J.D. Smith and D.B. Welbourn


Gearing Research in Cambridge has a long tradition. As early as 1827, G.B. Airy, known to mathematicians for his Airy Functions, published a paper "On the forms of the teeth of wheels". In view of the fact that by 1950, when the following story really starts, gear teeth were being cut with an accuracy of 5 microns, the last paragraph of his paper may be of interest.

"I am informed by engineers, that this question is now little more than one of mere curiosity. In consequence of the very extensive use of iron, where wood was formerly employed, the teeth of wheels are now made so small, that it is of little consequence whether they have, or have not, the exact theoretical form. Almost all teeth are now made with plane faces passing through the axis of the wheel, and are expected to wear themselves in a short time into proper forms. This is the case with nearly all the modern iron wheels that I have examined; in the wheels of clock and watch-work, some attention to the figure is however thought necessary."

April 30, 1825

This, however, was not the view of the Jacksonian Professor, the Reverend Robert Willis, who, 37 years before the appointment of the first Professor of Mechanism and Applied Mechanics, was already lecturing on mechanisms. He published in 1838 his classic paper in which he describes the invention of the odontograph, and also how to design cycloidal and epicycloidal gears so as to enable sets of change wheels to be made for lathes. In his book Principles of Mechanism published in 1841, he recommends the use of the involute tooth, today almost universally used in gearing. At the end of the 19th century, Sir Charles Parsons (q.v.), needing more accurate gears for turbine drives, invented the creep-type hobbing machine, which produced a dramatic improvement in the accuracy with which gears could be made.

Gearing Research and its Application 1947-1975

Shortly after the end of World War II, E.K. Frankl, a lecturer in the CUED, was investigating the measurement of strains using wire resistance strain gauges, which had been invented during the war, and was running short courses on their use for engineers from industry. This work was seen by D.B. Welbourn, later a member of the CUED staff, who then used such strain gauges at the firm of W.H. Allen Sons & Co. Ltd. to measure the strains in the annulus of an epicyclic gear. This was their first application to gearing.

Frankl was also investigating the measurement of stresses by the use of interference fringes, when perspex models were loaded. In order to observe how fringes were propagated under dynamic loading conditions, he had devised a rig consisting of two perspex gear wheels, arranged so that the tooth contact could be illuminated with a high intensity flash lasting only a few milli-seconds. Welbourn realised that Frankl’s gear rig might be used for measuring the loads when gear teeth came into contact, and persuaded the Admiralty to give the CUED a grant to start an investigation into dynamic tooth loads. This was a matter of great importance, since at that time various authorities maintained that when the speed at which a pair of gears was run was increased, the tooth loads (i) remained the same, (ii) increased, (iii) decreased!

The research, initially carried out by S.L. Harris, later to become Professor of Engineering at Lancaster University, proved to be extremely difficult, since it was soon discovered that two teeth could go past one another without touching. This was due to the teeth acting as springs, while the bodies of the gears provided inertias, with resultant torsional vibrations. With the aid of the first analogue computer to be installed in the CUED, Harris was able to calculate such vibrations, and to predict tooth loads. This resulted in a paper to the Institution of Mechanical Engineers which was so sharply and irrationally attacked by the "experts" that he decided to continue with the work.

Harris had come to realise that the exciting forces causing the gears to vibrate were dependent on the tooth errors, whether due to manufacture or to the bending of the teeth under load. He started work with a research student, R.G. Munro, later to become a Professor at Huddersfield University, on the measurement of transmission errors, a term which he coined, and whose concept lies at the back of all work done since then on gear dynamics and gear noise. This work was highly original and has formed the basis of understanding of gear meshing for the last 40 years. With a very elegant rig, dependent only on mechanical means of measurement, Harris and Munro were able to show how transmission error varies with load for different forms of gear profile, and in particular showed how correction of the involute could be used to reduce transmission error. This work was published jointly by S.L. Harris, R. Wylie Gregory and R.G. Munro in a paper to the Institution of Mechanical Engineers, and slowly achieved world wide recognition. A further paper followed, in which Harris maps, as shown in Fig. 1, were first used. Further work in this field was carried out by two research students, B.Oeppen and D.L.Seager.

The importance of this work cannot be over-emphasised, since even the manufacture of chips, let alone car gears, depends on gearing free from transmission error.

fig1.gif (4976 bytes)

Fig.1 Harris map of transmission error of a pair of gears at varying load
due to elastic deflections of the teeth designed with "long" tip relief

At no-load the shapes of the two gears meshing give the top curve labelled "n" which shows the variations (in the y-axis) of movement from the idealised horizontal dashed line which corresponds to perfect, rigid involutes meshing and with no Transmission Error. The tip relief sections from S to C on one pair of teeth and C to S on the next pair give a dip in the transmitted motion. The third curve, "h", is for half design load and shows how the elastic deflections combine with the original profiles. At the design load, curve "d", the elastic deflections have exactly cancelled the profile corrections and the Transmission Error is a straight line so no vibration would be generated. The sixth curve, labelled "o" is for 25% overload and gives a raised section where double tooth contact is occurring.

This work enabled gear designers to apply tip relief in a rational manner when designing gears.

Fig.2, ( see later ) demonstrates clearly another of Harris’ contributions to the thinking about gear dynamics. It shows that transmission errors give rise to steady state vibrations in gears, and not merely to transient ones as formerly supposed.

In parallel with this work, M.J. French, later to become the first Professor of Engineering at Lancaster University, had published a major contribution to the understanding of conformal gears, while A.L. Percival had done fundamental work on gear rolling as a manufacturing process.

In 1965, Welbourn had been approached by the firm of W.E. Sykes Ltd. to solve problems connected with its new shaping machine, and he had been able, with the backing of the Department of Trade and Industry and of the Science Research Council, to persuade the 3 leading manufacturers of gear-making machinery to collaborate in lending a hobber, shaper and shaver to the CUED so that the fundamentals of the gear cutting processes might be elucidated. This grant enabled the CUED to recruit J.D.Smith, a former undergraduate and Rex Moir Prize winner in the CUED, to work on the problem. He rapidly developed a new and simple type of 6-axis dynamometer which enabled the forces on the teeth to be measured continuously from the moment the cutter touched the wheel to the time that machining was finished. This was an enormous step forward compared with somewhat similar research at Aachen, where they could only measure over 2 teeth for lack of an adequate dynamometer. The main results were published at the Institution of Mechanical Engineering’s "Gearing in 1970" Conference, held in the CUED. The work on hobbing had largely been done by a research student D.A.D. Cooke, that on shaping by B.L. Tramontini, and that on shaving by J.A. Sharpe.

The work on forces in gear cutting and finishing enabled the three firms who had lent machinery to redesign their machines radically.

At this conference, Welbourn also published a paper on gear noise, showing the different sorts of frequencies which might be expected from gear errors, whether eccentricity, adjacent pitch error, or involute error, including tooth bending. This explained many published gear noise spectra which had previously seemed to be irrational. This paper was possible because .J.D. Smith, who had meanwhile become a lecturer, had started to work on the problems of gear measurement, and had designed a very precise checking device for adjacent pitch error.

Most of Smith’s work belongs to the period 1975-2000 (q.v. below). He had, however, with his research student F. Cunliffe, designed an epicyclic gearbox which could be instrumented so as to measure the loading on the planet pins, and a computer model which enabled the 13 possible modes of vibration to be calculated. Of these the most serious two could in fact be calculated on a slide rule, once the physical understanding was there. In addition, together with his research student M.W. Salzer, he had produced an analogue computer model of a car gear box which gave realistic noises through a loudspeaker, and with which could be shown the effect of introducing different types of gear error.

Gear Research and its Application 1975-2000

By this time, in parallel with the research work, there was an ever increasing requirement to assist industry with noise and vibration problems. This ideally required a robust, hand-luggage portable form of gear transmission error measuring equipment which could work at high speeds in the presence of vibration. J.D. Smith commenced work on the design and development of such equipment, and the work has progressed steadily ever since. Results are typically as shown in Figure 2.

fi2.gif (8761 bytes)

Fig.2 Variation of Transmission Error (T.E.) with speed due to internal dynamics

Sets of the equipment so developed have been sent to the USA, to Europe and to the far East as well as being essential on production lines in Britain. Methods were also developed to check encoders to high accuracy and to apply corrections if needed (though they rarely are).

Measurement of Transmission Error at speed and under load, with accuracies well below a micron, allowed exact knowledge of the excitation in a gearbox and hence the gears themselves could be used to carry out resonance testing to determine the internal and external resonances in the whole system. The overall path is shown in Figure 3.


Pinion distortion Wheel distortion

Gearcase deflection Gearcase accuracy

Pinion movement Wheel movement

Pinion tooth deflection Wheel tooth deflection

Pinion profile accuracy Wheel profile accuracy

Pinion pitch accuracy Wheel pitch accuracy

Pinion helix accuracy Wheel helix accuracy


Gear                  Support           Combined
Masses        Stiffnesses    Damping

Internal Dynamic Response


Casing    Casing    Casing
Masses  Stiffnesses      Damping


Antivibration Mounts


Sound Radiating Panel


Fig. 3 Overall vibration path

Papers were published on the various topics and the techniques and possibilities were summarised in a book ,"Gear Noise", in 1999. This also discussed the simple computing techniques that could be used to predict the sensitivity of designs to manufacturing errors both for peak stresses and for noise generation.

Some gear drives in diverse fields such as wheelchairs and printing presses suffer from problems associated with severe non-linearities when teeth come out of contact and subsequently impact with high noise and force levels, especially when misaligned. Simple computer predictions can be tied to practical testing to give understanding and solutions.

Arising accidentally from the Transmission Error work Smith made a discovery that it was possible to have very sensitive monitoring of any form of surface contact between mating gear flanks with detection of individual asperity contacts. This technique, though difficult to apply, can give very early detection of scuffing, long before it is visible on the gear teeth, and has been demonstrated on both small and large gearboxes.

Fig. 4 Progress of scuffing during test

Current work (AD 2000) is aimed at further reducing the size and increasing the robustness of Transmission Error measuring equipment and also at educating industry on noise reduction alongside the routine solving of failure problems which are arising from increased use of computerised design methods by inexperienced designers and also from increased loadings on rolling bearings in gear drives.

J.D. Smith and D.B. Welbourn
Michaelmas term 2000

Cambridge Gearing Publications

1. Airy, G.B., 'On the Forms of the Teeth of Wheels', Cam. Phil. Trans., 1827, vol.-II, p.279.

2. Willis, R., 'On the Teeth of Wheels', Trans. Inst. Civil Eng., vol. II, 1838.

3. Willis, R., 'Principles of Mechanism', 1841.

4. Bashforth, F. 'On the Formation of the Teeth of the Drivers of Pin Wheels', Proc.-Inst. Mech. Eng., April 1848, p.2.

5. Welbourn, D.B., 'Die Zahnbelastbarkeit von Getrieben’ Fachtagung Zahnradforschung 1950 (VDMA) and disc. on paper by Merrit, Proc. Inst. Mech. Eng., 1950, vol. 163, pp.168-169.

6. Harris, S.L., 'Dynamic Loads on the Teeth of Spur Gears', Proc. Inst. Mech. Eng., vol. 172, 1958, p.87.

7. Munro, R.G., 'Dynamic Behaviour of Spur Gears', Thesis Cambridge University, 1962.

8. French, M.J., 'Kinematics and Mechanics of Epicyclic and other Multiple Layshaft Gearing', The Engineer, 26 Oct 1962, p.207.

9. Gregory, R.W., Harris, S.L. and Munro, R.G., 'Dynamic Behaviour of 'Spur Gears', Proc. Inst. Mech. Eng., vol. 178, Pt. 1, 1963, pp.207-218.

10. Gregory, R.W., Harris, S.L. and Munro, R.G., 'A Method of Measuring Transmission Error in Spur Gears of 1:1 Ratio', Journal of Scientific Instruments, 1963, vol. 40, pp.5-9.

11. Gregory, R.W., Harris, S.L. and Munro, R.G., 'Torsional Motions of a Pair of Spur Gears', Proc. Inst. Mech. Eng. (Appl. Mech. Convention), vol. 178, Pt.3J, 1963-64, pp.166-173.

12. Oeppen, B., 'Torsional Vibration of Spur Gears', Thesis, Cambridge University, 1964.

13. French, M.J., 'Contact Curvature of Circular Arc Gearing', The Engineer, Tech. Contributors Section, Dec. 1966.

14. Seager, D.L., 'Some Elastic Effects in Helical Gear Teeth', Thesis, Cambridge University, 1967.

15. Smith, J.D. and Welbourn, D.B., 'Optimum Infeed Rates in Shaping Spur Gears', Machinery and Prod. Eng., 28th June 1967, vol. 110, No. 2850, pp.1404, 1405.

16. Cooke, D.A.D. and Welbourn, D.B., 'Forces in Gear Hobbing - 1', Machinery, 7-Feb 1968.

17. Cooke, D.A.D. and Welbourn, D.B., 'Forces in Gear Hobbing -2'.

18. Welbourn, D.B. and Smith, J.D., 'Gear Shaping, Hobbing and Shaving Machines', B.G.M.A. Meeting, 25 Sept 1968.

19. Percival, A.L., 'Preplasticization - a New Dimension in Gear Rolling', Metalworking Production, 2 April 1969.

20. Welbourn, D.B., 'Why this Curious Wear on Gears?', Engineering, 10 Oct 1969, p.395.

21. Smith, J.D. and Welbourn, D.B., 'Spur Gear Shaping Forces', Machinery and Prod. Eng., 10 Dec 1969, vol. 115, No. 2978, pp.936-939.

22. Welbourn, D.B. and Smith, J.D., 'Aspects of Shaping, Hobbing and Shaving Machines for Medium Sized Gears', I.Mech.E., Gearing in 1970 Conference, Sept 1970.

23. Smith, J.D., 'Application of Vibrometer 8 kc/s Carrier Frequency Amplifier to Gear Cutting Research at Cambridge University', Vibrometer News 4, pp.9,10.

24. Welbourn, D.B., 'Gear Errors and Their Resultant Noise Spectra', Inst. Mech. Eng. Gearing in 1970 Conference, pp.131-139.

25. Cooke, D.A.D., 'The Dynamic Behaviour of Gear Hobbing Machines', Thesis, Cambridge University 1970.

26. Sharpe, J.E. and Smith, J.D., 'Accurate Pitch Checking', Journal Mech. Eng. Science, vol. 13, No. 2, 1971, p.116.

27. Sharpe, J.E. and Smith, J.D., 'Pitch Information from Single Flank Tests', Journal Mech. Eng. Science, vol. 13, No. 4, 1971, p.227.

28. Cunliffe, F., Smith, J.D. and Welbourn, D.B., 'Epicyclic Gear Vibrations', Inst. Mech. Eng. Conference on Vibration in Rotating Systems', Feb 1972, p.5.

29. Welbourn, D.B., 'Forcing Frequencies Due to Gears', Inst. Mech. Eng. Conf. on Vibration in Rotating Systems, Feb 1972, pp.25-36.

30. Cunliffe, F., 'Epicyclic Gear Vibrations', Thesis Cambridge University, 1973.

31. Miller, R.B., 'Metal Deformation in Finish Rolling of Involute Gears', Thesis Cambridge University, 1973.

32. Cunliffe, F., Smith, J.D. and Welbourn, D.B., 'Dynamic Tooth Loads in Epicyclic Gears', Trans. A.S.M.E., Journal of Eng. for Industry, May 1974, vol. 96, No. 2, p.578.

33. Smith, J.D., 'Effects of Eccentricity in Gear Shaving', Machinery and Prod. Eng., vol. 125, No. 3225, Sept 1974, p.330.

34. Salzer, M.W. and Smith, J.D., 'Real Time Simulation of Gearboxes', 4th World Congress on Theory of Machines and Mechanisms, Sept 1975, Publ. by Inst. Mech. Eng. pp.175-177.

35. Smith, J.D., 'Involute Measurements on Large Gears', 4th World Congress on Theory of Machines and Mechanisms, Sept 1975, publ. by Inst. Mech. Eng., pp.215-218.

36. Smith, J.D., 'Toothed Gear Coupling Problems', Conference on Mechanisms 76, Mechanical Drive Systems, Sept 1976, UMIST, paper 7.

37. Salzer, M.W., Smith, J.D. and Welbourn, D.B., 'Simulation of Noise from Gears when Varying Design and Manufacturing Parameters' World Congress on Gearing, SYNECOT/AGMA/ASME, Paris, June 1977, pp.297-308.

38. Salzer, M.W., 'The Dynamics of a Layshaft Gearbox'', Thesis, Cambridge University, 1977.

39. Smith, J.D., 'Gear Tooth Coupling Thermal Instability', ASME 1977 International Power Transm. and Gearing Conference, Chicago, Sept 1977.

40. Welbourn, D.B., 'Gear Noise Spectra - a Rational Explanation', ASME 1977 International Power Transmission and Gearing Conference, Chicago, 28-30 Sept 1977.

41. Smith, J.D., 'Measurement of Gears in Situ Using Gratings', United Kingdom Machinery Health Monitoring Group No. 21, Leicester, Jan 1979.

42. Welbourn, D.B., 'Fundamental Knowledge of Gear Noise - A Survey', Inst. Mech. Eng. Conf. on Noise and Vibration of Engines and Transmissions', Inst. Mech. Eng. Aut. Div., Cranfield 1979 (10), pp.9-14.

43. Daly, K.J. and Smith, J.D., 'Using Gratings in Driveline Noise Problems', Conference on Noise and Vibration of Engines and Transmissions, Inst. Mech. Eng. Aut. Div., Cranfield 1979 (10), pp.15-20.

44. Daly, K.J., 'Gear Drive Transmission Errors', Thesis, Cambridge University, 1979.

45. Smith, J.D., 'Effects of Dynamics in Gear Tooth Contact', Tribology International, vol. 13, No. 3, June 1980, pp.133-135.

46. Smith, J.D., 'Gear Impact Noise in Diesel Drives', Proc. Inst. Acoustics, Diesel Engine Noise Research Conference, Loughborough, Sept 1980, pp.151-160.

47. Furley, A.J.D., Jeffries, J.A. and Smith, J.D., 'Drive Trains in Printing Machines', Inst. Mech. Eng. Conference, Vibrations in Rotating Machinery, Cambridge, 1980, pp.239-245.

48. Smith, J.D., 'Some Trends in Gear Metrology', NELEX 80 Metrology Conference, Oct 1980, NEL, East Kilbride, paper 3.3.

49. Daly, K.J. and Smith, J.D., 'Estimation of Excitation and Transmissibility from Output Measurements with Application to Gear Drives', Journal of Sound and Vibration 1981, 75(1), pp.37-50.

50. Daly, K.J. and Smith, J.D., 'Measurements with Rotary Gratings', Journal Mech. Eng. Science, vol. 22, No. 6, 1980.

51. Smith, J.D., 'Tackling Gear Noise Problems', Automotive Engineer, vol. 7, No. 3, July 1982, pp.13-15.

52. Smith, J.D., 'Gears and Their Vibration', Marcel Dekker, New York and MacMillan, London, 1983.

53. Smith, J.D., 'Energy Loss in Gear Tooth Impact', Proc. Inst. Mech. Eng., vol. 198C, No. 12, pp.205-208.

54. Smith, J.D., 'Transmission Error Measurements in Gearbox Development', AGARD-CP-369 Gears and Power Transmission Systems for Helicopters and Turboprops', Lisbon, Oct 1984, Paper 33 (available from B.L.L.).

55. McFadden, P.D. and Smith, J.D., 'An Explanation for the Asymmetry of the Modulation Sidebands about Tooth Meshing Frequency in Epicyclic Gear Vibration', Proc. Inst. Mech. Eng., 1985, vol. 199, No. C1, pp.65-70.

56. Smith, J.D., 'Involute Checking on a C.M.M.', Quality Today, Nov 1985, p.84.

57. McFadden, P.D. and Smith, J.D., 'A Signal Processing Technique for Detecting Local Defects in a Gear from the Signal Average of the Vibration', Proc. Inst. Mech. Eng., Dec 1985, vol. 199, No. C4, pp.287-292.

58. Smith, J.D., 'Measuring Gear Pitch on a C.M.M.', Quality Today, Feb 1986, p.17.

59. McFadden, P.D. and Smith, J.D., 'Effect of Transmission Path on Measured Gear Vibration', ASME Journal of Vibration, Acoustics, Stress and Reliability in Design, July 1986, vol. 108, pp.377-378.

60. Smith, J.D., 'Gear Helix Checking on a Manual C.M.M.', Quality Today, March 1987, pp.30-32.

61. Smith, J.D., 'Gear Transmission Error Accuracy with Small Rotary Encoders', Proc. Inst. Mech. Eng., vol. 201, No. C2, 1987, pp.133-135.

62. Smith, J.D., 'The Uses and Limitations of Transmission Error', American Gear Manufacturers Assoc. Fall Meeting, Paper 87 FTM 5, and Gear Technology July/August 1988, pp.34-39.

63. Smith, J.D., 'Identification of Small Amplitude Resonances in Rotary Digital Systems', Proc. Inst. Mech. Eng., vol. 202, No. C1, 1988, pp.63-65.

64. Smith, J.D., 'A Modular System for Transmission Error Testing', Proc. Inst. Mech. Eng., vol. 202, No. C6, 1988, p.439.

65. Smith, J.D. and Echeverria-Villagomez, J.S., 'Using an Encoder as a Torsional Vibration Transducer', Proc. Inst. Mech. Eng., vol. 203, No. C3, 1989, p.219.

66. Smith, J.D. and Echeverria-Villagomez, J.S., 'Comparing Encoder and Accelerometer Measurement of Transmission Error or Torsional Vibration', Proc. Inst. Mech. Eng. Gearbox Noise and Vibration Conference, Cambridge, 1990-5, p.43.

67. Smith, J.D., 'Practical Rotary Encoder Accuracy Limits for Transmission Error Measurement', Proc. Inst. Mech. Eng., 1991, 205 (C6), p.431-436.

68. Smith, J.D., 'Alias Errors in Precision Rotary Encoder Calibration', Proc. Inst. Mech. Eng., 1992, 206 (C ).

69. Smith, J.D.,'A Gear Test Rig with a Variable Torque Actuator.', Proc. Inst. Mech. Eng., 1992, vol 206 C, pp 311-315.

70. Smith, J.D.,'A New Diagnostic Technique for Asperity Contact.' Tribology International, 1993. Vol 26 No 1.p 25.

71. Smith, J.D.,'Generation of Smith shocks in gears during oil starvation', Proc. Inst. Mech. Eng., 1993, vol 207 C, pp 279-285.

72. Smith, J.D.,'Monitoring the running-in of gears using Smith shocks.', Proc. Inst. Mech. Eng., 1993, vol 207 C, pp 315-323.

73. Smith, J.D., 'Using Smith shocks for gearbox monitoring.' A.S.M.E. Proc. 1994 Eng. Systems Design & Analysis Conference London July 1994 Vol.8 Part B, PD-Vol.64-8.2 pp 355-361.

74. Smith, J.D., 'Helical gear vibration excitation with misalignment.' Proc. Inst. Mech. Eng., Vol 208C, 1994 pp 71-79

75. Smith, J.D., 'Transmission of Smith shocks through rolling bearings.' Journal of Sound and Vibration, 1995. 181 pp 1-6

76. Smith, J.D., 'Continuous monitoring of Smith shocks after lubrication failure.' Proc. Inst. Mech. Engrs, Vol 209C,1995 pp 17-27.

77. Smith, J.D., 'Estimation of the static load distribution factor for helical gears' Proc. Inst. Mech. Engrs, Vol 209C, 1995 pp 193-199.

78. Smith, J.D., 'Modelling the dynamics of misaligned helical gears with loss of contact.' Proc. Inst. Mech. Engrs, Vol 212C, 1998, pp 217-224.

79. Smith, J.D., 'Gear noise and vibration.' Marcel Dekker 1999.