Double pendulum

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In horology, a double pendulum is a system of two simple pendulums on a common mounting which move in anti-phase.

In mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior. The motion of a double pendulum is governed by a set of coupled ordinary differential equations. Above a certain energy its motion is chaotic. See also pendulum (mathematics).

An example of a double pendulum.
An example of a double pendulum.
A double pendulum with 1 second shutterspeed.
A double pendulum with 1 second shutterspeed.

Contents

[edit] Analysis

Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be simple pendulums or compound pendulums and the motion may be in three dimensions or restricted to the vertical plane. In the following analysis, the limbs are taken to be identical compound pendulums of length \ell and mass M, and the motion is restricted to two dimensions.

In a compound pendulum, the mass is distributed along its length. If the mass is evenly distributed, then the centre of mass of each limb is at its midpoint, and the limb has a moment of inertia of I=\frac{1}{12} M \ell^2 about that point.

It is natural to define the coordinates to be the angle between each limb and the vertical. These are denoted by θ1 and θ2. The position of the centre of mass of the two rods may be written in terms of these coordinates. If the origin of the Cartesian coordinate system is assumed to be at the point of contact of the wall and the first pendulum, then the centre of mass is located at:

x_1 = \frac{\ell}{2} \sin \theta_1,
x_2 = \ell \left ( \sin \theta_1 + \frac{1}{2} \sin \theta_2 \right ),
y_1 = -\frac{\ell}{2} \cos \theta_1

and

y_2 = -\ell \left ( \cos \theta_1 + \frac{1}{2} \cos \theta_2 \right ).

This is enough information to write out the Lagrangian.

[edit] Lagrangian

The Lagrangian is given by

L = \frac{1}{2} m \left ( v_1^2 + v_2^2 \right ) + \frac{1}{2} I \left ( {\dot \theta_1}^2 + {\dot \theta_2}^2 \right ) - m g \left ( y_1 + y_2 \right )

where the first term is the linear kinetic energy of the center of mass of the bodies, the second term is the rotational kinetic energy of each rod, and the last term is the potential energy of the bodies in a uniform gravitational field.

Substituting the coordinates above and rearranging the equation gives

L = \frac{1}{6} m \ell^2 \left [ {\dot \theta_2}^2 + 4 {\dot \theta_1}^2 + 3 {\dot \theta_1} {\dot \theta_2} \cos (\theta_1-\theta_2) \right ] + \frac{1}{2} m g \ell \left ( 3 \cos \theta_1 + \cos \theta_2 \right ).

There is only one conserved quantity (the energy), and no conserved momenta. The two momenta may be written as

p_{\theta_1} = \frac{\partial L}{\partial {\dot \theta_1}} = \frac{1}{6} m \ell^2 \left [ 8 {\dot \theta_1} + 3 {\dot \theta_2} \cos (\theta_1-\theta_2) \right ]

and

p_{\theta_2} = \frac{\partial L}{\partial {\dot \theta_2}} = \frac{1}{6} m \ell^2 \left [ 2 {\dot \theta_2} + 3 {\dot \theta_1} \cos (\theta_1-\theta_2) \right ].
Motion of the double compound pendulum (from numerical integration of the equations of motion)
Motion of the double compound pendulum (from numerical integration of the equations of motion)

These expressions may be inverted to get

{\dot \theta_1} = \frac{6}{m\ell^2} \frac{ 2 p_{\theta_1} - 3 \cos(\theta_1-\theta_2) p_{\theta_2}}{16 - 9 \cos^2(\theta_1-\theta_2)}

and

{\dot \theta_2} = \frac{6}{m\ell^2} \frac{ 8 p_{\theta_2} - 3 \cos(\theta_1-\theta_2) p_{\theta_1}}{16 - 9 \cos^2(\theta_1-\theta_2)}.

The remaining equations of motion are written as

{\dot p_{\theta_1}} = \frac{\partial L}{\partial \theta_1} = -\frac{1}{2} m \ell^2 \left [ {\dot \theta_1} {\dot \theta_2} \sin (\theta_1-\theta_2) + 3 \frac{g}{\ell} \sin \theta_1 \right ]

and

{\dot p_{\theta_2}} = \frac{\partial L}{\partial \theta_2} = -\frac{1}{2} m \ell^2 \left [ -{\dot \theta_1} {\dot \theta_2} \sin (\theta_1-\theta_2) + \frac{g}{\ell} \sin \theta_2 \right ].

[edit] Chaotic motion

Graph of the time for the pendulum to flip over as a function of initial conditions
Graph of the time for the pendulum to flip over as a function of initial conditions

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 10\sqrt{g/\ell\ } (green), within 100 (red), 1000 (purple) or 10000 (blue). Initial conditions that don't lead to a flip within 10000\sqrt{g/\ell\ } are plotted white.

The boundary of the central white region is defined in part by energy conservation with the following curve:

3 \cos \theta_1 + \cos \theta_2 = 2. \,

Within the region defined by this curve, that is if

3 \cos \theta_1 + \cos \theta_2 > 2, \,

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.


[edit] References

[edit] External links