Non-stationary Ambient Response Data Analysis for Modal Identification Using Improved Random Decrement Technique

Transcription

1 9th International Conference on Advances in Experimental Mechanics Non-stationary Ambient Response Data Analysis for Modal Identification Using Improved Random Decrement Technique Chang-Sheng Lin and Tse-Chuan Tseng Precision Mechanical Engineering Group, Instrumentation Development Division, National Synchronization Radiation Research Center, Hsinchu, Taiwan 3-5 September 213 Cardiff School of Engineering, Cardiff University Cardiff, Wales United Kingdom

2 Outline Introduction Engineering Problems (System Analysis) Ambient Vibration Modal-Identification Method Literature Review Non-stationary Random Decrement Algorithm Theoretical Development Practical Treatment Extended RDD Algorithm Ibrahim Time Domain Method (ITD) Simulation and Discussions Conclusions

3 Introduction Direct Analysis Input S Output System Identification Input S Output

4 Introduction Direct Analysis Input S Output Modal System Identification Excitation Input S Output Response It is desirable to develop techniques for modal-parameter identification only from ambient vibration data, i.e., without the need for input measurement.

5 In-operation testing Ambient Vibration In-flight measurement Most ambient excitations are random in nature Stationary (white) process probability distribution is independent of time Non-stationary process

6 Literature Review (1/2) The assumption of ambient excitation is stationary white Natural Excitation Technique (NExT) (James et al. [1993,1994, and1995]) Correlation Technique Random Decrement Technique (RDD) (Cole [1971], Vandiver et al. [1982], and Bedewi [1986]) Modal identification from ambient vibration data Extracting Dynamic Characteristic of a building from Ambient Vibration Measurement (Ventura et al.[23]) Ambient vibration-based seismic evaluation of a continuous girder bridge (Ren et al.[24])

7 Literature Review (2/2) Ambient Vibration Data Analysis for Structural Identification and Global Condition Assessment (Gul and Catbas [28]) System Identification of Suspension Bridge from Ambient Vibration Response (Siringoringo and Fujino [28]) Ambient Vibration Testing and Condition Assessment of the Paderno Arch Bridge (1889) (Gentile and Saisi [211]) EMD-based RDD for Modal Parameter Identification of an Existing Railway Bridge (He et al. [211])

8 Many modal-identification methods are done only from impulse or free responses of a structure Usually, the assumption of ambient excitation is stationary white How to identify the major structural modes from nonstationary ambient responses?

9 Theoretical Development of Nonstationary Random Decrement Technique Assumption: non-stationary white noise in the form of a product model. Non-stationary random decrement signature E[ X ( t) X ( t )] RXX ( t, ) E[ X ( t)] E[ X ( t)] ( t, t ) X ( t) X 2 1( t) XX 1 1 (, ) XX tt 1 2 (, ) XX tt 1 2 is in direct proportion to the R under the assumption of Gaussian processes (, ) XX t 1 2 For any fixed time instant t, R (, ) XX t is a sum of complex 1 2 exponential functions, which is of the same form as the free-vibration decay or the impulse response of the original system. A () t n ir jr ij (, ) exp rnr sin dr r r1 mrdr R t 9

10 Practical treatment of non-stationary data (1/3) Usually very limited data samples are available in engineering practice To transfer the original non-stationary responses into stationary ones Ergodic process

11 Practical treatment of non-stationary data (2/3) Assumption: f ( t) ( t) w( t) Slowly time- varying The responses of the system can be approximately derived as a product model with the same amplitude-modulating function as that associated with the excitation itself. S

12 Practical treatment of non-stationary data (3/3) By using interval average and then applying curvefitting, we obtain the temporal root-mean-square function, and so the envelope function. We can then acquire the approximate stationary responses by dividing the non-stationary responses of each DOF with the same envelope function.

13 Extended RDD Algorithm Original RDD is only good for the assumption of stationary excitation. Practical treatment of non-stationary force-vibration data is only applicable for the non-stationary excitation in the form of a product model of white noise and a deterministic slowly-time-varying amplitude-modulating function. The error involved in transforming the original nonstationary responses into the approximate response of free decay would generally lead to a distortion in the modal parameters of identification.

14 Extended RDD Algorithm The standard matrix equations of motion Mx(t) Cx(t) Κx(t) f(t) Introduce the transformation: 2m x(t) Φq(t) φr q r (t) r1 Express q r (t) by combining the homogenous solution q rh (t) and the particular solution q rp (t) q (t) q (t) q (t) r rh rp nonstationary zeromean random excitation q +ξ ω q t r dr dr r ω r dr -ξrωrt r r r r T e q cosω t + sinω t φ f ( )h (t - )dτ (a)

15 Extended RDD Algorithm The modal velocity response q r(t) results from the calculation of the first differentiation of r with respect to t q +ξ ω q q + ξ ω q r r r r r r r r r r r r d r d r r d r d r d r d r rp ωd r ωd r The following equation can then be derived q (t) -ξ rω rt q (t) = e [-ξ ω (q cosω t + sinω t) + (-q ω sinω t+ ω cosω t)] + q (t) -ξrωrt r r r r =-ξ ω q (t) + e (-q ω sinω t+ ω cosω t) +q (t) r r r r d r d r d r d r rp ωdr q +ξ ω q -ξrωrt r r r r q (t) + ξ ω q (t) - q (t) = e (-q ω sinω t+ ω cosω t) r r r r rp r dr dr dr dr ωdr q +ξ ω q (b)

16 Substituting Extended RDD Algorithm t into t + q +ξ ω q q (t ) = e e [cosω t(q cosω + sinω ) -ξ rω r t -ξ rω r r r r r r d r r d r d r ωdr sinω t q + ξ ω q + (-q ω sinω + ω cosω )] + q (t ) ω d r r r r r r d r d r d r d r rp d r ωdr (c) Through insertion of Eqs.(a) and (b) into Eq.(c), the following equation can then be derived -ξrωr t q r +ξrωrq r -ξrωr t q rpsinωdrt q r(t ) = e qrcosωdr t + sinωdrt - e qrpcosωdr t + + q rp(t ) ωdr ωdr (d)

17 Extended RDD Algorithm Random-decrement averaging Through the ensemble averaging of N preselected sample segments of the response measurement, the following time function can be obtained as follows N 1 ( ) q r(t i + τ) N i=1 (e)

18 In the proposed method, since the input excitation Extended is assumed RDD to Algorithm be a nonstationary random process with zero-mean, the behavior due The to following force vibration equation will is obtained be vanished by by performing insertion of the Eq.(d) random-decrement into Eq.(e) averaging. 1 q +ξ ω q 1 q sinω 1 N ω N ω N N N N t r r r t r t rp dr -ξrωr i i -ξrωr i ( ) = e qtircosω dr + sinω dr + -e qtirpcosω dr + + q rp(t i + ) i=1 dr i=1 dr i1 B+ξ ω A -ξrωr r r r r = e A cosω + sinω + Θ( ) r dr dr ωdr N -ξrωrt i=1 i=1 Θ(τ) = -e q (t ) cosω τ + sinω τ + q (t + τ) implies that () contains the behavior of free-decay vibration ξω q (t ) - q (t ) N n rp i rp i N 1 1 rp i dr dr rp i N i=1 ωdr N i=1 N corresponds to the behavior due to force vibration

19 Extended RDD Algorithm It has been shown that an improvement to the random decrement algorithm is presented for modal identification from zero-mean nonstationary ambient vibration data. We can perform the random-decrement averaging over the nonstationary-responses samples to obtain good improved RDD signatures, as quasi free-vibration data, for further modal identification without any additional treatment of transforming the original nonstationary responses into stationary ones. Therefore, we avoid a distortion in the modal parameters of identification induced by the error involved in the approximate quasi-stationary response obtained through curve-fitting technique.

20 Ibrahim Time-Domain Method (ITD) ~Using free-decay responses of a structure to identify its modal parameters in complex form [X]: data matrix from free responses 2m rt j x x ( t ) e ij i j ir n1 [Y]: data matrix from time-delay sampling 2m 2m ( t t) t ij i j ir ir r1 r1 r j r j y x ( t t) e e ir ir e rt 2

21 Define Least square via pseudo inverse Characteristic root of the original vibration system: Eigenvalue of system matrix [A] : Natural freq.: A X Y T 1 T A Y X X X a ib r r r i r r r Damping ratio: 1 ln 2t 1 tan 1 r t r 2 2 a r r r 2 2 r nr ar br r 2 2 ar br b r a

22 Non-stationary responses in the form of a product model Approximate stationary responses Approximate free-decay responses Modal parameters

23 Simulation and Discussions (1/1) 6-DOF chain model The non-proportionally damped system Non-stationary excitation acts on 6 th mass point of the system Non-stationary excitation employs product model where () t wt () f ( t) ( t) w( t) : Envelope function; : stationary white noise Envelope function and Non-stationary excitation 23

24 6-DOF chain model M K N sec N m N sec m 1. 1 C M K 2 m 66

25 Simulation and Discussions (3/1) Displacement response in 1 st, 3 rd, 5 th D.O.F. Fourier spectrum of displacement response in 1 st, 3 rd, 5 th D.O.F. 25

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27 Simulation and Discussions (5/1) Results of modal-parameter identification ITD method in conjunction with the conventional RDD and curve-fitting technique Natural Frequency (rad/s) Damping Ratio (%) Mode Exact ITD Error(%) Exact ITD Error(%) MAC ITD method in conjunction with the improved RDD Natural Frequency (rad/s) Damping Ratio (%) Mode Exact ITD Error(%) Exact ITD Error(%) MAC

28 Simulation and Discussions (6/1) 8-DOF truss model, / sec m N M, / m N K. sec/.1.17 m N K M C

29 Simulation and Discussions (7/1) Results of modal-parameter identification ITD method in conjunction with the conventional RDD and curve-fitting technique Natural Frequency (rad/s) Damping Ratio (%) Mode Exact ITD Error(%) Exact ITD Error(%) MAC ITD method in conjunction with the improved RDD Natural Frequency (rad/s) Damping Ratio (%) Mode Exact ITD Error(%) Exact ITD Error(%) MAC

30 Simulation and Discussions (8/1) 7-DOF model (containing two pairs of close modes) M N s / m, K N / m, Natural Frequency (rad/s) Damping Ratio (%) Mode Exact ITD Error(%) Exact ITD Error(%) MAC C.2M.1K Ns / m

31 Simulation and Discussions (9/1) For higher DOF structural system excited by a sample of the seismic record Assume that each mass is 1 kg and all spring constants are 6 N/m. The damping matrix of the system is assumed to be Mode Natural Frequency (rad/sec) Damping Ratio (%) Exact ITD Error(%) Exact ITD Error(%) MAC

32 Simulation and Discussions (1/1) (E6XR3X) Mode (~1Hz) Natural Frequency (Hz) Damping Ratio (%) Shock Ambient Shock Ambient Error(%) Error(%) Vibration Vibration Vibration Vibration

33 Conclusions Innovative methods have been proposed in this research for determination of the modal parameters of a structure from its measured ambient response data only. It is shown theoretically that the nonstationary response signals can be converted into freevibration data via the random decrement technique by assuming the ambient excitation to be nonstationary white noise (product model).

34 A technique of curve-fitting is proposed in conjunction with the random decrement technique for modal identification. It can be done, in practice, under the assumption of slowly-time-varying amplitudemodulating envelope function. However, the error involved in the approximate freedecay response would generally lead to a distortion in the modal identification. If the ambient excitation can be modeled as a zeromean nonstationary process, without any additional treatment of transforming the original nonstationary responses, the nonstationary cross randomdec signatures of structural response are shown in the same mathematical form as that of free vibration of a structure, from which modal parameters of the original system can thus be identified.

35 The choice of the reference channel is significant to compute the random decrement signatures. The reference channel is chosen as a response channel whose Fourier spectrum has rich frequency content around the structure modes of interest. The richer frequency content the reference channel has, the better results of modal parameters identification can be achieved. Numerical simulations, including one example of using the practical excitation data, confirm the validity of the proposed method for identification of modal parameters from nonstationary ambient response data.

36 Thanks The for your kind attention End!

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