Keywords: frequency response function, half-power bandwidth, extreme value, frequency resolution, error estimation, interpolation, adjacent mode, modal analysis, damping ratio, beam.

A single degree of freedom (SDOF) method to estimate the structural damping by measured frequency response functions (FRFs) is presented and validated by simulation and experiment. The method is compared to the traditional half-power bandwidth method. Possible error sources are discussed. In particular, error estimation is established to assess the influence of the measurement frequency resolution. The effective frequency resolution is increased by interpolation. The proposed method and calculated error bounds are verified by a synthetic modal analysis carried out using analytical FRFs of a cantilever beam attached with piezoelectric patch actuators and sensors. In addition, an experimental modal analysis is done for the first ten bending modes and the first longitudinal mode of the beam. The results are benchmarked with the ones obtained by a multiple degree of freedom (MDOF) curve fitting algorithm.

Both SDOF methods discussed within the paper are based on the transfer function (TF) of a generic continuous system. The modal description used assumes the linearity of the structural behaviour and proportional damping.

The traditional half-power bandwidth method is revised first. The damping ratio is related to the resonance bandwidth of the magnitude of FRFs neglecting the response contribution of adjacent modes. The relation derived not only depends on the resonance bandwidth but additionally incorporates the resonance frequency. Next, a method is proposed to infer the damping ratio from the frequencies corresponding to the extreme value frequencies of the real part of FRFs.

The accuracy of the SDOF methods essentially depends on the exactness of the characteristic frequencies used as input. The determination of these frequencies is prone to errors due to measurement noise, nonlinearity, leakage, measurement frequency resolution and adjacent modes [1]. The maximum error to be expected due to the limited frequency resolution for the frequencies corresponding to the half-power bandwidth is the same as the frequency resolution. For the extreme value frequencies and the resonance frequency, it is half the frequency resolution. The total differential of the equations derived for the estimation of the damping ratio is used to relate these errors to the damping ratio errors to be expected.

The characteristics of the FRF are sufficiently reflected by the measured FRF and the effective frequency resolution can be increased by about one order of magnitude using e.g. cubic spline interpolation. Thus the errors caused by the limited frequency resolution are decreased. Moreover, the interpolation smoothes the FRF in the vicinity of the extreme values such that the influence of the noise is reduced.

First the methods discussed and the error estimates are verified using the simulated patch FRF of a cantilever beam. The theoretical formulation of the FRFs is presented in detail in [2]. The half-power bandwidth method yields large errors of more than 100% for the weaker low frequency modes, even when a high frequency resolution of, for example, 0.01Hz is used. In contrast, the extreme value method shows damping ratio errors smaller than 1% using the same frequency resolution. The damping ratio errors obtained for the extreme value method are well within the calculated worst case estimates.

In addition, an experimental modal analysis is performed for the first ten bending and the first longitudinal mode of the beam. The modal parameters are inferred both using a multiple degree of freedom (MDOF) curve-fitting algorithm provided by the LMS CADA-X software (SW) and by the proposed extreme value method. The two sets of modal data obtained match very well except for the first bending and longitudinal mode with deviations of about 60%.

An effective SDOF method to infer the modal parameters based on measured FRFs has been proposed. Error estimation has been presented in order to assess the influence of the measurement frequency resolution. The feasibility to increase the frequency resolution by about one order of magnitude using interpolation has been shown. The method and worst case error bounds have been successfully verified by the synthetic and experimental modal analysis of a cantilever beam. The paper provides the structural test engineer with a SDOF method that is easy to implement, computationally efficient and gives reliable results. The method requires a minimum level of initial information such that it is suited for automated procedures.

1

M. Lee and P.E.&.M. Richardson, "Determining the accuracy of modal parameter estimation methods", International Modal Analysis Conference, IMAC X, 1992.

2

J. Dennerlein et al., "Improved analytical modelling of smart piezoelectric beams and its experimental verification", Technische Mechanik, accepted.

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