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.
Wetting Tracing Paper—Fiber Porous Media Curling Behavior and Mechanisms
This research presents a novel approach to understanding the curling and uncurling behavior of tracing paper when exposed to water, identifying limitations in traditional diffusion-based models like Fick’s second law. While Fick's model adequately represents the uncurling phase, where water content is stable, it falls short during the curling phase due to its inability to account for dynamic changes in diffusivity. Our study identifies capillary action, modeled through Richards' equation, as the primary mechanism in the curling phase, where diffusivity varies with water content due to capillary-driven water movement through the paper's porous structure. Experimental data align well with the Richards' equation model, highlighting a saturation point where curvature peaks, governed by evaporation's impact on moisture balance. To simulate this phenomenon, we developed a finite difference approximation scheme based on Richards' equation, discretizing the spatial domain for detailed control over moisture dynamics and incorporating the Robin boundary condition with virtual points. This approach, combined with evaporation considerations, produces simulation results consistent with observed data, emphasizing evaporation’s role in steady-state moisture gradients and the subsequent deformation mechanics. Our findings further reveal that factors like paper thickness, temperature, and salt concentration significantly influence curling behavior. We established linear correlations between peak time and thickness reciprocal, as well as between peak curvature and thickness squared, supporting theoretical models. Temperature affects both peak curvature and curling rate due to changes in viscosity and surface tension, and higher temperatures prevent full uncurling due to sustained evaporation effects. Increased salt concentration heightens peak curvature without altering expansion ratio, suggesting additional variables in play.