74 M. Bertha and J.C. Golinval 0 5 10 15 −5 0 5 x 10 −3 0 5 10 15 0 20 40 Frequency [Hz] 0 5 10 15 −5 0 5 x 10 −3 0 5 10 15 −1 0 1 x 10 −3 Time [s] 0 5 10 15 −5 0 5 x 10 −3 0 5 10 15 0 20 40 Frequency [Hz] f 1 (t) f 2 (t) 0 5 10 15 −5 0 5 x 10 −3 0 5 10 15 −1 0 1 x 10 −3 Time [s] x 1 (t) x 2 (t) f 1 (t) f 2 (t) x 1 (1)(t) x 1 (2)(t) x 2 (1)(t) x 2 (2)(t) Fig. 7.2 Time responses and extracted instantaneous frequencies and components. First row: time response of each DoF. Second row: identified instantaneous frequencies for each DoF. Third and fourth rows: components corresponding to the identified instantaneous frequencies • m1 D3kg; • m2 D1kg; • k1 D20;000N/m; • k2 D25;000 &5;000N/m (linear decrease in the time span); • c1 D3Ns/m. The system is submitted to an impulse at DoFx2 and the response of the whole system is simulated during 15 s. A standard Newmark integration scheme was used for the time integration in which the stiffness matrix is updated at each time step to take into account the dependence of k2 with respect to the time. Applying the standard HVD method on the two channels separately leads to some undesirable behavior. Each time one intrinsic component becomes dominant in the signal, the method follows it and a mode switching phenomenon occurs during the extraction process. Moreover, as it can be seen in Fig. 7.2, the mode switching does not occur at the same time on each channel, which makes the correction of these switches more difficult. When the frequency curve jumps from one mode to the other, the corresponding demodulated component also follows this jump.
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