Chaotic Inverting Pendulum
The concept of the chaotic double and triple compound pendulum has been known for many years.
The general idea is that of a multi-stage compound pendulum whose suspension point can be oscillated vertically at an adjustable rate, and with variable amplitude. On small scale systems this
can be done with a speaker cone fed from a signal generator, and on larger scales by an electric
motor driving a linear shaft. We have built versions on both scales, each having a three stage pendulum.
Small Scale System
The small scale version is best driven from a commercial laboratory diaphragm vibrator, usually used for
setting up standing modes in strings. A loudspeaker can be used but it is more difficult to mount
a rigid support from which to hang the pendulum. The pendulum itself is constructed out of three 4cm lengths of
aluminium strips with bearing joints to one another, and to the vibrator shaft itself.
By driving the vibrator from a signal generator it is possible to obtain amplitudes of many mm up to frequencies
of 50Hz or so, which should cover the stability region for a pendulum of these dimensions.
In other words, it should be possible to get the pendulum to stand on end and exhibit inverted
pendulum motion at a frequency equal to its lowest mode natural frequency when hanging
downwards.
Large Scale System
The large scale system we have constructed consists of three 20cm (nominally) long pieces of steel capillary
tubing of 4mm diameter. Such tubing provides a strong pendulum segment whilst being extremely light weight. The bearings have to be mounted on metal fittings designed to push into the ends of each segment, so that each segment can rotate in a full circle without hitting the segment
above. It is better to make the segments of different lengths to avoid this problem. So for example
the first segment is 22cm long, the next 21cm and the final one 20cm. The motor used has to be
reasonably powerful since most mechanical rotational to linear drives will be rather lossy.
The load is effectively an unbalanced wheel and so large peak torques should expected.
We used a 50V DC motor whose speed was adjusted with a 0-60V power supply capable of
providing up to 2A.
The linear drive we used was a commercial unit, however people with access to a machine
shop may prefer to design their own. We arranged the drive to provide peak to peak motions of
up to 2cm, adjustable by relocating a piston shaft to a different setting on the drive wheel, as
shown in the diagram. The whole assembly is mounted to an aluminium plate and fixed to a heavy
stand. We use a hollow box which we weight with four lead bricks in order to prevent the
system from vibrating across a floor. The rotation speed for this system to provide an inverted mode
is about 10-20Hz with a peak to peak motion of about 2cm. This is a rather severe off-balance
system, and it is advisable to make all the parts that undergo strain from steel (i.e. avoid the use
of brass bolts or shafts where they are likely to shear).
Experiments
Both units can be used to demonstrate the same effects, just on different scales. If the systems
are driven so that the frequency is only a few Hz, then the pendula should go into chaotic motion.
Sometimes a regular periodic motion persists, but if given a slight perturbation, they should go
into chaotic motion.
The next regime of interest is the inverted mode whereby they need to be driven at much higher
frequency. If the maximum peak to peak amplitudes afforded by both systems are set (i.e around
1cm for the small system and 2cm for the large system) then the frequencies to look for are
around a few tens of Hz (for the small system) and around 15Hz (for the large system).
If the amplitudes are not as great as this, then the required drive frequencies will increase.
To obtain the inverted motion, the pendulum has to be manually set upright and gently let go.
When the stability regime is entered, the inverted pendulum should be able to tolerate quite
large perturbations from the vertical.