StickSlipOscillator¶
Systems / ODEs / Oscillatory systems
A harmonically forced dry-friction oscillator whose Stribeck velocity switch produces alternating stick and slip phases that can turn chaotic.
Definition¶
\[
\begin{aligned}
\dot{y_{0}} &= y_{1} \\
\dot{y_{1}} &= a y_{0} - b y_{0}^{3} + eps \left(\alpha y_{1} - \beta \left(- vs + y_{1}\right)^{3} + \gamma \cos{\left(y_{2} \right)} - t_{0} \operatorname{sign}{\left(- vs + y_{1} \right)}\right) \\
\dot{y_{2}} &= w
\end{aligned}
\]
Parameters¶
| Symbol | Default | Role |
|---|---|---|
a |
1 |
linear stiffness of the Duffing restoring force |
alpha |
0.3 |
linear velocity coefficient of the friction torque |
b |
1 |
cubic stiffness of the Duffing restoring force |
beta |
0.3 |
cubic velocity coefficient of the friction torque |
eps |
0.05 |
coupling weight of the friction/forcing term |
gamma |
1 |
amplitude of the harmonic drive |
t0 |
0.3 |
static (Coulomb) friction level |
vs |
0.4 |
reference Stribeck sliding velocity |
w |
2 |
angular frequency of the harmonic drive |
Properties¶
Lyapunov spectrum
$+0.02426,\; -0.02074,\; 0$
computed at build
computed at build
Kaplan–Yorke dimension
$D_{KY} = 3$
Divergence ∇·f
n/a — non-smooth right-hand side — divergence is piecewise (defined almost everywhere)
Equilibria
none found (no equilibria)
Define it in TSDynamics¶
import tsdynamics as ts
sys = ts.systems.StickSlipOscillator()
traj = sys.integrate(final_time=100.0, dt=0.01)
exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)
Reference¶
Awrejcewicz & Holicke (1999), Int. J. Bifurc. Chaos