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).
Detailed analysis of worm gearing, covering efficiency, design considerations, materials, and manufacturing processes.
| Identifier | ExFiles\Box 136\5\ scan0320 | |
| Date | 1st September 1937 guessed | |
| PAGE 4, COLUMN 1: 4 Worm Gearing—contd. It must be emphasised that this comparison has no pretension to rigour but it does bring out the fundamental reason for the relatively high torque capacity of a worm wheel in relation to the comparatively soft material from which it is usually made. Efficiency. Designers are sometimes reluctant to make use of worm gearing because of a fear that it will prove to be very inefficient. The reason for this fear is probably the knowledge that worm gearing can easily be designed to be self-locking and that for certain purposes worm gearing may be employed for this reason alone. It is wrong to argue from this, however, that all worm gears must be of low efficiency. It is true that they are fundamentally less efficient than spur gears or helical gears because the ratio of mean velocity of sliding to mean circumferential velocity of the wheel is comparatively high. To offset the apparent importance of this consideration there must be borne in mind the facts that : (a) The power loss by tooth friction in spurs or helicals is very small (less than 1 per cent. of the transmitted power) so that the performance of these gears sets a severely high standard of efficiency. (b) That the coefficient of friction between the normal spur gear materials is higher than that for the combination (bronze and case-hardened steel) employed in worm gears. (c) Even a very low efficiency is not of PAGE 4, COLUMN 2: high proportion of slow running (which means low worm gear efficiency) would preclude its use, if mechanical efficiency were of first importance. A rigorous calculation of the power loss by tooth friction in a pair of worm gears would demand a double integration of the product of rubbing velocity, normal pressure and coefficient of friction for all points of contact in all phases of engagement. Uncertainty in the value of the coefficient of friction would, however, nullify the apparent refinement of such detailed calculation and in practice a much simpler procedure is found to meet all requirements. The worm is regarded as equivalent to a rolled wedge of effective angle equal to the lead angle of the worm at mid-depth of the thread. The efficiency of the gears, taking into account tooth friction only, is then given by the formulae η₁ = tan λ / tan (λ + ϕ) (Worm driving) or η₂ = tan λ / tan (λ - ϕ) (wheel driving) where λ = angle of lead of worm and ϕ = Angle of friction between worm and wheel. It may be shown by differentiation of the first expression with respect to λ that the maximum value of η₁ is obtained when λ = 45° - ϕ/2. As ϕ is an angle of the order of 3 deg. it is seen that a lead angle of about 45 deg. gives the highest efficiency. At the same time it must be noted that the curve which expresses the TEXT UNDER GRAPHS ON PAGE 4: Fig. 6. Approximate R.A.T.* max. torque exerted on wormwheel by engine or motor. *When both R.A.T. and G.L.W. are known, use R.A.T. scale and work on an effective R.A.T. obtained by multiplying actual R.A.T. by 1000 / (333 + r/C . Wr) Where r = effective radius of rear tyre in inches. itself sufficient to forbid the use of a mechanism. The efficiency of a rear axle worm drive may be (say) 94 per cent. as against 98 per cent. for a bevel gear, but the overall efficiency of the engine is of the order of 25 per cent. so that improvement of engine design offers much more scope in the direction of fuel saving than does any possible gain in transmission efficiency. It is significant that automobile worm gearing has a present time its most extensive application in public service vehicles—just where the relatively Fig. 7. Approximate efficiency of worm and worm wheel. Approx. rubbing speed = 0.09 N.C.√(R²+1) N = speed of wheel (r.p.m.) C = centre distance of gears (in.) R = reduction ratio. relation between η₁ and λ slopes only slowly for a considerable range of values of λ in the vicinity of 45 deg. and considerations other than those of theoretical efficiency may advise use of a lead angle appreciably smaller than 45 deg. The coefficient of friction between a case-hardened steel worm and a phosphor bronze worm wheel ranges between 0.15 in the static condition and 0.015 at a rubbing speed of 4,000 feet per minute. If the velocity and the diameter of the worm wheel are fixed, the lead of the worm is determined and the greater the lead angle the smaller must be the diameter of the worm and therefore the greater the bending stresses and the deflection of the worm shaft under the influence of the tooth load. This relative displacement of worm and wheel adversely affects the contact conditions, tending to produce high local stresses and to prevent maintenance of the oil films at the lines of contact. Both these effects lead to a higher coefficient of friction than would otherwise apply, and so it is that in practice a lead angle considerably less than 45 deg. - ϕ/2 may give the best results. PAGE 5, COLUMN 1: Assuming therefore some particular minimum diameter of worm to be used with a given worm wheel, the higher the velocity ratio of the gears the smaller is the lead of the worm and therefore the smaller is the lead angle and the lower the efficiency. Working on the very rough rule that the pitch diameter of the worm should not be less than one-fifth of the pitch diameter of the worm wheel, it is possible to trace a relation between reduction ratio and lead angle, and therefore between reduction ratio and efficiency for a given rubbing speed. It is on this basis that the curves in Fig. 7 have been plotted. Efficiency in the case of a complete worm gear assembly will, of course, be lower than that indicated by the curves on Fig. 7, by an amount depending on the power lost in the bearings and by turbulence in the lubricating oil. At high speeds this last effect is considerable, so much so that an oil spray is used in preference to an oil bath, where the available space permits the addition of an oil circulating system. In the case of an automobile rear axle drive such a refinement is hardly practicable, and indeed the gain to be expected would not justify the extra cost and complication. The efficiency of a rear axle drive thus depends on a number of factors external to the gears themselves and it is not possible to give hard and fast figures for it The curves embodied in Figs. 8 and 9 must therefore be regarded as typical; they indicate the overall efficiencies of well-designed rear axle worm-gear assemblies for two distinct types of vehicle for ranges of speed and output, with the worm driving in all cases. PAGE 5, COLUMN 2: 5 Worm Gearing—contd. Materials and manufacture. Selection of materials for worm gears is a matter of more importance than in the case of helical or bevel gears. Not only must the contacting surfaces be adapted to resist abrasion and surface pressure, but the combination must also be such as to ensure a low coefficient of friction under all conditions of load. Experience has shown that to withstand heavy loading the worm should be made from case-hardened steel and the worm wheel from phosphor bronze. So far as surface hardness is concerned, a low carbon case-hardening steel might be used for the worm, but it is found that greater core strength is necessary to avoid lipping of the surface and steel of about 3½ per cent. nickel content finds considerable favour. Similarly, certain restrictions are placed on the composition of the bronze for the worm wheel, and an alloy which has proved reliable for this purpose contains 87.7 per cent. of copper, 12.0 per cent. of tin and 0.3 per cent. of phosphorus. Centrifugal casting of the worm wheel is always desirable both because it minimises any risk of unsoundness and gives to the bronze higher physical properties than would be obtained by chill casting. These remarks refer to the most highly stressed worm gears, of which automobile rear axle drives are an outstanding example. For less onerous duties heat-treated high carbon or nickel chrome steel may be used for the worm, whilst cast iron is the commonest alternative to bronze as a material for the worm wheel, although aluminium bronze or duralumin are occasionally employed. The worm is nearly always made solid with its shaft (so that a small pitch diameter is obtainable in conjunction with a stiff shaft) and is case-hardened only on the threads. Grinding of the thread profiles is effected on a machine in which the worm is simultaneously rotated and traversed axially past an abrasive wheel which has been trimmed to a section which will produce the required thread form. On the return stroke of the machine table the abrasive wheel is withdrawn from contact with the worm and indexing from one thread to the next is automatically effected during this idle stroke. Longitudinal motion of the machine table in unison with rotation of the worm is commonly effected by a lead screw and change gears as in a screw-cutting lathe, but in machines of comparatively limited scope use is sometimes made of other mechanism designed to give a higher degree of accuracy. As previously mentioned, adoption of the involute helicoid thread form facilitates the profile grinding operation inasmuch as the abrasive wheel may be flat-sided, if its axis is appropriately positioned in relation to that of the worm. In such cases the wheel is trimmed by a diamond moving in a straight line. The correctness of the profile of the worm is checked by mounting it between centres, and bringing into contact with it a straight edge lying in a plane tangent to the base cylinder of the worm and inclined to the axis of the worm at an angle equal to the base lead angle. In other words, the straight edge takes the place of a generator of the thread surface which accordingly should “ fit ” the straight edge at all points between the crest of the worm thread and the base cylinder. A highly polished thread surface is obtained by running the worm with a wooden worm wheel charged with lapping compound, the worm being given axial reciprocation so that the whole length of the thread is treated. Rear axle worm wheels are usually designed to fit between the two halves of the differential gear housing and are therefore of annular form. After straight-forward turning and drilling operations, the teeth are cut by the hobbing process in a “worm wheel generator.” The wheel is mounted on a circular table which rotates about its vertical axis and simultaneously the hob, which corresponds to the worm, rotates about a horizontal axis at the relative rate corresponding to the velocity ratio of the worm gear. The final position of the hob relatively to the worm wheel is that which will ultimately be occupied by the mating worm. At the commencement of the cutting operation the hob is displaced axially from this final position, the distance between the axes of worm wheel and hob being (very nearly) the centre distance of the worm gears. Whilst work and hob continue to rotate at proportional rates which give a suitable cutting speed to the hob teeth, the hob is slowly advanced towards its final position. To compensate for this the worm wheel is given a partial rotation, superimposed on that already mentioned. This auxiliary action is possible by virtue of the inclusion in the table drive of a differential gear assembly whose planetary carrier is given a rotation proportional to the axial movement of the hob spindle and inversely proportional to the diameter of the worm wheel. Where quantities justify it, separate hobs may be used for rough-cutting and finish-cutting the worm wheel teeth. For the first operation the hob is tapered in diameter at its entering end, and it has relief-ground teeth of comparatively coarse pitch. The finishing hob has a large number of finely pitched cutting edges, it is known as a serrated hob. A further refinement introduced into the finish-generating process is a small angular displacement of the hob spindle from its nominal position perpendicular to the axis of the work. This removes metal from the tooth profiles near the outer edges, leaving an “entry gap,” and also modifying the profiles in a way which can be made exactly to compensate for the relative deflection of worm and wheel which occurs under load. In other words, the departure of the worm from its nominal position when load is applied is anticipated by deliberately displacing the hob from its nominal position when finish-cutting the worm wheel teeth. Working on loads on worm and worm wheel. The dispositions of the lines of action of the point loads exerted on a worm wheel when torque is applied to the worm are so varied that an accurate calculation of the resultant loads on worm and worm wheel is virtually impossible. The conventional assumption made to avoid this difficulty is that contact occurs only at the pitch point and that the resultant load acts along the common normal to the contacting surfaces at that point. The resultant load on (say) the worm may be divided into three mutually perpendicular components lying one in each of the following directions: (a) Along the common perpendicular to the axes of the gears. (b) Parallel to the axis of the worm. (c) Parallel to the axis of the worm wheel. The efficiency of a rear axle drive thus depends on a number of factors external to the gears themselves and it is not possible to give hard and fast figures for it. The curves embodied in Figs. 8 and 9 must therefore be regarded as typical; they indicate the overall efficiencies of well-designed rear axle worm-gear assemblies for two distinct types of vehicle for ranges of speed and output, with the worm driving in all cases. TEXT UNDER GRAPHS ON PAGE 5: Fig. 8. Variation of overall efficiency with speed.* *Worm drive for private car, 4.375 in. centres, 4.85 to 1 ratio. Fig. 9. Variation of overall efficiency with speed.† Now let d = pitch diameter of worm. D = " " worm wheel. l = Span of worm shaft bearings. L = " " wheel shaft bearings. e = End thrust on worm. E = " " worm wheel. R = Number of teeth in worm wheel / Number of threads in worm. ψd = Axial pressure angle of worm. T = Torque on worm wheel. Then, ignoring the effects of friction between worm and worm wheel: Force in direction (a) = S = (2T/D) tan ψd Force in direction (b) = e = 2T/D (axial thrust on worm) Force in direction (c) = E = 2T/dR (axial thrust on worm wheel) Resultant force perpendicular to axis of worm = √(S² + E²) †Worm drive for trolley bus, 7 in. centres, 10.33 to 1 ratio. | ||