gordon s Posted October 19, 2014 Share Posted October 19, 2014 Still trying to get my head round this one. Perhaps you know the answer? I have two lines that run parallel, but some distance apart. They both start and finish at the same points. For ease of calculation I have shown the length or run as 4m and 5m for each track. Taking the outer track a 1:100 gradient means a climb of 10mm for each metre. Simple... Now this is the bit I can't get my head round. If they are on the same plane and that gradient is 1:100, does the inner line climb at 10mm every 800mm, because in my brain that would say the gradient is only 1:80 and not 1:100. How can that be? Does the gradient change as the radius reduces, even though they are on the same plane and that plane is set at 1:100? It's a long time since I was a school and I can't see the logic yet…. Any ideas? Edit: Figures corrected for typo/handwriting error. Link to post Share on other sites More sharing options...
Graham Walters Posted October 19, 2014 Share Posted October 19, 2014 The logic is that the inner line is reaching the same height over a shorter distance, so mathematically it's a steeper gradient, and its the curve that causes the problem. The easiest way to look at it is think of a running track, and why the runners are staggered at the start, it's so that they all run the same distance. So to calculate the gradient on the inner track properly to compare it to the outer track, you have to have the same distance, then the gradients are the same. Link to post Share on other sites More sharing options...
Fat Controller Posted October 19, 2014 Share Posted October 19, 2014 1/100 is equivalent to 10 mm/1000 mm, I think. Rising 50mm in one metre would be 1/20, whilst rising 50mm in 800 mm would be 1/16 However, if you mean rising 50mm over 5 metres, then that would be 1/100, whilst rising 50mm over 4 metres would be 1/80. The curve means that the inner track is shorter, so, in attaining the same elevation, the gradient would be proportionately steeper. Link to post Share on other sites More sharing options...
shortliner Posted October 19, 2014 Share Posted October 19, 2014 This is one of the problems with double track spiral climbs/helices - which is why the up-hill traffic should always use the outer track, because the gradient is less steep Link to post Share on other sites More sharing options...
gordon s Posted October 19, 2014 Author Share Posted October 19, 2014 Sorry for my error. A simple handwriting error which was then repeated. I did know what I was trying to say….:-) The running track is a good example and someone who follows sport, I can understand that. It was adding in the third dimension that was confusing me. OK, so we can start at the same place, but the inner line will have to run for another metre past the finishing point to maintain a 1:100 gradient. Thinking about it, I guess that's why it's easier to climb a slope by going across it diagonally rather than straight up. Thankfully the inner track will be downhill….The same line uphill has more run length on the other side of the room. Link to post Share on other sites More sharing options...
Ron Heggs Posted October 19, 2014 Share Posted October 19, 2014 The problem you have is that neither line is on the same plane, because they both on curves rising at a differing gradients to maintain the same elevation. They would only be on the same plane if they were straight lines Link to post Share on other sites More sharing options...
gordon s Posted October 19, 2014 Author Share Posted October 19, 2014 Not sure I understand that bit, Ron. Lets assume you start with a large flexible sheet of material and cut the curve shape out to accommodate both lines. You can put a straight edge across two lines and that is maintained at any point across the sheet. That to me says they are in the same plane. The bit that was confusing me was the change of gradient as the radius changed I'd be very happy to listen to your explanation why they aren't in the same plane when you can put a straight edge across at any point. Or maybe you can't do that with a straight edge after all…. Link to post Share on other sites More sharing options...
Graham Walters Posted October 19, 2014 Share Posted October 19, 2014 Sorry for my error. A simple handwriting error which was then repeated. I did know what I was trying to say….:-) The running track is a good example and someone who follows sport, I can understand that. It was adding in the third dimension that was confusing me. OK, so we can start at the same place, but the inner line will have to run for another metre past the finishing point to maintain a 1:100 gradient. Thinking about it, I guess that's why it's easier to climb a slope by going across it diagonally rather than straight up. Thankfully the inner track will be downhill….The same line uphill has more run length on the other side of the room. The other issue is that a loco can be quite happy climbing say a 1:50 on a straight line, but put a curve into the equation and it will struggle, climbing and making a corner adds to the difficulty, I've seen a video on YT that proves this, but can't find it now. The YT poster had to reduce the gradient by 10% to get his engines to cope with the curve being on a gradient. Link to post Share on other sites More sharing options...
Graham Walters Posted October 19, 2014 Share Posted October 19, 2014 Not sure I understand that bit, Ron. Lets assume you start with a large flexible sheet of material and cut the curve shape out to accommodate both lines. You can put a straight edge across two lines and that is maintained at any point across the sheet. That to me says they are in the same plane. The bit that was confusing me was the change of gradient as the radius changed I'd be very happy to listen to your explanation why they aren't in the same plane when you can put a straight edge across at any point. Or maybe you can't do that with a straight edge after all…. People often make this mistake, as I said earlier, you have to calculate the gradients over the same distance to get a comparable gradient, or you hit the head scratching problem you are at. Gradient is calculated as the amount of rise over distance traveled. So a 100% gradient would be for every foot traveled you would rise one foot, = a vertical climb = 1:1 In your scenario the curve is adding to the gradient when the end point is at the same place ( point B in your diagram) so the gradient appears to be steeper. the fact they are on the same plane has nothing to do with it, because the distance is compensating for the curve, ( as in the running track). It's very confusing maths, and can only really be explained by sitting next to someone, basically you have to stop thinking in 2 dimensions, ( as in your diagram) and start thinking in three dimensions. Link to post Share on other sites More sharing options...
shortliner Posted October 19, 2014 Share Posted October 19, 2014 Erm - isn't a 1 in 1 a 50% (45°) gradient? A vertical climb is surely 1 in 0. Sorry - not being picky, just confused! Link to post Share on other sites More sharing options...
gordon s Posted October 19, 2014 Author Share Posted October 19, 2014 Thanks guys. My two dimensional brain was struggling with the three dimensional concepts, but now it makes more sense. Link to post Share on other sites More sharing options...
Graham Walters Posted October 19, 2014 Share Posted October 19, 2014 Erm - isn't a 1 in 1 a 50% (45°) gradient? A vertical climb is surely 1 in 0. Sorry - not being picky, just confused! Yep My bad ! See it's even confusing for those of us who think we know what we are talking about ! Link to post Share on other sites More sharing options...
shortliner Posted October 19, 2014 Share Posted October 19, 2014 You really had me worried there - I thought I'd had a brainstorm! Link to post Share on other sites More sharing options...
Grovenor Posted October 19, 2014 Share Posted October 19, 2014 Note that the difference in length over 5m with the same start and finish points is nowhere near a full metre, even if it was a continuous curve, hence the difference in grade is quite small. The circumference of a circle is 2pi times the radius, so the difference in length over a complete circle is 2pi times (R1 - R2). So for tracks at 50mm spacing you are looking at a length difference of 314mm for a full circle, about 100 mm for your sketch. Keith Link to post Share on other sites More sharing options...
gordon s Posted October 19, 2014 Author Share Posted October 19, 2014 Now you've got me at it again. A gradient of 1:1 is 45 degrees, but that is a 100% slope percentage…. Edit: I was just using those figures as an example, Keith. They are not a double track mainline, but some 400mm apart and they are a very shallow transition curve, not a constant radius. It was the concept I was more interested in and just chose those figures to keep the maths simple. Link to post Share on other sites More sharing options...
themagicspanner Posted October 19, 2014 Share Posted October 19, 2014 Erm - isn't a 1 in 1 a 50% (45°) gradient? A vertical climb is surely 1 in 0. Sorry - not being picky, just confused! Not quite. A 1 in 1 gradient is actually 100%... You're right about a vertical face being 1 in 0 though. Mike Link to post Share on other sites More sharing options...
shortliner Posted October 19, 2014 Share Posted October 19, 2014 Oops! looks as though I got my percentage wrong! - Apologoosies! It's a loooonnnnggg time sinse I dun maffs at skule as any fuel no! Link to post Share on other sites More sharing options...
Memphis32 Posted October 19, 2014 Share Posted October 19, 2014 Going back to the plane concept, you knocked it on the head (without realising it) when you said "flexible" sheet. It was on the same plane before you bent it! If those curves were elongated, you might eventually cross over the lines again at a different height, so they can't be on the same plane! Link to post Share on other sites More sharing options...
DCB Posted October 20, 2014 Share Posted October 20, 2014 I think you will find railway gradients are usually quoted in height gained in the actual distance covered while roads are usually quoted as height gained in the horizontal distance covered. the former calculation would give 1 in 1 as vertical, the latter 1 in 1 as 45 degrees. On any normal railway gradient the difference between the two is negligible. The problem with spirals and I have one which does not work is while the tracks are in the same plane viewed as a cross section the rails curve up in the vertical plane such that a perfectly true rigid 4 wheel chassis will rock supported by 2 diagonally opposite wheels and one other, My 18" 2nd radius spiral proved unworkable because loco drive Pacifics could not cope with this twist, Diesels were fine but my sons Flying Scotsman and Mallard could not cope with it. I think Pecorama has the same problem as I note the train always goes down their HO spiral. Curves on gradients also affect haulage power and drag, I operate a OO layout with a 1 in 30 ish 19" ish radius 120 degree curve and code 100 steel track (set track on the curve,) long wheelbase 4 wheelers. Palethorpes vans etc are banned but the locos grip better around the curve than along the straights before and after the curve, overloaded trains usually slip to a halt either just before or just after the curve. Link to post Share on other sites More sharing options...
themagicspanner Posted October 20, 2014 Share Posted October 20, 2014 I think you will find railway gradients are usually quoted in height gained in the actual distance covered... Interesting. I wasn't aware of that. I makes sense when you think about it as it's much easier to measure the length along the track than it is to measure a length that projects into a hillside! Link to post Share on other sites More sharing options...
Freddie Posted October 21, 2014 Share Posted October 21, 2014 Ever climbed a spiral staircase? the difference between the inner and outer handrail is an extreme example. Fred Link to post Share on other sites More sharing options...
28XX Posted October 21, 2014 Share Posted October 21, 2014 Ever climbed a spiral staircase? the difference between the inner and outer handrail is an extreme example. Fred But a spirit level placed across the handrails on a radius would (theoretically) always show level. Link to post Share on other sites More sharing options...
Freddie Posted October 21, 2014 Share Posted October 21, 2014 But a spirit level placed across the handrails on a radius would (theoretically) always show level. Indeed, but they are not on the same plane. Fred Link to post Share on other sites More sharing options...
trisonic Posted October 21, 2014 Share Posted October 21, 2014 Now I remember why I like the landscape to rise and fall around my track which is dead level........................or why Florida is popular with US modelers. I also need to lay down for a spell, excuse me. Best, Pete. Link to post Share on other sites More sharing options...
Silver Sidelines Posted October 22, 2014 Share Posted October 22, 2014 Hi Gordon Your question came to mind when I read this mathematics item on the BBC news. I especially like the question (and answer) about the size of the hole in the middle of a washer when its heated up! Regards Ray Link to post Share on other sites More sharing options...
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