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4.7 Contact Ratio

To assure smooth continuous tooth action, as one pair of teeth ceases contact a succeeding pair of teeth must already have come into engagement. It is desired to have as much overlap as possible. A measure of this overlapping action is the contact ratio. This is a ratio of the length of the line-of-action to the base pitch. Figure 1.16 shows the geometry. The length-of-action is determined from the intersection of the length-of-action arid the outside radii. The ratio of the length-of-action to the base pitch is determined from:
                   m_p = (R_o² - R_b²) +(r_o² - r_b²) - Csin f                                                                        (18)  
                                          P_c COS f

       It is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1, calculated for all tolerances at their worst-case values.
        A contact ratio between 1 and 2 means that part of the time two pairs of teeth are in contact and during the remaining time one pair is in contact. A ratio between 2 and 3 means 2 or 3 pairs of teeth are always in contact. Such as high contact ratio generally is not obtained with external spur gears, but can be developed in the meshing of an internal and external spur gear pair or specially designed non-standard external spur gears.

4.8 Undercutting

From Figure 1.16 it can be seen that the maximum length of the line-of-contact is limited to the length of the common tangent. Any tooth addendum that extends beyond the tangent points (T and T') is not only useless, but interferes with the root fillet area of the mating tooth. This results in the typical undercut tooth, shown in Figure 1.17. The undercut not only weakens the tooth with a wasp-like waist, but also removes some of the useful involute adjacent to the base circle.

From the geometry of the limiting length-of-contact (T-T', Figure 1.16) it is evident that interference is first encountered by the addenda of the gear teeth digging into the mating-pinion tooth flanks. Since addenda are standardized by a fixed ratio (1/P_d) the interference condition becomes more severe as the number of teeth on the gear increases. The limit is reached when the gear becomes a rack. This is a realistic case since the hob is a rack-type cutter. The result is that standard gears with

T44