258 K. Vijayan et al. 72 73 74 75 76 77 0 5 10 15 20 25 Time(s) δ−(y2−y1) (mm) 0 100 200 300 400 500 600 −80 −75 −70 −65 −60 −55 −50 −45 −40 −35 Frequency (Hz) Force Amplitude (dB rel. to 1 N) a b Fig. 28.4 Variation in the displacement of beams and the force spectrum. (a) Variation in clearance between the beams for excitation at 14.45 Hz. (b) Force spectrum at 14.45 Hz clearances between the beams. It can be observed that for a clearance of 4 mm the response of the system at the resonance peaks are less sharp. However as the clearance increases the resonance peaks becomes more distinct and typical of a linear system behaviour. The system response for 4 mm clearance however does indicate an increase in response near 14 and 30 Hz, which are close to the second natural frequencies of the thin and thick beam respectively. In order to understand the effect on the system response better a simulation was performed by base exciting the system at 14.45 Hz. Variations in the clearance between the beams were analysed. The variation in the time response of the clearance between the beams is shown in Fig. 28.4a. A negative value indicatesı < .y2 y1/ and the beams are in contact. Figure 28.4b shows the spectrum of impact force. The spectrum is obtained by carrying out block averaging of the Fourier transform of response from 40 to 100 s using 146 sections. The block averaging was carried out to smooth the response spectrum. The general stochastic nature of the response could be due to the fact that the system contains both steady and transient responses since the impact can be considered as repeated impacts with an impulse hammer. 28.2.2 Excitation Bandwidth The advantage of using an energy harvester using impact is that the higher modes are excited and this feature is explored by exciting the system at 14.45 Hz, which corresponds to a maximum response. The response of the thick beam at the point of impact is analysed. Two cases with different excitation force and clearance are analysed. The response spectrum of the system obtained corresponding to two different clearances of 4 and 10 mm between the beams are shown in Fig. 28.5a. The spectrum is block averaged to smooth the response. It can be observed from the figure that for the high clearance case the response is linear with the peak amplitude occurring near the resonance. However for the system with reduced clearance more modes are excited as observed in the response spectrum. A similar observation can be made with the variation in base excitation amplitude. Figure 28.5b shows the frequency response function for a fixedı of 10 mm corresponding to two different base excitation amplitudes of 0.03 and 0.09 m. The spectrum is block averaged to smooth the response. It can be observed that for lower base excitation amplitude the response of the system is linear with a single peak, however for the higher base excitation amplitude multiple peaks can be observed indicating the excitation of higher modes. 28.2.3 Power Generation The next stage is to analyse the effectiveness of the power generation. Piezoelectric material can be added to a beam in either unimorph or bimorph configurations [18]. The mechanical deformation produces an electric current from the strain in the piezoelectric material. The model includes only the direct piezoelectric effect and the converse effect is not considered in
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