On noise generation and dynamictransmission error of gears

Mats Henriksson

Stockholm 2009

Royal Institute of TechnologySchool of Engineering Sciences

Department of Aeronautical and Vehicle EngineeringThe Marcus Wallenberg Laboratory for Sound and Vibration Research

TRITA-AVE 2009:89ISSN 1651-7660ISBN 978-91-7415-537-2

c©Mats Henriksson, November 2009

Printed in Sweden,Universitetsservice US-AB

PrefaceThe work in this thesis has been carried out at the department of Aeronautical andVehicle Engineering at the Royal institute of Technology (KTH). The funding fromScania CV AB and the Swedish Agency for Innovation systems - VINNOVA is gratefullyacknowledged. I would also like to thank my supervisor Leping Feng at KTH for helpand guidance in my research and my present manager Anders Absér and my formermanagers Mikael Pärssinen and Gunnar Strandell for allowing me to perform this work.I am grateful for all the help from my colleagues at both Scania and KTH. I would alsolike to thank my wife Monica and our two children, Alva and Gustav, as well as the restof my family for their support during this work.

Mats Henriksson

Stockholm, November 2009

AbstractNoise from heavy trucks is an important environmental issue. Several sources contributeto the total noise level of a vehicle, such as the engine, gearbox, tires, etc. The tonalnoise from the gearbox can be very disturbing for the driver, even if the noise level fromthe gearbox is lower than the total noise level. The human ear has a remarkable wayof detecting pure tones of which the noise from loaded gears consists of. To be allowedto sell a heavy truck within the European Union, the so called pass-by noise test mustbe completed successfully. The maximum noise level permitted is 80dB(A) and undercertain conditions, the gearbox can be an important contributor to the total noise level.Gear noise is therefore an important issue for the automotive industry.

In this thesis gear noise and dynamic transmission error is investigated. Traditionally,transmission error (TE) is considered to be the main excitation mechanism of gear noise.The definition of TE is ”the difference between the actual position of the output gearand the position it would occupy if the gear drive were perfect”. Measurements ofdynamic transmission error (DTE) and noise have been performed on a gearbox. Themeasurement object was a commercial truck gearbox powered by an electrical motor.The torque used was in the normal operating range of the gearbox and the correlationbetween gear noise and DTE, when the torque is changed, is investigated. The resultdiffers for different gear pairs and for the first gear stage, located close to the housing,the correlation is high for most speeds. The measured DTE and noise show a poorcorrelation with calculated transmission error. A minimisation of TE therefore does notnecessarily mean a minimisation of gear noise.

A transfer function can be employed to calculate the relationship between DTE andnoise. The general trend of the gear noise is an increase of 6dB per doubling of therotational speed together with fluctuations around the mean due to resonances of thesystem. The magnitude of the transfer function can be estimated using the amplitudesof the gear mesh orders and harmonics.

Two gear pairs with similar macro geometry but different profile modifications areinvestigated. Although the gear pairs have similar transmission error, the noise leveldisplay a significantly different trend, further strengthening the position that transmis-sion error is not the single most important gear noise excitation mechanism. Furtheranalysis concludes that shuttling forces and friction forces can be more important thanwhat is often suggested. A dynamic model including transmission error and shuttlingforces is used to investigate the two gear pairs. The bearing forces show that for somefrequency regions shuttling forces can be of the same order of magnitude as the forcescaused by transmission error.

This work highlights the importance of considering other excitations of gear noisebesides transmission error when designing quiet gears. The influence of transmissionerror can not be determined by investigating the gears only. A deeper knowledge of thegear system is needed in order to minimise gear noise for a specific gear design.

Doctoral Thesis

This doctoral thesis consists of this summary and five appended papers listed below andreferred to as Paper A to Paper E.

Paper AM. Henriksson: “Analysis of gear noise and dynamic transmission error measure-ments”. Presented at ASME IMECE04, paper 61077

Paper BL. Feng, M. Henriksson, M. Pärssinen: “On model of gearbox noise and dynamictransmission error”. Presented at ICSV12, paper 485

Paper CM. Henriksson, L. Feng: “Analysis of gear noise and dynamic transmission error usinga recursive Kalman filter algorithm”. Submitted to International Journal of Acousticsand Vibration

Paper DM. Henriksson, Y. Pang, L. Feng: “Transmission error as gear noise excitation”.Submitted to International Journal of Acoustics and Vibration

Paper EM. Henriksson, L. Feng: “A dynamic gear model including transmission error andshuttling forces”. Submitted to Journal of Mechanical Design

Contribution from the author of this thesis

Paper APerformed the measurement, wrote the paper.

Paper BPerformed measurement, Performed analysis together with L. Feng

Paper CPerformed analysis, wrote the paper

Paper DPerformed analysis, wrote the paper

Paper EDeveloped model, wrote the paper

Material from this thesis has also been presented at the following two conferences:

Henriksson, M. and Pang, Y., 2009, Transmission Error As Gear Noise ExcitationIDETC/CIE 2009, paper DETC2009-86695.Henriksson, M. and M. Pärssinen,Comparison Of Gear Noise And Dynamic Trans-mission Error Measurements, ICSV10,2003, paper 61077

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Gear noise 22.1 Transmission error . . . . . . . . . . . . . . . . . . . . . . 32.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Shuttling forces . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Dynamic gear modelling 5

4 Measurement of gear noise and DTE 6

5 Summary of main results 85.1 Paper A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.2 Paper B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 Paper C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 Paper D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.5 Paper E . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6 Conclusions and future work 13

References 15

1 Introduction

1.1 Background

Noise from heavy trucks is an important environmental issue. Several sources contributeto the total noise level, such as the engine, gearbox, tires, etc. In this work, gearboxnoise is investigated. The tonal noise from the gearbox can be very disturbing for thedriver, even if the noise level from the gearbox is lower than the total noise level. Thehuman ear has a remarkable way of detecting pure tones of which the noise from loadedgears consists of [1]. To be allowed to sell a heavy truck within the European Union,the so called pass-by noise test must be completed successfully. The maximum noiselevel permitted is 80 dB(A) [2, 3] and under certain conditions, the gearbox can be animportant contributor to the total noise level. Gear noise is therefore an important issuefor the automotive industry.

The decreasing time for product development also creates a need for modelling andoptimising the dynamic and acoustic properties of gear systems. The ability to accu-rately predict the dynamic and acoustic behaviour before a prototype is built can saveboth time and money. Also, understanding how a system behaves dynamically is crucialwhen trying to find a solution to noise and vibration problems.

Transmission error (TE) is considered to be the main excitation mechanism of gearnoise. The definition of TE is ”the difference between the actual position of the outputgear and the position it would occupy if the gear drive were perfect” [4]. In DudleysGear Handbook [5], prof. D. Houser state that ”Transmission error is the single mostimportant factor in the generation of gear noise.”, other authors also asserts the impor-tance of transmission error as the dominant gear noise excitation [6, 7, 8]. Additionalsources beside TE can also contribute to gear noise. Research show that effects such asfriction and shuttling forces can be important when analysing the dynamic behaviour ofgears [9, 10, 11]. This will be discussed more in the next section.

This thesis analyses the properties of the dynamic transmission error and the pos-sibilities of using it to predict gear noise levels and the importance of other gear noisesources besides transmission error.

1.2 Motivation

A review of the available literature revealed a need for experimental study of dynamictransmission error and noise, especially for two stage gearboxes. For example, Davieset al. [12] writes in a review on gear dynamic behavior that ”With the increasing powerof predictive methods there is a substantial lack of good correlative experimental data”.Measurement of dynamic transmission error and noise was therefore the first part of thiswork. The result shows large deviations from the predicted noise behaviour of the gearsystem based on a minimisation of transmission error. Understanding the reason for thedeviation therefore became a natural continuation of the thesis.

1

Figure 1: Truck gearbox similar to the one used in measurements

2 Gear noise

Gear noise is normally divided into two parts, gear whine and gear rattle [13]. Gearwhine is described as gear mesh frequency and their harmonics and is the main focusof this thesis. Lightly loaded gears have been shown to exhibit non-linear behaviourwhen the surfaces of the gear teeth loose contact and later collide. Rattle can be excitedby external speed fluctuations, for example in the timing gears of internal combustionengines where the irregular speed of the crank shaft can excite rattling of the gears. Theinternal excitation of the gears can also be a source of rattling and is often modelledas a transmission error excitation [14, 15]. Gear whine on the other hand originatesfrom loaded gears. Gear whine can also display non-linear behaviour due to for examplepartial contact loss, which is a result of the increased mesh stiffness for increased loads.Gear noise is often described as a source-path-receiver problem where the excitationoccurs in the gear mesh and is then transmitted via the gear body, shafts and bearingsto the gearbox housing where it is radiated as noise. In figure. 1, a two stage truckgearbox, similar to the gearbox used in the thesis, can be seen.

Transmission error, friction and shuttling forces will be discussed more in the fol-lowing sections, although other excitations also exist. Air and lubrication entrapment

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occurs for high speed gears when air or lubricants get trapped between the gear surfacesand is squeezed out. Windage noise can also have an influence for high speed gears.These excitations are only important for high speed gears and are not considered tohave an impact in the tested and modelled gears of this thesis.

2.1 Transmission error

Theoretically, for two gears with perfect involutes and an infinite stiffness, the rotation ofthe output gear would be a function of the input rotation and the gear ratio. A constantrotation of the input shaft would therefore result in a constant rotation of the outputshaft. Due to both intended shape modifications and unintended modifications, such asmanufacturing errors, gears will normally not have a perfect involute shape. Also, dueto a finite gear mesh stiffness, there will be a motion error of the output gear relative tothe input gear. This error in motion is the transmission error and is calculated accordingto:

TE(t) = Rb,pΘp(t) + Rb,gΘg(t) (1)

Where Rb,p and Rb,g are the base radius and Θp(t) and Θg(t) are the rotation ofpinion(p) and gear(g). In eq.1, transmission error is described as a length along the lineof action. The line of action can be seen in fig. 2. It can also be described as an angulardisplacement of the pinion or gear and the TE is then divided by the base radius of thepinion or gear. Measuring TE in a single flank measurement machine is often employedby gear manufacturers to verify the shape of the gear. The measurement gives noinformation of the gear strength, only the shape deviations. The gears are mountedwith adequate spacing, representing the working conditions of the gear pair with onlyone flank of the gears are in contact.

The transmission error and mesh stiffness variation is often considered to be theprimary excitation of gear noise and a minimisation of the transmission error is believedto minimise noise. Several comparisons show the correlation between TE and noise, forone speed and torque the correlation can be high, although changing the speed or torquecan reduce the correlation, making it difficult to see a relationship between TE and noise.Akerblom [16] compares TE and measured noise for several gear pairs with differentfinishing methods and profile modifications. For one torque there seem to be a correlationbetween noise and TE but for another torque the correlation is lacking. Houser et. al.also compared measured noise levels to calculated TE levels and a correlation was foundin some cases but not in others [17]. N. Yildiri et. al. [18] successfully reduces noise froma helicopter transmission by changing the profile modifications and thereby reducing TEfor high contact ratio spur gears. Åkerblom [16] presents an improved gear design withdecreased transmission error which also demonstrate a decreased noise level, althoughthe width of the new gear pair is increased and the module decreased thus increasingthe total contact ratio with 36%. An increase in contact ratio has for a long time beenknown as an important method of reducing gear noise [19].

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−30 −20 −10 0 10 20 3025

30

35

40

45

50

55

Base circel

Line of action

Pitch circle

Pitch point

Figure 2: Schematic view of a gear pair showing the base and pitch circles and the lineof action

2.2 Friction

In the gear contact, the sliding friction produces friction forces that acts perpendicularto the line of action and is characterised by a mixed sliding and rolling motion. At thepitch point, there is purely rolling motion although outside the pitch point, which canbe seen in fig 2, there is always sliding. As the gear contact passes the pitch point, thedirection of siding changes, thus creating time-varying forces.

Friction has been reported to have an effect on both the DTE and noise of a gearpair [20]. Velex and Cahouet [21] compares a dynamic model which includes slidingfriction with measurement result and show an important contribution from friction forthe dynamic bearing forces for speeds below 200rad/s (1900rpm) for the investigated gearpair. As the speed increases further, the influence of friction decreases rapidly. Rebbechi,Oswald and Townsend [22] performs measurements of dynamic tooth friction in order tocalculate a friction coefficient. The friction coefficient reported is approximately 0.04 to0.06 with increasing values at lower speeds. Many dynamic models has been presentedin order to calculate the influence of friction. [20, 23, 24]. S. He and R. Singh [25] reportsthat friction has marginal effects on the dynamic transmission error for helical gears, ascompared with spur gears where the effect is much larger.

2.3 Shuttling forces

Shuttling forces were examined by Borner and Houser [11], and is the result of themotion of the contact lines across the gear surface as the gears mesh. As the contactlines move across the teeth, the resulting force is not centred on the tooth but movesslightly back and forth along the width of the gear. This results in time varying bearing

4

TEs

kmcm

Rbp

Rbg

θp

θg

Ip

Ig

Figure 3: Singel degree of freedom gear model

forces which can be calculated in a quasi-static analysis. There has been little researchcovering the importance of the shuttling force although it is naturally included in modelsthat calculate the dynamic contact conditions and do not rely on an excitation such astransmission error [21, 26]. Velex and Cahouet [21] notices differences in the bearingreaction forces of the pinion and suggest the difference is due to the shuttling force.

3 Dynamic gear modelling

Calculating static transmission error is a standard procedure, commercial gear calcula-tion softwares are available to predict TE. As the rotational speed of the gears increase,TE changes from being a purely geometry and load dependant property to also includingthe dynamics of the system. As the mesh frequency and the harmonics coincide withdifferent eigenfrequencies of the system, the resulting DTE is a function the dynamicproperties of the system. Since the DTE describes the complete system , including gears,shafts, bearings and housing, this makes the DTE more complex to analyse. DTE isoften predicted by dynamic models of gear systems. Static TE is commonly used asexcitation in order to calculate bearing forces, enabling the prediction of vibration andsound radiation from the housing. To make a dynamic model of a gear system it is im-portant to have a correct model of the gear-mesh interaction. Models vary from simpleone-degree-of-freedom lumped models [27, 24, 28], which can be seen in fig. 3, to largeFE-models [29, 9, 14, 30]. The single-degree of-freedom model seen in fig. 3 describes

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the system as a purely rotational model without shafts. The system consists of twoinertias (I) connected with the mesh stiffness (km) and mesh damping (cm) at the baseradius (Rb) for pinion(p) and gear(g). The transmission error is treated as an angulardisplacement in the gear mesh. Despite the large range of complexity between publishedmodels, the aim is to obtain a model that describes the gear system accurately enough,depending on the properties of interest. Many models do not take the effect of axialforces, due to the helix angle or bending vibrations of the shafts, into account whenmodelling the gears. Since a gear system is a very complex system, many assumptionsare made to simplify the system.

Many dynamic gear models use TE as excitation and if the purpose of the model isto simulate rattle, no load TE can be used, since rattle in most cases occurs in lightlyloaded gears [14]. It has been shown that gear with low contact rations(typically spurgears), loss of contact can occur for higher torques at the gear mesh resonance. Insteadof using TE, the variation in mesh stiffness is sometimes treated as an excitation [31], butin most cases the variations in mesh stiffness is included in the TE. Gear whine modelsoften use measured or calculated quasi-static TE under load as the excitation [32].

Velex and Ajmi [33] use a theoretical approach to determine the possibility of usingTE as excitation when modelling gears. Their conclusion is that the major excitation isthe difference between the no load transmission error and the quasi-static transmissionerror under load. Previous research has often used quasi-static TE under load alone asthe excitation. Limitations of the use of TE as the excitation are also discussed, such asgear body deflections when the contact conditions between the teeth no longer can beconsidered to be quasi-static.

In paper E, a dynamic model of a single gear pair is presented. The model is atorsional model with flexible bearings including transmission error and shuttling forces.The shaft only has a torsional stiffness and is considered to have an infinite bendingstiffness. The rotation is modelled as a reference frame with superimposed flexibledegrees of freedom. Gear mesh forces and bearing forces are examined for two differentgear designs.

4 Measurement of gear noise and DTE

A large number of papers concerning gear dynamics have focused on dynamic models.Little work concerning the comparison between simulation and measurements has beenpublished. Research on multi-stage gearboxes are especially hard to find. Special equip-ment has to be used when TE is measured on a complete gearbox. Instead of just givinginformation of the shape deviations of the gear flanks, the measurement describes thecomplete gear system. Factors such as shape deviations, misalignment of shafts, eccen-tricities and clearances in bearings all have influence on the results. Deflections also haveinfluence if the system is measured under load. Since many gearboxes consist of morethan one gear pair, the measurements includes transmission errors from several gearpairs. A complete gearbox is a very complex system and the result can be very different

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when disassembling and reassembling the gearbox. Åkerblom and Pärssinen [16] reportsup to 7dB difference in overall noise level for certain speeds. Åkerblom also reportslarge differences in noise levels depending on bearing pre-load. In paper D, the noiselevel from two different gear pairs were tested. The test object was a two-stage gearboxwhere the investigated gears were located as the first gear stage. The noise level fromthe second gear stage, which was the same during all the test, was examined in order toinvestigate the dynamic properties when the first gear stage has been changed. The twodifferent measurements were surprisingly similar, indicating that the dynamic propertiesof the system remained the same when the first gear stage was changed.

The definition of TE and DTE also becomes more complicated for a complete gearsystem. Often encoders are used and the location of the encoders affects the result.Similar difference can be seen when comparing single-degree-of-freedom (SDOF) modelsand multi-degree-of-freedom (MDOF) models. In the case of a MDOF model, the resultsdepend on which nodes that are used to calculate the TE. In most cases, the measurementdetermine the definition of TE since the location of the encoders can be difficult tochange.

Measurements of dynamic transmission error have been performed in order to in-vestigate non-linear behaviour of lightly loaded gears. The measurement setup is oftendesigned in order to model the gear pair as a single degree of freedom system[14, 15]. Theexcitation used is transmission error and a good correlation with measurement result hasbeen obtained.

For paper A to C, DTE and noise were measured on a truck gearbox in a noise testrig. The gearbox is powered by an electrical motor, capable of speeds up to 2000 rpm.Another electrical motor is used for torque reaction. Each motor is located in a sepa-rate room next to the measurement room, in order to minimise the noise contribution.Walls and ceiling are covered with acoustic absorbents to reduce the acoustic reflections.Rotational encoders are located on the input shaft, side shaft and output shaft. Theencoder location can be seen in figure 4. This enables calculation of the DTE from thefirst and second gear stage separately, as well as for the complete gear system. Fourmicrophones and four accelerometers were used. Measurements were made at differenttorques and speeds, typical of the operating conditions of the gearbox.

The measurement object was a standard gearbox from Scania ,similar to the gearboxdisplayed in fig. 1. The gearbox has a three-speed main section, a range and a splitterfunction, resulting in 12 forward speeds. The range gear is a planetary gear with twopossible gear ratios. In all measurements, the planetary gear was set in high range,which means that the complete planetary gear rotates as one unit. Therefore, theanalysis does not include the planetary gear. Instead the properties of the helical gearsin the main section of the gearbox are investigated. All gears in the gearbox are rotatingand contribute to the total noise level. The noise from the unloaded gears is of broadband character and no gear mesh orders for the unloaded gears can be detected.

The main gearbox has three gears which can be used in high or low split, resulting insix gears. The torque is transmitted from the input shaft to the side shaft via the high or

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Rangeplanetary gear

2:nd gear mesh1:st gear mesh

3x encoder

Figure 4: Schematic view of the gearbox

low split gear pair. From the side shaft to the output shaft, the gear pair correspondingto the first, second or third gear is selected and locked to the output shaft. In fig. 4a sketch of the working principles of the gearbox can be seen with the location of theencoders included.

For paper D the measurement facility is the same and a similar measurement objectis used, although twelve microphones were used instead of the previous four and nomeasurement was made of the rotation of the shafts. Two different gear designs wereused in the measurements, details can be found in the paper.

5 Summary of main results

5.1 Paper A

Correlation between noise and DTE is examined. The method uses fixed speed andcalculates the correlation as the torque is increased. Eight different speeds, typicalof the operating conditions of the gearbox, were used during the measurements. Ateach speed, the torque was varied between 600 to 2000 Nm in 200 Nm increments.The correlation between noise and DTE varies for different gear pairs. The highestcorrelation is found for the low split gear pair, which is located closest to the gearboxhousing. The correlation also seems to increase as the applied torque increases. Therecan be a number of reasons for a reduction in the correlation between noise and DTE.Some resonances can be found where the correlation is low, but not all resonances resultin low correlation.

Comparing the measured DTE and calculated static TE for the gear teeth shows anincrease in DTE and noise but a decrease in calculated TE as the torque is increased,

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Figure 5: Calculated static p-p TE and measured dynamic p-p TE for three differentspeeds for the first gear stage using the fifth gear in low split

Figure 6: Calculated static TE and noise level (lp) for three different speeds for the fifthgear in low split

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Figure 7: Averaged DTE of one tooth 1: 600 Nm; 2: 800; 3: 1000; 4: 1200; 5: 1400; 5:1600; 7: 1800; 8: 2000Nm

as can be seen in fig. 5. If the system is linear and the static TE is considered to bethe excitation, the DTE should follow the same pattern. Non-linear effects and othersources can be an explanation to the lack of correlation between the calculated staticTE and measured DTE. Similar conclusions are made when comparing the noise andthe calculated TE, which can be seen in fig. 6. The correlation between noise and DTEis much better. As the torque is increased the same trend can be seen in both DTE andnoise.

5.2 Paper B

Dynamic transmission error has a close relation with the excitation of a gear system.Below a certain limit, it is independent of the speed of rotation but has a certain de-pendence on the torque applied, due to the limited stiffness of teeth. An average of theDTE for a gear mesh can be seen in fig. 7. The radiated sound from a gear system isgenerally increased with the increase of the rotating speed at the rate of 6 dB per dou-bled rotating speed, with the fluctuations due to the shifted transfer functions, as can beseen in fig. 8. The sound pressure is very dependent on the structure and how the gearpairs are mounted. Identical gear pairs may produce totally different sound power whenmounted in different systems or stages. It is possible to estimate the transfer functionbetween DTE and the sound pressure by using a speed sweep measurement.

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Figure 8: Sound pressure levels at the meshing frequency of the first gear stage Thecenter dotted line shows 6 dB per doubled speed.

5.3 Paper C

Measurements of DTE and gear noise have been used to assess the possibilities of usinga linear model to describe the system. An order tracking method using a Kalman filterwas employed to calculate the amplitudes of the gear mesh orders. Close orders andoverlapping sidebands limited the possibility of using a standard FFT order trackingmethod.

The magnitude of the transfer function between DTE and noise is calculated usingthe result from the Kalman filter. There seems to be a linear relationship between noiseand DTE for the higher torques measured, which is consistent with the result in paperA and paper B.

Many dynamic models of gear systems use a linear approach to calculate dynamicforces and the radiated noise. Assuming a dynamic model that is linear with rotatingspeed, the non-linear effects in the measurement will be the error of the model. For eachfrequency, the average of the normalised gear mesh order and harmonics are calculated.No information can be given the first part of the gear mesh order, since an overlappingorder is required to assess the difference between the orders. Therefore no informationcan be given below 1200 rpm for the gear mesh order. This results in an averagetransfer function for all the orders. To calculate the gear mesh order, the estimatedtransfer function is multiplied with a suitable constant, the same applies to the rest ofthe harmonics.

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0 20 40 60 80 100−1

−0,8

−0,6

−0,4

−0,2

0

0,2

0,4

0,6

0,8

1

Stiffness k(kN/m)

Cor

rela

tion

coef

ficie

ntWithout friction force (a=0)

0 20 40 60 80 100−0.6

−0.4

−0.2

0

0.2

0.4

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Stiffness k(kN/m)

Cor

rela

tion

coef

ficie

nt

With friction force (a=1)

Gear mesh order AFirst harmonic AGear mesh order BFirst harmonic B

Figure 9: Correlation coefficient between suggested excitation and measured noise levelfor different values of k

5.4 Paper D

Two gear pairs with different micro geometry but similar macro geometry has beentested using a complete truck gear box in a noise test rig. The difference in noise levelfor different torques can not be explained using transmission error alone as excitation.The measurement result indicate that for this gearbox, shuttling forces and friction forcescan be very important when assessing the noise properties of the tested gears. Minimis-ing transmission error does not necessarily minimise noise even though the transmissionerror multiplied with the mesh stiffness seems to be the dominating excitation force.The reason is that multiplying TE with the mesh stiffness significantly overestimatesthe importance of transmission error for a gear pair not operating above the critical gearmesh resonance. In a complex gear system, the deflection does not necessarily take placeat the gear mesh. The correlation coefficient is calculated between the measured noiselevel and the suggested excitation for different values of stiffness k, which is multipliedwith the transmission error. The correlation coefficient or gear mesh orders and firstharmonics for both gear pairs can be seen in figure 9, with and without the contributionfrom friction. Multiplying the transmission error with a much weaker stiffness signifi-cantly increases the correlation with the measured result, something which reduces theTE excitation to the same level as shuttling forces and friction forces.

5.5 Paper E

A dynamic model has been set up using transmission error and shuttling force as ex-citation. In the sub-harmonic region, the system behaves quasi-statically. If all forces

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acted along the line of action and a constant torque was applied by an engine and breakthere would not be any fluctuating gear mesh forces, only a constant gear mesh forcedue to the applied torque. Since the constant gear mesh force moves back and forthover the gear tooth there is a shuttling force at the bearings. In the region above thefirst resonances, dominated by the engine and break, but below the next resonance, thetime varying bearing force is controlled by the shaft stiffness and the loaded transmis-sion error together with the shuttling force. The reason is that the inertia of the gearand engine are vibrationally stationary, only rotating with a constant angular speed.Since the shafts in this model are considerably weaker than the gear mesh, which oftenis the case, the transmission error acts as a forced angular displacement. The weakestcomponent, here the shafts, then deflects which determines the reaction force in the gearmesh. Due to the small time varying force due to transmission error, shuttling forces canbecome very important and dominate the excitation in this frequency region. At supercritical frequencies, above the highest resonance frequency, the system is controlled bythe inertia forces. This means that the rotations of all inertias are a function of therotating speed and gear ratio. Another way of describing the motion is to say that thepeak-to-peak angular transmission error is zero between all inertias. The deflection ofthe gear teeth is then equal to the loaded static TE and the force equal to the deflection(TEs) multiplied with the active stiffness which is the gear mesh stiffness, km.

Some conclusions can also be drawn for the resonant areas. For the resonanceswhich primarily includes the torsional degrees of freedom, the amplitude of the gearmesh force and bearing force is depending on the transmission error. Depending on themode shape, different stiffnesses will be important. For the lower torsional resonances,the shafts stiffness are the most important and the trend of the gear mesh force is thatof the transmission error multiplied with a constant stiffness. For the critical gear meshresonance, the most important stiffness is the gear mesh stiffness and the trend of thegear mesh force therefore follows the trend of transmission error multiplied with themesh stiffness. The rocking mode of the shaft is excited by the shuttling force andtherefore following the trend of the shuttling force.

Although other sources of gear noise such as friction and bending moments has beenproposed, TE has often been seen as the dominating excitation [11]. One of the reasonsis that the TE excitation force has been approximated as kmTE [11, 34], but below thegear mesh resonance the gear mesh force is dominated by the TE and the shaft stiffness.Since the shaft stiffness in this case is significantly lower than the gear mesh stiffens, thecorresponding force will also be lower, increasing the importance of other sources suchas shuttling forces.

6 Conclusions and future work

The measurements of dynamic transmission error and noise show a correlation betweenthe two properties of the tested gearbox. The correlation was highest for the input gearstage located close to the gearbox housing and lower for the second gear stage, which

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is located closer to the centre of the gearbox. One reason for the lack of correlationbetween DTE and noise can be that other forces such as shuttling and friction forceshave little influence on DTE but a larger influence of emitted noise. This is evident forthe measurements presented in paper D where the explanation for the difference betweenthe measured gear pairs can be found in the friction forces, but most of all the shuttlingforces.

This shows the importance of including other forces beside transmission error andalso to have a dynamic model of the gear system in order to accurately predict the noisebehaviour of a gear system. Since different excitations have a different sensitivity todifferent resonances of the system, a dynamic model of the system is needed.

Much of the work done in gear dynamics has to do with gears without manufacturingerrors. In the future, the influence of production error such as runout, pitch errorand gear shape errors must be included in the dynamic models in order to predict thedynamic excitation for a gear pair with typical manufacturing errors. Depending on thechosen manufacturing process, it would be of interest to be able to predict the standarddeviation of bearing forces for a specific gear design and production method.

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References

[1] Houser, D. R., 1986, “The root of gear noise-transmission error,” Power Transmis-sion design, 86(5), pp. 27–30.

[2] Council Directive 70/157/EEC, 1970, “Permissable sound level and the exhaustsystem of motor vehicles,” .

[3] Commission Directive 2007/34/EC, 2007, “Amendment of directive 70/157/EEC,”.

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