- Email: [email protected]
Output-only analysis of structures with closely spaced poles
ARTICLE IN PRESS Mechanical Systems and Signal Processing 24 (2010) 1240–1249
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Output-only analysis of structures with closely spaced poles Alessandro Agneni, Luigi Balis Crema, Giuliano Coppotelli Dipartimento di Ingegneria Aerospaziale e Astronautica, Universita degli Studi di Roma ‘‘La Sapienza’’, Via Eudossiana, 18, 00184 Rome, Italy
a r t i c l e i n f o
abstract
Article history: Received 6 May 2008 Received in revised form 6 October 2009 Accepted 22 October 2009 Available online 30 October 2009
In the framework of the operational modal analysis, several approaches have been developed for estimating the modal parameters, i.e., natural frequencies, damping ratios, and mode shapes. Specifically, a technique capable to evaluate the biased (i.e., unscaled by a constant or an almost constant function) frequency response functions, FRFs, has been proposed. Assuming that only the responses of the structure are disposable, the technique allows one to estimate biased FRFs starting from the power spectral densities, PSDs, and applying the Hilbert transform. This paper deals with the estimates of the modal analysis parameters mentioned above. It is possible to obtain each single mode shape, from the singular vectors achieved by applying the singular value decomposition to the FRF matrix evaluated at the spectral line corresponding to the selected natural frequency. A special attention will be devoted to structures with coupled modes, i.e., closely spaced modes. Once the FRFs have been obtained, the natural frequencies and damping ratios could be achieved either in the frequency domain or in the time domain. Experimental tests, carried out on beams, plates and on the AB-204 helicopter blade, will be presented. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Output-only analysis Experimental modal analysis Structural dynamics
1. Introduction Approaches which study the dynamic behavior of a structure deriving the modal characteristics from the output responses, while the system is excited by natural or operative loadings, have drawn the researchers’ interest. The problem has been faced both in the frequency domain [1] and in the time domain [2]. Recently a further approach was proposed, even if it uses the output time sequences, then the complete matrix of the frequency response functions FRFs is derived, firstly applying the Hilbert transform on the amplitude spectra achieved from the autocorrelation functions of the outputs recorded at each point of measurement [3,4]. Obviously all the FRFs are biased because the spectral density of the input is unknown, but if the input can be considered as a noise with a power spectral density constant in the band of interest, all the FRFs are known except for a biasing constant. A further development of this last technique is presented in this paper. In particular, while the mode shapes were gained—in the previously cited papers—through the residue estimate, in this paper each single mode shape is obtained by applying the singular value decomposition to the FRF matrix evaluated at the spectral line corresponding to the selected natural frequency. This last can be evaluated either in the time or in the frequency domain, being disposable the biased FRFs [3,4]. Experimental tests were carried out on a cantilever beam, a free– free plate and a helicopter blade (AB-204). Excitations were distributed for all the structures: crawling pencil (beam),
Corresponding author.
E-mail address: [email protected] (G. Coppotelli). URL: http://www.diaa.uniroma1.it/docenti/g.coppotelli/index.html (G. Coppotelli). 0888-3270/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2009.10.013
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loudspeaker (plate), and wind (rotorcraft blade). Although, three coupled modes have been found in the last structure, the proposed approach was able to face the problem of estimating natural frequencies, damping ratios and mode shapes. 2. Theoretical background 2.1. FRF estimates using Hilbert transform The dynamic behavior of a vibrating system could be expressed in terms of its frequency response function matrix, FRF, that relates the input excitations to the output responses. Considering an uncorrelated random load on the ith degree of freedom of the structure, i.e., fi ðtÞ, and the corresponding output response, xi ðtÞ, the squared magnitude of its FRF, evaluated for each frequency line o, is given by jHii ðoÞj2 ¼
Gxi xi ðoÞ Gf i f i ð o Þ
ð1Þ
where Gfi fi ðoÞ and Gxi xi ðoÞ are the auto power spectral densities of the input and the output signals respectively. If the random input load can be assumed as white in the frequency band of interest, then Gfi fi ðoÞ is constant. Therefore, taking the natural logarithm and performing the Hilbert transform (H) of Eq. (1), and considering that the Hilbert transform of a constant is zero, it is possible to obtain H½ln Gxi xi ðoÞ ¼ 2 H½lnjHii ðoÞj
ð2Þ
Moreover, the phase, Fii ðoÞ, between the output response and the input force could be also achieved. Recalling that for causal signals [5,6]holds the following relationship: H½lnjHii ðoÞj ¼ Fii ðoÞ
ð3Þ
the phase is given by
Fii ðoÞ ¼ 12H½ln Gxi xi ðoÞ
ð4Þ
Eqs. (1) and (4) could be used to estimate the frequency response function as Hii ðoÞ ¼ jHii ðoÞjejFii ðoÞ
ð5Þ
Since only the power spectral densities from the responses are available in the output-only analysis, introducing Eq. (1) into Eq. (5), the following unscaled FRF is achieved: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ~ ii ðoÞ ¼ Gf f Hii ðoÞ ¼ Gx x ðoÞejFii ðoÞ ð6Þ H i i i i (where i ¼ 1; . . . ; N and N is the number of the output degrees of freedom). It is worth noting that the unscaled diagonal terms of the FRF are gained by recording the output responses only, regardless the space distribution of the excitation loading, provided the excitation signals are characterized by an almost constant spectra. The direction of the excitation must be instead such as to properly excite the modes of interest. Therefore, except for an unknown constant, Eq. (6) provides the diagonal terms of the FRF matrix. The presence of the constant does not affect the estimates of the natural frequencies and damping ratios gained from the above mentioned frequency functions. Indeed, the poles of the dynamic system are independent from the (unknown) excitation level. If the first M modes of the structure have been excited, the ith diagonal term of the FRF matrix could be expressed as Hii ðoÞ ¼
M X
Rkii okd
k¼1
ðs þ joÞ2 þ ðokd Þ2 k
ð7Þ
where, for the kth mode, sk is the decay rate, okd is the relative damped circular frequency, and Rkii is the residue. Introducing the modal mass, mk , it is possible to write pffiffiffiffiffiffiffiffi Gf f k ð8Þ R~ ii ¼ k i ki m od and so ~ ii ðoÞ ¼ H
M X
k R~ ii okd
k¼1
ðsk þ joÞ2 þ ðokd Þ2
ð9Þ
The generic element of the FRF matrix, Hji ðoÞ could be obtained from the knowledge of the previous identified driving point FRF, Hii ðoÞ[3]. The response of the jth degree of freedom when a loading is applied at the ith degree of freedom, for
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deterministic signals, is given by Hji ðoÞ ¼
Xj ðoÞ Fi ðoÞ
ð10Þ
that could be rewritten as Hji ðoÞ ¼
Xi ðoÞ Xj ðoÞ Xi ðoÞ Fi ðoÞ
ð11Þ
This equation is also equivalent to the following relationship: Hji ðoÞ ¼
Xi ðoÞ Xj ðoÞ X ð oÞ Fi Fi ðoÞ i Fi ðoÞ
ð12Þ
that is Hji ðoÞ ¼
Gxi xj ðoÞ Gfi fi Hii ðoÞ
ð13Þ
This last equation not only is valid for deterministic signal, but also for the random ones, for which the Fourier transform ~ ij ðoÞFequal to does not exist, but can be achieved by applying the Wiener–Kintchin theorem. Similar to Eq. (6), defining H ~ ji ðoÞ for the Betti–Maxwell’s reciprocity theorem—as H qffiffiffiffiffiffiffiffi ~ ij ðoÞ :¼ Gf f Hij ðoÞ ð14Þ H i i the relation between the off diagonal terms of the biased frequency response matrix and the cross-power spectral density of the output responses is finally given by ~ ij ðoÞ ¼ Gxi xj ðoÞ H ~ ð oÞ H ii
ð15Þ
In this way all the terms of the biased FRF matrix could be achieved by only the auto and cross-power spectral densities of the output signals. If roving accelerometers are used to cover all the needed experimental degrees of freedom, few sensors have to be kept in a fixed, reference, position during all the needed sessions. If p is the session index, and subscript r is relative to the reference position, then the FRFs could be obtained from Gprr ðoÞ ¼ Gpff jHrr ðoÞj2
ð16Þ
i i
It is worth noting that the session number affects only the biasing factor, since the FRF is an intrinsic function of the system. The complete column (or row) of the frequency response function matrix is achieved deriving the biasing factors ~ p ðoÞ from different sessions and the reference ones. The modal parameters could be then from the ratio among the H ii estimated from the biased FRF matrix by using common softwares either in the frequency or in the time domain. 2.2. Modal parameter estimates using SVD In Ref. [4] the natural frequencies, damping ratios have been obtained by the data least square fitting with a polynomial ratio (residue/pole method), while the modes have been achieved from the residues evaluated by the same fittings. On the other hand, it is possible to estimate the mode shapes by applying the singular value decomposition, SVD, to the FRF matrix, in a similar way as they are achieved by the frequency domain decomposition, FDD, technique [1]. Rewriting Eq. (7), for the generic i2j degree of freedom, in terms of the mode shapes, and arranging the denominator of the system in a diagonal 2 matrix, LðoÞ whose kth element lk ðoÞ ¼ ðsk þ joÞ2 þ okd , the FRF matrix is given by HðoÞ ¼ WL1 ðoÞWT
ð17Þ
where W is the modal matrix. The previous equation, evaluated at the rth natural frequency, becomes T
Hðor Þ ¼ wr lr wr þ 1
M X
T
k wk l1 k w
ð18Þ
k¼1;kar
The rth term, for r ¼ 1; . . . ; N, in Eq. (18) will dominate, if the structure behaves as a single degree of freedom system in the neighborhood of the rth natural frequency. Under these circumstances the FRF matrix has unitary rank, therefore, its singular value decomposition will give one great singular value, whereas the others are negligible (they should be theoretically zero). The matrix HðoÞ can be decomposed as follows: svd½HðoÞ ¼ URVH
ð19Þ
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where U and V are the orthogonal matrices formed by the N singular vectors uN and vN , i.e., UH U ¼ I and VH V ¼ I, whereas R represents the diagonal matrix of the singular values, sr . Due to the symmetry of the FRF matrix, the singular matrices U and V (where stands for complex conjugate) are equal, except for phasors, different for each principal vector [7]. Since the rank of Hðor Þ is equal to one, Eq. (19) could be written as: svd½Hðor Þ ¼ u1 s1 v1
H
ð20Þ
obtained neglecting the singular values that are practically zero with respect to the dominant one (numerical algorithms commonly arrange in decreasing order the singular values). Therefore, starting from the measurements of the responses of the structure, the biased frequency response functions could be obtained by using the ‘‘Hilbert’’ transformation, and then the SVD operation could be used to estimate the mode shapes when a SDOF behavior is considered. This approach could be also successfully applied when an MDOF system is analyzed. In this case, the frequency response function evaluated at o ¼ or is no more a unitary rank matrix, and then the corresponding R matrix has more then one non zero value. Nevertheless, the mode shape could be estimated from the singular vector corresponding to the maximum singular value since the dynamic behavior of the structure at the natural frequency or is almost dominated by the rth eigenmode. From this point of view, a high frequency resolution is the crucial point to face coupled mode structures. 3. Experimental investigation 3.1. Cantilever beam In order to check the capabilities of the proposed procedure, the estimates of the modal parameters of a simple aluminum cantilever beam, whose dimensions were 0:2 0:0154 0:00285 m, have been first considered, and the results have been compared with those achieved in Ref. [8]. The experimental test, carried out in the frequency range of 0–400 Hz by using 8192 sampling points, allowed one to measure four time histories corresponding to the vertical accelerations of the points equally distributed along the beam span. Estimates of natural frequencies, and damping ratios as results of the proposed method, denoted as ‘‘Hilbert transform method’’ (HTM), have been carried out by fitting the biased FRFs with a least square polynomial ratio algorithm and then compared with those achieved either from the least square fittings of frequency response functions (FRFs) or by other approaches, as the one based on the ‘‘frequency domain decomposition’’ (FDD), the core of the software ARTeMIS from SVS [9]. When the FRF measurements are considered, the modal parameters Table 1 Natural frequency estimates for the cantilever beam. Mode #
fnFRF (Hz)
fnFDD (Hz)
fnHTM (Hz)
1 2
50.98 318.07
51.18 318.83
50.93 320.39
Table 2 Damping ratio estimates for the cantilever beam. Mode #
zFRF (%) n
zFDD (%) n
zHTM (%) n
1 2
0.56 0.40
0.49 0.25
0.89 0.38
0.8
Mode #2 Mode #1
Displacement
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 1
2
3
4
Grid Location Fig. 1. Cantilever beam—estimate of the first and second mode shapes.
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(natural frequencies and damping ratios) have been estimated by a least square fitting of the time response with complex exponentials, part of the LMS-TEST.LAB software, whereas a least square fitting with polynomial ratios was used when the HTM approach has been considered. Instead, with the FDD approach, the natural frequencies were evaluated from the peaks of the singular values of the power spectral density matrix, as function of the frequency, the mode shapes from the corresponding singular vectors and the damping ratios have been derived by the logarithm decrement. The response of the structure, randomly excited with a pencil crawling on one side of the beam, was recorded for a time period of 40.96 s, and the auto and cross-power spectral densities were averaged using 16 data block records with 2048 sampling points each. As one can see from Table 1, the natural frequencies are practically the same, whereas the damping ratio estimates depend on the estimation procedure, Table 2. It is worthwhile to remark that using a different evaluation method leads to damping factors in order of 0.80% for the first mode evaluated on the FRFs. A good agreement is also confirmed when the mode shapes, obtained from the singular vectors of the biased FRF—evaluated at the natural frequency, Fig. 1, and those from the other approach, not reported here but shown in Ref. [8], are correlated. 3.2. Free–free plate A second structure has been considered for the purpose of validating the methodology. The aluminum plate, 0:25 0:25 0:0025 m, has been already tested in Ref. [4] , and here the results are considered as reference. The excitation of the structure has been provided by a loudspeaker positioned in front of the middle of the plate, at a distance of 10 cm, whereas nine vertical responses, uniformly distributed over the structure, have been measured using four roving accelerometers. The same good correlation, as the one reported for the cantilever beam, has been obtained when comparing both the natural frequencies and damping ratios, Tables 3 and 4. The estimate of the mode shapes from the singular value decomposition confirmed its efficiency: the mode shapes estimated with the proposed approach, depicted in Figs. 2–4, need the evaluation of the singular vectors associated with FRF matrix only when o ¼ on .
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 3 2.5 2 1.5 1
3
2
1
Fig. 2. Free–free plate—first mode estimate.
Table 3 Natural frequency estimates for the free–free plate. Mode #
fnFRF (Hz)
fnFDD (Hz)
fnHTM (Hz)
1 2 3
70.20 102.21 149.30
69.40 101.7 148.7
70.90 101.6 148.2
Table 4 Damping ratio estimates for the free–free plate. Mode #
zFRF (%) n
zFDD (%) n
zHTM (%) n
1 2 3
0.63 0.14 0.27
0.45 0.23 0.81
0.55 0.48 0.96
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0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 3 3 2
2 1
1
Fig. 3. Free–free plate—second mode estimate.
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 3 2 1
1
2
3
Fig. 4. Free–free plate—third mode estimate.
3.3. AB-204 helicopter blade Finally, the output-only experimental analysis has been performed on the AB-204 helicopter blade, Fig. 5a, which had a span of 6100 mm with a chord of 530 mm, Fig. 5 b, with the total weight of 88 kg. The main structural components of the blade, i.e., the skin, the sandwich core, and the main spar were in aluminum alloys. The helicopter blade has been tested in a free–free boundary condition obtained using elastic suspensions, four roving accelerometers have been used to measure the vertical deflection at the leading edge, the trailing edge, and the elastic axis for eleven sections along the blade span (for a total of 33 experimental degrees of freedom). The excitation was due to a natural event (wind) being the structure hung outdoor. The frequency range considered was 0–100 Hz, whereas the number of the sampling points were 215 . All the approaches used to estimate the modal parameters identified six elastic bending modes, three of which were very close, as reported in Fig. 6, where the magnitude of the biased frequency response function, obtained with the proposed approach at the blade root on the elastic axis, is reported. These three modes have been found out to vibrate at 42.91, 43.61, and 44.91 Hz: although all of them were fully coupled bending–torsion modes, the first is a prevalent in-plane bending mode, whereas the last two were out-of-plane bending–torsion coupled modes. A comparison between the natural frequencies and damping ratios for the out-of-plane bending modes of the blade is reported in Tables 5 and 6. These results seem to provide a good correlation except some discrepancies in the damping ratios. The same good correlation has been obtained for the predominant in-plane bending mode. In this case the FDD approach identified the natural frequency at 43.08 Hz with a damping ratio of 0.18%. These values are practically equal to those identified by the HTM approach: fn ¼ 42:91, and zn ¼ 0:21%. Finally, the estimates of the modal shapes for the helicopter blade are reported in Figs. 7–11. The reinforcing structures near the blade root seems to affect the shape of the first, Fig. 7, the third, Fig. 9, and the fourth mode, Fig. 10, and not the second and the fifth modes, Figs. 8 and 11.
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Fig. 5. AB-204 helicopter. (a) General view of AB-204 helicopter. (b) AB-204 helicopter blade.
|H22(ω)|
102
101
100
10−1 10
20
30
40 50 60 Frequency (Hz)
70
80
Fig. 6. AB-204 helicopter blade—frequency response function.
Table 5 Natural frequency estimates for the AB-204 helicopter blade. Mode #
fnFRF (Hz)
fnFDD (Hz)
fnHTM (Hz)
1 2 3 4 5
7.49 21.67 43.70 44.98 72.69
7.43 21.60 43.55 44.85 72.69
7.32 21.62 43.61 44.91 72.63
Table 6 Damping ratio estimates for the AB-204 helicopter blade. Mode #
zFRF (%) n
zFDD (%) n
zHTM (%) n
1 2 3 4 5
0.17 0.14 0.39 0.14 0.20
0.37 0.19 0.38 0.20 0.90
0.79 0.44 0.43 0.42 0.23
4. Concluding remarks In this paper, a methodology able to identify the modal parameters of coupled mode structures, using output-only data, has been developed. As shown, natural frequencies, damping ratios and mode shapes can be obtained from the FRFs, which in turn have been achieved from the output signal power spectral densities. Obviously, the input is unknown, but if it is
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7 6 5
1
4
0
3
−1
2
Bla
1 de 0.5 Ch ord 0 (m )
)
n (m
pa de S
Bla
1 0
Fig. 7. AB-204 helicopter blade—first mode estimate.
7 6 5
1
4
0 −1 Bla 1 de 0.5 Ch ord 0 (m )
3 2
)
n (m
pa eS
d
Bla
1 0
Fig. 8. AB-204 helicopter blade—second mode estimate.
7 6 5
1 4
0
3
−1 Bla 1 de 0.5 Ch ord 0 (m )
2
de
Bla
n
Spa
(m)
1 0
Fig. 9. AB-204 helicopter blade—third mode estimate.
possible to consider the input auto spectral density as constant—at least in the considered band—the estimated FRFs differ from the actual ones for an unknown constant. Actually, it is straight to evaluate the natural frequencies and the damping ratios by least square fittings, either in the frequency or in the time domain, also with commercial codes. Concerning the
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Fig. 10. AB-204 helicopter blade—fourth mode estimate.
7 6 5
1
4
0 −1 Bla 1 de 0.5 Ch ord 0 (m )
3 2
pan
eS Blad
(m)
1 0
Fig. 11. AB-204 helicopter blade—fifth mode estimate.
modes shapes, although they could be derived from the residues (as done in the past by the authors), a technique based on the decomposition of the FRF matrix into its principal values and vectors has been proposed and successfully applied to a beam, to a plate, and to a helicopter blade. Although coupled modes were present in the spectra of this last structure, the modal parameters along with the mode shapes were identified. The presented approach needs only to perform the singular value decomposition of the FRF matrices evaluated at the frequencies of the modes, and that are known from the least square fitting of the frequency functions mentioned above.
Acknowledgment This research has been supported by ‘‘Ricerca di Facolta : Stima Parametri Modali dalle Sole Risposte Dinamiche di Sistemi Aerospaziali’’, University of Rome ‘‘La Sapienza’’, 2003. References [1] R. Brincker, L. Zhang, P. Andersen, Modal identification from ambient responses using frequency domain decomposition, in: XVIII IMAC, vol. 1, San Antonio, TX, USA, Society for Experimental Mechanics, Inc., Bethel, CT, USA, 2000, pp. 625–630. [2] L. Hermans, H. Van der Auweraer, Modal testing and analysis of structures under operational conditions: industrial applications, Mechanical Systems and Signal Processing 13 (2) (1999) 193–216. [3] A. Agneni, L. Balis Crema, G. Coppotelli, Time and frequency domain model parameter estimation by output only functions, in: Proceedings of International Forum on Aeroelasticity and Structural Dynamics, Amsterdam, NL, 2003. [4] A. Agneni, R. Brincker, G. Coppotelli, On modal parameter estimates from ambient vibration tests, in: Proceedings of International Conference on Noise and Vibration Engineering—ISMA 2004, Leuven, B, 2004, pp. 2239–2248.
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[5] H.W. Bode, Network Analysis and Feedback Amplifier Design, Van Nostrand, 1945, pp. 303–336 (Chapter XIV). [6] E.A. Robinson, M.T. Silva, Digital Signal Processing and Time Series Analysis, Holden Day, 1978, pp. 283–290. [7] D. Otte, J. Leuridan, H. Grangier, R. Aquilina, Prediction of the dynamics of structural assemblies using measured FRF-data: some improved data enhancement techniques, in: Proceedings of the Ninth International Modal Analysis Conference, 1991, pp. 909–918. [8] G. Coppotelli, On the estimate of the FRFs from operational data, Mechanical Systems and Signal Processing 23 (2009) 288–299. [9] Structural Vibration Solutions, Getting Started Manual for all Versions of ARTeMIS Extractor, NOVI Science Park, Aalborg, DK, 2003.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Output-only analysis of structures with closely spaced poles Alessandro Agneni, Luigi Balis Crema, Giuliano Coppotelli Dipartimento di Ingegneria Aerospaziale e Astronautica, Universita degli Studi di Roma ‘‘La Sapienza’’, Via Eudossiana, 18, 00184 Rome, Italy
a r t i c l e i n f o
abstract
Article history: Received 6 May 2008 Received in revised form 6 October 2009 Accepted 22 October 2009 Available online 30 October 2009
In the framework of the operational modal analysis, several approaches have been developed for estimating the modal parameters, i.e., natural frequencies, damping ratios, and mode shapes. Specifically, a technique capable to evaluate the biased (i.e., unscaled by a constant or an almost constant function) frequency response functions, FRFs, has been proposed. Assuming that only the responses of the structure are disposable, the technique allows one to estimate biased FRFs starting from the power spectral densities, PSDs, and applying the Hilbert transform. This paper deals with the estimates of the modal analysis parameters mentioned above. It is possible to obtain each single mode shape, from the singular vectors achieved by applying the singular value decomposition to the FRF matrix evaluated at the spectral line corresponding to the selected natural frequency. A special attention will be devoted to structures with coupled modes, i.e., closely spaced modes. Once the FRFs have been obtained, the natural frequencies and damping ratios could be achieved either in the frequency domain or in the time domain. Experimental tests, carried out on beams, plates and on the AB-204 helicopter blade, will be presented. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Output-only analysis Experimental modal analysis Structural dynamics
1. Introduction Approaches which study the dynamic behavior of a structure deriving the modal characteristics from the output responses, while the system is excited by natural or operative loadings, have drawn the researchers’ interest. The problem has been faced both in the frequency domain [1] and in the time domain [2]. Recently a further approach was proposed, even if it uses the output time sequences, then the complete matrix of the frequency response functions FRFs is derived, firstly applying the Hilbert transform on the amplitude spectra achieved from the autocorrelation functions of the outputs recorded at each point of measurement [3,4]. Obviously all the FRFs are biased because the spectral density of the input is unknown, but if the input can be considered as a noise with a power spectral density constant in the band of interest, all the FRFs are known except for a biasing constant. A further development of this last technique is presented in this paper. In particular, while the mode shapes were gained—in the previously cited papers—through the residue estimate, in this paper each single mode shape is obtained by applying the singular value decomposition to the FRF matrix evaluated at the spectral line corresponding to the selected natural frequency. This last can be evaluated either in the time or in the frequency domain, being disposable the biased FRFs [3,4]. Experimental tests were carried out on a cantilever beam, a free– free plate and a helicopter blade (AB-204). Excitations were distributed for all the structures: crawling pencil (beam),
Corresponding author.
E-mail address: [email protected] (G. Coppotelli). URL: http://www.diaa.uniroma1.it/docenti/g.coppotelli/index.html (G. Coppotelli). 0888-3270/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2009.10.013
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1241
loudspeaker (plate), and wind (rotorcraft blade). Although, three coupled modes have been found in the last structure, the proposed approach was able to face the problem of estimating natural frequencies, damping ratios and mode shapes. 2. Theoretical background 2.1. FRF estimates using Hilbert transform The dynamic behavior of a vibrating system could be expressed in terms of its frequency response function matrix, FRF, that relates the input excitations to the output responses. Considering an uncorrelated random load on the ith degree of freedom of the structure, i.e., fi ðtÞ, and the corresponding output response, xi ðtÞ, the squared magnitude of its FRF, evaluated for each frequency line o, is given by jHii ðoÞj2 ¼
Gxi xi ðoÞ Gf i f i ð o Þ
ð1Þ
where Gfi fi ðoÞ and Gxi xi ðoÞ are the auto power spectral densities of the input and the output signals respectively. If the random input load can be assumed as white in the frequency band of interest, then Gfi fi ðoÞ is constant. Therefore, taking the natural logarithm and performing the Hilbert transform (H) of Eq. (1), and considering that the Hilbert transform of a constant is zero, it is possible to obtain H½ln Gxi xi ðoÞ ¼ 2 H½lnjHii ðoÞj
ð2Þ
Moreover, the phase, Fii ðoÞ, between the output response and the input force could be also achieved. Recalling that for causal signals [5,6]holds the following relationship: H½lnjHii ðoÞj ¼ Fii ðoÞ
ð3Þ
the phase is given by
Fii ðoÞ ¼ 12H½ln Gxi xi ðoÞ
ð4Þ
Eqs. (1) and (4) could be used to estimate the frequency response function as Hii ðoÞ ¼ jHii ðoÞjejFii ðoÞ
ð5Þ
Since only the power spectral densities from the responses are available in the output-only analysis, introducing Eq. (1) into Eq. (5), the following unscaled FRF is achieved: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ~ ii ðoÞ ¼ Gf f Hii ðoÞ ¼ Gx x ðoÞejFii ðoÞ ð6Þ H i i i i (where i ¼ 1; . . . ; N and N is the number of the output degrees of freedom). It is worth noting that the unscaled diagonal terms of the FRF are gained by recording the output responses only, regardless the space distribution of the excitation loading, provided the excitation signals are characterized by an almost constant spectra. The direction of the excitation must be instead such as to properly excite the modes of interest. Therefore, except for an unknown constant, Eq. (6) provides the diagonal terms of the FRF matrix. The presence of the constant does not affect the estimates of the natural frequencies and damping ratios gained from the above mentioned frequency functions. Indeed, the poles of the dynamic system are independent from the (unknown) excitation level. If the first M modes of the structure have been excited, the ith diagonal term of the FRF matrix could be expressed as Hii ðoÞ ¼
M X
Rkii okd
k¼1
ðs þ joÞ2 þ ðokd Þ2 k
ð7Þ
where, for the kth mode, sk is the decay rate, okd is the relative damped circular frequency, and Rkii is the residue. Introducing the modal mass, mk , it is possible to write pffiffiffiffiffiffiffiffi Gf f k ð8Þ R~ ii ¼ k i ki m od and so ~ ii ðoÞ ¼ H
M X
k R~ ii okd
k¼1
ðsk þ joÞ2 þ ðokd Þ2
ð9Þ
The generic element of the FRF matrix, Hji ðoÞ could be obtained from the knowledge of the previous identified driving point FRF, Hii ðoÞ[3]. The response of the jth degree of freedom when a loading is applied at the ith degree of freedom, for
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deterministic signals, is given by Hji ðoÞ ¼
Xj ðoÞ Fi ðoÞ
ð10Þ
that could be rewritten as Hji ðoÞ ¼
Xi ðoÞ Xj ðoÞ Xi ðoÞ Fi ðoÞ
ð11Þ
This equation is also equivalent to the following relationship: Hji ðoÞ ¼
Xi ðoÞ Xj ðoÞ X ð oÞ Fi Fi ðoÞ i Fi ðoÞ
ð12Þ
that is Hji ðoÞ ¼
Gxi xj ðoÞ Gfi fi Hii ðoÞ
ð13Þ
This last equation not only is valid for deterministic signal, but also for the random ones, for which the Fourier transform ~ ij ðoÞFequal to does not exist, but can be achieved by applying the Wiener–Kintchin theorem. Similar to Eq. (6), defining H ~ ji ðoÞ for the Betti–Maxwell’s reciprocity theorem—as H qffiffiffiffiffiffiffiffi ~ ij ðoÞ :¼ Gf f Hij ðoÞ ð14Þ H i i the relation between the off diagonal terms of the biased frequency response matrix and the cross-power spectral density of the output responses is finally given by ~ ij ðoÞ ¼ Gxi xj ðoÞ H ~ ð oÞ H ii
ð15Þ
In this way all the terms of the biased FRF matrix could be achieved by only the auto and cross-power spectral densities of the output signals. If roving accelerometers are used to cover all the needed experimental degrees of freedom, few sensors have to be kept in a fixed, reference, position during all the needed sessions. If p is the session index, and subscript r is relative to the reference position, then the FRFs could be obtained from Gprr ðoÞ ¼ Gpff jHrr ðoÞj2
ð16Þ
i i
It is worth noting that the session number affects only the biasing factor, since the FRF is an intrinsic function of the system. The complete column (or row) of the frequency response function matrix is achieved deriving the biasing factors ~ p ðoÞ from different sessions and the reference ones. The modal parameters could be then from the ratio among the H ii estimated from the biased FRF matrix by using common softwares either in the frequency or in the time domain. 2.2. Modal parameter estimates using SVD In Ref. [4] the natural frequencies, damping ratios have been obtained by the data least square fitting with a polynomial ratio (residue/pole method), while the modes have been achieved from the residues evaluated by the same fittings. On the other hand, it is possible to estimate the mode shapes by applying the singular value decomposition, SVD, to the FRF matrix, in a similar way as they are achieved by the frequency domain decomposition, FDD, technique [1]. Rewriting Eq. (7), for the generic i2j degree of freedom, in terms of the mode shapes, and arranging the denominator of the system in a diagonal 2 matrix, LðoÞ whose kth element lk ðoÞ ¼ ðsk þ joÞ2 þ okd , the FRF matrix is given by HðoÞ ¼ WL1 ðoÞWT
ð17Þ
where W is the modal matrix. The previous equation, evaluated at the rth natural frequency, becomes T
Hðor Þ ¼ wr lr wr þ 1
M X
T
k wk l1 k w
ð18Þ
k¼1;kar
The rth term, for r ¼ 1; . . . ; N, in Eq. (18) will dominate, if the structure behaves as a single degree of freedom system in the neighborhood of the rth natural frequency. Under these circumstances the FRF matrix has unitary rank, therefore, its singular value decomposition will give one great singular value, whereas the others are negligible (they should be theoretically zero). The matrix HðoÞ can be decomposed as follows: svd½HðoÞ ¼ URVH
ð19Þ
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where U and V are the orthogonal matrices formed by the N singular vectors uN and vN , i.e., UH U ¼ I and VH V ¼ I, whereas R represents the diagonal matrix of the singular values, sr . Due to the symmetry of the FRF matrix, the singular matrices U and V (where stands for complex conjugate) are equal, except for phasors, different for each principal vector [7]. Since the rank of Hðor Þ is equal to one, Eq. (19) could be written as: svd½Hðor Þ ¼ u1 s1 v1
H
ð20Þ
obtained neglecting the singular values that are practically zero with respect to the dominant one (numerical algorithms commonly arrange in decreasing order the singular values). Therefore, starting from the measurements of the responses of the structure, the biased frequency response functions could be obtained by using the ‘‘Hilbert’’ transformation, and then the SVD operation could be used to estimate the mode shapes when a SDOF behavior is considered. This approach could be also successfully applied when an MDOF system is analyzed. In this case, the frequency response function evaluated at o ¼ or is no more a unitary rank matrix, and then the corresponding R matrix has more then one non zero value. Nevertheless, the mode shape could be estimated from the singular vector corresponding to the maximum singular value since the dynamic behavior of the structure at the natural frequency or is almost dominated by the rth eigenmode. From this point of view, a high frequency resolution is the crucial point to face coupled mode structures. 3. Experimental investigation 3.1. Cantilever beam In order to check the capabilities of the proposed procedure, the estimates of the modal parameters of a simple aluminum cantilever beam, whose dimensions were 0:2 0:0154 0:00285 m, have been first considered, and the results have been compared with those achieved in Ref. [8]. The experimental test, carried out in the frequency range of 0–400 Hz by using 8192 sampling points, allowed one to measure four time histories corresponding to the vertical accelerations of the points equally distributed along the beam span. Estimates of natural frequencies, and damping ratios as results of the proposed method, denoted as ‘‘Hilbert transform method’’ (HTM), have been carried out by fitting the biased FRFs with a least square polynomial ratio algorithm and then compared with those achieved either from the least square fittings of frequency response functions (FRFs) or by other approaches, as the one based on the ‘‘frequency domain decomposition’’ (FDD), the core of the software ARTeMIS from SVS [9]. When the FRF measurements are considered, the modal parameters Table 1 Natural frequency estimates for the cantilever beam. Mode #
fnFRF (Hz)
fnFDD (Hz)
fnHTM (Hz)
1 2
50.98 318.07
51.18 318.83
50.93 320.39
Table 2 Damping ratio estimates for the cantilever beam. Mode #
zFRF (%) n
zFDD (%) n
zHTM (%) n
1 2
0.56 0.40
0.49 0.25
0.89 0.38
0.8
Mode #2 Mode #1
Displacement
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 1
2
3
4
Grid Location Fig. 1. Cantilever beam—estimate of the first and second mode shapes.
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(natural frequencies and damping ratios) have been estimated by a least square fitting of the time response with complex exponentials, part of the LMS-TEST.LAB software, whereas a least square fitting with polynomial ratios was used when the HTM approach has been considered. Instead, with the FDD approach, the natural frequencies were evaluated from the peaks of the singular values of the power spectral density matrix, as function of the frequency, the mode shapes from the corresponding singular vectors and the damping ratios have been derived by the logarithm decrement. The response of the structure, randomly excited with a pencil crawling on one side of the beam, was recorded for a time period of 40.96 s, and the auto and cross-power spectral densities were averaged using 16 data block records with 2048 sampling points each. As one can see from Table 1, the natural frequencies are practically the same, whereas the damping ratio estimates depend on the estimation procedure, Table 2. It is worthwhile to remark that using a different evaluation method leads to damping factors in order of 0.80% for the first mode evaluated on the FRFs. A good agreement is also confirmed when the mode shapes, obtained from the singular vectors of the biased FRF—evaluated at the natural frequency, Fig. 1, and those from the other approach, not reported here but shown in Ref. [8], are correlated. 3.2. Free–free plate A second structure has been considered for the purpose of validating the methodology. The aluminum plate, 0:25 0:25 0:0025 m, has been already tested in Ref. [4] , and here the results are considered as reference. The excitation of the structure has been provided by a loudspeaker positioned in front of the middle of the plate, at a distance of 10 cm, whereas nine vertical responses, uniformly distributed over the structure, have been measured using four roving accelerometers. The same good correlation, as the one reported for the cantilever beam, has been obtained when comparing both the natural frequencies and damping ratios, Tables 3 and 4. The estimate of the mode shapes from the singular value decomposition confirmed its efficiency: the mode shapes estimated with the proposed approach, depicted in Figs. 2–4, need the evaluation of the singular vectors associated with FRF matrix only when o ¼ on .
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 3 2.5 2 1.5 1
3
2
1
Fig. 2. Free–free plate—first mode estimate.
Table 3 Natural frequency estimates for the free–free plate. Mode #
fnFRF (Hz)
fnFDD (Hz)
fnHTM (Hz)
1 2 3
70.20 102.21 149.30
69.40 101.7 148.7
70.90 101.6 148.2
Table 4 Damping ratio estimates for the free–free plate. Mode #
zFRF (%) n
zFDD (%) n
zHTM (%) n
1 2 3
0.63 0.14 0.27
0.45 0.23 0.81
0.55 0.48 0.96
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0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 3 3 2
2 1
1
Fig. 3. Free–free plate—second mode estimate.
0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 3 2 1
1
2
3
Fig. 4. Free–free plate—third mode estimate.
3.3. AB-204 helicopter blade Finally, the output-only experimental analysis has been performed on the AB-204 helicopter blade, Fig. 5a, which had a span of 6100 mm with a chord of 530 mm, Fig. 5 b, with the total weight of 88 kg. The main structural components of the blade, i.e., the skin, the sandwich core, and the main spar were in aluminum alloys. The helicopter blade has been tested in a free–free boundary condition obtained using elastic suspensions, four roving accelerometers have been used to measure the vertical deflection at the leading edge, the trailing edge, and the elastic axis for eleven sections along the blade span (for a total of 33 experimental degrees of freedom). The excitation was due to a natural event (wind) being the structure hung outdoor. The frequency range considered was 0–100 Hz, whereas the number of the sampling points were 215 . All the approaches used to estimate the modal parameters identified six elastic bending modes, three of which were very close, as reported in Fig. 6, where the magnitude of the biased frequency response function, obtained with the proposed approach at the blade root on the elastic axis, is reported. These three modes have been found out to vibrate at 42.91, 43.61, and 44.91 Hz: although all of them were fully coupled bending–torsion modes, the first is a prevalent in-plane bending mode, whereas the last two were out-of-plane bending–torsion coupled modes. A comparison between the natural frequencies and damping ratios for the out-of-plane bending modes of the blade is reported in Tables 5 and 6. These results seem to provide a good correlation except some discrepancies in the damping ratios. The same good correlation has been obtained for the predominant in-plane bending mode. In this case the FDD approach identified the natural frequency at 43.08 Hz with a damping ratio of 0.18%. These values are practically equal to those identified by the HTM approach: fn ¼ 42:91, and zn ¼ 0:21%. Finally, the estimates of the modal shapes for the helicopter blade are reported in Figs. 7–11. The reinforcing structures near the blade root seems to affect the shape of the first, Fig. 7, the third, Fig. 9, and the fourth mode, Fig. 10, and not the second and the fifth modes, Figs. 8 and 11.
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Fig. 5. AB-204 helicopter. (a) General view of AB-204 helicopter. (b) AB-204 helicopter blade.
|H22(ω)|
102
101
100
10−1 10
20
30
40 50 60 Frequency (Hz)
70
80
Fig. 6. AB-204 helicopter blade—frequency response function.
Table 5 Natural frequency estimates for the AB-204 helicopter blade. Mode #
fnFRF (Hz)
fnFDD (Hz)
fnHTM (Hz)
1 2 3 4 5
7.49 21.67 43.70 44.98 72.69
7.43 21.60 43.55 44.85 72.69
7.32 21.62 43.61 44.91 72.63
Table 6 Damping ratio estimates for the AB-204 helicopter blade. Mode #
zFRF (%) n
zFDD (%) n
zHTM (%) n
1 2 3 4 5
0.17 0.14 0.39 0.14 0.20
0.37 0.19 0.38 0.20 0.90
0.79 0.44 0.43 0.42 0.23
4. Concluding remarks In this paper, a methodology able to identify the modal parameters of coupled mode structures, using output-only data, has been developed. As shown, natural frequencies, damping ratios and mode shapes can be obtained from the FRFs, which in turn have been achieved from the output signal power spectral densities. Obviously, the input is unknown, but if it is
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7 6 5
1
4
0
3
−1
2
Bla
1 de 0.5 Ch ord 0 (m )
)
n (m
pa de S
Bla
1 0
Fig. 7. AB-204 helicopter blade—first mode estimate.
7 6 5
1
4
0 −1 Bla 1 de 0.5 Ch ord 0 (m )
3 2
)
n (m
pa eS
d
Bla
1 0
Fig. 8. AB-204 helicopter blade—second mode estimate.
7 6 5
1 4
0
3
−1 Bla 1 de 0.5 Ch ord 0 (m )
2
de
Bla
n
Spa
(m)
1 0
Fig. 9. AB-204 helicopter blade—third mode estimate.
possible to consider the input auto spectral density as constant—at least in the considered band—the estimated FRFs differ from the actual ones for an unknown constant. Actually, it is straight to evaluate the natural frequencies and the damping ratios by least square fittings, either in the frequency or in the time domain, also with commercial codes. Concerning the
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Fig. 10. AB-204 helicopter blade—fourth mode estimate.
7 6 5
1
4
0 −1 Bla 1 de 0.5 Ch ord 0 (m )
3 2
pan
eS Blad
(m)
1 0
Fig. 11. AB-204 helicopter blade—fifth mode estimate.
modes shapes, although they could be derived from the residues (as done in the past by the authors), a technique based on the decomposition of the FRF matrix into its principal values and vectors has been proposed and successfully applied to a beam, to a plate, and to a helicopter blade. Although coupled modes were present in the spectra of this last structure, the modal parameters along with the mode shapes were identified. The presented approach needs only to perform the singular value decomposition of the FRF matrices evaluated at the frequencies of the modes, and that are known from the least square fitting of the frequency functions mentioned above.
Acknowledgment This research has been supported by ‘‘Ricerca di Facolta : Stima Parametri Modali dalle Sole Risposte Dinamiche di Sistemi Aerospaziali’’, University of Rome ‘‘La Sapienza’’, 2003. References [1] R. Brincker, L. Zhang, P. Andersen, Modal identification from ambient responses using frequency domain decomposition, in: XVIII IMAC, vol. 1, San Antonio, TX, USA, Society for Experimental Mechanics, Inc., Bethel, CT, USA, 2000, pp. 625–630. [2] L. Hermans, H. Van der Auweraer, Modal testing and analysis of structures under operational conditions: industrial applications, Mechanical Systems and Signal Processing 13 (2) (1999) 193–216. [3] A. Agneni, L. Balis Crema, G. Coppotelli, Time and frequency domain model parameter estimation by output only functions, in: Proceedings of International Forum on Aeroelasticity and Structural Dynamics, Amsterdam, NL, 2003. [4] A. Agneni, R. Brincker, G. Coppotelli, On modal parameter estimates from ambient vibration tests, in: Proceedings of International Conference on Noise and Vibration Engineering—ISMA 2004, Leuven, B, 2004, pp. 2239–2248.
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[5] H.W. Bode, Network Analysis and Feedback Amplifier Design, Van Nostrand, 1945, pp. 303–336 (Chapter XIV). [6] E.A. Robinson, M.T. Silva, Digital Signal Processing and Time Series Analysis, Holden Day, 1978, pp. 283–290. [7] D. Otte, J. Leuridan, H. Grangier, R. Aquilina, Prediction of the dynamics of structural assemblies using measured FRF-data: some improved data enhancement techniques, in: Proceedings of the Ninth International Modal Analysis Conference, 1991, pp. 909–918. [8] G. Coppotelli, On the estimate of the FRFs from operational data, Mechanical Systems and Signal Processing 23 (2009) 288–299. [9] Structural Vibration Solutions, Getting Started Manual for all Versions of ARTeMIS Extractor, NOVI Science Park, Aalborg, DK, 2003.