# Two-Degrees-of-Freedom (2 DOF) Magnetic Spring-Based Electromagnetic Energy Harvester
## 1 Introduction  
In light of the pressing energy crisis exacerbated by the challenges of the pandemic, the need to explore alternative energy sources has become more urgent than ever. One promising avenue that has gained significant attention involves harvesting energy from vibrating environments [1, 2, 3, 4, 5]. This surge in diverse vibration-energy-harvesting techniques has sparked a fervent pursuit for improved output performance and operational efficiency. In this context, electromagnetic-energy-harvesting systems utilizing magnetic levitation architectures offer unique advantages [6, 7, 8, 9, 10]. These include more superficial mechanical structures, higher output power, and reduced costs, making them particularly well suited for track and vehicle monitoring applications. Initial investigations predominantly concentrated on linear dynamic energy harvesters, which, unfortunately, were limited to operating within narrow bandwidths and struggled to amplify output power under low-energy excitation [11]. However, nonlinear energy harvesters have effectively mitigated these constraints, addressing the narrow working bandwidth issue inherent in their linear counterparts and ensuring robust output performance near the designated working frequency. Using real data, linear and nonlinear oscillator systems have been compared, and it has been demonstrated that nonlinear oscillators have the highest power output in most cases. Still, nonlinear oscillators have a wider frequency bandwidth for harvesting energy [12]. [PERSON] et al. found that nonlinear oscillating systems are better than linear oscillators for extending the frequency responsebandwidth [13]. Researchers have proposed and developed nonlinear oscillator (magnetic levitation)-based electromagnet energy harvesters to harness energy from vibrations such as handshaking, human motion, and the environment. [PERSON] et al. showed that a magnetic levitation-based electromagnetic generator could be an interesting alternative to generate energy from vibration sources, especially low-frequency sources [14]. The magnetic spring-based system has a straightforward construction structure and low maintenance costs. The magnetic spring system has been used in translator design to make the oscillator nonlinear, which is more effective in the broadband frequency range, especially in low-frequency ocean environments [15]. In a single-degree-of-freedom (SDOF) magnetic spring-based system, the magnetic spring works like a physical spring and is created when two magnets face each other in the same poles (North-North or South-South). When two magnets are placed with like poles facing each other (either N-N or S-S), they experience a repulsive force. If they are closer together, the repulsion becomes stronger and stronger, meaning the force increases nonlinearly with decreasing distance. This behavior mimics a mechanical spring, but unlike a linear spring (where force increases linearly with displacement), this magnetic repulsion increases nonlinearly. SDOF magnetic spring-based oscillator systems have been used in electromagnetic energy harvesters to harvest energy [16]. [PERSON] et al. presented a detailed review of the architectures of a single-degree-of-freedom (SDOF)-based magnetic levitation system-based electromagnetic energy generator [17]. The geometric and constructive parameters, optimization approaches, and energy-generating performances of twenty-one design configurations were compared. This review also looked at the most significant advanced models (finite element method, empirical, analytical, and semi-analytical) that were utilized to explain the physical phenomena of their transduction mechanisms. The SDOF magnetic spring-based system has only one degree of freedom and is limited to only one resonant frequency. The energy generator should pick up and resonate at every frequency present in the vibration source. One can use two resonant frequencies and achieve maximum power by employing a two-degrees-of-freedom (2 DOF) oscillator system.