Abraham and Shaw: Dynamics: Forced Vibrations

by Carson Reynolds

The forced vibration chapter looks at limit cycles in 3d. The systems are split into two distinct cases:
1) rest system subject to periodic force
2) self-sustaining oscillator subject to periodic force
The first results were obtained by Duffing in 1918.

We begin by looking at the ring model for force oscillators, we wish to understand periodic attractors (by examining free vibration of coupled systems (which are equivalent to forced oscillation of driven systems)). By combining the phase space of a damped pendulum with a driven pendulum we get a three dimensional invariant manifold. A scroll rapped around in a circle (somewhat like a rolled pant leg). The center of the scroll is an attractor. If the pendulum is undamped then we instead get a picture which looks like a concentrically layered donut (torus). By using stroboscopy experimentalists were able to take snapshots of the phase portrait. The strobed trajectory let’s us see what a section of the torus appears like.

Next we look into “forced linear springs” by playing around with a “bob” pendulum and driving it at different frequencies. We then look at a spring mass coupled to a motor that is driving it periodically. We see that these systems have an attractive limit cycle, which is an isochronous harmonic. That is one cycle of the motor causes on cycle of the pendulum. By playing around with the driving frequency we find that amplitude and phase depend on driving freq.

Hard springs that are forced exhibit some interesting behavior: one is a hysteretic loop formed by two periodic attractors. Different forcing frequencies cause braid separated by a separatrix. If we vary the amplitude instead of the frequency then we see a tendency toward cusp catastrophe.

We then move our attention to harmonics. We learn different modes are integer multiples of the fundamental frequency. Ultraharmonics are defined to be a harmonic ratio of P:1 where P is an integer. A subharmonic ratio is 1:Q. The general harmonic is described as P:Q.