# Dynamic modeling of a horizontal washing machine and optimization of vibration characteristics using Genetic Algorithms

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GA optimizationVibration characteristics

mothehems are m

signal processing. This study has two main contributions: (i) a new method for design improvement

nd cussign iseral twash

a certain arrangement that can be decided during design cycle.Then, in some of the more recent work [4] the dynamic modelwas derived to include more degree-of-freedoms and complexcoordinate space to examine the whirling motion of the tub morerealistically. In some of those models [5] the focus of the attentionwas details such as exible components and the noise characteris-

ppliances.ization ar

l of washinn be cons

to represent the functional parts of the machine in terms of ozation parameters in the model. Such amodel can be construmulti-body dynamic simulation programs such as Adams/View andpowerful optimization algorithms can be run on a cluster using par-allel computing [12]. Furthermore, a multistep approach in optimi-zation process can be employed in order to solve the multi-objective optimization of washing machines taking into accountseveral cost functions including kinematic, dynamic, noise leveland walking avoidance. This type of optimization is very benecialsince it takes into account all dimensions of the optimization and

Corresponding author. Tel.: +90 212 2931300.

Mechatronics 23 (2013) 581593

Contents lists available at

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eviE-mail address: pboyraz@itu.edu.tr (P. Boyraz).meeting the customer dened-market applied criteria on the oper-ation of these devices. The rst reported studies [13] focused onmodeling and experimental assessment of suspension-group ofthe washing machine which comprises of springs and dampers in

does not affect the manufacturing cost of these aThe efforts on the dynamic modeling and optim

cately connected. For example a detailed modechine using multi-body system formalism [11] ca0957-4158/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.mechatronics.2013.05.006e intri-g ma-tructedptimi-cted induced noise and minimized vibration during operation. However,these customer expectations make the washing machine moreprone to exhibit poor vibration characteristics, such as high ampli-tudes of vibration as well as noise, even tendency to stepping andtipping motions. Therefore, great effort have been taken to improvethe vibration characteristics of the washing machine while still

and allows the designer to test a new design option. In some stud-ies the passive suspension idea was replaced by an active systemusing either moving, controlled balance mass(es) [9] or active-sus-pension elements such as magneto-rheological dampers [10]. It isan expected improvement in washing machines to use activevibration control as long as the addition of actuation and control1. Introduction

As the modern life requirements aupgraded, the washing machine dethis trend. There have been a genweight, portable and high-capacityapplying GA to optimization of vibration characteristics for the horizontal-axis washing machines, and(ii) a novel measurement method yielding the displacement in 2D and instantaneous frequency of vibra-tion from acceleration data. While the GA is contributing to passive improvement methods in the eld,the novel measurement method opens the way for low-cost diagnosis and active-vibration control ofwashing machines.

2013 Elsevier Ltd. All rights reserved.

tomer expectations arealso being affected byendency toward light-ing machines with re-

tics of the machine. Deriving a dynamic model is the rst step ofthe design improvement cycle and further work is required to val-idate the model and parameterize the system variables so that itcan be recast as a well-dened optimization problem. There areseveral studies on the optimization of structural parameters [68] offering a short design cycle because the models are parametricDynamic modelingHorizontal washing machine

formed using piezo-transducers and a novel measurement scheme is used to obtain displacement valuesfrom acceleration data as well as estimating the instantaneous frequency of the rotation with appropriateDynamic modeling of a horizontal washiof vibration characteristics using Genetic

Pnar Boyraz , Mutlu GndzMechanical Engineering Department, Istanbul Technical University, Inonu Cd, No 65, Gu

a r t i c l e i n f o

Article history:Received 17 November 2012Accepted 17 May 2013Available online 17 June 2013

Keywords:

a b s t r a c t

In this work, a 2D dynamicplane in order to examineing a new optimization scsimulated and the outputincluding the drum and th

Mecha

journal homepage: www.elsmachine and optimizationlgorithms

suyu, 34437, Istanbul, Turkey

del of a horizontal axis washing machine is derived regarding the rotationvibration characteristics of the spin-cycle and improve the design propos-e based on Genetic Algorithms (GA). The dynamic model is numericallye validated using experimental vibration data acquired from a test-rigotor of a horizontal-axis washing machine. The measurements are per-

SciVerse ScienceDirect

onics

er .com/ locate/mechatronics

allows the researcher to nd the right cost function denition tond better washing machine designs [13].

practical method for obtaining the displacement and instanta-neous frequency from the measured acceleration data was applied.

and rotations are restricted in the model. Washing machine cabi-net is assumed to be rigid and xed to the ground. No gyroscopic

Nomenclature

Variable Meaningm mass of the tubk stiffness coefcients of springsc viscous damping coefcient for shock absorbersxs geometric place of spring on x-axisxd geometric place of damper on x-axiszs geometric place of spring on z-axiszd geometric place of damper on z-axisfc friction coefcient

fver vertical forces on machine cabinetfhor horizontal forces on machine cabinetFn normal force component caused by unbalanced massFt tangential force component caused by unbalanced massFd damping force of shock absorbersFs force supplied by springs

582 P. Boyraz, M. Gndz /Mechatronics 23 (2013) 581593The novel measurement scheme can be used as a part of activevibration control making it feasible and low-cost.

2. Dynamic model derivation for vibration behavior

Dynamic model derivation is perhaps the most important partof the work since the optimization algorithm uses the parametersof the model directly to improve the nal design. In this section,rst the model assumptions and general equations of motion aregiven, and then the modeling details of each term in the equationare explained.

In order to obtain a model which is simple enough to compre-hend and use in optimization or control design, however, detailedenough to represent the real dynamics of the washing machine,some assumptions are included in the model. The oscillation groupis assumed to move on xz-plane only and any movements on y-axisIn this work, a simple 2D dynamic model of the washing ma-chine is derived in a parametric way so that it could be used inoptimization process using a single CPU. The model is validatedusing experimental vibration data collected from a horizontal-axiswashing machine. As the model was seen to approach to the realvibration characteristics closely, it was decided to be used in opti-mization. The parameters of the washing machine used in themodel are converted into string formats to be used in GA and threedifferent tness functions were dened representing the transientand/or steady-state properties of the washing machine vibration.The study has two main contributions in the eld: (1) a new designimprovement method was introduced applying GA in optimizationof vibration characteristics of the appliance and (2) a novel andFig. 1. Washing machine physical model shoeffects are taken into consideration on drum movement and theunbalanced mass is assumed to be in the middle plane of the oscil-lation group, hence not moving along y-axis. Stiffness effects of theexible parts such as gasket and water inlet tub are neglected. Thespin speed of the drum is realistic and sweeps spin-speed from 0 to900 rpm as in the experiments. The model is derived based on thephysical model given in Fig. 1 showing the washing machine oscil-lation group in equilibrium state and when moved on x and z axespositive directions.

Equation of motion can be written as in (1) and (2) for x and zaxes based on the physical model given in Fig. 1.

mx Fs1x Fs2x Fd1x Fd2x Fx ext 0 1

mz Fs1z Fs2z Fd1z Fd2z Fz ext 0 2Now, each term in the equations of motion is explained in detailstarting from the external force components Fx ext, Fz ext and contin-uing with spring force terms Fsnx ; Fsnz and damper forces denoted byFdnx ;Fdnz , n being the number of the component. The external forceson the washing machine oscillation group are forces caused by theunbalanced mass rotating in the drum. This force depends on thedrum rotation/spin speed which follows an exponential curve givenin [3]. The ramp-up characteristic of the spin speed of the drumwhich changes from 0 to 900 rpm given by (3) versus the real valuesgiven in experimental measurement can be observed in Fig. 2.

b0d N1 e1=1;8t 3Forces caused by unbalanced mass have tangential and normal

components given by (4) and (5) and shown in Fig. 3.

Fn murb0d2 4wing the equilibrium and moved states.

imu

P. Boyraz, M. Gndz /Mechatronics 23 (2013) 581593 583Fig. 2. Spin speed ramp-up characteristics in sFt murb00d 5In order to see the effects of these forces on x and z-axes (4) and

(5) are used in xz-plane decomposition to give (6) and (7).

Fx ext Fn cosbd Ft sinbd 6

Fz ext Fn sinbd Ft cosbd 7The second important term in the equation of motion is the springforce calculated by taking the dynamic length change of spring andmultiplying it with spring constant, thus the spring is assumed to belinear; however, because of the angular attachment of the springs tothe tub the spring force term is nonlinear. The spring force is formu-lated using Fig. 4 as it shows the dynamic length of the spring intwo axes for right spring. Derivation of the nonlinear spring force

Fig. 3. Tangential and normal components of the force caused by unbalanced mass.

Fig. 4. Dynamic length denition, displacement vectocan be performed using Eqs. (8)(11). The same procedure for thespring on the left of the suspension block is given in Appendix forthe sake of completeness.

xs1dynt xs1 xt 8

zs1dynt zs1 zt 9

ls1 xs12 zs12

q10

ls1dynamict xs1 dynt2 zs1 dynt2

q11

Fs1t ls1 ls1dynamictk; with the components

Fs1x t Fs1txs1 dyntls1 dynamict ; Fs1z t Fs1t

zs1 dyntls1 dynamict 12

Finally, the third main term in the force equation comes from thedampers/shock absorbers. The shock absorbers are modeled as lin-ear elements with a coefcient to represent the viscous dampingand it is calculated from the forcevelocity curve then taking theaverage of the values found. The shock absorber on the right of sus-

lation (left) and in real measurements (right).pension block is shown in Fig. 5 with the positive displacement vec-tor, force components and the relevant equations are given in Eqs.(13)(18).

xd1dynt xd1 xt 13

zd1dynt zd1 zt 14

r r and nonlinear force components for springs.

ld1 xd12 zd12

q15

ld1 dynamict xd1 dynt2 zd1 dynt2

q16

the parameters to be used in 3D model upgrade in future work.The constant matrix is updated at each simulation with the newparameters except the mass. Those parameters are mainly stiffnesscoefcient of springs, damping coefcients of dampers and the geo-metric locations of suspension elements in x and z axes. All the rela-tionships between forces are coded in the function block as an m-le.

3. Experimental measurements, estimations and modelvalidation

For the validation of the dynamic system model derived, theoutputs of the real system to the inputs used in the simulationshould be measured. Vibration characteristics of a horizontal loadwashing machine are measured using a low-cost set-up which isshown in Fig. 7. The experimental set-up includes variable auto-transformers to adjust the voltage input for the electric motor inthe washing machine assembly, a simple 3-axis accelerometerarrangement is used for measuring the acceleration in 3-axes andnally a data acquisition card is employed to log the data on com-puter. Table 1 enlists the components used in the set-up.

In the experimental set-up, only 3 accelerometers are used asvibration sensors and arranged perpendicular to each other as

quency from acceleration measurements required appropriate sig-

Fig. 5. Dynamic length denition, displacement vector r and force components fordampers.

584 P. Boyraz, M. Gndz /Mechatronics 23 (2013) 581593m1 ld1 dynamict 17

Fd1t m1c; with the components; Fd1x t

Fd1txd1 dyntld1 dynamict ; Fd1z t Fd1tzd1 dyntld1dynamict 18

Damper forces for the left shock-absorber are given in Appendix.The full model of the suspension group under the effect of unbal-anced mass when the washing machine is ramping up the full speedrange from 0 to 900 rpm is formed in MATLAB/SIMULINK environ-ment. The full model is given in Fig. 6 taking the external force com-ponents as input and translations as the output. The constant matrixincludes the mass values and coefcients for the parameters thatare included in the optimization whereas variant matrix includesFig. 6. Full model for the suspension block of a washing machnal processing in order to estimate them without any distortion,phase shift and integration errors. Therefore, the signal processingalgorithms will be detailed here to clarify the measurement-esti-mation methodology.

3.1. Displacement data using acceleration measurements

In obtaining the displacement data from acceleration measure-ments a double integration should take place theoretically.shown in Fig. 7. Data acquisition board can measure up to350 Hz and the maximum frequency corresponding to maximumspin speed is 15 Hz. Considering the Nyquist criterion Fs P 2f maxfor sampling rate selection, the signal is sampled at 300 Hz withoutany aliasing. Obtaining displacement data and instantaneous fre-ine gathering all optimization parameters in one matrix.

Fig. 7. Experimental set-up components and 3-axis accelerometer arrangement.

P. Boyraz, M. Gndz /Mechatronics 23 (2013) 581593 585Table 1Technical specications of experimental set-up.

Component Technical specications

Accelerometer Capacitive, measuring range: 10 g, frequency range0350 Hz, sensitivity: 30 mV/gHowever, due to the noise in the signal and the effect of initial con-ditions on the integration a sequence of low-pass and high-pass l-ters are employed as in Fig. 8.

Since the harmonics with close frequencies to the noise fre-quency have the risk of elimination, a digital lter with sharpcut-off frequency is required. Innite Impulse Response (IIR) ltershave sharp cut-off frequency property and they also have the

DAQ card: spider8 ofHBM

Sampling rate: [19600], 8-channels, 9600 baud(serial connection)

Autotransformer Voltage range: [0250] V

Fig. 8. Signal processing ow for obtaining th

Fig. 9. Raw acceleration data (left) and lteredadvanta...

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