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)
“There’s not the smallest orb which thou behold’st
But in his motion like an angel sings,
Still quiring to the young-eyed cherubins;
Such harmony is in immortal souls,
But whilst this muddy vesture of decay
Doth grossly close it in, we cannot hear it.”
—William Shakespeare (1564–1616)