OK, I was checking out the method described by John Armstrong in “Track Planning for Realistic Operation” by laying it out in AutoCAD.
The dashed line “Original Layout Line” is a plain curve with a radius but no easement. I constructed the the two arcs, R and R + X as described in the book. The blue line is the easeement line using a cubic spline function. Everything is fine to this point. The problem is that the departing line coming off the easement - and arc R - no longer coincides with my original line.
What are you supposed to do at this point: Move the entire curve up until the new line coincides with the original? Adjust your layout line to the new one? ???
My second question regards easements and basic radii. Armstrong says that using easements allows tighter basic radii than could be used with un-eased larger radii to give the same “lurch”. Can anyone report how this has worked on their layout?
No, that’s what I have, it’s just that the straight section I “splined” to is vertical and off the bottom of the page.
I haven’t tried a second spline to the line heading off to the upper right, if that’s what it takes.
Ignoring that for a minute, if my track ended about where the red R arrow points, the easement coming up from the bottom would still be off my original line. What is typically done to keep things where you want them after you get past the curve?
To address your last question, you must first accept that you could easily make do with two tangents at, say, 90 deg to each other, and make a constant radius curve between their ends. That works fine in snap track section with fixed radii. If you desire the easements, especially because you believe and understand what John Armstrong is telling you about getting away with somewhat sharper curves than heavyweight passenger cars with weather shields between them would take normally, then you have to modify that curve so that the cars ease their trucks and couplers into the strain of the tight(er) curve.
So, back to the same tangents with their ends at 90 degrees from each other and separated by, say 24", as the crow flies. You slip in a bent section of flextrack, slide the joiners into position, and you have that unwanted constant radius…not really, but let’s say it is essentially a constant radius. If you could shore up those two joints so they can’t move sideways (track nails, bricks outboard of the rails, whatever) and then push the very apex, or centre of the arc, in along the radius line at that point about 1/4", maybe up to 1/2", you will effectively generate an easement, but not necessariy a cubic spiral. So, as the train moves past the joints, it encounters a slight acceleration along the radius, but that acceleration changes as the radius tightens towards where you have tighened the curvature. The initial curvature is insufficient to meet the other tangent end at 90 deg, but by inducing much greater curvature at the apex of the curve, you do eventually meet that other tangent end, and you accomplished it with a radius at the apex that is a bit sharper than the minimum recommended for the rolling stock.
That is my understanding of John’s description of the purpose, if not his process. I just never used a tramel or lath, and relied on spline roadbed to “figure it out” for me.
The “Lurch” is what causes locomotives or long rolling stock to pull other cars off the tracks as they enter a curve. I placed a 85’ Heavyweight passenger car on a 33-1/2" curve and noticed the coupler hanging over the outside of the outer rail. This coupler was body mounted and thus could not be pulled back beyond the inside of the outer rail. It also meant that, without an easement, that car could not remain coupled to any bodly mounted or talgo (truck mounted) car that was still on the tangent. With a full, gradual easement, two of these cars could remain coupled together through the apex of a curve where the the radius went down to 30" or less.
Putting it a nother way, a 33-1/2" radius curve would place two parallel tracks 67" apart at the centerlines. Maintaining this 67" separation with full easements and the sharpest portion of the curve staying at 30" radius, you could operate 85’ cars with body mounted couplers.
The tangent has to be offset from the curve to allow room for the easement. That’s the significance of the “X” in “R+X.” If you don’t want to change the location of your tangent, offset the center of the curve by dimension X (along a line perpendicular to the tangent), and R+X will then align with the original straightaway.
Thanks Andy, moving the whole thing (up in this case) would put the new exit in-line with the old ROW, but I was thinking that trains coming off the other tangent need an easement too.
It looks like you really need to apply the easement simultaneously to both ends of the curve to get it right. (Which is what is really happening when you put a splint between pegs on your layout.) However, trying some combinations of departure angles and curve radii on CAD indicates that the geometry doesn’t always suit the constants in Armstrong’s method. For example, a short curve means that the point SC for one easement ends up well into the opposite tangent. Also, if you establish the end points as he indicates and make the center tangent to the offset R curve, the easement line does not “fall halfway between circle and tangent at midpoint M” like it should.
So, lemme ask: Is the critical aspect that the easement line fall between the original ROW and the offset line at the midpoint of the easement? If so, I think that in actual practice - vice AutoCAD - it’s easier to do than theory would indicate.
Another observation is that with easements, it’s almost impossible to have a curve leading right into a switch, or coming right off the normal (straight) leg because the easement runs through the points/frog. Does that seem right?
Let me also keep the second question alive: Has anyone successfully used eased curves to replace a larger radius?
Oh, Andy, I just use KL as a sig. I figure having my name as my user ID saves me ten letters per post!
This is an excellent web site about calculating easements. I was on the other 'puter yesterday and couldn’t find this. I have it as a shortcut on this 'puter.
I always found John Armstrong’s explanation of easement curves to a be a rare example where his writing was not clear enough for me.
Actually Andy Sperandeo was perhaps too modest to mention that his article on passenger train operations in the November 2006 issue of MR has a very good and clear (and brief) discussion of easements on page 56. We are of course talking about approximations or simulations of the true mathematics behind railroad easement curves. But the drawing with Andy’s article shows the offset tangent he is talking about very clearly.
There are mechanical methods rather than mathematical. A flexible stick fastened at one end and pushed at the far end will bend in a gradually increasing radius. Trace the bend with a pencil and you have a template for an easement curve, of sorts.
I am working up an article for MR or RMC or maybe Scale Rails on yet another mechanical method for simulating easement curves that I do not think has been used before but I am still working out the kinks. Stay tuned.
Here’s the trick. Yes, the sample shows only one side of the curve. The tangent at the other end of the curved section needs to be eased in as well. The sample diagram in Armstrong shows moving the tangent track by the offset and then easing into the curve. But you can do the same thing by moving the center of the curve’s arc - probably more likely since the location of your tangent tracks can’t move that much.
Or you can cheat for drawing purposes - use one of the MRR CAD programs that will draw the curves with easements for you. 3rd PlanIt does, I assume Cadrail and XTrack do as well. In 3rd PlanIt, you specify that you want to use curves with easements, and snap a line between the ends of the tangent pieces, and it fills in the curve complete with easements.
Agreed. Imo, the only thing the “use equation x, with formula y, to get answer z” way of doing this, is to take something incredibly simple, and make it incredibly complex for no reason.
Actually the various formulas model railroaders use for easements are approximations of easements and are NOT complex compared to what actual railroad engineering works with. I have a book called, if memory serves, Allen’s Railroad Curves (it is at home and I am at a cyber cafe) and it is hundreds of pages of densely packed formulas and drawings. And the math that goes with it, Smolley’s Tables, is another several hundred pages of tiny type size numbers. No we have it easy it is just that we don’t do it for a living.
And I’m glad that explanation was helpful. For those who haven’t seen it, it’s included in the premium booklet, “Workshop tips: Design concepts for your next layout,” packaged with this year’s edition of “Model Railroad Planning.” Our layout design annual, “MRP,” is on its way now to stores and those who placed advance orders. In the booklet I also had room to include Armstrong’s “coefficient of lurch” diagrams that show why easements are worth all the trouble.
I have a track plan I drew in Pro/ENGINEER. In that software, you can write relationships to control dimensions. I’ve created the formulas to control the curve dimensions. I have a parameter for the length of the easement that I can use to adjust every single easement in the track plan. That is called “EASEMENT”. Below is copied directly from Pro/ENGINEER.
Choose your offset carefully. In AutoCad, you may or may not have trouble if you try to attach a SPline to an unbroken circle. Once you have it set up, simply insert the SPline snapped tangent to the straight, and then tangent to the radius. Trim, and then identify the ends of the SPline and you’re done.
