s(t) impulse response contaminated by a random noise T data block length (s) t time (s) estimator f(t) generic function

Transcription

1 The International Journal of Analytial and Experimental Modal Analysis v 7 n 4 p Ot 1992 by A Agneni, Universita degli Studi di Roma "La Sapienza" ABSTRACT A method of estimating natural frequenies and damping ratios from noisy impulse responses is onsidered Parameter evaluations are arried out, in the time domain, from the instantaneous envelope and the total phase, whih an be derived from the omplex signal formed by the autoorrelation funtion and its Hilbert transform This approah enables modal parameter estimates to be obtained with aeptable errors even when the level of the random noise, added to the impulse response, is very high ( r) omplex signal ' ( -r) omplex signal formed by approximated funtions "( r) approximated omplex signal for small s ' s E { * ) espetation operation on (*) tj Fourier transfonn F(T) funtion depending on the type of m(t) modulating funtion n index of samples n(t) random noise R residue magnitude s(t) impulse response ontaminated by a random noise T data blok length (s) t time (s) estimator f(t) generi funtion Greek j( H )(t) Hilbert transform ofj(t) fn natural frequeny (Hz) a( r;n phase varying with time dl Hilbert transform a' approximated initial phase h(t) impulse response funtion h< H )(t) Hilbert transfonn of h(t) M( r;t) amplitude modulating funtion M'( r;n approximated amplitude modulating funtion m spetral line index E* error on (*) s visous damping ratio p( r) autoorrelation funtion p( r; n autoorrelation funtion for a finite data blok length p' ( r; T) approximated autoorrelation funtion AlessandroAgneni,Assoiate Professor, Universitadegli Studi di Roma ula Sapienza,'' DipartimentoAerospazia/e, Via Eudossiana, ,Roma,/taly Final manusript reeived: June 16,

2 p" ( r; n approximated autoorrelation funtion for small 's p( H )( r;n Hilbert transform of p'( r;t) p"( H )( r;t) Hilbert transform of p"( r;t) p orrelation between the impulse l,j responses h ; and h j deay rate (rad/s) a n standard deviation of the added noise r time lag (s) m d mn damped angular frequeny (rad/s) natural angular frequeny (rad/s) W arrier angular frequeny (rad/s) ' ; ' ": ; '1,,,, Correlation funtions are widely used in mehanial vibration tests to obtain the system frequeny response funtion [1] In partiular, they have been studied with regard to lightly damped strutures [2], the response of whih an be highly distorted, resulting in an overestimation of the damping ratio if ahieved by the lassi half-power tehnique They have also been used to obtain improved estimates of the frequeny response funtions of strutures exited by transient signals and in the presene of extraneous noise [3,4] This paper desribes a method of estimating natural frequenies, damping ratios and also residue magnitudes from impulse responses orrupted by very high additive noise The autoorrelation funtion of the impulse response is obtained first Then it is Hilbert transfonned in order to form a omplex signal (the real part of whih is the autoorrelation funtion and the imaginary part is its Hilbert transform) from whih the instantaneous envelope and phase are gained The slopes of these last two funtions, ie, the deay rate and the damped angular frequeny respetively, allow the natural frequeny, the visous damping ratio, and hene the residue magnitude to be derived This approah an learly operate on a single mode Therefore, if a multi-degree of freedom system is to be studied the mode of interest must be piked out by filtering the total funtion The estimates on the envelope and on the phase are arried out by least squares fittings, whih, as pointed out in previous works [5-7], minimize the effets of the possible residuals of the random noise Thus good evaluations an be obtained, as shown by the numerial simulations presented, even when the noise level is very high For these reasons the method seems to be espeially useful when it is diffiult or impossible to repeat tests on the system, and the noise present annot be removed by averaging several data bloks 2 ' If the impulse response funtion of a single degree of freedom (SDOF) system h(t) = Re -en sin( mdt) t O (1) valid for a real mode, is ontaminated by an unorrelated additive random noise with zero mean and standard deviation a n, the autoorrelation funtion of the resulting signal 286 Otober 1992

3 s(t) = h(t) + n(t) (2) is given by the following relationship p s ( r) = E { s( t )s( t + )} = ( ) 2 r=o Ph +ern Ph(r) ;to (3) Sine the signal, (Eq 2), is available for a limited time interval T, its autoorrelation must be estimated by the time average (T-t) Ps( r;t) = F(T) s(t)s(t + )dt (4) where the funtion F(n assumes either the value (T- t 1 for the unbiased estimator or 11 for the biased one [8,9] Furthermore, Ps( r;n is an even funtion and for this reason only the part relative to r <Twill be onsidered The autoorrelation estimator, onerning the impulse response funtion only, is then written as follows (T-r) Ph ( r; T) = F(T) h(t )h(t + )dt (5) whih, after the substitution of h(t), provides the relation R 2e-ar P (, T) F(T) os( r \d,) h "' = a + md r\d 2 /a 1-e-2a(T-r)- Ul " Ul (6) It an be also expressed in the fonn of a modulated signal (7) where the time varying funtions are given by The International Journal of Analytial and Experimental Modal Analysis 287

4 - 1-e -2a(T-r) os rod (T-r) /2 l- e -2o- ( T-r ) (8) and a( r; T) = -artan rod -e -zo-( T - r ) [a sin 2rod (T-r) +rod os2rod (T-r)] w 1 a - e- 2a(r-r) ro + r 2 I r -aos 2od(T- r)+ rod sinod(r-r) (9) Provided that the time interval (T-r) is large enough, that is for 'l'«t, the estimated autoorrelation ph( r;d is a good approximation of ph( r) [8] Besides, if Tis assumed fairly long with respet to the system time onstant, then Eqs (8) and (9) redue to (1) and ' a' = -artan - ( 11) Introduing Eqs ( l ) and ( 11) into Eq (7) gives an approximated version of the estimated autoorrelation funtion, p'h( r;n However, as a onsequene of the modulating signal, (Eq (1)), the approximated autoorrelation and p' ) (r; T) = M' (r;t)sin(wdr +a') (12) are not a Hilbert transform pair (see Appendix A) The Bedrosian theorem (see Appendix B) annot be applied and so the omplex signal, formed by the autoorrelation as the real part and Eq ( 12) as the imaginary part ' ( r) = p' h ( r; T) + jp' H) ( r; T) = M' ( r; T)ej( rodr+a') (13) is not analyti Nevertheless, if OJd»Cf, that is, for the visous damping model, '2 suffiiently small ompared with unity [5], p'h( r;d and p'h (H)( r;n, as W f=wn( 1-,2)112 and r=,mn, are further simplified to (14) 288 Otober 1992

5 and they an be regarded as a Hilbert transform pair, while the omplex signal R 2 " ( r) = F(T) e -ar e lwdr 4" (15) if not stritly, at least in the limit sense [1], an be onsidered analyti In fat, under the onditions mentioned above, the spetrum of the modulating funtion falls off rapidly and its ontribution beyond m d is negligible Modal parameters, ie, the natural frequeny and the damping ratio, are derived from the omplex signal In fat the instantaneous envelope (16) if represented in a semi-log plane, is a straight line, the slope of whih is the deay rate, " Similarly, the slope of the instantaneous phase m d r = artan p" H ) ( r; T) p"h ( r; T) (17) is the damped angular frequeny, md It is now straightforward to evaluate the natural angular frequeny (18) and the visous damping ratio (19) Atually, the previous estimates together with the initial value of Eq (15) ould also allow the residue of the impulse response to be obtained from R= 4a" () F(T) (2) When several modes (M) are present within the onsidered frequeny band, the total impulse response an be written as follows M i=l (21) so the relative autoorrelation funtion is given by The International Journal of Analytial and Experimental Modal Analysis 289

6 M M p < M ) ( r) = p ij ( r) + (j; r = i=l j=l (22) - r >O i=l j=l After the substitution of the impulse responses, eah similar to the one given by Eq( 1 ), and under the above said onditions, it beomes M p( M )( r; T)::: F(T) _ R; e-ar r i=l 4 ROJd I, I 2 2 a a + md I I, l 112 os rod,i r- artan (j I m d, I + (23) where a =(a+a), ns=(m, +rod ) and Or:(ma 1 -m, ) with ma J >m,, In this ase, before applying the S I j u,l,) u, u,l u, identifiation tehnique, it is neessary to filter the mode of interest, beause the approah, based on the Hilbert transform, an be only employed on a single mode Therefore this method is partiularly suitable for strutures with lightly oupled modes Numerial tests have been arried out on impulse responses sampled over N =496 points The number of lags,however, was hosen equal to 512 in order to get a suffiiently good estimate of the autoorrelation funtion and also to have the possibility of employing a radix-2 fast Fourier algorithm, beause the Hilbert transfonns were obtained using a frequeny domain method In addition, sine the biased autoorrelation estimator has been adopted, the true autoorrelation is weighted by a triangular window, entered at the origin Therefore, the sampling period must be hosen suh that the values assumed by the instantaneous envelope, in the time interval where estimates are arried out, are smaller than the ones of the triangular window Thus the ontribution of this last funtion to the deay rate is negligible with respet to that of the exponentially deaying signal [ 11] The impulse responses have been orrupted by an additive, unorrelated noise with zero mean and standard deviations given in perent of R, whih is equal to 1 in all the examples Deay rates and damped angular frequenies were estimated by linear least squares fittings, so that the effets of the possible random noise residuals, due to the numerial evaluation of the autoorrelation funtion, have been minimized Eah instantaneous phase has been straightened [ 12] in order to eliminate the periodial jumps of the 29 Otober 1992

7 trigonometri funtion present in Eq ( 17) The intersetion of the regression line, whose slope is with r, the ordinate axis provides the initial value ofeq(15), from whih the residue magnitude an be estimated, Eq(2) In Table 1 modal parameter estimates of an impulse response relative to the natural frequeny fn= 111 (Hz) and the damping ratio '=1 are presented Sine the sampling period is equal to se (ie, approximately 512 samples per seond), the data blok length is almost 8 se, whereas the maximum time lag is 1/8 of it No filtering has been used, before perforn1ing the Hilbert transform, on data up to an=4% However from O'n=5% onwards the noise has been partially filtered out, beause otherwise it was impossible to get the straightening of the phase owing to the jumps aused by the high level of the noise Instantaneous phases, relative to unfiltered and filtered noise, are shown in Figs 1 and 2 In this way negligible errors have been obtained for the natural frequeny, while the damping ratio has been estimated with aeptable errors (lower than 1% exept for the last ase: ern= 1% ) Also, the TABLE 1 ESTIMATES FROM THE AUTOCORRELATION OF THE IMPULSE RESPONSE WHOSE MODAL PARAMETERS ARE: R=1, fn=111 (Hz) and =1 R o o o83 1 o-2 1 ooa2 1 o o32 1 o o o o o ,,, '"'1 ' -g, ",, M a, ;:, 1:, - ' -g, " =,,, Q, ;:,, 1: 1:, Time (s) nma (s) Fig 1 Instantaneous phase from the unfiltered autoorrelation funtion relative to an=5 (/o) Fig 2 Straightening of the instantaneous phase (an=5 /o) when the residual noise is partially filtered out The International Journal of Analytial and Experimental Modal Analysis 291

8 amplitude evaluations, obtained from Eq(2) through the estimates of a and " (), show small errors, and partiularly good results have been obtained when the noise has been partially removed The time history and its envelope, relative to an=7o/o, are shown in Figs 3 and 4 Nat ural frequeny and damping ratio estimations, for an impulse response with the same fn of the previous example, but with a smaller damping ratio ('=3), are shown in Table 2 Better estimates than the ones in Table 1 have been ahieved for the natural frequeny, while errors of the same order have been obtained for the damping ratio It is worth noting the effetiveness of this approah by omparing the previous estimates with the ones diretly derived from the impulse response and its Hilbert transfonn (see Table 3) In this ase, an error on s of the order of 1% is already reahed for O"n=1% Worse results, although aeptable up to O'n=7%, for damping ratio estimations have also been obtained when an impulse response of a system with low natural frequeny and light damping ratio, ie, a funtion highly trunated at the end of the time lag window (where the value of its envelope is 725 times the initial one), is onsidered (see Table 4) Finally, (see Table 5) modal parameter estimations from an impulse response ontaining two modes are presented One mode at a time has to be studied, so eah mode has been taken away from the spetrum by an adaptable osine tapered filter The width of the filter, used to extrat the first mode, has been redued TABLE 2 ESTIMATES FROM AN IMPULSE RESPONSE WITH THE SAME NATURAL FREQUENCY AS IN TABLE 1, BUT WITH s=3 TABLE 3 MODAL PARAMETERS DIRECTLY ESTIMATED FROM THE IMPULSE RESPONSE OF TABLE 2!, o26 1 o o o61 1 o-2 o386 1 o-2 29 o396 1 o-2 32 o o o14 1 o-1 o o o4o 1 o o o ' o92 1 o-1 o o o58 1 o o : : : : :': H : :': ' < ',: =::=:,: :,= g-, - 1 ""' -2 o-t ose 11s 213 Jet 488 see ee 781 e7t 111 Time (s) Fig 3 Impulse response with the standard deviation of the added random noise equal to 7o/o tis 213 Jit S Time (s) Fig 4 Instantaneous envelope of the autoorrelation funtion estimated from the impulse reponse presented in Fig Otober 1992

9 from ern =3% upwards, in order to limit the noise ontained within it In ontrast, the seond bandpass filter, at the beginning narrower than the other one, has been kept unhanged From the approximated variant of the autoorrelation funtion, weighted and distorted by the bandpass filter [13], it is possible to derive the modal parameters with the same approah desribed before Very good estimates have been ahieved for the natural frequeny, whereas damping ratio and initial amplitude estimations are good enough, maximum errors are of the order of 1%, for standard deviation values up to 4% At higher ern unaeptable errors have been notied However, as for the SDOF system, estimates gained from the autoorrelation funtions, TABLE 4 ESTIMATES FROM AN AUTOCORRELATION FUNCTION WITH: fn=5 Hz AND,= o o47 1 o o o-1 o 1 o o o o o o o TABLE 5 ESTIMATES FROM AN IMPULSE FUNCTION CONTAINING TWO MODES: R1:R2:1, fn1=9(hz), fn2=19(hz),,1=5 AND,2:3 a,l mode R 1st o62 1 o-2 o587 1 o nd o-3 o o st 2nd o-1 o o o-3 o325 1 o st 2nd o24 1 o-1 o o o-2 o o st 2nd o37 1 o o o-2 o o st 2nd o-2 o o st 2nd o6o 1 o-1 o o o-3 o o The International Journal of Analytial and Experimental Modal Analysis 293

10 espeially for high an, proved to be muh better than the ones that ould have been obtained diretly from impulse responses In Figs 5 and 6 the instantaneous envelopes, relative to C1n=4%, are shown Data within the vertial lines was used for the linear least squares fittings The possibility of estimating natural frequenies and damping ratios from very noisy impulse responses has been investigated Modal parameters have been evaluated from the instantaneous envelopes and phases of autoorrelation funtions derived from impulse responses ontaminated by unorrelated additive random noise Although the approah presented needs a data blok length long enough so that a meaningful estimation of the autoorrelation funtion is obtained, unlike other methods that similarly operate in the time domain, it does not require any trigger, whih ould be a partiularly deliate problem for multi-degree of freedom systems With this approah it is possible to avoid test repetitions that are usually required in order to average out the added random noise, whih is always present in experimental data For this reason the method ould be espeially useful whenever data aquisition annot be repeated Therefore, the autoorrelation of the response funtion, along with the Hilbert transform method (whih is insensitive to the effets of the signal trunation at the end of the time window), seems to be attrative for strutures haraterized by long time onstants, for instane some large spae strutures They generally present very low natural frequenies and light damping ratios, and therefore exhibit responses whih are trunated at the end of the observation window Besides, due to their peuliar harateristis, dynamial tests on suh strutures should be arried out in orbit There great diffiulties are enountered in exitation and in test repetition Although only one mode at a time an be analyzed, the method an provide aeptable results provided that modal densities are not exessively high This researh was sponsored by the Ministero della Pubblia Istruzione under grant: "Stima del fattore di smorzamento da segnali molto rumorosi",, >, = Time (s) Time (s) Fig 5 Instantaneous envelope of the filtered first mode ("":4 /o) Fig 6 Envelope of the filtered seond mode ( O"n=4o/o) 294 Otober 1992

11 1 Bendat, J S; Piersol, A G Engineering Appliations of Correlation and Spetral Analysis New York: J Wiley & Sons; Clarkson, B L; Merer, C A "Use of ross orrelation in studying the response of lightly damped strutures to random fores," AIAA J v 3 n 12 p Kandianis, F "Frequeny response of strutures and the effets of noise on its estimates from the transient response," J Sound Vib v 15 n 2 p Kandianis, F "Correlation tehniques in the analysis of transient proesses" J Sound Vib v 26 n 2 p Agneni, A; Balis Crema, L "Analyti signals in the damping oeffiient estimation" Proeedings of the International Conferene on Spaeraft Strutures and Mehanial Testing, Noordwijk, The Netherlands, 1988 ESA SP- 289 p Agneni, A; Balis Crema, L ''Damping measurements from trunated signals via Hilbert transform" Meh Syst Signal Proess v 3 n 1 p Agneni, A; Balis Crema, L "A time domain identifiation approah for low natural frequenies and light damping strutures" Dynamis of Flexible Strutures in Spae: Proeedings of the First International Conferenes, Cranfield, UK, May 15-18, 199 p Papoulis, A Signal Analysis Singapore: MGraw-Hill; 1985 p Bendat, J S; Piersol, A G Random Data:Analysis and Measurements Proedures New York: J Wiley & Sons; 1971 p Ville, J "The6rie et appliations de Ia notion de signal analytique" Cables Transm v 2 n 1 p Agneni, A; Balis Crema, L "Damping ratio estimates from autoorrelation funtions" Proeedings of DAMPING '91, San Diego, CA, Feb 13-15, 1991 p GCD Otnes, R K; Enohson, L Applied Time Series Analysis, vol I, Basi Tehniques New York: J Wiley & Sons; 1978 p Soovere, J; Drake, M L Aerospae Strutures Tehnology Damping Design Guide Vol I - Tehnology Review AFWAL-TR , 1985 p 6-8 to Triomi, F G Integral Equations New York: Dover, 1985 p Tithmarsh, E C Introdution to the Theory off ourier' s Integrals Oxford: Clarendon Press; 1937 p Otnes, R K; Enohson, L Applied Time Series Analysis, vol I, Basi Tehniques New York: J Wiley & Sons, 1978 p Bedrosian, E "The analyti signal representation of modulated waveforms," Pro IRE v 5 n 1 p Ot Bedrosian, E "A produt theorem for Hilbert transform" Pro IEEE v 51 n 5 p May 1963 Hilbert Transform The Hilbert transfonn of an integrable funtionf(t) is given by the Cauhy prinipal value of the integral fl4,15] +oo p v -oo!( -r ) d -r t- 'f (Al) The International Journal of Analytial and Experimental Modal Analysis 295

12 Beause the previous relation an also be regarded as the onvolution betweenf(t) and ( 1/n t), it is possible to apply the Borel (or onvolution) theorem and then to obtain the funtionj<hl(t) through the Fourier transform (A2) where sgn represents the signum funtion Atually the signal proessing is performed on sampled data sequenes and therefore the diret and the inverse transforms are ahieved by FFT algorithms In this ase null buffers have to be added at the end of the time sequenes in order to remove the irular effets and to obtain the desired linear onvolution [16] From the disrete Fourier transform (Fm), derived from the time sequene ifn), the disrete Hilbert transform is given by the following relationship (A3) where sm is the disretized ounterpart of the [-j sgn(w)] funtion, that is m = 1,2,, s = m m =, N 2 (A4) +j N m= +l,,(n -1) 2 where N represents the total points within the blok length: data sequene and added zeros Bedrosian Theorem The Bedrosian theorem [ 17, 18] is applied when the Hilbert transform of a produt of funtions has to be performed In partiular, when the signal is given by the produt of a modulation funtion and a pure sinusoid arrier f(t) = m(t) os Wt (B l) its Hilbert transform is equal to (B2) 296 Otober 1992

13 when the angular frequeny of the arrier is greater than (or at the most equal to) the highest angular frequeny omponent of the m(t) spetrum In our ase the modulation signal is an exponentially dereasing time funtion and therefore its spetrum is unlimited, so the Bedrosian theorem is not appliable Nevertheless, if the deay rate of the exponential funtion is muh smaller than the damped angular frequeny, the funtions (B3) and f (H) ( t) = Ae- (J 1 sin m dt (B4) an be onsidered as a Hilbert transform pair Furthermore, the omplex signal (B5) is almost analyti In fat, even if the negative frequeny spetrum of (t) is not ompletely suppressed, it an be onsidered negligible for pratial purposes The International Journal of Analytial and Experimental Modal Analysis 297