42 B. Peeters et al. compared to the “nominal” case. It is important to have at least an idea on the magnitude of these changes and their impact on the final product behaviour when assessing and optimising the design based on an “ideal” product model. While statistical product testing may reveal such performance spread, this is in general very hard to realize. The evaluation of the uncertainty of the product performance starting from uncertainty bounds on the model parameters is an alternative approach [1]. In [2], a stochastic model updating method is introduced. The idea is to propagate the uncertainty in the Test to uncertainty bounds on certain parameters of the Finite Element Model: the uncertain eigenfrequencies and mode shapes from the experimental modal analysis are utilised to identify model parameter means and covariance matrix. The application of stochastic model updating to an aerospace structure is discussed in [3]. Experimental Modal Analysis has evolved to a widely accepted methodology in the analysis and optimisation of the dynamic behaviour of mechanical and civil structures. Modal tests are a standard part of the analysis and refinement of physical prototypes or even operational structures. The modal model results are considered to be a deterministic system description, which can be used for multiple applications, ranging from a mere verification of the fulfilment of the design criteria, to the validation and updating of CAE models and the integration in hybrid system models. In reality, the modal results are just an estimation of the model parameters based on a series of input–output or output-only tests and hence subject to all related testing and modelling errors. These data errors can be just stochastic disturbances on the input/output data, but can also be caused by invalid model assumptions or data processing effects. Some of the main sources of errors are [1, 4, 5]: • Sensor location and orientation errors. • Test set-up loading and constraining effects. • Sensor loading effects on the test structure. • Sensor calibration and data conversion errors. • Disturbance and distortion in the test data measurement chain. • Signal processing errors. • Model estimation errors. In this paper, the studied source of uncertainty is related to the variance (noise) on the Frequency Response Function (FRF) measurements. Assuming that there are neither modelling nor non-linear distortion errors, the residual errors between the selected parametric model and the measured FRFs should be mainly due to the noise that polluted the measurement data (e.g. measurement noise, process noise, generator noise and digitization noise). Then, if the probability density function of the noise on the measured data is known, the Maximum Likelihood Estimator (MLE) can be constructed with the aim to maximize this probability function. In Sect. 5.2, FRF variance estimation techniques will be reviewed including some pragmatic approaches. Advanced system identification methods like the Maximum Likelihood Estimator (MLE) and PolyMAX Plus have the possibility to take the uncertainty on the measurement data into account and to propagate the data uncertainty to (modal) parameter uncertainty. This is discussed in Sect. 5.3. Finally, in Sect. 5.4, some typical structural testing and modal analysis cases are illustrating the discussed concepts. 5.2 Estimation of FRF Variance In this section, FRF and FRF variance estimation will be reviewed for the general multiple-input case. It is most common in experimental modal analysis (EMA) to use the so-called H1 estimator for calculating the FRFs. This estimator is consistent in case only output noise is present and can be written as: ŒH D Syu :ŒSuu 1 (5.1) with ŒH.!/ 2 CNo Ni the FRF matrix of a system with Ni inputs and No outputs and consisting of elements Hoi, the FRF between output ‘o’ and input ‘i’; Syu 2 CNo Ni the output–input cross spectrum matrix; ŒSuu 2CNo Ni the input spectrum matrix. For the single-input case (we drop the sub-index ‘i’ in the following), the H1 FRF Ho and coherence function 2 o can be written as: Ho D Syou Suu ; 2 o D ˇ ˇ Syouˇ ˇ 2 SuuSyoyo (5.2)
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