# Balance Wheel inertia/torque issue.

1. Sep 23, 2012

### horologist101

I am a watchmaker and need some help with the formula and correct units to derive the Frequency of the Balance wheel. I'm am trying to prove the formula with a known Balance Wheel/Spring and Frequency but using all known values still cannot get the formula to work. When I have a working formula then I can make changes to the wheel and spring and predict what the result will be.

Formula given:

T(Hz)= 2∏√[12.M.r.sup.2.l/E.h.e.sup.3]

Issue 1.
in the text
M is described as Mass
'r.sup.2' is described as Radius of Gyration (Rg)
however
M.r.sup.2 is the formula for 2nd Moment of Inertia (Mass x radius).....so I'm not sure if I should be finding Rg or 2nd Moment of Inertia then multiply by the Mass again (M.Rg.sup.2).

Issue 2.
E(Modulus of Elasticity) should be in the units Nmm.sup.2/rad but I can only find E in Gpa ....is there a difference?

Issue 3.
The (Hair)spring is a flat spiral spring (example: 0.30mm high x 0.12mm thick x 260mm long)but when comparing 2 formula for Elasticity 'e' and 'h' are confused so that it is not clear which dimension should be 'e' or 'h' (height and thickness)

0.30 x 0.12.sup.3 = 0.0005184
0.12 x 0.30.sup3 = 0.00324

Issue 4.
The first formula seems to ignore phi θ for some reason (because we are not exceeding the elastic limit of the spring?)
Modulus of Elasticity= 2∏√[12.l.θ/E.h.e.sup.3]

Issue 5.
The Frequency (F) in Hz is also derived from this formula:
F= 1/2∏√(K/I)
Where
K is the spring Constant (Nm/rad)
I is the Mass (Kgm.sup.2)

K is derived from the formula:

K= M/.phi.

Where M = Modulus of Elasticity (back to my problem of working out M)
Where phi θ is described as the 'unit angle of twist' (of the spring) but I've no idea what value this should be (Oscillation of the balance wheel is 270°) and in what units (degrees or radians).

Finally here are the known dimensions of my Balance Wheel and Hairspring:
F(Frequency) = 2.5Hz

Balance Wheel
Wheel Diameter = 17.4mm
R1 (Outside rim radius) = 8.6mm
r2 (Inside rim radius) = 6.8mm
Mass = 0.56gms
Rim Height = 0.75mm
Rim Thickness = 1.8mm
Density = 0.00825 (approx)

Hairspring
E (Modulus of Elasticity) = 221Gpa (this value is a guess based on a range of materials used within the industry for Hairsprings)
l (Length) = 260mm
h (height) = 0.30mm
e (thickness) = 0.12mm
θ (oscillation?) = 270°

I hope this doesn't give anyone a head ache! Any assistance would be appreciated.

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Last edited: Sep 24, 2012