# A Vibrational Energy Harvesting Sensor Based on Linear and Rotational Electromechanical Effects
## 1 Introduction  
Self-powered sensors and networks have become a tangible reality, thanks to the ability to harness ambient energy from the surroundings to power electronic devices. This advancement has a profound impact on the realization of intelligent and adaptable sensor networks. As microelectronics and sensing technologies continue to advance, the power requirements for these devices have reduced to the microwatt regime. At this level, the available ambient energy in the environment becomes an attractive resource to power these systems. Efficient energy harvesting technologies will greatly diminish the reliance on short-lived, battery-powered approaches and eliminate the need for direct power supply to the devices [1; 2; 3]. This paves the way for the development of truly autonomous wireless sensor networks that can operate indefinitely, independent of routine maintenance and infrastructure requirements. These networks will have diverse applications in structural health monitoring, security perimeter tamper detection, autonomous systems, and seamless integration with existing Internet of Things (IoT) technologies [4; 5]. A key challenge for persistent sensing is power supply. Even when power requirements are in the microwatt range, they need to be continuous. Many research efforts are focused on developing sensor technologies that leverage ambient energy sources. For systems operating on machinery or structures with well-defined and repeatable dynamic characteristics (e.g., rotating machinery), the energy harvesting mechanism is designed to target specific vibrational frequencies [6]. This approach achieves optimal performance when the harvesting design resonates at or near one of the fundamental operating frequencies of the monitored system. Even small levels of vibrational energy can contribute significantly to harvested power when the harvesting system operates near a resonance. However, linear systems face performance degradation as the operational frequency deviates from the harvester's resonance design point [7; 8; 9]. Single-resonance VEH devicesoperating in a linear regime typically have an FWHM resonance that is 2-3 Hz. Nonlinear effects have shown an improvement by a factor 5, or 10-15 Hz [7; 8; 9]. One approach to introduce nonlinearities is through opposing magnetic fields [10; 11]. [PERSON] provides a thorough review of many electromagnetic vibrational energy harvesting (VEH) designs [12]. In particular, designs utilizing multiple masses and magnets to generate favorable harvesting performance have been the focus of numerous investigations [4; 13; 14; 15]. These designs are based on modifying the electromagnetic coupling of the magnet and the coil with the introduction of additional magnets to generate nonlinear forcing functions that lead to local minima in the potential energy function. The local minima generate a multi-stable vibrational response that leads to improved bandwidth and conversion capabilities. The approach is not without challenges--large potential wells can \"trap\" the system, reducing the high-energy interwell oscillations. To mitigate these effects, several investigators have developed tri-stable approaches resulting in lower potential interwell boundaries, resulting in lower excitation energies to achieve the interwell boost for improved energy harvesting [16; 17]. In the following sections, we explore the use of nonlinear coupled multi-modal behavior to enhance bandwidth and increase energy conversion efficiency. Our focus is on developing simplified ordinary differential equation (ODE) approximations to capture the electromechanical behavior. Experimental measurements and comparisons to the ODE model are used to validate the effectiveness of the sensor design. The nonlinear ODE system is also used to explore parameter values for analyzing performance and \"tuning\" of the device. Here, model parameters can be fit to match empirical data. The device is described in Section 2. The state-space ODE theory, derived in Section 3, is based upon a Lagrangian formulation.