78 M. Bertha and J.C. Golinval Fig. 7.5 Beam travelled by the moving mass Table 7.1 Modal parameters of the simply supported beam only (LTI system) ModeNbr. Frequency (Hz) Mode shape Mode 1 11.54 First vertical bending Mode 2 46.05 First lateral bending Mode 3 46.15 Second vertical bending Mode 4 103.74 Third vertical bending Mode 5 182.72 Second lateral bending Time [s] Frequency [Hz] 0 20 40 60 0 50 100 150 200 1 2 3 4 5 6 7 x 10−4 Node 1 – Lateral direction Time [s] Frequency [Hz] 0 20 40 60 0 50 100 150 200 2 4 6 8 10 12 x 10−3 Node 1 – Vertical direction a b Fig. 7.6 Wavelet spectra of two degrees of freedom. The time varying behaviour of the structure is clearly visible. (a) Node 1—Lateral direction. (b) Node 1—Vertical direction induced by the moving mass does not produce any work on that mode at this time. Conversely, when the mass is located at an anti-node of vibration for a particular mode, the inertia effect is maximum and produces the maximum decay in frequency for that mode. 7.6.1 Identification of Instantaneous Frequencies The algorithm proposed in Fig. 7.4 is now applied on each set of recorded signals. Based on the time-frequency plots Fig. 7.6a,b it is chosen to extract two modes from the set of lateral measurements and three modes from the vertical ones. The evolution of the five identified instantaneous frequencies are shown using white dashed curves in Fig. 7.7a,b for the lateral and vertical modes respectively. Wavelet spectra of node 1 are used in the background of the plots and it can be seen that the identified instantaneous frequencies match very well the highest amplitude in the spectra. 7.6.2 Component Extraction and Calculation of Mode Deflection Shapes Once that the instantaneous eigenfrequencies are well identified, the next step is to extract the intrinsic components corresponding to each frequency in all the channels. This is done by the use of the Vold-Kalman filter and results in a set of mono-components and complex amplitudes. Referring to (7.9), these complex amplitudes are used as unscaled instantaneous mode-shapes.
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