Interaction Between a Vibration Exciter and the Structure Under Test

Transcription

1 Interction Between Vibrtion Exciter nd the Structure Under Test Pulo S. Vroto nd Leopoldo P. R. de Oliveir, University of São Pulo, São Crlos, Brzil In order to obtin good qulity dt in odl nd vibrtion testing, the experientlist should py ttention to interctions tht occur between the structure under test (SUT) nd the instruenttion used in the test. Mesureent errors cn rise fro nuber of possible cuses, including trnsducer ss loding effects, trnsverse stinger stiffness nd exciter- SUT interctions. This rticle is to study one of these sources of error, nely, the interction between the electrodynic exciter nd the SUT. The effects cused by the shker rture ss on the dynics of the SUT re ssessed in different testing conditions by using theoreticl odels nd experientl nlyses tht include the exciter dynic chrcteristics. Results fro these studies indicte tht the shker intercts significntly with the structure under test nd, under soe circustnces, exciter dynic effects cn be ccounted for in the process of iproving esured dt. The electrodynic vibrtion exciter hs been intensively used in odl nd vibrtion testing s ens to drive the SUT (Sructure Under Test). In experientl odl nlysis, coon prctice is to ttch the exciter to the SUT through flexible stinger nd force trnsducer. 1 The stinger is used to trnsit excittion signls to the SUT in single direction, reducing secondry fors of excittion (e.g., bending oents) due to possible islignents. The force trnsducer is used long with the exciter to esure the input force pplied to the SUT. In vibrtion testing, the SUT is ttched to the exciter tble through test fixture. In this cse, the SUT is driven by bse excittion signls tht re trnsitted to the SUT through the test fixture. In this type of test it is coon prctice to eploy closed loop test procedure where the exciter s tble is controlled so tht signl hving prescribed frequency content is pplied to the SUT. In both cses, it is well known tht the exciter intercts with the SUT nd tht in soe circustnces, distortions due to the rture s dynics cn significntly lter esured dt. Exciter dynics nd its interctions with the SUT hve been pproched by severl uthors. Tolinson studied the interction between the exciter nd the SUT, pying specil ttention to the force dropout tht occurs when the structure is excited in the vicinity of structurl nturl frequency. This work lso ephsizes the nonlinerity chrcteristics tht rise fro the electrognetic field tht is generted during the exciter working cycle. Olsen 4 studied the effects of the rture s ss nd suspension stiffness on esured dt nd pointed out tht the rture s ss effects on the esured Frequency Response Functions (FRF) cn be iniized by selecting n pproprite exciter for given test. Ro 6 followed the bsic developent by Olsen, but the dropout of the excittion force ws nlyzed in ore detil. According to his work the dropout phenoenon is due pririly to echnicl interction between the rture s ss nd the structure nd electrognetic chrcteristics of the exciter coil circuit. Exciter dynics s well s its interctions with the SUT hve been pproched in vrious text books on odl nd vibrtion testing. Ewins 1 forulted siple dynic odel tht explins the bsic echnicl interction between n exciter s rture nd the SUT. The uthor drws ttention to Bsed on pper presented t the 19th Interntionl Modl Anlysis Conference, Kissiee, FL, Februry 1. Arture X 1 Figure 1. Arture DOF dynic odel. F 1 X k c the fct tht the excittion force ust be directly esured during the test in order to obtin relible FRF esureents. McConnell developed n extensive study on exciter dynics nd its interction with the SUT. In his work severl nlyticl odels were developed to explin echnicl interctions between the exciter nd both free nd grounded SUT. In ddition, his work pproched the electric chrcteristics of the exciter-power plifier syste during the test, showing the bsic differences tht occur when the power plifier is used either in its voltge or current odes of opertion. Mi 9 lso developed n interesting study on the subject by using siple dynic odels to odel interctions between the exciter nd grounded structures. An insightful pper ws published by Lng 5 where severl siple tests re perfored in order to evlute exciter dynics. The objective of this work is to perfor n experientl study on two different vibrtion exciters, ttepting to evlute soe of their bsic dynic chrcteristics s well s their interction with the test environent. The results obtined in this study nd exhibited in this rticle constitute prt of project developed by one of the uthors towrds degree in Mechnicl Engineering. Review of Theory This section presents review of iportnt theoreticl spects tht re concerned with the dynics of vibrtion exciter. The theoreticl developent tht is described in this section is bsed on the work by McConnell tht presents coprehensive nlyticl developent of exciter dynics. The theoreticl developent described here considers the exciter s rture dynics s well s the electrodynic reltionships tht re needed to explin n exciter s electric behvior nd the two odes of opertion of the power plifier the voltge nd current odes. Arture Dynics. Figure 1 shows the rture dynic odel tht consists of the tble nd the electrognetic coil. In this cse, the exciter bse is considered to be rigidly ttched to the floor. Hence, we hve the DOF (Degree of Freedo) echnicl syste shown in Figure 1, where 1, k 1 nd c 1 represent the tble ss, stiffness nd viscous dping nd, k nd c the spider ss, stiffness nd viscous dping, respectively. As described in McConnell, the frequency doin ccelertions A 1 (ω) nd A (ω) exhibited by the tble nd the coil, respectively, cn be written in ters of the driving point nd trnsfer ccelernce FRFs, A 11, A 1 nd A, ccording to F Tble Coil k 1 / c 1 / c 1 / Bse 1 k 1 / SOUND AND VIBRATION/OCTOBER

2 11( ω) A A F A1( ω) 1 = 1 + F (1) 1( ω) A A F A( ω) = 1 + F where A 11 (ω) is the tble driving point FRF nd A 1 (ω) = A 1 (ω) is the trnsfer ccelernce FRF. The rture s FRFs present in Eq. (1) represent iportnt quntities since they give soe useful infortion bout rture behvior. The driving point nd trnsfer ccelernces A 11 (ω) nd A (ω) re expressed s A 1 r ( 1 M)( β r iβη ) A11( ω) = = + + F1 () r A r ( 1 M) ( 1 Mβ ) r i( η M 1 β η) A( ω) = = F () r A r ( 1 M)( β iβη ) A1( ω) = = + + (4) F () r where the uxiliry vribles ppering on Eqs., 3 nd 4 re defined s ω β = M = (5) ω11 11 ( r) = ( 1+ β M) r + jr( ζ + M)* 1 ζ β β r + jrζ β β + M β jrζ These three diensionless ccelernce FRFs re shown in Figure. They re identicl s the diensionless frequency rtio r rnges fro.1 to 1. In this frequency rnge it is observed the existence of nturl frequency tht is coon to ll FRFs nd tht it is the first resonnce of the two DOF syste. Above r = 1, the FRFs diverge nd ech one exhibits its own chrcteristics. The coil s ccelernce A (ω) decreses fter r = 1 up to r 31.6, where the tble presents dynic bsorber behvior for the coil. This ccelernce then increses up to r 15 (coil s resonnce) nd becoes constnt. The tble driving point ccelernce A 11 (ω) exhibits n ntiresonnce t r = β = 1 followed by its resonnce t r 15 nd reches plteu of 1.1 (i.e., 1+M) for higher frequencies (r >> 1). Finlly, considering the tble trnsfer ccelernce A 1 (ω), we hve resonnce t r 15 tht decreses t rtio of 4 db/ decde. This behvior shown by A 1 (ω) clerly indictes tht it is quite ipossible to control the tble fter r = 3, since its dynic response is ore ffected by externl forces thn by the coil s input. Therefore, the rture-coil syste hs n upper frequency liit for effective use of the exciter. In this cse this frequency liit is given by : ω r = β 1+ M = 1+ M = ω11 k( 1+ ) k 1 Electroechnicl Model. Figure 3 shows the electroechnicl odel used by Olse 4 nd McConnell to describe the electrognetic coupling on the rture-coil syste. This electroechnicl coupling is governed by severl preters: the coil resistnce R, inductnce L, input voltge signl E(t) nd the bck electrognetic voltge E bef. The equtions tht govern the echnicl nd electricl systes shown in Figure 3 re, respectively, given s: x + cx + kx = Fc() t (8) RI + LI + E = E() t (9) bef where Eq. 9 ws obtined using stndrd electric circuit reltionships. However, the electrognetic force F c (t), s well s the E bef voltge re hrdly dependent on the exciter s ode of opertion. The subsequent sections show the bsic equ- () (3) (6) (7) Tble nd Coil Accelernces Frequency Rtio [r] M-,1 β-1 η 1 -,5 η -,1 Figure. Diensionless ccelernce plot for n exciter rture. X Figure 3. Exciter electroechnicl odel. F c tions for both cses voltge nd current ode. The electrodynic phenoenon tht occurs on the exciter s circuits is inly due to the interction between the current nd the rture otion in the exciter s electrognetic field. The Apere lw reltes the electrognetic force F c (t) nd the current I(t) through the coil, ccording to Fc = ( nbl) I = KfI (1) where B is the gnetic field intensity generted by the current I(t) through n coils, ech one hving length equl to l. The constnt K f is the force-current constnt nd is equl to (nbl). Tolinson 3 developed theoreticl odel estblishing reltionship between the gnetic field B nd the coil position x given s: where B is the highest intensity tht the field B reches, x is the rture initil position, x is its oscilltion plitude nd x x is the xiu plitude. It cn be verified yet by Equtions 1 nd 11 tht the reltionship between the force nd the current is nonliner: x+ x Fc = nlb 1 I (1) x x In ddition, Lenz s lw gives the reltionship between the E bef nd the rture s velocity s: It could be noticed fro Eq. 13 tht the E bef lso presents nonliner fctor proportionl to the reltionship between excittion plitude nd rture xiu plitude nd its velocity, i.e., the excittion frequency. 3 Figure 4 shows the E bef behvior in ters of excittion plitude nd frequency. When the frequency is hlf of the originl excittion frequency, the E bef drops to new vlue tht is hlf of the originl one. The effects of the plitude of oscil- A A 11 A 1 k c E(t) E bef B d ψ x+ x = = B 1 dx x x x+ x Ebef = nblx = Kv x = Kv 1 x x x R I(t) L (11) (13) DYNAMIC TESTING REFERENCE ISSUE 1

3 ltion re relted to the nonliner behvior. The sller the plitude, the ore liner the E bef vrition. Power Aplifier Modes of Opertion. As stted erlier, the power plifier hs two odes of opertion, the current nd voltge odes, respectively. These odes estblish the voltge versus current reltionships during exciter opertion. The bsics of ech ode of opertion will be described. In the current ode of opertion, the reltionship between the input voltge to the plifier nd its output current is given by n eqution of the type E bef ω = ω 1 x x = 1 ω = 1/ ω 1 x x =.5 I( ω) = Gi( ω) V( ω) (14) The frequency doin versions of Equtions 8 nd 9 re given s ( k ω + jcω) X = KfI (15) ( R+ jlω) I + jk ωx = E (16) v where I nd E re reference plitudes for the current nd voltge, respectively. Notice tht the sipler reltionship between the E bef nd the tble s velocity (Eq. 13) is used in Eq. 16. The diensionless rture ccelernce in the current ode is given by: ( ω X) r A( ω) = = (17) Kf I 1 r + jζ r where r is the diensionless frequency rtio, now bsed on the rture nturl frequency, nd ζ is the rture viscous dping rtio. Thus, the voltge needed to intin the current gnitude is given s E ζ e E( ω) = = 1+ j β1 + RI (18) 1- r + jζ r where β 1 = ω / ω e nd ω is the rture s nturl frequency nd ω e is the brek frequency: R ω e = (19) L The electrognetic dping rtio ζ e is given by: C K vkf ζ e = = k R k () which is the result of the bck ef current being dissipted by the coil circuit. It is iportnt to reeber tht these equtions re vlid for low tble plitudes, which ke the nonliner ters vnish, s described by Tolinson. Siilr to the current ode, the voltge ode of power plifier opertion cn be odeled s gin, constnt in frequency up to given cutoff frequency, ccording to the following eqution E( ω) = Gv ( ω) V( ω) (1) where G ν (ω) is the plifier gin, V(ω) nd E(ω) re the plifier s input nd output voltges, respectively. Following siilr procedure, the diensionless rture ccelernce in the voltge ode of opertion is give s: ( ω X) r A( ω) = K E = ( + ML) r + j 1 1 ( ζ + ζe) + β( r ) f R 1 r () where the diensionless ss rtio M L = e / nd the inductive ss e is given by: Lc e = (3) R where it cn be noticed fro this lst eqution tht the inductive ss is dependent on rture dping. In order to gin dditionl insight to n exciter s dynic behvior when operted in either the current or voltge odes, siple siultion ws perfored with the physicl preters of n vilble vibrtion exciter. The exciter preters X / X x Figure 4. E bef behvior due to frequency nd plitude vrition. Arture Accelernce (diensionless) Figure 5. Diensionless bre tble ccelernce. Force Trnsducer Figure 6. Test on free-free steel be. Frequency Rtio (r) Current Mode Voltge Mode re shown below nd correspond to MB Dynics Modl 5 vibrtion exciter. These preters were obtined fro the exciter s operting nul. Force N Useful Displceent Mxiu Displceent Shker Mss kg Arture Mss....7 kg Arture Axil Stiffness N/ Coil Current (Mx) A (Low Ipednce) 4. A (High Ipednce) Coil Resistnce Ω (Low Ipednce) 5. Ω (High Ipednce) Figure 5 shows the bre rture ccelernce FRF behvior for the current nd voltge odes, ccording to Eqs. 17 nd, where it shows the differences between the two odes of opertion. In the current ode FRF (solid line), once the tble in Acceleroeter Signl Conditioning Spectrl Anlyzer out Shker Power Aplifier CPU SUT SOUND AND VIBRATION/OCTOBER

4 FRF Mgnitude, V 1 5 Mgnitude Force 1 7 Force Accelertion Voltge K Figure 7. Force nd current esureents for be test. psses through its echnicl resonnce, the FRF plitude becoes constnt. In principle, this iplies relible condition for exciter opertion s stted by McConnell. The other two FRFs depicted on Figure 5 correspond to the bre tble FRF in the voltge ode of opertion. There is cler distinction between these FRFs when copred to the current ode FRF. The bre tble resonnce is severely dped in the voltge ode FRF, ostly due to high electrodynic dping. Also, the two voltge ode FRFs shown in Figure 5 re different in the sense tht they use different vlues for the coil resistnce. It is seen tht the sller resistnce yielded gnitudes closer to the current FRF for higher frequencies. Accelernce, g/n b Accelertion Force SUT Experientl Results This section presents soe experientl results obtined fro tests perfored using two different exciters, B&K 481 with Power Aplifier B&K 77 nd MB Dynics Modl 5 with Power Aplifier SL5VCF. These tests were perfored in order to get soe prcticl understnding of the exciter s dynic behvior s well s the interction between the exciter nd the SUT. Results for Exciter-SUT Interction. This section shows sple ong ny results obtined in the developed project. The results shown here contin iportnt fetures concerned with the exciter-sut interction. Figure 6 shows the experientl setup used in one of the tests. This test used cold-rolled steel be ( ) ounted directly on the MB Dynics exciter tble. A Kistler 91 force trnsducer ws used to esure the input force to the be nd B&K 4371 cceleroeter ws used to esure the be s output ccelertion. Hnning windows were used in both the input nd output signls. The be ws excited with rndo signl in the -1 Hz frequency rnge. The SL5VCF power plifier ws djusted to operte in the voltge ode. Figure 7 shows the results obtined for the input force to the be nd the voltge for the voltge ode of the power plifier. Note tht dropout on the voltge vlues occurs t frequencies in the vicinity of the be s ungrounded nturl frequencies. These voltge dropouts coincide with the force dropout for the lower nturl frequency, but devite s frequency increses. Siilr behvior ws observed by McConnell in nuericlly siulted results. Figure 8 siultneously shows plot of FRF of n ircrft wing structure nd plots of the output ccelertion nd input force. These quntities were norlized so tht they could be plotted on the se grph. These experientl results were obtined using the B&K exciter nd power plifier. This figure clerly shows tht given structurl nturl frequency (notice the dshed lines on Figure 8) does not necessrily occur where the structure presents xiu vlues for the output c- Figure 8. () FRF, ccelernce nd force esureents of n ircrft wing structure. (b) Force dropout in shker testing. celertion. Rther, they occur where the input force drops out to iniu vlue. Agin, the se observtions were de by other uthors. 1,,6 Figure 8b shows essentilly the se behvior observed in Figure 8. In this cse, sipler structure ws used tht contins single nturl frequency in the -1 Hz frequency rnge. Once gin, the vlue of the structure s nturl frequency nd the frequency where the force dropouts occur re very close. Figure 9 shows n ccelernce FRF tht ws obtined using the B&K 481 exciter with rndo excittion nd the PCB ipct her. The experientl results re copred with results obtined fro n nlyticl odel of the SUT. The in feture of this test is tht even though two different excittion echniss were eployed to drive the test structure, the se vlue is obtined for the nturl frequency (bout 7.3 Hz), s shown in Figure 9. While not shown here, recll tht in ipct testing the input force uto spectrl density is constnt (or lost constnt) in the tested frequency rnge. Figure 9 still shows rigid body nturl frequency (bout 5 Hz) due to the SUT suspension syste tht is not present in the idel siulted FRF. Although the nturl frequencies re essentilly the se for both testing conditions, there is istch in the nti-resonnce tht occurs t frequency close to Hz. An interesting fct cn be observed in the nti-resonnce obtined in the exciter test. The nti-resonnce (dotted line) occurs t pproxitely 18 Hz, vlue tht is slightly lower thn the nti-resonnce vlue for the her test. The rture s suspension stiffness nd ss vlues for the B&K exciter re bout K = 1 kn/ nd =.454 kg, s stted in the operting nul. On the other hnd, the SUT hs ss of = 1.46 kg. These vlues llow us to get the frequency ω = [k /(+ )] 1/ = rd/ s = Hz! Hence, the nti-resonnce observed in Figure 9 is essentilly the nturl frequency of the rture-sut syste, but in DYNAMIC TESTING REFERENCE ISSUE 3

5 1 1 Off Accelernce, g/n Ipct Shker Siultion Accelernce, g/n 1 1 On F(t) (t) Figure 9. Anlyticl nd experientl FRFs (t) V(t) Figure 11. Electrognetic dping. 1 Mgnitude, g/v Frequency, khz Figure 1. B&K 481 bre tble FRF. Accelernce, g/n Tble H H 31 the SUT response it ppers s dynic bsorber ction. Finlly, the nti-resonnce plitude obtined with the exciter testing is higher thn tht obtined with ipct testing. This result, long with the frequency istch lredy discussed, indictes cler exciter-sut interction. Results for Arture s Dynics. This section shows sple of experientl results tht were obtined in tests tht ied to deterine n exciter s bsic dynic behvior. Figure 1 depicts the B&K 481 exciter s bre tble ccelertion FRF. This result ws obtined by esuring the exciter s tble ccelertion using the B&K 4371 cceleroeter while the excittion frequency ws vried in the -1, Hz frequency rnge. The ccelertion vlues were norlized by the input voltge fro the signl genertor. The result shown in Figure 1 exhibits pek frequency t 6885 Hz, while the exciter operting nul gives 7 Hz s the bre tble nturl frequency. Hence, lthough there is difference of bout 4%, this test ws considered effective in obtining the bre tble nturl frequency. Figure 11 shows results obtined when the shker tble is ipcted by n instruented her nd the tble ccelertion is esured in two different conditions. The solid line shows the FRF tht ws obtined when the power plifier is turned off nd the dshed line shows the se FRF with the power plifier turned on. In this condition no excittion signls were sent to the shker. The only difference between these test conditions is tht in the first cse the exciter s internl circuits re not electriclly energized while in the second the power plifier is turned on nd thus electricl energy is flowing through the coil circuit. The results shown in Figure 11 re suitble for observing the effects of the dditionl dping nd possibly stiffness induced by the coil electrognetic field tht is estblished during exciter opertion. Frequency (Hz) Figure 1. Effects of rocking otion on tble. Figure 1 shows results tht were obtined by ipcting the B&K 481 bre tble. The i of this test ws to investigte eventul rocking otions presented by the exciter s tble. For this purpose, two ccelernce FRFs were gthered, s shown in Figure 1. Accelernce H (ω) is nerly driving point FRF, where the exciter s tble ws ipcted t loction (Figure 1) nd the response ws esured close to the tble center. The plot of H (ω) shows single nturl frequency tht corresponds to the rture s suspension nturl frequency. The plot for ccelernce H 31 (ω) ws obtined by ipcting the tble t loction 1 nd esuring the ccelertion t loction 3. Both ccelernce FRFs show essentilly the se behvior fro -1 Hz, fter which they differ significntly. While H (ω) is nerly constnt up to the end of the frequency bndwidth, H 31 (ω) exhibits second resonnt pek t pproxitely 8 Hz followed by vlley t 33 Hz. This second nturl frequency indictes tht strong rocking otion is tking plce on the shker s tble when the ipct is pplied t point tht does not coincide with the rture s verticl xis. Soe other interesting results not reported here on dping differences were lso observed. This test tells us soething bout running bse excittion tests in the SUT tht re not syetricl with respect to the rture s verticl xis! A siple but interesting test ws perfored with the MB Dynics Modl 5 exciter in order to get n estite of the rture s suspension dping rtio. A sll ss with initure cceleroeter ws ounted on the top of the exciter tble. With the power plifier off, the rture ws plucked nd the free decy ccelertion ws e- 4 SOUND AND VIBRATION/OCTOBER

6 sured. These dt were used to estite the dping fctor through logrithic decreent, resulting in vlue of.6% for the rture s dping rtio. Sury nd Conclusions This rticle presents n experientl study on exciter dynic behvior s well s on its interction with the structure under test. Severl tests were perfored nd interesting chrcteristics were observed. The jor conclusion fro this work is tht the exciter represents n effective excittion echnis. However, it should be used with cre, since it intercts with the test environent. Also, the two power plifier odes cn distort test results. Acknowledgeents The uthors kindly cknowledge the finncil support provided by FAPESP (grnt # 99/9-6) to Mr. Leopoldo P. R. de Oliveir during the developent of the project nd to EESC-USP for providing the necessry lbortory fcilities. References 1. Ewins, D. J., Modl Testing: Theory nd Prctice, RPS, London, McConnell, K. G., Vibrtion Testing: Theory nd Prctice, John Wiley & Sons, NY, Tolinson, G. R., Force Distortion in Resonnce Testing of Structures with Electrodynic Vibrtion Exciters, Journl of Sound nd Vibrtion, Vol. 63, No. 3, 1979, pp Olse, N. L., Using nd Understnding Electrodynic Shkers in Modl Applictions, Proceedings of the 4th Interntionl Modl Anlysis Conference, IMAC 1986, Vol., pp Lng, G. F., ElectroDynic Shker Fundentls, Sound nd Vibrtion, April, pp Ro, D. K., Electrodynic Interction Between Resonting Structure nd n Exciter, Proceedings of the 5th Interntionl Modl Anlysis Conference, IMAC 1987, Vol., pp Instruction Mnul, B&K Vibrtion Exciter Syste V, Instruction Mnul, MB Dynics Modl 5A Vibrtion Exciter,. 9. Mi, N. M. M., Silv, J., Theoreticl nd Experientl Modl Anlysis, RSP, The uthors y be contcted t: vroto@sc.usp.br nd leopro@ sc.usp.br. DYNAMIC TESTING REFERENCE ISSUE 5