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.

Short answers:

(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

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