5 Nonlinear 2-DOFs Vibration Energy Harvester Based on Magnetic Levitation 43 Table 5.1 Physical and geometric properties of the proposed 2-DOFs vibration energy harvester rint Internal resistance ( ) 188 l Total coil length (m) 0:015 d0 Separation distance (m) 0:015 B Residual magnetic flux density (T) 1:18 c Mechanical damping [7] (Ns=m) 0:116 m Magnet density (kg=m2) 7800 Rj;j 2Œ1;3 External load resistance ( ) 10 4 Nj;j 2Œ1;3 The coil number (turns) 2000 M1 Mass .kg/ 0:125 M2 Mass .kg/ 0:0195 M3 Mass .kg/ 0:02 M4 Mass .kg/ 0:01 Fig. 5.4 Variation of the frequency responses in term of harvested power with respect to the excitation amplitude for the proposed device. Black and gray lines denote respectively stable and unstable branches Fig. 5.5 A typical frequency response of the proposed device showing a hardening behavior for an excitation amplitude Y0 D2:7mm, the frequency bandwidth is BW D14:4%. Solid and dashed lines denote respectively stable and unstable branches 4 5 6 7 8 9101112 0 50 100 150 200 250 f(Hz) P (mW) f=7.2 Hz P=170 mW BW = 14.37 % Y0 = 2.7 mm f=8.2 Hz P=222 mW 5.4.2 Bandwidth Enhancement Several numerical simulations have been performed for the set of design parameters listed in Table 5.1. Figure 5.4 displays the evolution of the frequency response in term of harvested power for different values of amplitude excitation Y0 up to 3mm. It is shown that bistability takes place for large excitation amplitudes (Y0 > 2mm) for which the dynamic response has two solutions for a given frequency inside a range separated by two bifurcation points. Hence, one can take advantage of the nonlinear spring hardening effect in order to enlarge the frequency bandwidth of the VEH. Interestingly, Fig. 5.5 shows that the frequency bandwidth of the proposed device can reach 14:4% for an excitation amplitude Y0 D2:7mm.
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