What is the derivation of the torque formula for a piston engine?

[cjmdjm]

I am working on programming a simulation/model of a piston engine (steam, gas, etc). I have run into some trouble with the geometry. I need to calculate the torque T on the crankshaft as a function of the force F on the piston, the crankshaft angle theta, the connecting rod length L, and the crank length R (which is equal to half the stroke length). I should have no trouble getting the force from the pressure difference across the piston, butting getting the torque from that is more difficult. Heres an image that should clarify:

http://web.mit.edu/~j_martin/www/pistonphysics.bmp

The thing is, I actually already found the answer online on page 4 (1118) of this document:

http://www.iop.org/EJ/article/0143-0807/26/6/020/ejp5_6_020.pdf?request-id=9d55429d-8fc3-428a-959e-33173d288def

But I am really curious how this formula is derived. I can't seem to prove that formula myself. I can get a formula for it, but it is messy and involves lots of arcsin and arctan etc. Any help or insight would be greatly appreciated, thanks.

 

[gmax137]

Well your second link doesn'twork for me so I can't see the equation the book gets. The mathematical description of the motion of a crank-rod-piston device is indeed a messy beast full of arc-functions. And it isn't symmetrical between top & bottom (like a sine function) because the motion as the big-end bearing goes over the top is different than as it comes around the bottom. Maybe the (simpler??) formula I can't see is an average over the power stroke, or something like that? The "standard" torque equations typically have terms like "BMEP" which is a kind of average cylinder pressure.

 

[cjmdjm]

Ok, I attached the formula that they come up with in that document. I am just curious how it is derived, because it is somewhat simpler than the formulas I can come up with. Lol, in the document they say, "From simple geometry." I wouldn't really call it simple though.