# A Vibrational Energy Harvesting Sensor Based on Linear and Rotational Electromechanical Effects
## 2 Device Description  
The vibrational energy harvesting (VEH) device is pictured in Figure 1. An idealized diagram of the sensor is shown in Figure 2. The prototype sensor uses dual cylindrical magnets each attached to its own planar spring. Threaded rods are utilized to adjust the separation distance of the magnets and their angular orientation. One magnet is at the center of a cylindrically wrapped copper coil. The magnets are oriented with similar poles counterposed. The repulsive magnetic coupling introduces an additional magnetic force in addition to the back electromotive force (EMF) resulting from the coil/magnet interaction. The effect introduces additional nonlinear force behavior to the response of the VEH. Both magnets experience the same base excitation. Another characteristic of the design which results in rotational motion can be attributed to the intentional asymmetries caused by the threaded rods adjustments and unintentional asymmetries due to device imperfections [17]. A result of the asymmetrical repulsive force is to induce rotational motion between the upper magnet and the lower magnet-coil assembly, which adds an additional degree of freedom to the system. It was observed that the planar springs do not operate in a simple vertical (up/down) motion but rather experience out-of-plane tilting as the opposing magnetic forces push the magnets away from their vertical orientations. Consider holding two bar magnets, one in each hand, arranged with identical poles facing each other. As the magnets are moved closer to each other, the opposing force will not only push them apart along their axes but also deflect them with a force felt in the hands. Thus, the planar springs are modeled as bars with springs on their ends as opposed to a single vertically restoring spring. The center of rotation is not necessarily the center of the bar. In deriving the theory, attention must be paid to the sign of the tilt angles, \(\theta_{1}\) and \(\theta_{2}\). Following the diagram of Figure 2, \(\theta_{1}\) is negative. Thus, \(\sin\theta_{1}\) is negative and \(-\sin\theta_{1}\) is positive, making \(x_{1}-l_{11}\sin\theta_{1}\) a positive displacement and \(x_{1}+l_{12}\sin\theta_{1}\) a negative displacement. The diagram shows \(\theta_{1}\) is positive. Thus, \(\sin\theta_{2}\) is positive and \(-\sin\theta_{2}\) is negative, making \(x_{2}-l_{21}\sin\theta_{2}\) a negative displacement and \(x_{2}+l_{22}\sin\theta_{2}\) a positive displacement. Referring to Figure 2, the planar spring radii are 10 mm. Thus, the bar lengths are 10 mm. The center of rotation is located at \(l_{11}\) from the edge of spring 1 and \(l_{21}\) from the edge of spring 2. By definition, \(l_{12}=10-l_{11}\) and \(l_{22}=10-l_{21}\). The measured linear planar spring constants are listed in Table 1. However, the spring constants, \(k_{11},k_{12},k_{21}\) and \(k_{22}\), which permit both vertical and tilt motion, are unknown. Additionally, the center of rotation as given by \(l_{11},l_{12},l_{21}\) and \(l_{22}\), the damping coefficients, \(c_{1}\) and \(c_{2}\), and the moments of inertia, \(J_{1}\) and \(J_{2}\), are also unknown. The unknown parameters are identified by fitting the empirical output voltage spectrum to that of a lumped parameter nonlinear ordinary differential equation (ODE) model of the device. The following section develops the ODE model. Section 4 describes how the unknown parameters are fit. Figure 1: (**Left**): The dual magnet device. Threaded rods are utilized to adjust the separation distance of the magnets and their angular orientation (**Right**): The planar spring facilitates out-of-plane displacement and rotation of the supported upper magnet and the lower magnet/coil assembly.