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Physics-based simulation of a vibrating pendulum with a pivot point is shaking rapidly up and down. Surprisingly, the position with the pendulum being vertically upright is stable, so this is also known as the inverted pendulum.

Click and drag near the pendulum to modify its position. The anchor can also be moved. Enable the "show controls" checkbox to set gravity, frequency of oscillation, magnitude of oscillation, and damping (friction).

A regular non-vibrating pendulum is stable only when it is hanging straight down. In this simulation, the support pivot point of the pendulum is oscillating rapidly up and down. When the oscillation is rapid and of small amplitude, there is a second stable position, with the pendulum standing straight up, in a vertical upright "upside-down" position.

You can build a physical version of the inverted pendulum using a jigsaw to power the oscillations, see references below. This is a popular demonstration used in university physics or math classes.

Experiments to try:

- Disturb the pendulum from its stable inverted position. Can it recover? How far can you disturb it?
- What is the slowest frequency that still has the stable inverted position?
- What is the range of amplitude that still has the stable inverted position?
- For a faster or slower frequency, does the range of amplitude change?
- Does the strength of gravity affect whether the inverted pendulum is stable?
- How does friction (damping) affect the inverted pendulum?
- Make the time step smaller to get more calculations per second, which will give a smoother graph.

The time rate adjustment makes the simulation run slower than real-time so that smaller time steps are used, which gives better accuracy and smoother graph plots. The physics and math is all the same, it is only displayed at a slower rate than it would be in real time.

The mathematics of this simulation is given at the Moveable Pendulum web page. Also available: source code, documentation and how to customize.

There are actually two simultaneous simulations happening here: the pendulum, and the anchor block. The anchor block is modelled as a point mass and can be moved by a force such as that applied when dragging with the mouse near the anchor point. The anchor block is not affected by the pendulum. See Anchor Block Dynamics on the Moveable Pendulum page for the math. The oscillatory motion of the anchor is caused by applying a time varying force to the anchor point. This means that you can also drag the anchor point with your mouse, which applies another force. An alternative method would be specify where the anchor point is in space at any given time with an equation, so that the anchor point could not move in any way other than this equation of oscillation.

- Video of inverted pendulum with jigsaw and explanation of how to set up the experiment and why it happens.
- A Wikipedia page covers several kinds of inverted pendulum. See the section about Pendulum with oscillatory base which refers to the Mathieu equation as describing the motion.
- Bibliography for the Pendulum is a list of 70 academic articles collected by the math faculty at California State University, Fullerton.
- Jig Saw Puzzle by Evelyn Sander, 1995, gives a simple explanation of the inverted pendulum.
- The Flying Circus of Physics has many links to videos and papers about the inverted pendulum.
- (Paywalled) Stroboscopic study of the inverted pendulum by M. M. Michaelis, 1984, describes how to use a jigsaw "to provide an inexpensive large-scale demonstration of the inverted pendulum experiment." Also covers the Kalmus and Kapitza theories used to mathematically analyze the behavior.
- (Paywalled) Stabilization of the inverted linearized pendulum by high frequency vibrations, Mark Levi and Warren Weckesser, SIAM Review, 37(2), 219-223 (1995).
- (Paywalled) Experimental study of an inverted pendulum, Smith and Blackburn, American Journal Physics 60 (10), October 1992 p. 909. "the ranges of displacement frequency and amplitude for which the inverted state is stable have been experimentally determined and compared to theoretical calculations."