COLLECTED BY
Organization:
Internet Archive
The Internet Archive discovers and captures web pages through many different web crawls.
At any given time several distinct crawls are running, some for months, and some every day or longer.
View the web archive through the
Wayback Machine.
Wide crawls of the Internet conducted by Internet Archive. Access to
content is restricted. Please visit the Wayback Machine to explore
archived web sites.
TIMESTAMPS
Parametric Resonance
Parametric resonance is a resonance phenomenon different from normal
resonance and
superharmonic resonance because it is an instability phenomenon.
1. The instability
The vertically driven pendulum is the only driven pendulum in the lab which has the
same stationary solutions as the undriven pendulum, namely
= 0 and
= 180°. In the undriven case, these
solutions are always stable and unstable, respectively. But vertical driving can change stability into instability
and vice versa. On this page, mainly the destabilization of the normaly stable equilibrium of the pendulum
(i.e.,
= 0)
will be dicussed. Stabilization of an unstable upside-down pendulum will be mentioned
only briefly.
In order to investigate the stability of a fixed point, you have
to linearize the equation of motion around a fixed point.
For
= 0 one gets the (damped) Mathieu equation:
.
The same equation holds for
= 180° except that
02 has a minus sign. Thus, one studies the Mathieu
equation for positive as well as negative values of
02.
There is a simple intuitive understanding of the parametric resonance
condition.
And maybe you already have such intuitive knowledge! Imagine you are on a
fair and you want to swing a swing boat (did you ever try?).
Standing in the boat
you have to go down with your body when reaching a maximum because you
want to speed the boat up by putting an
additional acceleration to the acceleration of gravity.
If you do that near the forward and the backward maximum of oscillation,
you are just realizing first-order parametric resonance because you are
moving periodically up and down with a
frequency just twice the frequency of the swing boat. If you go down
only every nth maximum you will have
parametric resonance of order n.
The onset of first-order parametric resonance can be approximated
analytically very well by the ansatz:
![](Parametric%20Resonance_files/parres0.gif)
This ansatz is a truncated Fourier series of the periodic function c(t)
of the Floquet theorem. It neither decays nor grows. Thus the amplitude a
of driving has to be just the critical one ac. Put this ansatz into (1) and
neglect the terms
. Now, gather all terms
having the factor
and you will get
![](Parametric%20Resonance_files/parres1.gif)
These two equations have a nontrivial solution for c+
and c- only if
(3) |
![](Parametric%20Resonance_files/parres2.gif) |
This instability threshold has a minimum just at the parametric resonance condition f =
0/
(plus a correction of second order in
). The minimum reads ac=2
f
.
2. Parametrically excited oscillations
QUESTIONS worth to think about: |
- Does parametric resonance also appear for other driven pendula in the Pendulum Lab?
- Imagine a media where waves (e.g. sound waves) can
propagate. What might happen if the speed of wave is modulated periodically?
|
© 1998 Franz-Josef Elmer,
Franz-Josef doht Elmer aht unibas doht ch,
last modified Monday, July 20, 1998.