Dynamics of Transition Regime in Bistable Vibration Energy Harvesters

Schematic of the experimental setup and the stability configurations of the resonator: (a) schematic of the experimental setup and (b) monostable and bistable configurations

The experimental setup of the piezoelectric energy harvesting, cantilever harvester to the right, and PUMA spectral dynamics analyzer to the left

Schematic for the total magnetic force acting on the tip mass

The tip mass static response with the separation distance, d, between the two magnets. Threshold value, d_th, found to be 20 mm. Two values for monostable and bistable regions are selected for analysis.

First normalized mode shape plotted at different distances between the two magnets, where d = 20 mm is the threshold distance

Variation of the first natural frequency with the distance, d, between the two magnets. Threshold value match static profile with value of d_th = 20 mm.

Experimental and simulated forward sweep frequency and voltage responses for monostable resonator at d = 40 mm, A = 0.5 g, and damping μ = 0.038: (a) tip mass displacement and (b) output voltage

Experimental frequency forward swept responses for monostable resonator at d = 40 mm at different excitation levels: (a) tip mass displacement and (b) output voltage

Experimental frequency forward swept responses for monostable resonator at d = 22 mm at different excitation levels: (a) tip mass displacement and (b) output voltage

Experimental frequency responses for bistable resonator at d = 5 mm: (a) tip mass displacement and (b) output voltage

Experimental and simulated frequency responses for bistable resonator at d = 18 mm, A = 0.5 g, and damping μ = 0.038: (a) tip mass displacement and (b) output voltage

Bistable simulated results for the nonlinear energy harvester at d = 18 mm, A = 1.0 g, and damping μ = 0.038: (a) tip mass displacement and (b) output voltage

The basin of attraction for the resonator for (a) d = 18 mm at 1.0 g, Ω = 18 Hz, and μ = 0.038 and (b) d = 18 mm at 1.0 g, fold frequency Ω = 16.7 Hz, and μ = 0.038. Light area is the basin of attraction for lower branch, while the dark area is the basin of attraction for the upper branch.

Experimental frequency responses for bistable resonator at d = 18 mm: (a) tip mass displacement and (b) output voltage

Simulated frequency responses with shooting methods for bistable resonator at d = 18 mm: (a) tip mass displacement and (b) output voltage, damping μ = 0.038

Simulated frequency responses with long time integration and shooting methods for bistable resonator at d = 18 mm: (a) tip mass displacement and (b) output voltage, damping μ = 0.038

Maximum output voltage with increasing the resistance up to 10 MΩ for an excitation level of 0.3 g, separation distance of 18 mm, and damping of μ = 0.038

The bifurcation diagram of the actuator constructed from a force (excitation level) sweep at the threshold distance (d_th = 20 mm) and its natural frequency of Ω = 12.59 Hz, single-sided Poincaré sections obtained at the period of excitation

Chaotic response for the harvester at d_th = 20 mm, Ω = 12.59 Hz, and 0.2228 g excitation level: (a) phase portrait and (b) Poincaré map

The bifurcation diagram sweeping the frequency of excitation at d_th = 20 mm and Amp = 0.2 g, single-sided Poincaré sections obtained at the period of excitation

(a) Maximum output voltage as the distance between magnets varies at the excitation level of 1.0 g and (b) dynamics behavior for energy harvester