Un péndulo accionado amortiguado es un sistema caótico. En un sistema caótico el comportamiento futuro
depende en gran medida del valor exacto de las condiciones iniciales. Un pequeño cambio en
las condiciones iniciales pueden causar grandes cambios después de un corto período de tiempo.
Haga clic en la pestaña "Simulador" para acceder a parámetros como: fuerza motriz,
amplitud, frecuencia, masa, gravedad y amortiguamiento. Puedes arrastrar el péndulo con tu
ratón para cambiar la posición inicial.
Los botones a continuación establecen parámetros para que la simulación esté en un bucle o en un
estado caótico; la amplitud de excitación es diferente en cada caso y se muestra entre paréntesis.
Se tarda alrededor de un minuto en producirse un bucle.
The math behind the simulation is shown below. Also available are:
open source code,
documentation and a
simple-compiled version
which is more customizable.
The Damped Driven Pendulum
The pendulum is subject to frictional damping, meaning that it will slow down over
time if there is no driving force. To see this try setting the drive amplitude to zero,
so the only forces are damping and gravity.
The driving force is shown as a curved arrow in the simulation. The length of the
arrow corresponds to the amount of torque (twisting force) at a given moment. The
direction of the arrow, clockwise or counter-clockwise, shows the direction of this
applied force. You can change the amplitude (strength) of the driving force or its
frequency (how often it switches directions).
Not all combinations of the parameters (length, gravity, drive amplitude, drive
frequency, damping) will lead to chaos. Many combinations result in simpler repeating
behavior. Click the "single loop" button above to see this. You'll need to wait about a
minute for the simulation to settle into the loop.
There is a phenomenon known as period doubling or bifurcation where
slowly increasing one of the parameters, such as drive amplitude, causes the number of
loops in the repeating behavior to double. Some of the above buttons change the drive
amplitude gradually so that you see this period doubling occur. For example, you get a
single loop with drive amplitude of 1.35, a double loop with 1.45, a quadruple loop
with 1.47, and eventually you get chaos at 1.50.
After enough period doublings occur, the system becomes chaotic. Click one of the
buttons labelled "chaos" to set the parameters and see this. A curious thing about this
chaotic behavior is that it is not entirely random. While you can't predict the exact
state of the system at a given time in the future, it is possible to show that the
system will follow an elaborate pattern, These patterns are fractals, which
are patterns that repeat themselves when you magnify them.
Physics

pendulum variables
The pendulum is modeled as a point mass at the end of a massless rod. The damping
(friction) is proportional to the angular velocity of the pendulum. There is also an
external driving force which provides a periodic torque (twist). Define the following
variables:
-
θ =
angle of pendulum (
0 =
vertical)
-
ω = θ' =
angular velocity
-
R =
length of rod
-
m =
mass of pendulum
-
g =
gravitational constant
-
b =
damping (friction) constant
-
A =
amplitude of driving force
-
k =
constant related to frequency of driving force
-
t =
time in seconds
We will derive the equation of motion for the pendulum using the rotational analog
of Newton's second law for motion about a fixed axis, which is
-
∑ τ =
sum of applied torques
-
I =
rotational inertia
-
α = θ'' =
angular acceleration
The rotational inertia about the pivot is
I = m R2
. Torque can be
calculated as the vector cross product of the position vector and the force.
- torque due to gravity is
τ = −R m g sin θ
.
- torque due to friction is
τ = −b ω
.
- torque due to driving force is
τ = A cos(k t)
.
So equation (1) becomes
m R2 α = −R m g sin θ −
b ω +
A cos(k t)
which we can write as
θ'' = − g⁄R sin θ +
|
−b θ' + A cos(k t)
|
m R2
|
This is the equation of motion for the driven damped pendulum.
Numerical Solution
To solve the equation of motion numerically, so that we can run the simulation, we
use the Runge Kutta method
for solving sets of ordinary differential equations. First we define a variable for the
angular velocity
ω = θ'
. We also add an equation for time because time
appears explicitly in the equations in the driving force
A cos(k t)
. Then
we can write the second order equation above as three first order equations.
t' = 1
θ' = ω
ω' = − g⁄R sin θ +
|
−b ω + A cos(k t)
|
m R2
|
This is the form needed for using the Runge-Kutta algorithm.
This web page was first published September 2001.