Improvement of the Non-periodic Energy Harvesting Behavior of a Non-ideal Magnetic Levitation System Utilizing Internal Resonance

Abstract

Introduction

This research paper investigates the dynamics and control of a non-ideal magnetic levitation (Maglev) system, with its potential for energy harvesting. The system in view consists of a center body suspended by magnetic forces on the top and bottom with a shaker at the base.

Purpose

The study aims to explore the behavior of the Maglev system under varying conditions and its potential to enhance energy harvesting performance.

Method

Approximate solution of the nonlinear Maglev system oscillation is obtained by implementing a perturbation technique referred to as the method of multiple scales. A fourth-order numerical method is applied to obtain the time response and the outcomes are visually presented through Poincare maps, phase plane, bifurcation analysis and parametric variations.

Result

This research considered capacitance adjustment to induce internal resonance, yielding pronounced periodic oscillations with varying excitation voltage. Detailed analyses demonstrate periodic motion for the electric shaker's charge and for the middle block displacement, affirming analytical predictions.

Conclusion

The study emphasizes internal resonance potential for enhancing average power output and highlights a direct proportional relationship between the middle block and shaker’s velocity at low frequencies. Under internal resonance, average harvested power increases by approximately compared to no internal resonance case, showcasing its effectiveness in enhancing energy harvesting performance.

Appendices

Appendix A

The components of the square matrix for the stability criteria of Eqs. (25a, 25b) and (26a, 26b) are given below.

$${\eta }_{11}=-\frac{{\alpha }_{1}}{2}$$

$${\eta }_{12}=-\left({\sigma }_{1}+\sqrt{{W}_{4}}-1\right){a}_{10}+\frac{3{W}_{3}}{8}{a}_{10}^{3}$$

$${\eta }_{13}=\frac{{\alpha }_{1}}{2}\frac{{a}_{10}}{{a}_{20}}$$

$${\eta }_{14}=0$$

$${\eta }_{21}=\frac{{\alpha }_{3}\left({\sigma }_{1}+\sqrt{{W}_{4}}-1\right)}{{W}_{4}{\alpha }_{2}}\frac{{a}_{10}}{{a}_{20}^{2}}-\frac{3}{8}\frac{{\alpha }_{3}{W}_{3}}{{W}_{4}{\alpha }_{2}}\frac{{a}_{10}^{3}}{{a}_{20}^{2}}+\frac{\left({\sigma }_{1}+\sqrt{{W}_{4}}-1\right)}{{a}_{10}}-\frac{9{W}_{3}}{8}{a}_{10}$$

$${\eta }_{22}=\frac{{\alpha }_{1}{\alpha }_{3}}{2{W}_{4}{\alpha }_{2}}\frac{{a}_{10}^{2}}{{a}_{20}^{2}}-\frac{{\alpha }_{1}}{2}$$

$${\eta }_{23}=-\frac{\left(2{\sigma }_{1}+\sqrt{{W}_{4}}-1\right)}{{a}_{20}}+\frac{3}{8}\frac{{W}_{3}}{{a}_{20}}{a}_{10}^{2}$$

$${\eta }_{24}=\frac{{\alpha }_{4}}{2}+\frac{{\alpha }_{1}{\alpha }_{3}}{2{\alpha }_{2}{W}_{4}}\frac{{a}_{10}^{2}}{{a}_{20}^{2}}$$

$${\eta }_{31}=-\frac{{\alpha }_{1}{\alpha }_{3}}{2{W}_{4}{\alpha }_{2}}\frac{{a}_{10}}{{a}_{20}}$$

$${\eta }_{32}=\frac{{\alpha }_{3}\left({\sigma }_{1}+\sqrt{{W}_{4}}-1\right)}{{W}_{4}{\alpha }_{2}}\frac{{a}_{10}^{2}}{{a}_{20}}-\frac{3}{8}\frac{{\alpha }_{3}{W}_{3}}{{W}_{4}{\alpha }_{2}}\frac{{a}_{10}^{4}}{{a}_{20}}$$

$${\eta }_{33}=-\frac{{\alpha }_{4}}{2}$$

$${\eta }_{34}=-{\sigma }_{1}{a}_{20}+\frac{{\alpha }_{3}\left({\sigma }_{1}+\sqrt{{W}_{4}}-1\right)}{{W}_{4}{\alpha }_{2}}\frac{{a}_{10}^{2}}{{a}_{20}}-\frac{3}{8}\frac{{\alpha }_{3}{W}_{3}}{{W}_{4}{\alpha }_{2}}\frac{{a}_{10}^{4}}{{a}_{20}}$$

$${\eta }_{41}=-\frac{{\alpha }_{3}\left({\sigma }_{1}+\sqrt{{W}_{4}}-1\right)}{{W}_{4}{\alpha }_{2}}\frac{{a}_{10}}{{a}_{20}^{2}}+\frac{3}{8}\frac{{\alpha }_{3}{W}_{3}}{{W}_{4}{\alpha }_{2}}\frac{{a}_{10}^{3}}{{a}_{20}^{2}}$$

$${\eta }_{42}=-\frac{{\alpha }_{1}{\alpha }_{3}}{2{W}_{4}{\alpha }_{2}}\frac{{a}_{10}^{2}}{{a}_{20}^{2}}$$

$${\eta }_{43}=\frac{{\sigma }_{1}}{{a}_{20}}$$

$${\eta }_{44}=-\frac{{\alpha }_{4}}{2}-\frac{{\alpha }_{1}{\alpha }_{3}}{2{\alpha }_{2}{W}_{4}}\frac{{a}_{10}^{2}}{{a}_{20}^{2}}$$

Appendix B

The square matrix components for the stability criteria of Eqs. (40a, 40b) and (41a, 41b) are given below.

$${\delta }_{11}=-\frac{{\alpha }_{4}}{2}$$

$${\delta }_{12}=\frac{{W}_{4}{\alpha }_{2}{\alpha }_{3}}{2\sqrt{{W}_{4}}\left({W}_{4}-1\right)}{a}_{2}-{\sigma }_{1}{a}_{2}-\frac{{\alpha }_{4}}{2}{a}_{2}$$

$${\delta }_{21}=\frac{{\sigma }_{1}}{{a}_{2}}-\frac{{W}_{4}{\alpha }_{2}{\alpha }_{3}}{2\sqrt{{W}_{4}}\left({W}_{4}-1\right){a}_{2}}$$

$${\delta }_{22}=-\frac{{\alpha }_{4}}{2}$$

Appendix C

This mechanical analysis explains the energy dynamics within the energy harvesting device. It comprises of the kinetic and potential energy components, energy dissipation mechanisms, Lagrangian function formulation, and Euler–Lagrange equations. This mathematical analysis lays the groundwork for designing an effective energy harvesting strategy. The kinetic energy can be expressed as follows:

$$T=\frac{1}{2}m{\dot{x}}^{2}+\frac{1}{2}{L}_{s}{\dot{q}}^{2},$$

The potential energy can also be written as:

$$V=mgx+\frac{1}{2}{x}^{2}\left({K}_{ma{g}_{b}}-{K}_{ma{g}_{t}}\right)+S\dot{q}x+\frac{1}{2}\frac{{q}^{2}}{{C}_{s}},$$

The energy dissipation is written as:

$$D=\frac{1}{2}\left[\left({c}_{m}+{c}_{e}\right){\dot{x}}^{2}+{R}_{s}{\dot{q}}^{2}\right],$$

The Lagrangian of this system can be written as

$$L=T-V=\frac{1}{2}m{\dot{x}}^{2}+\frac{1}{2}{L}_{s}{\dot{q}}^{2}-mgx-\frac{1}{2}{x}^{2}\left({K}_{ma{g}_{b}}-{K}_{ma{g}_{t}}\right)-S\dot{q}x-\frac{1}{2}\frac{{q}^{2}}{{C}_{s}},$$

The Euler–Lagrange equations $\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{x}}_{i}}\right)-\frac{\partial L}{\partial {x}_{i}}+\frac{\partial D}{\partial {\dot{x}}_{i}}={F}_{ex}$ (for $i=\mathrm{1,2}$) such that ${x}_{1}=x$ and ${x}_{2}=q$ are modified by including additional terms that capture the energy dissipation and the energy harvesting mechanism. This can lead us to Eqs. (1) and (2).