Page 3 – Some Maths

Mimesis 30×40 – PUlley

I need to modify the chart I posted just above as since I created that I scaled the design for a 30″ dia roller which ended up 40″ long. Rather than simply making it shorter to match the 36″ length for the largest breaker I simply adjusted numbers. In any case let’s look at the pulley.

When I created this I simply “eyeballed” the thing. What I ended up with after the scaling was a pulley with the following dimensions (I design for O scale/1:48 so I will start with the dimensions in mm and show them in inches and then full scale.)

 mminFS in

I have no real idea of where I am going with this .. so I will just go and see where it ends up … the following is taken from the book – A Treatise on Belts and Pulleys by John Howard Cromwell pub. 1888

Page 19 – “Power – By the power of a pulley we mean the force with which the circumference of the pulley turns: it is equal to that force which, if applied to the pulley-circumference in a direction opposite to that in which the pulley rotates, would be just sufficient to stop the motion of the pulley.” – to me this the braking force required to stop the pulley.
Page 23 and 24 – “If we represent the horse-power of a pulley by H, and the circumferential force or power in pounds by P, then H X 33,000 pounds lifted one foot high per minute will represent the power of the pulley. If therefore we denote by v_m the circumferential velocity of the pulley in feet per minute, we have have, for the power in pounds, the expression –

P = 33000H/v_m

From the chart on page one for the Mimesis 30×40 we find it requires 45 HP to run (minimum) at 100 RPM. That then gives us – H=45
From the chart above we see that the pulley is 32.5" dia. The Circumference of a circle is C=\pi D. Therefore the Circumference of this pulley is 102.1" or 8.51'. Since the pulley is running at 100 RPM then the v_m = 850.8

Sticking these numbers back into our formula we get P = 33000 * 45/850.8 = 1745.4

Belt Width Table

Soooo .. P = 1745.4 – which is nice and all and exactly how does that help? Had no idea but was fun playing with the numbers. Ah ha. I ran across this chart which shows belt width for different P. (from Page 110).

I can see that with riveted joints I would still need a 20″ belt for that P of 1745 .. twice what the pulley currently is.

I can now start to play a bit with the design. Fun stuff.

Note: Where it says a=180^{\circ} it is talking about two pulleys aligned horizontally to each other. Where it says \acute{o}=7/32" .. this is the belt thickness

Calculating P for 100 RPM and 45 HP

I created a spreadsheet in Excel. Yeah – I know I could have reduced the columns 2/3 to one but spreading the formulas out makes it easier to catch mistakes. JMO.

This is using the afore mentioned 100 RPM and 45 HP.

  • Column 1 : Pulley Dia. In.
  • Column 2 : Circumference in In.
  • Column 3 : Circumference in Ft.
  • Column 4 : vm
  • Column 5 : P

This was set up so that the breaker runs at 45 HP at 100 RPM per the earlier chart. P decreases as the dia of the pulley increases. A 6 ft dia / 72 inch pulley running at the stated HP and RPM has a P of 787.82 and would require a 9 in belt. A 4 ft dia / 48 inch pulley would need to have a P of 1181.73 which would require a 14 inch belt (riveted joint). For any pulley much larger than the 32.5 inch version in my drawing would require that either the breaker be held off of the mounting surface or a slot for the larger pulley part of the design. In any case .. fun times!

On a side note. In order to keep the original pulley diameter with a 1719 P (rounded) it would require a 20 inch belt. A combination of a larger dia pulley and wider belt are all possible using the data to Engineer the pulley .. not just eyeball what looks correct.

Visually, here is the difference. The 32″ dia pulley has a P= 1773 from the Calculating P chart. We can see from the chart above that .. call it the “Leather Belt” chart a 20″ riveted belt has a P=1802. The 64″ dia pulley has a P=886 from the Calculating P chart. From the “Leather Belt” chart a 10″ belt has a P=901.

Keep in mind that for this design, any pulley larger than about 32″ dia will require you either sit the breaker high enough for the larger pulley to clear or engineer a slot in the”Ground Line”. Since you have to design for the exit hopper in an case this is more of a “keep it in the back of your mind”. I think that mounting the breaker on concrete risers would make sense in that you could make the risers high enough either for a conveyor belt or even a truck – you have to have a way to get the crushed coal out even if it is simply men with wheelbarrows and shovels.