Double pendulum: Definition from Answers.com

Long exposure of double pendulum exhibiting chaotic motion (tracked with an LED light)

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. For certain energies its motion is chaotic. See also pendulum (mathematics).

1 Analysis
- 1.1 Lagrangian
2 Chaotic motion
3 References
4 External links

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 (also called complex 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.

An example of a double pendulum.

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 $\textstyle I=\frac{1}{12} m \ell^2$ about that point. The moment of inertia of a rod rotating around an axis attached to one of its ends equals $\textstyle I=\frac{1}{3} m \ell^2$ .

It is convenient to use the angle between each limb and the vertical as the generalized coordinates defining the configuration of the system. These angles are denoted θ₁ and θ₂. The position of the centre of mass of each rod may be written in terms of these two coordinates. If the origin of the Cartesian coordinate system is taken to be at the point of suspension of the first pendulum, then the centre of mass of this pendulum is at:

$x_1 = \frac{\ell}{2} \sin \theta_1,$

$y_1 = -\frac{\ell}{2} \cos \theta_1$

and the centre of mass of the second pendulum is at

$x_2 = \ell \left ( \sin \theta_1 + \frac{1}{2} \sin \theta_2 \right ),$

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

This is enough information to write out the Lagrangian.

Lagrangian

The Lagrangian is

$\begin{align}L & = \mathrm{Kinetic~Energy} - \mathrm{Potential~Energy} \\ & = \frac{1}{2} m \left ( v_1^2 + v_2^2 \right ) + \frac{1}{2} I \left ( {\dot \theta_1}^2 + {\dot \theta_2}^2 \right ) - m g \left ( y_1 + y_2 \right ) \\ & = \frac{1}{2} m \left ( {\dot x_1}^2 + {\dot y_1}^2 + {\dot x_2}^2 + {\dot y_2}^2 \right ) + \frac{1}{2} I \left ( {\dot \theta_1}^2 + {\dot \theta_2}^2 \right ) - m g \left ( y_1 + y_2 \right ) \end{align}$

The first term is the linear kinetic energy of the center of mass of the bodies and the second term is the rotational kinetic energy around the center of mass of each rod. The last term is the potential energy of the bodies in a uniform gravitational field. The dot-notation indicates the time derivative of the variable in question.

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)

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

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

These last four equations are explicit formulae for the time evolution of the system given its current state. It is not possible to go further and integrate these equations analytically, to get formulae for θ₁ and θ₂ as functions of time. It is however possible to perform this integration numerically using the Runge Kutta method or similar techniques.

Chaotic motion

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

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.

References

Meirovitch, Leonard (1986). Elements of Vibration Analysis (2nd edition ed.). McGraw-Hill Science/Engineering/Math. ISBN 0-07-041342-8.
Eric W. Weisstein, Double pendulum (2005), ScienceWorld (contains details of the complicated equations involved) and "Double Pendulum" by Rob Morris, Wolfram Demonstrations Project, 2007 (animations of those equations).
Peter Lynch, Double Pendulum, (2001). (Java applet simulation.)
Northwestern University, Double Pendulum, (Java applet simulation.)
Theoretical High-Energy Astrophysics Group at UBC, Double pendulum, (2005).

External links

Animations and explanations of a double pendulum and a physical double pendulum (two square plates) by Mike Wheatland (Univ. Sydney)

Video of a double square pendulum with three (almost) identical starting conditions.

Double pendulum physics simulation from www.myphysicslab.com

Simple explanation of chaos theory and double pendulums at chaoticpendulums.com

Simulation, equations and explanation of Rott's pendulum

Comparison videos of a double pendulum with the same initial starting conditions on YouTube

Double Pendulum Simulator - An open source simulator written in C++ using the Qt tookit.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

	pendulum
	metronome (mechanical device – in music)
	Double inverted pendulum

	If you want to double the period of a pendulum by how much do you need to change the length? Read answer...
	What happens to the period of a pendulum when the length is doubled? Read answer...
	What does the pendulum symbolize in pit and pendulum? Read answer...

	A simple pendulum has a frequency of oscillation f In order to double f the length of the pendulum should be?
	What will be effect on time period of pendulum when amplitide becomes double?
	If you double the frequency of a pendulum what happens to the period of one cycle?

		Sci-Tech Dictionary. McGraw-Hill Dictionary of Scientific and Technical Terms. Copyright © 2003, 1994, 1989, 1984, 1978, 1976, 1974 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Sports Science and Medicine. The Oxford Dictionary of Sports Science & Medicine. Copyright © Michael Kent 1998, 2006, 2007. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Double pendulum". Read more

Double pendulum

Contents

Analysis

Lagrangian

Chaotic motion

References

External links