Chaotic Motion
The double pendulum undergoes chaotic motion, and shows a sensitive dependence on initial conditions. The image to the right shows the amount of elapsed time before the pendulum "flips over," as a function of initial conditions. Here, the initial value of θ1 ranges along the x-direction, from −3 to 3. The initial value θ2 ranges along the y-direction, from −3 to 3. The colour of each pixel indicates whether either pendulum flips within (green), within (red), (purple) or (blue). Initial conditions that don't lead to a flip within are plotted white.
The boundary of the central white region is defined in part by energy conservation with the following curve:
Within the region defined by this curve, that is if
then it is energetically impossible for either pendulum to flip. Outside this region, the pendulum can flip, but it is a complex question to determine when it will flip.
The lack of a natural excitation frequency has led to the use of double pendulum systems in seismic resistance designs in buildings, where the building itself is the primary inverted pendulum, and a secondary mass is connected to complete the double pendulum.
Read more about this topic: Double Pendulum
Famous quotes containing the words chaotic and/or motion:
“The attitude that nature is chaotic and that the artist puts order into it is a very absurd point of view, I think. All that we can hope for is to put some order into ourselves.”
—Willem De Kooning (b. 1904)
“The moments of the past do not remain still; they retain in our memory the motion which drew them towards the future, towards a future which has itself become the past, and draw us on in their train.”
—Marcel Proust (1871–1922)