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If the tooth thickness of either or both mating gears is less than the zero backlash value, the amount of backlash introduced in the mesh is simply this numerical difference:    j = Sstd - Sact = DS                (14-1) where:    j = linear backlash measured along the pitch circle (Figure 14-2(b))    Sstd = no backlash tooth thickness on the operating pitch circle, which is the standard tooth thickness for ideal gears    Sact = actual tooth thickness    Backlash, Along Line-of-Action = jn = jcosa    When the center distance is increased by a relatively small amount, Da, a backlash space develops between mating teeth, as in Figure 14-3. The relationship between center distance increase and linear backlash jn along the line-of-action is:    jn = 2 Da sina                             (14-2)    This measure along the line-of-action is useful when inserting a feeler gage between teeth to measure backlash. The equivalent linear backlash measured along the pitch circle is given by:    j = 2 Da tana                         (14-3a) where:    Da = change in center distance     a = pressure angle

Hence, an approximate relationship between center distance change and change in backlash is:    Da = 1.933Dj for 14.5º pressure angle gears  (14-3b)    Da = 1.374Dj for 20º pressure angle gears     (14-3c)    Although these are approximate relationships, they are adequate for most uses. Their derivation, limitations, and correction factors are detailed in Reference 10.    Note that backlash due to center distance opening is dependent upon the tangent function of the pressure angle. Thus, 20º gears have 41% more backlash than 14.5º gears, and this constitutes one of the few advantages of the lower pressure angle.    Equations (14-3) are a useful relationship, particularly for converting to angular backlash. Also, for fine pitch gears the use of feeler gages for measurement is impractical, whereas an indicator at the pitch line gives a direct measure. The two linear backlashes are related by:       j =   jn                                            (14-4)            cos a    The angular backlash at the gear shaft is usually the critical factor in the gear application. As seen from Figure 14-2(a), this is related to the gear's pitch radius as follows:       jq = 3440 j  (arc minutes)                (14-5)                    R1        Obviously, angular backlash is inversely proportional to gear radius. Also, since the two meshing gears are usually of different pitch diameters, the linear backlash of the measure converts to different angular values for each gear. Thus, an angular backlash must be specified with reference to a particular shaft or gear center.    Details of backlash calculations and formulas for various gear types are given in the following sections. 14.2 Backlash Relationships    Expanding upon the previous definition, there are several kinds of backlash: circular backlash Jt, normal backlash jn, center backlash jr, and angular backlash Jq(º), see Figure 14-4.    Table 14-1 reveals relationships among circular backlash jt, normal backlash jn and center backlash Jr. In this definition, Jr is equivalent to change in center distance, Da in Section 14.1.

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