Phase - Evergreen State College



DRAFT of Lab 2.4: Mass + Spring + Rubber band

Differential Equations, Fall 2002, TESC

Text = Differential Equations (2002, ed.2) by Blanchard, Devaney, and Hall (pp.221-223)

10.Jan. 2003 - E.J. Zita

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Overview:

We investigate a mass on a spring with a rubber band which supplies no compressional restoring force:

(1) Simple harmonic oscillator with equilibrium shifted by gravity: [pic]

(2) Harmonic oscillator with damping: [pic]

(3) Oscillator with rubber band and no damping: [pic]

(4) Oscillator with rubber band and damping: [pic]

For each case, we numerically investigate various intial conditions (y0,v0), and damping constant b.

Methods:

We write each second-order differential equation as two first-order equations,

dy/dt = v and v = f(y,t). We then let v=x in the "HPG System Solver" software on the DETools disk, and approximately solve each system numerically and plot timeseries and phase plots.

Short answers:

1) The simple harmonic oscillators have sinusoidal solutions of frequency [pic]with constant amplitude, and the phase plot is a limit cycle, as expected.

2) The damped harmonic oscillator has sinusoidal oscillations with a lower frequency and exponentially decaying amplitude. The plase plot spirals in to zero. We should find a bifurcation between 1 ................
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