| Vibration : | Lab Frame： Pendulum Frame |
|
Pendulum Angle:
$\varphi$ = deg
|
Vibration Direction:
$\theta$ = 0deg
|
Dissipation:
$\alpha$ = 0.02 $\omega_0$
|
|
Amplitude:
$f$ = 0.06 $\ell$
|
Frequency:
$\Omega$ = 70 $\omega_0$
|
$\Omega f = $ 1.8 > $\sqrt{2}\;\omega_0\,\ell$ |
Using this simulator, you can simulate the Kapitza pendulum. In the initial settings, there is no vibration, and a small amount of dissipation (friction) is included. When the simulation starts, the pendulum behaves like a normal simple pendulum. Pressing the vibration button initiates vibrations at the pivot, and depending on the timing, the pendulum may start rotating or oscillating. If the dissipation is increased, the pendulum usually comes to rest in the downward position. However, if the pendulum angle is moved upward, it naturally stabilizes in the inverted position.
In this simulator, the vibration
\[
f(t):=f\sin\Omega t
\]
is imposed on the pivot point in the direction $\theta$ from the
vertical direction.
The equation that is numerically solved is
\[
\ddot\varphi = {g\over\ell}\sin\varphi + {\ddot
f(t)\over\ell}\sin\big(\varphi-\theta\big) -\alpha\,\dot\varphi
\]
for the angle $\varphi$ from the upward vertical direction.
Here, $\ell$ is the length of the arm, and $\alpha$ is the parameter
for the dissipation introduced phenomenologically.
In the simulation,
the values of the parameters are given in the length unit
$\ell$ and the time unit $\sqrt{\ell/g}:=1/\omega_0$
with $g$ being the gravitational acceleration.
This pendulum tends to stabilize in the direction of the vibration.
If you rotate the direction of vibration with larger value of
dissipation, the direction of pendulum will follow the direction of
vibration.
With larger values of the product $f\Omega$ of the amplitude $f$ and
the frequency $\Omega$, the pendulum follows the direction of
vibration more faithfully.
It is shown that the pendulum is stabilized in the inverted
state when
\[ \Omega f > \sqrt{2}\; \omega_0\,\ell , \]
in the case where the amplitude of the vibration $f$ is much smaller
than the length of arm $\ell$ and the angular frequency of the imposed
vibration $\Omega$ is much larger then the angular frequency of the
pendulum $\omega_0:=\sqrt{g/\ell}$.
When the vibration is in the vertical direction, the pendulum is
stabilized not only in the upward direction, but also in the downward
direction. Therefore, in order to stabilize the pendulum in the
upward, the arm has to be held beyond the horizontal direction
initially.
Without dissipation, the pendulum keeps on rotating or swinging around
the stable direction and never settles in a stationary state.
If the vibration is imposed in a diagonal direction with larger
dissipation, the pendulum also stabilized in the diagonal direction.
Theory
Theoretical analysis on the pendulum with a rapidly oscillating pivot
has been given by Kapitza ("Mechanics" by L.D. Landau and E.M.
Lifshitz, Sec. 30.)