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Mechatronics 23 (2013) 581–593

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Mechatronics

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

0957-4158/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.mechatronics.2013.05.006

⇑ Corresponding author. Tel.: +90 212 2931300.E-mail address: [email protected] (P. Boyraz).

Pınar Boyraz ⇑, Mutlu GündüzMechanical Engineering Department, Istanbul Technical University, Inonu Cd, No 65, Gumussuyu, 34437, Istanbul, Turkey

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:Dynamic modelingHorizontal washing machineGA optimizationVibration characteristics

a b s t r a c t

In this work, a 2D dynamic model of a horizontal axis washing machine is derived regarding the rotationplane in order to examine the vibration characteristics of the spin-cycle and improve the design propos-ing a new optimization scheme based on Genetic Algorithms (GA). The dynamic model is numericallysimulated and the outputs are validated using experimental vibration data acquired from a test-rigincluding the drum and the motor of a horizontal-axis washing machine. The measurements are per-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 appropriatesignal processing. This study has two main contributions: (i) a new method for design improvementapplying 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 field,the novel measurement method opens the way for low-cost diagnosis and active-vibration control ofwashing machines.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

As the modern life requirements and customer expectations areupgraded, the washing machine design is also being affected bythis trend. There have been a general tendency toward light-weight, portable and high-capacity washing machines with re-duced 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 stillmeeting the customer defined-market applied criteria on the oper-ation of these devices. The first reported studies [1–3] focused onmodeling and experimental assessment of suspension-group ofthe washing machine which comprises of springs and dampers ina 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 flexible components and the noise characteris-

tics of the machine. Deriving a dynamic model is the first 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-defined optimization problem. There areseveral studies on the optimization of structural parameters [6–8] offering a short design cycle because the models are parametricand 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 controldoes not affect the manufacturing cost of these appliances.

The efforts on the dynamic modeling and optimization are intri-cately connected. For example a detailed model of washing ma-chine using multi-body system formalism [11] can be constructedto represent the functional parts of the machine in terms of optimi-zation parameters in the model. Such a model can be constructed inmulti-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 beneficialsince it takes into account all dimensions of the optimization and

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Nomenclature

Variable Meaningm mass of the tubk stiffness coefficients of springsc viscous damping coefficient 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 coefficient

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. Gündüz / Mechatronics 23 (2013) 581–593

allows the researcher to find the right cost function definition tofind better washing machine designs [13].

In 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 fitness functions were defined representing the transientand/or steady-state properties of the washing machine vibration.The study has two main contributions in the field: (1) a new designimprovement method was introduced applying GA in optimizationof vibration characteristics of the appliance and (2) a novel andpractical method for obtaining the displacement and instanta-neous frequency from the measured acceleration data was applied.The 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 final design. In this section,first 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-axis

Fig. 1. Washing machine physical model sho

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

m€xþ Fs1x � Fs2x � Fd1x þ Fd2x � Fx ext ¼ 0 ð1Þ

m€zþ Fs1z þ Fs2z þ Fd1z þ Fd2z � Fz ext ¼ 0 ð2Þ

Now, 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 ¼ Nð1� e�1=1;8tÞ ð3Þ

Forces caused by unbalanced mass have tangential and normalcomponents given by (4) and (5) and shown in Fig. 3.

Fn ¼ murðb0dÞ2 ð4Þ

wing the equilibrium and moved states.

Fig. 2. Spin speed ramp-up characteristics in simulation (left) and in real measurements (right).

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

P. Boyraz, M. Gündüz / Mechatronics 23 (2013) 581–593 583

Ft ¼ murb00d ð5Þ

In 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 ¼ Fn cosðbdÞ � Ft sinðbdÞ ð6Þ

Fz ext ¼ Fn sinðbdÞ þ Ft cosðbdÞ ð7Þ

The 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. 4. Dynamic length definition, displacement vecto

can 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.

xs1dynðtÞ ¼ xs1� xðtÞ ð8Þ

zs1dynðtÞ ¼ zs1� zðtÞ ð9Þ

ls1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixs12 þ zs12

qð10Þ

ls1dynamicðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixs1 dynðtÞ2 þ zs1 dynðtÞ2

qð11Þ

Fs1ðtÞ ¼ ½ls1 � ls1dynamicðtÞ�k; with the components

Fs1x ðtÞ ¼ Fs1ðtÞxs1 dynðtÞls1 dynamicðtÞ

; Fs1z ðtÞ ¼ Fs1ðtÞzs1 dynðtÞls1 dynamicðtÞ

ð12Þ

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 coefficient to represent the viscous dampingand it is calculated from the force–velocity curve then taking theaverage of the values found. The shock absorber on the right of sus-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).

xd1dynðtÞ ¼ xd1� xðtÞ ð13Þ

zd1dynðtÞ ¼ zd1þ zðtÞ ð14Þ

r r and nonlinear force components for springs.

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


ld1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixd12 þ zd12

qð15Þ

ld1 dynamicðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixd1 dynðtÞ2 þ zd1 dynðtÞ2

qð16Þ

m1 ¼ ld1 dynamicðtÞ ð17Þ

Fd1ðtÞ ¼ m1c; with the components; Fd1x ðtÞ

¼ Fd1ðtÞxd1 dynðtÞld1 dynamicðtÞ

; Fd1z ðtÞ ¼ Fd1ðtÞzd1 dynðtÞld1dynamicðtÞ

ð18Þ

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 coefficients for the parameters thatare included in the optimization whereas variant matrix includes

Fig. 6. Full model for the suspension block of a washing mach

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 stiffnesscoefficient of springs, damping coefficients 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-file.

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 andfinally 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 asshown 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 max

for sampling rate selection, the signal is sampled at 300 Hz withoutany aliasing. Obtaining displacement data and instantaneous fre-quency from acceleration measurements required appropriate sig-nal 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.

ine gathering all optimization parameters in one matrix.

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

Table 1Technical specifications of experimental set-up.

Component Technical specifications

Accelerometer Capacitive, measuring range: ±10 g, frequency range0–350 Hz, sensitivity: 30 mV/g

DAQ card: spider8 ofHBM

Sampling rate: [1–9600], 8-channels, 9600 baud(serial connection)

Autotransformer Voltage range: [0–250] V


However, 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 fil-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 filter with sharpcut-off frequency is required. Infinite Impulse Response (IIR) filtershave sharp cut-off frequency property and they also have the

Fig. 8. Signal processing flow for obtaining th

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

advantage of realizing the filtering operations with lower ordersrequiring less processing time. Although IIR is the right choice forthe application given here it has a drawback of introducing phasedistortions in the signal. To eliminate the phase distortion, the timesignal is reversed and filtered again. The change in the amplitudedue to this double filtering is corrected as a final step. As thenumerical integration method, trapezoidal approximation is em-ployed. The signal processing flow includes a low-pass filter first(Fig. 8) since the high-frequency noise in the signal must be elim-inated. The high-frequency noise is caused by oversampling sincethe DAQ device has a sampling rate of 300 Hz whereas the maxi-mum frequency for the washing machine vibration is around15 Hz. Therefore, the low-pass filter at start is a fifth order Butter-worth filter with a cut-off frequency of 20 Hz eliminating all thehigher frequency terms. The second filter is a high-pass filter withcut-off frequency of 1.5 Hz to eliminate the DC term in the acceler-ation caused by the capacitive nature of accelerometer. High-passfiltering does not lead to serious frequency information loss sincethe frequency range we are interested in is not around 1.5 Hz.

e displacement values from acceleration.

acceleration data without DC term (right).

Fig. 10. Displacement data calculated without high-pass (left) and with high-pass (right).

Fig. 11. Phase angle of the signal in [�p, +p] (left) and unwrapped phase angle (right).

Fig. 12. Calculated instantaneous frequency (left) and smoothed curve (right).


The elimination of DC term from the raw acceleration data can beobserved in Fig. 9.

After the integration, DC term problem arises again and an addi-tional high-pass filter is used to correct this. The displacement datafound by integration with and without high-pass filtering is givenin Fig. 10.

3.2. Instantaneous frequency estimation

During the experiments, the input voltage to electric motor issupplied using a variable autotransformer to adjust the spin speedmanually. However, the direct measurement of spin speed by anencoder is not always possible due to shaft accessibility problems.Therefore a practical way of indirect measurement is necessary.Among many frequency estimation methods [14–17], method usedin [17] is selected. Hilbert transform is given in (19) in proper form.

H½gðtÞ� ¼ 1p

lim�!0þ

Z t��

t�1=�

gðsÞt � s

dsþZ tþ1=�

tþ�

gðsÞt � s

ds !

ð19Þ

Hilbert transform converts a real-valued signal into its complexsequence with the same length. The resulting analytic signal isused in calculation of instantaneous attributes of the signal. Theinstantaneous phase angle of the input sequence is the angle ofthe analytic signal, so taking its rate change yields the instanta-neous frequency. In implementation, DC term is removed with ahigh-pass filter of very low cut-off frequency as it was performedin displacement data estimation. Then the Hilbert transform is ap-plied to calculate the phase angle of the signal. The calculatedphase angle of the signal and its unwrapped version is given inFig. 11.

Next, the time derivative of the unwrapped phase angle is cal-culated to give the instantaneous frequency. In the measurementset-up to obtain this data, the spin speed of the washing machinestarts from 0 rpm ramps-up to 900 rpm and then falls down to0 rpm again. Calculated instantaneous frequency from this experi-ment using the method here can be seen in Fig. 12. There are twoimportant observations about the calculated values: (i) when thefrequency is about 1.5 Hz or below the frequency estimates arenot correct since these terms were removed from the signal

Fig. 13. Spin speed measurement by encoder and estimation by instantaneous frequency method.

Fig. 14. Dynamic system model predictions and real measurements comparison: x-axis (left), z-axis (right).


previously, and (ii) the signal has to be smoothed in the mid-regionas well. After the removal of the noise by Savitsy–Golay filtering[18] we obtain the second curve given in Fig. 12.

After the smoothing and excluding the incorrectly estimatedlower frequency terms the ramp-up of spin speed can be obtainedas it was given in Fig. 2. In order to validate the instantaneous fre-quency measurements, an experiment was run using the sameramp-up and deceleration inputs and the frequency was measuredby encoder at first run and on the second run it was estimated bythe method detailed here. The spin speeds measured by encoderand by the instantaneous frequency measurement here are com-pared in Fig. 13. As it can be seen they are in good agreement ex-cept the low-frequency values.

3.3. Validation of the model

In order to validate the dynamic model, two main magnitudesrepresenting the steady-state behavior and transient responseare compared between the model prediction and real measure-ments. These magnitudes are (i) maximum amplitudes of transientvibrations and (ii) maximum amplitudes of steady-state vibrations.One of the comparisons can be seen in Fig. 14 and the relevantmetrics can be seen in Table 2.

The absolute errors between the model and the measurementsare caused by the omitted parts of washing machine assembly.Considering the simplification applied to reduce the model com-plexity and the model flexibility in terms of parametric representa-tion, the model is considered to be sufficient for the optimizationpurposes. Although the model is validated experimentally and isadequate for the optimization purposes, it is necessary to statethe full capability of the validated 2D model especially comparedto 3D dynamic models. Modeling the vibration dynamics of awashing machine using a 3D model would give the possibility toexamine the rotational motion around the x-axis, which might givethe opportunity to examine the rotational slip as in [9]. In additionto this, any side effects caused by the motion of the laundry massalong the y-axis may cause a wobbling motion and this could beexplored if a 3D model is formed. Including the y-axis in the model,we could also optimize an additional design parameter such as theangle of the tub to prevent the wobbling between the front andrear planes of the machine or the rotation around x-axis. However,this type of in-depth exploration is out of the scope for this paper.In the 2D simple model validated here, it is assumed that the seg-ment along the y-axis where the laundry mass is located can be as-sumed stationary. In other words, the laundry mass does not movealong the y-axis during spinning cycle it sticks to a region in theinner surface of the tub. Furthermore, the critical speed for

Table 2Comparison of transient and steady-state key magnitudes between model andmeasurements.

# Simulation Measurement Absolute error

x Transientmax (mm) 10.93 8.9 �2.03min (mm) �10.46 �9 1.46

z Transientmax (mm) 7.75 7.3 �0.45min (mm) �7.13 �7.5 �0.37

x Steadt-state (mm) ±4.45 ±5.8 1.35z Steady-state (mm) ±4.6 ±5.1 0.5


translational speed that is examined in our paper along x-axis ismuch higher than the rotational slip condition. Thus, the 2D modelcan account for the worst conditions although neglecting less crit-ical slip conditions that happen before the max speed is reached.

Table 3Fitness function definitions, optimization criteria and relevant system dynamics.

Fitness function Optimization criteria

f 1 ¼ 1maxð

ffiffiffiffiffiffiffiffiffiffiffix2þz2Þp Minimize the wobbling movement of the oscillation group

xstspan ¼ xmax � xmin Minimize steady state vibration span

zstspan ¼ zmax � zmin

Spanst ¼ xstspan þ zst

span

f 2 ¼ 1Spanst

xstspan ¼ xmax � xmin Minimize both steady state and transient region vibration

spanzstspan ¼ zmax � zmin


span

xtrspan ¼ xmax � xmin

ztrspan ¼ zmax � zmin

Spantr ¼ xtrspan þ ztr

span

f 3 ¼ 1Spanst

þ 1Spantr

4. Optimization with Genetic Algorithms

Genetic Algorithms are selected for the optimization because oftheir attributes such as being independent from the function eval-uation, continuity, and differentiability. In other words, GA can beapplied to all kinds of problems including discrete, continuous,non-differentiable and nonlinear systems because the emphasisof the algorithm is not on the exploitation but on the explorationof the solution space. However, important part of GA implementa-tion is in recasting the optimization problem in GA terminology. Inorder to do that the problem parameters must be refined as chro-mosomes and the performance of each chromosome should beevaluated in a fitness function defined based on the original prob-lem parameters. In addition, any optimization constraint should beencoded as an inhibition mechanism in GA. In a generic GA, theapplication of the algorithm includes following steps: initial popu-lation generation, evaluation of the members in initial populationaccording to fitness function, selection of members based on their

Relevant system dynamics

Table 4GA optimization runs and vibration characteristics of the given solutions for all fitness criteria.

Fitness function Fitness values/termination condition Vibration characteristics of solution

f 1 ¼ 1maxð

ffiffiffiffiffiffiffiffiffiffiffix2þz2Þp Kept running for 100 generations. The highest value was

obtained at 92th generation

xstspan ¼ xmax � xmin Terminated on first stage at 71th generation



span

f 2 ¼ 1Spanst

xstspan ¼ xmax � xmin Terminated on first stage at 14th generation



span

xtrspan ¼ xmax � xmin

ztrspan ¼ zmax � zmin

Spantr ¼ xtrspan þ ztr

span

f 3 ¼ 1Spanst

þ 1Spantr

Table 5Comparison of simulation results and improvement by GA.

Original Fitness f1 Fitness f2 Fitness f3

Design vector k = 6155 N/m k = 6071.4 N/m k = 7142.9 N/m k = 5000 N/mc = 308.6 Ns/m c = 400 Ns/m c = 150 Ns/m c = 364.29 Ns/mxs = 0.075 xs = 0.0871 m xs = 0.0729 m xs = 0.0729 mxd = 0.075 xd = 0.0871 m xd = 0.0871 xd = 0.0871 mzs = 0.19 zs = 0.1513 m zs = 0.2126 m zs = 0.1717 mzd = 0.205 zd = 0.1153 m zd = 0.1 m zd = 0.1459 m

x Transient (mm)Max 10.93 6.74 6.93 6.6Min �10.46 �5.2 �6.3 �4.8

z Transient (mm)Max 7.75 7.47 11.89 7.01Min �7.13 �6.3 �9.45 �6.2x Steady state ±4.45 ±4.43 ±4.4 ±4.43z Steady state ±4.6 ±4.5 ±4.5 ±4.5



fitness, and reproduction to generate new members to enter intothe GA cycle again. There are two separate algorithms running inevaluation and reproduction parts. During evaluation, the solutioncandidates are converted into their phenotypes and evaluated inobjective functions to determine the fitness values. In reproduc-tion, the genotypes are used in mutation and crossover processesto generate new members.

In minimizing the vibrations of washing machine, the problemis recast in terms of GA. First, the design vector to be used in opti-mization is defined to comprise of stiffness and damping coeffi-cients of elements and the geometric places at which thoseelements are assembled to the tub. The suspension block is sym-metrical; therefore there are six independent parameters in designvector given in (20). Here, k is stiffness, c is damping coefficient, xsand zs are the coordinates of the springs whereas xd and zd arecoordinates of dampers.

d ¼ ðk; c; xs; zs; xd; zÞ ð20Þ

The fitness functions are designed to represent three differentoptimization criteria as given in Table 3.

During GA optimization a well-known constraint in washingmachine vibration called stepping condition is used. Stepping

Fig. 15. Sensitivity analysis including all fitness values and all variables: the % c

conditions may occur when horizontal forces acting on the wash-ing machine exceeds the friction force between the floor and thewashing machine. In order to avoid the solution candidates violat-ing the stepping condition, the constraint given in (21) is used,where fver and fhor are vertical and horizontal forces acting onwashing machine cabinet respectively. fc is the friction coefficientand it is assumed to be 0.2 which is the coefficient of plastic footsand a smooth surface such as a bathroom surface.

ðfver þMgÞ � fc � fhor > 0 ð21Þ

The members of the population violating (21) constraint are as-sumed to be dead-members in GA and they are assigned zero fit-ness value to inhibit their reproduction.

Another important consideration in GA optimization is to deter-mine the criteria for termination. GA does not guarantee a globaloptimization result but it can always converge to a sub-optimalsolution given the appropriate mutation and cross-over probabili-ties. In our study, a two-stage termination condition is applied:First, total fitness value of generations are checked and comparedto each other and if it did not change significantly (i.e. less thana) for a pre-determined number of generations (i.e. 5) then theGA evaluation is terminated. However, in the probabilistic nature

hange in fitness values versus % change in the variables from the optimum.


of GA this condition may not occur, so there is an upper bounddetermined to halt the evaluation, selected as 100 here. The solu-tion is chosen as the best individual amongst the individuals ofthe last generation if the evaluation stops at the first stage. If theevaluation continues until the upper bound, then the individualwith the max fitness value in all populations is declared as theoptimum solution. Table 4 shows the termination conditions foreach definition of fitness value criteria and the resultant vibrationcharacteristics with the given solution (i.e. the design vector).

In order to observe the improvement gained by GA optimizationa comparison is shown in Table 5 including the results obtainedfrom the simulation using original design parameters of the wash-ing machine, and the design vectors obtained from GA for each fit-ness criterion.

In order to observe the effects of the parameters on theoptimization results a sensitivity analysis is performed. The resultsshowing the variation around the optimum values for all the vari-ables changing one variable at a time are given in Appendix. How-ever, from the sensitivity analysis it can be briefly said that fitnessfunctions f2 and f3 are the most robust functions since the changein the optimum variable affects them the least. In addition to that ifthe fitness functions f2 and f3 are taken as a comparison basis, themost effected variable is the z position of the damper and the dam-per coefficient itself. These effects can be seen in Fig. 15 clearly.

5. Conclusions and future work

In this work, a washing machine suspension block is dynami-cally modeled in 2D with the spin-speed as the input and the trans-lations in x–z plane as the outputs. The model is validated using thevibration characteristics measurements performed in a test-rigincluding a real washing machine, variable autotransformer asthe voltage source, the accelerometers as main sensors, encodersas redundant sensors, a DAQ card and a PC for recording the data.To obtain displacement data in x–z plane from the measured accel-eration values, a filtering sequence is employed to eliminate theoversampling, DC term and initial condition based problems inintegration. Instantaneous frequency of the rotation is also esti-mated using Hilbert transform and validated by encoder readings.After the dynamic model was found to be sufficient to representthe vibration characteristics the optimization was performed usingGA. Three different fitness criteria is used in simple GA using rou-lette wheel as selection mechanism and a user defined two-stagetermination criterion. All fitness functions performed well, how-ever the third fitness function yielded the most successful solutionimproving both the transient and steady-state vibration character-istics. The framework presented in this report can offer a fastdevelopment/improvement method for washing machine designcycle as the previous work in the area offered. However, the contri-bution of this work is in its flexibility to define the objectives of theoptimization without considering any linearity or differentiabilityconstraints. In addition, the measurement methods developed indata acquisition phase can offer a low-cost method to obtain theinputs for active vibration control applications in washing ma-chines. As seen in the sensitivity analysis the damper coefficientand its position are the most sensitive parameters affecting thecost or fitness functions. Therefore, it will be meaningful to designa separate system including active-suspension control changingthe damper coefficient dynamically to suit the particular condition.In future work, it is considered to extend the model in 3D and in-clude the neglected parts. In addition, a new active vibration con-trol scheme using the automotive-based semi-active suspension oractive suspension control [19] will be applied still keeping the GA

optimization in the loop for obtaining adaptive parameters underdifferent conditions and employing the measurement set-updeveloped in this work.

Appendix A. Models for spring 2 and damper 2

Spring 2 model:

xs2dyn(t) = xs2 + x(t)zs2dyn(t) = zs2 - z(t)

ls2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixs22 þ zs22

pls2dynamicðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixs22

dynðtÞ þ zs22dynðtÞ

qFs2(t) = [ls2 � ls2dynamic(t)]k

Fs2x ðtÞ ¼ Fs2ðtÞ xs2 dynðtÞls2dynamicðtÞ

Fs2z ðtÞ ¼ Fs2ðtÞ zs2 dynðtÞls2dynamicðtÞ

Damper 2 Model:
xd2dyn(t) = xd2 + x(t)zd2dyn(t) = zd2 + z(t)
ld2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixd22 þ zd22

pld2dynamicðtÞ

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixd2 dynðtÞ2 þ zd2 dynðtÞ2

qm2 = ld2dynamic(t)Fd2(t) = v2c

Fd2xðtÞ ¼ Fd2ðtÞ xd2 dynðtÞ

ld2�dynamicðtÞ

Fd2zðtÞ ¼ Fd2ðtÞ zd2 dynðtÞ

ld2dynamicðtÞ

Appendix B. Sensitivity analysis full results

Deviation fromoptimum (%)

K_spr
f1 f2 f3
�50
2500 61.6085 54.2238 18.3965 �40 3000 65.5312 44.9791 17.8138 �30 3500 65.8311 45.3975 17.8123 �20 4000 73.2542 55.0170 17.8332 �10 4500 76.2343 55.0276 18.8074 �5 4750 77.1554 55.0714 18.5669
Kopt (%) 5000 79.5182 55.1677 18.9945
5 5250 80.6161 55.2641 19.1193 10 5500 82.8090 55.5328 18.8669 20 6000 87.5260 56.3109 18.6149 30 6500 84.0599 50.1199 17.2157 40 7000 80.2365 48.9195 16.7296 50 7500 81.1408 47.6735 16.5506



C_damp
f1 f2 f3
�50
182.1450 69.6326 52.3279 15.2329 �40 218.5740 71.4901 50.3224 16.1906 �30 255.0030 73.1351 48.8887 17.1351 �20 291.4320 75.4889 53.8359 17.8320 �10 327.8610 77.4635 54.8938 18.1210 �5 346.0755 78.1096 54.9781 18.2772
Copt (%)
364.29 79.5182 55.1677 18.9945 5 382.5045 79.7344 53.5699 18.9523 10 400.7190 82.2146 56.1880 19.5854 20 437.1480 81.3614 52.9521 19.4387 30 473.5770 83.1674 45.2792 18.5484 40 510.0060 86.0851 46.1224 18.7601 50 546.4350 88.0307 46.1898 19.1336

X_spr
f1 f2 f3
�50
0.0365 53.2141 54.6499 16.7260 �40 0.0437 58.5445 55.1280 17.3341 �30 0.0510 63.0254 55.2713 17.8567 �20 0.0583 67.6372 55.2022 18.4031 �10 0.0656 72.9139 55.1888 18.7622 �5 0.0693 76.3199 55.1913 18.7153
Xs_opt (%) 0.0729 79.5182 55.1677 18.9945
5 0.0765 82.7353 55.1941 18.7796 10 0.0802 86.0480 55.1420 18.9328 20 0.0875 89.6216 55.1033 18.8448 30 0.0948 77.7404 55.1408 18.9618 40 0.1021 70.8614 55.0443 18.6221 50 0.1094 63.5231 55.1184 17.6940

Z_spr
f1 f2 f3
�50
0.0859 67.5764 47.6900 18.3874 �40 0.1030 68.5217 54.7428 18.1314 �30 0.1202 70.2937 55.1483 18.5122 �20 0.1374 84.5210 55.3227 18.8724 �10 0.1545 86.6495 55.2343 18.9295 �5 0.1631 82.7201 55.2249 18.7955
Zs_opt (%)
0.1717 79.5182 55.1677 18.9945 5 0.1803 76.7890 55.1920 18.7257 10 0.1889 74.2533 55.1827 18.4682 20 0.2060 70.2846 55.1906 18.5474 30 0.2232 65.9450 54.7624 18.2078 40 0.2404 64.0877 55.1761 18.0041 50 0.2576 62.1459 55.1477 17.7662

X_damp
f1 f2 f3
�50
0.0436 66.8306 45.8639 15.8951 �40 0.0523 68.8640 46.1163 16.1166
Sensitivity Analysis Full Resultsanalysis full results (continued)


X_damp
f1 f2 f3
�30
0.0610 72.3324 49.8935 17.3665 �20 0.0697 76.0906 49.3524 17.5102 �10 0.0784 76.3755 55.9765 18.7070 �5 0.0827 78.0843 55.5268 18.8373
X_damp (%)
0.0871 79.5182 55.1677 18.9945 5 0.0915 80.0792 56.1022 19.1193 10 0.0958 80.9425 55.9034 18.8329 20 0.1045 85.6951 54.6803 18.9495 30 0.1132 88.2477 52.8166 18.9153 40 0.1219 88.8615 48.0593 18.3304 50 0.1307 92.5167 48.1945 18.4371

Z_damp
f1 f2 f3
�50
0.0730 91.2087 53.3683 18.7540 �40 0.0875 87.5323 48.6241 18.6074 �30 0.1021 87.4362 47.4453 18.2248 �20 0.1167 85.1749 50.4640 18.4956 �10 0.1313 81.6661 55.7756 18.8315 �5 0.1386 80.2932 56.0651 18.7183
Z_damp (%)
0.1459 79.5182 55.1677 18.9945 5 0.1532 78.1869 55.6216 18.8546 10 0.1605 77.4393 56.2400 18.7616 20 0.1751 75.8834 53.7766 18.4606 30 0.1897 74.6872 54.3648 18.1445 40 0.2043 73.0979 50.8689 17.4311 50 0.2189 73.6208 54.9471 17.6554
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