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Rolling resistance is a measure of the amount of effort required to pull a train. It is usually expressed as a ratio of the total train weight. This whole topic has been the subject of many theoretical papers, but basically reduces down to how heavy the train is, how fast it is going and how steep a hill it is ascending, with other secondary effects like bearing and air resistance.
This rolling resistance must be overcome by any locomotive hauling a train, and is determined by the following factors: the type of track, the curvature of the track, the speed of the train and the gradient of the track. The Rolling Resistance Coefficient, or RRC, is usually expressed in kg/1000kg (of train weight). For example, if the RCC is 0.02 and the train weight is 1,000kg, it will require a force of 0.02 x 1,000kg = 20kg to move that train. The RCC factor has components for straight track, curved track, gradient and speed. It is expressed as a coefficient or ratio, not an absolute number.
For most purposes the type of track is constant, and usually steel (or aluminium) rail on sleepers. The RCC (straight track) is typically around 0.005~0.008.
When there are curves in a track, it requires more effort to move the train around the curve, and the sharper the curve, the more effort required. The RCC for curves is around 0.004 to 0.007 per degree of curvature.
The RCC gradient component obviously depends on the gradient of the track. Gradient is specified as the distance the track rises per unit of distance travelled. Refer to the diagram below:
The steeper the grade, the more effort required to lift the train vertically that distance. Gradient is expressed as 1 in X, or in percent. For example if the height (H) is 1, and the distance (D) travelled is 50, the grade is 1 in 50 (or 1:50), or as a percentage 1/50 x 100% = 2%. The gradient component of RCC is usually around 0.01 times the grade in percent.
The final part of RCC is due to the speed of travel. As speed increases, things like air resistance increase as the square of the speed. Other things like rolling friction of bearing etc also increase with speed. The speed component of RCC around 0.0002 times the speed squared. Although for models, air resistance is negligible, the other speed related factors are not.
Another factor to be taken into account is that as things get bigger, it is generally more efficient, or conversely as things get smaller, they require more effort in proportion to their size. This is due to things like bearing resistance does not scale, and other factors as well. Suffice to say that model trains usually present a drag figure a little higher that expected. It is a fudge, but assume say a 20% loading for models.
To sum up all of the above, it is possible estimate the amount of effort required to move a train as a desired speed (and hence the minimum tractive effort required from a locomotive to do this). Refer to the calculator below. If you input the total train weight, steepest grade, sharpest curve and desired speed, you can calculate the drawbar pull required.
Note the accuracy of these calculations is not necessarily exact, and figures calculated here may vary from actual results.
Results to be used at your own risk. Calculator provided for your personal use only.
These figures give an estimation of the effort required to pull a train and keep it moving. The effort required to start a train from rest is approx 15~20% higher than that required to keep it moving.
[Updated: 27 Jan 2013] 12 |
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