An error in pitch virtually increases the effective diameter of a bolt or screw and decreases the effective diameter of a nut. The meaning of the virtual change in effective diameter is that if any screw is perfect except for pitch error, it will not screw easily into a perfect ring gauge of same nominal size until its effective diameter is reduced.
For Whitworth thread, if Sp is the error in pitch then the virtual increase (decrease) in the effective diameter of the thread in case of bolt (nut) is given by the relation :
Virtual change in effective diameter = 1.921 x op. Similarly errors in flank angles also require a corresponding reduction in the effective diameter if the screw is to fit a perfect ring gauge of the same nominal size.
It 80! and 882 are the errors in flank angles in degrees (regardless of sign), the corresponding virtual change (increase or decrease) in effective diameter of the thread in case of a bolt or nut is given by (for Withworth thread) dE – 0.0105 x p (8Q1 + 882), where p is the normal pitch.
Thus it means that the error in pitch and angle can be accounted for by a suitable alternation of the effective diameter. There is however a limit to the magnitude of these errors for accurate work. This is done by fixing a maximum limit for the equivalent of the combination of pitch and angle errors in terms of the effective diameter of the gauge. Thus for Whithworth
The virtual effective diameter could now be defined clearly as the sum of the simple effective diameter and the effective diameter equivalent, as it is the effective diameter of the smallest nut of perfect form and pitch which the screw will enter e.g., for Whitworth threads :
Mathematical Derivation for Effect of Pitch Errors.
Let us imagine a perfect bolt, having some pitch error and it has to enter a nut of perfect form and pitch (Fig. 13.6). It will not be possible without experiencing a lot of strain, as the error has to be accommodated by the strain.
Fig. 13.6. Effect of pitch error.
The other way out is to increase the effective diameter of the nut. Let 6p be the cumulative pitch error over the length of the engagement.
By increasing the effective diameter of the nut and retaining the same pitch, the two threads can assemble without interference as shown in Fig. 13.7.
It is assumed that the maximum pitch error over the length of engage-
ment is equally distributed at each end of engagement. Increase in effective diameter will obviously be the vertical movement of flanks necessary to produce coincidence.
It may be mentioned here that effect of long or short pitch will be same, i.e. increase of the interference between the mating threads, so each will lead to increase in effective diameter of nut.
Fig. 13.7. Pitch error is accommodated by increasing virtual diameter.
Since cot 55°/2 = 1.921 (for Whitworth), its effect is nearly doubled when the equivalent increase in effective diameter is calculated.
Similarly the effect of pitch error will be to reduce the effective diameter of the screw.
Angle errors on threads may be either due to errors on one or both flanks. Any error in angle of thread results in interference between the bolt and nut and to accommodate it, the effective diameter of nut has to be increased. Thus like pitch errors, the angle errors also increase the virtual effective diameter of a bolt and decrease that of a nut. Assuming that one of the pairs is correct, it is possible to satisfactorily assemble the thread pairs by modifying the effective diameter. The effective diameter of an incorrect bolt must be decreased to permit a correct mating thread to mate and similarly the effective diameter of an incorrect nut must be increased.