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Balance Wheel inertia/torque issue.

  1. Sep 23, 2012 #1
    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.
     

    Attached Files:

    Last edited: Sep 24, 2012
  2. jcsd
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