Double pendulum with external drive

The simulation demonstrates a frictionless, mathematical double pendulum that either:

  1. starts passively from given initial conditions,
  2. or is driven periodically by an external source.

The massless, rigid pendulum rods are shown as straight lines. Masses are concentrated in the center of the pendulum bobs.

When the simulation file is opened, the pendulum is in horizontal position. The length of the secondary pendulum 2 (yellow) is one half that of the primary one 1 (blue): L2/L1 = 0.5. The mass of both is equal: m2/m1 = 1. Their initial speed is zero. There is no external drive: A=0.

Start initiates the calculation of movement under the influence of gravity. The path of the secondary yellow pendulum bob (2) is shown in red for a limited time period. Stop freezes the movement until a new start. Clear deletes traces, Reset reestablishes the default initial conditions.

The primary blue bob can be drawn with the mouse to create different initial angular positions, with stretched pendulum and zero initial velocity.

In the right window a phase space diagram y1´(y1) is shown for the blue bob of the primary pendulum (for a simple pendulum this would be a closed periodical curve, and would be a circle for small arcs of oscillation). The length of the traces is limited to 2500 points of calculation.

The speed of the animation can be varied with slider Speed.

The length L1 of the primary pendulum is kept constant. Slider L2/L1 changes the length of the secondary pendulum. The window size is adjusted to twice the maximum size of the double pendulum. L2/L1 = 0 results in a single pendulum, with both bob masses coinciding.

Slider m2/m1 determines the mass relation of the pendulum bobs. m2/m1= 0 results in a single primary pendulum. In the simulation the orientation of the fictive massless secondary pendulum stays constant, while it will briskly oscillate for any finite mass.

The movement of the double pendulum is chaotic and shows no periodicity. Yet it is strictly deterministic: after reset the same path will be resumed (start without clearing the trace).

In a real pendulum friction would diminish the amplitude of oscillations. Friction is neglected in this model.

With A > 0 a periodic external torque is acting upon the primary pendulum, with amplitude and direction as shown by the black arrow. Slider delta changes the drive frequency. With delta =1 the drive frequency is equal to that of the single primary pendulum (m2 = 0) at small arcs. For all initial conditions the driven double pendulum shows a rich variety of deterministic chaotic movements.

The 3D-Phase space button opens another window with two rotating 3D frames for the three dimensional phase spaces of both pendulums.

y1´´(y1 , y´1) und y2´´(y2 , y2´)

The scalings of the 3D frames are adjusting automatically to the amplitudes.