Fig. 137.
An improved form of template odontograph, designed by Professor Robinson of the Illinois School of Industry, is shown in [Fig. 137].
In this instrument the curved edge, having graduated lines, approaches more nearly to the curves produced by rolling circles than can be obtained from any system in which an arc of a circle is taken to represent the curve; hence, that edge is applied direct to the teeth and used as a template wherefrom to mark the curve. The curve is a logarithmic spiral, and the use of the instrument involves no other labor than that of setting it in position. The applicability of this curve, for the purpose, arises from two of its properties: first, that the involute of the logarithmic spiral is another like spiral with poles in common; and, second, that the obliquity or angle between a normal and radius sector is constant, the latter property being possessed by this curve only. By the first property it is known that a line, lying tangent to the curve c e h, will be normal or perpendicular to the curve c d b; so that when the line d e f is tangent to the pitch line, the curve a d b will coincide very closely with the true epicycloidal curve, or, rather, with that portion of it which is applied to the tooth curve of the wheel. By the second quality, all sectors of the spiral, with given angle at the poles, are similar figures which admit of the same degree of coincidence for all similar epicycloids, whether great or small, and nearly the same for epicycloids in general; thus enabling the application of the instrument to epicycloids in general.
To set the instrument in position for drawing a tooth face a table which accompanies the instrument is used. From this table a numerical value is taken, which value depends upon the diameters of the wheels, and the number of teeth in the wheel for which the curve is sought. This tabular value, when multiplied by the pitch of the teeth, is to be found on the graduated edge on the instrument a d b in [Fig. 137]. This done, draw the line d e f tangent to the pitch line at the middle of the tooth, and mark off the half thickness of the tooth, as e, d, either on the tangent line or the pitch line. Then place the graduated edge of the odontograph at d, and in such a position that the number and division found as already stated shall come precisely on the tangent line at d, and at the same time so set the curved edge h f c so that it shall be tangent to the tangent line, that is to say, the curved edge c h must just meet the tangent line at some one point, as at f in the figure. A line drawn coincident with the graduated edge will then mark the face curve required, and the odontograph may be turned over, and the face on the other side of the tooth marked from a similar setting and process.
Fig. 138.
For the flanks of the teeth setting numbers are obtained from a separate table, and the instrument is turned upside down, and the tangent line d f, [Fig. 137], is drawn from the side of the tooth (instead of from the centre), as shown in [Fig. 138].
It is obvious that this odontograph may be set upon a radial arm and used as a template, as shown in [Fig. 126], in which case the instrument would require but four settings for the whole wheel, while rolling segments and the making of templates are entirely dispensed with, and the degree of accuracy is greater than is obtainable by means of the employment of arcs of circles.
The tables wherefrom to find the number or mark on the graduated edge, which is to be placed coincident with the tangent line in each case, are as follows:—