## PSN-L Email List Message

Subject: Re: Other thoughts on an inverted pendulum
From: John & Jan Lahr johnjan@........
Date: Sun, 26 Jan 2003 11:19:12 -0700

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Hi Ed,
The way to think about the inverted pendulum is that we are balancing two
forces.
Gravity wants to tip the pendulum over, and the greater the angle the
greater this
force is.  The spring at the base wants to keep the pendulum
upright, and the
greater the angle the greater the force of the spring is in this
direction.
If I have the math correct, then the problem I had with instability may
have
been due to my setup.  My inverted pendulum was made from mounting
a
thin metal rod vertically and putting a mass at the top.  The entire
rod bent.
When I tried to achieve a long period, the rod would bend to one side and

stay there!
The CSM student design has a solid rod with a spring at the base.
http://jjlahr.com/science/psn/epics/reports/sens/index.html
I think for that design, the gravity toppling force is:
Fg = mg sin(theta), where theta is the angle of tilt.
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 would be:
d (Fnet) / d theta = k - mg cos(theta)
For very small theta, cos(theta) approaches 1.0, so
Effective spring constant = k - mg
It then increases with amplitude.
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 just slightly larger than mg.
See:
http://jjlahr.com/science/psn/inverted/
I hope Chris Chapman will check this for errors!

Damping is another thing that will need to be worked out.  When the
pendulum
is properly damped, if it is displaced, for example 1 mm to one side and
released,
it will move through the center position and continue for about 1/20 mm,
turn around,
move through the center and continue for 1/400 mm, etc.  In other
words, the
amplitude will die out very quickly.
My suggestion would be to first work on getting as long a period as
possible and then
work on the damping.
Cheers,
John

At 07:40 AM 1/26/2003, you wrote:
John,
some other thoughts......I am attaching a drawing......I hope you receive
it. I will describe it to you anyway.......

I am thinking of possibly extending the brass
rod down into the spring while it still protrudes from the top. I would
also attach a very small level at the bottom of the brass rod to insure
leveling and possibly increase damping. Instead of placing four wooden
dowels (as shown in the drawing) around the spring to possibly prevent
the magnet/weight from getting to the point where it falls/hangs over the
spring I would replace them with a transparent (to view the level)
glass/plastic circular column that reached the elevation of the magnet.
Please remember one thing....I don't know what I'm talking