Fig. 62.
Fig. 63.
Fig. 64.
Internal or annular gear-wheels have their tooth curves formed by rolling the generating circle upon the pitch circle or base circle, upon the same general principle as external or spur-wheels. But the tooth of the annular wheel corresponds with the space in the spur-wheel, as is shown in [Fig. 61], in which curve a forms the flank of a tooth on a spur-wheel p, and the face of a tooth on the annular wheel w. It is obvious then that the generating circle is rolled within the pitch circle for the face of the wheel and without for its flank, or the reverse of the process for spur-wheels. But in the case of internal or annular wheels the path of contact of tooth upon tooth with a pinion having a given number of teeth increases in proportion as the number of teeth in the wheel is diminished, which is also the reverse of what occurs in spur-wheels; as will readily be perceived when it is considered that if in an internal wheel the pinion have as many teeth as the wheel the contact would exist around the whole pitch circles of the wheel and pinion and the two would rotate together without any motion of tooth upon tooth. Obviously then we have, in the case of internal wheels, a consideration as to what is the greatest number (as well as what is the least number) of teeth a pinion may contain to work with a given wheel, whereas in spur-wheels the reverse is again the case, the consideration being how few teeth the wheel may contain to work with a given pinion. Now it is found that although the curves of the teeth in internal wheels and pinions may be rolled according to the principles already laid down for spur-wheels, yet cases may arise in which internal gears will not work under conditions in which spur-wheels would work, because the internal wheels will not engage together. Thus, in [Fig. 62], is a pinion of 12 teeth and a wheel of 22 teeth, a generating circle having a diameter equal to the radius of the pinion having been used for all the tooth curves of both wheel and pinion. It will be observed that teeth a, b, and c clearly overlap teeth d, e, and f, and would therefore prevent the wheels from engaging to the requisite depth. This may of course be remedied by taking the faces off the pinion, as in [Fig. 63], and thus confining the arc of contact to an arc of recess if the pinion drives, or an arc of approach if the wheel drives; or the number of teeth in the pinion may be reduced, or that in the wheel increased; either of which may be carried out to a degree sufficient to enable the teeth to engage and not interfere one with the other. In [Fig. 64] the number of teeth in the pinion p is reduced from 12 to 6, the wheel w having 22 as before, and it will be observed that the teeth engage and properly clear each other.