Nonlinear Dynamics of Rolling Eccentric Mass Systems

Abstract

This thesis investigates the nonlinear dynamics of rolling eccentric mass systems. This investigation combines a theoretical, computational, and experimental approach using a geared track system. In this setup, a circular gear with an eccentric mass creates a system where the center of gravity of the system is offset from its geometric center. This system exhibits damped oscillatory motion when released from rest on a mating gear track. Key characteristics of this motion include the oscillation frequency, damping rate, equilibrium position, equilibrium orientation, and stability. These characteristics are dependent on several factors including the size of the eccentric mass, placement of the eccentric mass, and initial orientation of the eccentric mass system relative to the track. The governing equations of motion are derived using Lagrangian mechanics. The derivation of the curved track case results in a second order nonlinear ordinary differential equation. The driving forces in this equation are two gravitational torques: one from the curvature of the track acting on the whole gear system and one from the eccentric mass rotating about the center of the gear. Additionally, the friction in this system is modeled using linear viscous damping. This theoretical model predicts equilibrium behavior, stability, and bifurcation as the initial orientation of the system varies. A total of 2273 experiments are conducted on this curved track system. Data collection and processing uses a YOLOv8 convolutional neural network trained for simultaneous detection of the gear’s geometric center, eccentric mass, and track boundaries for each frame of the 60 frame per second data. The machine learning pipeline following initial detection uses three stages. First, geometric filtering logic to avoid physically unreasonable detections. Next, a temporal de-swap algorithm using intersection-over-union tracking to correct any swapped trackers in the multi-object detection. Finally, an automated pixel to physical measurement calibration using the track length. Post-processing of the data includes Hampel filtering and velocity-gated outlier removal, model-aided motion onset detection, and various metrics extraction. The experimental results show that oscillation frequency increases monotonically with the eccentric mass radial position on the gear, ranging from 0.80 Hz at the innermost position to 1.35 Hz at the outermost position when using a maximum mass of 12.4 g. The maximum mass produced higher frequencies than the reduced mass of 6.9 g with initial orientations near zero degrees, but this relationship is inverted as the initial orientation approaches 180 degrees. The initial orientation of the system introduces a complex frequency modulation that spans up to 0.40 Hz of change within a single position of the eccentric mass. Exponential decay rates extracted from the peak amplitude envelopes show a similar but inverted dependence on the eccentric mass weight and radial position to that of frequency but are less clear due to the contributions of both linear viscous damping and dry friction. A bifurcation analysis shows that the system transitions from a single stable equilibrium to a double-well potential energy landscape as the initial configuration nears placing the eccentric mass at 180 degrees from vertical at the bottom of the curved track. The exact point of this bifurcation is dependent on both the magnitude of the eccentric mass and its radial position. The experimental frequencies and equilibrium positions show strong qualitative agreement with the theoretical predictions.