Nonlinear resonance in strongly damped systems

In considering asymptotic procedures for nonlinear dynamics, the focus is typically on systems that are close to an integrable system. However, systems with complete or partial strong damping are usually found in applications, especially in control systems. The strong recent trend toward light-weight designs makes mechanical structures extremely sensitive to any type of vibration excitation. Therefore, tightly focused damping has to be applied to reduce vibration amplitudes to a certain acceptable level without significantly decreasing the efficiency of a mechanism as a whole. The usual situation in which the vibration amplitudes have to be reduced is the classical Sommerfeld effect, which was first described by Arnold Sommerfeld and later explained by Ilya Blekhman. The main effect described by Sommerfeld concerns the possibility of capture into the resonance of an unbalanced rotor mounted on an oscillating carrier system and driven by an induction engine.
The objectives of this talk are as follows:
• to provide a simple justification for the analysis methods of the Sommerfeld effect in the case of strong damping;
• to demonstrate the close relationship of these methods to systems with singular perturbations 9;
• to demonstrate the efficiency of the averaged equations for the design of active strategies enabling reliable passage through resonance;
• to suggest a simple passive device enabling avoidance of capture into resonance;
• to investigate capturing into the resonance of a system with a self-balancing device;
• to demonstrate how this system is related to phase transitions in dynamics; and finally
• to suggest a mechanical solution enabling the reduction of rotor vibrations while passing though resonance without decreasing the performance of the balancers in the overcritical domain.