Worm Contact.*
Page 12
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Again us the co-tangent of the angle included between the axis and the tangent to the curve of section at the point whose co-ordin, ates are xy. Or again, x is the tangent of the angle included between the normal to the curve of section at zy and the pitch line (which is parallel to the axis).
In Fig. 27 CP2D is the plane section of the helix and P2S is the normal at P. S is therefore the pitch point, and if RST are fresh axes of co-ordinates, and X and Y the co-ordinates of P2 referred to them, we have Xx (to
which is the equation of the path of contact belonging to the plane section whose equation is (6).
In using (in) we must remember that
y = y r (rx) where r .= the pitch radius of the worm. And hence (to) can he ssed in terms of X and Y and constants.
APPENDIX II.
If Fig. 28 represents a plane section of the curved surfaces of a orm and worm-wheel, with a thin film of oil or grease sustaining the load pressing them together, it is reasonable to suppose that the intensity of pressure will he greatest at c where they are nearest together, and that it will tend to become zero at points a and b, where the surfaces are so separated that the oil-film breaks down. The distribution of pressure may in fact be represented by the curve MQN, ordinates from the base line MN, representing intensity of pressure. The width MN may he taken as representing the " effective breadth " of contact for the section. It is clear that this "effective breadth" varies Ye( with the curvature of the sur e.. faces, the nature of the lubricant and the surfaces themselves, and the relative or rubbing velocity. Whilst the effect of the last three of the variables enumerated can he only dealt with experimental 1 y, the effect of curvature alone can be treated mathematically. Thus for constant conditions of lubricant, nature of surfaces and rubbing velocity, there should be some thickness of film t at which the power of sustaining load In this investigation the surfaces are supposed to be actually touching at one point, a condition not quite in accordance with what one might anticipate. but it may reasonably be assumed that the film's thickness at the minimum distance of the surfaces is small compared with its maximum thickness, so that equation (7) may be regarded as approximately true.
A PPJ2ND IX III.
The expression given for the effective breadth of contact is capable of consider able simplification when contact takes place at thepiich line.
The theorem upon which this depends is to be found in any work which treats of the curvature of the envelope of a curve carried by a rolling curve and invariably connected with it, and is attributed to Chasles (see J. 355, Williamson's Differential Calculus," 6th edition).
Let AB, Fig. 30, be a straight line (the pitch line of a worm), and Cl) a curve connected by it and carried by it (a worm
tooth section). Let AB roll on the circle MN (the pitch line of the worm-wheel), and in so doing let CD generate the envelope ST (which is therefore the section of the mating worm-wheel tooth). Then, when contact takes place at the pitch line, if re and r2 are the radii of curvature of the surface sections in contact at P Fig.23 where R is the radios of MN (and therefore the pitch line radius of the worm-wheel), and a is the angle of inclination of the contact path at the pitch line. Hence at the pitch line b = K „v/le cos a. This expression for —is a good average value for the contact of any two sections or ri +F51
worm and worm-wheel throughout their contact.
The following Table shows that the average value for V cos a does not greatly alter for very considerable variations of the ratio of worm-thread angle. The pitch radius of the worm is throughout 6 inches.
Refer to Figs. rg, zo, and zoa.