CHARACTERISATION OF VIBRATION ISOLATORS USING VIBRATION TEST DATA

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1 Mustafa E. Levent, Page number 1 Abstract CHARACTERISATION OF VIBRATION ISOLATORS USING VIBRATION TEST DATA Mustafa E. Levent, KenanY. Sanlturk 1 Istanbul Techncal Unversty, Faculty of Mechancal Engneerng, Gumussuyu, 80191, Istanbul, Turkey 1 sanlturk@tu.edu.tr Faruk Bayraktar Arcelk A.S., Tuzla, 81719, Istanbul, Turkey Vbraton solators, usually made from vscoelastc materals, are used to suppress excessve vbratons n many applcatons. However, the optmsaton of vbraton solators s a dffcult process manly due to nherent dependency of the propertes of vscoelastc materals to many factors, ncludng temperature, frequency and stran. It s therefore necessary n many cases to dentfy the characterstcs of vbraton solators usng expermental technques. There are well-establshed expermental procedures for the determnaton of the dynamc propertes of vbraton solators or vscoelastc components n general. However, these procedures mostly requres expensve testng equpments ncludng hydraulc testng machnes equpped wth varous controllers and sensors. Ths paper presents two approxmate methods for the determnaton of dynamc propertes of vbraton solators or smlar components exhbtng elastc and dampng nonlneartes wthout the need for dedcated hardware. INTRODUCTION Vbraton solators are used n many applcatons to mnmze the transfer of vbratons from machnes to foundatons. Ths s almost always the case n domestc applances, a typcal example beng refrgerators where vbraton solators are used to mnmze the structural borne nose due to the vbratons of compressor. For ths partcular case compressor grommets are the vbraton solators whch are used to mnmze the transfer of vbratons from compressor to refrgerator body. However, the optmsaton of such solators s a dffcult process, one aspects of the dffculty beng the dependency of vscoelastc materals, from whch most solators are made, to many 1 Author to whom all correspondence should be addressed

2 Erdem Levent, Page number 2 factors ncludng frequency, temperature and stran. Therefore, t s often necessary to determne the dynamc propertes of vbraton solators before any optmsaton process. There are well-establshed expermental procedures for the determnaton of the dynamc propertes of vbraton solators. Drect Method [1, 2] s a well-known tradtonal method. Other methods for characterzng vbraton solators have also been developed: Indrect Method [1, 3, 4], Transfer Functon Method [5] and others [6, 7]. A common feature of these methods s that they all need expensve hardware wth sophstcated actuators and control systems. It s desrable to have some alternatve means for estmatng the dynamc propertes of vbraton solators wthout havng to resort to such systems, or when such systems are not avalable to engneers. Ths paper proposes two smple methods that can be used for estmatng the dynamc behavour of vbraton solators usng basc vbraton testng equpment. It s shown n ths paper that lnearsed propertes of vbraton solators can be estmated wth acceptable accuracy for varous applcatons by measurng the nput and output sgnals of a system comprsng a rgd mass and one or more vbraton solators. The second method ams to estmate non-lnear stffness and dampng propertes of vbraton solators by measurng the response of the system only. OVERVIEW OF THE PROPERTIES OF VISCOELASTIC ISOLATORS Dynamc propertes of vscoelastc solators exhbt temperature, frequency and stran (statc and dynamc) dependence [8-9]. It s also known that such solators show softenng behavor, known as creep [10] under statc preload and Mullns [11] effect under constant stran cycles. It s obvous that measurng the propertes of vscoelastc solators as a functon of all these varables s a complex and expensve task. Therefore, dependng on the objectves of the test, some of the varables mght be gnored or kept constant durng the measurement. In some cases, further steps are taken to elmnate some of the varables. For example, t s suggested n [12] that softenng behavor of vscoelastc components can be removed f hgh stran dynamc loadngs are appled to such components before the measurements are made. All the measurements presented n ths paper are performed at room temperature, hence the temperature effect s not nvestgated. However, t s amed at nvestgatng the effect of stran and frequency on the dynamc propertes of vbraton solators. Furthermore, the measurement technques presented n ths paper are demonstrated usng a specfc type of compressor grommets as shown n Fg. 1. Also shown n the same fgure s the structural model assumed for such solators. 1 m 2 Fg. 1 Compressor grommet and the structural model k*

3 Erdem Levent, Page number 3 ESTIMATION OF DYNAMIC PROPERTIES OF VISCOELASTIC VIBRATION ISOLATORS Estmaton of Isolator Propertes usng Frequency Response Functon f(t) k* M a(t). Fg. 2 Sngle Degree of Freedom (SDOF) model (M>>m) Ths approach ams to estmate frequency-dependent (but ampltude-ndependent) stffness and dampng propertes of the solator by establshng a SDOF system shown n Fg.2, comprsng a vscoelastc solator (k*) loaded by a rgd block wth mass M where the solator mass m s neglgble compared to M. The approach here s to measure the Frequency Response Functon (FRF) of the system by measurng the response - say acceleraton a(t) - and the exctaton force wthout any control, and processng them to obtan the system FRF. The measured FRF can then be processed usng modal analyss technques to obtan the natural frequency and the dampng of the system. Fnally, the modal parameters are related to the solator stffness and dampng propertes as: k =, 2 k real = M f, m k real η k * = + (1) 2 2 k real km where f s the natural frequency, η s the loss factor, k real and k m are the real and magnary parts of the solator stffness respectvely. It s obvous that the ampltude non-lnearty cannot be measured usng ths approach although the procedure s very fast and can provde a quck, yet qute approxmate propertes of the solator. It must be noted however that the frequencydependent propertes can be estmated by repeatng the measurements usng varous rgd blocks wth dfferent mass propertes. It s also worth notng that measurements are made under statc stran due to the preloadng of block mass M. Estmaton of Isolator Propertes Usng Log Decrement Method The method of estmatng the dynamc propertes of solators descrbed n ths secton s based on the same SDOF system shown n Fg.2. However, a dstnct feature of the method proposed here s that t does not requre the measurement of the force appled to the system. It s therefore only necessary to measure the response sgnal, e.g., acceleraton a(t), hence the measurement process s even smpler than that of the prevous secton although processng the sgnal usng the Log Decrement Method s somewhat more complcated.

4 Erdem Levent, Page number 4 Fg.3 Transent response of a)lnear SDOF system, b)nonlnear SDOF system The use of the Log Decrement Method s a well-establshed technque for the estmaton of the modal propertes (natural frequency and dampng) of a lnear SDOF. Ths s llustrated n Fg.3a n the case of a typcal transent sgnal where the sgnal ampltude decays exponentally and the perod T between the consecutve cycles remans constant. These propertes lead to well-known relatonshps between the logdecrement (δ), structural dampng (η) and the natural frequency f of the system: 1 A( t ) δ = ln, n A( t + nt ) 2δ η =, 2 2 4π + δ f 1 = (2) T where A(t ) s the modulus of the peak response at tme t (.e., A(t )= a(t ) ) and n s number of cycles after t. If the system s lnear, the same modal propertes are obtaned rrespectve of whch cycle or how many cycles are used n the evaluaton of the above equatons. When the system s nonlnear, however, the arguments stated above are no longer vald; the modal propertes estmated usng Eq.(2) wll vary dependng on the ampltude at whch these propertes are estmated. Ths s precsely the property that wll be utlzed to extract as much nformaton as possble from the transent sgnal about the ampltude- and frequency-dependent propertes of vscoelastc solators although the frequency and the ampltude wll be nterrelated. The dea behnd ths approach can be descrbed n connecton wth a transent sgnal n Fg.3b for a nonlnear SDOF system. As ndcated n Fg.3b, the perod T wll be dfferent for each cycle as the transent response decays and the peak ampltudes of ndvdual cycles A and the correspondng tme t can be determned usng expermental data. Ths makes t possble to estmate the dynamc propertes of vscoelastc solators as a functon of ampltude and frequency by usng the logdecrement method n a pece-wse fashon as summarzed n Eqs.(3-4). Once the natural frequency and the dampng factor are known as a functon of dsplacement ampltude X, the real and magnary parts of the solator stffness are calculated accordng to Eq. (5). It should be noted that the partcular mplementaton adopted n ths paper makes use of all the peaks and t s well suted for nosy data as some knd of averagng s nvolved. It s also worth restatng that the measurement procedure summarzed here provdes stffness and dampng propertes of vscoelastc components

5 Erdem Levent, Page number 5 as a functon of ampltude and frequency. However the ampltude and the frequency are not ndependent from each other. δ A 2 + A = ln A + A + 1 1, η 2δ = (3) 2 2 δ + 4π f = 1 0.5( t + t ) 0.5( t t ), X 0.5( A + A = ( 2πf ) 2 1 ) (4) k = M f, km _ = kreal _ η (5) real _ 2 Expermental Setup EXPERIMENTAL STUDY M c) Fg.4 a) Setup for Estmaton of Isolator Propertes usng Frequency Response Functon. b) Setup for Estmaton of Isolator Propertes Usng Log Decrement Method. c) Mathematcal model of the system The expermental setup used n ths study s amed at obtanng an equvalent SDOF system n Fg.2. As llustrated n Fgs. 4a and 4b, the test rg conssts of two blocks of mass, one at the base glued to the ground and the other on the top actng as a rgd mass M. In between these two blocks, there are 3 compressor grommets, propertes of whch are assumed to be dentcal. The mathematcal model of ths SDOF system can be represented as n Fg.4c. Ths system can be excted usng an nstrumented hammer (Fg.4.a) or usng an ordnary hammer (Fg.4b). In both cases, specal care was taken n selectng the hammer tps so as to ensure that only the SDOF mode of vbraton s excted. Results All the results presented here are obtaned usng a rgd mass of 10 kg,.e., M=10 kg. As stated before, Estmaton of Isolator Propertes usng Frequency Response Functon requred measurng both the mpact force and the response level usng the

6 Erdem Levent, Page number 6 setup n Fg.4a. A sample FRF s llustrated here n Fg.5 and the repeatablty of the measurement results are tabulated n Table1 for three ndependent measurements. A typcal acceleraton measurement llustrated n Fg.6 demonstrates the applcaton of the so-called Estmaton of Isolator Propertes Usng Log Decrement Method. In Fg.6, only the peaks marked wth arrows are used n the calculatons snce the ntal peaks ndcated buld-up perod. Processng the data n Fg.6 yelds the dynamc stffness and loss factor as presented n Fgs.7 and 8 respectvely. Repeatng ths process for many measurements can also provde statstcal nformaton, as llustrated n Fgs.9 and 10 for the case of 50 ndependent measurements. The results of the proposed methods are also compared aganst those obtaned va Drect Method measurements accordng to ISO standard usng MTS 831 materal testng system at the same preload and preloadng tme condtons (same statc stran). The comparsons are gven n Fg.11 at a specfc response level (.e., 0.05 mm). It seems that the approxmate methods proposed n ths paper slghtly overestmate the dynamc propertes of the solators compared to those of the MTS system. However, the results can be qute acceptable for many applcatons consderng the varatons of the propertes of nomnally dentcal solators. Table 1 Results of repeatablty tests Meas. No Natural Frequency (Hz) Dynamc Stffness (N/m) Loss Factor η Fg.5 Sample results for Estmaton of Isolator Propertes usng FRF 12 Acceleraton (m/s^2) Tm e(s) Fg. 6 The use of the peak ampltudes for the Estmaton of Isolator Propertes Usng Log Decrement Method

7 Erdem Levent, Page number 7 CONCLUDING REMARKS Two approxmate methods have been presented n ths paper for the measurement of the dynamc propertes of vscoelastc components wthout the need for expensve and dedcated hardware. The frst method s, n effect, based on lnearsng the system, hence provde a quck, yet average values for the component propertes. The socalled Estmaton of Isolator Propertes Usng Log Decrement Method, on the other hand, ams to provde ampltude-dependent parameters for the component under test. Another advantage of the second approach s that t requres perhaps the smplest test rg snce the response of the system only needs to be measured. The test results are verfed usng a dedcated Materal Testng System. It s found that the estmated parameters can be acceptable for many purposes. It s however worth notng that the use of the proposed methods for wder frequency range requres performng repettve tests, the nomnal natural frequency of the SDOF system beng adjusted each tme. REFERENCES [1] ISO Acoustcs and vbraton Laboratory measurements of vbro-acoustc transfer propertes of reslent elements Part 1: Prncples and gudelnes. [2] ISO Acoustcs and vbraton Laboratory measurements of vbro-acoustc transfer propertes of reslent elements Part 2: Dynamc stffness of elastc supports for translatory moton Drect Method. [3] ISO Acoustcs and vbraton Laboratory measurements of vbro-acoustc transfer propertes of reslent elements Part 2: Dynamc stffness of elastc supports for translatory moton Indrect Method. [4] D.J. Thompson, W.J. Van Vlet and J.W. Verhej, Developments of the ndrect method for measurng hgh frequency dynamc stffness of reslent elements, Journal of Sound and Vbraton, 213(1), (1998). [5] T. Prtz, Transfer functon method for nvestgatng the complex modulus of acoustc materals: sprng-lke specmen, Journal of Sound and Vbraton 72(3), (1980). [6] T.K. Ahn and K.W. Km, Dynamc stffness estmaton of pre-deformed elastomers, 8 th Internatonal Congress on Sound and Vbraton, (2001) [7] C.M. Rchards and R. Sngh, Characterzaton of rubber solator nonlneartes n the context of sngle- and mult-degree-of-freedom expermental systems, Journal of Sound and Vbraton, 247(5), (2001) [8] J.C. Snowdon, Vbraton and Shock n Damped Mechancal Systems. (John Wley & Sons, New York, 1968) [9] A. D. Nashf, D. I. G. Jones and J.P. Henderson, Vbraton Dampng. (John Wley & Sons, New York, 1986) [10] E.I.Rvn and B.S. Lee, Expermental study of load-deflecton and creep characterstcs of compressed components for vbraton control devces, Journal of Mechancal Desgn, 16, , (1994) [11] L. Mullns, Softenng of rubber by deformaton, Rubber Chemstry and Technology, 42, , (1969) [12] M. Sjöberg, Dynamc behavor of a rubber component n the low frequency range measurements and modelng, 7 th Internatonal Congress on Sound and Vbraton, , (2000) Dynamc Stffness (N/m) 0.00 Ampltude (mm) Dynamc Stffness (N/m) Fg.7 a) Modulus of dynamc stffness aganst ampltude (sngle measurement) b ) Modulus of dynamc stffness aganst frequency (sngle measurement)

8 Erdem Levent, Page number 8 Loss factor Ampltude (mm) Loss Factor Fg.8 a) Modulus of loss factor aganst ampltude (sngle measurement) b ) Modulus of loss factor aganst frequency (sngle measurement) Dynamc Stffness (N/m) a) Ampltude (mm) Dynamc Stffness (N/m) Fg.9 a) Modulus of dynamc stffness aganst ampltude (50 measurements) b ) Modulus of dynamc stffness aganst frequency (50 measurements) b) Loss Factor (N/m) Ampltude (mm) Loss Factor Fg.10 a) Modulus of loss factor aganst ampltude (50 measurements) b ) Modulus of loss factor aganst frequency (50 measurements) Dynamc Stffness (N/m) MT-1 MT-2 MT-3 LD DM-g1 DM-g2 DM-g Loss Factor 0.40 MT-1 MT-2 MT-3 LD DM-g1 DM-g2 DM-g Fg.11 Comparson of equvalent results wth those obtaned usng Drect Method. (Ampltude:0.05 mm, MT-1,2,3: Based on FRF, LD: Based on Log-decrement, DM-g1,2,3: Based on Drect method for ndvdual grommets)