by

**Allen Hazen**Timz:

You say

"When Alco/UP said the 4-8+8-4 had 135,375 lb TE (which is what you get using 0.85-- cyl 23.75 by 32, drivers 68, pressure 300) they probably figured everyone could apply whatever roller-bearing correction they wanted to that figure."

I think you're right. By the time these locomotives were built, everyone who mattered knew that nominal tractive effort wasn't really a very important figure-- that at best it was a very rough indication of locomotive performance at low speeds-- so they didn't agonize about finding a better formula for its calculation!

... As to the N&W A class: My bad. I didn't check the arithmetic and thought the A.R.E.A. formula LeMassena referred to was the one we'd been discussing.

Thanks for the information about the A's maximum cut-off. That (see below) ought to cut the nominal t.e. a bit, but maybe not all the way down to 105,000.

---

And now, some fun numbers. Consider the Pennsylvania Railroad's I1s and I1sa 2-10-0. Cylinders 30.5" diameter, 32" stroke, boiler pressure 250 lbs/sq.in., driver diameter 62". The formula (if I've done my arithmetic right: you might want to check) with a "coefficient" of .85, gives a t.e. of 102,000 pounds, which would probably have been too much to be useful, given an I1's weight on drivers...

Published t.e., however, was 90,024 pounds for the original I1s design and 96,026 for the modified I1sa. I think the differences are attributable to limited cut-off: virtually the ONLY difference between the I1s and the I1sa was that the I1s had 50% cutoff and the I1sa had 78%. The numbers make sense to me (on my thoroughly amateurish understanding of the physics involved).

(((I think real-world effective cut-offs for locomotives without "limited cut-off" as a stated design feature would be a bit under 100%, and that the .85 coefficient in our formula takes this, among other things, into account. My description of the coefficient = 1 version of the formula -- steam exerting full boiler pressure on piston for full length of travel -- describes an idealized, probably unattainable, 100% cut-off. For 50%... think of steam as going into the cylinder right as the piston starts to travel (probably in the real world this isn't QUITE what happens, but suppose we get as close as practically possible to this ideal). For the first 50% of its travel, then, the piston has full-pressure steam pushing it. Then the admission of steam is cut off (to coin a phrase). The piston continues on its way, still pushed by the steam that is already in the piston, but pressure drops as the steam expands to fill the increasing volume: at the end of piston travel, the volume is twice what it was when steam admission was cut off, so the pressure is down to half of full pressure.

Now imagine a graph of the the force exerted on the piston: x-axis represents the motion of the piston, y-axis the pressure. From 0 to the 50% point, the graph will be level at the [[boiler pressure times piston area]] value, after which it will slope down, reaching half that value at the 100% point. Tractive effort is an average over the whole cycle, and what matters is the area under the graph. So a locomotive with 50% cut-off ought to have a t.e. of 87.5% that of a locomotive with -- practically unattainable -- 100% cut-off. I haven't worked out numbers in detail, but I think the differences between the I1s, the I1sa, and the imaginary 102,000 pound t.e. locomotive are about what this model suggests.)))

If someone with real engineering knowledge is reading this and I've made a bad blooper, I hope they'll correct me!

You say

"When Alco/UP said the 4-8+8-4 had 135,375 lb TE (which is what you get using 0.85-- cyl 23.75 by 32, drivers 68, pressure 300) they probably figured everyone could apply whatever roller-bearing correction they wanted to that figure."

I think you're right. By the time these locomotives were built, everyone who mattered knew that nominal tractive effort wasn't really a very important figure-- that at best it was a very rough indication of locomotive performance at low speeds-- so they didn't agonize about finding a better formula for its calculation!

... As to the N&W A class: My bad. I didn't check the arithmetic and thought the A.R.E.A. formula LeMassena referred to was the one we'd been discussing.

Thanks for the information about the A's maximum cut-off. That (see below) ought to cut the nominal t.e. a bit, but maybe not all the way down to 105,000.

---

And now, some fun numbers. Consider the Pennsylvania Railroad's I1s and I1sa 2-10-0. Cylinders 30.5" diameter, 32" stroke, boiler pressure 250 lbs/sq.in., driver diameter 62". The formula (if I've done my arithmetic right: you might want to check) with a "coefficient" of .85, gives a t.e. of 102,000 pounds, which would probably have been too much to be useful, given an I1's weight on drivers...

Published t.e., however, was 90,024 pounds for the original I1s design and 96,026 for the modified I1sa. I think the differences are attributable to limited cut-off: virtually the ONLY difference between the I1s and the I1sa was that the I1s had 50% cutoff and the I1sa had 78%. The numbers make sense to me (on my thoroughly amateurish understanding of the physics involved).

(((I think real-world effective cut-offs for locomotives without "limited cut-off" as a stated design feature would be a bit under 100%, and that the .85 coefficient in our formula takes this, among other things, into account. My description of the coefficient = 1 version of the formula -- steam exerting full boiler pressure on piston for full length of travel -- describes an idealized, probably unattainable, 100% cut-off. For 50%... think of steam as going into the cylinder right as the piston starts to travel (probably in the real world this isn't QUITE what happens, but suppose we get as close as practically possible to this ideal). For the first 50% of its travel, then, the piston has full-pressure steam pushing it. Then the admission of steam is cut off (to coin a phrase). The piston continues on its way, still pushed by the steam that is already in the piston, but pressure drops as the steam expands to fill the increasing volume: at the end of piston travel, the volume is twice what it was when steam admission was cut off, so the pressure is down to half of full pressure.

Now imagine a graph of the the force exerted on the piston: x-axis represents the motion of the piston, y-axis the pressure. From 0 to the 50% point, the graph will be level at the [[boiler pressure times piston area]] value, after which it will slope down, reaching half that value at the 100% point. Tractive effort is an average over the whole cycle, and what matters is the area under the graph. So a locomotive with 50% cut-off ought to have a t.e. of 87.5% that of a locomotive with -- practically unattainable -- 100% cut-off. I haven't worked out numbers in detail, but I think the differences between the I1s, the I1sa, and the imaginary 102,000 pound t.e. locomotive are about what this model suggests.)))

If someone with real engineering knowledge is reading this and I've made a bad blooper, I hope they'll correct me!