# DRAFT of Lab 2.4: Mass + Spring + Rubber Band

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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:

(2) Harmonic oscillator with damping:

(3) Oscillator with rubber band and no damping:

(4) Oscillator with rubber band and damping:

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.

(1) The simple harmonic oscillators have sinusoidal solutions of frequency 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<b<10.

(3) Undamped SHO + rubber band has a stronger restoring force when stretched (y>0) than when compressed (y<0). The result is …

(4) Undamped SHO + rubber band …

Key points:

Case (1) Simple harmonic oscillator (SHO with no damping):

,

These simple harmonic oscillators have sinusoidal solutions of frequency with constant amplitude, and the phase plot is a limit cycle, as expected. The phase plot has the same size and shape for any value of (A=k1), and it cycles a little more rapidly for higher A. In the time-plots, the longest period corresponds to A=12 and the shortest period to A=13. The apparent increase in amplitude of x with A is spurious.

Case (2) Harmonic oscillator with damping:

, .

These damped harmonic oscillator have sinusoidal oscillations with a lower frequency and exponentially decaying amplitude. The plase plot spirals in to zero.

Should be a bifurcation between 1<b<10…

Mathematica plots for k1=12.5, for b=1-10. These are consistent with HPGSolver plots

Bifurcation diagram vs b for dt=0.1
/ Bifurcation diagram vs b for dt=0.01

(3) Oscillator with rubber band and no damping:

, for y>0, for y<0

Use h = heaviside function = UnitStep where

Surprisingly, this has nice ordinary oscillations for 1<y0<6, for 4.5<k2<5.0 (x0=0)

y0=1 / y0=6
Phase plot
/ Phase plot

(4) Oscillator with rubber band and damping:

,

Vary damping for y0=4
/ Y0=2