Chaotic Resonances Research Articles

Non-linear dynamics of a spur gear pair

Non-linear frequency response characteristics of a spur gear pair with backlash are examined in this paper for both external and internal excitations. The internal excitation is of importance from the high frequency noise and vibration control viewpoint and it represents the overall kinematic or static transmission error. Such problems may be significantly different from the rattle problems associated with external, low frequency torque excitation. Two solution methods, namely the digital simulation technique and the method of harmonic balance, have been used to develop the steady state solutions for the internal sinusoidal excitation. Difficulties associated with the determination of the multiple solutions at a given frequency in the digital simulation technique have been resolved, as one must search the entire initial conditions map. Such solutions and the transition frequencies for various impact situations are easily found by the method of harmonic balance. Further, the principle of superposition can be employed to analyze the periodic transmission error excitation and/or combined excitation problems provided that the excitation frequencies are sufficiently apart from each other. Our analytical predictions match satisfactorily with the limited experimental data available in the literature. Using the digital simulation, we have also observed that the chaotic and subharmonic resonances may exist in a gear pair depending upon the mean or design load, mean to alternating force ratio, damping and backlash. Specifically, the mean load determines the conditions for no impacts, single-sided impacts and double-sided impacts. Our results are different from the frequency response characteristics of the conventional, single-degree-of-freedom, clearance type non-linear system. Our formulation should form the basis of further analytical and experimental work in the geared rotor dynamics area.

Dynamics of the Uranian and Saturnian satelite systems: A chaotic route to melting Miranda?

We argue that the anomalously large inclination of Miranda, the postaccretional resurfacing of both Miranda and Ariel, and the anomalously large eccentricities of the inner Uranian satellites indicate that resonant configurations once existed in the Uranian satellite system that have been since disrupted. Similar anomalies that cannot be accounted for by the present resonant configurations also exist in the Saturnian satellite system, and we suggest that temporary resonances existed in the past in that system as well. Using classical methods of analyzing the dynamics of resonance, we show how temporary capture into a second- or higher-order resonance can produce large increases in e and I on comparatively short time scales. However, these methods may not provide a complete description of resonances in the Uranian satellite system. Since values of J 2( R p a ) 2 for the inner Uranian satellites are small while their mass ratios, m M , are large, resonances in the Uranian system are not always well separated. For resonances that are not well separated, it is not possible to analyzed the dynamics using a disturbing function that is truncated to the extent that it contains only a single resonant argument. We have made some progress with this problem using the Cornell National Supercomputer to simulate the dynamics numerically. We find that capture into resonance may result in chaotic motion. We discuss two mechanisms that can be invoked to disrupt high-order resonances: the “spontaneous” disruption of chaotic resonances and the disruption of resonances due to the tidal damping of a satellite's eccentricity while the satellite is in a nonsynchronous spin state.