The nternatonal Journal of Analytcal and Expermental Modal Analyss v 7 n 2 p129149 Apr 1992 TUTORAL: SGNAL PROCESSNG ASPECTS OF STRUCTURAL MPACT TESTNG by M W Trethewey* and J A Cafeo, The Pennsylvana State Unversty ABSTRACT Spectra estmated from structural mpulse tests va fast Fourer transform (FFT) algorthms are susceptble to data acquston related dffcultes The problem stems from vbraton response sgnals whch decay suffcently slow that t s mpractcal to capture the waveform n ts entrety The problem s further compounded by the data acquston and processng constrants mposed by FFT equpment Ths paper revews and nvestgates varous aspects of FFT based sgnal processng n relaton to commonly encountered stuatons n structural mpact testng The analyss s based on tme and spectral relatonshps developed for a sngle degreeoffreedom (SDOF) system excted by a half sne force mpulse The truncaton problem s nvestgated and the results show that provded at least sx tme constants of data are captured the error n the estmated frequency response functon magntude and phase s less than 5% Next, the colorng effects of exponental wndowng on spectra from an dealzed mpact test are examned The results ndcate that the wndowng effects can be usually compensated for and accurate modal parameters extracted Fnally, the "double ht" phenomena s examned to llustrate the effects on the spectra and the mplcatons wth respect to modal analyss procedures c vscous dampng constant (Nsec/m) E [ ] expectaton operator F ampltude of half sne forcng functon for sngle ht (N) F1 ampltude of frst half sne forcng F 2 functon durng double ht event (N) ampltude of second half sne forcng functon durng double ht event (N) F(t) forcng functon F( ro) Fourer transform of forcng functon F(w,T1) fnte Fourer transform of forcng analyss analyss frequency (Hz) sample dgtal sample rate (Hz) lf frequency resoluton (Hz) GF ( w, ) auto spectrum of transent force G1x ( w, T) auto spectrum of vbraton response sgnal Gx ( w, T) cross spectrum of force and vbraton response sgnal G ( ro, T) auto spectrum of general transent sgnal f n functon natural frequency (Hz) H(ro) theoretcal frequency response functon *Portons of ths work were perfonned whle the author was on leave at: nsttute of Sound and Vbraton Research, The Unversty of Southampton, Southampton, Hampshre S9 5NH, Unted Kngdom Martn W Trethewey (SEM member), Assocate Professor and John A Cafeo (SEM member), Research Assstant, Department of Mechancal Engneerng, The Pennsylvana State Unversty, Unversty Park, PA 1682 Fnal manuscrpt receved: January 7, 1992 129
13 A rl 1 992 H ( w,1) estmated frequency response functon y(t) general tme sgnal k sprng constant (N/m) Greek m mass (kg) a ntegraton varable n number of spectral lnes R modal resdue snc(f) sne(f)/f T capture tme (sec) T 1 half sne nput force pulse duraton (sec) Yo tme delay between mpulses (sec) dampng rato cor corrected dampng rato lw exponental wndow tme constant (sec) fs sngle degree of freedom system tme t tme (sec) constant u( t y) unt step functon, actve when argument OJ frequency (rad/s) W BOX W EXP s boxcar data capture wndow exponental data capture wndow X(m) Fourer transfonn of system response m d damped natural frequency (:rad/s) m 1 half sne pulse forcng frequency ( 1t/T 1) m n (rad/s) undamped natural frequency (rad/s) X( m,1) fnte Fourer transform of system response * complex conjugate operator x system response (m) Y( ro,n fnte Fourer transform of a general tme sgnal mpact exctaton s one of the most common methods used for expermental modal testng Hammer mpacts produce a broadbanded exctat on sgnal deal for modal testng wth a mnmal amount of equpment and set up Furthermore, t s versatle, moble and produces relable results Although t has lmtatons wth respect to precse postonng and force level control, overall ts advantages greatly outwegh ts dsadvantages makng t extremely attractve and effectve for many modal testng stuatons [ 1 ] The use of mpulse testng wth FFT sgnal processng methods presents data acquston ondtons whch must be consdered to ensure that accurate spectral functons are estmated Problems ste1n from the avalablty of only a fnte duraton sample of the nput and output sgnals When a structure s lghtly damped the response to the hammer mpact may be suffcently long that t s mpractcal to capture the entre sgnal The truncaton effect manfests tself n terms of a spectral bas error havng the potental to adversely affect the estmated spectra The sgnal truncaton problem s further compounded n practce by the computatonal andl hardware constrants of the FFf processng equpment Typcally the equpment has a lmted number of data capture lengths or frequency ranges whch are avalable for an operator to select Normally a user s more concerned wth useable analyss frequences and less wth the data capture length Therefore, t s concevable that an napproprate data capture duraton could be used whch truncates the vbraton sgnal and ntroduces errors n the estmated spectra To suppress the truncaton a common practce s to artfcally force :t to decay wthn the data capture wndow [1,2,3] Ths artfcal reducton s obtaned by multplyng the slowly decayng vbraton sgnal by an exponental functon However, the applcaton of the exponental wndow must be consdered carefully snce t may also adversely affect the estmated spectra
A phenomena commonly encountered durng mpact testng s the so called "double ht" The "double ht" apples two mpulses to the structure, one ntally and one tme delayed Both the temporal and spectral characterstcs of the "double ht" nput and output are sgnfcantly dfferent than a "sngle ht" The nput force spectrum for the "double ht" no longer has the wde band constant type characterstcs of a sngle ht f not handled properly, the estmated spectral quanttes may be sgnfcantly affected creatng testng problems The purpose of ths paper s to examne the use of mpact vbraton testng n relaton to the constrants mposed by typcal FF sgnal processng technques The characterstcs of the mpact testng procedure are examned wth analytcal tme and spectral functons developed for an dealzed test: a sngle degreeoffreedom system excted by a half sne mpact force Once an understandng of the fundamental characterstcs s developed t s appled to examne the specfc stuatons encountered n structural mpact testng The relatonshp of the systems parameters wth respect to data capture requrements s evaluated The effects of exponental wndowng s developed to examne the effects on the estmated spectra and modal parameters Fnally, the "double ht" phenomena s examned by combnng the results from the sngle degreeoffreedom system excted by two mpulses, one of whch s tme delayed The results from these related studes are combned to provde nsght nto data acquston gudelnes for struc tural mpact testng The response of a sngle degree of freedom vscously damped system to an mpact can be detennned by modelng the hammer mpulse as a half sne pulse [4]: D 1f rosn t f(t) = 1 1J < t (1) The vbraton response of the SDOF system to ths force s: StS1j cm1f 2 = + 2:: 2 km1m + cm1 cos( m dt) (2) + The nternatonal Jo rn fan
cm1f + 2 2, 2 cos( m dt) k w1m + cm1 A typcal hammer mpact and sy stem response usng these relat onshps are shown n Fg 1 The smlarty of the half sne pulse to actual mpact hammer force sgnals s apparent along wth the slow transent vbraton response The Fourer tran sfonns of the Eqs ( 1) and (2) provde a means to develop frequency doman spectral functons F(m) = f(t)e_,mrdt oo X(ro) = x(t)em1dt (3) oo x(t) Note the n tegraton can be pe rfonned only when analytcal expressons exst for andf(t) n pract ce, these ntegrals cannot be evaluated because only fnte duraton dgtal representatons of the waveforrns exst The duraton, T, over whch the samples are acqured can have a very profound effe ct on the Fourer tran sfouns Typcally the ch oce of a capture duraton, T, s not a parameter drectly consdered For example, the sgnals would be dgtzed over a part cular capture duraton at a sample rate se le cted by the operator Usually, the operator se le cts the desred frequency range whch automatcally defnes the sample rate and capture duraton For a baseband analyss, the relatonshp between the analyss frequency, bandwdth (/), number of spectral lnes (n) and the data capture duraton (D s: MODAL HAMMER FORCE ACCELEROMETER /\ ACCELERATON " (\ 1\" v ANALYZER SNGLE DEGREEOFFREEDOM SYSTEM Fg 1 dealzed mpact test of a sngle degreeoffreedom system 132 Aprl 1992
T = analyss n = 1 at (4) Equaton (4) shows that the hgher the frequency range ( analyss Hz), the shorter the data capture du mton, T The effects due to truncaton can be evaluated by mposng a boxcar data capture wndo\y to the force and response sgnals A boxcar data capture wndow of duraton T s expressed as Wsox (t ) = 1 OS:tS: T T<t (5) The Fourer trans fonn for a fnte duraton sample of a tme sgnal, y(t), s expressed as Y(co,T) = W 8x(t)y(t)em1dt oo (6) Equaton (6) may be smplfed by changng the lmts of ntegraton Y(m, T) = T y(t)em1dt (7) The effects of sgnal truncaton on the frequency doman functons for the dealzed mpact test can be evaluated by applyng Eq (7) to Eqs (1) and (2) Assumng that the duraton of the mpact s small (T 1 < n the force sgnal s entrely captured Applyng Eq (7 ) to Eq ( 1) the fnte Fourer transform of the mpact force S (8) Equaton (8) shows that the force energy spectrum s not a functon oft and s unaffected by the wndow duraton provded the entre mpact sgnal s captured Smlarty, the tme response, Eq (2), may be evaluated by Eq (7 ) a A le [{ron +( rod ro)]r = 1 [con + (cod co)] + = +e (9) A 2e;p += ;(co,+ co) ( ro +ro )rt e 1 The nternatonal Journal of Analytcal and Expermental Modal Analyss 133
134 Aprl 1992 where Fm1 m]m+cmn k + mdcm1f \, 2 1 At = 2 2 {3 = tan 1 km]m The fnte Fourer transform of the vbraton response s a functon of the dynamc system parameters (m,c,k), the force characterstcs (F, T1) and the capture duraton (T) The Fourer transfonns n Eqs (8) and (9) can now be used to estmate spectra and the frequency response functon Snce the force and vbraton response sgnals are transent n nature the auto spectra may be estmated n terms of energy [5] G(m,T) = 2E r*(m,t)y(m,t) (1) For detennnstc sgnals Eq ( 1 ) collapses to G = 2Y*(m,T)Y(m,T) (11) Applyng Eq (1 1) to Eq (8) the auto spectrum of the half sne pulse sgnal s (12) The auto spectrum of the vbraton response sgnal s G(m,T) = 2X*(m, T)X(m,T) (13) The operatons n Eq ( 13) do not produce a compact for rn and wll be evaluated numercally n SlUbsequent sectons The Frequency Response Functon (FRF) can be fonned to descrbe the causal relatonshp between the force (nput) and vbraton response (output) n practce, the frequency response functon s estmated by the rato of the cross spectrum between the two sgnals and the nput auto spectrum Snce the sgnals n ths study are nose free and determnstc the functon may be detennned drectly fr om the Fourer transforms n Eqs (8) and (9)
H(mT) GFx{m,T) X(m,T) = = GFF ( T) F( m, T) W, (14), " Due to the resultng complexty of Eq (14) t wll be evaluated numercally To provde a comparson bass to evaluate truncaton effects, t s necessary to express the unbased spectral functons Provded the force sgnal s not truncated, Eq ( 12) represents an unbased spectrum The unbased vbraton spectrum can be evaluated by combnng Eqs (9) and (13) wth the varable T (the data capture duraton) set to nfnty The actual frequency response functon can be developed drectly from the sngle degreeoffreedom system parameters 1 1, 1:1 1: : ; (15), The functons developed n ths secton provde the foundaton to evaluate the effects of truncaton n an dealzed structural mpact test The expressons developed n Secton 2 can be used to examne effects on spectra estmated from mpact tests where the vbraton decays suffcently slowly that t can not be captured n ts entrety Consder a sngle degreeoffreedom system, Fg 1, wth a low natural frequency and dampng rato (fn =1 Hz, s=oo 1) The tme constant for ths second order system s gven by 1 2 1 r A FORCE AUTOSPECTRUM z " 8 r 6 r 4 r u 2 7 8 r 9 r 2 8 6 E 4 c " E " u 2 2 1 Tme (s) 2 B 3 1 " " :J c " ::::; 11 12 " 13 1 / N 14 15 / \ 4 " Cl 6 8 5 1 1 5 2 25 3 16 17 ll 5 1 1 5 2 25 3 35 4 Tme (s) Frequency (Hz) Fg 2 a) Half sne mpact force, T,=1 (sec) b) Vbraton response of a sngle degreeoffreedom (fn=1 Hz, (=1) to the mpact force n Fg 2a Fg 3 Auto spectrum for half sne mpact force n Fg 2a, T1:1 (sec), F:1 (unts of force) The nternatonal Journal of Analytcal and Expermental Modal Analyss 135,
] 1 1 = system (l)n (1}(1[2n]) = 1 59 sec l (16) Assume the system s excted by an mpulse of duraton 1 ms wth a force level arbtrarly set to 1 N The tme response sgnals are llustrated n Fg 2 The entre mpulse sgnal s easly captured, whle the lght dampng causes the system response to decay slowly requrng over 15 seconds to elapse before the ampltude s neglgble Typcally a capture duraton of ths length s mpractcal To examne the effect of truncaton on the spectral functons, consder the applcaton of Eqs (13) and (14) to ths data wth two dfferent capture duratons Frst, consder a capture duraton equal to 1/2 system tme constant, T=795 sec Fgure 2 llustrates the truncaton of the vbraton sgnal s sgnfcant n practce, processng of sgnals truncated ths severely would be jud g ed unacceptable, however they wll be used to accentuate the nduced bas errors The force spectrum s shown n Fg 3 and s not subjected to any truncaton effects because t s captured n ts entrety The spectrum has a decayng lsnc(f)l type characterstc The frst null occurs at frequency of 3/2T1 and subsequent nulls at nteger multples of 1/T1 Provded the test frequency range of nterest s approxmately less than l{f 1 (ampltude decreased by approxmately 1 db) a broadband force s produced The resultng broadband force characterstc from an mpulse s one of the most attractve features of mpact exctaton The theoretcal (Eq (15)) and estmated (Eq (14)) frequency response functons are compared n Fg 4 The estmated functon exhbts a bas n both the magntude and phase The magntude has the proper basc shape, however the truncaton causes a sne( f) type functon to be supermposed on the theoretcal functon The values around the resonant frequency are most sgnfcantly affected The characterstcs of the peak are smeared nto adjacent frequences and the estmated phase oscllates around the theoretcal value The phase agrees well at the natural frequency, but devates at surroundng frequences, Next consder a more realstc capture duraton correspondng to 2 tme constants of vbraton data, T =21 run =3 18 sec The force spectrum remans unchanged from Fg 3 The estmated frequency response FREQUENCY RESPONSE FUNCTON FREQUENCY RESPONSE FUN(,,ON 18r 18 Q> ( CJ 9 9 ; A " 1 t _ \ 1 c Q 18 1 1& 1 L& 1= 1 2 A : =, 4 ao :_ = _ = : _t, 1 1, Theory Truncated cao Q> Q t) ( cs c g 9 1 9 18 L 1 2 l A Theory Truncated l cao 3 N 3 4, u t B 1 o o 1 U ll 5o """L " 2 4 6 8 1 12 14 16 18 2 22 3 N 3 4 so 2 4 6 8 1 12 14 16 18 2 22 B Frequency (Hz) Frequency (Hz) Fg 4 Frequency response functon estmated wth 1/2 of a tme constant of data ( fn=1 OHz, =1) [Theoretcal, Estmated ] Fg 5 Frequency response functon estmated wth 2 tme constants of vbraton data ( fn=1 Hz,,=1) [Theoretcal, Estmated 136 Aprl 1992
functon s shown n Fg 5 The magntude shows a lsnc(/)1 as before, however the functon s closer to the theoretcal The correlaton s sgnfcantly mproved around the resonant frequency n both magntude and phase By comparng Fgs 4 and 5 t s apparent that the capture tme plays an mportant role n producng an accurate estmate of the frequency response functon Establshng a relatonshp between the capture duraton and the nduced bas error would provde a crtera to mantan the truncaton error below an acceptable level Examnaton of Fgs 4 and 5 shows that the error s sgnfcant at frequences near the natural frequency Furthennore, snce data must be most accurate n ths regon for modal analyss applcatons, t s logcal that any analyss concentrate on these frequences The frequency response functon magntude error at the natural frequency was evaluated wth ncreasng data capture duratons wth the results shown n Fg 6 The percent error n the frequency response functon magntude (at/,) s presented versus the number of tme constants of data captured The relatonshp shows large error values exst for hgh degrees of truncaton As the amount of data captured ncreases to around fve tme constants,lhe error decreases dramatcally f at least 6 tme constants of data are captured the estmated response functon magntude s wthn 5% and the phase s wthn 25 degrees of the respectve theoretcal values Both of these values are vald for a baseband frequency range equal to twce the systems natural frequency (e, fanalyss = 2fn) The number of tme constants of data captured can be drectly related to the analyss frequency selected by an analyzer operator The sx tme constant capture crtera mples the nequalty (17) The capture perod Ts related to the samplng tme va the FFblock sze (T = N at) or the samplng frequency fs a mp l e= l/ &) (18) n order to satsfy the Nyqust samplng crtera, the maxmum frequency sampled,fsample must be at least twce the maxmum desred frequency,analyss n practcefsample s usually more than twcefanalyss G ven 1 a o LL 2 4 8 8 1 12 Tme Conatents Captured Fg 6 Percent error of the frequency response functon magntude at the natural frequency of a sngle system versus tme constants of response data captured The nternatonal Journal of Analytcal and Expermental Modal Analyss 137
a 4 lne FFT analyzer wth 124 pont blocksze, the analyss frequency s related to the samplng frequency by N 124 /sample = 4 a nalyss = 4 analyss (19) Substtutng Eq (19) nto Eq (18) yelds (2) Solvng Eq (2) for the dampng rato yelds > 6 analyss = O 24 analyss {27r}4 n n (21) Eq (21) shows that the rato of the analyss frequency to the system natural frequency determnes the dampng rato necessary to satsfy the data capture crtera For a frequency rato fanalys slfn) of 1, a dampng rato of at least 24 s requred to capture at least sx tme constants of data f the frequency rato s 11 only a dampng rato of 27 s requred to meet the crtera Snce the force mpulse s entrely captured n the data wndow, the force autospectrum s free: from any truncaton bas error The bas error n the frequency response functon s solely nduced by an nsuffcent capture of the slowly decayng vbraton sgnal Condtons for the error to be sgnfcant exst when the dampng s lght and the rato of the selected analyzer analyss frequency range to the syste:m natural frequency s large When the data capture crtera developed n Secton 3 can not be satsfed an exponental data capture wndow may be appled to the system response to force t to decay faster An exponental wndow s defned by t W xp(t)= e rw OtT T<t (22) where the tme constant, rw, s selected by the operator Often, rw s selected so the vbraton s forced to decay to 2% of ts ampltude wthn the capture duraton T = = 2556T 1" w ( ) (sec) ln 2 (23) Ths wndow s referred to as the 2% exponental wndow Fourer transfonns of the wndowed samples are affected by ths wndowng operaton Consder Eq (6), wth the boxcar wndow replaced by the 2% exponental wndow 138 Aprl 1992
J$S4 l _Ul lu1 SS42ES 222 ==ss & =aacsa aaa aaa aaatsss caaaa a_&_s_j_s_s aaaaaaaa aaa;jjsscs& aaa sss4ssss 4SSSS4SSSS X(w,T) = W xp(t)x(t)e wrdt oo T t = e o2556tx(t)ezwtdt (24) Substtuton of the sngle degreeoffreedom system vbraton tme response, Eq (9), nto Eq (24) provdes a means to evaluate the frequency doman effects of the exponental wndowng operaton An alternatve to performng the resultng ntegraton nvolves the varable substtuton shown n Eq (25) nto the unwndowed Fourer transfoun [3] l m = m (25) Equaton (25) was appled to Eq (9) n subsequent numercal evaluatons of exponental wndowng operatons The wndowng effects on the Fourer transfonns can also be vewed n the frequency doman as a convoluton of the actual sgnal transfonn wth the wndow transform [6] Y(m,T) = W(a)Y(w a)da (26) oo where Y( a) s the Fourer transform of the system response, W( a) s the wndow Fourer transfonn and a s an ntegraton varable Equaton (26) mples the spectral effects of exponental wndowng can be examned wth the respectve wndow Fourer transfonns Fgure 7 compares the Fourer transfonn magntudes of a boxcar wndow (Eq (5)), and an 2% exponental wndow (Eq (22)) both wth a duraton T ( rw=2556d The fgure shows the characterstc jsnc(f)l leakage lobes of the boxcar wndow wth nulls occurrng at frequences equal to nteger multples of 1 rr The 2% exponental wndow has a exponental 1 \ r r r Boxcar Exponental () Q) Q c en Q) r:: 1 :3 r:: " " 1\ A 1\ 1\ " 1\ 1\ f, 1\ 1 & 2 4 6 8 1 12 Frequency ( 1/T) Fg 7 Fourer transform magntudes of a boxcar (duraton=n and an exponental wndow (duraton:t, r = T ln{2}) The nternatonal Journal of Analytcal and Expermental Modal Analyss 139
type behavor wth a small rpple at the hgher frequences and smlar roll off characterstcs The use of an exponental wndow to reduce truncaton of a slowly decayng vbraton sgnal rases several questons Frst, to what level may the orgnal sgnal be truncated and artfcally forced to decay whle not colorng the spectral nfonnaton to preclude accurate modal parameter extracton? Secondly, because of the severe leakage characterstcs of the exponental wndow, can the responses of closely spaced modes be obscured suffcently such that they can not be separated? (a) Truncaton Effects n Exponentally Wndowed Data l The truncaton/wndowng queston was nvestgated numercally wth the same sngle degreeoffreedom system fn=1 Hz,,=1) n Secton 3 The system frequency response functon was estmated for dfferent capture tmes selected to accentuate the wndowng effects Another set of responses was generated by applyng an exponental wndow to the truncated responses The modal parameters were then extracted from each of the respectve FRFs usng a complex exponental modal extracton algorthm The modal parameters extracted are the natural frequency fn), the dampng rato (C), and the modal resdue (R) The estmated parameters were then compared to the theoretcal values Fgure 2 llustrates the half sne force pulse and the correspondng system response wth a capture tme of 318 seconds (two system tme constants) The same response wth a 2% exponental wndow appled s shown n Fg 8 The decay envelope of the response ncludes both the nherent system and artfcal wndow decay behavor The more rapd decay of the exponentally wndowed vbraton response s apparent The frequency response functon estmated wth a 318 second boxcar wndow s shown n Fg 5 and the exponental wndowed data results n Fg 9 The wndowng smoothed the lsnc(f)l effect across the entre frequency range and reduced the magntude around the resonance by about 1 db To further accentuate the wndowng/truncaton effect the capture tme was decreased to one system tme constant, 159 sec The frequency response functon estmated wth a 2% exponental appled to the vbraton response sgnal s shown n Fg 1 The ampltude at the resonance relatve to the theoretcal value decreases by about 16 db The phase at the natural frequency s dentcal but shows a lower gradent slope, ndcatve of the artfcal dampng nduced by the wndow FREQUENCY RESPONSE FUNCTON j ; ; j ; ; 6 4 s 2 N _,, c 4) e u cs V) Q 2 4 ll 6 5 1 v 15 ",, vv,, _ 2 25 3 35 Q) Q _, Q) () cs c: D 18 9 9 18 u 1 3 c 2 cs ::1 _, 3 3 4 (\) 5 2 4 6 8 l j_ 1 12 14 A Theory Truncated B 16 18 2 22 Tme (s) Frequency (Hz) Fg 8 Two tme constants of vlbraton response of a sngle (fn=1 Hz, =1 ) to the mpact force n Fg 2 after the applcaton of a 2fo exponental wndow Fg 9 Frequency response functon estmated wth 2 tme constants of data wth a A, exponental wndow (fn=1 Hz, =1) [Theoretcal, Estmated ] 14 Aprl 1992
Clearly, these two examples show that the estmated FRFs dffer from theory The queston remanng s, how much does ths affect the modal parameter extracted from these functons? A complex exponental routne was used to estmate the modal parameters for four cases: wndowed and unwndowed data wth capture duratons of one and two tme constants respectvely The results are lsted n Table 1 Resonance frequences extracted for all the cases are wthn 1% of the theoretcal value of 1 Hz The dampng values for the unwndowed FRFs, cases 1 and 3, are also reasonable compared to the theoretcal value ofoo 1 The dampng s slghtly hgher for case (1 tme constant) than that for case 3 (2 tme constants) as a result of leakage The dampng values for the wndowed FRFs, cases 2 and 4, are at least three tmes larger than the theoretcal value Ths s caused by artfcal dampng nduced by the exponental wndow A corrected dampng value can be calculated wth knowledge of the exponental wndow tme constant [3] (27) r Q _, Q) (f) cs c:: g G), = c QO cs : _, 3 N 18 9 9,_ 18 1 2 3 4 FREQUENCY RESPONSE FUNCTON l J t A Theory Truncated B 5 L_ 2 4 6 8 1 12 14 16 18 2 22 Frequency (Hz) Fg 1 Frequency response functons estmated wth 1 tme constant of data wth a At exponental wndow (fn=1 Hz, =1 ) [Theoretcal, Estmated] TABLE 1 EXTRACTED MODAL PARAMETERS FROM A SDOF SYSTEM MPULSE TEST WTH BOXCAR AND EXPONENTAL WNDOWS Case Wndow # Type Theory None Mode fn T l R corr # (Hz) (s) (s) 1 1 1 1 1 Boxcar 1 995 133 11 159 2 2 /o Exp 3 Boxcar 4 2 /o Exp 1 1 1 1 495 16 99 159 49 1 14 1 318 1 299 14 1 318 818 The nternatonal Journal of Analytcal and Expermental Modal Analyss 141
Applyng Eq (27) to the dampng rato estmates obtaned from the wndowed data produces corrected values wthn 1% of the theoretcal All the modal resdue (R) values are wthn 1% of the true value n practce, dffcultes can be encountered when tryng to utlze Eq (27) wth actual expermental data The numercal values of the equatons tern1s can be small whch can nduce large errors n the corrected dampng rato f the orgnal dampng estmate s naccurate Therefore, use of Eq (27) must be handled carefully (b) Exponental Wndowng wth Closely Spaced Modes Another queston concernng exponental wndows s the ablty to dstngush closely spaced modes because of ncreased leakage Due to the number of parameters and complexty of the relatonshps t s dffcult to completely evaluate the problem Therefore, the problem wll be llustrated wth several numercal examples where the effects are pronounced The examples wll descrbe the results from a two DOF system wth close natural frequences To generate the system FRFs, decoupled modal coordnates are used allowng two SDOF FRFs to be supermposed wth Eqs (8), (9), (14) and (25) The frst mode was specfed at 1 Hz wth =1 and R= 1 whle the second mode was at 12 Hz wth =5 and R=1The theoretcal FRF s compared to one estmated by a boxcar wndow (T=318 seconds: two tme constants of data for the 1 Hz system, frequency resoluton = 5 Hz) n Fg 11 "A TABLE 2 EXTRACTED MODAL PARAMETERS FROM A TWO DOF SYSTEM MPULSE TEST WTH BOXCAR AND EXPONENTAL WNDOWS Case Wndow # Type Mode fn # (Hz) carr R T f lf (s) (s) (Hz) Theory None 1 1 1 1 5 1 Boxcar 2 2 /o Exp 3 Boxcar 4 2 /o Exp 5 Boxcar 6 2 /o Exp 7 Boxcar 8 2 /o Exp Theory None 9 Boxcar 1 2 /o Exp 11 Boxcar 12 2 /o Exp 13 Boxcar 14 2o/o Exp 15 Boxcar 16 2 /o Exp 1 989 642 1 13 51 1 11 16 1 13 49 1 13 23 1 1 12 32 1 11 14 1 12 299 2 12 5 2 12 99 2 12 376 2 12 5 1 2 12 37 2 12 7 2 12 214 2 12 5 1 2 12 2 13 156 159 5 11 987 159 49 5 1 159 25 1 978 159 49 25 14 318 5 17 1 318 818 5 961 318 25 14 995 318 818 25 1 5 111 159 5 52 99 159 49 5 58 159 25 46 99 159 49 25 16 318 5 52 997 318 818 5 754 318 25 5 1 995 318 818 25 142 Aprl 1992
smlar comparson s presented n Fg 12 wth the frequency resoluton ncreased by a factor of fve to 25 Hz The leakage effects are more pronounced Fgure 13 shows the FRF estmated wth a frequency resoluton of 25 Hz when an 2% exponental wndow s appled to the vbraton response sgnal The peak ampltudes are lower than theory and they tend to blend nto one another as a result of the exponental wndow leakage To examne these effects on the modal parameters extracted under these condtons a complex exponental algorthm was appled to 16 dfferent cases wth the followng parameters vared; 1) T the capture tme; 2) tf the frequency resoluton; 3) wndow type boxcar or 2% exponental The results are summarzed n Table 2 The modal extracton routne predcts the resonance frequences for all of the cases Q ) cs = Q 16 9 9 18 3 2 FREQUENCY RESPONSE FUNtON 1 1, r 4 =::: :, _ A Theory Truncated, Q G> fl) cs c: Q 18 9 9 18 3 2 FREQUENCY RESPONSE FUNCTON 1 1 t l /,, ; " \ Theory Truncated 1 1 1 o L _ t: 2 3o 2 4 6 8 1 12 14 16 18 2 22 Frequency (Hz) B J N 1 2 1, B t, 3 2 4 6 8 1 12 14 16 18 2 22 Frequency (Hz) Fg 11 Two DOF FAF estmated wth 2 tme constants of data of mode 1 ( fn=1 Hz, =1) and L\1::5 Hz [Theoretcal, Estmated ] Fg 12 Two DOF FAF estmated wth 2 tme constants of data of 1 ( fn=1 Hz,,:1, and Af=25Hz [Theoretcal, Estmated ] G> Q ) cs c: Q 18 9 9 18 3 2,, = FREQUENCY RESPONSE FUNCTON 1 c:: to e ::1 j_ A Theory Truncated J 1 N 2 B 3 2 4 6 8 1 12 14 16 18 2 22 Frequency (Hz) Flg 13 Two DOF FAF estmated wth 2 tme constants of data of mode 1 (fn=1 Hz, =1) wth a k exponental wndow and Af=25 Hz (Theoretcal, Estmated ] The nternatonal Journal of Analytcal and Expermental Modal Analyss 143
wthn 2% of the theoretcal values The dampng values for the truncated unwndowed data (cases 1,5,9, 13) dffer from the theoretcal values sgnfcantly The frequency resoluton of these FRFs was 5 Hz The dampng values for the truncated unwndowed data (cases 3,7,1 1,15) all compare to wthn 1% of the true values These cases have a frequency resoluton of Af25 Hz The corrected dampng values for the wndowed data (cases 2,4,6,8, 1, 12, 14, 16) are all wthn 1% of the true values n general the modal resdues for the wndowed data (cases 2,4,6,8, 1, 12, 14, 16) all compare well (wthn 1%) wth the true values for f equal to both 5 and 25 Hz The modal resdues for the boxcar wndowed data wth tf25 Hz (cases 3,7,11, 15), dffer from the true values by as much as 5% Some observatons can be made from the data presented n Table 2 The modal extracton routne predcts the resonance frequences accurately for all the cases of truncaton, wndowng, and frequency resoluton studed Modal resdues and corrected dampng values for all exponentally wndowed data are accurate The two dfferent frequency resolutons studed dd not sgnfcantly nfluence the modal parameter extracton Both resolutons were fne enough to adequately separate the closely spaced modes Although dffcult to make explct statements about sgnal truncaton and n relaton to the applcaton of exponental wndows, some general observatons can be made Ths analyss has demonstrated that spectral functons estmated from mpulse tests are affected by the wndowng/truncaton of slowly decayng vbraton response data However, the results also ndcated that modal parameters estmated from ths spectral data are rather robust and not sgnfcantly affected by the truncaton Ths apparent contradcton s answered by realzng that the complex exponental modal extracton routne used n ths study was able to recover accurate modal parameters from even spectral data whch s vsbly based Therefore, the agreement obtaned n ths study must be attrbuted to the capabltes of the modal extracton routne and not the robust nature of the data Snce smlar results may not be obtanable wth other algorthms, t s mportant to estmate the most accurate FRFs possble Therefore, t s better to ncrease the capture duraton to capture at least sx tme constants of data rather than apply an exponental wndow f the wndow must be appled one must be aware of consequences assocated wth wndowng, specfcally: 1) colorng and leakage of the spectra nduced by the convoluton of the wndow transfonn; 2) possble numercal dffcultes n back calculatng the true dampng rato When physcally mpactng a structure wth an nstrumented hammer, a so called "double ht" occasonally occurs Under ths condton two pulses are appled to the structure, as llustrated n Fg 14 The cause of the second mpact can be from many sources related to the test procedures and system dynamcs A common occurrence s when the structures response to the ntal mpact s rapd and large, such that a second contact s made wth the hammer before t s fully retracted Expermentalsts usually reject "double hts" from enterng the ensemble average set used to estmate the spectral quanttes These precautons are mplemented because past experence shows that the qualty of a FRF estmate s degraded when a double ht s ncluded n the averagng ensemble To examne the double ht phenomena, consder the applcaton of a double ht type force, as llustrated n Fg 14, to the sngle degreeoffreedom system n Fg 1 Snce the system s lnear, the expressons presented n Secton 2 can be combned to produce both the temporal and spectral quanttes The vbraton tme response sgnal s x(t) = xl (t) + x2 (t ro)u(t ro) (28) 144 Aprl 1 992
wherex 1 (t), andx 2 (t) are obtaned from Eq (2) wthf and T 1 equal to the respectve characterstcs of each of the two pulses, u(t(>) s a unt step functon and ro s the tme delay between the mpulses Smlarly, the Fourer transforms of the force and response can be evaluated by combnng Eqs (8) and (9) wth the relatonshp Y( OJ, T) = 1f (OJ, T) + Y2 ( m, T)e o (29) where Y( m,n represents the respectve characterstcs from ether the frst or second mpulse 1 Q) CJ & CJ Cl Q 8 6 4 2,,,,, 1,,,,,,,,, 2 4 6 8 1 12 Tme (s) Fg 14 Typcal double ht experenced n a structural mpact test 18 FREQUENCY RESPONSE FUNCfON ao DOUBLE HT FORCE SPECTRUM, Q G,c g,, ::s c, : 3 N 9 9 18 1 2 3 4 \ A Theory Truncated B, c CO 9 cg :: 1 r J N 11 5 L 2 4 6 8 1 12 14 16 18 2 22 Frequency (Hz) 12o 1 2 3 4 5 6 7 8 9 1 Frequency (Hz) Fg 15 FAF estmated wth 6 tme constants of data (f,=1 Hz, =1 ) excted by the double ht n Fg 14 [Theoretcal, Estmated ] Fg 16 Auto spectrum for a double ht half sne mpact forces n Fg 2a, 7;:1 ( F,=1, F2:5 (unts of force) tme delay = 5 sec The nternatonal Journal of Analytcal and Expermental Modal Analyss 145
Consder the force n Fg 14 wth T1=1, y=5 wth the second mpact magntude half the frst, appled to the sngle degreeoffreedom system used n the prevous Sectons fn= 1, =1) The capture duraton was set to sx system tme constants after the applcaton of the second mpulse (T=6/Jn y + ) The frequency response functon calculated under ths set of condtons s compared to the theoretcal values n Fg 15 There s no vsble dfference between the two curves ( 25% magntude error atfn), ndcatng an accurate estmaton of the response functon Ths mples that provded that a suffcent length of response data s captured, at least sx tme constants after the second mpact, accurate FRFs can be detennned wth double ht mpacts The result n Fg 15 should be readly expected from lnear system theory but does not explan the apparent degradaton the expermentalst experences under smlar condtons nsght nto the queston s produced through examnng the auto spectrum of the double ht n Fg 16 The characterstcs of ths spectrum are sgnfcantly dfferent from those produced when only a sngle mpact s appled to a structure, Fg 3 The sngle mpact produced a broadband spectrum wth the magntude gently rollng off and eventually decreasng by approxmately 1 db at a frequency of 1/T 1 or 1Hz On the contrary the double ht spectrum s oscllatory n nature wth ampltude dfferences of approxmately 1 db every 1Hz The sngle pulse spectrum decreases by less than 5 db over the same frequency band The double ht has greatly dmnshed a major characterstc of mpact testng whch makes t so attractve; a flat, broadbanded exctaton The characterstcs of the double ht force spectrum can be more readly examned by combnng Eq (8) wth Eq (29) When the only dfference between the frst and second mpacts s the pulse tnagntude, the Fourer transform of a double ht s gven n Eq (3) 1 + F2 tmr e o (3) where F 1 and F 2 represent the respectve peak ampltudes of the two mpulses and F 1 ( m,t1) s the Fourer transfonn of the half sne pulse gven n Eq (8) Substtutng Eq (3) nto Eq ( 11) yelds the autospectrum of a double mpact 2 F cos( my) + 2 2 (3 1) where the frst tenn on the rght sde of the equaton s the autospectrum f only the frst mpact was appled nspecton of Eq (3 1) shows parameters controllng the characterstcs of the double ht spectrum The oscllatory nature s controlled by the tme between the mpulses, the longer the delay ro, the greater the number of oscllatons The ampltude dfferences are controlled by the relatve magntudes, F:lF 1, of the two mpulses The greater the magntude of the second mpulse, the greater the magntude dfferences n the spectrum From a theoretcal standpont, these spectral characterstcs are mmateral and wll not affect an FRF estmated wth them, as demonstrated by the data n Fg15 However, pragmatcally they nduce expermental dffcultes f the tme between mpacts s a sgnfcant porton of the capture duraton, truncaton problems are lkely to arse Furthennore, the oscllatory nature of the force spectrum can 146 Aprl 1992
potentally ntroduce severe nstrumentaton sgnal to nose problems Ths problem can be partcularly severe as the magntude of the two mpacts approach each other, producng nulls n the spectrum The double ht stuaton dramatcally ncreases the number of ensemble averages requred to produce hgh confdence spectral estmates n an actual modal test as few as fve mpacts are ensemble averaged to estmate the spectra Ths low number of ensemble averages s feasble as a result of the very repeatable nature of the nput and output responses n other words the nput and output data can almost be consdered detennnstc, negatng the need for frequency doman ensemble averagng However, averagng s stll utlzed to mnmze the effects of any extraneous nose wthn the nstrumentaton or systent When the double ht s encountered the repeatablty characterstc s lost Because of the many factors whch control the creaton of the second mpact, ts characterstcs can often change sgnfcantly between subsequent mpacts Partcularly varable s the delay between mpacts, y Consder Fg 17 whch presents the force spectrum obtaned wth three dfferent delays, y = 4, 5, 6 seconds, whle holdng all other parameters dentcal to those used n Fg 16 The varaton n each of the spectra wth respect to one another s qute apparent From a spectral estmaton standpont each of these curves represent one event whch may be subsequently averaged to produce the force autospectrum Because of the transfonn dfferences the varance of the spectral estmates also ncreases Therefore to decrease the varance the number of averages must also must be ncreased t s dffcult to state exactly how many ensembles are necessary to produce low varance spectral estmates, however, at least an order of magntude ncrease would be needed Ths mples that to nclude double ht exctaton n the ensemble, the total number of events s approachng that whch s deemed acceptable for random exctaton Obvously the ease and smplcty assocated wth usng only fve or fewer events to obtan an accurate FRF s lost Typcally expermentalsts select mpact exctaton because of the relatve ease n whch rnpacts tests are perfonned However when double hts are encountered, many of the attractve aspects of the method are lost: flat broadbanded exctaton and low number of ensemble averages CL), ::s c bo as :: j N FORCE SPECTRUM 8 9 1 1 1, tj 6 6 6 6 4 1 41 6 6 4 6 6 6 4,} 6 6 4 6 6 o o 6 6 1 6 1 41 6 6 6 4 4 6 6 6 6 s n f fṫjj e 4 e ee 4 4 W 6 6 6 " to e 6 6 6 6 6 4 6 e 6 4 e 6 14 6 6 Tme Delay = 6 : f Tme Delay = 5 Tme Delay = 4 12 1 2 3 4 5 6 7 8 9 1 Frequency (Hz) Fg 17 Auto spectrum for a double ht half sne mpact forces wth 7;=1, F,=1, F2:5 (unts of force) wth, tme delays:4, 5, 6 sec compared to the sngle ht spectrum of Fg 3 The nternatonal Journal of Analytcal and Expermental Modal Analyss 147
Ths paper has examned several aspects of FF sgnal processng assocated wth ham1ner mpact exctaton The method s qute robust, beng suffcently nsenstve that accepted testng procedures usually keep any adverse effects to a mnmum f appled properly, the method s capable of exctng a broadband of frequences wth mnmal equpment whle provdng hgh qualty spectral estmates wth only several ensemble averages Obvously these aspects are what make the method so attractve and so wdely used Accurate results can be obtaned by mpact tests provded the practtoner s cognzant of the hammer mpact and vbraton response characterstcs n relaton to the data acquston specfcatons The study has shown that the spectral data can be subjected to bas errors assocated wth an nadequate capture of the slowly decayng vbraton sgnal Analyss has ndcated that provded at least sx tme constants vbraton response data s acqured wth a boxcar wndow the estmated FRF wll be wthn 59o of the true value at a natural frequency f the vbraton decay s suffcently long that ths crtera can not be satsfed, an exponental wndow can be employed to artfcally reduce any truncaton effects The exponental wndow, however, ntroduces another set of potental problems n tenns of sgnal truncaton and spectral leakage The leakage s most detrmental because of the smearng effects between closely spaced modes For the cases examned n ths work, modal parameters estmated from FRF accentuated suffcently that the leakage smearng and truncaton characterstcs were vsble, proved to be very accurate The accurate modal parameter estmaton was attrbuted to the capabltes of the extracton algorthm n actual testng stuatons one cannot always rely on the extracton routne to compensate for defcences n the spectral data Therefore, t s prudent to strve to obtan the hghest qualty spectral data possble The results from ths study can be used to establsh practcal tme capture crtera to assure that the wndowng/truncaton bas error s mnmzed for mpulse test data acqured under more realstc condtons n general, the greater the duraton of data captured the lower the error For mpulse response data acqured wth a boxcar wndow, an accurate (wthn 5 % of the theoretcal values over a baseband frequency range double the natural frequency) response functon s estmated f 6 tme constants of data s captured t s advsable to ncrease the data capture duraton rather than apply an exponental wndow to slowly decayng vbraton response Data acqured wth an exponental wndow s colored by the process potentally makng the modal parameter extracton more dffcult From a practcal standpont t may be dffcult to back calculate the modal dampng rato because of numercal naccuraces assocated wth the computatons The examnaton of the double ht phenomena n mpact testng demonstrated that ther presence do not preclude the accurate determnaton of spectral functons However, n order to do so potental sgnal to nose problems must be overcome and the number of ensemble averages used n the estmaton process be ncreased dramatcally Therefore, n practce t s usually more convenent to smply reject a double ht from the ensemble n general, the entre mpulse testng, data acquston, and modal extracton process s rather forgvng and robust By followng accepted testng procedures the practtoner can usually obtan hgh qualty spectral functons through mpact testng Corell, D; Brown, D L "mpact testng consderaton" Proceedng of the 2nd nternatonal Modal Analyss Conference, Orlando, FL, Feb 69, 1984 v 2 p 7357 42 148 Aprl 1992
2 Sohaney, RG; Neters, JM "Proper use of weghtng functons for mpact testng" Proceedngs of the 3rd nternatonal Modal Analyss Conference, Orlando, FL, Jan 283 1, 1985 v 2 p 1121 16 3 Clark, RL; Wcks, AL; Becker, WJ "Effects of an exponental wndow of the dampng coeffcent" Proceedngs of the 7th nternatonal Modal Analyss Conference, Las Vegas, NV, Jan 3Feb 2, 1989 v 1 p 8386 4 Trethewey, MW "Truncaton and tme delay bas spectral estmaton errors n structural testng" Proceedngs of the 4th nternatonal Modal Analyss Conference, Los Angeles, CA, Feb 3 6, 1986 v 1 p 123129 5 endat,js; Persol, A G Random Data: Analyss and Measurement Procedures New )" ork: Wley nterscence; 1986 6 Bendat, J S; Persol, A G Engneerng Applcatons of Correlaton and Spectral Analyss New York: Wleynterscence; 198 The nternatonal Journal of Analytcal and Expermental Modal Anctlyss 149
DNAME 93 Symposum on Dynamc Problems of Mechancs Hotel Plaza Caldas da lmperatrz Santo Amaro, Santa Catarna, Brazl March 15, 1993 :1 Abstract deadlne: July 3, 1992 Sponsored by: Brazlan Socety for Mechancal Scences For addtonal nfonnaton contact: Eduardo MO Lopes, Laborat6ro de Vbrafes e Acustca, Departamento de Engenhara Mecanca, CP 476, Floran6pols, SC, Brazl, CEP 8849 STRUCTRURAL D AMCS MODELNG Test, Analyss and Correlaton Cranfeld, UK July 79, 1993 Abstract deadlne: August 31, 1992 CoSponsored by: Dynamc Testng Agency (DT A) and Natonal Agency for Fnte Element Methods and Standards (NAFEMS) For addtonal nfonnaton contact: Mr Nel Harwood, DT A Conference Offce, NEL, East Klbrde, Glasgow G75 OQU, UK Phone: 44/ 355272363 Fax: 44/ 35527247