7 Identification of a Time-Varying Beam Using Hilbert Vibration Decomposition 79 1 2 Time [s] Frequency [Hz] 0 10 20 30 40 50 60 0 50 100 150 200 1 2 3 Frequency [Hz] Time [s] 0 10 20 30 40 50 60 0 50 100 150 200 a b Fig. 7.7 Identified instantaneous frequencies. (Numbers on the right of each time-frequency plot indicates the sequence of the modes extraction.) (a) Instantaneous frequencies of lateral bending modes. (b) Instantaneous frequencies of vertical bending modes For seek of illustration, the three first vertical bending modes obtained at three different times are given in Fig. 7.8 along with the mode-shapes of the beam subsystem. Note that the lateral bending modes show the same behavior but are not reported here. In the first row of Fig. 7.8, the considered time is 15 s. At this time, the mass is located at a quarter of the length of the beam, which corresponds to an anti-node of vibration for the second mode. As seen in Fig. 7.8b, the second mode is highly perturbed by the moving mass. In Fig. 7.8f,h, the deformed shapes corresponding to modes 3 and 2 at 20 and 30 s respectively are shown. In both configurations, the mass is located at a node of vibration for the considered mode and the instantaneous mode-shape is similar to the corresponding mode-shape of the beam subsystem. Finally, Fig. 7.8a,d,g reveal that the first bending mode is not very sensitive to the presence of the moving mass. From the set of identified instantaneous mode-shapes, it is possible to perform a correlation with the mode-shapes of the beam subsystem using the classical modal assurance criterion (MAC). As the instantaneous modes are identified at each time step, the time dimension has to be taken into account. So for each time step, the MAC matrix between the identified mode-shapes and the mode-shapes from the finite element analysis (FEA) of the beam subsystem is reshaped in a column vector. In Fig. 7.9, the instantaneous MAC values are shown between identified bending modes in the lateral Fig. 7.9a and vertical Fig. 7.9b directions respectively. The previously observed perturbations of the mode-shapes due to the presence of the moving mass at vibration nodes or anti-nodes are also visible in Fig. 7.9a,b. As for the identified instantaneous frequencies, the time-varying MAC values drop periodically when the mass is passing on an antinode of vibration and come back close to unity when the mass is passing on a node of vibration. 7.7 Conclusion In this paper, two well known techniques (the HHT and the HVD methods) used for non-stationary signal decomposition were considered and their limitations in the case of crossing frequencies or amplitudes of intrinsic components were highlighted. As these methods work on single signals, it was shown that, in the case of a multiple degree-of-freedom system, applying them separately on each channel can lead to non-unique frequency curves and consequently to non-corresponding demodulated components on all the channels. To alleviate this problem, a source separation technique was introduced into the algorithm of the original HVD method. It was shown that both limitations were removed as a single frequency curve was calculated for each mode. Thanks to the better estimation of the instantaneous frequencies, mono-components and instantaneous mode-shapes were calculated.
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