13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- cantilever beam would be excited. As seen in Fig. 2(a), while the beam vibrates, the impact body may hit PZT-T-IMP and PZT-B-IMP and induce resonant vibration mode in the top and bottom frames. Due to the piezoelectric effect, AC voltage is generated across all piezoelectric components. At the same time, the copper coil vibrates relatively to the permanent magnets, as shown in Fig. 2(b), causing a variation of magnetic flux through the coil. According to the Faraday’s law, AC voltage will be generated in the coil. To generate voltage in the electromagnetic module efficiently, it is so designed that the net magnetic flux through the coil is zero when the middle cantilever remains static. For low frequency (several to tens of Hertz) excitation, we only take into consideration the fundamental vibration mode of the middle cantilever beam, which can be simplified into an equivalent single-DOF spring-mass-damping system. As indicated in Fig. 2(a), the displacement of the vibration energy harvester and the tip of the cantilever beam are denoted by xs and x, respectively, with respect to the inertial reference frame of the earth. Setting the relative displacement between the cantilever tip and VEH frame be y = x − xs, and the source excitation of the shaker be harmonic, i.e. xs = Assinωst, the steady-state solution of the differential vibration equation of the cantilever beam is [16] sin( ) s y Y tω ψ = − , (1) 2 2 2 2 1 2 s s cn s s cn cn A Y ω ω ω ω ζ ω ω ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎢ ⎥ − + ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ g , (2) 1 2 2 tan 1 s cn s cn ω ζ ω ψ ω ω − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ −⎛ ⎞ ⎥ ⎢⎣ ⎜ ⎝ ⎟ ⎠ ⎥⎦ g . (3) where ωcn is the natural angular frequency of the cantilever beam, m, ζ and k are the equivalent mass, damping ratio and spring constant, respectively. Hence, the condition for impact is Y D≥ . (4) For a given low excitation amplitude (As < D), there exists an excitation frequency range beyond which the impact cannot happen. Denoting ωsl and ωsu as the lower and upper limits of the excitation frequency range, it is derived that ( ) ( ) 2 2 2 2 2 2 2 1 2 1 2 1 1 s sl s s A D A D ζ ζ ω ω ⎛ ⎞ − − − − ⎜ − ⎟ ⎝ ⎠ = − , (5)
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