High-Energy Orbit Harvesting with Torsionally Coupled Mistuned Pendulums

Purpose This article demonstrates the possibility of energy harvesting by mistuned pendulums with torsional coupling. Two pendulums of different lengths with coils and magnets at the pivots are used as electromagnetic harvesters. The ambient energy source to the system is considered in the form of harmonic base excitations. Torsional coupling is achieved by connecting the pendulums with a torsional spring. The non-linearity of the underlying dynamics arises due to mechanical coupling and forcing amplitude. Numerical results are presented to analyze the performance of the pendulum energy harvester under different torsional coupling values. Method The mathematical model of the electro-mechanical system with torsionally coupled pendulums under harmonic excitation is developed. The mistuning is introduced by two pendulums with different lengths l 1 and l 2 . The coupling between them is achieved by a torsional spring of stiffness k c . These pendulums are connected to the shafts of the electromagnetic generators with a magnet attached to the pendulum as a rotor and coil as stator. Current i is generated due to electromagnetic induction whenever pendulum oscillates. The numerical and harmonic balance method analysis were carried out. Runge-Kutta integration was used for numerical analysis. The analysis was carried out to determine the effect of coupling, length and resistive coefficient. The bifurcation diagram and Poincare plots were used for dynamic analysis and cross recurrence plots for analysis of relations between pendulum harvesters. Results The effect of coupling on harvester performance was carried out, the variation of maximum power and frequency band with coupling can be categorized into 3 zones. The ranges for Zone-I, II and III of maximum power are [0-0.059], [0.06-0.11] and above 0.11 respectively. In Zone-I total power and the total band are dominating over pendulum-1 and pendulum-2. In Zone-II, the performance of pendulum-2 is comparable with the total performance. In contrast, the performance of pendulum-1 is low. Saturation of maximum power and bandwidth can be observed in zone-III. Uncoupled pendulum-1 and 2 exhibits a periodic oscillation. A low amplitude quasi-periodic response of pendulum-1 and high amplitude quasiperiodic response of pendulum-2 can be observed for coupling of β = 0.04. Both pendulums exhibit high amplitude chaotic response at β = 0.07. The introduction of coupling induced nonlinearity in both pendulums. the amplitude and velocity also increased due to the quasi-periodicity and chaos induced in pendulums due to the coupling and better energy harvesting capabilities can be expected. The harmonic balance analysis indicates that there is a possibility of obtaining a high amplitude current at lower frequencies provided proper initial conditions are chosen. Conclusion A twin electromagnetic pendulum energy harvester with torsional coupling is analyzed in order to optimize the efficient energy harvesting and study the energy harvester dynamics. The paper categorized the three distinct zones based on the coupling ratio β and length ratio α 2 . One can select parameters from these zones based on need. Zones-II and III are the most promising zones. If one wants to harvest broader energy Zone-II is preferable with harvesting only from the energy harvester with a low resonant frequency which in turn saves material cost. To harvest peak power one can, select Zone-III and harvest from both energy harvesters. The performance in the respective zones can be further enhanced by using optimal initial conditions to obtain high energy orbits at lower frequencies. The probability of obtaining high energy orbits decreases with an increase in the coupling ratio. The effect synchronization with coupling on energy harvesting plays an important role form magnitude and bandwidth point of view. The analytical results obtained by the Harmonic Balance Method are in good agreement with numerical results.