[jtr1962]

WaltN, on 16 September 2018 - 09:13 AM, said:

I agree with your first sentence, except that plot is for one radius -- 100 meters. The elevation discrepancy will grow proportionately with radius.

My preliminary research on this suggests 10 meter sections, regardless of radius, for a really steep grade, such as the 3° maximum of MSTS. You can relax that proportional to the grade. For example, 100 meter sections on an 0.5% grade should give good results.

[WaltN]

jtr1962, on 16 September 2018 - 10:49 AM, said:

My preliminary research on this suggests 10 meter sections, regardless of radius, for a really steep grade, such as the 3° maximum of MSTS. You can relax that proportional to the grade. For example, 100 meter sections on an 0.5% grade should give good results.

What's your criterion for acceptability?

[jtr1962]

WaltN, on 16 September 2018 - 11:18 AM, said:

What's your criterion for acceptability?

~0.1% or smaller deviation from the theoretical elevation change. See post #116 where I did an experiment using 5 degree sections to make a 120 meter radius curve at 3° elevation. Those sections turn out to be 10.472 meters long. The deviation from theoretical is only 0.12%.

I haven't subjected any of this to rigorous analysis to determine the maximum track length versus radius and gradient. I'm just extrapolating from my findings. If ~10 meter sections work on an ~5% gradient, then 100 meter sections should work on an 0.5% gradient.

[WaltN]

Just to close out my particular contributions to this topic, I'd like to develop the proper equations for circular arc curves on a grade. We start with the three equations we developed to start the Rigid-Body form, x’, y’, and z’, except we can drop the “prime” embellishment because we won’t need it.

x = -R (1 - cos ϴ)
y = 0
z = R sin ϴ

We’re going to modify the second equation for y to reflect a constant grade. So far, it represents the projection of the curve in the x-z plane.

First, we need to develop some notation to describe the grade. As you know, a grade is defined as a ratio, rise / run. Let’s call that fraction G. The rise at arbitrary point P is just y. The run at P is just R ϴ meters, but that’s only true when ϴ is expressed as radians. As well, the total run is R ϴ_T when ϴ_T is expressed as radians. Of course, R and ϴ_T are what MSTS uses to characterize circular arc curves.

Let’s take a couple of examples illustrating different ways of specifying grades. First up is the MSTS way, using the grade angle in degrees. Let’s consider 3°. G = rise / run = tan 3° = 0.0524.

Another way is as a percentage, say 3%. G = 0.03. It’s that simple!

Now let’s develop the function that generates the change in elevation, y, as a function of parameter ϴ. We know that at the end of the curve y_T = G R ϴ_T. Note that factor G, the grade parameter, scales R ϴ_T, the length of the curve in meters (in the horizontal plane).

But what we want is a linear increase in y as we progress around the curve. We can achieve that by weighting the arc length progressed with respect to the total arc length –

 Untitled-1.gif (5.63K)
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Note that the two factors (R ϴ_T) cancel out, leaving:

y = G R ϴ

As simple as that!

Plugging this function into our set of three equations:

x = -R (1 - cos ϴ)
y = G R ϴ
z = R sin ϴ

Note that these three equations are a lot simpler than the rigid-body equations, perhaps even elegant.

Let’s see if they yield correct results. Here’s a plot of the same fashion as before. I plotted points and lines so that you could see the plot interval –

 Helix.gif (4.09K)
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Note that the slope of the points is the same throughout, and, overall, the curve achieves its full specified elevation. And there is no twist, unless superelevation is superimposed.

Q.E.D.

[MLProject]

Man, when I saw those Blender made-tracks, damn... they look good, will be there any tutorial of how to make it? (Really could be helpful)