Calculating Tractive Effort for steam locomotives
When modeling a set of train vehicles, an important information needed for the dynamics of a locomotive is "(max) tractive effort" (TE, resp max TE), which is determining the ability of a locomotive to pull up a train. This characteristic detail is somewhat mysterious, and is unfortunately missing from locomotive descriptions most of the time, be them derived from the internet or even from railway books.
The main problem with TE is that it cannot be derived automatically by TTDPatch, although there´s a "default value" for it, calculated from the weight of the locomotive. But this is only a very over-simplified model, taking into account only the maximum transferable force between driving wheels and rails for a fixed factor of friction (µ = 0.3). It neither takes into account the TE generated by the locomotive nor the special limits put in effect by the characteristics of the engine´s drive or its "adhesive weight". Moreover, special building schemes of steam locomotives, like Mallets, Fairlies, Kitson-Meyers, ... cannot be taken into account in that way.
Now, since a couple of months, I´m able to calculate correct TE from steam locomotive data detail, which is given e.g. on J.D.H. Smith´s steam locomotive tables
This calculation is very accurate. I´m owning parts of the official technical manuals of the DRG (steam locomotives) which contain both calculated and measured TE values for german steam locomotives, and I have repeated that calculation with the needed technical data (cylinder dimensions, boiler pressure, ...). The outcome is very accurate (error < 1%), so it should be a feasible method to calculate TE for every
So, what´s the benefit for .grf set authors?
I´ve already calculated some french steamers for Snail´s French Set and some serbian steamers for Wile´s Serbian Set and apparently, most of the data for TE (which had been derived from the internet before), have been shown to be wrong.
Moreover, the neat effect of the calculation is that every steam locomotive of every .grf set would get comparable numbers. No more "behemoths", which in real life shrink down to only moderate engines, and vice versa. Then, we have those american engines, where TE is given in pounds and, in addition to wrong numbers for TE at all, the conversion to kN introduces even more uncertainty.In fact, we´re now able to calculate correct TE for every steam locomotive!1. For those who like to do it yourself:
Tractive effort (TE) (or "tractive force") is in principal determined by boiler pressure, cylinder proportions, and size of driving wheels:
TE = D² * S * const * p / d
with D = cylinder diameter, S = piston stroke, p = boiler pressure, and d = driving wheel diameter. ("const" being a heuristic constant in the range 0.75 ... 0.85, depending on speed.)
Because of the nature of the limited friction (µ) between wheel and rail, TE is limited by the weight on the driving wheels ("adhesive weight", W_adh), i.e. non-driven and slipping driving wheels cannot transmit force. In addition, friction depends heavily on the condition of the rails: clean, dry or sanded rails increase friction, but grease, ice, mud, leaves, etc. will all cause the locomotive to slip before nominal TE is reached. Thus, to transmit TE as best as possible, the quotient W_adh / W_tot should be as large as possible, which would be best achieved by a locomotive having only driving axles (0-x-0 scheme, W_adh / W_tot = 1).
[This is an excerpt from http://www.tt-forums.net/viewtopic.php?p=573658#573658
Let´s have an example (SNCF 141TD):
I´ll use a slightly different formula which the DRG used for "plain steam" in contrast to "compound steam" in its technical manuals:
TE = 0,8 * p * D² * S * z / 2d (for plain steam)
("z" being the number of cylinders)
According to J.D.H. Smith´s tables, numbers for the french 141TD are as follows:
d = 1420 mm -> 1.42 m (driving wheel diameter)
D = 510 mm -> 0.51m, D² = 0.2601 m² (cylinder area)
S = 660 mm (3) -> 0.66 m (cylinder stroke)
p = 16 atm -> 1621200.384 N/m² (boiler pressure)
TE = 0.8 * 1621200.384 N/m² * 0.2601 m² * 0.66 m * 3 / (1.42 m * 2) -> 235 kN
This is the "nominal maximum tractive effort".
I.e., for an adhesive weight of 75t and a friction coefficient µ between 0.25 (wet, slippery) and 0.33 (dry, clean) we get:
TE_max = 75 t * 9.81 m/s² * [0.25 ... 0.3 ... 0.33] -> 183 ... 220 ... 242 kN
[This is an excerpt from http://www.tt-forums.net/viewtopic.php?p=573863#573863
Let´s have an example (MAVAG 601):
This one is a four-cylinder compound engine, hence:
TE = (2 * 0.85 * p * D_hp² * S) / (D_hp²/D_lp² +1) * d
d = 1440 mm -> 1.44 m (driving wheel diameter)
Dhp = 520 mm -> 0.52 m, Dhp² = 0.2704 m² (hp cylinder area)
Dlp = 850 mm -> 0.85 m, Dlp² = 0.7225 m² (lp cylinder area)
S = 660 mm -> 0.66 m (piston stroke)
p = 16 atm -> 1621200.384 N/m² (boiler pressure)
TE = (2 * 0.85 * 1621200.384 * 0.2704 * 0.66) / (0.2704 / 0.7225 +1) * 1.44
TE = (491854.04) / 1.9728 -> 249 kN
TE_max = 94.7 t * 9.81 m/s² * [0.25 ... 0.3 ... 0.33] -> [232 ... 278 ... 306] kN
[This is an excerpt from http://www.tt-forums.net/viewtopic.php?p=709546#p709546
]2. For those who don´t like to calculate TE yourself
Simply post the country and name resp. class number of the steamer in question here and I´ll do the calculation for you.
Please note that calculation of TE for diesels and electrics isn´t possible in this way.