GEAR DYNAMICS, NOISE AND VIBRATION

Work undertaken under BGA Contracts Nos. R&D/002 and R&D/003 dated 26 August 1987

H. Kohler - December 1988

This bibliography has been compiled to complement the BGA State-of-the Art surveys on Gear Dynamics, and Gear Noise and Vibration. The literature in these two areas has so much in common that separation is hardly possible, and certainly not sensible.

There are thousands of references on gear dynamics in the international literature, of which only hundreds have been seen, and of which a further reduced selection is given here. So this selection inevitably reflects the personal interests and opinions of the compiler.

It may be thought that the bias is towards rather elderly contributions, but it is believed that the historical development of the topics remains important. Much of the more recent literature, with some notable exceptions, does not offer any great advance over the work of previous decades. Also, gear noise work has defence applications.

Much is therefore known that has not been published in the open literature. Commercial considerations also limit the publication of knowledge. A common feature, sadly, of many otherwise interesting and potentially useful publications is that insufficient detail is given to apply the reported work. This may be due to the drive for conciseness in publication, but it could be that there is sometimes a wish not to reveal all.

The reader will be aware that no abstract can do more than give a brief impression of a publication. These abstracts are intended to assist him in selecting those he should read, and to remind a reader of the content of material he has read in the past.

Particular thanks are due to the Design Unit, and the Library, of Newcastle University, for conducting computer searches and providing lists of references.

References are cited in the main texts in the form - surname, year date - a fuller reference is given only when this would be insufficient identification.

Some abstracts express opinions of the work described. These opinions are entirely the compiler's own, and not the responsibility of the BGA. Likewise, errors of fact or interpretation are the compiler's, it is too much to hope that there are none.

Most papers not seen, but measurement and control of transmission error was stated as one of three major topics.

approx. 1968

Gearcase within gearcase construction for reducing radiated/transmitted noise and vibration. AEI Trafford Park demonstration rig, dramatically effective. Unpublished HK notes and diagram.

JSME "Gearing", Semi-international symposium, 2967 p 151, paper 317

Experiments on open gears in anechoic chamber. Variable stiffness differential equation is discussed. Strain gauge, noise and vibration results (very like other contemporary work) are shown. Gears considered as noise radiators (vibrating pistons), influence factors, directivity, helicals, effect of errors. Good, standard work for the period.

NEL Report No. 507 April 1972

Power circulator tests, in an anechoic chamber, are reported. Acceleration and noise was measured. A large number of gears from different manufacturing processes (relevant only in terms of the errors produced) were tested, speeds varied, and mesh frequency and low order harmonics measured. The rig possessed a typically complex response behaviour. AGMA quality numbers characterised the gears. Extensive regression analysis showed that profile error had the major, 'if not the most' effect on noise and vibration. The BS grading system was also used, no difference between it and AGMA as a predictor of effects was found. Correlations found were that profile and undulation had most effect, alignment and runout came next, pitch and swash last. All-recess gears tested were found to be noisier, but those tested had a contact ratio less than unity!

IMechE Con f. paper C122/79 p 47-1979

Vehicle noise from gears considered. Subjective ratings discussed, single tone effects easy to define, assessment of complex tone levels less so. Rigs are described, and spectral analysis considered. Offending signals, including sidebands, are identified, but the difficulties are also discussed. Component tests include audit and production test rigs. Noise is preferred over vibration, because of its averaging properties. Comparison between rig tests and production test signatures used for acceptance testing. Subjective tests not satisfactory. Signature analysis is better, also aural comparison technique using narrow band filters. Experience enables useable standards to be set. Many figures present rating methods, and typical measurements, also some illustrations of effects of gearbox modifications.

VDI Ber. 524, Dusseldorf VDI Verlag,1984

not seen

Diss. RWTH Aachen,1985

not seen

Materiaux Mecanique Electricite No. 416 May-June 1986

Deals with system component parameter value estimation from test results. Introduces new 'instrumental variables' method. The problem is formulated as a non-linear least squares optimisation. Applied particularly when many parameters are known with sufficient accuracy, but others need to be found. Illustrated by special pseudo random excitation of one degree of freedom damped system. Results for several methods, including classical modal analysis, show effect of measurement noise on estimate quality. A three rotor system with flexible, damped bearings is also examined. 

Third European rotorcraft and powered lift aircraft forum paper No. 50 Ass.Aero.et Astro. de France 1977

A Wessex tail rotor gearbox was modelled, with 13 stations each with 6 degrees of freedom. Model includes the bearing and easing stiffnesses, and includes input based on finite element analyses. Excitation was by transmission error. Full modal analysis, (linearised system). Only few higher frequency modes cause high mesh deflections. Comparison with a torsional only model is made. Close grouping of natural frequencies in zones noted. Modal Q in range 8 - 22 used (based on limited measured data). At frequencies > 3 kHz, I000 LB tooth load results from 0.001 in excitation. Transmission error spectra were taken from similar gears, and the response calculated. General order of magnitude agreement confirmed model. Parameter studies are proposed to improve performance, after more experimental verification of the method. Only low - 3 to 5% - dynamic increments found in the running range, but these are important for noise.

Doctorate thesis, University of Sheffield, Department of Applied Mechanics, 1956

Perhaps the first experimental study of dynamic tooth deflections. An adapted IAE 5 in centres gear test machine was used. Dynamic tooth deflection was measured, using as a relative displacement transducer a strain gauge bridging the gap between a block attached to the side of a tooth and a reference disc attached to the gear body. Static transmission error was measured both by sine bar and by using a tape unwinding from one base circle onto the other, via a jockey pulley. The jockey pulley movement represents transmission error. Tooth stiffness was investigated on an enlarged Perspex model. Dynamic tooth deflection traces clearly showed dynamic increments appearing with increasing rotational speed, and the presence of vibrations.

ASME paper No. 58-SA-32,1958

A summary of the work described in reference 'Attia, 1958' above.

ASME paper 63-WA-141963

Early references described. Finds components of tooth and rim deflection, tooth proportion effects. Shows suspect graph with discontinuity at pitch point - (due to inconsistent treatment of rim deflection component?). Tooth size effect, neighbouring tooth contributions, deflection in action, experimental comparisons, including dynamic results. Analytical appendices.

JSoundVib. (1970) Vol.11 (4) p 383-

Noise radiation from open Novikov gears in an anechoic chamber was measured. Polar patterns are given. Some increase with load and speed found. Noted that noise increases when number of teeth in contact is a whole number -1 or 2. Assumed to be the result of the sudden jump in load location across the facewidth of the gear. Many graphs giving results. Comparison with previous similar work - same rig and conditions - on involute helicals shows that Novikov is 6 - 8 dB louder. Radiation pattern is more uniform. Major component is mesh frequency for Novikov, 2nd harmonic for involute.

JSoundVib. (1971) 18(1) p 129-

Introduces a theory which claims that there is free vibration during tooth engagement, excited by impact on the commencement of tooth contact. Highest tooth loads, and hence noise, occurs during the free vibration segment of contact. Change of tooth pitch (higher diametral pitch, smaller teeth) reduces the duration of this critical contact segment, and thereby reduces the radiated noise. Tests of the direct noise radiation of unenclosed gear pairs, similar in all respects except pitch, show a substantial reduction in radiated noise, typically 10 dB for a change from 3 dp to 14 dp. A theory leads to calculated noise reductions which equal those actually found.

ASME Design Eng.Tech.Conf. paper 84-DET-1611984

The classical strain gauge experimental technique (fillet gauges and sliprings) is used to measure tooth strain, and obtain tooth loads via calibration curves. Traces typical of gears are obtained, with evidence of resonance effects. Transmission error is varied slightly by meshing different pairs of teeth. Modest dynamic increments of perhaps 25% are mostly found, all appear to be at speeds well below primary resonance, (though a maximum of 140% at high load and error is reported). Finite Element modelling of Hertzian tooth impact response to sudden loads at various application rates appears to agree well with measurements.

Tech.Report 70-12 (AD 869822) US army air mobility research and development laboratories Ft. Eustis, Virginia 219 pages 1970

Abstract from paper - "A method of computing helicopter gearbox noise from design and operating data is verified by a comparison of calculated and measured gearbox noise spectra. Measurements on CH-47 helicopters are used to provide experimental data.

Positive identification of the source and mechanism of gearbox noise energy has been made. The most objectionable noise originates in the meshing action of the gear teeth. The gear tooth deflection, together with tooth profile variations due to manufacturing errors excites torsional vibrations in the helicopter power train. Each gear mesh produces noise at frequencies corresponding to the tooth meshing rates and their higher harmonics. This gear induced vibration is transmitted to the gearbox structure and thence to the aircraft structure. Results indicate that the CH-47 gearbox mounts transmit a significantly higher level of vibration to the aircraft structure than do the UH-1 D mounts.

Calculations on the CH-47 gearbox indicate that spur gear tooth profile variations can be used to reduce gearbox noise levels but manufacturing tolerances do not yet appear to be low enough for application of this principle. Studies further indicate that variation in planet carrier torsional compliance will result in only modest changes in noise level over the range of compliance studied."

Modelling is crude, with square wave stiffness variation and empirical factors for the transmission paths. Only 6% and third octave frequency resolution was used. Despite the hopeful summary, prediction of absolute levels was very poor.

Tokyo Conference? Paper ref 852288, p 859-

A small spur gear system mounted on a complex, open frame is investigated experimentally. Sinusoidal and random excitation at the stationary gears is used to find acceleration transfer functions between the gear mesh and various parts of the rig frame. The gears, of accuracy grade 7 (no standard referenced), when running give an excitation also used to measure transfer functions. Results show that transfer functions are generally uniform at low frequencies, change from point to point once structural resonances are reached.

ASME Des.Eng.Tech.Conf.1984 paper 84-DET-208

A difficult paper. A fixed annulus epicyclic is dynamically modelled in the high speed regime, - mass controlled - as single planet and sun, both having bearing clearance. It appears that the shaft is always required to contact the bearing, at a contact point defined by a geometrical constraint, derived apparently by considering all gears to engage on their pitch (working?) circles at all times. Computed results show either instabilities or very rapid 'rattle' around the clearances. Tooth loads appear up to four times those which are calculated with zero clearance.

Bruel & Kjaer, Denmark 1984

Previous techniques reviewed, types of bearing defects. Detection of impulses. Envelope of time signal shows 'bumps' at impulses. These may be constant or varying in amplitude. Frequency analysis of the modulation envelope reveals periodicity of the impulses, enabling the fault source to be traced. Uses Hilbert transform. Zoom analysis allows band pass filtering of time signals. Method described. Case history, references.

Trans.ASME. J.Mech.Des. Vol 100 Jan 1978 p 26

The time-varying mesh stiffness simple gear system is analysed for steady, and oscillatory external loading. Magnification factors and instability regions are found. Square wave and sinusoidal stiffness variation are considered. Interesting experimental method used gear segments (Perspex) driven through an oscillatory angle to simulate rotation and time-varying contact conditions. Regions of sub- and unity frequency ratio resonances where amplitudes built up to destructive levels were demonstrated. An appendix illustrates the properties of the describing equation - Mathieu?

Trans.ASME. J.Mech.Des. Vol 103 April 1981 p 372-

Techniques for analysis of multiple reduction drives, with time variable stiffness, are given. The form and amplitude of the stiffness function determines stability. Rectangular and sinusoidal forms are covered. Instability regions are given as functions of stiffness range and tooth contact frequency. Torque response spectra are included. Dynamic load diagrams are given, and maps of potential resonant zones. Input and output torques have an imposed sinusoidal ripple. Parameters involved in the response are the ratios of external to resonant, and tooth contact to resonant frequencies. Concludes that in most systems, each gear pair can be isolated, and analysed as driven by the outputs from the other pairs. Information given enables stability of each pair to be found. Notes that mesh stiffness is difficult to calculate. Regions of instability even with constant input and output are identified.

Proc. Noise, Shock and Vibration Conf. Monash Univ. Melbourne, Aust. May 1974 p 426-

Only part reprint available. Conclusions suggest this to be a typical tooth dynamic loading investigation, with a primary resonance excited by transmission error harmonics.

JSoundVib; Vo136 No 2;1974' p 179-

The Tuplin model is extended to continuous excitation, of more complex systems. Modal analysis methods are used for the sample analysis of two multi-rotor torsional sub-systems coupled by a gear pair. Co-ordinate vectors and Lagrange's equations, produce the equations of motion, with transmission error, both transient and continuous, at the tooth pair contact. The analogous cam-follower system is also cited. An example has a five inertia, three stiffness model, with harmonic transmission error excitation. Q = 30 is used to determine the modal co-ordinate amplitudes.

ASME Des.Eng.Tech.Conf. September, 1979

On PT6 turbines, casing vibration levels indicate quality of manufacture and assembly of the gears; spacing error effects, transmission error accepted as source, effects of error phasing on spectra; big variations found.

H. Stam N.V. Netherlands approx. 2960

Chapters on tooth coupling design and manufacture, tooth proportions, misalignment effects, design for dynamic loading of floating component, loads when idling at speed, crowned teeth, failures, surface hardening, design stresses, references. Design of machine elements in relation to damping, vibration and noise - gear vibrations in relation to surface hardened steels, case and intruded - (un)damping in subsurface, noise production mechanism, references.

H. Stam N.V. Technical Publications, Netherlands 1961

Design, classification societies and rules, detailed consideration of deformations, tooth, rim, wheel stresses. Dynamic loading (p I47) Zeman method, heat treatment, couplings, fluid couplings, others, torsional frequency calculations for gas turbine locked train design. Bearings, lubricants, gear oil film thickness (p 232), materials. Manufacture, with error discussion, tooth surface conditions, hobbing considerations, facets, feeds, active teeth etc., load distribution by photoelasticity, detailed analysis of hobbed tooth surface, heat treatment, measurements, machine error influences, research and testing, includes statistical failure analysis, tooth surfaces, ultimate loads, vibration and noise, pinion torsionals, vibration of nested (multi-path) gears, sound production and vibrations.

Basis for specifying evaluation standards for rotating machines with operating speeds from 10 to 200 revolutions/second. Uses vibration severity - the rms velocity in the band 10 -1000 Hz. Annex gives six classifications of machine types, and a chart relating vibration severity with four quality judgements (AD) for each machinery class. Based on work by Rathbone in the USA. General feeling of tentativeness about the recommendations. Some advice on what and where to measure; suitable instruments are covered in Part II of the standard.

Analytical mechanics of gears - McGraw-Hill 1949

Presents results of ASME research committee, in particular ASME research publication "Dynamic loads on gear teeth", 1929, with additions. Considers errors to act mainly at load transfer from one tooth pair to the next. Errors (including deflections due to load) act by changing relative velocities. Acceleration loads would vary as the square of velocity for rigid gears. At transfer, relative velocity can cause separation, followed by impact which causes the major dynamic load. Acceleration load is independent of error magnitude, as time of introduction is related to magnitude! Hertzian compression of a tooth pair is constant for constant load throughout path of contact. Detailed single error analysis including impact. Toothpair compliances see page 444. Effect of fine pitch and high speed is to prevent full effect. Page 454 has important insights.

Product Engineering June 5th, 1964

Advocates recess action gearing to reduce noise. Notes limit to driving/driven pair, applications include Mt. Palomar 200 in telescope, many other satisfied users. Gives geometric theory, with design methods. Gives friction graphs. Shows servo gear designed for almost no lubrication. Promotes Buckingham gear tables.

ASME Trans. J.Mech.Tran.Auto. in Design Vol.120 March 1988 p 100-

The theory of conjugate surfaces (due to Chen) is used, with vector/matrix algebra, to reformulate the dynamic loading problem. Paper incomprehensible without prolonged study. Author states that 'Equations for determination of the dynamic loads on gear teeth are derived, which can be solved numerically by an iterative method with the aid of computer and the finite element method.'

Proc.7^th World Congress, Theory of machines and mechanisms; Seville, Sept. 1987 pp 815 -

The frequency of free torsional vibration of a spur gear pair with backlash is considered. By including the 'interpenetration' - the Hertzian deflections and natural damping a frequency formula is derived. The instantaneous frequency is amplitude dependent, rapidly reducing as the oscillation decays. The formula is simple, giving the natural frequency for a given gear pair when the initial relative angular velocity at impact is specified. High accuracy is claimed for the formula, but this is in comparison with a more complex computer simulation, not with real gears. The work of Wang (q.v.) has been ignored.

Trans.I.Mar.E.(TM) Vol. 99 Paper 18 (1986)

Reviews general requirements, current plant types, flexible couplings, SSS clutches, quill shaft designs, clutches and brakes, lock-in. Gear design, materials, NAVGRA, load intensity expressed by K; gear problems, classed by failure type. Nitrided, carburised, induction and through hardened gears.

HMS Torquay, old Whitby class frigate, extensive failure and reasons. Research and development, epicyclics (noisy), bearings, thrust cones, CVS arrangement, CVS details, fluid couplings, transmission error, tooth contact program (from DBGI ?), gearbox explosions.

ASME J.Mech.Des. Vol. 100 No.1 Jan. 2978 (Review copy from NASA Conf.Pub. 2210 1983)

Uses a two inertia, mesh stiffness coupled model, transformed (as in Tuplin) into the form of a travelling wedge driving springs and a single inertia. This easily adapts to variations in contact ratio. The increased compliance, and greater sensitivity to error distribution, of high contact ratio gears is noted. Programming methods are considered. Profile modifications are included, and determined by error and deflection considerations. Increased dynamic effects were found for high contact ratio gears with respect to low contact ratio. Extent of profile modification was found to be important. Maximum stresses at resonance are almost the same for the two types of gears.

ProcIMechE, presented Jan.1943, published with discussion and communications 1944

This excellent paper describes the isolation of 'ghost' errors, from the hobbing machine table, as the prime source of noise in large helical gears. The production of improved master wormwheels is described, and the corresponding improvements in gear form and errors illustrated. The transfer of table errors to the cut gear is solved in detail, and the location of contact points in service determined. After the new master wheels had been introduced,'...this...was used for twelve years, and gear noise troubles were forgotten...'. The extensive discussion and communications are also well worth reading, though too disparate to summarise here. H. Walker observed that the worm, as well as the wormwheel, contributes to the table error.

Round table discussion on Marine reduction gears, Delft University of Technology,1958. Ed. Blok, H. p 130 -

Historically interesting, with full discussion of creep hobbing, and fine pitch table wheel hobbing. Actual tooth surface effects are analysed. Accelerometer measurements on pinion shafts are shown. Photoelestic tooth stress pictures. Effect of hob tooth flank straightness on normal or axial section of cut tooth. Some spectra, and noise and vibration results. Dynamic measurement instruments described. Engagement shock due to various tooth reliefs considered.

IMechE Aerospace Ind.Div. Seminar Nov 1985, Transmissions technology for propane and geared fan engines. Paper 5

Design aims, optimisation of configurations, various schemes, planetary and multipath. Detail design - conformal gears; Finite Element methods, dynamic modelling, materials, lubricants, manufacture CADAM. Testing, research needs. Good figures, many schemes. Maths of configuration optimisation using specific torque capacity.

Doctorate Thesis, Cambridge University, 1973

General dynamic model of epicyclic gears. Not seen recently.

ProcIMechE Vibrations in rotating systems February, 2972

A star gear with three planets, 'stiff' or 'flexible' elements, was tested and modelled as a system with 13 degrees of freedom. Instrumentation was on bearings and gears. Either average, or variable tooth stiffness was used in analogue computer models. Tip relief was included. Good results up to 1000 Hz. Noted that 'even quasi-static load sharing cannot be expected' with normal manufacturing errors. Stiff and flexible planet pins considered. Short paper, see PhD thesis (Cambridge) for more details.

A3MF (Design Eng.Tech.Conf·.) paper 84-DET-1741984'

Discusses sidebands m gear noise and vibration spectra. Frequency and amplitude modulation described and analysis given. Transmission error excitation, results from tests showing intermodulation sidebands as well. Effect of deliberate run-out (0.002 in). Complex modulation with multiple modulating components also analysed, and asymmetry of resulting sidebands confirmed.

IMechE. C118l79 1979

Transmission error measurements on hypoid drives are used to show errors, using time averaging to separate out the contributions from pinion and wheel. Dynamic errors are measured at 48 r.p.m. Problems due to grating couplings resonating are discussed.

JSoundVib., 75(1) 1981 p 37-

With particular reference to gearing, it is shown that if an excitation function with many harmonic components of known frequency (known function of shaft speed) exists, then by tracking the corresponding components in the output a series of estimates of transmissibility result, which can be combined to give the system transmissibility. Details are discussed. An alternative method, where only a small running speed range is available, involves calculating the slope of the transmissibility at the harmonic response frequencies and deriving transmissibility and static error from that. Valid for linear systems, nonlinearities show up as scatter. Method relies on some results being available well below the system first resonance to obtain a calibration baseline. Tests on a special flexible gear arrangement illustrate the methods.

JMechEngSci. Vol. 22 No. 61980

The standard grating based transmission error system is used dynamically. Speeds to 6000 r.p.m., and frequencies to 600 Hz have been analysed. The limits of multiplication and division of grating signal frequencies are considered, also non-integer ratios (e.g. needed for belts). Grating heads were tested to obtain response characteristics. A practical system is described, applications are referred to.

Diss. RWTH Aachen 1981

not seen

Machine Design, Nov. 20, Dec. 11 1980, and Jan. 8,1981

An introductory three article series, oriented to helicopter transmissions, which covers theory, measurement and control of transmission noise. Considers full computer analysis of structure, transmission error, shafts, instrumentation, including FFT, noise transmission mechanisms, accuracy, contact ratio, trapped air effect, helix angle - many design charts - special purpose gearing and materials. Other aspects are included, for example phasing in epicyclics, profile modifications and crowning, damping, inserts, absorbers. Shaft alignment, dynamic bearing loads, housing modifications, bearing types. Acoustical treatment of transmissions, isolation, gear isolation and limitations. Damping systems, vibration absorption, impulse phasing. Good on ideas, short on hard data

Masters Thesis, Department of Applied Mechanics, Sheffield University 1960

A substantial literature survey showed that there was already a considerable body of work at this date. Much of the thesis is devoted to rig and anechoic chamber construction. A Ford 204E gearbox was run in a power circulator. All the ratios were examined, and noise and vibration spectra measured (6% bandwidth). Also summary curves showing noise as a function of speed and load are given. Noise from various arrangements of shafting and bearings was discussed. It was observed that major audible pure tones, associated with tooth contact frequencies, did not show up as dominant in the spectrum measurements. Little discussion, and no measurement, of transmission error or tooth errors.

University College Cardiff, Department of mechanical engineering and energy studies, Rep. No. 1166, Nov. 1984

A detailed 'exact' analysis of the contact conditions in W-N gears with realistic profiles. Uses vector analysis methods. Gears are rigid. Many contour plots of the contact zone. Shows possibility of interference. Appendix considers generation of tooth forms.

ASME Trans, J.Mech.Des. Vol.101 July 1979 p 369-

Brief review, headings are Excitation; Gear System Modelling; Response of lightly Ioaded gear systems; Parametric excitation/instabilities. 15 references.

AGMA P 139.041982 Fal1 meeting, New Orleans.

Collects together the views of ten members of an AGMA committee, including C.C. Wang, W.D.Mark, R.J. Drago. Most comments are fairly superficial. Contributors are generally not satisfied that the method is an advance. Trenchant objections from Mark and Wang.

NEL Report No. 46, DSIR, Aug. 1946

Describes tests on dimensionally similar case-hardened ground helical involute, and unhardened, lapped, Novikov gears. No difference in efficiency was found, but the Novikov gears were up to 20 dB higher in noise and vibration. For these, tooth contact and its second harmonic were the main noise components. No indication of the accuracy of the gears of either type is given. 

South African mechanical engineer Vol. 25 April 1975

Review of Kohler paper at vehicle noise conference, Hatfield Poly., April 1974. Advocates transmission error testing of complete vehicle gearboxes, as well as of component gears.

IAE Automobile research committee, No. 1942/9

A collection of abstracts compiled by the IAE Research Department. Seven major classes, of which one is noise. Fairly well cross referenced. Dates are from the early Nineteen-twenties, to the date of publication. Abstracts of particular interest are no. 246, A. Graf Soden, VDIZ 1933, which discusses pitch point friction impulse, and 249, W.E. Sykes, 'Steel', May 1936, which reads well even by current standards. No. 255, G. Dietrich,

Kraftfahrtforschung, March, 1939 describes early use of piezo-electric force transducers to measure gear shaft forces and friction. Various accuracies, and deliberate errors, were investigated.

Int.Symp.Design and Synthesis, Tokyo July 1984 p 259-

In a servo train - a telescope aerial steering device is illustrated - it is shown which frequency components of the overall gear transmission error can affect the performance of the servo. This depends on which path, direct or feedback, the gears are in. The various error components of each gear pair, runout, mounting slackness, tooth errors, are statistically described and the appropriate frequency components summed. Rational selection of error limits is then possible.

Financial Times, Unknown date

Publishes Eaton's "Multimesh" concept, and outlines new range of Eaton Transmissions.

ProcIMechE. Vol. 178 Part 1 No. 81963

One of the few classic papers. The transmission error as a function of load, with load sharing and static effects. Monroe test rig and results, showing non-linear resonant behaviour. Dynamic transmission error records, loss of contact demonstrated. Jump phenomena. Equations of motion. Early references. 

THE paper. Essential reading. Develops the Transmission Error concept for spur gears, including effects of load and profile modifications. Steady deflection included. Shows existence of an optimum load and design of profiles to suit. Vibration modes - periodic errors, stiffness variation, running off-design condition, stiffness variation induced intermittent contact. Damping. Test rig with photoelastic gears. Dynamic photoelasticity used for extensive results, of static and dynamic transmission error, and loads, equivalent dynamic system, tooth details, including tip relief, many results, transmission error curves under many conditions. Conclusions - transmission error, stiffness variation, loss of contact. Phase-plane development of non-linear vibration from stiffness variation.

Bull. J.S.Prec.Eng. Vol. 9 No. 3 Sept 1975

Classifies profile errors into 9 types, with 25 subdivisions. Corresponding static transmission error from each is expanded as Fourier series, and the component amplitudes and phases are plotted as 2D vectors. When in the low speed (rigid tooth assumption) area of the simple gear pair dynamic model, tooth separation is calculated, and associated with the magnitudes of the error vectors. Extremely condensed and unclear paper.

Int.Symp Gearing and Power Transmissions; 1981 Tokyo paper b-41(p 497 )

This describes a claimed new approach to the measurement of dynamic transmission error. By placing heavy flywheels on either side of the gear pair being; tested, the four torques at either side of the flywheels can be used to determine transmission error. The flywheel shafts are located in air bearings, strain gauge bridges on the shafts give signals amplified locally before transmission to the data analysis unit via slip rings. Digital data processing is used. Thin shafts to the flywheels keep system torsional resonances low. From the published results, the measurement resolution appears to be perhaps one micron, the accuracy a few microns, based on gears of about 100 mm centres. Method seems to be a research tool only.

Experimentally considers noise of lightly loaded spur gears using 'S2' tester (no refs) with gears 120 mm dia. of 2 mod and 5 mm face. Compares impact energies from analysis of measured dynamic transmission error and loss of contact detection with overall sound pressure level as a function of speed, 300 - 3600 rpm. Various loads and backlash values examined. Overall correlation between square of impact velocity and SPL has scatter of ± 7 dB for low SPL and impact velocities with ground gears, reducing to ± 5 dB at high velocities and SPL on hobbed gears. Not clear whether measurements are within or outside gearcase.

BuII.Res.Lab.Prec.Mach.Elec. Tokyo Inst. of Tech. Vol. 40 Sept. z977 p 37-

The standard single degree of freedom model is excited by transmission error, which is represented by the vector plot of its Fourier components. The static error vectors are modified by the complex response of the system to give the output vectors of the dynamic error. No indication of what damping values were used is given. Errors are principally profile, of ramp form. Mesh frequency is well below primary resonance, and many harmonic resonances are considered. An analysis of tooth separation due to relative velocity leads to comparisons between prediction and measurement for three types of profile error, ramp up, ramp down, and convex. The amplitude of the tooth separation is found to increase with speed almost to the fourth power. Conversion of the dynamic error to tooth load does not seem to have been made.

IMechE Conf. paper C132l79 p 47-1979

General experimental description of vehicle driveline analysis. Relevant to dynamic systems analysis. Back axle tooth contact frequency found to be a strong exciting influence. Other gear contacts not in evidence.

VDI Berichte 332 Munich Conf. Zahnradgetriebe 1979 p 165 -

Paper includes dynamic factor calculation, with design charts. Based on simple one degree of freedom model. Resonance zone from 0.8 to 1.2 of geartooth-gear system. Error effects on K considered.

Doctorate Thesis, RWTH Aachen, February 1963

Extensive theoretical and practical investigation of the properties of gearcases. Dynamics of structure are considered, as is radiation efficiency, and measures to control vibration amplitudes. Extensive tests on ribbed plates. Many references. Published also as the two following references.

Ind. Anz. Vol. 85 No. 89 Nov.1963 p 1989-

Describes a gearcase noise test rig. in detail, including many acoustic considerations for the test room. Considers the paths by which noise is transmitted and radiated, including the properties of the gearcase as an enclosure. Explains a statistical exercise to determine the best gearcase sizes for experiments. Considers the wave propagation properties of gearcase 'walls' and the effects of ribbing.

Ind. Anz. Vol. 86 No. 2 Jan.1964 p 25-

Trans.I.Mar.E.(TM) Vol. 99 Papers 14-18 (1988)

Reviews standards, includes application and dynamic factors, lead corrections, accuracy, etc.

Proc.7th World Congress, Theory of machines and mechanisms, Seville, Sept. 1987 p 1319 -

A comparison between the Standards, both in their current (1987) forms, and in their previous versions, also Niemann, 1965. (TGL 10545 is the East German equivalent of ISO/ DIN). A typical large industrial gear pair (710 mm centres) is considered. Both through hardened and case hardened material is considered, with conventional or long addendum (transverse contact ratio 2.03) tooth forms. Application, dynamic, and misalignment factors are taken as identical for all cases, their effects are therefore not included. The comparisons therefore show the different influences of the geometric variations on the stress as found between the various standards. Large differences are found. The future task is to reconcile these, particularly in the assessment of transition from spur to helical gears.

Ohio State University, Dept. of Mech. Eng. 1987

Comprehensive bibliography, (no abstracts) mainly English language sources (UK, USA, Japan); very few German sources.

Power Transmission Design, No. 5,1986 p 27-

Popular exposition of gearbox vibration and noise, specially referring to results from a spur gear transmission error calculation program. Also shows some results for helicals, and the effect of face width.

SAE Conf. SP-523 Transmission and driveline noise, paper 821066,1982

Summary of work. Topics in progress include production of practically useful versions of research results for dynamic load and noise; optimum profile modifications and effect of error on transmission error, transmission error prediction program, including spectral analysis; load distribution and transmission error for helical gears, tooth compliance estimation; Finite Element models of gear teeth; bearing transmissibility studies; acoustic intensity measurements. Good references.

Int.Symp.Gear. and Power Transmissions, Tokyo 1981 paper c-6 p 32-

Theory and tests based on an inertia controlled simple gear pair model, with rigid tooth assumption. An epicyclic power circulator, in which the instrumented test gear is stationary, and the mating gear orbits around it, was used. The test gear is restrained by a long, damped flexible shaft with strain gauges. Low speed tests give static transmission error. Quasi-static transmission error resolved into torque variation due to errors, and to friction effects. These components are proportional to the first and second derivatives of transmission error, and to the friction torque. They are used in the equations of motion to produce the output waveform, with dynamic effect of speed added. Tooth separation is permitted in the analysis, but not actually found experimentally. Good agreement between experiment and theory found. Concluded that the torque variation gives excitation mainly below system primary resonance, that the tooth friction effect takes over there and at higher speed.

IMechE 1987

Includes software references, BGA details, standards, conference papers, other papers and books. Also databases. Classified references include Noise and Vibration, Design. Research at British Universities - incomplete - ? SERC only. References 90 - 97 on noise, mostly recent Japanese. 94 Linear dynamic coupling in geared rotor systems (English), 95 Nature of gear vibration and the total vibrational excitation (in Japanese) may be of interest.

AGMA 219.25 Semi-annual meeting, Toronto, 1981

Reviews and contrasts methods. Includes section on dynamic factor. Compares evaluations by the two methods, giving factor ratios where appropriate. Examples show interesting and significant differences.

ASME Lubrication conference, October, 1984 Paper No. 84-Trib-28

New approach to fatigue, involving statistics of small volumes, a fatigue failure model, tests to get data, empirical power law basis, Lundberg-Palmgren based. Stress plots, S-N curves, accounts for differences in the material through a fatigue limit, and detailed differences in subsurface stress fields. Expresses need to evaluate standard coefficients further.

JMS "Gearing", Semi-international symposium 1967 p 161-, paper 318

A disk machine, modelling the friction and pitch line impulse conditions is tested, with an anechoic enclosure around the disks. Eccentric drive gears give a complex contact sliding pattern. Friction noise is theoretically analysed. Many results, with narrow band spectra. Gear tests also made. Concludes that friction noise is 5 to 15% of all gear noise, and pitch line impulse is 10 to 40% when tooth meshing frequency is low. States that engagement impulse, and pitch circle impulse, are the main sources of gear noise.

Chinese J. of MechEng.Vol. 22 No. 2. June 1986

Uses the classical gear pair model, with transmission error with eccentricity and ramp form involute error. Results are qualitatively the same as many older studies (Kohler, Rettig, Japanese.). Sidebands appear to be shown in the output spectral analysis.

ProcIMechE International conference on gearing September 2958 p 18 -

The receptance of the tooth mesh is excited by the transmission error. Types of error and the resulting spectra are considered. Possible vibration modes, resonance, and tooth separation are discussed.

Proc. Int.Conf. on Prod. Eng. 1974 p 439- (no other details)

Stepanek mentioned. Magnetic recording of pulses is considered and mathematically modelled. Optimum recording conditions for a sinusoidal signal are determined, and a special recording head designed. Recorded wavelength is 200 microns. Static positioning during recording is possible with the method. Master scales can then be transferred. Errors are measured by optical scales, accuracy is better than one arc second. Large scales made on a hobbing machine table for indexing. Applications described are tooth profile and lead measuring, single flank testing, automatic dividing machine, and a digitally controlled gear hobber.

Int.Symp. Gearing and Power Transmission, Tokyo 1981 p 491-

Describes single flank error and profile measurement systems based on applying an accurate magnetic scale to the rotating table of a hobbing machine. For profile error, the tooth drives a stylus attached to a linear magnetic scale. For single flank error a master pinion head is driven by the gear on the machine table, with a further magnetic rotary scale on the axis of the master pinion. A computer-based digital system controls the measurement process and analyses the raw data. High frequency components due to the drives of the hobbing machine are eliminated by filtering. A good selection of results shows resolution better than 1 micron, some comparisons with other techniques on the same gears, and good repeatability. A new system. particularly suited to large gears, of economic cost, is claimed.

ASME Trans. J. of Mech.Des. Vol. 103, April 1981 p 398-

A highly detailed tooth profile model was used to calculate contact points and mesh stiffness. Typical profile errors and effects due to pitting are considered Contact ratio above 2 allowed. Strong mesh stiffness effects of errors and pits are found. The model appears to be a 'thin gear' model, that is, no variations across the tooth faces. Hub flexibility is assessed, and found to contribute significantly to load sharing. Non-involute contact incorporated by instantaneous line of contact and variable transmission ratio. Dynamic analysis of many conditions is made, and graphically presented. Factors above 3 are easily found. Many parameter combinations are examined. Conclusions include: Optimum design for minimum dynamic load is possible. Torsional flexibility in hubs is desirable. Dynamic factors reduced by increasing damping and contact ratio. High contact ratio gearing has reduced dynamic loads and Hertz stresses.

International conference on gearing, IMechE, London, 1958, p 374-

Brief presentation of dynamic tooth deflection measurements on spur gears, and corresponding experimental transmission error

Doctorate thesis, University of Sheffield, Department of Applied Mechanics, 1959

An experimental and theoretical study of dynamic loads. The system model is a damped single degree of freedom. system, excited by transmission error (called velocity ratio error). The excitation is expressed as a Fourier series. Experimental measurements include transmission error, and its spectral analysis, and tooth dynamic loads. The latter were measured at lower speeds by tooth deflections, using an inductive transducer, and at higher speeds by piezo-electric strain gauges on the sides of the test gears close to the tooth root. Rig limitations enabled only modest specific loads to be reached, and the test gears were narrow facewidth spurs, of nominal involute form. The transmission error amplitude was about 0.001 in. Test results were obtained for speeds up to and beyond primary resonance, and the resonance of successive harmonics of the tooth contact frequency with the primary resonance at about 2,300 Hz clearly demonstrated. The maximum dynamic increment observed was 3. The results were analysed to show the damping of the resonant response to each tooth contact frequency harmonic, and this was found to be very amplitude sensitive. For low amplitude error components about a twentieth of the static deflection corresponding to the steady load, the resonant magnifier Q was of the order of 30. Q fell to about ten for relative errors up to about 0.4, and for higher error magnitudes the response was amplitude limited to a maximum of 3. It was presumed that the loss of contact shown experimentally at high incremental loads ultimately caused detuning of the resonances and a self-limiting of amplitudes. At a fixed speed the process was essentially a steady state forced vibration, 'sampled' by individual teeth as they passed through mesh.

'Engineering; Feb. 51960, p 209-

A brief description of dynamic tooth load measurements using piezo-electric strain gauges, and of transmission error measurement using the relative movement of wires unwrapping from base cylinders. The relationship between transmission error spectral components and tooth loads was shown.

ASME 63-WA-141963

ASME

rejects Attia figure 7 - which shows jumps in deflection as pitch point is traversed. Explanation offered.

IMarE JBSCA The recording and interpretation of engineering measurements Conf. April, 1972

Discusses the use of the FFT, and data-logging, in the investigation of noise and torsional vibration signals from a test gearbox. Compares analogue tuned filter with DFT methods, and considerations involved in analysing time series data from deterministic, periodic sources. Gives a measured transmission error to noise transfer function for a 3:1 reduction gearbox. Interest now mainly historical.

ProcIMechE Gearing in 1970 Vol.184 part 3 O.

Excitation and system dynamics of a single stage 3:1 helical gearbox are considered. Transmission error is analysed. Synthetically generated transmission error using systematic and random errors, shows features characteristic of real gear transmission error. A dynamic model of a gearbox, together with results for static transmission error and dynamic transmission error and noise, is detailed, and experimental and analytical results compared. The complexity of the problem is emphasised. Many references.

ProcIMechE 61/85 Vol.199 No. C31985

Pitch error data taken from Mark, W.D., 1979, is used to show that in otherwise ideal spur gears, pitch errors lead to tooth frequency components in the transmission error spectrum, except in very special circumstances. Effects due to contact outside the theoretical involute action, and timing shift due to pitch error, are identified, but not analysed. The interaction between contact ratio and the transmission error spectrum is demonstrated. The validity of superimposing transmission error components from various elemental errors to obtain total transmission error is questioned, for real cases, as these can be expected to display non-linear summation effects shown. Good summary of technique, but insufficient detail as so often the case.

Bull. JSME Vol. 23 No. 183 Sept. 1980 p 1536-

First sets up the matrix equations for a helical pair, with shaft and bearing stiffness included. Tooth pair stiffness evaluated from integral equations which allow for convection between 'slices'. Actual source of values is not clear. Four characteristic tooth error forms are considered, pressure angle, convex, concave and wavy profile. Helical and spur comparison brings out averaging effect of helix. Low order error harmonics are shown to correlate with the dynamic loads when in resonance. Non-linear resonances suggest that quite low tooth loads were used. Contact ratio, helix angle effects shown. In particular, harmonic amplitudes for the various errors show minima at specific loads as expected. Stiffness can be obtained from equivalent spur gear. Useful conclusion summary covers point made above. Up-to-date Munro & Kohler.

Proc. 7th World Congress, Theory of machines and mechanisms, Seville, Sept. 1987 p 1355 -

It is observed that '..only those investigations concerning the relative rotation vibration of a spur gear pair whose shafts have flexible enough twisting rigidity and solid enough transverse rigidity have shown good agreement between theoretical results and measured results'. One reason for this is the difficulty of successfully incorporating time variable parameters, particularly mesh stiffness, in the analysis. By examining the second order differential equation with time varying stiffness, and excitation with a constant plus a time variable component, for various proportions of constant to varying components, the requirements for adequately good solutions using constant coefficients are found. A special way of expressing the excitation due to transmission error and variable mesh stiffness, so that the constant coefficient equation is an adequate model, is specified. Good experimental justification is presented.

JSME Int. Jnl. Ser.3 Vol. 31 No. 2 June 2988 p 426-

not seen) Published abstract - '...a new method to estimate the state of vibration of gears at arbitrary operating conditions from the data of vibrational acceleration measured at some definite reference operating condition when the manufacturing and alignment errors of gears are unknown.'

IMechB Conf Vibs in rotating machinery, Sept. 1984 Paper C317/84 p 81-

Uses an 8 degrees of freedom. model for a gear pair, with time-dependent mesh stiffness, a tooth error function, and variable external torques. Backlash and contact across the backlash is included, also load dependent contact ratio. Some linearisation for small amplitude solutions is made, but parametric and combination excitation is allowed. Solution is by numerical integration. Typical theoretical and measured results (from the Munich test rigs) show better agreement than usual in such work. Experimentally, parametric and combination resonances did not occur.

Springer Berlin, 1987 ISBN 3-540-17121-8

(in German) Subtitled 'models, methods, and behaviour', this is a review volume, covering complex dynamic models of simple, through multistage, to epicyclic, transmissions. Detailed mathematical models are derived, and solution methods given. Nonlinear and stability problems are tackled, and backlash induced 'rattle' in unloaded elements of transmissions described. Somewhat over compressed, with matrix formulations. Careful study should be worthwhile.

Bull. JSME Vol. 9 No 38196f p 829-

Crowning, tip and root relief were investigated. Dynamic stresses (strain gauges) and noise were measured, as functions of peripheral velocity. Optimal values are deduced. 'Maps' of the deviation of the tooth face from involute are derived, and load distribution, and factors, on the tooth faces calculated. The deflection component of transmission error was determined, in a similar way to Harris and Munro. Similarities with spur gear relief calculations are noted.

Bull. JSME Vol. 9 No3f 196b p 81b-

Wide cantilever plate theory, with point loading, is used to calculate stresses in helical gears under static loading. Theory is developed, simplified formulae given, and experimental verification (strain gauge measurements) reported. Equations obtained are similar to the Lewis formula, true stresses are therefore calculated with the further aid of Dolan and Broghamer photoelastic results.

ASME Trans. J.Vib.Aco.Str.Rel. in Des. Vol. 208 July l986 p 363-

The single degree of freedom model, with loss of contact and backlash, is excited by transmission error, described as an oscillatory random process (justified on the basis of a typical transmission error record showing eccentricity and variable tooth to tooth errors). A Markov process as a narrow band shaping filter is introduced. Mesh stiffness is piecewise linear (3 parts). Predicted results for dynamic force resulting from given transmission error are very similar to the deterministic model, except that the ordinate is variance of the force, and the error is expressed by its standard deviation (? variance). Force is plotted as a function of position along the contact path. Primary resonance has the usual effect. Finally, the variance of the variance of dynamic load is given, and shows wild fluctuations at resonance. Stated that results conflict with the Tobe and Sato analyses.

VDI Fortsckr. Ber. VDIZ Series T1 No. 54 2983

not seen in full. Appears to consider the optimum stiffness and deflection variation through the meshing process, and to present calculation programs to minimise these. Shows typical gearbox noise and vibration spectra. Examines statistics of a large number of epicyclic gearboxes, also other gear configurations.

Doctorate Thesis RWTH Aachen 1970

Considers the noise and vibration of high-speed spur gears, and the appropriate dynamic models. A power circulating test rig is described. Speed-noise characteristics are determined and explained. Resonance is considered, Fourier analysis of the transmission error, derived from contact conditions and stiffness, is given. Noise at speeds beyond the resonance regime is next considered. Various dynamic models are used to illustrate the experimentally determined effects of tooth errors, helix angle, and other parameters, particularly possible damping methods. Dynamic models of dampers are created, and experimental dampers tested. Large reduction of dynamic load at resonance is found.

Third Int.Conf.Vibrations in rotating machinery, IMechE 1984 Paper C318/84

Describes theory and experimental investigation of a turbine driven pump with intermediate gearing. A system analysis is based on the departure from true involute form of each gear element, rather than on gear pair transmission error. Qualitative results are supported by experience. Applicable to the lower frequencies, rather than primary resonance. Essentially torsional vibration investigations, uses a four degrees of freedom model, with rigid teeth, but with tooth errors. Much in common with Wang, C.C. in approach. Non-linear terms occur. Matrix formulation is used. Experiment includes torque telemetry. Detaining and forcing modification by altering coupling stiffness was effective. Contains the suggestion that, as the fundamental period of tooth phenomena is very low, excitation from harmonics of this can drive what appears to be a speed independent constant frequency resonance. Agreement between theory and measurement is barely adequate as presented

I.Mech.E 2nd conf.Vib.in Rot.Sys.1980 paper C265/80 p 103-

A pulverising mill drive is analysed, as a five degrees of freedom torsional system, with 'standard' error excitation at the higher speed gear mesh. High wear at the final girth gear of the mill is common, but large teeth have long life. However, resulting dynamic effects may be important, and excitation from this source is also considered, though the low speed behaviour is straightforward. In the Lees excitation model, sinusoidal errors at the mesh interact non-linearly in the compatibility condition necessary to ensure continuous contact across the mesh, to introduce system resonant responses at multiples of the error frequency. Double helical error mis-match is shown to give complicated frequency spectra. Agreement with measurements is qualitative, but development of the methods is proposed.

ProcIMechE Vol. 174 N.174N. 23 1960 p 689-

Wheelsets with close-coupled axle-suspended motor moving over railjoints. Corresponding dynamic tooth loads analysed. Traction motor drive as two mass system. Gear tooth separation. Dynamic increments are calculated. Resilient/non-resilient drives, if inherent resilience governs, then added resilience is advantageous.

ISVR TR-124 86 p. June 1984 (Gearbox vibration)

not seen

ASME Trans. J.Mech.Tran.Auto. in Design Vol.110 June 1988 p 221-

The dynamic loading problem is restated yet again. Effects included are mesh stiffness, friction at the tooth contact, damping. Difficult to see much new. Step-by-step solutions proposed.

ASME Trans. J.Mech.Tran.Auto. in Design VoI.110 June 1988 p 226-

The procedure for solving the model introduced in Part 1 (see above) is given. The effects of varying parameters in the model of a particular system are shown, with elegant presentations. The principal source of excitation is the stiffness variation between single and double tooth contact. Discussion is 'out of the Ark' for a 1988 paper. Transmission error appears to be ignored, perhaps this accounts for the conclusion that higher tooth stiffness lowers the dynamic factor.

IAVD Vehicle Design, Geneva, March 1985

not seen

Precision engineering Butterworth, April 1987 Vol. 9 No. 2 p 83-

A ingenious single flank tester for single gears, which uses a master worm to drive the gear, is described. The worm has a 'protrusive measuring thread' which enables engagement with the complete flank of each tooth of the gear under test. A grating system is used for measurements. Various forms of profile relief (spur gears) are shown to lead to various degrees of jumping at the transition between successive tooth contacts. OBM (out of bound meshing) describes the condition. Jumps cause angular acceleration and noise. Direct radiation from gears in an open gear noise tester is measured. The gears with the gentlest transition from tooth to tooth were found to be quietest. Contact ratios near integer - either 1 or 2 were best. Reviewer suspects that results are for very light loading only.

ASME Trans. J.Mech.Trans.Auto.in Des. March 1985 Vol.107/5 ASME paper 84-DET 54

A PT6 turboprop engine epicyclic is considered. A gear stage is modelled, and typical transmission errors simulated. These are mostly of once-per-rev plus tooth mesh frequency. Dynamic loads are calculated (Lagrangian method outlined in an appendix) and related to transmission error amplitudes. Root stress is then considered. It is shown how the disposition of the error between various gears gives different dynamic loads, and how annulus flexibility is advantageous. Sun gear misalignment effects are demonstrated.

Inter-noise 83 - Inst. of Acou. Edinburgh 1983

Briefly describes a transmission error measurement system using two toothed wheels, waveform shaping of the signals from magnetic transducers, and phase measurement. Also has method using one transducer, and a phase comparison with an average signal derived from a phase-locked loop. Best resolution is 4 arc seconds. Results shown for gears (using gratings in this case) and for a misaligned constant velocity joint transmission shaft, and a six cylinder IC engine.

JAcoustSocAm 63(5) May 1978

The most complex theoretical analysis of transmission error and its spectra. Highly mathematical, assumes linear systems theory will work for the gear mesh. Gear errors are divided into systematic and random, with differing effects, arbitrary zone of contact is allowed. Harmonic analysis relies on transforms of the error sequences, and a 'Mesh transfer function' (a difficult concept). References.

Continuation of 83(5) above. Uses Legendre polynomials to isolate various error contributions to overall transmission error. Stiffness and load-dependent effects. Contact lines for helicals; examples of spectra - easily misunderstood - use of the DFT on pitch error data. Mesh attenuation concept, with mesh transfer functions. Extensive discussion of effects of various error components. helicals and spurs compared, design parameter effects. References. See 'Noise and Vibration' survey for fuller discussion.

NASA-CR-159088 February 1979

not seen

NASA Contractor report 3626 - Contract NAS 3-21978, Oct. 1982

Applies Mark theory to a particular gearbox. General system dynamics are considered. Profile design to minimise transmission error is developed, and dynamic loads calculated, as are bearing loads and tooth stresses. Differences between conventional and specially calculated tooth reliefs are calculated. Optimum profile leads to marginally bigger stresses. Difficult to see wood for

Plenum, New York - Engine noise, excitation, vibration and radiation (Ed. Hickling F Kamal),1982

The JASA papers in more digestible form. Transmission error is decomposed into elementary errors. Legendre polynomials used to obtain spectral representations. Spectra of pitch error and mesh transfer functions derived, Static transmission error and mesh transfer function examples, effect of design parameters, predictions for trends with contact ratio etc. Amplitude modulation sideband interpretation of tooth face deviations. Zone of contact modulation effects, References, discussion. See 'Noise and Vibration' survey for fuller discussion.

ASME Trans, J.Mech.Trans.Auto. in Des. Vol.105 Dec. 1983 p 725-

Profile and lead measurements are used to define the surface of an unloaded tooth. Gaussian quadrature is used to obtain a 2-D expression covering all points on the tooth face. Alternatively, polynomial interpolation is shown to give equivalent results. Heavy maths involves the Legendre polynomial approximations introduced in Mark's major papers. 'Typical sets of profile and lead measurements possess no common origin of ordinates. Moreover, each such measurement generally contains non-insignificant errors.' The formulation enables such measurements to be used to give functions applicable over the whole tooth faces.

ASME Trans, J.Mech.Trans.Auto. in Des. Vol.109 June 87 p 269

First describes cumulative and adjacent pitch errors in complicated mathematics. Defines positive error as lack of tooth material. Considers the collation of a grid of measurements of 'pitch error' over each tooth face to give total deviations. Uses the Legendre polynomial approximations introduced in Mark's major papers. The measurement values are derived from profile and lead measurements, and the effect of incomplete agreement between practical measurements is allowed for. The essence appears to be a formulation that interpolates the point measurements to give functions applicable over the whole tooth faces. Decomposition into mean and random components again used.

ProcIMeckE Vol. 201 C31987 p 230-

Reply to criticisms of original Mark papers contained in Kohler 8z Regan, 1986. Corrects misunderstandings of the mathematical derivations in the Mark papers. Also comments on the effects of elastic deformation under load in contributing to the validity of a linear analysis. Author's reply (K 8z R), admits misunderstandings, but expresses continued doubts that elastic deflections in real gears are sufficient to ensure that the full geometric zone of contact exists for all tooth error conditions

ProcIMechE 42l87 Vol. 201 No. C31987

ASME Trans, J.Mech.Trans.Auto. in Des. Vol. 109 June 1987 p 283-

Generalised transmission error for spiral bevels in terms of mesh loading and geometric tooth surface deviations is considered. Force-deformation and force-transmission error relations are established. Matrix analysis derives the general equations of motion. Integral equations for localised tooth pair stiffness are given.

ASME Trans; J.Mech.Trans.Auto. in Des. Vol. 109 June 1987 p 275-

Notes first that the assumption implied in most parallel axis gear transmission error analysis, that the shafts themselves remain parallel in action, does not hold for bevel gears. The kinematics of nominally spherical involute bevel gears is described, and a three component transmission error defined.

ASME Trans. J.Mech.Tran.Auto. in Des. Vol.110 June 1988 p 203- '

For straight or spiral bevel gears of nominal spherical involute design, the resultant total force vector transmitted by the gear mesh, in the absence of friction, lies in the plane of tooth contact. This force vector can be characterised by three scalar components, two orthogonal force components lying in the plane of contact and the resultant moment taken about the nominal centre of the zone of contact. Equations for these three generalised force components are derived. The equations are expressed in terms of tooth pair/gear body stiffnesses, bearing/bearing support flexibility influence coefficients, the shaft input torque, deviations of the tooth running surfaces from perfect spherical involute surfaces, and bearing centreline offsets from the positions occupied by the base cone axes of the perfect involute bevel gear counterparts to the actual gears under consideration. Inertial forces arising from transverse and axial vibrations of the gear bodies are assumed to be negligible in comparison with the bearing support reaction forces.' (Author's summary)

ASME Trans. J.Mech.Tran.Auto. in Des. Vol.110 June 1988 p 211- '

For a given set of forces transmitted by the gears, each of the three components of the generalised transmission error of spiral bevel gears is shown to be stationary with respect to small independent variations in the positions of the endpoints of the lines of tooth contact about their true values. The tangential generalised transmission error component is shown to take on a minimum value at the true endpoint positions. A computational procedure based on the method of steepest descent is described for computing the true line of contact endpoint positions and the three components of the generalised transmission error. A method of computing the Fourier series coefficients of the tooth meshing harmonics of the three generalised transmission error components is also provided' (Author's summary)

Symposium on internal noise in helicopters, Southampton, July 1979 Not seen

ProcIMechE 24l851985

Unusual distribution and cancellation of sidebands in epicyclic gearing is explained by the variable source-transducer distance, causing phase/time effects. Effect of multiple planets can both add and subtract components. The Sea King gearbox is used to show complete cancellation of the tooth contact frequency, but strong sidebands.

ProcIMechE 76l85 2985

Defects of the type of root cracks are considered. Time signal averaging of vibrations gives a result, which is band-pass filtered around the most significant tooth contact frequency component area. Reconstruction of the time signal after removal of the single major tooth frequency component reveals a local impulse, envelope of this signal can be used to locate the defect quite closely. The method obtains the modulation caused by the defect, kurtosis can be used to detect the presence of local peaks. Case study shows that it works.

Pitman, London,1954, 3rd. Ed.

One of the few major texts. Section 25.12 describes the use of profile modification (relief) to mimimise deviations from uniform motion, to control dynamic loads, vibration and noise. Dynamic loading is mentioned in section 28.10, which describes the underlying ideas of BS 436:1940. The possibility of large dynamic loads and high noise occurring when there is coincidence between tooth contact frequency or the frequency of error components with system resonances is mentioned. Noise is dealt with in sections 21.18 and 21.19, recognised as the effect of non-uniform tooth load, due to non-uniform motion, and the inertias of the components. Tooth errors may produce tooth contact frequency excitation, ghost frequency errors due to the cutting machine are mentioned. Control measures include better accuracy, inertia reduction, vibration isolation, enclosure. The effectiveness of a measure '...may puzzle even the most experienced'. Worm gears are quietest, helicals are preferred, ground gears have better accuracy, parallel drives are easier to mount, but '...accuracy is the greatest factor.

Pitman, London, 2972

'Section 5.20 explains the successive concepts of kinematic error, static transmission error and dynamic transmission error, where the dynamic transmission error reflects vibrations superimposed on the static transmission error.'

IMechE Aerospace Ind.Div. Seminar Nov. 1985, Transmissions technology for propfan and geared fan engines. Paper 4

Transmission requirements, health monitoring. Little text, many illustrations, gear configurations, sizes, reliability of Tyne, tooth forms, transmission knowledge required.

ASME Prod.Eng.Div. PED Vol. 81983 p 102-

Reviews basic theory of gear vibration components, harmonics, ghost, bolt holes, modulation. Distinguishes 'nicks' which appear more prominently in time domain averages. For a dedicated on-line test machine angular velocity measurements are chosen, and the high resolution needed calculated. Connection problems in a test stand are considered. A special transducer (see SAE 820054) gives angular acceleration and velocity (incremental shaft encoder) and data references. A high inertia is coupled to the output, a low stiffness drive to the input, which also has the transducer. System analysis selects best running speed. Calibration is elaborate, based on system analysis. Signal processing to give spectra with array processors, whole system has 30 seconds floor-to-floor time, 5% standard error.

SAE Conf.SP-523 Transmission and driveline noise, paper 821065,1982

Reviews older work, stating deficiencies. Excitation, system dynamics. Opitz classification with modifications, more results and classification suggestions. Describes new system analysis, which includes gyroscopic moments, housing, variable mesh stiffness. Finite Element methods to give casing response, surface velocity to radiated noise conversion. New analysis represents mesh as 36 component stiffness tensor, makes modulated sidebands calculable. Transfer matrix technique used. Branching allowed. Shows that forces that result due to stiffness at one frequency, and a forced response at another, interact to give sum and difference components. Analysis very complex, given in outline only. References to work at Virginia State University. (reviewer comment - transmission error excitation almost ignored, simple error shown in example. Not clear how accurately system parameters need to be known)

Trans.I.Mar.E.(TM) Vol. 99 Papers 14-181986

Discusses basis for the BS and DBGI approach to surface hardened material stress specification, including residual stresses, sub-surface stresses, review of gear tests for NAVGRA, AVGRA, with results summary.

DBGI for AGMA tech conf. 801980?

The DBGI transmission error and load distribution computer program, computer outputs given, as published elsewhere, intensity of load distribution, misalignment effects, pitch errors, end relief, variation in meshing deflection. Conclusions- load distribution very sensitive to errors. Contact marking an inadequate guide. Wear and plastic flow makes gears of softer materials work. High accuracy required of surface hardened gears. Reliefs can increase surface loading, use with care.

IMechE C258/83 1983

Instrumentation for transmission error under load, encoder(grating) based, plus micro. Uses MAGRAC/NAVGRA rigs, 560 centres, case hardened, open gears with temporary bearings. Frequency analysis shows some expected results, a few error values for the gears given, some general conclusions on the effect of manufacturing error on tooth contact frequency components. Shows - change in transmission error with load, no-load, transmission error useful as a quality control tool, transmission error - bearing cap vibration correlation, system dynamics influence on bearing vibrations.

Doctorate thesis, University of Cambridge, January 1962

Describes work between 1957 and 1960, supervised by R.W. Gregory, and influenced also by S.L. Harris. A spur gear power circulating rig, not very highly loaded, was run, with static and dynamic transmission error measurement. The dynamics were modelled, reduced to a 2 degrees of freedom. system. An 'exact' equation was idealised to simplify it, but nonlinearities were not ignored. Instabilities at primary and sub-harmonic frequencies were predicted and found experimentally Dynamic tooth loading, and true transmission error after the Harris pattern, were investigated. The work was reported in several IMechE papers.

ProcIMechE Vol. 184 Pt. 3 O, 'Gearing in 1970' p 79 -

Transmission error is defined and discussed. The effects of a variety of profile error pairings is shown. single and double flank testing is compared. Single flank composite error testing is advocated for the future.

BGMA half yearly technical meeting May 1988

Discussion of error types, leading to single flank error. Effect of profile, pitch. Sample results, dual flank measurement. Comparison of information from various methods, bevel, spiral bevel and hypoid examples. Effect of errors in mounting of hypoids on tooth to tooth composite error.

AGMA 239.10, August 1967

Development of transmission error from single pair error curves, error effects, design load effect for spur gears, some dynamic results (all as elsewhere). Optical grating measurement systems. Goulder single flank tester. Some results, references.

Institute of Petroleum Gear Lubrication Symposium. (mechanical tests and lubricants panel) Brighton, Oct. 1964 - 'Lubrication evaluation with gear machines' p 73-

Weber results used on IAE standard gear form, effect of pure involute, and tip and root reliefs. Contact beyond nominal path. Predictions for the IAE rig, tooth load cycles (static), oil film thickness predictions, loading effects. Concludes that tip relief on the test gears is needed for the IAE machine to give valid results.

Proc. 6th Japan Nat.Cong.for App.Mech. 2956 paper IV 15

Used strain gauges in the root fillet for measurements to compare with theory. A square wave form of tooth stiffness is adopted and the system treated as piecewise linear, with compatibility conditions governing the transition between regimes. Different resonant frequencies cause corresponding vibrations at the zones of single and double tooth contact. Spur teeth appear to have no reliefs. Remarkable agreement between measurement and theory.

JSME 'Gearing; Semi-international symposium, 1967 p 171- paper 319

Noise radiation is considered as from a circular plate. General forcing-response model follows mainstream thought at this time, that is, tooth error forcing, and possible resonant response. Subharmonic vibrations, with tooth separation, are recognised. Extensive theory and calculations of the vibration form of disks on shaft. Gear considered as circular plate driven harmonically at a point on its circumference. Power circulator in anechoic room is tested with overhung thin web gears in open. Near and far field results discussed, resonance between disk modes and tooth contact frequency recognised.

Proc. 7th World Congress, Theory of machines and mechanisms, Seville, Sept. 1987 p 1295 -

The interaction between support (bearings) vibrational input, and gearing vibrational modes is studied. Shaped random vertical excitation at the supports is applied to a single-reduction gearing coupled motor-generator set. The excitation is seen to cause responses influenced by the gearing dynamics.

VDI Fortschr. Ber. VDIZ Series I No.104 (RWTH Aachen) 2983

Extensive modern analysis of tooth loading. Includes state of the art survey. Aim - to develop a universal computation technique for contact loading and stressing of all gear forms. Uses finite elements, with selection of methods to make the necessary three-dimensional analyses of manageable length. So - (1) General development of calculation methods for elastic bodies in contact. (2) Transfer of the technique to the special requirements of gearing, including modelling of errors and casing deflections. (3) Checks by other calculations and measurements (4) Demonstration applications. All this appears to have been achieved, and there are many comparisons with other published data. Super references.

Maschinenelemente, Vol. 2, 'Getriebe' Springer Berlin, 1960

Two important sections

A designer's summary of the results of work, largely at FZG, Munich. The emphasis with respect to noise generation was on the tooth engagement impulse. The characteristics of noise levels and spectra with respect to speed and load are given as Campbell diagrams, thereby recognising system resonances. The impulse resulting from insufficient profile relief, or errors, is shown. Other parameters specifically considered include pitch error, profile relief, surface roughness, contact ratio, and helix angle. For contact ratio, low values, (near unity) are bad, there is a step improvement of about 3 dB as contact ratio reaches 2. For helix angle, the effect is load dependent, zero at low loads, rising to about 8 dB less noise at the highest loads for a helix angle change from zero to 45o.

An interesting section describes modifications, of which a conical outer diameter is typical, to straight spur gears which can diminish the engagement impulse. Resilient gear mounting and the effect of oil viscosity are also considered.

The behaviour of large gearsets is discussed, and the levels and spectral characteristics grouped, as follows:

Dynamic load (pp 78 - 81).

Transmission error is clearly recognised as the appropriate excitation, of the gear inertia-mesh stiffness system. Simplified diagrams give the effective loads as a function of specific loading, pitch line velocity, and pitch error, and a summary of results from Rettig. A limited calculation method is given. (See 'Noise and Vibration' survey for fuller discussion.)

VDIZ 112 (1970) No. 4 February (Part 1)

Straight spur gear tooth contact and stiffness. Description of test rig and instrumentation. Details of test gears. Transmission error measurements as a function of load. Effects of design parameters. Tip and root reliefs, and the corresponding load transfer diagrams. See part II below. Long, excellent papers.

VDIZ 112 (1970) No. 8 April (Part II)

Continuation of above. Considers load lines, stiffness of helical contacts, buttressing effects of excess tooth width, contact and overlap ratios. Examines the influences of parameters - excess width, helix angle, overlap ratio, profile modification x, module, on transmission error. Discusses relationship between transmission error and noise. Many graphs and experimental results, but for only one basic size of gear.

Konstruktion Issue No. 41966 p 129-

The definitive early paper on helical gear noise. Concludes that transmission error is paramount in determining noise, though concentrating on the tooth to tooth components. Tests in an anechoic chamber of a number of 91.5 mm centres gears, with various geometric and error properties are exhaustively reported. Contact line positions and mesh stiffness are examined. Results encompass the effects of speed, torque, helix angle, overlap ratio, addendum change, reliefs, module, addendum modification, machining process, DIN error rating, and these are summarised by empirical equations. Frequency spectra are also shown, with sidebands, and a breakdown of the contributions to each spectral area from various factors. The results are perhaps 'too good to be true' (reviewer's comment), and specific to the gear system tested. English translation available, from H. Kohler.

VDIZ Vol. 99 No.3 p 89- Jan.1957, and No.4 p 131- Feb.1957

Contemporary with Harris, Attia, Kohler. See IMechE 1958 gear conference. Displacement transducer used to measure tooth deflection during passage through mesh in a power circulator. Calibration, static tooth loads, running in effects of the instrumented tooth, which had deliberate pitch error. Tooth load waveforms show that most running was below primary resonance, but also that loss of contact was found. Additional inertias used to reduce primary resonance for some tests. Extensive analysis showing effect of load, speed and errors on tooth loads. Tooth load was found to range from 1.03 to as much as 3.3 times the nominal value. A variety of gear forms, materials and addendum corrections were tested. Use of the results for practical purposes to be treated in a future publication. Transfer to large gearing is to be assessed by sample testing.

ProcIMechE International conference on gearing September 1958 pp 31-42

Straight spur gear tests of deflection relative to the gear body during tooth engagement are used to demonstrate the load cycle on individual teeth. Static tooth deflection as a function of load and phase of engagement is shown. Some of the earliest dynamic tooth deflection records are given. Load capacity influences were separately tested. Running in effects are carefully considered. Conclusions refer specifically to transmission error as the fundamental variable for dynamic load calculations.

'Maschmenelemente' Vol. 2, 'Gefriebe...' Second Edition; Springer Berlin, 1983

Extensively reworked by Professor Winter, this is an update of the first edition, by Niemann.

Much expanded from the previous edition. Typical noise levels for gearboxes of many types as a function of power are graphed. The gearbox is considered as a dynamic system, and various effects examined. These are: variable mesh stiffness, engagement and end-of-engagement impulse, excitation due to the change of direction of friction at the pitch point - friction impulse, rubbing noise due to the relative sliding of the tooth flanks, instrument gearing.

The effects on noise of the following parameters are considered: Load, module, tooth number, contact ratio, addendum modification, tip relief, spur and helical gearing, helix angle and overlap ratio, crowning. The spectral content of gear noise is described. Production is considered, the effects of material, surface condition and running-in, surface treatments, quality errors considered are pitch, base pitch, pressure angle or base circle, lead, tumbling of the gear axis, profile errors and relieves, eccentricity and cumulative pitch, machine tool errors, all are included. A brief section gives a list of other influences such as oil properties. Plain bearings act as damping elements for structure borne vibrations, rolling contact bearings should be preloaded. The radiation properties of the gearcase are included, with recommendations.

There are also sections on the load distribution in tooth contacts, tooth pair and mesh stiffness, transmission error.

BulI. Japan Soc. of Prec.Engng. Vol. 17 No. 2 1983

Variability of the tooth pair transition contact points is calculated for eccentrically mounted 'perfect' gears. Uses concepts published in UK well before. Modulation effects due to variability in contact conditions give calculated modulation index for FM. Very interesting but does not explain how the variability in contact phasing causes transmission error.

Int.Symp.Gearing and Power Trans. Tokyo 1981 p 91-

A pendulum damper in the form of six rubber covered rollers in holes in a flywheel, tuned to the gear tooth contact frequency, was tested. The pendulum damper frequency is directly speed proportional, so is always tuned. In the tests, dynamic transmission error was recorded. Low speeds, (174 rpm) and low tooth mesh frequencies (50 and 190 Hz.) were investigated. Some reduction in tooth contact frequency vibration amplitudes was found, rollers without rubber coating were found less effective. About 10 dB reductions reported, further development required.

Phil. Trans. Roy. Soc. Vol. 263 Dec. 1968 p 369-

Definitive summary paper of its time. Shows correlation between transmission error and noise. Has graphs giving transmission error versus load; noise vs. load; noise vs. speed; noise vs. torque/face width; noise vs. contact ratio; examples of pitch, profile, and alignment error effects on noise. Also considers control measures, bearings, flexibilities, gearcase construction. Contains results of measurements on a large number of gear sets, with classification by sound radiation efficiency, size, quality of manufacture, and noise level at a standard distance. (See 'Noise and Vibration' survey for fuller discussion.)

JSME 'Gearing' Semi-international symposium, 1967 p 199- paper 322

Analogue computer solutions for a single degree of freedom. model includes variable tooth stiffness and backlash. Error free gears are first considered, resonant phenomena established. Errors ('incorrect gears') are introduced, calculations are compared with measurement (excellent agreement!), helicals are considered. Comparisons between measured valued of load increment and existing formulae -Niemann, Tuplin, Buckingham, show formulae in error. Good critique of formulae, which are dismissed as only giving coarse approximation.

Sales information leaflet, Klingelnberg No.1156e 1956

Gives data of the original Opitz seismic transducers, and associated analysis equipment. Lowest useful frequency 1 Hz. Range ± 17 mins of arc. 'release' (= stiction?) 0.1 arc second. FM carrier-based analogue electronics. Size -180 mm dia. 258 mm high. Can be run at reduced accuracy with axis horizontal.

Round table discussion on Marine reduction gears, Delft University of Technology, 1958. Ed. Blok, H. p 36-

Basic noise theory, measurement technique and results from a number of ships. Vibration and noise correlation observed. Qualitative association with manufacturing quality observed. Noise control by easing treatment. Early narrow band spectra, Campbell diagrams of excitation/resonance field. Discussion with many results from Renk, reports that ghost is the major noise component.

Doctorate thesis; DIK, LM, TU Munich (FZG) 1982

Discusses EHD theory as applied to gear contacts. Film thickness, temperature distributions, loading, film geometry. Computing procedure, results. From the pressure distribution, to stresses, Hertzian, thermal, application to surface failure.

JSoundVib Vol. 121 No. 3 p 383-1988

A useful summary review with 188 references, covering the history and present state of dynamic models of gearing. Only very brief descriptions of individual models. Classes considered are - history and simple dynamic factor models; models with tooth compliance; models for gear dynamics; models for geared rotor dynamics; other mathematical models.

JSoundVib Vol. 125 No.1 p 71-1988

An initial general survey is followed by modelling considerations. The differential equation model using static transmission error forcing is discussed. Computation techniques are described, for various degrees of completeness of formulation, including loss of contact. A number of published experimental results by Kubo were compared with calculations, with good results. Some other theoretical solutions were also compared with the new computations. It is claimed that an accurate calculation scheme has been developed, which uses the loaded static transmission error as the excitation function. Many useful observations on the reliability of other methods are made, for example it is stated that there is no use in introducing second order effects in calculations if the damping is only guessed.

Noise-Con 85, Ohio State University, Columbus June 1985

The effect of involute crowning (actually tip and root relief) on noise and vibration was examined to see if the signal characteristics could be used to measure crowning. A vehicle transmission with all but two gear pairs removed was driven in open loop at light loads (just sufficient to prevent detected tooth separation) and gears with modest and excessive 'crowning' tested, for shaft angular acceleration, linear vibration of the casing, and airborne noise. By examination of Campbell diagrams a best running speed was chosen, and the power ratio of the signal component in a narrow band around tooth mesh frequency to that in a bandwidth covering the first three harmonics was chosen as the discriminating parameter. Only angular acceleration of the output shaft of the gear pair produced clear discrimination. Static transmission error measurements confirmed that the 'crowned' gears had higher tooth to tooth amplitude..

Konstruktion, 371985 Issue 11 p 417

A dynamic model of an epicyclic gear system is established, using all relevant degrees of freedom. System parameters are considered, it is noted that accuracy is required here. Periodic stiffness variation is considered important, analysis gives mode shapes, load time variation at critical points, dynamic increment to speed relationships for the tooth meshes. Load distribution factors, representing for example. the variation between planets, are determined. Errors are briefly considered. No experimental verification.

Antriebstechnik, 201981 Nr. 7-8

Concise description of drive line system modelling by the 'Aachen' methods. Includes motor inertias, backlash, non-linear stiffnesses, gear contact models with variable mesh stiffness, cardan shaft with backlash. Shows mode shapes, frequencies and torques during gear changing. Clutch operation modes included. Experimental verification is promised.

RWTH Aachen internal presentation to IMechE Gearing investigation Committee

Illustrates an investigation on an electric motor driven geared turbocompressor. Laboratory rig using misaligned Hookes joints used to give cyclic excitation to a gear power circulating rig. Large motor compressor torsional model with backlash. Simulation results showing high impact loads. Damping within the backlash contact investigated by simulation. Big effects.

Konstruktion 341982 No. 6 p 229-

Finite element treatment of gearcase design, with particular attention to the deflections of the bearing bosses. Modelling considerations, stiffening ribs, extensive tables of relative effects for various arrangements. Also covers effect of angular misalignment on fluid film hearing performance, and includes the moments applied by the bearing itself. Matches casing stiffness to the shaft misalignment under load to produce best bearing performance. Complements Widyanata RWTH thesis.

Proc. 7th World Congress, Theory of machines and mechanisms, Seville, Sept. 1987 p 417 -

A new method for treating dynamic systems with backlash (plays) is presented. It consists in expressing the general dynamic behaviour of each of the system sections between the points at which backlash occurs separately, and linking them by impact (considered as instantaneous) boundary conditions, which determine the 'free flight' behaviour of the subsystems. Any form of backlash functions, including time dependency, is allowed. As well as solving for displacements, energy losses during the impacts may also be estimated. One illustration of the method is by application to a five-speed automotive gearbox, when it is shown that some non-loaded meshes rattle in harmony with the engine torsional input, but other meshes exhibit quasi random behaviour, which is actually deterministic (chaotic?). The effect of backlash and oil damping changes is shown, and it is found that a small amount of gear backlash may give lower excitation than none at all.

Trans.I.Mar.E.(TM) Vol. 99 Papers 24-28 2986

Lloyds men - survey data for marine gears, recent service experiences, defects analysis tables. Design factor discussion, includes application and dynamic factors. Discussion of stress values from ISO, accuracy effect, inspection and testing. Reliability concepts. IACS (association of clas. socs.) is considering adoption of ISO 6336

Doctorate thesis, University of Sheffield, Department of Applied Mechanics, 1971

Describes the development of a back-to-back 3:1 gear test rig. Measurements include transmission error, static and dynamic, by specially made seismic torsional transducers; tooth strain measurements, by root strain gauges. A multiple degrees of freedom theoretical analysis (linearised) was used to predict system modes, several of which were experimentally confirmed. Specific tooth loading was relatively low. A major investigation, unpublished in detail elsewhere. The test rig and gears, and further results, were used in Kohler, Pratt 8z Thompson, 1970.

Doctorate Thesis RWTH Aachen Fakultaet fuer Maschinenwesen, 1967

Discusses contact conditions in bevel gears with and without alignment errors. The influence of crowning, and helix and pressure angle errors is investigated. An experimental rig, allowing misalignments, is described, and results for noise and errors, including spectra, given. Results analysis leads to the usual graphs giving the influence of various parameters on noise levels.

Bruel & Kjaer Technical Review, No. 31982 p 3 -

The theory of the cepstrum, which is the spectrum of a logarithmic amplitude spectrum, is presented. There are also application examples, and instrumentation techniques. Excellent introduction.

IMechE Conference. Vibrations in rotating machinery 1984 Paper C305/84

Considers gear noise and vibration spectrum attributes. Cepstrum analysis, which is described, produces a cepstrum in which zones due to excitation are confined to well defined bands, the remainder of the cepstrum can be used to estimate the structural response components by the curve fitting methods used for experimental modal analysis. Transfer function information is mostly in the low quefrency area. This concept is examined by theoretical example, and the process of reconstructing the transfer function by modelling with a finite number of poles and zeros shown. It is concluded that the method warrants further work.

Proc. 6th World congress on theory o f machines and mechanisms 1983 p 802-

Differs from the usual analysis of gear pair dynamics by admitting departure of the position of the instantaneous point of contact between the teeth from the nominal line of action. Tooth sliding velocity and friction effects are incorporated. Simplifications of the problem, to obtain general insights, are extensively discussed. System parameters are identified as the coefficients in the describing differential equation of the displacements (stiffness-like) and velocities (damping-like). The physical influences on these coefficients are described. Concludes that time-dependent damping has negligible effect, and in particular that damping at the mesh and from angular rotation at the bearings, is insufficient to account for practical values observed. Other sources - additional modes, support flexibility, must provide additional damping not considered in this analysis.

J Inst.Eng.Indza Part ND Vol. 64 Nov. 1983 p 24'-

The vibration problems of a typical turbine - gear reduction - shaft - propeller ship's installation are described. They are mostly low frequency, and propeller induced. The gearing, introduces high frequencies from meshing, and unbalance is said to cause modulation. Source identification by signature analysis is mentioned.

ASMB Trans J.Mech.Dss. Oct. 1978 VoF.100 p 713-

With application to the standard gear dynamic response single degree of freedom model, performs Fourier analysis of transmission error (called linear meshing error). Ideal profile reliefs (straight taper, long relief, and superimposed single pitch errors, are considered, with load effect included. Amplitude modulation effects in transmission error are analysed. Some idealised spectra are calculated. Concludes that excitation spectra can be minimised to control noise and dynamic loads.

Proc.Int.Fed.of Theory of Machines and Mechanisms, 4th World Congress, Newcastle, U.K.1975 p 263 -

Based on computations, mainly analogue, the characteristics of gear vibrations are classified. A six mass element analysis includes instabilities with non-linear jumps. The influential factors are - mesh stiffness characterised by the single-double contact 'jump' and influenced by corrections, expressed by the transmission error (Drehwinkeldiagramm); damping, of critical importance, but usually sufficient to give almost linear behaviour; shaft and casing deflections, which greatly affect the system. Simplified calculations are proposed, as the primary resonance usually dominates, and additional calculations for others are possible. Method refers to region of single pair contact only, that is during maximum. Individual tooth loads; includes root stress in K_v. Describes the derivation of the methods given in DIN 3990 and ISO 6336. Includes a method for resonances at N/2 and N/3. Experimental results include comparisons for ground, single adjacent pitch error, hobbed, gears. Agreement claimed as satisfactory, but noted that subcritical resonances, and high supercritical behaviour not well modelled.

VDI Berichte 332 Munich Con f. Zahnradgetriebe 1979 p 273 -

Considers the generation of gear noise, states that tooth stiffness variation is the most important effect in precision gears. Not transmission error oriented. Many analogue computer acceleration/speed results, also test results from a Munich test rig. Order analyses are given for the test results. The usual parameters are individually assessed including long teeth, high contact ratios. Waterfall type diagrams show complex excitation - response properties. Important conclusions -1 gearbox noise has two components, due to the toothing, and due to bearings, etc. 2 Fourier components of the stiffness variation cause the main dynamic response. Transmission error not mentioned. Helical with integer overlap particularly advantageous. Teeth and load only affect noise in the sub- and critical resonance area. Response is also a function of errors and damping. 3 Radiation efficiency of the gearcase also controls noise.

A.I.Aero.Astro. Propulsion conf. June 1987 paper AIAA-87-1811

Applies to large double helical marine gears the methods of Conry and Seireg, using compliance matrix with components due to tooth bending, gear rim distortion, pinion bending and torsion and Hertzian deformation. Also includes thermal, centrifugal, misalignment effects and axial sliding of the pinion. 3-D finite elements used for the compliance matrix. Models shown, complexities considered. Results given for compliance components shown on tooth flanks, and for the effect of load variation. Claims that 'the high level of detail... also provides a basis for realistic analysis of the noise characteristics of gear meshing.'

French or Canadian conference

Schemes for constant length of engagement by laterally tapering the teeth of spur and helicals. Load position and distribution analyses, involute and noninvolute gearing. Good analysis of contact lines and disposition.

Materiaux Mecanique Electricite No. 417 July-October 1986

(edited copy of database abstract) Errors of transmission and friction effects in gears of robots experimentally studied. Harmonic drives, epicyclics and a flat version of the harmonic drive (distortable gears - model type A.I.D. R100, no manufacturer given) all showed 'chaotic' (random?) fluctuations in output angular acceleration for uniform input. Worst with flat harmonic drive. Friction effects depend weakly on speed, strongly on transmitted torque. From results, system resonances appear to dominate effects. No apparent mention of transmission error of gearing itself.

SAE SP-523 Millwaukee, Sept. 1983

see Mitchell, Houser, Clark

Doctorate thesis, Cambridge University 1977

A Ford Capri gearbox was examined. The excitation components were identified, as tooth mesh frequency harmonics and shaft orders. An eight degrees of freedom dynamic model, including nonlinearities, was investigated by analogue computing. Extensive transfer function measurements are reported. Time domain averaging as well as spectra were used. Transmission error was measured and its analysis compared with the recorded excitation functions. Pitch error effects were found to be more than expected. Bearing stiffness showed great influence.

Proc.World Congress on Gearing, Paris Vol.1 p 297- June 1977

Describes the Cambridge dedicated analogue computer simulation of gear noise, which enables the effects of parameter changes to be aurally assessed. A simple layshaft gearbox is modelled, with eight degrees of freedom. The importance of comparing model and system, using measured transfer functions from, for example, order tracking (transmission error orders) is emphasised. Involute, pitch, dropped tooth and eccentricity errors can be modelled. Mesh stiffness is considered constant. Some parameters, damping is particularly mentioned, are set by adjustments to give good correspondence between the real sound of the gearbox, and the synthesised computed noise, or vibration. The realism of the synthetic sound is said to be impressive.

Bull. JSME Vol. 28 No. 240 June 1985

Considers whether jump phenomena associated with time varying stiffness and with backlash also occur with true random excitation. Calculations are given, where the system output is the mean square response. Because of the link to tooth engagement, the excitation is a narrow band random transmission error. General behaviour found is similar to that already known, with weakening spring characteristics, and large hysteresis loops. Extensive and complicated analysis leads to conclusions that jumps occur with narrow band, but not with wide band excitation. In particular, some responses show initial weakening spring characteristics, transferring to stiffening spring at high amplitudes of response (and excitation?). Idealised system equations are used, but modelled loads are realistic at 250 N/mm. No experiments.

Bull. JSME, Vol. 29, No. 2511986 p 1586-

A simple one degree of freedom model has a periodic mesh stiffness of rectangular form, and sinusoidal transmission error excitation with the addition of a random component, modelled as a random external torque. Extensive analysis leads to the mean and variance, as a function of time (rotary position), of the dynamic displacement across the gear pair. Result difficult to interpret, as relative magnitudes of transmission error and random torque not dear, but seem to show that the contribution from the random component is small.

Machine Design, Feb. 271964

Based on helicopter work, reviews mechanisms, engagement impulse, pitch line impulse, errors. Surveys gear types, tooth design. States that Novikovs much noisier, but have simpler spectra. Noise due to air ejection at sonic speed. Profile modifications, added compliance, damping. Noise transmission control. Further articles promised.

Report No. USARTL-TR-78-2A US Army research technology laboratory (A VRAD COM)

not seen - referenced in Mitchell and Houser

Report No. USARTL-TR-78-2B US Army research technology laboratory (A VRAD COM)

not seen - referenced in Mitchell and Houser

ASME 69-VIBR-161969

The Munro test rig re-used with helical gears. The existing back-to-back rig had helical gears of 1.1 overlap ratio fitted. The dynamic model allowed the shafts to tilt, the symmetrical system reduced to three degrees of freedom. Added shaft inertias gave a primary resonance of 500 Hz, transverse mode 1330 Hz and rocking mode 1540 Hz. Transmission error exhibited a best load. No tooth separation, jumps or subharmonic vibrations were found In comparison with the Munro spur gears large reductions in dynamic effect were found. Damping at resonance was about 0.06 critical, compared with 0.02 for spurs. Dynamic increment ranged from 0.l to 0.5 of applied load at primary resonance. Transverse motion was small, attributed to high bearing damping. Rocking was not found. Transmission error is reported as less than for the spurs (which is not given). Appendix analyses the effect of load shuttling in helical gears, and shows this is negligibly small in this case.

ASLE Trans, Vol. 13, No.11970 p ,p 66-77

Thesis material expounded clearly. Thin slice theory modified to include convection for bending only. Uses Weber basics, transmission error concept, differential operators for convective effects. Maths difficult, but results fairly easy to compute. Only modest experimental confirmation, though much work done. Profile modifications, for example reliefs and crowning, included. Effects of convection clearly demonstrated, and particularly significant on load distribution. Non-dimensional plots of relative effects for one gear design. Notable that transmission error for 1.1 overlap ratio shows same effects as spur gears, with optimum load and 10 - 30% fluctuation for off-design cases. Concludes that relaxation length is twice module, Weber's stiffness about 25% high.

AGMA 109.26 October,1965

Tip and root reliefs on the basis of preventing engagement shock, and tooth tolerances. Longitudinal corrections for spur and helicals on a basis of pinion deformation, temperature effects, multiple tooth contacts, helix angle. Case studies - 'field experience' Good diagrams.

Automotive Engineer June/July 1983

Concise review paper of Cambridge gear noise work. Describes transmission error, back axle tests, system responses, use of time averaging with respect to various shafts to isolate individual contributions to overall error. Discusses limitations of dynamic transmission error measurement under load. Includes mention of low torque loss of contact. Reinforces view that estimation of system parameters remains as a substantial obstacle to accurate analysis.

Marcel Dekker, 1983. Book ISBN 0-333-35045-6

Essential reading. This book considers the kinematics and geometry of gears, manufacturing methods, deflections and transmission error, all from the standpoint of gear vibration and particularly noise. There is extensive discussion of the effects of many parameters Modelling of geared systems, elastic analysis of gearbox components, and in particular-the analysis of transmission error, noise and vibration data in time and frequency domains is considered. There is material on non-parallel axis; non-involute, and epicyclic gearing, and brief consideration of failures and light loading.

Cambridge University, Engineering Laboratories 1986

Accelerometers and gratings as torsional vibration/transmission error transducers. Daly work (q.v.), gratings used as check, accelerometers proved difficult - x-axis sensitivity important, and response to gravity gives a once-per-rev component of huge proportions. Not bad technique for high frequencies.

Cambridge University, Engineering Laboratories 1986

Comments on NEL abandonment of transmission error testing, in particular. Needs of industry. Suggestions for taking over - associated costs. Review of Cambridge work, with bibliography.

IMechE 33l87 2987

Back-to-back calibration shows that the higher frequency components of transmission error, as measured by rotary encoders, are seen with greater accuracy that might be expected. Dynamic, though fairly low speed, tests are given. Results show errors of less than 0.1- 0.25 arc seconds at harmonic orders of shaft revolution greater than 5 - 7. Concludes that small encoders, nominally of about 7 arc seconds peak-to-peak accuracy, give results at tooth contact frequencies reliable to 0.03 arc seconds.

AGMA Fall Meeting, 1987, paper B7FTM5 also Gear Technology, July/Aug. 1988 p 34-

Surveys the concepts of transmission error. Observes that there is no connection between smoothness of drive and tooth stresses. Discusses the mechanism of vibration generation in gears, and gives a diagrammatic summary of the factors involved in vibration generation and transmission. This demonstrates the complexity of the process. Whilst there may be some hope of predicting, or better, measuring, system response to excitation, (damping being the most difficult of the many factors to evaluate), there is no practical expectation of any accurate calculation of transmission error from the elemental factors which cause it. These cannot, with present methods, be measured to sufficient accuracy. Transmission error can only feasibly be obtained by measurement. Current methods for transmission error measurement are reviewed; accuracy to be achieved with FIaidenhain grating units is discussed.

ProcIMechF 1988l89 (to be published)

Describes the modules needed to assemble a transmission error test system from two rotary grating head transducers. The interconnection of modules to satisfy different shaft speed and harmonic order resolutions is considered. Limitations due to the properties of the rotary grating heads, and the electronic modules, are listed, as are sampling considerations. Examples of correct interconnection for given circumstances are given.

NEL report No. 249 September 1966

Describes the large NEL grating unit, with discussion of precision and use. Notes that 43,200 line grating with phase detection has a repeatability of 1 / 100 of a line spacing and an apparent absolute accuracy of 0.5 seconds of arc. (system still working in the 1980's when at NEL)

SERC bulletin, Vol. 3 No. 5 Summer 1986

Brief summary of Cardiff conformal gear work.

Machinery, Sept. 21959 p 395 ; & March, 1960 p 646-

Describes the use of a magnetic scale for angular measurements. (Not recently seen)

Design Unit, Newcastle University, December 1987, ref. GR/C 96067

Describes in outline finite element approach to general meshing analysis and tooth stiffness of spur gears, with experimental comparisons. Work has shown that, especially for large tooth numbers, the conventional stiffness formulations have significant uncertainties.

IMechElProcAutoDiv 1954-55 No.10 p 267-

Review paper. Stages of gearbox development. Includes sliding, dog, synchro, automatic details.

AGNAP/DOC/4O J.Am.Soc.Naval Eng. Vol. 65 No. 4 Nov.1954 p 829-

Discusses tooth design, allocates pitch line impulse as the noise generating mechanism, also need for low errors. Other sources include oil pumping, air entrapment. Wide review of previous work, little consistency reported. Accuracy the only agreed factor. Pre-1940 measurements on many US navy ships showed little consistency, except that increasing structural rigidity was good, higher pitch line speed bad. Post-war experience includes major tests with a submarine section with rubber isolated gearbox, which only showed slight improvement, and little test consistency. Lists factors which probably cause noise. Presents current research program, 300 kW power circulator, and isolation systems. No mention of transmission error.

Doctorate thesis, RWTH Aachen 1969 Fakultaet fuer Maschinenwesen

Treats the detail of tooth engagement in straight spur gears with respect to load and stiffness, as reflected in movement of the engagement point on the path of contact. Calculates the corresponding impulse and its rate of application. Considers the effects of tooth errors, and reliefs. Discusses the correlation between engagement impact and the radiated noise. An ingenious experiment in which two gears collide rotationally at various speeds is described, also some measurements on running gears.

Konstruktion 371985 Issue 5 p 193-

Design advice for turbine and turbo-compressor gears, especially where the impeller is directly carried by the pinion shaft. Considers shaft bending, bearing effects, load distribution on gear teeth, whirling of shafts, excitation by unbalance, vibration modes. Torsional vibration models, tooth stiffness effects, dynamic shaft torques, multi-element fluid film bearings. Thrust cones for helical gears, power losses, use of CAD. Good references.

Proc. 6th World congress on theory o f machines and mechanisms 1983 p 859-

A nodding donkey oil pump drive gearbox is analysed experimentally to measure connecting rod forces and drive torques. A multi-component torsional system, including backlash in the gears, with either periodic or constant tooth stiffness assumptions, is studied and results for the impact loads of the various gear stages, as functions of the backlash, given. The results show that in terms of drive torque, the effects are minimal. However, in terms of gear loading, effects are more important. Maximum increments appear to be about equal to the load corresponding to the nominal torque.

NELEX 80; NEL East Kilbride, 7-9 October, 1980

Discusses frequency modulation, then applies modulation index due to cumulative pitch error to several transmission error spectra, and shows that the modulation sidebands are of the correct predicted magnitude. Derivation of the modulation index expression is not given, and HK and AMT have not been able to reproduce it. If true, the result is very important.

Bruel & Kjaer Technical Review, No.1 1988

Describes the use of measured modal shapes of plate elements and their correlation with acoustic intensity measurements. Experimental work on a simple structure shows that the two techniques complement each other.

Doctorate thesis, RWTH Aachen, February 1965

Effect of manufacture, and bearings, and loads on gear shaft alignment. Tests using eccentric bearing bushes. Theoretical and practical measurements of tooth deflections in the misaligned condition. Noise change with misalignment. Modulation effects of 'tumbling' errors. Concludes that misalignment has the effect of reducing the effective facewidth, thereby increasing tooth deflections and relative loads. There was a close dependence between tooth deflections and noise. Many references.

NEL Report No. 227 April 1966

Describes the original NEL single flank grating based machines, with discussion of accuracy and electronics for gear ratio adjustments. Contains comparisons between transmission error deduced from profile measurements, and as measured, with excellent agreement to about 0.0001 in.

VDI -Berichte No. 47,1961

Noise is due to the impulse at the engagement of teeth, resulting from the tooth-to-tooth errors. Other effects are the gear material, size, speed, etc. The change in velocity from one tooth pair to the next is considered as a geometric result of pressure angle error. Deflection due to load also contributes to relative velocity. The dynamic load is found from a nonlinear Volterra differential equation with a Poisson kernel(?), solution by approximation is very briefly outlined. Experimental results which support the calculations are given.

JSME 'Gearing; Semi-international symposium, 1967 p 189- paper 321

Considers the dynamic load resulting from tooth impact consequent upon tooth separation 'after the gear teeth have separated because of the relative change in velocity imparted to the gears by the acceleration load'. Numerical integration gives many results, and the dynamic load factors are graphed. Load and stress are considered. Little indication as to why teeth separate.

Bull. JSME Vol.19 No.133 July 1976

Develops previous work by including inertias and shafts to either side of the gear pair. Transmission error is divided into periodic and random components, Markov processes(?) used in the analysis. Overall, it is shown that the shafts have negligible influence when stiffness is less than about 1/25  of the mesh stiffness. Test gears are I25 mm dia. and 5 mm module, 10 mm facewidth. Many calculation results show the variation of dynamic load with phase of engagement over many tooth cycles, load factors up to 4 in resonance ranges are found. Gears have a periodic error of 5 microns, and a random component of 4 microns rms. approximately (if paper figures have been correctly understood)

Bull. JSME Vol. 20 No.145 July 1977

Transmission error is divided into a periodic (once per tooth) and a random error, the latter described by its autocovariance function. M.B. Priestly's theorem is used to handle the derivation of the components from the known transmission error. An example has a tooth to tooth variation of transmission error of about 20 microns, a once per rev amplitude of 1000 microns, and a rather 'ragged' general form. It is shown that consistent decomposition into periodic and random parts is possible. The gear system with adjacent coupled inertias is analysed, for the 10 mm face, 125 mm dia. gears used in all this series of papers, and results extensively graphed. Dynamic load ratios are up to 4. Analogue computer simulations agree with the calculations. Whilst the general dynamic loads are the same as for a deterministic excitation, the variance increases rapidly with the (ratio running speed/resonance speed) (?).

Bull. JSME VoI. 20 No.148 Oct.1977

Further paper in series 120 mm centres power circulator, with flexible loading shafts, used tooth strain and torque measurements, up to high loads. Multiple samples were taken. 8 and 10 mm face gears used, primary resonance at 1600 Hz, both good and poor gears tested. Transmission error was measured, after removal of periodic components, a consistent autocorrolation function was obtained (? language obscure). Often results are very repeatable with each gear revolution, but at some speeds, random behaviour appears. Analogue computer simulation was used. Results show some agreement, in particular the random appearance of dynamic loading at non-linear resonant zones. However, there is reasonable consistency between the maximum dynamic increments irrespective of individual tooth loading patterns. Results clearly show dynamic load factors exceeding 10 at resonances, but it is not clear to what the factor refers. See perhaps other papers in the series.

Doctorate thesis RWTH Aachen, 1966

Describes classical test rig work, using torsional accelerometers for transmission error measurement, the simple tooth stiffness/gear inertia vibration model, and dynamic transmission error acceleration/speed tests. Some spectra, many experimental graphs showing the influence of parameters on noise. Material effects are considered, as are pitch errors, hob wobble, profile errors, ghost effects. Reduction of noise by helical teeth, effect of tip reliefs on transmission error, practical tests, conclusions.

AGMA Washington, 1963

not seen

JSME 'Gearing - semi-international symposium -1967; .Tokyo, Paper 332 p 279-

Brief note, showing use of tangentially mounted accelerometers to measure torsional accelerations of gears in action. Used on straight bevels.

Stahl und Eisen 991979 issue 22 p 1232 .-

Detailed system analysis demonstration of the Aachen software developments. Includes classical dynamic model of gear pairs, variable stiffness mesh, branched systems, transient analysis. Very complex multi-branched arrangement, with 8 similar branches. System parameters, mode shapes and frequencies. Conclusions - the importance of parametrically excited tooth forces.

Stahl und Eisen 991979 issue 24 p 1372-

Detailed system analysis including classical dynamic model of gear pairs, variable stiffness mesh, branched systems, transient analysis. System considered in previous Stahl & Eisen ref. above. Simulation results given in detail, effect of including flexible couplings and altered tooth numbers demonstrated. Campbell plots, many time domain torque traces. Conclusions - the previously established model permitted the examination of various proposed changes to the system. Possible 'cures' were: translation of the 8 highest natural frequencies to lower values, smaller teeth to give higher excitation frequencies, combinations of these two. Flexible couplings would not be effective, but tooth number increase would.

VDI Berichte 332 Munich Conf. Zahnradgetriebe 1979 p 241-

Concentrates on the dynamic modelling of large drive systems - for example rolling mills - with particular reference to the effect of time variable tooth stiffness. Gives many computed results for transient behaviour during startup, including non-linear couplings, and backlash effects. Includes the 8 - stand wire mill results given in other similar papers. Conclusions are essential reading.

Proc. 6th World congress on theory of machines and mechanisms 1983 p 936-

Systems analysis using the RWTH methods is applied to a motor-turbocompressor and an IC engine - generator. The gear pair is modelled with ten degrees of freedom, tooth stiffness is time and phase-of-engagement variable. Backlash included. It is also found that the oil film needs to be carefully modelled. Extensive results are presented which show the influence of damping at the tooth contact on maximum torques, with backlash as the independent variable, and the time taken to run through a given speed range as a parameter. Comparisons with previously published approximate methods show significant differences

ProcIMechE International conference on gearing September 1958 p 24-

The effect of the introduction of a single wedge form error into a mass spring system is used as an analogy of tooth engagement to obtain limiting values of dynamic load increment. The fundamental resonance of a gear pair is found. A simple treatment of continuous engagement of spur gears leads to further approximate limiting values. Resonant amplitudes are limited by tooth separation. Post-resonant behaviour is considered. Identification of the most serious single error in a gear is considered.

Bull. JSME Vol. 29 No. 251 May 1986 Paper 251-33 p 1605-

Straightforward one degree of freedom analysis of relatively narrow helicals, includes parabolic stiffness functions and apparently empirical values and equations. A simple open gear pair is tested, and torsional effects measured by accelerometers at 180 degrees, and a differential amplifier. Results of torsional acceleration are exactly as expected, showing resonance of transmission error harmonics at appropriate speeds. Primary resonance as measured is compared to calculations, and satisfactory agreement obtained after some fiddling. Concludes that the theoretical stiffness values are acceptable, and that the primary resonance can be predicted from simple one degree of freedom model. Stiffness calculation is interesting, otherwise nothing new.

JSME Int. Jnl. Ser.3 Vol. 32 No. 2 June 1988 p 357-

(not seen) Published abstract - 'The main aim of gearing technology at its present state of maturity,... should be a continued effort to minimise noise and vibration. . . A simplified means of analysing low vibration parallel axis gear pairs through computer-aided-design, vibration-performance diagrams is presented.

Proc.6thJapan Nat.Cong. for App.Meclt.1956 paper IV 14

Strain gauges in the root fillet of gears on a power circulator were observed. Calibration to give tooth loads is considered. Tooth strain signals typical of contemporary work - showing evidence of impact-excited decaying vibrations.

Bull. JMS, Vol. 4 No.16, 2962 p 706-

A large high speed test rig (700 h.p., 4,500 r.p.m., 270 mm. centres) was instrumented to show tooth stresses. Detailed reconstruction of the tooth pair meshing conditions, as static transmission error, is compared with dynamic measurements. Deliberate tooth errors are introduced. Good agreement between calculation and measurement is demonstrated. Tooth separation is experienced, and shown analytically. Primary resonance effects appear.

Bull. JSME, Vol. 5 No.18,1962 p 374-

The same large high speed test rig (700 h.p., 4,500 r.p.m., 270 mm. centres) was used as in Utagawa, 1961 above. Similar work is presented, but linked to deliberate pitch error rather than tooth profile as previously. Again, detailed reconstruction of the tooth pair meshing conditions, as static transmission error, is compared with dynamic measurements. Good agreement between calculation and measurement is demonstrated. Tooth separation is experienced, and shown analytically. An appendix analyses tooth edge contact, beyond the nominal zone of involute contact.

Lund Technical University, Lund, Sweden. Trans. of machine elements division. 2981

Comprehensive, clear, analysis of the geometry of helical teeth, and modelling by finite elements. Geometry includes exact relief, root form, meshing details. Loading is Hertzian and bending. Model is of one tooth pair only, mesh is fairly coarse. Many results for bending and contact stresses. Spur gear loading diagrams unconvincing, showing dip in central loading. May be due to mesh, a four slice model is implied. Experimental verification not very close, but test rig is unimpressive. Very limited references.

Diss. Univ. Bochum 1982

not seen

ASME Lubrication conference, October, 1984 Paper No. 84-Trib-2

Bearing studies, which may have gear contact stress implications. X-ray diffraction, retained austenite/depth relationships, residual stresses, texture studies of steels, micrographs, failed and unfailed rings, V shaped orientation in spall regions. Development of texture in the advanced state of material instability prior to bearing fatigue failure. Scatter in life due to stress raiser distribution but also microcrack extension characteristics, altered by anisotropy.

Engineer 1661938 p 409-, 434- & 170;1940 p 102-

Original work on tooth stiffness

Trans. ASME J.Mecli.Des. Paper 77 DET-1051977

First of two papers giving the detailed analysis of the numerical calculation of gear vibrations in lightly loaded gears which allow loss of contact and also contact across the backlash. Steady state results from limit calculation. Experimental investigation - drawtwister gear drives with crossed helicals - is given. Comparisons between theory and measurement. Results include impact torque as a function of backlash and speed. Conclusions include that operation of lightly loaded gear drives is a series of impulses, not a smooth motion.

Trans. ASME J.Mech.Des. Paper 80-C2lDET 611980

Continues the exposition of Part 1(ASME 77-DET-105) of a drawtwister drive. A three mass system is considered. Further analysis shows non-linear effects, with sub-harmonic and harmonic resonances. Also multiple critical speeds for a single degree of freedom system are demonstrated. Computer solutions with piecewise linear integration have been made, and show good agreement with experimental results, also given. General conclusion - severe tooth loading in nominally lightly loaded gears is demonstrated, experimentally and analytically. Solutions are possible with grouped linear equations, steady state response is obtained when input and output energy balance stabilises.

Trans. ASME J.Mech.Des. Paper 80-C2lDET 551980

Simple experiments to measure Hertzian impact between crossed cylinders illustrate the subsequent analysis of crossed helical gears with backlash. Proposes that his (Wang's) more exact formulation of contact geometry is superior to that of Buckingham, but admits to results from Buckingham being similar to those from the newer theory. Wang's theory gives best values for the shape of the contact patch under impact.

ASME 84-DET 53, Design Eng.Tech.Conf. Sept. 1984

Review of dynamic factors, Sheffield Hertzian impact tests, rigid body gear trains. Analyses dynamic loading with rigid system elements, with known or estimated transmission error input. Interesting solution strategy involving curve fitting to transmission error graphs, step-by-step calculation using second order derivatives of transmission error. Mathematical derivation, substantial numerical examples, leading to dynamic loads and factors. Good references. Mention of AGMA gear rating committee.

DSIR Sponsored research, Report No. 3 1949

Definitive work on gear tooth stiffness. See also Weber et al. below, and Walker, H. above.

Schriftenreihe Antriebstechnik, Heft 11, Braunschweig, Vieweg, 1953

Includes detailed analysis of tooth and mesh stiffness.

VDI Berichte Nr. 3541979

Claims to have found a method for replacing gear with disc testing. Describes disc testing, residual stresses, through and case hardened materials, pitting distribution, tooth residual stresses by X-ray technique, correlation of location of tooth failure with running parameters - slide/sweep etc. Hertz stress distribution, subsurface stress variation with slip, evaluation of comparative stresses, computer model.

VDI Berichte Nr. 488 2983

Size effect, and disc - gear data relationships. Grinding method - working orientation, pitting distribution on flanks, results comparison, definition of size effect - statistically distributed cracks, quality of defects and inclusions, diameter/surface stress relations, effect of inhomogenious stress distributions, variable residual stress effects with size. Derating by radius of curvature graph. High quality of experimental results.

VDI Berichte 332 Munich Conf. Zaknradgetriebe 1979 p 13-

Reviews many aspects, induding dynamics. Several dynamic models of bevel, spur, and complex gear trains are presented, mode shapes, natural frequency calculations, modelling considered. Dynamic effects are calculated, and compared with measurements. Review with good references.

VDI Berichte 332 Munich Con f. Zahnradgetriebe 1979 p 263 -

Discusses the requirements in industrial gear noise applications. Gives review of idealised behaviour, extracts from VDI standards, calculation flow diagrams; use of coherence method in FFT to eliminate background noise from measurements. Gives selection matrices for noise reduction parameters. Considers load distribution as a function of tooth design, stiffness variation along path of contact. Describes control of radiation from the gearcase, finite element methods included.

ProcIMechE Vol.184 Pt 3 O, Gearing in 1970 p 131-

Good short bibliography. All gears have errors. Fourier analysis of components of gear errors and superposition is assumed possible. Computed spectral components are found for eccentricity, pitch, dropped tooth, involute, tooth bending, helix angle errors. Spur and helical gears are compared. Experimental transmission error from Goulder equipment is considered. Gaussian random errors are generated and analysed, and real errors shown to have the same characteristics. First appearance of the idea that pitch errors by themselves cannot cause tooth contact frequency harmonic components in the transmission error.

IMecltE Vibrations in rotating systems, Feb. 2972

Effects on noise/vibration frequencies of eccentricity, pitch errors (case studies) are discussed with experimental details. Observations are based on Gaussian distribution of errors, and some computations are given. Early paper with much to stimulate thought.

IMechE Con f.Pubs 1979 -10. C117/79

Reviews the main stream of gear noise research, from Opitz/Harris on. Abstract here impractical, but concentrates on German results, as confirmed elsewhere. Important reference lists.

ProcIMechE Vol. 200 No. C3 - discussion

The main conclusion of the K 8z R paper, that pitch errors themselves in real gears can cause transmission error spectral contributions at tooth contact frequencies, is disputed.

Considers a tooth as a cantilever plate of finite lateral width. Shows edge effects, and experimental confirmation of the simplified theory presented. Inclined load lines in helical gearing well modelled. (Reviewer suggests the techniques may be applicable to tooth stiffness estimation.)

Doctorate thesis IMM, RWTH Aachen, 1981

Applied particularly to gearcases, the finite element method, including automatic mesh generation, is used to extensively analyse a single-reduction gearcase. The deflection considered is the tilt of the bearing axis. The effect of stiffening ribs and their location is found, extensive tables rank order and give the relative performance of various rib arrangements. The effect of misalignment on plain hydrodynamic bearing performance is discussed, and suggestions for optimising gearcase design given. Refined treatment.

Strain gauge readings Vol V, No.4 Oct/Nov 1962, also Arizona State University SGR (V-3) ? Chapter 17 p 633

Work prompted by the occurrence of beats around 4000 Hz. frequency components in a turbine drive. Noise test signals were interpreted by filtering and Lissajous figures, and by demodulation and analysis of the result. No correlation with simple tooth contact frequency or blade passage frequencies. Assumes noise caused by alternating shear forces at perfect gear contact, worsened by pitch diameter errors (this seems to be untrue as described) It is supposed that the pitch diameter varies with rotation and this modulates the mesh frequency. Frequency doubling can occur, and unsymmetrical error can cause all shaft speed multiples to modulate. Hence frequency components at all sum and difference frequencies between mesh, and shaft, and shaft multiples, are caused. Effective complete suppression of the carrier can occur if sufficient energy is redistributed into the sidebands.

IMechE Aerospace Industries Division Seminar Dec.1986

Seven papers, safety standards for helicopter transmissions- statistics, recommendations. Efficiency of advanced propeller transmissions - propfans, high bypass ratio fans, Tyne as basis, epicyclic, star and multipath transmissions, contra-rotating schemes, efficiency maps. Spacecraft transmissions - relative performance, nitriding, titanium- nitride.... Bearing material advances - new steels, special production techniques, SAE 52100; M50, 5156. Better helicopter gearboxes - new designs, lubricants, monitoring, rig details. High conformity gearing - load contours, EHD lubrication conditions, design recommendations. Failure recognition - error analysis, early signs of failure, check lists, particle detection, surface finish related failures, micropitting.

Int.Symp.Design and Synthesis, Tokyo July 1984 p 253-

Errors due to tooth error, eccentricity and slack mounting are assumed to have appropriate distribution, for example Rayleigh, and the statistics of the total calculated. Effect in a compound train is studied. Also backlash is developed in a similar way from the contributory factors. Based on tests on a large number of samples, excellent agreement between theory and practice is found.

ASME Trans, J.Mech.Trans.Auto. in Des. Vol.109 June 1987 p 189-

Impact across the backlash of gears is analysed taking into account energies of Hertzian compression and tooth bending and compression energies. Damping and coulomb friction are modelled, multiple tooth contacts allowed, sudden and sinusoidal excitation considered. Computer simulation claims analytical derivation without simplifications. Energies when know convert to tooth loads or stresses. Essential result is that bending energy predominates, in worst case other energy components only reach 30% of it.

YARD Memorandum No. 4524 M110 January 1985 - a commissioned study for MOD, Bath.

A careful analysis of the W.D. Mark papers. Studies the implications of the theory, in particular the absence of tooth contact frequency components resulting from pitch errors. Discusses profile correction design, and use of integral contact ratios. Observes that Randall R.B. also misinterprets the Mark mesh transfer function. Further gives an analysis of the linear and non-linear models of tooth contact under load, and suggests the possibility of combining them. Comments on lack of experimental verification. Excellent references.

ProcIMechE Vol. 169 p 611-1955

Torsional vibration. Impedance network equivalent system. Branching. Excitation by gearing errors (brief). Damping. Examples with propeller excitation, eccentric gear excitation. Choice of Q. Torsiographs. Examples from real practice, including gear excitation, (PAMETRADA). Extensive discussion, contributions from Tuplin, Archer, Downs, Duncan, King. Long reply.

Trans.I.Mar.E.CTM) Vol. 99 Papers 14-181986

Mostly epicyclics, service performance assessment, MTBF, spares commitments, failure histories, quality assurance methods, performance checks in the factory, back to back tests. Gear accuracy, meshing, CMM, vibration and dynamic loads, system considerations, research and development and Allen test rigs. Development of Stoeckicht designs.

Proc. 7th World Congress, Theory of machines and mechanisms, Seville, Sept. 1987 p 1347 -

Fatigue tests on surface hardened gears, taken to failure, are reported. Of interest is that tooth profiles, tooth dynamic root stresses, and sound and vibration were all recorded as the tests progressed The associated changes which occur as the tooth profiles change are shown. It is found that for surface failures, dynamic behaviour changes 'remarkably', but for root bending fracture changes are 'not remarkable'. Of interest in condition monitoring.

ASME Trans.J.Meck.Des. April 1979 Vol.102l269 (77-DET 51)

Open loop rig tests, and transmission error tests with an ingenious epicyclic single flank tester, which uses three master gears and one test gear, are described. Test gears with five forms of profile error, all wavy. Tooth fillets were strain gauged, and noise spectra, both narrow band and 1 /3rd octave, taken. Results relate the error types with noise spectra, tooth strain, and speed at medium and low loads. At low loads, results are as expected, for errors below 10 microns, the error form is unimportant. Overall, slight convex error is best. As so often the case, results apply to test set-up only, and are not generalised.

Diss. RWTTi Aachen 1972

not seen

VDIZ Vol. 99 No. 6 Feb. 1957

A classic expression of the single error insertion analysis of the one degree of freedom gear pair model. Negative and positive sine ramp and half sine error insertions, called contact disturbances, analysed. Error insertion causes free vibration, and loss of contact and impacts. Results analysed in terms of maximum tooth spring deflection, and thereby load. Also considers continuous errors, as leading to a steady state forced vibration, and handles by Fourier decomposition. Ends with a graph comparing the results with methods proposed by Buckingham, Niemann, Tuplin. A wide range of predictions is obtained for the sample data. Concludes that current methods (1957) are very approximate, but no better viewpoint is to hand.

Proc. 3rd. International modal analysis conference, Schenectady, USA 1985 p 668-

Uses modified Niemann formula, reports that great differences are found in practice. Many results given. Resonances often occur within the running range. Adding in the simple radiation properties of sources and the dynamic magnification factor, gives a further simple addition to the formula, but this requires the damping ratio. Noise reduction by the use of damping rings (the rings are not described) is shown to be effective for initially low damping ratios (say 0.01 or less) but not useful for higher values. Some experimental confirmation of the modified formula, and the prediction of the effect of damping rings is claimed.

M. Phil thesis, University o f Sheffield, 1986

Spur gear (thin) transmission error is analysed, incorporating all envisaged effects. These are pitch, profile, eccentricity errors, tip/root reliefs, noninvolute contact beyond nominal contact path, tooth tip chamfering/rounding. Loading is included, but using only typical approximate (phase of contact dependent) stiffness data taken from Walker. The effects on the transmission error spectrum, up to high harmonic orders are demonstrated. Some aspects of W.D. Mark's theory are considered, and linear/nonlinear regions of effects considered. In particular, summation of the individual error contributions to transmission error, to give total transmission error is show to be true only for special cases. A PRlME FORTRAN program to generate transmission error and transmission error spectrum from arbitrary, defined errors and gear data was completed.

Int.Symp.Design and Synthesis, Tokyo July 1984 p 265-

Deals with the analysis of transmission error in a gear train of a 600 mm hobbing machine. The FFT is described. By expressing the 512 point DFT spectral components of the measured overall transmission error as correctly phased 2D vectors (as Hayashi), insight is gained into the relative importance of the components.