374 M. Song et al. Fig. 40.1 The tested wing-engine structure in which ™ denotes the updating parameters, d refers to the measured data, p(™j d) is the posterior distribution function, p(dj ™) is the likelihood function, p(™) is the prior function. By assuming an uninformative prior function p(™) (uniform distribution), the posterior distribution function will be proportional to the likelihood function which can be derived as: L.™jd/ D Nm YiD1 1 .2 / N N i NNs i exp2 64 1 2 N XjD1 0 B@ Nm XiD1 i .™/ Q j i 2 2 i Q j i 2 C Nm XiD1 i .™/ i Q j i 2 2 i i Q j i 2 1 CA 3 75 (40.2) in which N is the number of datasets, Nm is the number of modes, Ns is the mode shape components, D! 2 where ! is circular frequency, is mode shape, and is the statandard deviation of and mode shapes respectively, is mode shape scaling factor. Markov Chain Monte Carlo (MCMC) algorithms (Metropolis-Hastings and Transitional MCMC) are used to sample the derived posterior distribution function and evaluate the maximum-a-posteriori (MAP) estimator b ™ MAP . This Bayesian model updating approach is also performed to calibrate another FE model with significant modeling errors for the purpose of verification and comparison. The two calibrated models are then used to predict the structure response (NNMs and time history) by accounting for model parameters uncertainties and modeling errors. The model predictions are then compared with a set of experimental results which were not used in the calibration process and only considered for validation purpose. References 1. Renson, L., Noël, J., Barton, D., Neild, S., Kerschen, G.: Nonlinear phase separation testing of an experimental wing-engine structure. In: Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry, vol. 8, pp. 115–117. Springer (2017) 2. Noël, J.-P., Renson, L., Grappasonni, C., Kerschen, G.: Identification of nonlinear normal modes of engineering structures under broadband forcing. Mech. Syst. Signal Process. 74, 95–110 (2016) 3. Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes, Part II: toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23, 195–216 (2009)
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