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Journal of Sound and Vibration

Journal of Sound and Vibration 340 (2015) 343–353

http://d0022-46

n CorrTel.: þ8

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journal homepage: www.elsevier.com/locate/jsvi

Mode selection of modal expansion method estimatingvibration field of washing machine

B.K. Jung, W.B. Jeong n

School of Mechanical Engineering, Pusan National University, Busan 609-735, Republic of Korea

a r t i c l e i n f o

Article history:Received 10 February 2014Received in revised form10 December 2014Accepted 13 December 2014

Handling Editor: H. Ouyang

However, there have been few studies about selecting the eigenvectors with respect to the

Available online 6 January 2015

x.doi.org/10.1016/j.jsv.2014.12.0160X/& 2014 Elsevier Ltd. All rights reserved.

espondence to: Department of Mechanical E2 51 510 2337; fax: þ82 51 517 3805.ail address: wbjeong@pusan.ac.kr (W.B. Jeon

a b s t r a c t

This paper is about a study estimating the vibration and radiated noise of a washing machineby using a mode selection-applied modal expansion method (MEM). MEM is a technique thatidentifies the vibration field from a portion of eigenvectors (or mode shapes) of a structure,and thus, the selection of the eigenvectors has a big impact on the vibration results identified.

structural vibration and radiated noise estimation. Accordingly, this paper proposes the use ofa new mode selection method to identify the vibration based on the MEM and then calculateradiated noise of a washing machine. The results gained from the experiment were alsocompared. The vibration and noise results of numerical analysis using the proposed selectionmethod are in line with the measured results. The selection method proposed in this papercorresponds well with the MEM and this process seems to be applicable to the estimation ofvarious structure vibrations and radiated noise.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Advanced mechanical engineering has led to an improvement of people's attitude toward life and an increase in people'sinterest in improving living conditions. Consequently, noise and vibration issues related to household electrical applianceshave been rising. The noise and vibration of home appliances are likely to primarily shorten the lifespan of products, andsecondly, bring physical and psychological damage to indoor residents and users. Against this backdrop, the home applianceindustry has been recently carrying out studies on noise and vibration reduction [1–4], and as a part of those studies, thispaper analyzed the noise and vibration of household washing machines.

There are many experimental or numerical studies about the vibration and noise characteristics of washing machine. Agnaniresearched the dynamic characteristics of the machine by using the multi-body analysis and compared the numerical resultswith the experiment results [5]. Chiariotti researched the noise source of washing machine by using sensor array technique andnear field acoustic holography [6]. Jung mentioned from his study that the noise of the washing machine can be divided into thestructure-borne noise caused by tub and cabinet vibration and the air-borne noise caused by fluid behavior within a drum anddrain pump [7]. Commonly, the air-borne noise is more dominant than the structure-borne noise in the washing machine.However, the home appliance industry often makes an effort to reduce the structure-borne noise through the structuralimprovement in the product development phase. Thus, this paper estimates the washer's vibration and structure-borne noise atthe time of dehydration without fluid by using a finite element method (FEM) and a boundary element method (BEM).

ngineering, Pusan National University, Jangjeon-dong, Kumjung-ku, Pusan 609-735, Republic of Korea.

g).

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353344

FEM refers to the calculation of approximate solutions numerically by dividing into finite small areas the naturalphenomenon influenced by a specific rule. This is one way to reproduce static and dynamic phenomenon. However, it is veryimportant to implement an accurate exciting force so as to reproduce the exact dynamic phenomenon. The exciting forcewhich causes the vibration and noise of a structure is significant information in dynamic analysis, but it is often impossibleto directly measure it owing to geometric limits. Accordingly, Otsuka [8] and Jung [9] were involved in a study thatestimated the exciting force from vibration signals, which are relatively easy to measure, based on a numerical analysis. But,in the case of a study on the exciting force identification [10,11], it is important to implement the exact transfer function inproducing reliable exciting force. Considering this, it is very difficult to implement precisely the transfer function in the caseof a complex shaped structure composed of various single pieces. On the other hand, if it is properly materialized in ageometric perspective, it will be relatively easier to reproduce the eigenvector of a structure compared to the transferfunction. As a result, there have been studies about the modal expansion method (MEM), which estimates the structuralvibrations by using the structure's eigenvectors and vibration response signals [12–14]. The MEM refers to a way ofestimating the modal participation factor (MPF), which is the level of contribution of the eigenvector, can express thevibration field in a faster manner and has relatively less computation in comparison to the force identification method.However, given that the MEM is needed to estimate the vibration by using some of the numerous eigenvectors, the MEM hasan issue with selecting the eigenvectors. One of the representative mode selections includes the MSC. Nastran's selectionmethod, which uses effective mass [15,16], but the method has not been verified in terms of MEM and radiated noise.

Accordingly, this paper intends to describe a newmode selection method that considers the radiated noise and the MEM,and estimates, based on the MEM, the vibration field that occurs when a washing machine is operated at the maximumdehydration rate of 1370 rev/min. To that end, in this paper, the vibrations on the surface of a washing machine weremeasured through an experiment, and the eigenvalue problem of the FE model was performed numerically. Additionally,principal eigenvectors were selected with the utilization of the proposed selection. Based on this, this paper is designed toestimate the vibration results on the surface of a washing machine by using the MEM, and additionally, the radiated noise isestimated with the utilization of the acoustic direct BEM at the time of dehydration. These numerical results were comparedwith the experimental results.

2. Vibration and noise identification theory and mode selection

2.1. MEM theory

The MEM is a method to estimate the MPF by using eigenvectors and the measured signals obtained from the surface of astructure. This MEM theory originated from Eq. (1), which is the linear vibration system of the multidegree of freedom.

M €xþC _xþKx¼ F (1)

where,M refers to the mass matrix of a structure, C the damping matrix, K the stiffness matrix, F the external force vector. x,_x and €x mean vibration displacement, velocity and acceleration vector, respectively. In order to calculate the forced vibrationresponse of Eq. (1), the free vibration should be first calculated based on an eigenvalue analysis of

Kφr ¼ω2rMφr (2)

where, ωr refers to the r-th natural frequency, and φr to the r-th eigenvector. Using these factors, the eigenvector matrix Φis represented as

Φ¼ φ1;φ2;…;φr

� �(3)

When the equation solution of Eq. (1) is represented in a modal coordinate using the above factors, the result is as seen inthe following equation:

x¼Φu (4)

where, u called MPF which means the eigenvector's contribution level or displacement of the modal coordinates.The process of calculating the forced vibration response by using these factors can be seen, as shown below. First, afterEq. (4) is substituted into Eq. (1), the equation of the physical coordinates is converted into the modal coordinates.

�ω2ΦTMΦþ jωΦTCΦþΦTKΦ� �

u¼ΦTF (5)

The modal coordinates go through the matrix diagonalization and then are represented as independent coordinates asbelow.

�ω2Iþ2jωζrωrþω2r

� �u¼ qr (6)

Where, I refers to the identity matrix, ω the frequency, and ζr to r-th the damping ratio. Based on the above equation, theMPF vector, u, is defined as

u¼ �ω2Iþ2jωζrωrþω2r

� ��1qr (7)

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353 345

MEM that calculates the MPF by using the known response signals and the eigenvector matrix is as follows.

xm ¼Φmu (8)

Where, xm refers to the vibration response vector of the point measured during an experiment, and Φm the eigenvectormatrix corresponding to the measured points. The method of estimating u is shown in the following equation:

u¼Φmþxm (9)

In the above equation,þ refers to pseudo-inverse, indicating that Φm may not be a square matrix. Generally, in the casewhere the pseudo-inverse is used for a problem, a more accurate solution can be obtained when the number of equationsoutnumbers the unknowns. In other words, a more accurate and reliable u cannot be estimated until the number of sensorsmeasured during an experiment outnumbers the eigenvectors used.

Eq. (10) shows the process of calculating the normal direction velocity of all points for radiated noise calculation with theutilization of the calculated MPF vector.

Vn ¼ jωTx (10)

where, Vn refers to the normal-directional velocity vector, and T represents the transformation matrix for converting x into anormal-directional response vector.

2.2. Mode selection theory

MEM uses a limited number of eigenvectors in order to estimate the response of overall points from vibration responsesof several points. However, as the structure has a theoretically unlimited number of eigenvectors, there is an arising issue ofwhich eigenvectors should be selected and used. Generally, the more complex or bigger finite element model of thestructure, the bigger the degree of freedom. As a result, a number of eigenvectors exist within the frequency bands ofinterest in the case of the eigenvector analysis of the finite element model. Therefore, it is necessary to select principaleigenvectors for applying MEM.

A representative selection method is one that uses the effective mass offered by MSC.Nastran. This method is composedof that shown in Eq. (11), based on a theory that any eigenvector with bigger effective mass makes a bigger contribution tothe vibration response.

EMr ¼X6D ¼ 1

φTrMφD

�� ��2 (11)

where, φD refers to the eigenvector corresponding to rigid-body motions. By using the above equation, EMr which is the r-theffective mass, can be calculated. By aligning in the order of biggest to smallest based on the calculated value of EMr , theprincipal eigenvectors with a higher participation factor can be selected. It is a suitable mode selection method for theestimation of the vibration, but it is not verified for the estimation of the radiated noise. Accordingly, this paper proposednew mode selection method.

The vibration response is composed of the superposition of modes (or eigenvectors). So, using global modes has anadvantage to represent the vibration field rather than using local modes. Also, the radiation efficiency of global modes ishigher than that of local modes. Especially, bending modes of global modes have very high radiation efficiency. The newmode selection method was based on these facts mentioned above. The index is primarily split into global modes and localmodes by dividing the average value of the normal-directional vibration by the maximum value. The transformation matrixT is used to transform it into a normal-directional vibration and to consider bending modes which have high radiationefficiency. This can be expressed, as shown below

GLr ¼E Tφr

�� ��� �max Tφr

�� ��� � (12)

where, GLr refers to a value of the proposed index corresponding to the r-th eigenvector, ‘E’ to a process of averaging and‘max’ to a function that seeks the maximum value. In the above equation, whose index value ranges from 0 to 1, if it is closerto 0, the index means the local vibration that has less contribution to the noise. On the other hand, if it is closer to 1, theindex means the global vibration that has a large contribution level to the noise.

This paper used GLr and EMr to select the principal eigenvectors corresponding toΦm, and then MEM was implementedto estimate the vibration noise of a washing machine.

2.3. Radiated noise theory

In this paper, acoustic BEM was used to analyze the external radiated noise of a washing machine at the time ofdehydration. The acoustic BEM is split into a direct method [17,18] and an indirect method [19]. For the radiated noiseanalysis of a closed space, the direct method is used, while for the radiated noise analysis of an open space, the indirectmethod is used. In this paper, the direct method was used to predict the radiated noise from overall vibrations of a washingmachine. The method utilizes the surface pressures and normal velocities to calculate the sound pressure away from the

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353346

surface of the structure. It has the following governing equation:

C xð ÞP xð Þ ¼ZS

P yð Þ∂ψ x; yð Þ∂n

þ jρωVnðyÞψ x; yð Þ

dS (13)

In the above equation, xmeans any location of field point, y to one location on the surface of an acoustic grid, ω to frequency,ρ the density of a medium, n the normal vector, S the surface, P the acoustic pressure, Vn the normal velocity, and ψ x; yð Þ theGreen function, as shown below

ψ x; yð Þ ¼ ejkR

4πr(14)

where, R refers to distance between x and y (R¼ x�y�� ��), and k to ω=c with the meaning of a wavenumber. According to

Eq. (13), C xð Þ means a function dependent on the location of field point. In the case of the external radiated noise issue, it isdefined as

C xð Þ ¼1 ðin domainÞ1�R

S∂ 1=4πRð Þ

∂n dS ðon boundaryÞ0 ðout domainÞ

8>><>>:

(15)

This paper uses the MEM to calculate Vn yð Þ and then uses Eqs. (13)–(15) to obtain the surface pressure P yð Þ. Additionally, theradiated noise analysis was implemented to estimate the acoustic pressure P xð Þ at the field point by using the estimatedsurface pressures and normal velocities.

3. Vibration field visualization

3.1. Finite element model and eigenvector analysis of a washing machine

The finite element model was built in order to realize numerically the vibrations occurring at the dehydration of awashing machine. In this case, the outer face of the structure was accurately modeled in order to properly realize the surfacevibrations important in calculation of radiated noise, while internal structures were simplified with equivalent mass. Theequivalent mass models of the internal structures were connected with outer face of the washing machine by using twospring elements and three damper elements. Boundary condition of the finite element model was the clamped condition onthe legs of the washing machine. In the case of the finite element analysis, a commercial program, MSC.Nastran, was usedfor the eigenvector analysis, and the numbers of nodes and elements used for the analysis were 101,766 and 100,642,respectively. Fig. 1 displays the finite element model of a washing machine modeled based on what was previouslymentioned. Fig. 2 shows the comparison of the natural modes and Fig. 3 shows the comparison of the frequency responsefunctions between the FE analysis and the experiment. These figures represent the validation of the finite element model.

3.2. Frequency vibration response acquisition experiment

It is necessary to measure the frequency responses at some points of surface so that the vibration field of a washingmachine at the time of dehydration can be predicted based on MEM. Thus, LMS.Scadas, FFT analyzer, was used andacceleration signals were measured at an interval of 1 Hz from 0 Hz to 2048 Hz, when a washing machine reached the toplevel which runs at the maximum dehydration velocity of 1370 rev/min. Hanning-windowed response signals with 400times averaging to reduce noise were gained. When it comes to the measurement location, a total of 34 points ofacceleration responses were measured with a composition of 8 points in front of the cabinet, 5 points on the right, 6 pointson the left, 10 points in the back, and 5 points on the top plate of a cabinet so as to properly represent the overall vibrationsof the structure. Fig. 4 shows those points. The two verification points also shown in the figure were designed to verify thevalidity of the predicted analysis results.

3.3. Mode selection and MEM application

This paper sets the frequency bandwidth of interest from 63 to 1000 Hz in the form of a 1/3 octave band, where 719eigenvalues were present. Then, the two mode selection methods previously introduced in Section 2.2 were used to select30 principle eigenvectors. The numbers of 30 eigenvectors are attributable to the fact that more reasonable results can beobtained when the number of response points (34 points) outnumbers the number of eigenvectors used in the MEM. Table 1refers to the order of the principle eigenvectors selected based on GLr or EMr , as well as relevant index values.

When taking a closer look at the selected eigenvectors identified in Table 1, it can be seen that those eigenvectors belowthe 30th are selected in a similar manner, but others show different tendencies. These selected principle eigenvectors andmeasured acceleration signals at the response points were utilized in the MEM. As the MEM of Eq. (9) uses the displacement

Fig. 1. The finite element model of a washing machine.

Fig. 2. Comparison of the natural modes of the cabinet of a washing machine: (a) numerical modes and (b) experimental modes.

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353 347

50 100 150 200 250 300 350 40010

-4

10-3

10-2

10-1

100

101

Frequency [Hz]

Acc

eler

ance

[(m

/s2 )/N

]

Experimental FRFPredicted FRF

Fig. 3. Comparison of the experimental and predicted FRF (accelerance) of a washing machine.

Fig. 4. Measurement and verification points.

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353348

vector, the relational equation shown in Eq. (16) was used for transforming from acceleration to displacement.

xm ¼ � €xm

ω2 (16)

Using the displacement based on the above equation and the principle eigenvectors, the MPF was calculated from theequation shown in Eq. (9). Fig. 5 illustrates with the 5th and 9th MPF, which are commonly shown in the selection of GLrand EMr , where a bar with solid pattern refers to the MPF calculated by GLr , and a bar with slanted-line pattern refers toMPF calculated by EMr . The 5th and 9th MPF originated from EMr are generally bigger than the MPF from GLr in overallfrequencies.

Based on the estimated MPF vector of u, the acceleration responses of the all nodes were calculated, as shown in Eq. (17),by using the eigenvector matrix Φ corresponding to the overall system of the washing machine.

€x ωð Þ ¼ �ω2x ωð Þ ¼ �ω2Φu ωð Þ (17)

Fig. 6 illustrates with the 1/3 octave band a comparison result between the experimental vibration results and theacceleration responses at the verification points calculated by Eq. (17). The dB reference of acceleration used at this time is1�10�6 m s�2. The bar with solid pattern in Fig. 6 refers to the acceleration response of the experiment, and bars with

Table 1Results of two mode selection methods.

Order of eigenvectors GLr Order of eigenvectors EMr

5 0.5729 5 0.31342 0.5600 11 0.24423 0.3031 10 0.24154 0.1860 14 0.2323

15 0.1072 9 0.22679 0.0992 19 0.2144

10 0.0825 15 0.212348 0.808 34 0.196435 0.0427 2 0.169813 0.0373 4 0.16866 0.0278 44 0.1666

50 0.0267 6 0.155611 0.0264 7 0.154347 0.0232 24 0.154030 0.0188 3 0.147214 0.0172 12 0.135458 0.0143 151 0.133985 0.0140 48 0.132654 0.0124 33 0.132519 0.0116 28 0.129646 0.0107 20 0.124561 0.0104 43 0.124340 0.0104 116 0.122812 0.0099 8 0.121562 0.0098 18 0.117438 0.0097 17 0.114792 0.0089 22 0.111327 0.0088 31 0.109428 0.0086 45 0.108837 0.0080 65 0.1073

Fig. 5. Comparison of the MPFs estimated by MEMs using the mode selections: (a) 5th MPF and (b) 9th MPF.

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353 349

Fig. 6. Comparison of the accelerations at verification points between measured and estimated results: (a) verification point # 1 and (b) verification point # 2.

Table 2Errors and standard deviation of the acceleration at verification points.

Frequency Error at point.1 (dB) Error at point.2 (dB)

GLr EMr GLr EMr

63 �5.23 �11.79 �1.45 �0.4880 �1.19 �13.75 0.17 0.04100 0.81 0.27 �8.14 �6.83125 �4.39 �19.35 3.77 �6.07160 �2.66 �2.33 0.42 �3.43200 0.06 0.04 �1.47 �2.63250 �1.40 �2.37 �2.88 �3.80315 �2.46 �5.80 5.39 5.31400 �1.24 �6.27 6.89 7.33500 �0.53 �0.69 3.27 4.80630 �0.55 �3.88 3.29 3.20800 �2.19 �1.99 6.29 7.111000 �2.77 �2.58 7.74 8.11Standard deviation 2.45 7.91 4.73 5.18

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353350

slanted-line pattern and crisscross pattern refer to the acceleration responses in the MEM by using GLr or EMr , respectively.For the numerical comparison, Table 2 describes the acceleration errors between the experiment and analysis and standarddeviation of those errors. The standard deviation σ was calculated by using Eq. (18) below.

σ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi ¼ 1ðErrorÞ2i

� �N

vuut(18)

where, N means the number of center frequencies in a 1/3 octave band between 63 and 1000 Hz. The acceleration vibrationresults estimated by the MEM based on the two mode selection methods corresponded well to the experimental results, asshown in (a) and (b) of Fig. 6. However, according to the numerically implemented Table 2, the acceleration result from

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353 351

using GLr has relatively fewer errors and a less standard deviation in most frequencies, compared to the result of EMr ,thereby being thought to be a better one in terms of reproducing the experimental values.

3.4. Radiated noise analysis and verification

Using all of the surface vibration responses at the time of dehydration of a washing machine, which were previouslyestimated in Section 3.3, an external radiated noise analysis was conducted based on the LMS.Sysnoise's acoustic direct BEM[20]. Fig. 7 illustrates the acoustic boundary element model of a washing machine. The field point related to the numerical

Fig. 7. The boundary element model of a washing machine and a reflection plane.

Fig. 8. Comparison of the acoustic pressure at the field point between measured and estimated results.

Table 3Errors and standard deviation of the acoustic pressure at field point.

Frequency Error at the field point (dB)

GLr EMr

63 4.11 �4.3280 �12.69 �18.65100 1.21 �3.49125 �2.70 �12.89160 0.55 �2.11200 5.52 �3.57250 �1.28 �11.85315 �2.13 �14.38400 �9.16 �19.95500 7.47 �9.40630 0.02 �10.23800 �2.42 �10.521000 �3.69 �11.50Standard deviation 5.43 11.58

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353352

analysis was placed 1.0 m ahead of the machine and above the ground. A reflection plane was also installed on the ground toreproduce real experimental conditions. In addition, a chief point was temporarily set with an aim to eliminate the internalresonance. In case of the experiment, the microphone was set at the same position of the field point in the numericalanalysis. The acoustic signal was measured by using LMS.Scadas at an interval of 1 Hz from 0 Hz to 2048 Hz. A-weightingwas applied to the signal. Fig. 8 illustrates a graph which compares with the experimental pressure result (solid pattern) andthe acoustic pressure at the field point calculated by the radiated noise analysis using GLr (slanted-line pattern) or EMr

(crisscross pattern). At this time, the acoustic pressure's dB reference is 2�10�5 N m�2. Fig. 8 indicates that the acousticpressure at the field point estimated by the mode selection GLr proposed in this paper represents well the experimentaltendency in comparison to that of the other selection method. Table 3 depicts the relative errors and standard deviation σmentioned above about the acoustic pressure at the field point.

According to Table 3, it can be verified that the acoustic pressure resulted from using GLr has fewer errors and lessstandard deviation compared to that of EMr . In the case of the overall SPL shown in Fig. 8, GLr has an error of about 1 dBAwhile EMr of about 10 dBA indicate a significant difference from the experiment. The selection method proposed by thispaper is more reasonable not only in the vibration estimation, but also in the radiated noise prediction compared to theexisting method. Therefore, this method can be regarded as a more suitable mode selection for estimating the vibration andnoise of a washing machine using the MEM.

4. Conclusion

This paper used the MEM and the acoustic direct BEM in order to identify the vibration and noise of a washing machineat the time of the dehydration. The MEM is a method to calculate the modal participation factors by using the experimentalvibration responses and the eigenvectors obtained from the finite element analysis. The vibration results estimated by theMEM are affected by the eigenvectors used for the calculation of the MPFs. Therefore, the suitable mode selection method isneeded. This paper used two types of the mode selection methods, the well-known EMr of MSC.Nastran and the new indexGLr proposed in this paper. EMr generally used in the vibration analysis was calculated based on the effective mass, but GLrwas calculated based on the index of global/local modes. In spite of this difference between two methods, both estimatedvibration results calculated by using the GLr and the EMr corresponded well to the experimental results at the verificationpoints. However, estimated noise result calculated by using the GLr showed the more accuracy prediction result than that ofthe EMr . Because the GLr mainly considered the bending modes which generally have the high radiation efficiency. In thecase of the numeric comparison using the errors and standard deviation, the values of the GLr were less than those of EMr .In other words, the mode selection proposed in this paper was found to display reasonable results regarding the vibrationand noise estimation of a washing machine based on the MEM and the radiated noise analysis. Additionally, this process isexpected to be utilized for the vibration and noise analysis and design improvement of a washing machine.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2014R1A1A2054372).

References

[1] N. Qi, B.M. Gibbs, Circulation pumps as structure-borne sound sources emission to semi-infinite pipe system, Journal of Sound and Vibration 264 (2003)157–176.

[2] H.S. Kim, J.E. Oh, The analysis of noise contribution about drum washer under dehydrating condition using multidimensional spectral analysis,Transactions of the Korean Society for Noise and Vibration Engineering 17 (11) (2007) 1056–1063.

[3] K.H. Kang, J.E. Oh, The analysis of structure-borne noise due to cabinet vibration from operational drum type washing machine using TPA, in:Proceedings of the KIEE Annual Autumn Conference, 2008, pp. 295–296.

[4] H.W. Chen, Q.J. Zhang, S.Y. Fan, Study on steady-state response of a vertical axis automatic washing machine with a hydraulic balancer using a newapproach and a method for getting a smaller deflection angle, Journal of Sound and Vibration 330 (2011) 2017–2030.

[5] A. Agnani, F. Cannella, M. Martarelli, G. Merloni, E.P. Tomasini, Dynamic characterization of a washing machine: numerical multi-body analysis andexperimental validation, in: Proceedings of the IMAC XXVI. Orlando, USA, 2008.

[6] P. Chiariotti, M. Martarelli, Noise source localization on washing machines by conformal array technique and near field acoustic holography, in:Proceedings of the IMAC XXVIII. Jacksonville, USA, 2010.

[7] J.E. Jung, J.E. Oh, Sound quality evaluation for laundry noise by a virtual laundry noise considering the effect of various noise sources in a drumwashing machine, Transactions of the Korean Society for Noise and Vibration Engineering 22 (6) (2011) 564–573.

[8] T. Otsuka, T. Okada, T. Ikeno, K. Shiomi, M. Okuma, Force identification of an outboard engine by experimental means of linear structural modeling andequivalent force transformation, Journal of Sound and Vibration 308 (2007) 541–547.

[9] B.K. Jung, W.B. Jeong, Estimation of vibration source and sound radiation of a refrigerator fan by using measured acceleration signals, Transactions ofthe Korean Society for Noise and Vibration Engineering 21 (9) (2011) 834–841.

[10] E. Parloo, P. Verboven, P. Guillaume, Force identification by means of in-operation modal modes, Journal of Sound and Vibration 262 (2003) 161–173.[11] Y.E. Lage, A.M.R. Ribeiro, Force identification using the concept of displacement transmissibility, Journal of Sound and Vibration 332 (2013) 1674–1686.[12] P. Guisset, M. Brughmans, Modal Expansion of Experimental Vibration Data for Acoustic Radiation Prediction, SAE Technical Paper 951090, 1995,

10.4271/951090.[13] P. Avitabile, J. O'Callahan, Frequency response function expansion for unmeasured translation and rotation DOFs for impedance modeling applications,

Mechanical Systems and Signal Processing 17 (4) (2003) 723–745.

B.K. Jung, W.B. Jeong / Journal of Sound and Vibration 340 (2015) 343–353 353

[14] B.K. Jung, W.B. Jeong, Estimation of vibration and radiated noise of a compressor by using modal expansion method, in: Proceedings of the KSNVEAnnual Spring Conference, 2011, pp. 554–555.

[15] M. Razeto, J.C. Golinval, M. Geradin, Modal Identification and Determination of Effective Mass Using Environmental Vibration Testing on an Electra-Dynamic Shaker, PTDC. XII IMAC. Honolulu, Hawaii, 1994, pp. 648–654.

[16] V.L. Tom, B. Marc, Using MSC/Nastran and LMS/Pretest to find an optimal sensor placement for modal identification and correlation of aerospacestructures, LMS International, 2010.

[17] R.J. Bernhard, B.K. Gardner, C.G. Mollo, Prediction of sound fields in cavities using boundary element methods, AIAA Journal 25 (1987) 1176–1183.[18] Z. Tong, Z. Zhang, Dynamic behavior and sound transmission analysis of a fluid-structure coupled system using the direct-BEM/FEM, Journal of Sound

and Vibration 299 (2007) 645–655.[19] J.P. Coyette, K.R. Fyfe, Solution of elasto-acoustic problems using a variational finite element/boundary element technique, numerical techniques in

acoustic radiation, in: Proceedings of the Winter Annual Meeting of the ASME, 1989, pp. 15–25.[20] LMS, LMS Virtual.Lab REV9 acoustic training, LMS International (2011).