Resistance
Locomotive and Train Resistance
A locomotive’s tractive force is required to overcome the resistance to motion of both locomotive and train. When the tractive force is greater than the resistance, then the train will accelerate in accordance with Newton’s law of motion: Force = Mass x Acceleration or Acceleration = Force ÷ Mass. If the tractive force is equal to the resistance, then the train will travel at constant speed. If the tractive force is less than the resistance, then the train will slow down.
There are various forms of resistance that need to be considered.
Locomotive Resistance

Starting resistance, associated with static friction which is usually higher than dynamic friction – i.e. it needs to be overcome before the locomotive will start moving;

Internal (mechanical/frictional) resistance from the locomotive’s motion etc;

Rolling resistance deriving from axle bearings and wheels on rail – it increases in proportion to the speed;

Wind resistance increases in proportion to the square of the speed;

Gravitational resistance when on an incline – equals the locomotive weight x the gradient;

Rail curvature resistance applies only to curved track – being proportional to the curvature of the track (or inversely proportional to the radius of curvature).
Train Resistance

Starting resistance, associated with static friction which is usually higher than dynamic friction – i.e. it needs to be overcome before the train will start moving. In the case of roller bearing stock, this value is close to zero, but can be significant where journal bearings are fitted to wagons;

Rolling resistance deriving from axle bearings and wheels on rail – it increases in proportion to the speed;

Wind resistance increases in proportion to the square of the speed;

Gravitational resistance when on an incline – equals the train weight x the gradient;

Rail curvature resistance – being proportional to the curvature of the track (or inversely proportional to the radius of curvature), but applying only to the length of train on the curved track.
In addition, inertial resistance has also to be considered (e.g. when a train is being accelerated).
The term “specific resistance” means the resisting force per tonne of (train) weight. It is pertinent to note that the specific resistance of empty freight trains and of passenger trains is much higher than that of loaded freight trains. This is largely because the wind resistance of an empty or lightly loaded train is the same as or higher (see below) than that of a full train. Since the wind resistance becomes dominant at higher speeds, the total resistance per tonne of train weight becomes higher. In the case of opentopped wagons, the wind resistance of empty wagons is higher than of full wagons because of the increased in air turbulence inside and around the empty wagon bodies. This is illustrated in the graphs at the bottom of this page.
It might also be mentioned that wind resistance is higher when the wind comes from an angle to the direction of the train than when the wind is directly headon to the train. This is because the wind has more impact on the ends of each wagon or carriage than when the wind is directly in front of the train where slipstreaming from one wagon or carriage to the next reduces drag.
Train resistance can be measured by the force in the coupling connecting the locomotive to its train (e.g. using a dynamometer car, or in latter days using an electronic load cell). Locomotive rolling resistance is harder to measure.
There are no exact methods for estimating locomotive or train resistance and reliance has to be placed on empirical formulae based on measured values. Almost every country’s railway has its own formulae for estimating rolling resistance, all (presumably) being based on recorded measurements.
Most resistance formulae are divided into three sections:

a fixed constant, representing a fixed part of the rolling resistance,

a term proportional to the train speed, representing a variable part of the rolling resistance, and

a term proportional to the square of the train speed representing wind resistance.
One of the more recent national train resistance formula to be developed was the Serbian one that was formulated in 2005 based on tests undertaken by M. Radosavljevic from the Department of Mechanical Engineering, the Institute of Transportation in Belgrade. He wrote a paper on his findings entitled “Measurement of Train Traction Characteristics” published in the Proceedings of the Institution of Mechanical Engineers Vol 220 Part F: J. Rail and Rapid Transit. A copy of the formulae and their graphical representation is copied below. (See below for further comparisons)
The above formulae are compared with additional train resistance formulae in the Excel graph below [click on the image for a fullsized PDF image.] The additional formulae covered are as follows

China National Railways (full wagons)

China National Railways (empty wagons)

Canadian National Railway (full wagons)

An (unnamed) Australian mineral railway (full wagons)

An (unnamed) Australian mineral railway (empty wagons)

Koffman’s formula for BR carriages as used by Wardale in FDC 1.1.
It will be readily seen that (as explained above) the specific resistance of empty wagons is much higher than that of full wagons and that for BR Carriages is also high by comparison with loaded wagons (due to their relatively light loaded weight). On the other hand the majority of formulae give very similar estimates for the specific rolling resistance of full wagons, with the exception of the (unnamed) Australian mineral railway figures which show much lower figures. This is probably because the wagons are much larger and more heavily loaded than those in the other countries listed.
Wardale’s Resistance Calculations
In FDC 1.1 (line 24) and FDC 1.2 (line 2) Wardale uses the following formulae for calculating locomotive and carriage rolling resistence (respectively):

Locomotive rolling resistance: R ≈ (45 + 0.24v + 0.0036v2) N/Tonne

Carriage rolling resistance: R = (1.1 + 0.021v + 0.000175v2) kg/tonne – from Koffman applying to BR coaches.
where R is the specific rolling resistance in N/Tonne of weight and v = speed in km/h.

In FDC 1.3 line 164, Wardale quotes Koffman to assume a specific starting resistance on level tangent track for roller bearing stock of 7.0 kg/tonne (= 6.99 N/kN).

In FDC 1.3 line 166, Wardale uses a figure of 100 N/tonne (=10.2 N/kN) for the specific starting resistance for a locomotive.
Chinese National Railways Resistance Calculations
For the record, various resistance formulae used by Chinese National Railways are summarized as follows, using the notation:

R = resistance in Newtons

W = weight in kN (not tonnes)

V = speed in km/h

Grade = slope in ^{o}/_{oo} [e.g. 1 in 100 = 10^{o}/_{oo}]

Rad = curved track radius in metres.

L_{t} = length of train in metres.

L_{c} = length of curve in metres; L_{c }= L_{t} where L_{t} ≤ L_{c}
Rolling Resistance of Locomotives – in Newtons:

QJ with 6 axle tender; R= W x [(1.09 + 0.0038 V + 0.000586 V^{2}) + Grade + 600/Rad]

QJ with 4 axle tender: R= W x [(0.70 + 0.0243 V + 0.000673 V2) + Grade + 600/Rad]

JS: R= W x [(0.74 + 0.0168 V + 0.000700 V2) + Grade + 600/Rad]

SY: R= W x [(0.74 + 0.0168 V + 0.000700 V2) + Grade + 600/Rad]
Rolling Resistance of Wagons – in Newtons:

Roller Bearing Wagons: R= W x [(0.92 + 0.0048 V + 0.000125 V2) + Grade + 600/Rad x L_{c}/L_{t}]

Journal Bearing Wagons: R= W x [(1.07 + 0.0011 V + 0.000236 V2) + Grade + 600/Rad x L_{c}/L_{t}]

Empty Wagons: R= W x [(2.23 + 0.0053 V + 0.000675 V2) + Grade + 600/Rad x L_{c}/L_{t}]
Starting Resistance

Locomotives: 8 N/kN (weight)

Roller bearing wagons: 3.5 N/kN

Journal bearing wagons: 5 N/kN
Canadian National Railway Resistance Formulae
Chapter 2.1 of AREMA (American Railway Engineering and MaintenenceofWay Association) Manaul for Railway Engineering, dated 1999 and titled “Resistance to Movement” provides data and formulae used by Canadian National Railways. The document can be downloaded from this website by clicking here.