From the Rolls-Royce experimental archive: a quarter of a million communications from Rolls-Royce, 1906 to 1960's. Documents from the Sir Henry Royce Memorial Foundation (SHRMF).
Analysis of worm gear design, contact mechanics, load capacity, and comparison with bevel gears.
| Identifier | ExFiles\Box 136\5\ scan0319 | |
| Date | 1st September 1937 guessed | |
| [Page 2] Worm Gearing—contd. change its shape and its length, and this it continues to do until tooth and thread lose contact altogether, the line of contact having gradually diminished to zero length during the final stages. Other typical positions of the line of contact are indicated by CD and EF. Further rotation of the gears brings a tooth δ adjacent to α successively into the positions occupied by α. Simultaneously the line of contact on tooth δ undergoes the same changes as those experienced by A1,A2,A3,A4,A5. Zone of contact. The cycle of events may be summarised by saying that whatever the angular positions of the gears, the lines of contact lie on a certain three-dimensional surface, defined by the paths of the various points of contact of worm and wheel on planes perpendicular to the axis of the worm wheel. This surface is known as the “zone of contact.” In any angular position of the gears, contact between worm and wheel is confined to a line or lines contained in the zone of contact. As the gears revolve the line (or lines) of contact sweep over the zone of contact. They originate near one end of the zone and move along it in unison with the movements of worm and worm wheel, finally disappearing at the opposite end of the zone to that at which they started. To ensure continuity of tooth action, with uniform transmission of angular velocity, there must be at least one line of contact in every angular position of the gears. Fig. 2 gives some indication of the shape of the zone of contact in the case of a typical automobile worm drive. Motion of contact lines. When the shape and position of the zone of contact are known the motion of the contact lines along it can easily be found and from this the motion of the contact line over the worm threads and over the wheel teeth can be determined. What happens on the worm wheel is usually the more important because it is normally made of softer material than the worm. The most suitable condition for maintaining the oil film at the line of contact requires that : (a) The motion of the contact line over the worm wheel tooth should be in approximately the same direction as the sliding velocity of the worm on the wheel. (b) The contact line should be approximately at right angles to its direction of motion. If the gears are correctly designed and manufactured, the contact lines sweep the whole surface of the flank of each worm-wheel tooth (except for a small area near the “entering” side of the wheel) and as this often shows itself, after a little running, as a uniform polish on the teeth, the impression may be formed that all the polished area touches the worm simultaneously. From what has been said it will be understood that such is not the case. Contact between each worm-wheel tooth and the worm is confined to a single line which changes its position continuously as the gears rotate in operation. Fig. 2. Three views of zone of contact in typical automobile worm drive. Choice of thread form. An early practice in the design of worm gears was to use a worm thread which is straight sided on a plane section containing the axis of the worm. This probably is of practical importance in connection with spur or helical gears because it makes possible the generation of such gears by means of cutters having straight-sided teeth. In the case of a worm the advantage is not so marked because although the worm thread might be chased in a screw-cutting lathe by means of a straight-edged cutting tool, the thread profile of nearly all precision worms are finished by grinding, and an abrasive wheel of non-linear profile would be required to produce threads which are straight-sided on the axial section. Furthermore, it is not practicable to adopt a standardised axial pressure angle for worms in the same way as for spur and helical gears. The reason is that, especially in worms of high lead angle, it is necessary to adopt a high pressure angle in the central plane if undercutting of the worm wheel teeth is to be avoided in other parallel planes contained within the width of the worm wheel. Also it is found that the disposition and motion of the contact lines in worm gears of this type tend to be far from ideal from the point of view of oil film maintenance. A form of thread which largely avoids these disadvantages has been in use in automobile worm gears for many years and has been adopted, comparatively recently as the British Standard. It is known as the “involute helicoid,” because the transverse section of the worm shows involute gear teeth. Fig. 3. Transverse, normal and axial sections of involute helicoid worm-thread. Involute helicoid. Fig. 3 shows the transverse, axial and normal sections of the thread of a typical involute helicoid worm. It will be noted that not one of these sections is straight-sided so that, on the face of it, the worm is less simple than the older type, based on a straight-sided axial section. On the other hand, the thread surface does contain a straight line longer than any of the three profiles shown, and this characteristic of the involute helicoid worm makes manufacture and measurement much simpler than is the case in a worm whose axial section shows a straight-sided profile. Referring to Fig. 4, it will be seen that an involute is described by any point on a sheet of paper unrolled from a cylinder and that transverse sections of the cylinder are the base circles from which originate the various involutes. The edge of the paper sweeps out a surface which is an involute helicoid, since successive transverse sections of it are all involutes originating on the straight line AB marked out by the edge before unrolling commenced. The edge AB is the “generator” of the involute helicoid surface which contains accordingly an infinite number of straight lines. Fig. 4. Development of involute helicoid. [Page 3] Worm Gearing—contd. The thread profile of an involute helicoid worm can therefore be made to give full contact with a straight line set at a distance from the axis of the worm equal to the “base radius,” and inclined to a transverse plane of the worm at an angle equal to the “base lead angle.” This forms the basis of a simple method of checking the profile. Since the profile of the involute helicoid worm is everywhere convex, it is possible to set a flat-sided abrasive wheel in such a position as to touch the worm along a generator. Consequently the worm thread profile may be precision-ground by such a wheel. Other thread forms. It will be seen that the main points in favour of the involute helicoid worm are facilities which it offers for manufacture and inspection. Apart from this it has no unique claim to distinction, and in fact other thread forms are in use. The first essential in designing a worm gear on any such basis is to make sure that there is no appreciable interference to lead to undesirable worm wheel tooth profiles in any part of the face width. Worm wheel teeth. From Fig. 1 it will be observed that the working surface of a worm wheel tooth is of complicated shape inasmuch as the sections of any tooth on planes parallel to the axis of the wheel are, in general, all different one from another. This leaves only one broad basic method of cutting worm-wheel teeth and that is by “generation.” The cutter (a hob) is essentially a gashed and relieved edition of the worm and it is gradually brought into the same position relatively to the worm-wheel blank as that to be occupied by the worm when the gears are assembled, blank and cutter being meanwhile rotated at relative angular velocities corresponding to those of worm and worm wheel and giving suitable cutting speeds to the working edges of the cutter. This process automatically produces a worm wheel which will mesh correctly with the worm provided only that no section of the worm thread on a plane perpendicular to the worm-wheel axis is such as to lead to “interference” in the generating process. This proviso imposes a limit on the face width of the worm wheel for any given worm ; otherwise it might be thought that the tip radius of the worm wheel could be equal to the centre distance of the gears, and the face width equal to the tip diameter of the worm (see Fig. 5). The regions of no interference are bounded (in a central plane containing the axis of the worm) by two straight lines OA, OB, symmetrically disposed about the common perpendicular to the axes of worm and wheel, and inclined to it at an angle depending on the normal pressure angle and lead angle of the worm. These lines therefore define the useful face width of the worm wheel from the point of view of surface contact. Worm-wheel teeth rarely fail by breakage, but there is sometimes a tendency for crack to originate at the outer extremities of the roots of the teeth. For that reason it is useful in heavily loaded gears to extend the face width of the worm wheel so that the length of the root section of the tooth is increased and in order that the tendency to stress concentration at the exposed ends of the root may be diminished. Load capacity. At any particular instant the load exerted by a worm on its mating worm wheel is transmitted across a number of lines of contact whose number, length and disposition depend on the instantaneous angular positions of the gears. The contact conditions continuously change as the gears rotate and any reliable estimate of load capacity must be based on the least favourable combination of circumstances. It is common, at least in talking of spur gears to refer to a “safe load per inch width” for some specified material at a certain speed of rotation. Such a criterion is misleading, even for the simplest type of gear, and it is very much more so in the case of worm gears. It is first necessary to appreciate that what may be a “line of contact” when the gears are unloaded becomes a “band of contact”—containing the primitive line within its boundaries—when a normal working load is applied. Furthermore, the lubricant, if correctly selected and used, forms a film which is sufficiently tenacious to resist rupture even under comparatively high local pressures, and by virtue of its viscosity and oiliness causes a distribution of transmitted load over an area even wider than the band formed by the mutual flattening of the mating thread and tooth. The surface stress at any point on a “line of contact” therefore depends primarily on the intensity of line loading at that point, and the width of the contact band which, in turn, is determined by the relative radius of curvature of the contacting profiles. The permissible surface stress depends on the materials of the mating gears, on the frequency of repetition of load on each tooth and on the rubbing speed. When it is realised that the number of phases of engagement of a worm and worm wheel is infinite, and that in each of these phases the conditions at all points on the lines of contact are different from each other, it will be understood that rigorous analysis of worm gear contact with a view to calculating load capacity is too laborious to contemplate. What has been done is to investigate fairly thoroughly a number of different cases and from the calculated values of relative load capacity to deduce some approximate rules which could be checked by experiments in the laboratory and by observing the behaviour of gears in actual service. From this it has been found that the torque capacity of a worm wheel, for a given permissible surface stress, is approximately proportional to the effective face width and to the 1.8th power of the pitch diameter. The effective face width of the worm wheel is equal to the actual face width, but is subject to an upper limit depending on the dimensions of the worm. Fig. 6 shows the approximate rear-axle worm gear centre distances for four distinct types of vehicle on the basis of maximum rear axle torque. The curves are subject to slight variation up or down with the laden weight of the vehicle in relation to the maximum rear axle torque. Fig. 5. Limits of effective face width of worm-wheel. Wormwheel toothform in shaded areas is made ineffective by interference. Relative load capacity of worm gears and bevel gears. In comparing the dimensions of worm gears and bevel gears for the same duty, for example, the rear axle drive of an automobile, one may be surprised at first to find that a bronze worm wheel is rated as capable of transmitting a greater torque than a case-hardened steel bevel gear of the same diameter and face width. It is not perhaps immediately obvious why the much harder material of the bevel gear does not permit greater loading. The explanation lies in the fact that the mean relative radius of curvature of the contacting surfaces in the worm gear is much greater than that in the bevel gear. To illustrate this, consideration may be given to the contact between a pair of spur gears. There, the reciprocal of the relative radius of curvature is proportional to the sum of the reciprocals of pitch radii of the two gears. The value of this sum is dependent more on the diameter of the smaller gear (the pinion) than on that of the wheel. The same applies in a general sense to bevel gears and worm gears. In a high ratio bevel gear, the pinion is of small diameter, and therefore the torque capacity of the wheel is relatively low in spite of the high surface stress permitted by the case-hardened material of the gears. It has already been pointed out, however, that a worm wheel, when meshing with the worm, is virtually meshing with a rack, which is a gear of infinite diameter. Very approximately the comparison between the relative radii of curvature in worm gears and bevel gears of ratio 7 to 1 and wheel diameter 4 in. is thus : Worm gear. Worm wheel Pitch radius = 7in. Worm (a rack) „ „ = ∞. Relative radius of curvature ∝ 1 / (1/7 + 1/∞) = 7in. Bevel Gear. Bevel wheel Pitch radius = 7in. Bevel pinion „ „ = 1in. Relative radius of curvature ∝ 1 / (1/7 + 1/1) = 7/8 in. | ||