Model Description
This model describes the dynamics of a waterwheel. The wheel has 10 cups that are filled when as they pass over the top of the arc of the wheel (imagine a Ferris wheel where instead of seats there are buckets. Water is flowing down on the Ferris wheel filling the buckets). The weight of the water makes the wheel turn. However, in this model, the buckets leak. So if the flow of the water from the top is too slow, some interesting motion can occur.
This waterwheel model is a perfect physical representation the Lorenz model. The water wheel exhibits a chaotic motion. That is, the direction and speed of spinning is unpredictable.
Reference
German Guillermo Theler (2004). Poster presentation titled, "Lorenzian chaotic waterwheel". Link: http://ib.cnea.gov.ar/~thelerg/pdf/wheel2004.pdf.
Default Dynamics
(Please describe default dynamics for this model.)
Waterwheel
(Please describe this animation.)
$\ddot{\theta}=\sum ^{N-1}_{i=0} \rho v_{i}gR \ \mbox {sin}\left ( \theta+i \frac{2\pi}{N} \right )-k\dot{\theta}$ |
| Acceleration of waterwheel angular position. |
$\dot{\theta}=\ddot{\theta}$ |
| Velocity of waterwheel angular position. |
# the waterwheel ala Lorenz but discrete par p=0, vi=0, g=0, R=0, k=0 ff(u)=heav(cos(u)-cos(pi/n)) flow[0..9]=flow*ff(theta-2*pi*[j]/n) cp[0..9]=flow[j]-mu*c[j] m[0..9]=c[j]+mc/n c[0..9]'=cp[j] mdot=sum(0,9)of(shift(cp0,i')) m=sum(0,9)of(shift(c0,i'))+mc theta'=thetap thetap'=(-nu*thetap-l*l*mdot*thetap+l*sum(0,9)of(shift(m0,i')*sin(theta-2*pi*i'/n)))/(m*l*l) par flow=.5,mu=.1,n=10,l=.15,mc=2,nu=.1 init theta=.05 ### some stuf for animation x[0..9]=.3*sin(theta-2*pi*[j]/n)+.4 y[0..9]=.3*cos(theta-2*pi*[j]/n)+.4 yc[0..9]=.3*cos(theta-2*pi*[j]/n)+.4+.1*c[j]/2 @ total=200,dt=.05,meth=cvode,tol=1e-5,atol=1e-4 done
# waterwheel animation file # here is the faucet frect .39;1;.41;.95;$BLACK frect .395;.95;.405;.93-.03*ran(1);$BLUE # # here is the waterwheel line .4;.4;x[0..9];y[j];[j]/10 line x[0..9];y[j];x[j];yc[j];$BLUE;4 done
Anderies JM, Arizona State University.
Bozicevic M, Arizona State University.
