SwingingAtwood

Systems / ODEs / Physical systems

The swinging Atwood machine — a mass on a pulley that swings into chaotic motion.

continuous · ODE4 dimensionsconservativechaotic

projection (r, th, pr)

projection (r, th, pth)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{r} &= \frac{pr}{m_{1} + m_{2}} \\ \dot{th} &= \frac{pth}{m_{1} r^{2}} \\ \dot{pr} &= 9.82 m_{1} \cos{\left(th \right)} - 9.82 m_{2} + \frac{pth^{2}}{m_{1} r^{3}} \\ \dot{pth} &= - 9.82 m_{1} r \sin{\left(th \right)} \end{aligned} \]

Parameters

Symbol	Default	Role
m1	1	swinging pendulum mass
m2	4.5	non-swinging counterweight mass (ratio m2/m1 controls the dynamics)

State variables: r, th, pr, pth

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 0$
constant

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.SwingingAtwood()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

Tufillaro, Abbott & Griffiths (1984), Am. J. Phys. 52, 895-903
doi:10.1119/1.13791

BibTeX

@misc{swingingatwood,
  title = {SwingingAtwood system},
  note = {Tufillaro, Abbott & Griffiths (1984), Am. J. Phys. 52, 895-903},
  doi = {10.1119/1.13791}
}