I hate these things, they keep me up at night. :)
Lets do some math. I will state some constants for the sake of ease.
20HP @ 5000 Engine RPM's
1st gear ratio of 15:29(1.93:1 ratio) which means at 5000rpm our CS is turning 2591rpm's (5000/1.93)
To figure torque of shafts, we need to first figure chain speed and working load.
The formula for figuring chain speed is:
Number of Teeth x RPM's / Pitches per Ft.
(I will figure for an 11t x 44t [4:1 ratio] and a 13t x 52t [4:1 ratio] on a 520 chain)
11t x 2591= 28,501
28,501 / 19.2(# of pitches per ft.) = 1484 Ft. per Min.
13t x 2591= 33,683
33,683 / 19.2 = 1754 Ft. per Min.
Now that we know the Ft. per Min of each sprocket we can now figure the Working Load of the two.
Working load is figured as follows:
HP x 33,000 / Ft. per Min.
(11t) 20 x 33,000 =660,000 / 1484 = 444.74 in. lbs. working load
(13t) 20 x 33,000 =660,000 / 1754 = 376.28 in. lbs. working load
Now we know the working load of each sprocket, we can figure the shaft torques.
Shaft torque for driving shaft (or CS) is as follows:
HP x 63,000 / RPM = CS shaft torque (expressed in in. lbs.)
20 x 63,000 = 1,260,000 / 2591 = 486.3 in. lbs. (this remains constant )
Shaft torque for driven shaft (or rear) is as follows:
Working Load x Pitch Radius = Driven Shaft Torque
(11t) 444.74 x 4.3805 (pitch rad. of 44t) = 1948.18 in. lbs. torque
(13t) 376.28 x 5.1755 (pitch rad. of 52t) = 1947.43 in. lbs. torque
What we see when we do the math using engineering formulas is that torque on the CS remains constant with a constant output of horse power. We also see that the torque on the rear shaft remains virtually the same between the two drive set-ups at the same ratio.
What we see different is the working load that each set-up creates. THIS is where the problem lies in going to a smaller CS. The additional working load is directly transferred to the chain links, and as a result, a smaller CS sprocket puts a greater load on the chain. We also have a lesser number or sprocket teeth to accomplish the same amount of work, so both the chain and the sprocket teeth have to work harder, and as a result, wear out faster with the smaller set-up.
I don't have an engineering degree, I'm just a geek who reads and thinks too much, so if any engineers want to correct me, please feel free. :)