How do you accurately measure gradient on a curve?

[Capt Bob Johnson]

If at all possible you would want to keep your grades in the one to 1.5% range.  Anything more becomes very limiting as to train length, and 3.5% is just about the max you can get awway with.

[NeO6874]

Puckdropper has it dead on, with all the math for converting from any angle to any gradient.

 

I took that and applied it to a range of angles to give a rough "line" of where things come into play.

 

While Puckdropper's method will give you nearly an exact angle to grade conversion (allowing, of course, for any calculator round-off error), it can get tedious.

 

If you don't like the math so much, you could probably use my table to get a rough approximation of grade.  For example:

• You take your level, and determine that you have a 35º angle.
  
• Take a quick look at my table, it's between 22.5 and 45º
• that corresponds to a gradient of something between 50 and 100% (it's actually a 70% grade)

So, as you see, you can "cheat" in a way, and approximate how steep of a gradient something is just by using that table.  I don't know much about Garden RRing, but I would like to believe that similar rules apply as with smaller scales regarding grades.  as such, you'll want to keep your angles roughly 1.5º (which is a grade of 3.125%) or less.

 

Hope this helps some. 

 

[Capt Bob Johnson]

imrnjr, Yes, I agree with your method; however, the question was if you had a spirit level that you could adjust or rotate the vials in the frame, and the amount of rotation is calibrated in degrees, what was the math required to determing how many degrees will result in what percentage of grade.

So far, the math majors who have expounded on the subject are giving answers far beyond my mathematical abilities.   Heck, I have to unzip my fly to count over 20!

[imrnjr]

If you know the verticle rise from the begining  of the curve (point a)  to the end of the curve (point b) and the distance (outside, centerline, inside, PICK ONE) from a to b (I use a walking wheel to measure) then it is the same calculation used for a straight line.  e.g. 3" rise in 100" of curve equals 3%. 

 

A Walking wheel to measure distance cost $40 to $120 depending on features... mine cost $99 about 2 years ago...puchased for some row work I was involved with now a good tool for track length and grade analysis.  A laser level and yardstick...meter stick for Ian... to develop the vertical differences point a to b....well $30 to $400  (it's a level versus a transit and 30 feet versus 300 to 600)...

arcs and tangents and cosines ..... just to complicated for me

 

mr 

 

[NeO6874]

It's very easy really. A gradient is between 0 and 90 degrees, therefore 90degrees is 100% gradient

 

actually a vertical rise (90º to the plane of the ground) has an unsolvable gradient (that, or it's infinite...don't remember which at the moment) - so a grade is measured between 0º and 45º.

(example below)

Angles and their corresponding gradients: (rise:run) 

• 90.00º angle has an unsolvable gradient (1:0)
  
• 45.00º angle has a gradient of 100% (1:1)
  
• 22.50º angle has a gradiant of 50% (1:2)
  
• 12.25º angle has a gradient of 25% (1:4)
  
• 6.125º angle has a gradient of 12.5% (1:8)
• 3.063º angle has a gradient of 6.25% (1:16)
• 1.531º angle has a gradient of 3.125% (1:32)
• 0.766º angle has a gradient of 1.563% (1:64)

so by that, if you had a grade of 1.5%, you'd be rising 1 foot for every 64 feet of run.

[kimbrit]

It's very easy really. A gradient is between 0 and 90 degrees, therefore 90degrees is 100% gradient, a 1 in 1 gradient is 45 degrees or 50%, a 1 in 2 is 22.5 degrees or 25%. 1% is therefore 0.9 degrees. I think the majority of garden railroaders want a gradient between 2 points rather than set out at a fixed degree of climb to get the right percentage from a fixed point to somewhere that becomes the right height. The real life surveyors did this but they had miles of countryside to play with and they could go where the terrain allowed their planned gradient. (a very rough description but you get the gist)

I plan a gradient so that the climb between the 2 fixed points is equal throughout, once the loco enters the gradient it pulls at a constant rate without meeting unexpected dips or bumps. Once the rough groundwork is done I set my spirit level to the requiered degrees, a full curve of 8' diameter is roughly 20', a climb of 6" makes it 1 in 40 or 2.5 degrees on the spirit level. I then run the spirit down the grade - no track as yet - and make adjustments as I go, when that's done the track is added and the spirit is mounted on the flat car and the bubble shows level. The car is run down the track slowly and if the bubble moves out of level adjustments are made. This works equally on straights and curves and the percentage or degree can be worked out once length and height of climb is known. Please remember that this is a hobby and not an exact science where steepness of grades means loss of revenue etc etc.

Cheers,

Kim

[Puckdropper]

Important note:

When using a calculator to do the conversions, make sure it's in degree mode when working with degrees!  tan(45) = 1 and atan(1)=45, if you get any other answers it's not in degree mode.     

 

Ready for some trig?

tan {theta} = rise / run

where {theta} = the angle

  rise = the distance elevated

  run = the distance between the start of the grade and measured elevation

 

For a 1% grade, tan {theta} = 1/100  solving for {theta} we get {theta} = 0.5729 degrees

(Plug that in to your calculator using either the atan or inverse tan (tan**-1) function: atan(1/100) 

 

Going the other way,  tan (0.5729 degrees) = grade proportion.  The grade proportion = 0.0099 in this case, which is very close to 1/100 = 0.01

 

Multiply the grade proportion by 100, and you have your grade percentage. 

 

---

If you want to use the length of the track instead of the distance traveled from the start of the grade to the point measured, use sine rather than tangent.  However, you won't get the grade proportion back, you'll get the ratio between the rise and track run.

 

You can do more math to get all the numbers you could want, but by now you've probably got the information you want. ;-)  (If your head hurts, just forget this stuff and go run trains for a while.) 

[Anonymous]

Thank you fo the advice and if you knew me you would know the answer would not be that easy.

I don't believe a laser will do it as they don't go round corners. And a small level will not do it as although the reading is accurate it is too localised. DannyS even has a device that you can buy for a very modest sum that will show it to you on a scale.

The problem is the curve and using stepping blocks and a 1 metre long straightedge with a spirit levl is pretty simple and roughly correct. But again the length will not be quite right unless the straight edge's length is the same as the length of the gradient being measured. So in my opinion a curved straightedge of the correct length may do the job.

Using a jigsaw (saber saw) you measure out 1 metre length of plywood of roughly the ciorrect curve, cut a 4 cm wide length and use this as a straight edge.

The idea of using stepping blocks and a spirit level you will get to the right conclusion. a secondary stepping block of half the size of the main one to  be placed under the centre, will greatly assist stability and accuracy. I am also looking at using flextrack or similar, bent to the right curve as wel,l for the straightedge.

If you use a 1 metre length straight edge, then the stepping blocks size in centimetres will be the same as the %. a 2 cm rise in a metre will be 2 % and a 1 cm will be 1 %.

Now getting on to converting from % gradient to degrees, for the likes of me one is pretty close to the other but if you must; multiply the % by 1.1 and it will come to very close to the degrees. I know this isn't 100 % but for practical purposes it should work.

Rgds Ian  

PS, I have not actualy done this and i was wondering what others thought?  

[John Busby]

Hi Bob

The easiest way to work a 1% grade is for every 100 mm the train travels it rises 1mm.this works for every measurement that the user finds easy to use whether its cubits or inches or mm.

Since the dia of the circle is known  it should be a simple matter of marking the diameter and for every 300mm rasing the line 3mm or 1/8" roughly for every 12"

But do check the manufacturer has measured the track diameter the correct way using the center line of the track.

regards John

 

[Capt Bob Johnson]

OK Kim, what's the formula to convert degrees of grade into percentage?

If a 1% grade is the ideal we should be aiming to achieve, how many degrees of gradient does it take to get that percentage?   I had to take algebra 2 twice just to squeak through it, so am not very up on that math!

[kimbrit]

Hi Ian,

I'm sure there are lots of ways for this. On my gradient, which does a full circle in 8' diameter and climbs 6" I use a small level that has an adjustable dial. I put the level onto a flat car at the top of the gradient, on the slope, set the adjustable dial so it reads horizontal and then go down the gradient slowly adjusting the track as I go to keep the level at the horizontal. I do this 2 or 3 times until I'm happy with it.

Cheers,

Kim

Just come back to this Ian, forgot to add that the adjustable dial is marked off in degrees for gradient measurement.

[pimanjc]

See my thread at http://www.trains.com/trccs/forums/965564/ShowPost.aspx for the easiest way to determine grade using a digital level from Sears.

JimC.

[Anonymous]

I wish to do this over a distance say about a metre. The method i use is ok but pretty crude and not very accurate. I use a stepping block and a 1 metre long piece of wood and a spirit level. This is ok but it is just point to point and doesn't allow for the extra distance of the curve. 

Rgds ian