Abraham and Shaw: Dynamics: Vibrations

by Carson Reynolds

The topic of this section is self-sustained oscillations, in a variety of different systems: wind instruments, bow instruments, radio transmitters, and plants. In each of them we are interested in finding limit cycles that arise under certain conditions.

Here the model is not a frictionless one, but one where energy is added to the system. In the case of a reed, we idealize it as a weight attached to a pendulum. When energy (inverse friction) is applied we alter the phase portrait so that it now contains a cycle limit. Stiffer reeds change the shape (timber) of the cycle limit, blowing harder changes the amplitude and tone of the cycle.

Bowed instruments are modeled as mass-spring system on a conveyor belt. The belt adds energy to the system. The interaction between the conveyor and the dampening friction causes “inverse friction” which in turn creates a runaway oscillation. The phase portrait of the system involves a focal point repllor, a critical point. The repellor and the larger vortex pull trajectories toward a limit cycle, a self-sustaining oscillation.

Radio transmitters exhibit similar behavior. They are a kind of relaxation oscillator, where the system may longer at one state before snapping to another, somewhat like a human heartbeat.

The chapter closes with a model for plant leaf growth based on a periodic time series. Using a model for morphogen and interaction between cells, the authors find a system that mimics patterns observed in plant growth.