I got a PM recently asking if the above tables were finalized. Actually I had planned to post about an alternate way to calculate the parameters P1 & P2. The formulas are a lot simplier because we don't have to calculate the minimum radius that is associated with a given allowed track speed and superelevation/cant angle. You will have to determine that some other way if you are following prototype design and operation practices.
The following is the post that wasn't included when I posted the 2 main tables. This approach ignores any reference to track speed, though allowed track speed is the main reason for adding superelevation. The accelerations that affect safety and comfort as a train travels around a curve are directly related to train speed and track curvature. But if you just want to apply a specific superelevation to a curve, which in Trainz is expressed as a cant angle, it is possible.
I'll add my definition of the 2 parameters for what it's worth:
P1 is the ratio of the cant angle to the horizontal curvature the track. Units are radians-meters
P2 is the upper limit for the cant angle that Trainz will apply to a local track section. Units are radians.
If we know a curve has a specific value of superelevation based on track data of the prototype railroad or we just want to set a specific superelevation value without any regard to prototype speed on the curve, we can set the P1 and P2 parameters using the following:
Start by calculating the P2 parameter:
Code:
P2 = Ed/g Ed is the desired superelevation on the curve
g is the track gauge
Use the same units for both Ed and g: in and in or mm and mm. The result is a ratio that approximates the cant angle in radians. Strictly speaking P2 is the arcsin(Ed/g) but for small angles measured in radians the approximation is accurate enough. The supplementary tables in the above 2 posts show values of P2 for various superelevations on standard gauge track (see last comment in post above for info on distance between contact points vs track gauge).
Next calculate parameter P1 by multiplying P2 by the radius of the curve (in meters) that we want to have the cant angle P2. In Trainz Surveyor module we can measure the radius of the curve in question at a few points and pick a radius, Rmax, that is somewhat larger than the largest one we measure:
Code:
P1 = P2xRmax P2 is an angle in radians
Rmax is a radius in meters
The following is provided for information but is not necessary to calculate the 2 parameters.
Spline track in Trainz doesn't create constant radius curves- mathematically it can't. It can approximate them but the radius will vary in some fashion along the curve. Hence the suggestion to measure the radius at a few points and pick a max value. Trainz calculates the cant angle to apply by multiplying parameter P1 by the curvature of the track at that point. Since curvature is equal to 1/R (in meters), for any point where the radius is less than the one used to calculate P1 the calculated cant angle will be greater than P2 and Trainz will therefore use the limiting value, P2 for the cant angle at this point.
Using P1 & P2 as described above will assign a cant angle P2 to any curve that has a radius equal to or less than Rmax. Using these P1 & P2 values on curves with radii larger than Rmax will result in superelavations that range from Ed at Rmax down to 0 on tangent (straight) track. That might be ok for your needs or on larger radius curves you can just recalculate appropriate P1 & P2 values to apply and Trainz will calculate the cant angle you want it to use.
Bob Pearson
PS. An example: I have a curve in my route that I know the prototype used a superelevation value of 5.0 inches and I want to have Trainz use that same value on the curve.
I measure a few values of the radius and get the following values: 225, 250, 260, 240 & 220 m. I set Rmax to 270 m.
P2 = 5.0 in / 56.5 in = 0.0885 radians
P1 = 0.0885 x 270 m = 23.90 radian-meter
Checking the tables in post 1 above I'll note that the allowed speed on a curve with 270m radius and Ea = 5.0 in would be approx 42 mph.
Example 2: I have a curve in my route that I want to have a superelevation value of 180 mm.
I measure a few values of the radius and get the following values: 1975, 2000, 2015, 2005 & 1988 m. I set Rmax to 2020 m.
P2 = 180 mm / 1435 mm = 0.1254 radians
P1 = 0.1254 x 2020 m = 253.30 radian-meter
Checking the tables in post 2 above I'll note that the allowed speed on a curve with 2020m radius and Ea = 180mm would be approx 208 km/h.
(P2 =0.1254 -> Ke = 1.1487 -> P1 = 253.3/1.1487 = 220.51 -> V = 208 or by formula in note3)