4 Nonlinear Dynamic Behavior of Cantilever Piezoelectric Energy Harvesters: Numerical and Experimental Investigation 39 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.8 b a 0.6 0.4 0.2 0 15 20 25 30 35 40 45 d = 10 mm d = 20 mm d = 30 mm d = 40 mm d = 30 mm d = 20 mm d = 10 mm 50 55 60 0 15 20 Frequency [Hz] Amplitude [Volts] Amplitude [Volts] Frequency [Hz] 25 30 Fig. 4.8 Test results for variable magnet spacing: (a) repulsive; (b) attractive the results for the repulsive and attractive configurations of the tip magnets when the input excitation voltage is varied. It can be noticed in both cases that increasing values of the input voltage leads to an increase in the harvester’s output voltage. However, for the repulsive case in Fig. 4.6 as the input voltage increases the FRF narrows what is clear indication of a reduction on the harvester working frequency range. For the case of attractive magnets it is seen from Fig. 4.7 that increasing values of the excitation signal leads to more electrical output power being generated and the working bandwidth of the harvester also tends to increase, what in principle points to an advantage over the repulsive configuration. Finally, Fig. 4.8 shows experimental results obtained when the distance between the tip magnet and the fixed magnet is varied, for both repulsive and attractive testing scenarios. It is observed that in the case of repulsive magnet force, Fig. 4.8a as the magnet distance increase the harvester natural frequency and power peak amplitude are affected. Similar results were obtained for the attractive case, as shown in Fig. 4.8b. 4.5 Concluding Remarks In this paper we presented the modeling and the analytical investigation of a nonlinear piezoelectric energy harvester. The device consisted of a cantilever beam partially covered by piezoelectric material with a magnetic tip mass. In this model was considered the nonlinearity inherent comes from the piezoelectric constitutive equations besides the nonlinearity comes from the magnetic tip mass. The electromechanically coupled equations were solved numerically, through the initial value problems for ordinary differential equations. The electrical power output was investigated varying the amplitude of the base acceleration, the distance between the magnets and the load resistor. We also concluded that the parameters investigated influence in the frequency range of operation of the device and the nonlinear effects present on the energy harvester extend the useful frequency range of studied device. Experimental data supported the results obtained from numerical simulations. Acknowledgements Authors are grateful to CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brazil) for the financial support received through graduate scholarships and to EESC-USP for the laboratory support received.
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