HOW CURVE FRICTION HAPPENS ON MODEL RAILROADS
Geometry of the rails:
For a rigid, un-coned, wheelset on a circular curve, the inner wheel will travel a distance of 2 x Pi x the radius of the inner rail, per each full circle of turn. Similarly the outer wheel will travel a distance of 2 x Pi x the radius of the outer rail, per each full circle of turn. The distance for a smaller angle of turn is found by multiplying the result by angle/360.
The difference in distance traveled must be the distance one of the wheels ended up slipping, or the combination of each wheel sharing the slipping some of the time. So the total distance slipped by the equivalent of a single wheel is 2 x Pi x (Outer rail radius minus Inner rail radius) x angle of turn / 360.
The first conclusion is that the radius difference between the rails for a railroad is of course is the track gauge, so the overall radius of the turn becomes irrelevant. The slipping distance for each wheel is only proportional to its total angle of turn!
Applying this to the slipping wheels situation and still assuming no wheel coning:
The wheel slip will occur on all wheels of that/those part(s) of a train instantaneously travelling around any curved portion or portions of track. The amount of slip is proportional to the total angle of turn, or the sum of the individual amounts of slip, of those parts of a train on multiple separated angles of turn. Since friction force occurs regardless of slip direction, the slip on any combination of multiple and/or reverse curves and different parts of a train is a cumulative.
Considering train speed, rate of change of turn and deriving force by:- work done = force times distance.
So the total wheel slippage of a train is the sum of the individual wheel slippages, but that should be calculated over a defined distance, bearing in mind that not all wheels may be experiencing the same rate of change of angle of turn, at the same time. Parts of the train may be traveling on curves of different radii, which means the rate of turn per distance traveled of those parts is a variable.
For example the extra work overcoming slipping friction done by a short train that entirely fits on an 18” radius 180 deg curve is likely the same extra work done by the SAME TRAIN being pulled around a 36” radius 180 deg curve. But the first work was done over half the distance. Which would infer that the sliding frictional force on the train due to slippage would be twice as much on the 18” radius curve, even though the amount of physical slippage should be the same
Moving to my assumptions for friction:
For our case, static friction is the force needed to start something stationary and heavy on a “rough” flat surface, moving horizontally. Friction on a rough surface is a constant determined by the roughness of the materials involved, but is then proportional to the weight of the object. For multiple objects, the frictional force is the sum of the friction of the individual objects. Usually, once moving, the initial sliding (slipping) friction force is less than the stationary force, but does not necessarily increase in direct proportion to the speed of movement. (or sometimes apparently much at all). Experimental testing, rather than calculation, is often the simpler way to estimate or measure the moving friction forces.
This complicates things:
It may be, that the force to move a slipping wheel may become constant, once the wheel is slipping fast enough to easily slide over the rail. However, that can be relatively easily measured by pulling an experimental train with locked wheels along straight track, using a spring balance or strain gauge to determine the extra pulling force needed compared to the same train with unlocked wheels. Since only one wheel of each axle slips on a curve, the locked test train, with both wheels of each axle locked and slipping, needs only half the number of cars to generate the same force as the same train on a curve.
However, the rate of slip will be different, so the measurement must be done at the same rate of slip. This means the work is measured, over the same effective slipping distance and the equivalent force derived, rather than just measuring the instantaneous force. We do know the slip distance from the geometry analysis, as above, so in practice a reasonable estimate of the slip friction force can be found.
Finally, considering the effect of coning.
For the prototype, with its usually very large curve radii, wheel coning virtually eliminates wheel slip, except where the terrain forces tighter radii. And these effects are usually documented and allowed for in terms of reduced maximum train weight/length restrictions set for every US railroad route.
For model railroads, we need to calculate minimum radius for coning to work, in order know at what point wheel slippage becomes a problem, even with coning.
Just considering NMRA Standard HO and code 110 wheels, with a 1:20 prototypical coning. The amount of sideways wheel movement (gauge - BB - 2 x flange width) is (0.649-0.635) or 0.014”.
A wheel diameter equivalent to 33” in HO is 0.379”. So the effective diameter difference between the two wheels when running fully sideways on a curve is 0.379 x 0.014 / 20 or 0.00027”.
Since 0.00027” in represents the around the curve distance difference for a wheel of size 0.039, that will be the same ratio as the difference between the track gauge to the inner rail curve radius at minimum radius. So dividing 0.649” by 0.00027” = 1404” as the minimum HO radius for coning to be 100% effective at canceling out wheel slippage. For the common 24” radius, then coning will only fix 24 / 1404 = 1.7% of the slippage effect.
Other factors such as wheel bearings:
I have not considered here the possible effect of running heavy and or long trains around curves such that the coupler tension pulls the cars strongly towards the inner rail and thus may have some effect on the friction of the inner side pin-point bearings. There are a lot of unknown variables just in the understanding of the friction and construction variations of the bearings themselves. So it is not obvious whether such effects might be significant or not compared to better understood wheel slippage friction.
I would think that it would be best to attempt to measure the relative extent any such effects, before spending time creating a workable theoretical predictive model.