced load in

horizontal washing m,achiness”, IEEE / ASME International Conference on Advanced Intelligent

Mechatronics, Singapore, 14-17 July , pp.118-123, 2009.

[7] T. Nygrads and V. Berbyuk, “Pareto optimization of a washing machine suspension system”, 2nd

International Conference on Engineering Optimization, Lisbon, Portugal, 6-9 September 2010, 10 pages.

[8] T. Nygrads and V. Berbyuk, “Multibody modeling and vibration dynamics analysis of washing

machines”, Multibody System Dynamics, December , 57 pages, 2011.

[9] S. Solhar and D. Patel, “Optimization of a drum type washing machine by analytical and computational

assessment”, International Journal of Scientific and Engineering Research, vol.4, issue 6, pp.2759-2763, 2013.

[10] C. Mukerje and A. Bhardwaj, “A study of vibration control mrthods for front loaded washing machine”,

International Journal on Recent Technologies in Mechanical and Electrical Engineering, vol.1, issue 2,

September , pp.47-48, 2014.

[11] C. Zhang, X. Wu and X. Shen, “ Vibration signal analysis of washing machine based on method of

Lissajou’s figure”, Journal of Applied Science and Engineering Innovation, vol.2, issue 2, pp.46-49, 2015.

[12] X. Ma, F. Hu and J. Liu, “Dynamic characteristic simulation of drum washing machine rigid-flexible

coupling model”, International Journal of Control and Automation, vol.8, issue 5, pp.167-176, 2015.

[13] B. Yakin and H. Erol, “Semiactive vibration control for horizontal axis washing machine”, Shock &

Vibration, vol.2015, ID 692570, 10 pages, 2015.

BIOGRAPHY

Prof. Galal Ali Hassaan: Emeritus Professor of System Dynamics and Automatic Control.

Has got his B.Sc. and M.Sc. from Cairo University in 1970 and 1974.

Has got his Ph.D. in 1979 from Bradford University, UK under the supervision of Late Prof. John

Parnaby.

Now with the Faculty of Engineering, Cairo University, EGYPT.

Research on Automatic Control, Mechanical Vibrations , Mechanism Synthesis and History of

Mechanical Engineering.

Published more than 100 research papers in international journals and conferences.

Author of books on Experimental Systems Control, Experimental Vibrations and Evolution of

Mechanical Engineering.

Chief Justice of International Journal of Computer Techniques.

Member of the Editorial Board of a number of International Journals including IJAETMAS..

Reviewer in some international journals.

Scholars interested in the author’s publications can visit:

http://scholar.cu.edu.eg/galal