A two-mass cantilever beam model for vibration energy harvesting applications

While vibration energy harvesting has become a viable means to power wireless sensors, narrow bandwidth is still a hurdle to the practical use of the technology. For conventional piezoelectric or electromagnetic harvesters, having multiple proof masses mounted on a beam is one way to widen the effective bandwidth. This is because the addition of proof masses increases the number of resonant modes within the same frequency range. Based on the assumptions of the Euler-Bernoulli beam theory, this paper presents a continuum-based model for a two-mass cantilever beam. First, the equation of motion is derived from Hamilton's principle. Next, the modal analysis is presented and a steady state solution for harmonic base excitation is derived. The two-mass beam is considered as two serially connected beam segments. In the derivation, emphasis is given to the transition conditions, which would otherwise not appear in the traditional single mass beam model. Experimental validation on a stainless steel beam confirms that the model can accurately predict both natural frequencies and the frequency response of an arbitrary point along the beam. The derivation procedure presented in this paper is applicable to a beam with any number of proof masses. Lastly, it is demonstrated how the model can be applied to a piezoelectric energy harvester.