- Joined
- Feb 18, 2016
- Messages
- 451
Ok, I thought this was going to be easy and productive but it's becoming a nightmare. The numbers I'm getting just aren't realistic.
Here's my thinking:
I was going to simply calculate what the inner and outer diameters of a tapered wheel would need to be to make both wheels turn the same number of times on a 24" radius track. It turns out to be significantly different diameters. The larger diameter I would choose to be 1.25" (same as on the Bachman model I'm using). But then when I calculate what the smaller diameter would need to be it comes out to only be about 0.86", that would be one whale of a tapered wheel flat! Either I'm doing something wrong or this just isn't going to work out as nice as I thought it might.
Here's My Actual Calculations (done in a sleepy state of mind, so I may have made some major mistakes hopefully)
First, I decided to just arbitrarily choose a 90 degree curved section of track for the calculation. This makes calculations easy since it's just 1/4 of a circle.
So here's where I start:
The center line of curved section of track is 24" radius. The distance between the rails is 1.750"
Using those numbers the radius of the outer rail is just 24" + 1/2(1.750") = 24.88" radius
The radius of the inner rail is 24" - 1/2(1.750") = 23.13" radius.
Now armed with those radii I can calculate the length of those rails.
The circumference of a circle just Pi times the diameter. Since I'm only working with 90 degree turn I include a factor of 1/4. (i.e. I'm only working with 1/4 of the circle).
So to find the length of a rail I use the formula:
1/4 (one-forth of a circle) times (2 x the radius i.e. the diameter) times Pi.
So the length of the outer rail on a 90 degree turn is 1/4 x (2 x 24.88) x Pi = 39.07"
And the length of the inner rail on a 90 degree turn is 1/4 x (2 x 23.13) x Pi = 36.32"
So the difference in length of travel over the outer rail compared with the inner rails is 39.07" - 36.32" = 2.75"
This means that the outer wheel must travel 2.75" (almost 3 inches) further when going around these tight bends.
Now I choose an arbitrary large diameter for the bogie wheel. I'm using 1.25" since this is what the wheels on my Bachman train set measure to be. By the way, the smaller diameter is only about 0.015" smaller. Not much difference at all.
So onward with my calculation!
Now I figure out how many times the wheel will need to turn to traverse the outer rail.
First I calculate the circumference of the wheel. That's just Pi times the diameter. I choose to the diameter to be 1.25" so the circumference of the wheel is 1.25" x Pi = 3.93". In other words every time this wheel rotates one revolution it travels 3.93" down the rail (assuming no slippage).
Now I just figure out how many times the wheel needs to turn to complete a 90 degree curve. And that is just the distance of the rail divided by how far the wheel travels per revolution. Or 39.07" / 3.93" = 9.95 turns.
So the wheel has to make about 10 revolutions to go around a 90 degree turn on the outside rail.
So how to find out what the diameter of the smaller radius would need to be to keep the inside wheel turning the same number of turns we just use that 9.95 number of turns to figure that out.
The inner wheel must turn 9.95 turns to travel over the length of the inner rail which we had previously calculated to be 36.02". So we just divide "36.02" / 9.95 turns = 3.56".
So the smaller diameter of the bogie wheel needs have a circumference of 3.56"
Now we can calculate what diameter it must have.
The circumference of a circle is given by Pi times the diameter, so the diameter must be given by Pi divided by the circumference.
So we have Pi / 3.56" = 0.86"
WOW! That a lot smaller than the larger diameter of 1.25" In fact it's 1.25" - 0.86" = 0.39
That's a big difference. Way bigger than what Bachman is using. In fact, this difference would create a wheel that has a seriously slanted surface that rides on the rails.
~~~~~
Conclusion, either I've made some major mistakes here, or a 24" radius curve really is so sharp that it would require this steep of an angle to make the wheels run true.
If these calculations are right, I'd be better off allowing the individual wheels to just spin freely on the axles and not use a solid axle at all.
I may have made a major mistake in my calculations and methodology. I'm tired. I'm going to bed.
I was hoping to get a number I could work with, but a 0.39" difference between the large and small radius of a 1/4" wide wheel flat would be enormous. Seriously not good. Like I say, this Bachman model only has a difference in diameter across the flat of about 0.015". Hardly noticeable.
But now that you've mentioned this whole thing I'm seriously thinking of just making the wheels independently free to rotate however they like. That way the outer wheels can just turn more times with no problem.
Here's my thinking:
I was going to simply calculate what the inner and outer diameters of a tapered wheel would need to be to make both wheels turn the same number of times on a 24" radius track. It turns out to be significantly different diameters. The larger diameter I would choose to be 1.25" (same as on the Bachman model I'm using). But then when I calculate what the smaller diameter would need to be it comes out to only be about 0.86", that would be one whale of a tapered wheel flat! Either I'm doing something wrong or this just isn't going to work out as nice as I thought it might.
Here's My Actual Calculations (done in a sleepy state of mind, so I may have made some major mistakes hopefully)
First, I decided to just arbitrarily choose a 90 degree curved section of track for the calculation. This makes calculations easy since it's just 1/4 of a circle.
So here's where I start:
The center line of curved section of track is 24" radius. The distance between the rails is 1.750"
Using those numbers the radius of the outer rail is just 24" + 1/2(1.750") = 24.88" radius
The radius of the inner rail is 24" - 1/2(1.750") = 23.13" radius.
Now armed with those radii I can calculate the length of those rails.
The circumference of a circle just Pi times the diameter. Since I'm only working with 90 degree turn I include a factor of 1/4. (i.e. I'm only working with 1/4 of the circle).
So to find the length of a rail I use the formula:
1/4 (one-forth of a circle) times (2 x the radius i.e. the diameter) times Pi.
So the length of the outer rail on a 90 degree turn is 1/4 x (2 x 24.88) x Pi = 39.07"
And the length of the inner rail on a 90 degree turn is 1/4 x (2 x 23.13) x Pi = 36.32"
So the difference in length of travel over the outer rail compared with the inner rails is 39.07" - 36.32" = 2.75"
This means that the outer wheel must travel 2.75" (almost 3 inches) further when going around these tight bends.
Now I choose an arbitrary large diameter for the bogie wheel. I'm using 1.25" since this is what the wheels on my Bachman train set measure to be. By the way, the smaller diameter is only about 0.015" smaller. Not much difference at all.
So onward with my calculation!
Now I figure out how many times the wheel will need to turn to traverse the outer rail.
First I calculate the circumference of the wheel. That's just Pi times the diameter. I choose to the diameter to be 1.25" so the circumference of the wheel is 1.25" x Pi = 3.93". In other words every time this wheel rotates one revolution it travels 3.93" down the rail (assuming no slippage).
Now I just figure out how many times the wheel needs to turn to complete a 90 degree curve. And that is just the distance of the rail divided by how far the wheel travels per revolution. Or 39.07" / 3.93" = 9.95 turns.
So the wheel has to make about 10 revolutions to go around a 90 degree turn on the outside rail.
So how to find out what the diameter of the smaller radius would need to be to keep the inside wheel turning the same number of turns we just use that 9.95 number of turns to figure that out.
The inner wheel must turn 9.95 turns to travel over the length of the inner rail which we had previously calculated to be 36.02". So we just divide "36.02" / 9.95 turns = 3.56".
So the smaller diameter of the bogie wheel needs have a circumference of 3.56"
Now we can calculate what diameter it must have.
The circumference of a circle is given by Pi times the diameter, so the diameter must be given by Pi divided by the circumference.
So we have Pi / 3.56" = 0.86"
WOW! That a lot smaller than the larger diameter of 1.25" In fact it's 1.25" - 0.86" = 0.39
That's a big difference. Way bigger than what Bachman is using. In fact, this difference would create a wheel that has a seriously slanted surface that rides on the rails.
~~~~~
Conclusion, either I've made some major mistakes here, or a 24" radius curve really is so sharp that it would require this steep of an angle to make the wheels run true.
If these calculations are right, I'd be better off allowing the individual wheels to just spin freely on the axles and not use a solid axle at all.
I may have made a major mistake in my calculations and methodology. I'm tired. I'm going to bed.
I was hoping to get a number I could work with, but a 0.39" difference between the large and small radius of a 1/4" wide wheel flat would be enormous. Seriously not good. Like I say, this Bachman model only has a difference in diameter across the flat of about 0.015". Hardly noticeable.
But now that you've mentioned this whole thing I'm seriously thinking of just making the wheels independently free to rotate however they like. That way the outer wheels can just turn more times with no problem.