A double pendulum is a pendulum attached to a pendulum.
In physics and mathematics, in the area of dynamical systems, a double pendulum, also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a complex physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a pair of coupled ordinary differential equations and is chaotic.
Long exposure of double pendulum exhibiting chaotic motion (tracked with an LED) By George Ioannidis – Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=7920826
Here’s a very creative video that maps double pendulums to graphs, sounds, and spirograph curves.
The double pendulum lends itself to some beautiful time lapse photos.
The Make Magazine web site shows you how to build a double pendulum at home or office cubicle.
Devices that demonstrate true chaotic behavior (in a strict mathematical sense) are rare. Even rarer are chaotic devices that are easy enough for the typical maker to build at home and are interesting and beautiful. But one device nicely fits the bill: the double pendulum. […]
[ed. note: a Rott pendulum is also known as a triple pendulum.]
Involving a number of daunting differential equations[1], the mathematics that describe what’s going on in a Rott’s pendulum are extremely complex. But the upshot is that at small amplitudes (when given a gentle push) the 2 pendulums will remain in motion for a far longer time than either pendulum would move alone.
If you crunch through the math you’ll find that when the ratio of the length of the cross-member of the L-shaped pendulum to the length of the other 2 legs is 1.283567 to 1, the resonant frequencies of the 2 pendulums are integer ratios of one another, and the 2 parts, although much different in shape, are resonantly coupled.
While it’s difficult to make a Rott’s pendulum precisely enough to exhibit perfect resonant coupling, with enough care, you can demonstrate the phenomenon. Give a well-made Rott’s pendulum a small shove and it swings on and on and on; give it a big push and it will exhibit the wild, chaotic behavior characteristic of any well-made, low-friction double pendulum. It’s the best of both worlds.
William Gurstelle is a contributing editor of Make: magazine. His new book, ReMaking History: Early Makers is now available.
You can Do It Yourself!
[1] And Therefore
Just to give you a flavor, here is a portion of the equations from the Double Pendulum Wikipedia page. My favorite part is “and therefore….”
Dispelling Pendulum Darkness
We were disturbed to see this nonsensical Google result for “pendulums”:
To counter this darkness, WLBOTT has allocated funds from its bountiful coffers to build a Pendulum Museum.
WLBOTT Pendulum Museum
The Hall of Metronomes
The Hall of Metronomes, WLBOTT Pendulum Museum
A vast and elegant interior gallery known as the Hall of Metronomes inside the WLBOTT Pendulum Museum. The space resembles a grand cathedral of rhythm and motion, with high vaulted ceilings, polished stone floors, and warm oak display cases lining the walls. Easy access to restrooms and the gift shop.
Newton’s Cradle
Newton’s Cradle is an example of a plethora of pendulums that have brought hours of joy to office workers world-wide.
To commemorate Newton’s Cradle, the WLBOTT Pendulum Museum has a special exhibit, staffed by unpaid interns:
Newton’s Cradles conserve momentum better than most corporate employees conserve enthusiasm.
As Elder G explains…
Imagine a row of identical metal spheres suspended by thin wires from a frame. When you lift one ball and release it, the following happens:
The lifted ball swings down like a pendulum.
It collides with the row of stationary balls.
The impulse passes through the middle balls.
The ball on the far end swings outward.
To the eye it looks almost magical, as though the energy tunneled through the row.
Why It Works
Two key physical principles are at play:
Conservation of Momentum
The total momentum before and after the collision stays the same.
Conservation of Kinetic Energy
Because the steel balls collide elastically, kinetic energy is largely preserved.
The middle balls hardly move because the forces pass through them almost instantly.
Where the Pendulums Come In
Each sphere in the cradle is actually a tiny pendulum:
suspended from two wires
swinging under gravity
oscillating around a stable equilibrium
So the system combines pendulum motion + elastic collisions.
The pendulum behavior itself comes from the restoring force of gravity, which for small angles behaves approximately like simple harmonic motion.
Mythbusters took the cradle to its illogical conclusion on season 9, episode 14.
Don’t miss the Museum’s Chaos Swing!
Joseph-Louis Lagrange
Joseph-Louis Lagrange developed the mathematics used to analyze, among many things, the double pendulum.
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia 25 January 1736 – 10 April 1813) was an Italian and naturalized French mathematician, physicist and astronomer. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
It was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley [of comet fame] from 1693 which he came across by accident. Alone and unaided he threw himself into mathematical studies; at the end of a year’s incessant toil he was already an accomplished mathematician.
We decided to dedicate a wing of the Pendulum Museum to honor Joseph-Louis Lagrange. He had many, many admirers among to top French opera singers, and we included their portraits in the wing.
We commissioned several portraits of the romance of Renée-Françoise and Joseph-Louis
It was about [1792], that the unaccountable sadness of his life and his timidity moved the compassion of 24-year-old Renée-Françoise-Adélaïde Le Monnier, daughter of his friend, the astronomer Pierre Charles Le Monnier. She insisted on marrying him and proved a devoted wife to whom he became warmly attached.
Possibly for the first time a romance made it into a calculus text book (Larson Calculus)!
Lagrange was lifted from his lethargy in 1789 by the violent French Revolution, but his love of mathematics was rekindled by an extraordinary romance with the bright young daughter of his colleague, the astronomer Pierre Charles Le Monnier. Renée Françoise Adélaîde Le Monnier fell in love with Lagrange and was quite persistent in her advances. Lagrange, appalled by the forty-year difference in their ages, was at first reluctant to involve himself in a relationship with her, but in 1792 they were married. His wife’s devotion and support restored his emotional stability, and gave him great happiness.
