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Something I have been thinging about for quite some time now is the
possibility of building a two cyclinder Stirling type engine that
uses the above ground atmosphere to provide the warm temperature for
one cyclinder and then bury the other cylinder far enough into the
earth to create the low temperature cyclinder. I envision the
finished product looking like a grain silo with some type of power
collection building attached to it (energy barn). The top cyclinder
could even have a coating like solar panels or magnifying glass to
ensure a higher temperature.

Up until several days ago, I did not know how a Stirling engine
woked, but I have been tinkering around with this idea for a couple
of years now. I have since ordered a book from this website to learn
the principles behind this type of engine and am also researching
different geographic locations that would provide a suitable
environment.

Any comments or suggestions would be great. Especially if you think
it is a crazy idea that would not work.

An innovative idea! There may be a problem with the temperatures available from the heat source(s) and sink(s) suggested. At a few feet depth the ground temperature remains relatively constant the year around which makes it ideal for use in connection with a heat pump. At grater depths it increases. Use "Google" and other search engines to learn more about the earth's temperature gradient.
Build or assemble a model Stirling engine.
Use the sitemap on this URL. There is a great deal of information on Stirling engines, how they work and why, readily available.

So in order to get a large enough temperature variance, the hot cyclinder would need to go far enough in the ground to find a sufficient source of heat? Seems like it may work but that would be pretty big.

I have another idea about a single cylinder focused windmill type engine (more like a generator). It involves a large ferris wheel like hollow loop that is large enough to bury into the ground to either a natural or artificial heating source with the surface exposed end high enough to not be affected so much by the heating source.

With several well placed one-way valves and a couple of focused cones with an alternator type device mounted in the middle (for power harvesting), a current of air (nitrogen, or the like) could be created by the warm air expanding through to the cold and looping back.

The finished product would stick out of the ground similar to the Arch in St. Louis.

If anyone sees a serious flaw in my thinking here, please speak up. I hope to build a model of some sort in the spring once I get a few things drawn up. I am an Engineer, but not the type of engineer for this field, so I have alot of research to do to get up to speed.