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19.4 Use of Strength Formulas

The load carrying ability of gears was defined in section 13.1 by means of the Barth revision of the Lewis Formula.
The same formula can be applied to the metric module type gearing using the following approach:
Express the module size gear in terms of diametral pitch. The diametral pitch obtained in this manner will be a decimal value, All other parameters will be inch-system units, In other words, the module-size gear is treated as an odd-diametral pitch inch-size gear.
For the Lewis formula without the Barth modification, we illustrate the metric formulation explicitly*. The classic Lewis equation, as given in section 13.1, was as follows:

Wt = FSY                                                       (67)
        Pd

where: F = Gear-face width, inches
       Pd= Diametral pitch
            S = Stress, lbs/in2
        Y = Lewis tooth form factor
            Wt= Transmitted tooth force, lbs.
For use with metric units it is convenient to introduce the Lewis factor based on circular pitch, Yc where the two Lewis factors are related by:

    Yc  =                                                             (68)
             p

Then the metric formulation is (expressed in SI units):

Wt =   p mFSYc                                                        (69)

and for metric units:

       Wt  = Transmitted tooth force, in Newtons (N)
        F = Face width in millimeters
     m = Module, in millimeters
 
     Yc= Lewis factor, dimensionless
      S = Stress, megapascals (MPa)**

Associated with inch gearing the same value of Lewis factor, Yc can be used. However, this is limited to metric gears that have addenda and dedenda proportioned in the same values as the American gear standards for which the Yc factor applies. Fortunately, the metric ISO gear standard and American inch gears are compatible in this regard. Nevertheless, this may not be true of all metric gears (non-ISO standard) and one must carefully watch for this possible difference, then use the proper Lewis factor, Yc if a difference exists.
Other rating factors require great care in conversion to SI units—especially if the formulas involved are not dimensionally uniform—and cannot be transformed as readily as the Lewis formula. Further discussions of gear ratings for metric gears can be found in ref. 4.
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*Except for minor changes in wording, this paragraph is quoted with the permission of the Industrial press Inc. ‘World Manic Standards for Engineering" K.O. Kvemelarrd (Editor), Industrial Press, New York, N.Y., 1978; Section on ‘Mechanical Power Transmission Systems", by G.W. Uchalec and F. Buchsbaum, pp. 13.21 to 13.23. 
**One megapascal =1 MPa= 1 N/mm2 =1/9.806650 kilogram force per square millimeter or approximately0.1 kgf/mm2. For reference, I 000psi = 8.894757 MPa 7MPa.

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