Double-Versus Triple-Potential Well Energy Harvesters: Dynamics and Power Output

The present datasets and figures focus on the analysis of BEH and TEH systems where the corresponding depth of the potential well and the width of their characteristics are the same. The efficiency of energy harvesting for TEH and BEH systems assuming similar potential parameters is provided. The basic types of multistable energy harvesters are bistable energy harvesting systems (BEH) and tristable energy harvesting systems (TEH). Providing such parameters allows for reliable formulation of conclusions about the efficiency in both types of systems. These energy harvesting systems are based on permanent magnets and a cantilever beam designed to obtain energy from vibrations. Starting from the bond graphs, we derived the nonlinear equations of motion. Then we followed the bifurcations along the increasing frequency for both configurations. To identify the character of particular solutions, we estimated their corresponding phase portraits, Poincare sections, and Lyapunov exponents. The selected solutions are associated with their voltage output. The results in this numerical study show clearly that the bistable potential is more efficient for energy harvesting provided the corresponding excitation amplitude is large enough. However, the tristable one could work better in the limits of low-level and low-frequency excitations. 

Series information

Potential characteristics of energy harvesting systems caused by magnetic field of distributed permanent magnets:

• Fig4a_b_V_y1.txt
• Fig4a_r_V_y1.txt

Potential characteristics of energy harvesting systems caused by magnetic field effect as in the previous case and an additional change in the stiffness of the flexible cantilever beam:

• Fig4b_b_V_y1.txt
• Fig4b_r_V_y1.txt

Series of steady states of the system against frequency for two potential wells. 

• Fig6a_p005.txt
• Fig6c_p025.txt
• Fig6e_p05.txt
• Fig6g_p085.txt

Series of steady states of the system against frequency for three potential wells

• Fig6b_p005.txt
• Fig6d_p025.txt
• Fig6f_p05.txt
• Fig6h_p085.txt

The excitation amplitude increases downwards from 0.05 to 0.85 and its values are listed in the corresponding description. The results were obtained for zero initial conditions. ω and x are dimensionless.

ig 2.jpg - A graph of bonds representing the dynamics of the tested design solutions of energy harvesting systems.

Fig 3.jpg - A Lagrangian bond graph, with causality conflicts intentionally introduced.

Fig 4.jpg - Potential characteristics of energy harvesting systems caused by: (a) magnetic field of distributed permanent magnets (Fig. 1); (b) magnetic field effect as in previous case and an additional change in stiffness of the flexible cantilever beam (to satisfy equal potential barriers V₂ = V₃ ) used in further calculations.

Fig 6.jpg - Bifurcation diagrams (stroboscopic) of steady states of the system against frequency for: (a) Two potential wells; (b) three potential wells. The excitation amplitude increases downwards from 0.05 to 0.85 and its values are listed in the corresponding sub-figures. The results were obtained for zero initial conditions. ω and x are dimensionless.

Fig 7.jpg - Exemplary solutions showing the geometrical structures of chaotic phase flows and the corresponding Poincaré cross-sections of a BEH. Dc denotes the corresponding correlation dimension. ω, p, x, and x' are dimensionless.

Fig 8.jpg - Examples of periodic responses of a BEH system identified for dimensionless mechanical vibration amplitudes: (a) p = 0.05; (b) p = 0.25; (c) p = 0.5; (d) p = 0.85. ω, p, x, and x' are dimensionless.

Fig 9.jpg - Example solutions showing geometric structures of chaotic phase flows and corresponding Poincaré cross-sections, which were identified for a system with three potential wells (TEH). Dc denotes the corresponding correlation dimension. ω, p, x, and x' are dimensionless.

Fig 10.jpg – Influence of external load characteristics on periodic induced solutions in a TEH. Trajectory shapes are plotted for selected frequencies ω. ω, p, x, and x' are dimensionless.

Fig 11.jpg - Multicolored maps of the values of effective energy harvesting systems (RMS voltage outputs) with the potential: (a) two-well (BEH); (b) three-well (TEH) for zero initial conditions. ω and p are dimensionless, while U_RMS is expressed in Volts.

Fig 12.jpg - Comparison of RMS voltage outputs for two and three wells potential systems versus amplitude and frequency. ω is dimensionless while U_RMS is expressed in Volts.