Modal frequencies in a nonlinear beam-magnet coupled oscillator system
In this paper, I investigated the motion of a nonlinear coupled oscillator system consisting of two leaf springs secured to a non-magnetic base with magnets attached to the upper ends such they repel and are free to move. My results showed that the system exhibits the beats phenomenon, and interestingly that the frequencies show a dependence on initial conditions. I hence hypothesized this sensitivity is due to two sources of nonlinearities: geometric nonlinearity during large deflections of the leaf springs and the nonlinearity in the magnetic force. To test this hypothesis, a nonlinear mathematical model was developed, accounting for nonlinear beam effects up to third order and fully solving the nonlinear magnetic force using a current cylinder model, accounting for the tilting of the magnets. An approximate linear model was also developed for comparison. The theoretical models were validated experimentally by investigating the dynamic motion of the springs through time, as well as how the modal frequencies in the system depend on the initial displacement, the length of the spring, and the distance between the springs. The more accurate nonlinear model I derived shows good agreement with experimental results while the linear theory does not, highlighting the importance of nonlinearities in this system. An improved understanding of these nonlinear systems could lead to advancements in design and efficiency, and safety in various applications such as energy harvesting.