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Equations of Motion

Lagrangian

EQ of Motion

The Lagrangian of a dynamical system is a function that summarizes the dynamics of the system. The Lagrangian is named after Italian-French mathematician and astronomer Joseph Louis Lagrange.

Equations of Motion

Positions

Cartesian coordinates of M and m with respect to the pivot point of the first pendulum

Lagrangian

Velocities^2

small angle approximation

Velocities^2

Coordinates

Sensitive to Initial Conditions

Finding the Frequencies and

Normal Modes

Normal Modes

Through some algebra and simplification the normal modes can be obtained

Frequencies

Putting the equations in matrix form and take the determinant to find the eigenvales - the frequencies

Substitution

Convert the coupled differential equations into a system of linear algebraic equations by substitution

Chaotic Motion in Double Pendulums

By: Anna Wetterer

Special Cases

L = l

L=l but M>>m

In both modes, the heavier upper mass essentially stands still while the lighter lower mass oscillates, and even flips around the upper pendulum.

Image by Michael Devereux

L=l but M<<m

L=l and M=m

In the first mode (the positive one) the lower heavier mass is essentially standing still while the upper lighter mass vibrates back and forth at a high frequency due to the tension in the rods. In the second mode, the rods form a straight line and the system is essentially a simple pendulum with length 2 l

M = m

Long exposure of 2 LEDs, attached to the end of each pendulum of a double pendulum.

M=m but L>>l

In the first mode the masses are moving equal distances in opposite directions at a high frequency. In the second mode the rods form a straight line, similarly to when M was less than m, and the system acts like a single pendulum.

M=m but L<<l

In the first mode the bottom mass is basically standing still while the top mass is oscillating at a high frequency. In the second mode the system acts like a pendulum of length l because the tangential force on the top mass is zero.

What is a double pendulum?

Chaotic Motion

A chaotic system is one that exhibits aperiodic long term behavior in a determinist system with sensitivity to initial conditions

Single Pendulum

Double Pendulum

Sources

  • Beard, W. C., B. Vest, T. Warn, and M. J. Madsen. "Analyzing SHO of a Double Pendulum." Wabash Journal of Physics 4.2 (2010): 1-9. PHY381. Web. 29 Mar. 2014.
  • Johnson, Porter Wear. Classical Mechanics with Applications. Hackensack, NJ: World Scientific, 2010. Print.
  • Kibble, Tom W. B., and Frank H. Berkshire. Classical Mechanics. 5th ed. London: Imperial College, 2009. Print
  • Marion, Jerry B., and Stephen T. Thornton. Classical dynamics of particles and systems. 4th ed. Fort Worth, Texas: Saunders College Publishing, 1995. Print.
  • Morin, David J. Introduction to Classical Mechanics. Cambridge, UK: Cambridge UP, 2008. Print.
  • Wolfram Alpha LLC. 2010. Wolfram|Alpha. Double Pendulum (access 12 April 2014)
  • A special thanks to Allison and AJ Lawrence who assisted me in making my double pendulum system
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