7 Experimental and Numerical Study of the Second Order Moment. . . 41 Fig. 7.2 Experimental set-up 7.4 Model Reduction In order to fit into the particular framework under which the analytical results have been derived, the multi-degree-of-freedom equation of motion is reduced to a set of single-degree-of-freedom equations of motion by re-writing it in the modal basis. The coupling between these equations come from the parametric excitation. The dynamics of the structure at a single point can therefore be described by a unique single-degree-of-freedom equation when the parametric excitation is kept sufficiently small in amplitude and when the forced and parametric excitations are defined as narrow-band processes exciting only one bending mode of the structure. This work aiming at the experimental validation of a single-degree-of-freedom model, narrow-band excitations are therefore considered. 7.5 Experimental Study The forced excitation Fw is defined as a narrow-band process of constant power spectral density ˜Sw = 5 · 10−3 N2 /Hz on the frequency interval [0.87f0; 1.13f0] = [34; 44] Hz. The parametric excitation Fu is defined as a narrow-band process of constant power spectral density constant power spectral density ˜Su =5· 10−3 N2 /Hz on the frequency interval [0.77f0; 2.57f0] = [30; 100] Hz. The definition of these excitations ensures that only one mode of the structure is excited. The equivalent dimensionless parameters corresponding to the dimensionless Mathieu Eq. (7.1) are Sw = 2.5 · 10−10, Su =1.8 · 10−4 and a=8ξ/Su =132. The results are summarized with a first passage time map, representing the contours of equal mean square first passage times as a function of the reduced initial energy and the reduced energy increase in the considered system (Fig. 7.3). The Hamiltonian Hof the system is defined by Eq. (7.2). Figure 7.3(a) shows the influence of the narrow-bandedness of the excitations: even if the behavior is qualitatively the same, quantitative differences are observed between the curves. Figure 7.3(b) compares the maps obtained experimentally and numerically (by Monte Carlo simulations of the system subjected to the narrow-band excitations of the experiment). The good match between the first passage time maps confirms the accuracy of the finite element model updating as well as the relevance of the theoretical model for this type of problem. The three foreseen behavioral regimes are also observed. For small initial levels of energy and sufficiently large energy increments, the first passage time is independent from the initial
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