l b w t1 t2 INTERNAL RIBS FIXED END TOP PLATE BOTTOM PLATE SENSOR LOCATION FORCE LOCATION (b) (a) Mode 2: 45 Hz Mode 1: 12.7 Hz Mode 4: 90.4 Hz Mode 3: 71.2 Hz Figure 4: (a) Box-beam model used for analytical study of the expansion algorithm along with typical set of sensor locations; (b) First four mode shapes of the structure. The finite element model of the box-shaped structure contains a total of 434 nodes. The natural frequencies and mode shapes of the beam are shown in Figure 4b. The frequencies typically are well separated. The model has an assortment of modes containing transverse and lateral bending and torsion modes which will be excited using a time pulse where, primarily, the first four frequencies are excited. Modal damping typically seen in these types of structures was added to the model. The excitation used was a time pulse applied vertically downwards, 10 inches from the bottom right corner of the box-beam. The input time pulse is shown in Figure 5a. The time pulse is a combination of two triangular functions which when seen in frequency domain show a very uniform excitation over the frequency range of interest as shown in Figure 5b. The time pulse primarily excites the first four modes of the structure as seen in FFT of the output response as shown in Figure 5c, but there is also some additional response of smaller magnitude from several higher frequencies. -170 -160 -150 -140 -130 -120 -110 -100 -90 -80 0 100 200 300 400 500 Mode 1: 12.7 Hz Mode 2: 45 Hz Mode 3: 71.2 Hz Mode 4: 90.4 Hz DISPLACEMENT MAGNITUDE (dB) FREQUENCY (Hz) FFT of Output Response at one sensor location 0 0.005 0.01 0.015 0.02 -4 -3 -2 -1 0 1 Input Pulse Time (sec) Amplitude 0 100 200 300 -60 -40 -20 0 FFT of Input Pulse Frequency(Hz) Magnitude(dB) (a) (b) (c) Figure 5: Details of the excitation force and response of the structure. 192
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