La simulación basada en la física de un péndulo que vibra con un punto de pivote que se mueve rápidamente
arriba y abajo. Sorprendentemente, la posición con el péndulo verticalmente hacia arriba es
estable, por lo que esto también se conoce como el péndulo invertido.
Haga clic y arrastre cerca del péndulo para modificar su posición. El ancla también se puede mover.
Habilite la casilla de verificación "mostrar controles" para establecer la gravedad, la frecuencia de oscilación, la magnitud
de oscilación y amortiguamiento (fricción).
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 are:
open source code,
documentation and a
simple-compiled version
which is more customizable.
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.
References
-
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."