Weights, Gradients & Tractive Effort

As I run a 00 garden railway with gradients of up to 1 in 50, I have developed a bit of a preoccupation with the weight of rolling stock and the pulling power of locomotives. This has led to the building of a simple locomotive test rig, and the generation of a spreadsheet to convert the test results into "number of coaches up 1 in 50" for each of my locomotives. I have later compared the results with the actual performance achieved in the garden, and - purely out of scientific interest - the tractive efforts  and weights of full-size locomotives and stock. It also allows me to try to improve poor performers by trial and error, wihout having to keep taking 10 coaches into the garden to see if it has worked.

First the test rig. A short length of track (1m is more than enough) is attached to a level base with a vertical drop of at least 25cm at one end. The edge of the drop is rounded, and covered with smooth tape. A length of cotton running over the edge has a loop at one end, and to the other end is attached a cradle weighing between 5 and 10 grammes. (I used a plastic lid from an aerosol can.) You also need up to 10 identical weights of between 5 and 10 grammes. (I used 8g steel balls from a bearing.)

Place the loco on the track, and attach the cotton loop to its coupling hook. Start it running so that it lifts the cradle off the ground. Keep repeating, each time adding another ball, until the loco can't lift it. Take out  one ball, and the weight of the cradle plus balls is the loco's maximum tractive force. In some models this can be improved by adding or moving weight within the loco, or adjusting bogie springing, to optimise the weight on the driving wheels

To convert this figure into "coaches up 1 in 50", first decide on the average weight of a coach. I use a figure of 150g per coach, which seems fairly typical. The number the loco can pull up a 1 in 50 gradient is then

[(tractive force x 50) -(weight of loco & tender)] / coach weight

assuming that losses due to rolling resistance and track curvature can be ignored. If your steepest gradient is 1 in 80, replace the 50 with an 80. I have found in practice that the results usually agree to within one coach with what is achieved in the garden, though I suspect that curves of radius less than about 4ft might start to have a significant effect.

Before proceeding to a comparison with full-size equipment, let me introduce the concept of "scale weight". If a full-size item weighs 35 tons, the "scale weight" of its 00 model is

35 x 1000000 / (76 x 76 x 76) grammes, = 80g.

This assumes 1 ton = 1 tonne, and 00 is 1/76 scale, both of which are true within the accuracy of our measurements. Thus the scale weight of an 87 ton 9F 2-10-0 is 198g, and the scale weight of a 55 ton Midland 2P 4-4-0 is 126g. This compares with an actual weight of 395g for the Bachmann 9F, and 170g for the Hornby (loco-drive) 2P, which is 2x the scale weight for the 9F, and 1.3x the scale weight for the 2P.  Similarly, a typical 35 ton loaded mk1 coach weighs about 35 tons, giving a scale weight of 80g, yet a typical 00 model weighs 150g, almost twice the weight.

Why are the models so much heavier? There are several reasons.

1. A model has to survive handling by "giants", so the thickness of materials is much greater.
2. Stock must stay on rough track with no wheel springs or compensation, so weight is added.
3. Uncompensated locos need more weight to prevent wheel-spin with heavy trains.

The 2P can afford to be proportionately lighter than the 9F because it has rubber traction tyres.

The final concept to get our heads round is Tractive Effort. The figure we measured earlier using the test rig is effectively the tractive effort of the model loco - the force with which it can pull a train. Tractive Effort in the world of real railways, as anyone who has owned an Ian Allan ABC of steam locomotives will know, has the same meaning, but is usually quoted as a theoretical figure calculated from certain characteristics of a class of loco. I won't go into any more details here; it is explained in full in the front of the said ABC books, and the tractive effort of each class is given in the data tables in the same books. I will however use the same two locos, the 9F and the 2P, to make a comparison between the "scale" of the theoretical T.E. of the full-size engine, and the measurements made on my test rig. The 9F has a T.E. of 39670lb, which to 00 scale is 41g, compared with 35g, 39g, and 43g for the three Bachmann 9Fs I have measurements for. The 2P has a T.E. of 17730lb, which to scale is 18g, compared with 23g for the Hornby model. Not quite so close, but the model does have the advantage of rubber tyres. A key observation I haven't mentioned before is that I have yet to see the tractive effort of a modern 00 model limited by the power of its motor. In every case, it is because the wheels start to slip, and lose traction on the rails.

Well, what does all that prove? Was it worth the effort of writing it down? I'll leave it for you to judge.