does anyone have the radius easement formula? I'm starting a new layout and it's been so long since I've used it I've forgotten it..old age
I have the formula for the cubic spiral programmed into an EXCEL spreadsheet. It's on the computer at home; will post it later.
I found the formula for the cubic spiral in a rather ancient civil engineering text book. It is given as: dp -- = k * s ds s = the distance along the curve measured from a point of zero curvature p = the angle phi k =constant, where: ......1 k = ------- .....R * L Integrating the first equation yields: ......1........2 P = --- k * s .....2 My apology for the dots; they are "place holders" that I've used to keep everthingthing "line-up" The geometry of an infinitesimal triangle gives: dy = ds * sin(p) and dx = ds * cos(p) Fortunately the text performed the calculus for me. Mine’s very rusty. So to getting to the nub of things, and if I haven’t fat-fingered anything, here’s what the results looks like in EXCEL spreadsheet format f(x)=((C4)-(((POWER(C4,5))/((40)*(POWER(A4,2))*(POWER(B4,2)))+((POWER(C4,9))/((3456)*(POWER(A4,4))*(POWER(B4,4))))))) f=(((POWER(C7,3))/(6*A7*B7))-((POWER(C7,7))/(336*(POWER(A7,3))*(POWER(B7,3)))+((POWER(C7,11))/(42240*(POWER(A7,5))*(POWER(B7,5)))))) In the attached spreadsheet, a cubic spiral is calculated for a curve with a radius, R, of 48” and a spiral length, L, of 18 inches. I have selected one row from the spreadsheet to illustrate what the values represent. At point, s=9”, along the spiral, the distance from the TS along the X-coordinate=8.9980” and the offset (the y-coordinate) = 0.14060”. One can cut and paste values into the spreadsheet to create a spiral specific to ones needs. I haven’t figured out how one might create a scaled drawing of the “chart” that EXCEL creates. No doubt, there are other application that will. Way back in my MAC SE days, I used an application called Cricket Graph that would generate scale drawings. Otherwise, it might be possible to create some styrene or cardboard templates from the dx/dy pairs. On the prototype, spirals are used to build/lose elevation. At the TS, the elevation of the “outside rail” will be 0”, and at the SC, the elevation of the “outside rail” will be at its designed maximum. The Frisco used a max superelevation of 6”, while the MoP used 7”. So one of the determinates with regard to the spiral length is the superelevation of the curve, and of course the amount of elevation in a curve depended upon the degree of curvature, and the speed that the railroad wished to operate through the curve. Secondly, the spiral is used to change the lateral direction of the train gradually for the comfort of the passenger, to reduce damage to lading, to help “ease” long, wheel-based steam locomotives into the curve, etc. Since model railroads don’t have to deal with the same mass as the prototype, it isn’t necessary to place spirals and super elevations on our scale curves. However, the esteemed layout designer, the late John Armstrong argues in “Track Planning for Realistic Operation” that the use of spirals on model railroad layouts improves operations, and will allow the use of smaller radius curves. He uses the notion of the “Coefficient of Lurch” to prove his point. He defines the “Coefficient of Lurch” as the Delta-Z, where Z is the distance from the track center line to the center line of two railroad cars' intersection at their couplers. Armstrong demonstrates that rate of change in the value of Z or Delta-Z is less on an 18” radius curve with a spiral, than it is for a 24” radius curve without a spiral. I have seen several layout that use spirals and super elevations on curves, and it’s cool to watch a HO scale passenger train cant into curve. It’s an effect that’s worth the effort. In HO scale, to achieve 4 scale inches of rail elevation place 0.06” shims at the end of a scale 9 foot tie. To obtain an elevation of 6 scale inches use 0.09" shims.
Professor Brand has done it again! Thanks. Lets see - where are my calculus books? On my original HO layout, built at home in the 1960's while I was at MSM, I had a long 22" radius curve on the prime viewing end of the layout, probably about a 120 degree curve. I superelevated the curve using small sections of flat-end toothpicks placed under outside end of the ties. It looked pretty neat when a train went through the curve, but since there was no transition into either the curve or into the super elevation, I had quite a "lurch factor" I'm sure. I don't think my early Frisco passengers enjoyed being slammed around as my version of the "Will Rogers" hit that part of the layout! Ken
In Karl's post the numbers confused me so I didn't read it. Basically there is 4 ways I can think of. 1. Example: If you want 18R curve and have a 36" piece of flex track, you use 2 pieces to make a 180. It would take 4 to make a full circle. Basically the formula is Distance of flex track/Radius=number of pieces of flex track needed for 180 degree turn. 2. get a radius tool. I can't remember the name of the company that makes them, but there is a company that makes a radius tool. Just slip is between the rails and slide. Never used one though. 3. guess. 4. sectional track
IB - Uh, no, not correct. Go back and check your geometry book. In order to make a 18" radius turn into a full circle, you need to know the circumference to determine how many sections of 36" flex track you will need. The formula for circumference of a circle is pi x D, i.e., 3.1416 times the diameter of the circle (36" dia for an 18" radius circle). Thus the circumference is 3.1416 x 36" = 113.1" of flex track. So, you would need about 3.15 lengths of 36" flex to bend into a 36" diameter circle. Karl was talking about the complex formula to smooth the transition into the turn progressively from straight tangent into the full maximum radius of the turn such that the train, passengers and lading would not suddenly lurch into the turn, but enter it gently. He was also talking about super-elevating, i.e., banking, the outer rail in a turn such that the train can take the curve easier at speed - just like the turns of a race track are banked. He mentioned the SLSF used a maxium super-elevation of 6". Ken
I must admit that your formula stumped me also. I don't even bother with worrying about all that and just lay the track until I get a smooth transition. If I remember correctly the NMRA has a very good explanation on transistions in their data sheets. Lay the track and run the trains, thats my policy! El Bob Oh (Math Idiot)
I'm not in geometry, I'm in 8th grade algebra I, which for Lamar that's a year early, normally you can't get in till your a freshman. I came up with my calculations from the right track software. Basically the software shows that bending one piece of flex track 90 degrees, it make 18R. I don't know much about transition curves. I like El Bob Oh's idea
Bob, that's what I used to do until I did easements on my current layout. I found an article in a MR that explained the technique in laymans terms and I tried it. I must say the difference is noticeable Thanks for the formula Karl and especially the diagram..I'm a picture person!
Flexible track facilitated easements in a previous life. A couple pieces of 1/16 balsa wood under the outside piece of cork roadbed gave me super-elevation, too. Passenger trains would indeed fly around the 22" radius and I am sure the 1/87-size riders felt little lurch. I know I calculated the spirals and laid the roadbed to put them in, but in HO, the difference, even for 22" was very subtle.
John Armstrong also summed it up well in his books. I like the bent-stick method used in conjunction with the NMRA recommended offsets - about a half inch in HO if I recall correctly.
Nice thread ... good discussion. I have heard of the "carrot and stick" approach, but, what is a "bent-stick" method? ""When the "carrot and stick" approach did not work, my mother used the "bent-stick-accross-my-butt" method.""
The basic idea is you draw out the center-line for your straight track, then you lay out your curve radius with an offset of 1/2" or so. Next, take a yard stick and place it on edge along the straight track center-line, and bend it until it intersects the curve. The curve the yardstick takes becomes the center-line of your transition. Paul
Thanks, Paul. Sort of thought that might be what was talked about. I had lots of easements on my earlier layout; much of the track base was made using 1x12 boards ripped to 3/16 strips, laminated together with glue and homemade clamps. Really made nice curves - and I could elevate the curves by using a planner and sander to shape the top. Just had a funny thought: Lots of my co-pilots (mostly ex-military fighter jocks) could really "Lurch" into turns. Used to drive me nuts.
IB - In a sense, you are correct. It does take four sections of 36" flex-track to make an 18" radius/36" diameter circle. Except - you only need 15% (about 5.4") of the fourth section to make a 36" diameter circle. If you made a circle out of all four full 36" sections of flex track, you would have a circle of 144" in circumference. (4 x 36") That circle would have a diameter of 45.8", or a radius of 22.9". (144"/3.1416 = 45.8" diameter). Ken
Model Railroader magazine carried 7 different spiral templates in its October 1969 issue. The article and the pdf files of the templates are available at: http://www.trains.com/mrr/default.aspx?c=a&id=290
