# Inverted Pendulum

 The pendulum consists of two plastic-coated metal rods mounted with epoxy in a plywood base.  Two pairs of magnets held magnetically to the rods provide the mass.  The coil surrounding the upper pair senses motion of the pendulum.

 Two nested plastic bottles provide a shield from air currents.  The lower pair of magnets is moved  up and down to adjust the period.

This setup turned out to be quite noisy and not very sensitive.  Damping was provided by the coil, as a small resistor was placed in parallel with it.  The sensitivity would have been improved by using more turns on the coil, and having less clearance between the coil and the upper magnets.

A major problem with this design was the difficulty adjusting it to obtain a long period.  For any period longer than a second or two the pendulum was unstable and would lean to one side.  I think this could be because the design had a flexible vertical rod rather than a stiff rod with a spring at its base, as in the Colorado School of Mines SENS design.

### Equations for an Inverted Pendulum:

The gravity toppling force is:
Fg = mg sin(theta), where theta is the angle of tilt, m is the mass and g is the acceleration of gravity.

The spring restoring force is:
Fs = k theta, where k is the spring constant.

The net restoring force is then:
Fnet = Fs - Fg = k theta - mg sin(theta)

The effective spring constant is:
Effective Spring Constant = d (Fnet) / d theta = k - mg cos(theta)

For very small theta, cos(theta) approaches 1.0, so

Effective Spring Constant = k - mg

If I've not made a mistake, this means that the system will be stable as long as k is greater than mg and that a very long-period small-motion natural period can be obtained if k is made just slightly larger than mg.

k is the spring force per radian of deflection, which is the spring force for a deflection of 57 degrees from the vertical.

 This may be a better design for such a pendulum.

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