In this paper, a non-linear tri-stable piezoelectric cantilever energy harvester with a novel-type dynamic magnifier was proposed to achieve more effective broadband energy harvesting under low-level ambient excitations. According to the generalized Hamilton principle, a mathematical distributed parameter model of the piezoelectric energy harvester was proposed. The novel-type dynamic magnifier is a system consisting of two spring masses, one placed between the fixed end of the piezoelectric beam and the L-shaped frame, and the other, between the L-shaped frame and the base. The harmonic balance method was adopted to work out the analytical expressions of the steady-state displacement, steady-state output voltage and power amplitude of the energy harvester system. The effects of the distance between the magnets, the spring stiffness of the dynamic magnifier, and the load resistance on the performance of the system were also investigated. The results show that different from that of the conventional tri-stable piezoelectric energy harvester, the frequency response curve of the proposed novel-type energy harvester system with a two-spring-mass dynamic magnifier exhibits two peaks as a result of the interactions of the coupled elastic system, where the left peak stands for the resonant value of the tri-stable piezoelectric energy harvester, while the right one the resonant value of the dynamic magnifier. It is able to achieve higher output power over a broader frequency band under low-level environmental excitations, and the harvested power can be significantly strengthened if the mass and stiffness of the dynamic magnifier are selected properly.

A distributed parametric mathematical model of a new-type dynamic magnifier for a bistable cantilever piezoelectric energy harvester is proposed by using the generalized Hamilton principle. The new-type dynamic magnifier consists of a two-spring-mass system, one is placed between the fixed end of the piezoelectric beam and the L-shaped frame, and the other is placed between the L-shaped frame and the base structure. We used the harmonic balance method to obtain the analytical expressions for the steady-state displacement, steady-state output voltage, and power amplitude of the system. The effect of the distance between the magnets, the spring stiffness ratio and mass ratio of the two dynamic magnifiers, and the load resistance on the performance of the harvester is investigated. Analytical results show that compared with the bistable piezoelectric energy harvester with a typical spring-mass dynamic magnifier, the proposed new-type energy harvester system with a two-spring-mass dynamic magnifier can provide higher output power over a broader frequency band, and increasing the mass ratio of the magnifier tip mass to the tip magnet can significantly increase the output power of the BPEH + TDM system. Properly choosing the stiffness ratio of the two dynamic amplifiers can obviously improve the harvested power of the piezoelectric energy harvester at a low excitation level.

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In order to further improve the vibration energy harvesting efficiency of piezoelectric energy harvester under low frequency environmental excitation, this paper, based on the traditional magnetic tri-stable piezoelectric energy collector model, proposes a tri-stable piezoelectric energy harvester (TPEH+DEM) model with two elastic amplifiers which are installed between the U-shaped frame and the base and between the fixed end of the piezoelectric cantilever beam and the U-shaped frame respectively. Based on Hamilton principle, the motion equation of electromechanical coupling of TPEH+DEM system is established, and the analytical solutions of displacement, output voltage and power of the system are obtained by harmonic balance method. The effects of the mass of elastic amplifier, spring stiffness, magnet spacing and load resistance on the dynamic characteristics of energy harvesting of TPEH+DEM system are analyzed. The result shows that there are two peaks in the response output power of TPEH+DEM system in the operating frequency range. By adjusting the mass and stiffness of the elastic amplifier reasonably, the system can move into the inter-well motion under low external excitation intensity, and produce high output power. Compared with the traditional model which only has an elastic amplifier on the base of piezoelectric energy harvester, TPEH+DEM model has better energy harvesting performance under low frequency and low intensity external excitation.

Given the geometric nonlinearity of the piezoelectric cantilever beam, this study establishes a distributed parameter model of the nonlinear bi-stable cantilever piezoelectric energy harvester, following the generalized Hamilton variational principle. The analytical expressions of the dynamic response were obtained for the energy harvesting system using Galerkin modal decomposition and the multi-scale method. The investigation focuses on how the performance of the energy harvesting system is influenced by the gap distance between magnets, external excited amplitude, mechanical damping ratio and external load resistance. The calculation results were compared with those obtained neglecting the geometric nonlinearity of the beam. The results show that the system responses contain jump and multiple solutions. The consideration of the geometrical nonlinearity significantly amplified the peak displacement and peak output power of the intra-well and inter-well motions. There is an evident hardening effect of the inter-well motion frequency response curve. By reasonable adjusting the parameters, it is possible to improve the output power of the piezoelectric energy harvesting system and broaden the operating frequency of the system.