ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 21 (2007) 74–97 www.elsevier.com/locate/jnlabr/ymssp The role of anti-resonance frequencies from operational modal analysis in finite element model updating D. Hansona,Ã, T.P. Watersb, D.J. Thompsonb, R.B. Randalla, R.A.J. Forda aUniversity of New South Wales, Sydney, Australia bInstitute of Sound and Vibration Research, Universtiy of Southampton, UK Received 13 September 2005; received in revised form 29 December 2005; accepted 4 January 2006 Available online 15 March 2006 Abstract Finite element model updating traditionally makes use of both resonance and modeshape information. The mode shape information can also be obtained from anti-resonance frequencies, as has been suggested by a number of researchers in recent years. Anti-resonance frequencies have the advantage over mode shapes that they can be much more accurately identified from measured frequency response functions. Moreover, anti-resonance frequencies can, in principle, be estimated from output-only measurements on operating machinery. The motivation behind this paper is to explore whether the availability of anti-resonances from such output-only techniques would add genuinely new information to the model updating process, which is not already available from using only resonance frequencies. This investigation employs two-degree-of-freedom models of a rigid beam supported on two springs. It includes an assessment of the contribution made to the overall anti-resonance sensitivity by the mode shape components, and also considers model updating through Monte Carlo simulations, experimental verification of the simulation results, and application to a practical mechanical system, in this case a petrol generator set. Analytical expressions are derived for the sensitivity of anti-resonance frequencies to updating parameters such as the ratio of spring stiffnesses, the position of the centre of gravity, and the beam’s radius of gyration. These anti-resonance sensitivities are written in terms of natural frequency and mode shape sensitivities so their relative contributions can be assessed. It is found that the contribution made by the mode shape sensitivity varies considerably depending on the value of the parameters, contributing no new information for significant combinations of parameter values. The Monte Carlo simulations compare the performance of the update achieved when using information from: the resonances only; the resonances and either anti-resonance; and the resonances and both anti-resonances. It is found that the addition of anti-resonance information improves the updating performance for some combinations of parameter values, but does not improve the update in significant other regions. The simulated results are verified using resonance and anti-resonance frequencies measured on a steel beam test rig. The investigation is extended to include the updating of parameters of a petrol generator set. It is found that the contribution of the anti-resonances to the model update is heavily dependent on the geometry of the model and the choice of variables to be updated, suggesting that, for some models, the pursuit of anti-resonance information through expensive operational modal analysis may be inappropriate. r 2006 Elsevier Ltd. All rights reserved. Keywords: Finite element model updating; Anti-resonance frequencies; Operational modal analysis ÃCorresponding author. Tel.: +61 2 9385 6256; fax: +61 2 9663 1222. E-mail address: [email protected] (D. Hanson). 0888-3270/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2006.01.001 ARTICLE IN PRESS D. Hanson et al. / Mechanical Systems and Signal Processing 21 (2007) 74–97 75 1. Introduction Finite element models of many engineering structures, for example passenger vehicles, are generally poor predictors of dynamic properties. It is not uncommon for resonances predicted by such models to differ by 10% or more from those obtained from measurements. An accurate knowledge of the modal properties of the vehicle is important for passenger comfort, ride and handling, and fatigue considerations. Finite element models can be updated, based on measured modal properties, so as to more closely match the actual dynamic properties of the vehicle. These modal properties, such as resonances and mode shapes, are usually obtained from a modal test in the workshop where the response to a known input is measured. However, many vehicle components, such as the suspension and glass reinforced polymer components, exhibit significant non-linear behaviour. Therefore, the behaviour of the vehicle in the workshop may be significantly different from the behaviour in service. It is generally not possible, however, to measure the input forces when the vehicle is in service. In-service modal properties can be obtained through operational modal analysis (OMA), whereby the system modal properties are estimated without knowledge of the inputs. Such techniques incorporate one of a number of blind system identification techniques, such as the popular frequency domain decomposition [1], stochastic subspace [2], maximum likelihood [3], and cross correlation and cross power spectra based methods [4]. Recently, we introduced a new blind system identification technique, which has particular relevance to passenger vehicles excited by internal combustion engines [5]. This technique exploits the cyclostationary properties of the engine combustion signal and so can be employed when the vehicle is in service. It identifies the resonances and anti-resonances of the transfer function between the engine and each measurement location, effectively reducing a multiple-input multiple-output (MIMO) system of a vehicle in operation to a single-input multiple-output (SIMO) system. The spectrum of the engine excitation is not white however, and so any mode shape information extracted by the blind identification process will be polluted and have arbitrary scaling. However, by curve fitting in the cepstrum domain, it has been shown that the anti-resonance frequencies can be accurately identified as long as the log spectrum of the input is relatively smooth so that its cepstrum is short [5,6]. Therefore, the anti-resonance frequencies can be used to provide extra information to an updating process. However, blind system identification represents a significant increase in cost and complexity relative to identification of the resonances alone, e.g. from the power spectra of just a few well chosen responses. The motivation for this paper then was to answer the question of whether the anti- resonances contribute to the updating process sufficiently to justify the expense of blind system identification. There have recently been several examples in the literature of the use of anti-resonances in finite element model updating. In [7], Jones and Turcotte compare the performance of updating based on resonances only and based on resonances and anti-resonances on an antenna array. They report an improved model update using the anti- resonances compared to the resonances alone, even when using an over-determined system of 11 resonances and only seven update parameters. D’Ambrogio and Fregolent [8], however, report only a slight improvement in results when using anti-resonances in the updating process, again using an over-determined system with nine resonances and eight update parameters. They note one particular parameter to which the resonance exhibits negligible sensitivity and the anti-resonance exhibits reasonable sensitivity, and in this specific case, anti-resonances can be seen to contribute significant new information to the updating procedure. The relative sensitivities of the resonances and anti-resonances to the updating parameters are not reported in [7], but this might have contributed to the improvement in results obtained by the use of anti-resonances. D’Ambrogio and Fregolent [9] further show that robust FE model updating using anti-resonances requires point FRFs because the anti-resonances in these measurements do not appear and disappear abruptly due to small parameter changes. A modal test consisting of point measurements, which would add significantly to the cost of a modal test, is not necessary though, as they also explain how transfer measurements can be converted to point measurements. It was previously shown by Mottershead [10] that the sensitivities of update parameters to anti-resonances can be expressed as a summation of sensitivities to natural frequencies and mode shapes. Thus anti-resonance frequencies do not introduce any new information to the updating process, which was not offered by the traditional natural frequency and mode shape data. However, the anti-resonance frequencies are much more accurately and easily measured than mode shapes and could therefore provide a more reliable means of collecting the same data. ARTICLE IN PRESS 76 D. Hanson et al. / Mechanical Systems and Signal Processing 21 (2007) 74–97 The anti-resonances can be defined in terms of the eigenvalues of reduced mass and stiffness matrices. These matrices are formed from the original mass and stiffness matrices, but with a row and column, corresponding to the measurement and excitation degrees-of-freedom (DOFs), removed. The anti-resonance sensitivities are likewise defined in terms of the eigenvalue and eigenvector sensitivities of this reduced system. Mottershead showed that the sensitivities of the eigenvalues and eigenvectors of the resonances that are