Oscillations and Waves > Oscillations > Nonlinear Systems
DCS# 3A60.64

VERTICALLY-DRIVEN INVERTED PENDULUM

 

APPARATUS
Pasco driver
202-17-A3-1
function generator/amplifier
202-17-A3
rod on pivot
202-17-E5
hardware to secure driver
202-01-E6
 
DESCRIPTION
The vertically-driven pendulum is stable in the inverted position for (a/L') (ω/ωo) >2, where a is the driving amplitude, L' is the effective length, ω is the driving frequency, and ωo is the natural frequency of the undriven pendulum for small angles.

This pendulum has a length of 6.0 cm, and an effective length, L' = I/md = (mL2/3) / (mL/2) = 2L/3.  The distance d, from the center of mass to the pivot point, is L/2 if we ignore the small (0.6 cm) offset between the pivot and the end of the rod.

The natural frequency is
ωo = (mgd/I) = √[(mgL/2) / (mL2/3)] = (3g/2L) = 19 s-1.

The condition for stability is a
ω > √2 L'ωo= 2 (2L/3)(3g/2L) = (4Lg/3) = 0.72 m/s.

For a driving amplitude of about 4 mm, the necessary driving frequency is 30 Hz.

NOTES

The driver needs to be secured in place, otherwise it will creep as it vibrates.

The amplitude of the driver decreases as the frequency increases which makes it difficult to demonstrate the onset of stability by varying the frequency.

Use the newer Pasco driver, which gives the pendulum less lateral motion.

The coffee stirrer prototype also works nicely:

An interactive simulation can be found here:  http://demonstrations.wolfram.com/VibratingInvertedPendulum/.

REFERENCES

AJP 38, 874, (1970),
AJP 69, 755-768 (2001),
TPT 40, 356-357 (2002).