Keywords: vector ARMA method, nonlinear stochastic dynamics, linearized model, PSD, state representation, simulation.

In nuclear power plants the high rate of fluid flow can induce vibration on many components causing dynamic interactions due to impacting and sliding that can result in wear and loss of friction controls with serious consequence. This paper presents a method to model the nonlinear vibratory behaviour of interacting components. This is a challenging task because of several particularities of this industrial problem.

The interaction of mechanical components includes several causes of nonlinearity as intermittent contact with friction and damping, wich are pourly known, and consequently hard to model.

Uncertainties due to the random nature of the external excitation generated by flow turbulence greatly perturb the response system.

This response is only partially known through experimental tests performed on real scale models leading to vector values    R,    N, measured at times , ,    N, evenly distributed on a finite time interval    R.

To be usefull to industrial applications, the nonlinear random dynamical system behaviour must be described by means of a linearized model able to approximate accurately the second order statistics of the stationary response.

The proposed method is an approximate procedure of spectral linearization type [1,5]. In this approach, the PSD of the centered stationary response is assumed to be known. Practically, this second order characteristic is derived from the given experimental samples by using suitable statistical techniques of signal processing [3]. In what follows, the centered stationary response is denoted by    R. It is a second order R-valued zero-mean process on R assumed to possess the following properties : it is stationary in the wide sense, mean-square continuous, physically realizable [4,2] and its PSD    R   C is known and such that rank n for almost all    R. By sampling this process at times    Z, according to the Shannon's rule [3], where is the experimental time increment, we obtain a time discrete process    Z, which is a physically realizable stationary (wide sense) R-valued zero-mean process on Z whose PSD is a    C-valued 2-périodic function such that : . This new PSD is the basis datum of the study. It is important to note here that the physical realizability property of is equivalent for to the existence of a factorization of the form :

where belongs to the Hardy class NR, denotes the -transform [3], NR, and the subscript * denotes the conjugate complex transposed matrix.

Our aim is then to construct a physically realizable R-valued zero-mean stationary time discrete process    Z whose PSD is close to (in a sense to be specified) and admitting a linear state representation of the form :

where    Z (the state) is a R-valued (m>n) zero-mean stationary process on Z,    Z is a R-valued discrete standard Gaussian white noise, and    R,    R, and    R are matrix coefficients to be determined. In addition, the pair (Y,W) satisfies the following property due to the underlying mathematical structure of (33):    Z, is independent of the family . It should be noticed that (33) defines X as a Gaussian process while the target process is not Gaussian. However, it is of no importance here because we are only interested in the spectral approximation.

The application considered here concerns the two-dimensional nonlinear stationary dynamical response of structural elements of a nuclear power plant. It is a typical example of a complex nonlinear random dynamical behaviour which is very difficult to model. The application of the presented method to weakly nonlinear, moderately nonlinear and strongly nonlinear cases shows the relevance of the method.

1

P. Bernard and M. Taazount, "Random dynamics of structures with gaps : simulation and spectral linearization", Nonlinear Dynamics ,313-335, 1994. doi:10.1007/BF00045340

2

R. Azencott and D. Dacunha-Castelle, "Séries d'observations irrégulières Modélisation et prévision", Masson, Paris, 1984.

3

C. Soize, "Méthodes mathématiques en analyse du signal", Masson, Paris, 1993.

4

P. Krée and C. Soize, "Mécanique aléatoire", Dunod, Paris, 1983.

5

M. Alaoui Ismaili, "Méthodes d'approximation en dynamique stochastique et fiabilité des systèmes", Ecole Doctorale SF, Université Blaise Pascal, Clermont-Ferrand, France, 1995.

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