Frictional Torque of a Spinning Top

[MR SPIN]

Hello,

This is not a home work question but this seems to be the most appropriate spot to post this. I am trying to design a top that will spin for as long as possible. (Google "spinning top" if you're not sure what I a mean by a top) . I've been experimenting with different materials and top designs and have managed to get a top to spin for about 20 min with the assistance of electric starter.

I need help calculating the frictional torque after the top is released . Can you check my analysis and tell me if I am right?

Assumptions
Top initial av = 125.6 rad/sec
Air resistance is zero
The top has 2 main parts
1 - Metal flywheel
2 - "non flywheel" part of the top (inside section, stem, pivot ball have an immaterial mass and MOI.)
TOP 1
Flywheel
Stainless Steel Hoop
Density 8 g/cc

Outside Diameter 62 mm
Wall thickness 1 mm
Outside Radius 31 mm
Inside Radius 30 mm
Height 13 mm

vol = 2.5 cc
mass = 20 gI = 1/2 m (r1^2 + r2^2)
= .5 * 0.02 kg * (.031^2 + 0.030^2)
= 1.85 x 10 ^ -5
The top spins on a 4mm diameter tungsten ball attached (fixed, glued on) to the bottom of a straight stem that goes directly through the central axis of the flywheel. The top is spinning on glass surface with a 0.2 coefficient of friction. The estimated radius of the contact area (spherical cap) the ball makes with glass is 0.2 mm.

Here's how I got 0.2mm:

Base surface area (contact area) of 4 mm ball
= pi * h (2r-h)
= 3.14159 * 0.01 mm * ( 2*2mm - 0.01mm)
= 0.125349

I just picked h= 0.01 mm out the air (who know how much of the tip actually contacts the surface, h =0.01 is a "constant" I will use when comparing designs)

A=pi*r ^2

r = sqrt (0.125349/pi)
= 0.2 mmfrictional torque = 2/3 * mass * g * cof * r (contact radius)
= 2/3 * 0.020 g * 9.8 * 0.2 * 0.0002 m
= 5.2 x 10^-6

spin time = 125.6 (I/torque)
= 125.6 (1.85 x 10 ^ -5 / 5.2 x 10^-6)
= 447 secI am not too concerned about finding the exact COF, friction contact surface area , spin times. I will

use COF and contact surface area as the same "constants" in all my designs. I am more concerned with

the making comparisions with other designs on a relative basis. Eg top design 1 spins for x sec, top

design 2 spins for 2x seconds. I know air drag/resistance plays a very large role in overall spin

times but for now I just want to isolate the impact of tip friction only.

Am I doing this correctly? If I am I have some very interesting results I would like to get your

opinion on. Eg It does not matter what the flywheel height is. It could be 1000 mm or 1 mm , the

theoretical spin time with always be the same. Also it does not matter what the density of the material

is , the spin time will always be the same as long the the hoop dimensions stay constant. You could make

the top out of tungsten or wood and still get the same theoretical spin time.

Bottom line: Are the above calculations correct?

 

[haruspex]

Am I doing this correctly?

Looks good to me.

it does not matter what the density of the material is

Yes, until you consider drag.

 

[MR SPIN]

Looks good to me.

Yes, until you consider drag.

Thanks for the confirmation. Will work next on a drag analysis . Appears drag consists of 2 main components 1 ) pressure drag 2) viscous drag. Off to the books.

 

[Dr.D]

Your 4 mm ball looks rather large to me. By reducing the ball diameter, you automatically reduce the friction radius. This was commonly done in the watch industry using pointed pivots running in pockets in jewel bearings. You can do a lot better, I suspect, than the 4 mm ball.

 

[MR SPIN]

Your 4 mm ball looks rather large to me. By reducing the ball diameter, you automatically reduce the friction radius. This was commonly done in the watch industry using pointed pivots running in pockets in jewel bearings. You can do a lot better, I suspect, than the 4 mm ball.

Yes, my next top will have a smaller ball. My analysis shows a 2mm ball increases spin time by about 40% and 1 mm ball almost doubles the spins time (in theory). Will try 2 mm ball in my next design.

 

[Dr.D]

Have you applied Hertz contact stress theory to look at the size of the contact zone? This would also give you the pressure distribution over the contact zone and therefore enable you to calculate the friction torque, at least numerically if not in closed form. I am certain that the smaller you can make the tip (and thus the smaller the contact zone), the lower the friction torque will be.

 

[MR SPIN]

Have you applied Hertz contact stress theory to look at the size of the contact zone? This would also give you the pressure distribution over the contact zone and therefore enable you to calculate the friction torque, at least numerically if not in closed form. I am certain that the smaller you can make the tip (and thus the smaller the contact zone), the lower the friction torque will be.

I googled "Hertz contact stress theory" and definitely looks interesting but way over my head. My next big concern is finding the most aerodynamic shape.