Round the Bend

[What About The Bee]

This discussion is how to permit a multi axle system go around a curve.  I will be using my S&DR Experiment and an R2 (438mm curve), but the equations are suitable for any system in which all axles can move side to side.

As decision time nears for a commitment to print, I have been giving a tremendous amount of thought to going around a curve.

To be explicit, this defines how thin the chassis must be to accommodate the side to side travel of the axle.  This is so because the axle and wheels are fixed dimensions; to work as a system with the track.  The thing we adjust is the chassis, not the wheels.  

In particular, the axle bushings must be slightly proud of the plastic  chassis, so in fact, we define the location of the bushings to permit travel round the bend, and fit the chassis to the bushings. 

It should be clear that if the axles are not permitted to travel laterally (side to side) then the wheels will not fit into the track gauge on a curve.  They will beautifully fit on a straight section of track.  Without lateral axial compliance, they will not fit.  Lateral axial compliance is the same as side to side play.

So how do I get from this

To this

How much compliance?  How to define Cₛ'?

There are several factors to consider.

One factor is the gap between the gauge of the track and the Front to Front dimension of the wheel set.  That is, when the axle and wheels are placed on a track, they can be moved side to side (laterally). 

Another factor is the flange of the wheel as it faces the track.  The drum touches the track at a point, but the flange inside the rail extends forwards and backwards from that point.  This is a chord of a circle, a function of the radius of the drum and the flange height.

Another factor is the chord of circle formed by the track and the wheelbase.  The wheelbase must include consideration of the flange of the wheels.  Each practical pair of axes must be considered.  

Another factor is the distribution of the lateral compliance.  Is the chassis permitted to shift off center, or must the full lateral compliance be on both sides of the chassis?

Another factor is a practical one.  Once the mathematical minimum is met, can the parts manufactured and assembled meet that minimum solution or should we expect build up of tolerance?

Since this is a fairly complex topic, I have broken this up into chapters. Each chapter will focus on an individual factor.  Finally, there will be a chapter tying it all together. 

Note: all equations were developed on a blank sheet of paper.  It is very likely that there are other ways to solve this.  I am not saying that this is the way to solve the problem, rather, it is how I solve the problem.  

All errors are mine alone.  

Bee

[What About The Bee]

Chapter One: Track Gap

This factor is the gap between the gauge of the track (Gₜ) and the Front to Front (F2F) of the wheelset.

The gap is required, since both wheels are hard coupled to the axle.  The wheel drum is conical in section.  Thus, lateral motion permits different diameters of the drum presented to the track, thus the axle and wheels go around a turn without wheel slip.   This was a fundamental patent from the early days of railways. 

Gₜ is obvious for OO, 16.5 mm.

What is not so obvious is the F2F dimension, since the profile of the tire of the wheel involves conical and curved sections.  F2F is the dimension from the front side of one wheel to the front side of the other wheel, at the flange.

I will be using Scale Link Bogie wheels for S&DR 'Experiment'.  SW1412B, to be specific. I can set the B2B to anything I like, here 14.54 mm, which is identical to the Romford locomotive drive axle dimensions, with the wheels mounted.

Yet we know that the B2F is critical for going through points.  Using a set of Peco points (SLE-192), the Scale Link SW1412B wheelset and a feeler gauge, I can determine the check rail gap to within 0.0005 inches or  0.0127 mm (12.7 microns).  Once I know the B2F and the B2B, algebra will permit the determination of one flange or front.  From this, I measure F to be 0.906 mm.  Therefore I find the F2F to be 14.54 mm + .906 mm + .906 mm = 16.352 mm.  I will emphasize that this dimension is for Scale Link wheels, not to be used for Hornby wheels & etc.

Thus, the first factor (Q) is merely the difference between the track gauge (Gₜ) and the F2F. 

Q = Gₜ - F2F

Q= 0.148 mm

[What About The Bee]

Chapter Two:

This factor is the extension of the wheelbase, as a function of the drum radius and flange height.  In an automobile, the wheelbase is simple, it is the distance between the first and last axles.  Railway wheels have flanges, which extend the wheelbase.  What is that extension?

For SW1412B, the drum radius R𝒹 7mm and the height of flange (H) is 0.66 mm.

Therefore the distance from the axle centerline to the flange top of rail intersection (X) is 

X = sin(cos⁻¹(R𝒹 / (R𝒹 + H))) × (R𝒹 + H)

For SW1412B, X = 3.111 mm

[What About The Bee]

Chapter Three:

We must fit the wheels on their axles within the curve.  Each axle will be shifted laterally.  I just need to slide each axle, from side to side, such that they fit into the track.

When this is done, you should see that the first and last axles form a chord to the curve of the track.

We know the given radius of the track  R2=438 mm in OO.  This, however, is the the centerline of the track.  The true radius (Rₜ) of the curve we care about is to the outside of the track.  This is easy to find

Rₜ = R2 + Gₜ/2 = 446.25mm

The nominal wheelbase is the distance from the first axle centerline to the last axle centerline.  This is merely a design criteria, so obtain it from the CAD.  In my case, the axle wheelbase (Wₐ) for S&DR 'Experiment' is 65.791 mm.  

The true wheelbase (Wₜ) is the axle wheelbase (Wₐ) plus the flange extension (X) for the leading and trailing axles

Wₜ = Wₐ + X + X = 72.013 mm

So to fit those leading and trailing SW1412B wheelsets into the R2 curve, we have required lateral compliance (Y).

Y = (1-cos(sin⁻¹((Wₜ/2)/Rₜ))) × Rₜ

For SW1412B and R2 in OO, Y = 1.455 mm 

[What About The Bee]

Chapter Four:

Distribution of Compliance 

Experiment is running from a straight section of track onto a 438 mm radius curve in OO.

Because the drums are a section of a conic, all the axles will be centered.  For the purposes of this discussion, I will assume that the chassis is also centered to the track, although we will find this does not really matter.

Nothing happens until ½ of Q is taken up by the first axle.  The wheel sets are centered and so until the flange touches the rail in the curve, no adjustment in lateral displacement need occur, except by the conic section of the drum

As Experiment moves forward from the touch point, the axle is shoved over, relative to the chassis, by another ½Q.  That is, the entirety of Q is consumed by the curve, the wheel on the outside of the curve will have no gap to the track.  All of Q will be allocated to the inside wheel v track.

Thus we can reduce the required lateral compliance (Y) by the gap (Q)

Y' = Y - Q = 1.455 - 0.148 = 1.307 mm

Y' is now the defined lateral distance the first and last axles must be shoved over.

What of the interior axles, numbered 2 and 3?  

As Experiment continues forward, the identical process occurs.  The second axle utilizes ½Q and then another ½Q, and then lateral shift due to the chord of the curve.

Simply define the wheel base as a function of X and the distance between the 2nd and 3rd axles.  Again axle 2 to axle 3 is CAD data Wₐ₍₂₊₃₎ = 21.930 mm

Wₜ₍₂₊₃₎ = Wₐ₍₂₊₃₎ + X + X = 28.150 mm

From this, the lateral distance (Y₍₂₊₃₎) can be found

Y₍₂₊₃₎ = (1-cos(sin⁻¹((Wₜ₍₂₊₃₎/2)/Rₜ))) × Rₜ
 
Y₍₂₊₃₎ = 0.222 mm

Deducting the gap, as before

Y'₍₂₊₃₎ = Y₍₂₊₃₎ - Q =  0.222 - 0.148 = 0.074 mm

Y'₍₂₊₃₎ is now the defined lateral distance the second and third axles must be shoved over.

So at last, we know the maximum travel (T) for the axles.  

Y' to Y'₍₂₊₃₎   is 1.307  - 0.074 

T = 1.233 mm  

T is a critical value, as it defines the total requirement of gap .  T is the Total Travel! The gap between the left wheel and its bearing + the gap between the right wheel and its bearing.  

Because track can bend to the left or to the right, it is best to centralize the chassis relative to the permitted  lateral axle play.  Thus, in the initial condition, the gap on the left should be equal to the gap on the right.  The axle bearings are fixed to the chassis.  The outside of the bearing defines the limit of side to side travel, until we permit the chassis itself to move over.  Put another way, the first and last axles get shoved over, relative to the chassis to fit into the curve by Y'.  But what happens if the inside of the wheel touches the outside bearing surface and continues to shove over?  It moves the chassis over!

Thus, we can say:

Cₛ= T/2

Awesome Sauce!  Except that is completely impractical in the real world.

Edited August 3 by What About The Bee

[What About The Bee]

Chapter Five: Build up of Tolerance

Cs will be fed from the equations into the CAD model, as this is the line to line, precise minimum value for a OO R2, 438 mm curve.  

The technical term used is "Build up of tolerance".  Can I set the back to back (which achieves the F2F) to within 1 micron?  No.  Can the chassis be 3D printed to within 1 micron (0.001 mm)? No.  Can I place both bearings relative to the chassis to within 1 micron?  No.  Even if I could, there is 3 microns of uncertainty, right there.

Each object's dimensions has a tolerance.  For example, the flange of a 2mm axle bushing is measured at 0.24 mm.  But plus or minus what?

Now add up my inability to position each component with the inability of the manufacturer to make any part precisely to the print (CAD).  

Shazzam!  Build up of Tolerance.

Cₛ' = Cₛ + BoT

Bot is an arbitrary, but small number.  I selected 0.1016 mm (0.004") per side.  I may regret this, or relax it even more as I commit to print, depending upon my level of panic
 

[What About The Bee]

Chapter Six

If you have made it this far, I congratulate you on your perseverance.  This topic is by necessity a trigonometric exercise.  

But can we put this all together?  Yes we can

Here is the closed form series of equations and values for Experiment.  Each equation relates to a previous chapter and will help us to define the value to be used in CAD

And that is how we go from this

To this

Okay, but that is for an even number of axles.  Suppose the number of axles is an odd value.  The above solution still works, selecting pairs of axles from the outside in.  The final axle will be just itself, and:

Wₐ=0  
Wₜ = Wₐ + X + X

That is, twice the flange extension.  The chord is merely across the flange extension in both directions, the nominal wheelbase doesn't exist, it is a point.

What is the shift in chassis?

You will note that Cₛ' is less than T.  So when we require the axles to shift by more than Cₛ', the chassis will also shift by T - Cₛ.   This means that the first and last axles, on the inside of the curve, will come very near to the chassis.  

And that, lads, is why back to back opposing curves are not recommended.  You have no idea what lateral compliance a designer has put in.  I've put in just enough to get around a 2nd radius curve, but if they go back to back, then the front and rear of the chassis is forced in opposite directions, leading directly to derailment 

 

[What About The Bee]

And that is why this is titled

Round the Bend

Its driven me there! 🤪

Bee

 

[Peter Stiles]

You *are* Carol Vorderman and I claim my five pound prize!

 

Or/and: good, interesting work. I now know, mathematically why I'll never try to model in Fine Scale.

[atom3624]

Dear Carol ....  you missed something!

Cleaning and light lubrication. 

I always check a multi-wheeled locomotive's freedom of lateral movement before I start using it again.

I love formulae, but must admit I glossed over much of this obviously well researched and informed material.

An excellent reference point for anyone wondering if a certain locomotive can make it around their layout.

Al.

[threelink]

Another learned post, Bee. Maths and trig were never my strong points so my solutions to such problems tend to be trial and error - mostly error. 

[LTSR_NSE]

Fascinating posts (as always) Bee.

The detailed descriptions (and pretty pictures) explaining what you are wanting to achieve are particularly helpful.

Especially for those of us who don’t remember being taught or learning either trig or algebra! 🙃

 

[What About The Bee]

19 hours ago, Peter Stiles said:

You *are* Carol Vorderman and I claim my five pound prize!

16 hours ago, atom3624 said:

Dear Carol ....  you missed something!

My only familiarity with Countdown are the episodes with Jimmy Carr.  Which leads to my preference of Rachel Riley 😉

÷÷÷÷

The problem so lends itself to trigonometry, as most of the solution is triangles within a circle.  The flange extension (X) and the lateral wheel compliance (Y) are essentially trigonometric identities with a radius of wheel drum and track radius respectively.

I moved OO Experiment from a straight to an R2 curve in my mind's eye, literally hundreds of times, checking and re-checking.  And then formulated a set of equations and trialed them against the CAD.  Repeatedly.  The set of equations presented meet requirements.  

Bee

 

[What About The Bee]

LT&SR_NSE pointed out that diagrams are helpful.  

This diagram illustrates chapter two, where we find the flange extension to the wheelbase.

It should be evident that the distance X, from the axle centerline, the flange and rail meet geometrically.  From this pure side elevation, you can see the exact point.

For OO Twin Sisters, the center axle is captive by the enveloped gears.  Therefore, it cannot shift side to side.  How much lateral compliance does this axle need?

Find X, the flange extension, 3.316 mm, using the process described in this post.  The wheelbase, Wa is 0, as this is a single axle.  Solve for Y, the chord formed by twice the flange extension and a 2nd radius curve.  Answer?  0.012 mm.

The track gap Q, chapter 1, is 0.148 mm.  Thus, whatever lateral compliance we need for this axle will be easily accommodated by Q.  There is no issue!

Bee

[Rallymatt]

Beautifully illustrated with mathematics 😁. When I did a video on why re-gauging wheelsets improves the running for the TT:120 models, I pre-empted the question why manufacturers opted for 10.2mm B2B when the scale Target B2B is 10.5mm. Except I did it with a sheet of paper, a magic marker and very un-technical drawing 🤣

@What About The Bee, your diagrams are a great explanation why models need to have undersize B2B , we are expecting vehicles to negotiate almost impossible curves, I’m still surprised some ‘experienced modellers’ still don’t appreciate why models ‘wobble’ especially with steam locos.
Since I started banging on about B2B in TT:120 and recommending setting it to 10.4- 10.43mm, Hornby is delivering new releases with wider B2B and run much better for it 😁

[What About The Bee]

Hi @Rallymatt

Thank you for your kind compliment.  I used the illustrations to confirm the equations I wrote for myself.  All mistakes in the equations are my own.   

I thought to share the equations and diagrams so that other modelers could chime in with additions and questions.  It is a fantastic topic!

For the enthusiasts attempting to make their own, the equations are design considerations.  Hornby, or indeed any manufacturer, is unlikely to make the more obscure locomotives, carriages or waggons.    If an enthusiast wants those, there is but one choice. Make them yourself.  Why wait?  Understanding the system comprised of wheels, points and curves is a pre-requisite to making them yourself without a tremendous trial and error effort.

Turning to your discussion about the B2B.  I agree that making the B2B smaller helps in curves.  It increases Track Gap (Q in chapter 1) and mathematically relaxes requirements. 

There is a caveat to reducing the B2B, to wit: points.  With the wheel near the stock rail constrained by the check rail, the B2B controls precisely where the other wheel will be, as it rolls through the wing and frog rails.  I recognize that you understand this, but state the obvious for a somewhat more casual reader.  Making the B2B smaller does help in turns, but we must stay within a fairly small range of values.

Our community has an extremely broad span.  I do not expect even the most experienced modelers to know everything.  Further, experience, in and of itself, is a poor substitute for actual understanding of theoretical aspects.   And so, if an enthusiast does not understand the reasons why a model can wobble, its forgivable. 

I also recognize that the vast bulk of the people visiting the Forum are looking for a basic answer to basic questions, and that simple technical help is all they require.  Practical experience, especially if it is decades of that, will provide the quick answer to their queries.  Its a valuable commodity and the reason why the experienced correspondent is so treasured.  None need to understand why that thing is so, it just must be.  Do this, don't do that.

The equations presented in this thread are far beyond what is needed to "make a layout work".  However, for those who wish to know why that thing is so, the insight is extraordinary.  

Bee

[Rallymatt]

Perhaps when I mentioned ‘experiencing’ modellers my tongue was firmly in my cheek. When covering the issue of less than long lasting motors with a solution, a few critics were talking about loco wobble without realising a certain amount is just how it is, although I did have a recent loco that had a waddle that would have embarrassed a duck, I think it was an twisted chassis casting and Hornby changed it immediately. 
As an engineer I appreciate your efforts in presenting the calculations and diagrams to modellers, models run because of the same mathematics as full size machines. I must admit in my hobby life, I just make it work though 🤣

[Gordonvale]

A note from the blue beyond.

The Hornby standard track centre is 2⅝ inches (2.625), which is 66.675mm. R2 is 17.25 inches which is 438.15mm.

Beware of rounding errors!

Edited August 6 by Gordonvale

[What About The Bee]

Hi @Gordonvale

You pose a valid and quite interesting point.  It speaks to standards and specification.

For the purposes of the lateral compliance computation, there is little appreciable difference.  For OO Experiment, using 438 mm, Y' is 1.306997 mm [rounded up to 1.307 mm in the equations above].  Using 438.15 mm as the track radius, Y' is 1.306507 mm.  This makes sense, of course, a slightly larger radius requires slightly less lateral compliance.  The delta is 400 nanometers or less than 20 millionths of an inch.  My cutoff for model railway computations is typically 1 micron, making this below notice.

Yet, in the interests of mathematical rigor, the point must be pursued.  I think this conversation must begin with what Hornby themselves state

https://support.hornby.com/hc/en-gb/articles/10406079411612-00-Gauge-Track-Geometry

Hornby specifies a 2nd radius curve as 438 mm.  There is no imperial conversion, no indication of inches, anywhere.  Just metric.  

Peco also sells 2nd radius curves.  Here's one

https://peco-uk.com/products/double-curve-2nd-radius-2?variant=7435751620642

Peco specifies a 2nd radius curve as 438 mm.  There is no imperial conversion, no indication of inches, anywhere.  Just metric.  

Here is the support I could find for a 17¼" specification, but these specs are all very old and superceded 

https://www.brightontoymuseum.co.uk/index/Category:Hornby_Dublo_two-rail_system

States 17¼"  for Hornby Dublo 2 Rail.

https://www.classiccollectmodels.co.uk/category/model-railway-history/

Series 3 track, 17¼". 

Super 4 track, 17¼" (43.8 cm).  Both metric and imperial units are displayed.  Apparently the root cause, the sloppy unit conversion starts at the inception of Super 4.

I would like to know when Hornby only states metric. 

Those with old catalogs can best tell us this.  

Bee

Edited August 7 by What About The Bee

[What About The Bee]

Correction to above.  The Series 3 track, large radius curve is specified as 17⅛" (43.5 cm).  My apologies for mangling the data point.

The question of when Hornby switched to a metric specification only remains.  Should be post Super 4 track, so the 70's or 80's.

Bee

Edited August 7 by What About The Bee
Words

[Going Spare]

I think the changeover would have been around 1970, or slightly later, with the introduction of System 6 trackage (R6xx) which was made in Austria (with steel rail) for Hornby.

[Rallymatt]

Was that the Roco made track? 

[Going Spare]

There was nothing on the track sections themselves other than the Hornby name but yes it was of Roco manufacture.

[Rallymatt]

I remember the Made in Austria stamp, and always assumed it was Roco making it, was it the same geometry as Roco’s at the time? That’s a brand that’s really fallen. It’s all made not very well in Vietnam these days ☹️

 

[Going Spare]

I presume it was standard domestic-use Roco track - I can't see it having been economic for Hornby to adopt it if it meant special production just for them.