The achievable limits of operational modal analysis. * Siu-Kui Au 1)

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1 The achievable limits o operational modal analysis * Siu-Kui Au 1) 1) Center or Engineering Dynamics and Institute or Risk and Uncertainty, University o Liverpool, Liverpool L69 3GH, United Kingdom 1) siukuiau@liverpool.ac.uk ABSTRACT Operational modal analysis oers an economical way or identiying the modal properties (natural requency, damping, mode shape) using output-only vibration data collected under working condition o the structure without artiicial loading. As the modal properties are identiied without using explicit inormation o the loading, they oten have signiicantly higher uncertainty than those identiied in ree or orced vibration tests. It is o both scientiic and engineering signiicance to understand the undamental scaling law o their uncertainty, which governs the achievable limits o operational modal analysis. For example, how much data is needed or identiying the damping ratio to within 30% coeicient o variation? In a Bayesian probability context, this paper gives a undamental answer or small damping, suicient data duration and wellseparated modes. 1. INTRODUCTION Ambient vibration (output-only) tests have gained increasing popularity in both theory development and practical applications (Brincker 001; Brownjohn 003; Au 011a). This is to a large extent attributed to its economy in implementation. Ambient vibration data are obtained when the structure is under unknown working load assumed to be random with broadband spectral characteristics. Ambient modal identiication, oten called operational modal analysis, allows the extraction o modal properties under such context. In the absence o speciic loading inormation, the uncertainty o the identiied modal parameters is oten signiicantly larger than that using orced vibration (known input) or ree vibration tests where the signal-to-noise (s/n) ratio can be managed to an adequate level. One question that arises requently in perorming ambient vibration tests is: How much data do we need? 1) Chair o Uncertainty, Reliability and Risk 797

2 The answer to question is by no means simple. In this paper we provide a undamental answer to this question or well-separated modes under some asymptotic conditions, namely, small damping, high modal signal-to-noise ratio and suiciently long data. These conditions are typically met in ield testing applications o civil engineering structures and so the results can provide useul guidelines or perorming ambient vibration tests in practice. In what ollows, we shall irst reason logically the context where the question and answer should be placed. We shall then outline the answer, reerred as uncertainty laws. Practical implications shall also be provided.. BAYESIAN IDENTIFICATION FRAMEWORK Suppose we have a digital record o acceleration time history data { x ˆ R n : j 1,..., N} at n measured degrees o reedom (dos) o a structure under j ambient condition. From this data set we would like to identiy the modal properties o a given mode o interest. The modal properties include primarily the natural requency, damping ratio and (incomplete or partial) mode shape Φ (an n -by-1 real vector). It is assumed that the structure is classically damped and the mode o interest is wellseparated rom other modes, in the sense that it dominates the requency response (e.g., in terms o power spectral density, PSD) in its resonance band (see more later). It is also assumed that the data is o good quality, in the sense that the contribution o modal response is large compared to the channel noise level in the resonance requency band o the mode. Ambient condition or broadband excitation here reers speciically to the modal excitation having a constant PSD in the resonance requency band. The modal excitation only need to be locally white within the resonance band, rather than the whole sampled requency band. The ambient assumption here is much easier to justiy that one might typically perceive. In the presence o uncertainties or lack o inormation associated with the measurement (channel noise, inite data length) and modeling assumptions (e.g., classical damping, ambient excitation) one cannot expect to determine the modal properties exactly even in the presence o the data. The remaining uncertainty associated with the modal parameters given the data can be quantiied undamentally in a Bayesian identiication perspective (Jaynes 003; Beck 01). Let the modal parameters be collected in a vector θ and let the measured data be denoted by D. In the presence o data all inormation about θ is encapsulated in the posterior distribution p( θ D). According to Bayes Theorem, it is given by p( 1 θ D) p( D θ) p( θ) p( D) (1) The RHS in (1) should be viewed as a probability distribution o θ and so only its variation with respect to θ matters. The last term on the RHS does not depend on θ and so it is irrelevant to the knowledge o θ. The term p (θ) is the prior distribution we mentioned beore. The term p ( D θ) is called the likelihood unction, which is the most 798

3 important term because it dictates the mechanism by which the inormation in D can be utilized to iner θ. It can be (and must be) derived based on the identiication model that relates θ and D, which corresponds to a orward (rather than inverse ) problem. Assume that the data is suicient to narrow down the posterior distribution to having only a single peak, which is oten valid in modal identiication. In this case the uncertainty o each parameter in θ can be conveniently characterized by the most probable value (MPV), which is where the posterior distribution is peaked; and a posterior standard deviation, which is related to the spread o the posterior distribution about its peak. For convenience an equivalent dimensionless measure called the posterior coeicient o variation (c.o.v.) is used, which is deined as the ratio o the posterior standard deviation to the MPV. Clearly, the posterior MPV and c.o.v. depend on the measured data. They can be obtained rom the posterior distribution or given data. The process is primarily a computational problem which has been solved eiciently, typically in a matter o seconds (Au et al. 013). 3. ASYMPTOTIC UNCERTAINTY LAWS The oregoing discussion shows that the remaining uncertainty o the modal parameters can be quantiied in terms o their posterior c.o.v.. This, however, does not provide much insight about how the posterior c.o.v. depends on various test conigurations because it can only be calculated point-wise or a given set o data. The exact dependence o the posterior c.o.v. on test conigurations and the measured data is expected to be extremely complicated and is unlikely to be described in a close-orm explicit expression. However, a recent study (Au 013a,b) shows that it is possible to obtain close-orm expressions or the leading order term o the posterior c.o.v. or wellseparated modes and under some asymptotic conditions, namely, small damping, high modal s/n ratio and suiciently long data duration. The latter conditions are oten encountered in ambient vibration tests o civil engineering structures and so the results can provide useul guidelines in practice. The results are reerred as uncertainty laws. The derivation is quite lengthy (omitted here) but the results are remarkably simple. The ull set o uncertainty laws also includes results on the mode shape, excitation intensity and channel noise but or practical purposes they are omitted in this paper. See Au (013a, b) or a ull story. Let and denote respectively the posterior c.o.v. o the natural requency and damping ratio. For a well-separated classically damped mode, small damping, high modal s/n ratio and suiciently long data duration, it can be shown rigorously that ( ~ reads asymptotic to or to the leading order ) ~ N B c ( ) () 1 ~ (3) N c B ( ) 799

4 where and are respectively the natural requency and damping ratio o the mode o interest; Nc Td (4) is a normalized data length equal to the duration o data T d divided by the natural period ( 1 ); and 1 B ( ) (tan ) 1 (5) 1 1 (tan ) B ( ) tan 1 (6) are data length actors that are monotonic increasing unction o the bandwidth actor. The latter relects the amount o inormation that can be utilized in the data or identiying the mode o interest without incurring signiicant modeling error. More speciically, the uncertainty laws have been derived assuming that the mode is identiied using inormation o the FFT o ambient vibration data within a requency band o ( 1 ). 4. PRACTICAL IMPLICATIONS The uncertainty laws capture undamentally the eect o test conigurations and they have important implications on perorming ambient vibration tests. First, in Eq. () is proportional to, while on the contrary in Eq. (3) is inversely proportional to. Ignoring the data length actors, / 1. For small encountered in applications, say, 0.5%~5%, this means that the damping ratio has much larger posterior uncertainty than the natural requency or mode shape, and so it is likely to govern the required data length. This is consistent with common indings (Tamura et al. 1996; Au et al. 01). The dependence o the uncertainty laws on can be explained intuitively but it is omitted here; see Au (013b). Governed by the uncertainty o the damping ratio, the required data length as a multiple o the natural period to achieve a given posterior c.o.v. is given by, using Eq. (3), N c [B ( ) (7) 1 ] To give a rule-o-thumb, consider a damping ratio o 1% and a bandwidth actor o 6, which gives B (6) ~ 60%. The required data length is then N c 7.5/, say, 800

5 30 N c ( 1%, 6 ) (8) This means that 300 natural periods are required to achieve a moderate posterior c.o.v. o 30% ; 750 periods or 0% ; and 3,000 periods or 10%. The corresponding c.o.v.s o the natural requency are 0.67%, 0.7% and 0.067%, which are negligible. Smaller damping or bandwidth requires longer data length. The value suggested in Eq. (8) is the minimum data length based on accuracy requirement and assuming good modal s/n ratio. In practice it will need to be traded o with other practical constraints. When little is known about the existence o a mode in a requency band one may increase (e.g., double) the data duration to get a clearer picture o the spectrum or deciding the number o modes. On the other hand, there are situations that limit the data duration and hence the identiication accuracy. Super-tall buildings (height >300m), or example, have a natural period in excess o 5 seconds. Assuming 1% damping, it requires over 4 hours to achieve a posterior c.o.v. o 10%. This duration is too long that signiicantly weakens the stationarity assumption in the stochastic load and the time invariance assumption o modal properties, giving rise to modeling errors that may invalidate the ormulation. Wind loads during typhoons can change by orders o magnitude in a matter o an hour. The damping ratio can change signiicantly over such period as a result o amplitude dependence. In view o this, or super-tall buildings a c.o.v. o 30% would be a reasonable accuracy to aim at, requiring about hal an hour data. This may put a limit on the precision o ield evidence or wind eects on long-period structures. 5. CONCLUSIONS Uncertainty laws Eq. () and Eq. (3) have been presented or the natural requency and damping ratio, respectively. They govern the achievable limits o accuracy in the natural requency and damping ratio identiied using ambient vibration data. They are derived undamentally based on a Bayesian identiication ramework under asymptotic conditions o small damping and long data duration. Our discussion reveals clearly that the identiication precision is primarily related to the spectral inormation in the data in the resonance band o the mode o interest. Inormation or complexities in other bands are irrelevant. As the Bayesian approach processes undamentally the usable inormation in the data or given modeling assumptions, the uncertainty laws represent the lower limit o uncertainty that can be achieved by any method, including Bayesian and non-bayesian methods. In the latter, uncertainty is interpreted as the ensemble variability o the estimates in a requentist sense when there is no modeling error (Au 01a). 801

6 6. ACKNOWLEDGMENTS This paper is supported by General Research Fund (CityU 11001) rom the Research Grants Council o the Hong Kong Special Administrative Region, China. REFERENCES Au, S.K. (011a), Fast Bayesian FFT method or ambient modal identiication with separated modes, Journal o Engineering Mechanics, ASCE, 137 (3), Au, S.K. (013a), Uncertainty law in ambient modal identiication. Part I: theory, Mechanical Systems and Signal Processing, To appear. Au, S.K. (013b), Uncertainty law in ambient modal identiication. Part II: implication and ield veriication, Mechanical Systems and Signal Processing, To appear. Au, S.K., Zhang, F.L. and To, P. (01), Field observations on modal properties o two tall buildings in Hong Kong, Journal o Wind Engineering and Industrial Aerodynamics, 101, 1-3. Au, S.K., Zhang, F.L., Ni, Y.C. (013), "Bayesian operational modal analysis: Theory, computation, practice," Computers and Structures, In print. DOI: /j.compstruc Beck, J.L. (01), Bayesian system identiication based on probability logic, Structural Control and Health Monitoring, 17 (7), Brincker, R., Zhang, L. and Anderson, P. (001), Modal identiication o output-only systems using requency domain decomposition, Smart Materials and Structures, 10(3), Brownjohn, J.M.W. (003), Ambient vibration studies or system identiication o tall buildings, Earthquake Eng. Structural Dynamics, 3, Jaynes, E.T. (003) Probability Theory: The Logic o Science, Cambridge University Press. Tamura, Y., Suganuma, S.Y. (1996), Evaluation o amplitude-dependent damping and natural requency o buildings during strong winds, Journal o Wind Engineering and Industrial Aerodynamics, 59,