Applied Acoustics 68 (2007) 1003–1025

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Experimental study of gear rattle excited by amulti-harmonic excitation

M. Barthod a,*, B. Hayne a, J.-L. Tebec a, J.-C. Pin b

a Laboratoire de Mecanique Vibratoire et d’Acoustique – Ecole Nationale Superieure d’Arts et Metiers 151,

bd de l’hopital, 75013 Paris, Franceb RENAULT Direction de la Mecanique Centre Technique de Lardy, 1, allee Cornuel 91510 Lardy, France

Received 26 April 2006; accepted 26 April 2006Available online 13 July 2006

Abstract

This paper deals with the rattle noise, caused by the fluctuations of the engine torque (acyclic exci-tation) which, under special conditions, can cause multiple impacts inside the gearbox. Its aim is toexperimentally describe the rattle phenomenon in a gearbox. First, a fully instrumented test rig con-sisting of a simplified gearbox was designed in order to recreate the rattle noise phenomenon for amulti harmonic excitation imposed to the input shaft of the gearbox. Second, different gearbox con-figurations were used to characterize the rattle threshold and the rattle noise evolution, in relation toexcitation parameters and mechanical gearbox parameters. Third, a simplified model of the rattlephenomenon is drawn, using a Kelvin–Voigt model, aiming to determine the most significant param-eters influencing the rattle noise.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Gear noise; Teeth impacts; Gearbox; Experiment

1. Introduction

1.1. Context of the project

Driving comfort, especially acoustic comfort, has now become a marketing issue. Theglobal reduction of emitted noise level causes the emergence of noises that had previously

0003-682X/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apacoust.2006.04.011

* Corresponding author. Tel./fax: +33 1 44246229.E-mail address: muriel.barthod@paris.ensam.fr (M. Barthod).

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been masked. This is the case of the ‘‘rattle noise’’, caused by fluctuations of the enginetorque which, under certain conditions, can cause multiple impacts inside the gearbox.The rattle noise problem is purely perceptive since the impacts on gear teeth due to rattledo not affect the mechanical behaviour of the gearing and do not lead to breakage. Rattlenoise is considered as particularly annoying and has a negative influence on vehicle inte-rior sound quality.

1.2. Rattle phenomenon

Rattle is an impulsive phenomenon that occurs on unloaded gears which do not trans-mit any power.

These unloaded gears, free in rotation, can knock each other under some operating con-ditions and thus cause rattle noise. Fig. 1 illustrates in a simplified way the backlash cross-ing phenomenon.

Several theoretical model have been described, assuming different kind of impacts (elas-tic or inelastic) [1]. Some authors [2,3] have taken into account axial impact due to theaxial play of the gear on the shaft.

These impacts will be also highlighted in Section 5.4 of our study.The acyclic excitation on the input shaft of the gearbox is function of the engine tech-

nology (4 or 6 cylinders, in line or in V), of the design of the driveline (design of clutch anddrive shafts), and of the vehicle running conditions (load conditions and engine speed). Inthe case of a four-stroke and four cylinder engine, since there are two explosions per rev-olution, the spectrum of the angular acceleration is in theory composed of engine speedharmonics H2n.

1.3. Literature review

There have been many studies of gearbox rattle noise.Rattle noise can be studied with a global point of view or with a more local phenom-

enon point of view. In these two cases, there are experimental and numerical studies; theconsidered excitation being more or less simplified.

1.3.1. Rattle studied with a global point of view

In global studies, the rattle noise problem is considered as a driveline design problem.With this point of view, whether studies are experimental or numerical, the objective is to

Fig. 1. Backlash crossing phenomenon.

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determine the influence of the driveline design choices on rattle noise. The dynamic of thegearing is not well defined, so we will not detail more this kind of study.

Experimentally, the objective is to study the torsional dynamic behaviour of the drive-line (i.e. to measure torsional eigen mode of the kinematics driveline) which corresponds toan excitation amplification and so can lead to rattle noise amplification [4,5,1].

In numerical models [6–9], non linearity in gearing stiffness, clutch hysteresis and widthof backlash are taken into account. Generally, a rattle noise reduction follows from clutchdesign optimization [10–14]. Nevertheless it seems to be necessary to work on both theentire driveline design and the gearbox design [10]. On the other hand, some studies arefocused either on a gearbox or only on one gear pair.

1.3.2. Rattle studied with a more local phenomenon point of view

1.3.2.1. Oil influence. Oil in the gearbox is not negligible since it influences drag torquesapplied to unloaded gears. Meisner [15] and Weidner [16] have measured drag torquesinside a gearbox. Influence of temperature, viscosity and quantity of oil (linked to theunloaded gear splashing) [17] and so the influence of the gearbox orientation [3] have beenstudied. Overall, the rattle noise sound pressure level decreased when the drag torqueapplied to the unloaded gears increased.

1.3.2.2. Rattle threshold. Rattle threshold has been studied in two ways.The simplest way is to define rattle threshold as the possibility of contact loss between

two pieces [18,19,16]. Rattle threshold index is defined either with clutch parameters [20],or by comparison of the gears acceleration to drag torque applied to the unloaded gear[11]. That definition is theoretical and supposes that the acceleration imposed on inputshaft is sinusoidal.

In experiments, rattle threshold is detected by listening, or by measuring vibrations ofthe gearbox case [5,21], or by measuring root mean square angular acceleration of theunloaded gears, or by visualising a contact loss between teeth [17,1].

Some authors study the rattle threshold from an auditory perceptive point of view.Thus, backlash crossing phenomenon does not necessary lead to a rattle noise that canbe heard or that is annoying [17,21,13]. Even so, the acyclism amplitude is not necessarilycorrelated to the subjective perception of rattle noise [1]. Although such conclusions are inagreement with our results, previous studies still consider a sinusoidal excitation.

1.3.2.3. Dynamic of gears under rattle – Models. With regard to the prediction of the gear-ing dynamics with backlash crossing phenomenon, most of studies deal with simplifiedmodels with only one degree of freedom (in translation, on the line of action).

Some models take into account the gearing backlash and use a mean gearing stiffness,constant in time with meshing [20,16,18,22]. Other models take into account non linearitydue to backlash and to gearing stiffness variations [23,15,6]. Dogan [3] and Lang [2] pro-posed a model of teeth impacts where axial impacts are distinguished from backlashimpacts in a gearbox. Pfeiffer presents three methods making it possible to know thedynamics of one or several gears excited in rattle [24]: the ‘‘patching method’’, the ‘‘pointmapping method’’ and the ‘‘stochastic method’’ which is a probabilistic approach.

1.3.2.4. Dynamic of gears under rattle – Experiments. The evolution of the produced rattlenoise in relation to excitation frequency and excitation amplitude has been studied in the

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form of ’’sensitivity curves’’ (in the case of a sinusoidal angular excitation).Overall, thesound pressure level increases with the engine speed, corresponding to excitation fre-quency, and with the acyclism level, corresponding to excitation amplitude.

Weidner [16] studies a ball which moves in a U-shaped part. He gives (by experimentsand a mathematical model) the evolution of the sound pressure level in relation to the exci-tation amplitude (sinusoidal).

With regard to only one gearing, Pfeiffer [10] links the rattle noise level to the geomet-rical parameters of the gearing (unloaded gear inertia and to gears radius) and to excita-tion parameters. Chae [25] works on real gearboxes excited by a sinusoidal acyclism. Hesuggests that gearbox sensitivity to rattle is function of the gearbox design, even if the glo-bal shape is the same for all gearboxes. Dogan [3] and Forcelli [26] measure the rattle noiselevel produced on real gearboxes under sinusoidal or multi-harmonic acyclism. But theinfluence of harmonics has not been observed in detail.

Weidner [16] and Lang [2] show that amplitude of the impact on unloaded gear (and sorattle noise level) increases when the backlash increases, even if the temporal shape of theimpact signal changes. With a 3-degrees-of-freedom model, Wang [22] suggested that therattle noise level increases when backlash increases and decreases when the unloaded gearinertia increases. Theses results are in agreement with our experimental results. Wang alsoshows that there is an interaction between these parameters.

1.3.2.5. Unsteady behaviour. The rattle phenomenon is not always stable. With regard to aball moving in a U-shaped part with a sinusoidal displacement, Weidner [16] distinguishesdifferent kind of relative movement (periodic, chaotic). Pfeiffer [23] uses a model of onegearing under a sinusoidal excitation and gives diagrams of the unloaded gear positioninside the backlash with impact phase. He underlines periodic, quasi-periodic or chaoticdynamic and bifurcations [10]. Dai [27] defines the ‘‘periodicity ratio’’ (based on the num-ber of points that overlap on Poincare diagram) to distinguish periodic and chaoticdynamic comportment of the system. Blazejczyk [28] defines intermittency as a chaoticdynamic comportment characterised by a periodic comportment interrupted by short cha-otic phases. That comportment can occur in a gearbox and explains the irregular charac-teristic of the rattle noise sometimes perceived.

Some studies [27,29,30] deal with jumping phenomenon and with a branching link tothe non linearities of the mechanical system. Other studies deal with problems due tonumerical resolution of the dynamic equations [31,32]. But in all these models, the excita-tion is supposed to be sinusoidal and the mechanical system under study is very simplifiedcompared to a real gearbox.

1.3.2.6. Test rig. In experiments, in most case, the incoming acyclic excitation on an iso-lated gearbox is applied by means of a universal joint assembly.

The oscillating part of rotational speed can be generated around a constant rotationalspeed delivered by an electrical motor [4,5,21,33]. In that case, the excitation imposed tothe gearbox input shaft is sinusoidal.

Other studies [2,3,34] have used a synchronous tree-phase motor to impose an acyclismcomposed of several harmonics, representative of a four or six cylinder engine. But theinfluence of the harmonic composition of the excitation was not quantified. In these stud-ies, only the global sound pressure level or the root mean squared casing acceleration weremeasured to quantify the rattle noise, the gearing dynamic was not precisely measured.

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Otherwise, some authors have used more simplified experimental test rigs which giveaccess to the gearing dynamics, but the system is still very simplified compared to a realgearbox. Pfeiffer [24,35] worked on one gear with only one tooth, excited with an eccentricdevice. Crocker [17] and Weidner [16] have measured the restitution coefficient of a toothimpact while Azar [36] has studied the contact between two teeth.

1.4. Objectives

On a vehicle, the gearbox is more or less ‘‘sensitive’’ to rattle phenomena. This sensitiv-ity is function of 3 parameters: the excitation (angular acceleration) imposed to the inputshaft of the gearbox (i.e. the acyclism) [35,37], the dynamic response of the internal gear-box architecture and then, the vibration transfer to the casing. Here, we are interested onthe influence of the two first parameters on rattle noise.

Note: Acyclism is the general name of the excitation imposed on the input shaft of agearbox. In fact, on vehicle (in the case of a four-cylinder motor, 4 strokes), this acyclismis mainly composed of harmonics of even order of the engine speed (H2, H4, H6. . .). Forour experiments, the acyclism is simplified and corresponds to a multi harmonicexcitation.

(1) Until now and in most studies, the torsional excitation is very often simplified: onlythe 2nd order harmonic (H2) of the engine speed is considered. A measure on a realvehicle [38] clearly show that the acyclism on input shaft of the gearbox is far fromsinusoidal.Our objective is to observe whether temporal and spectral characteristics of a realacyclism have to be taken into account. For that, we work with an acyclism com-posed of the 2nd order and 4th (and eventually 6th) order harmonics of the enginespeed. In other word, what happens when the excitation imposed to the gearbox ismulti-harmonic? In the case of a single gearing, we have previously showed [39] thattaking into account a multi-harmonic excitation has an important influence on thesonority of rattle noise produced.

(2) To measure the dynamic response of the internal gearbox architecture, parametersthat have to be taken into account include: gear inertia and backlashes (betweenunloaded and gearing gears), position of the unloaded gears (on primary or on sec-ondary shaft), gear reductions and drag torques.

Gearbox rattle sensitivity can be obtained analytically by modelling the transmission.However it is very difficult to consider all the influencing components such as gears,synchronizers, bearings, case, oil and nonlinear properties such as meshing stiffness, vis-cosity. Integrating all plays and backlashes is very difficult. Even though analytic mod-els have been fully developed, the validity of results has not been checked. Ourcontribution is an experimental investigation on the influence of unloaded gears inertiaand backlash.

In order to characterize the influence of excitation parameters as well as the influence ofsome geometrical parameters on the gearbox sensitivity to rattle noise, two characteristicsof the rattle noises are observed. Hence, we define the rattle noise threshold and the rattlenoise evolution in relation to amplitude and frequency of the excitation (these parametersare in theory linked to engine working conditions).

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One of the objectives is to assess the validity of a very simplified model of the rattle phe-nomenon to estimate the sensitivity of a simplified gearbox. The model used is a Kelvin–Voigt model, usually used to analyse the backlash crossing phenomenon between gears[40,41]. We have to show the limitations of such a model and underline the parametersthe most important on rattle noise.

2. Gear model

According to Pfeiffer [35], rattle in a real gearbox is a cascade process. Such phenom-enon is difficult to analyse since interaction between gears have to be taken into accountfor the resolution of dynamical equations.

The model used is a Kelvin Voigt model (Fig. 2). It is a simple model with two degreesof freedom (Fig. 2). It is made of a driving gear and an unloaded gear, whose motion arelinked during contact phases, or are independent during free-flight phases, when theunloaded gear moves within backlash. Some modifications of the Kelvin Voigt model havebeen made (i.e. following R.C. Azar et F.R.E. Crossley [36]) in order to avoid numericaldiscontinuity problems. We have introduced a non linear parameter in the expression forthe damping during impact.

The angular position of the driving gear (primary shaft of the gearbox) is given by theangle h1, the position of the unloaded gear is given by the angle h2 ( _hi is the angular veloc-ity in rad/s and €hi is the angular acceleration in rad/s2). The radii of the driving andunloaded gears are R1 and R2 (in m) respectively, their inertia around their rotation axesbeing I1 and I2 (in kg m2), j is the backlash, k is the contact stiffness and c is the contactdamping.

The hypotheses of the model are:

� in neutral, no average torque is transmitted, there is only oscillating torque,� gears are spur toothed,

Fig. 2. Diagram of the two degrees of freedom model used (modified Kelvin–Voigt model). hi: angulardisplacement (rad), R1 and R2: radiuses of the driving and unloaded gears (m), I1 and I2 their inertia around theirrotation axes (kg m2), j: backlash (m), k: contact stiffness, c: contact damping.

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� drag torques are assumed to be constant (Cdrag),� the imposed torsional excitation (Cexcitation) is not influenced by the dynamics of the sys-

tem (driving gear/unloaded gear).

Modelling is carried out under MATLAB version 6, we have chosen to use theNewmark method to solve the dynamic equations. Values of the parameters used inour model are estimated by simple mechanical calculation, or experimentally measuredor are derived from comparison between numerical results and experimentalmeasures.

3. Description of the experimental setup

3.1. Design specifications and realisation of the test rig

Our aim is to reproduce a rattle phenomenon with a perfect control of the excitation(angular acceleration) imposed to the input shaft of the gearbox. Since universal jointassembly do not seem to be adapted, we have to design a new type of test rig. Relativeharmonic amplitudes and phases of the excitation have to be adjustable at will, so as toexplore all the possible excitation configurations. We also need a good access of gearsto study their dynamics: sound pressure level, impact amplitude and relative motion ofgears have to be measured, which requires instrumentation on the gears.

In our study, the input shaft does not rotate, we chose to work on neutral and do nottake into account the oil influence: contacts are oiled but there is no splashing.

As a gearbox is an assembly of numerous mechanical parts with backlashes betweeneach others, there is a potential of many noise sources. We study the multiple impacts

Fig. 3. Simplified gearbox used in our experiments – opened gearbox case.

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between gear teeth: all other noise sources should be excluded. Our tests are carried on asimplified gearbox whose gear forks and synchronization mechanisms have been removed,only one pair of cylindrical gears with helical teeth remains, and there is no oil (Fig. 3).The gear is maintained by a ring force-mounted.

Rattle phenomenon is due to teeth impacts after backlash crossing, so we have to studythe relative motion of gear pairs. Working with a gearbox on neutral allowed us to cancelthe average excitation, and to impose only an oscillating torque. In that case, excitationapplied to the gearbox is equivalent to angular oscillations applied to the primary shaftof the gearbox. An electrodynamic translation exciter is used, attached to the gearbox witha crank and driven by a signal generator. As the gearbox input shaft does not rotate, theinstrumentation is easier.

We should be able to impose angular accelerations to the gearbox input shaft, similarin amplitude and frequency with those measured on a vehicle. It is necessary to ensurean excitation from 0 to 1200 rad/s2 root mean squared (RMS), on a frequency rangegoing from 30 to 180 Hz. This range correspond to engine speeds from 900 to1800 rpm, where rattle noise is significant, engine noise being too weak ‘‘to cover it’’.So, the whole excitation mechanism has to be carefully designed. In particular, transmis-sion system has to be without backlash and rigid enough to avoid resonances in our fre-quency range of interest.

3.1.1. InstrumentationThe input shaft and the unloaded gear are equipped with an accelerometer and a non-

contacting displacement sensor (eddy current). An accelerometer is attached to the gearboxcase. A sound level meter near the gearbox is used to compare the sound pressure signal ofrattle noise from different excitation configurations.

All the excitation parameters (frequency, harmonics amplitudes and phases) can beseparately adjusted and allow continuous sweep. For example, an excitation deviceenables us to carry out progressive continuous sweeps of the global excitation amplitudeimposed to the input shaft, whatever the composition of this excitation. That allow us toobserve the evolution of rattle in relation to the acyclic excitation amplitude for a givenengine speed.

The electrodynamic translation exciter is controlled in order to impose an angularacceleration (in rad/s2) on the input shaft of the gearbox. H4 and H6 harmonics amplitudeare expressed in %, relative to the 2nd order harmonic amplitude (e.g. it is a relative ampli-tude), their phases (u4 and u6) are in relation to H2. In the case of a composite excitation,the frequency known as the excitation frequency corresponds to the frequency of the 2ndorder harmonic (H2).

3.2. Gearbox configurations used

Six different configurations of gearbox were used (Table 1). Configurations 1, 2 and 3(named ‘‘with modified inertia’’) are obtained from the same unloaded gear which succes-sively undergoes an increase of inertia (by addition of a disc on a side) then a reduction ofinertia (by machining). The modifications of inertia are about 50% of initial inertia. Con-figurations 4, 5 and 6 (named ‘‘with modified backlash’’) are obtained with 3 differentunloaded gears. Precise measurements of the backlash were taken using noncontacting dis-placement sensor. The backlashes were 75, 83 and 100 lm.

Table 1Gearbox configurations used in our experiments

No. config. Corresponding value of inertia and backlash

1 Configuration {backlash; inertia} initial2 Unloaded gear with increased inertia, initial backlash3 Unloaded gear with decreased inertia, initial backlash4 Unloaded gear with minimum size backlash, initial inertia5 Unloaded gear with medium backlash, initial inertia6 Unloaded gear with maximum size backlash, initial inertia

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4. Study of the rattle NOISE threshold

In our study, rattle threshold is defined as the angular acceleration amplitude imposedto the input shaft of the gearbox (in rad/s2 RMS) from which the rattle phenomenonoccurs and is maintained (i.e. stable in time; that definition of threshold allows us to avoidthe influence of the relative position of gears before the first impact).

4.1. Rattle threshold in relation to excitation parameters

Here, we observe the influence of the spectral composition (2nd, 4th and 6th order har-monic amplitude) and of the temporal shape (harmonic phases) of the excitation imposedto the gearbox input shaft on the rattle threshold. In other words, is the presence of har-monics of order 4 and 6 in the excitation signal stimulating or not to the appearance ofrattle?

We have previously showed [42] that the rattle threshold mainly evolves with the fre-quency of the imposed H2 harmonic: the higher is the excitation frequency, the higheris the acceleration amplitude from with rattle appears.

Then, in the case of a multi-harmonic excitation, there are numerous possible excitationconfigurations. We have used the experiment design method in order to estimate the influ-ence of the amplitude and phase of the 2nd, 4th and 6th order harmonics on the rattlethreshold.

Thus, we have observed that for a given excitation frequency, the threshold is obtainedfor a nearly constant amplitude of the 2nd order harmonic, whatever the 4th and 6th har-monics amplitude are. It seems that the spectral composition of the acyclic excitation hasfinally little influence on the occurrence of rattle.

This result can be explained by the fact that the 4th and 6th order harmonics have littleinfluence on the kinetic energy. For example, when the amplitude of the H4 harmonic goesfrom 20%o 80%, the global root mean squared value of the acceleration vary of 25%, andthe global root mean squared value of the speed only vary of 7%.

4.2. Rattle threshold in relation to gearbox parameters

Rattle threshold has been measured for sinusoidal excitations with frequency at 30, 45and 60 Hz, and for the 3 different unloaded gears inertia (in that case, the backlash is con-stant and equal to 120 lm). We show that the evolution of the threshold with the unloadedgear inertia is linear.

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Fig. 4 shows the evolution of the rattle threshold in relation to excitation frequency (inthe case of a sinusoidal excitation) and in relation to the unloaded gear inertia. For con-fidential reasons, experimental data are normalized (the reference is the threshold obtainedwith an excitation at 30 Hz and an initial unloaded inertia). The weaker the inertia, themore significant is the acceleration necessary to start the rattle phenomenon.

The threshold is obtained for a constant kinetic energy imposed on the input shaft ofthe gearbox. With regard to 2 gears with inertia noted I+ and I� (Eqs. (1) and (2)):

1

2Iþ _h2

Iþ¼ 1

2I� _h2

I�ð1Þ

so : Iþ€h2Iþ ¼ I�€h2

I� ð2Þ

with _h angular speeds measured at threshold, in rad/s, €h corresponding angular accelera-tions, in rad/s2, x the excitation pulsation, in Hz.

Threshold values obtained on our test rig suggest that the relationship is kinetic. Thisconfirms the fact that the rattle threshold (for one gearing) is obtained for a constantkinetic energy introduced into the system, the value of this energy depending on the gear-box architecture.

Then, rattle threshold has been measured for sinusoidal excitations with frequency at30, 45 and 60 Hz, and for the 3 different values of backlash (in that case, the unloaded gearinertia is constant and equal to 0.00098 kg m2).

Fig. 5 shows the evolution of the rattle threshold in relation to excitation frequency (inthe case of a sinusoidal excitation) and in relation to the backlash. For confidential rea-sons, experimental data are normalized (the reference is the threshold obtained with anexcitation at 30 Hz and a backlash of 100 lm).

The more significant the backlash, the higher is the rattle threshold. To start rattle, it isnecessary that the displacement imposed on the input shaft corresponds at least to thebacklash: the larger the backlash is, the higher the imposed displacement must be and thusthe more significant the corresponding acceleration must be.

Note: In this article, we considers backlash with values within manufacture tolerances.Studies on gearings noise have showed that there is an optimum backlash for concerning

Fig. 4. Evolution of the rattle threshold in relation to excitation frequency and in relation to the unloaded gearinertia. Experimental data are normalized with respect to a reference threshold obtained with an excitation at30 Hz and an initial inertia.

Fig. 5. Evolution of the rattle threshold in relation to excitation frequency and in relation to the backlash.Experimental data are normalized with respect to the reference threshold obtained with an excitation at 30 Hzand a backlash of 100 lm).

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the noise produced by gears. For a backlash larger or smaller than this optimum one, it isknown that the noise is more significant.

5. Study of the rattle noise evolution

We suppose here that rattle started, we are interested in the evolution of rattle, in func-tion of the frequency and the amplitude of the acceleration imposed to the input shaft ofthe gearbox, then in function of gearbox geometrical parameters.

5.1. Transfer path between unloaded gear and gearbox case

For each experiment, we simultaneously record the rattle noise signal (with a micro-phone near the gearbox), the gearbox case vibration, the unloaded gear acceleration andthe input shaft acceleration. We have previously observed [39] that, for an excitationwith constant spectral and temporal parameters and during a progressive sweep of theRoot Mean Squared excitation amplitude, the Root Mean Squared (RMS) accelerationof the unloaded gear is well correlated by a linear relationship with the acoustic pressureof the rattle noise produced. In other words, the rattle noise sound pressure can berather simply estimated from the RMS value of the impacts on the unloaded gear,and vice versa, as shown in Fig. 6. That result can also be found in Fujimoto works[43], or in Pfeiffer [35] and Swadowski [20] works: a good estimation of the rattle noisesound pressure level is given by the mean impulse force measured on the unloaded gearsof the gearbox.

Curves of Fig. 6 are obtained by post processing of the data recorded during progres-sive and continuous sweeps of the excitation amplitude for sinusoidal excitation at 30 Hz,45 Hz or 60 Hz. For each sweep (0 to more than 1000 rad/s2 RMS), we calculate the RMS

Fig. 6. Evolution of the RMS value of the sound pressure of the rattle noise, measured near the gearbox (in Pa) inrelation to the RMS unloaded gear acceleration (in m/s2, on the unloaded gear primitive radius). Data recordedduring progressive and continuous sweeps of the excitation amplitude for sinusoidal excitation at 30 Hz, 45 Hz or60 Hz.

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values of the unloaded gear acceleration (expressed in m/s2, on the unloaded gear primitiveradius) and the RMS value of the sound pressure of the rattle noise, measured near thegearbox. We combine these two evolutions to build Fig. 6.

5.2. Influence of excitation parameters on rattle

5.2.1. Rattle produced in the case of a sinusoidal excitation

Results obtained for a sinusoidal excitation are presented on Fig. 7. We give the evo-lution of the RMS acceleration of the unloaded gear (impact due to backlash crossingof teeth, expressed in m/s2) in relation to the RMS acceleration imposed to the input shaft(expressed in rad/s2). The different curves correspond to several progressive sweeps inRMS excitation amplitude (increasing and decreasing) for sinusoidal excitations at 30,45 and 60 Hz.

The dispersion of measurements is weak enough (about 15% for an excitation at800 rad/s2 RMS) to clearly release the influence of the excitation amplitude and frequency.

Overall (by excluding the beginning of the recordings to 30 Hz), for a constant excita-tion frequency, the RMS acceleration of the unloaded gear increases with the RMS exci-tation amplitude. Besides, for the same excitation amplitude, the higher is the excitationfrequency, the higher is the rattle noise. It means that an excitation level which is not ‘‘crit-ical’’ (i.e. giving a low rattle noise) to weak driving regime can become ‘‘critical’’ if theengine speed increases.

Remark: On a vehicle, engine speed and acyclism amplitude (i.e. excitation level) arecoupled. In our study, we voluntarily uncouple the two parameters so as to observe theirrespective influence.

Fig. 7. Evolution of the RMS acceleration measured on the unloaded gear in relation to the RMS accelerationimposed to the input shaft. The different curves correspond to several progressive sweeps in RMS excitationamplitude (increasing and decreasing) for sinusoidal excitations at 30, 45 and 60 Hz.

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5.2.2. Rattle produced in the case of a multi-harmonic excitation

We extend here the study to the case of a more realistic excitation. So as to limit thenumber of parameters, we work with an excitation signal composed of the 2nd and the4th order harmonics.

Trying to understand how the harmonics play a part on rattle phenomenon, we havemeasured the evolution of the RMS impacts amplitude (unloaded gear acceleration, inm/s2) according to the global RMS excitation amplitude (input shaft acceleration, inrad/s2) (Fig. 8), either according to the global peak-to-peak excitation value or accordingto the H2 RMS excitation value.

For example, curves presented here are obtained for various global amplitude sweeps(H2 at 30 or 60 Hz, with various percentage of harmonic H4 and with a phase = 0). OnFig. 6, every curve is indicated by two numbers: the first one corresponds to the frequencyof harmonic H2 of the excitation imposed to input shaft, the second corresponds to therelative amplitude of H4 (in % with regard to that of H2). The phase of H4 with regardto H2 is nil. Curves in dotted line remind for comparison the average measures obtainedfor sinusoidal excitations at 30 and at 60 Hz.

For example, the curve named ‘‘60–40’’ on Fig. 6 was obtained during a progressivesweep of the global amplitude of the excitation composed of a harmonic H2 at 60 Hzand a harmonic H4 (at 60 · 2 = 120 Hz) with an amplitude equal to 40% of the harmonicH2 amplitude.

We can see that the introduction of harmonics H4 do not modify the global evolution ofcurves connecting the answer of the unloaded gear to the excitation introduced on theinput shaft (compared to the case of sinusoidal excitation). However, the dispersion isgreater than with a sinusoidal excitation.

Fig. 8. Evolution of the RMS unloaded gear acceleration in relation to the global RMS acceleration imposed tothe input shaft of the gearbox – Case of multi-harmonic excitations. Every curve is indicated by two numbers: thefirst one corresponds to the frequency of harmonic H2 of the excitation imposed to input shaft, the secondcorresponds to the relative amplitude of H4 (in % with regard to that of H2). The phase of H4 with regard to H2 isnil.

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The presence of harmonic H4 favours the appearance of a ‘‘jump phenomenon’’ in thebehaviour of unloaded gear. These jumps will increase the rattle noise perception sincethey correspond to sudden variations of sound pressure level (near 2 or 3 dBA on our testrig) or sudden variations of the ‘‘sonority’’ of the rattle noise.

Overall we have observed the important influence of spectral and temporal parametersof the acyclism imposed to the gearbox on the produced rattle noise (level and sonority)[39,44].

5.3. Influence of geometrical gearbox parameters on rattle

5.3.1. Unloaded gear inertia influence

We compare gearbox configurations no. 1, 2 and 3 (initial, increased and decreasedinertia). Fig. 9 gives for these 3 configurations, the evolution of the RMS unloaded gearacceleration according to the RMS acceleration imposed on the input shaft of the gearbox,for excitations at 45 Hz. The same data were obtained for excitations at 30 and 60 Hz.

That clearly suggests that for the same excitation amplitude, a reduction in theunloaded gear inertia gives more significant impact RMS amplitude. In other words,decreasing the unloaded gear inertia tends to increase the ‘‘sensitivity’’ of the gearing toacyclism.

To explain this result, we visualize the temporal shape of accelerations, speeds and dis-placements of the input shaft and unloaded gear for a constant excitation (Fig. 10). Forthe same excitation amplitude, the impact speed of the unloaded gear is smaller in the caseof the initial inertia configuration than in the case of the decreased inertia configuration,

Fig. 9. Influence of the unloaded gear inertia: evolution of the RMS unloaded gear acceleration according to theRMS acceleration imposed on the input shaft of the gearbox, for excitation at 45 Hz. Curves obtained for 3gearbox configurations (initial, increased and decreased inertia).

Fig. 10. Temporal shape of accelerations, speeds and displacements of the input shaft and unloaded gear for aconstant excitation. Two gearbox configurations are observed: Initial inertia and decreased inertia of theunloaded gear.

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(whatever the unloaded gear is or not axially maintained i.e. Section 5.4) this explains thelouder noise produced [45].

5.3.2. Backlash influence

We now compare gearbox configurations no. 4, 5 and 6.Fig. 11 gives the evolution of the unloaded gear RMS acceleration according to the

RMS acceleration imposed on the input shaft, for an excitation at 30 Hz. The same datawere obtained for excitations at 45 and 60 Hz.

It appears that the wider the backlash, the higher is the RMS value of the impact (evenif the effect is weak). This is explained by the fact that an increase in the backlash gives alonger free flight phase, and, actually, a more significant speed difference between gears,just before the impact.

5.4. Description of an unsteady behaviour

Actually, for a strong level of excitation imposed on the input shaft, a very ‘‘irregular’’rattle noise is sometimes obtained. The passage to an ‘‘unsteady’’ behaviour seems to berandom and can be observed by listening to a rattle noise. We notice an increase of thenoise level and different rhythms of impacts, on the unloaded gear acceleration signal,or on the casing vibration even if the excitation parameters are constant. Fig. 12 gives tem-poral signals of the unloaded gear acceleration measured for a ‘‘steady’’ then an‘‘unsteady’’ behaviour, the excitation amplitude being the same.

Measurements taken with a triaxial accelerometer have allowed us to better observe thedynamics of the unloaded gear. The presence of axial impacts can be checked and could be

Fig. 11. Influence of the Backlash: evolution of the RMS unloaded gear acceleration according to the RMSacceleration imposed on the input shaft of the gearbox, for excitation at 30 Hz. Curves obtained for 3 gearboxconfigurations (backlash = 75, 83 and 100 lm).

Fig. 12. Visualization of a ‘‘usual’’ and then an ‘‘unsteady’’ behaviour. Temporal signals of the unloaded gearacceleration.

M. Barthod et al. / Applied Acoustics 68 (2007) 1003–1025 1019

explained by the axial backlash of the unloaded gear on the secondary shaft of the gearboxand by the helix angle (forces applied to the unloaded gear are decomposed in axial andradial component). This confirmed observations of [3,2].

The uncontrolled occurrence of the parasitic axial impacts led us to control the axialmovement of the unloaded gear during our recordings.

We chose to work with the most stable gearbox configuration in order to optimizerepeatability of measurements: the axial play has been suppressed.

6. Comparison of experimental and numerical results

6.1. Comparison of temporal shape

Figs. 13 and 14 compare the temporal signal obtained on our test rig with thoseobtained by calculation excitation at 30 Hz, 700 rad/s2 RMS on the input shaft). We suc-cessively visualize: accelerations of the input shaft and unloaded gear, (in m/s2), relativedisplacement between gears (in mm), input shaft and unloaded gear speeds (in m/s), thenrelative speed between gears (in m/s). We have also checked that this correlation is verygood on all the measuring range excitation.

6.2. Influence of excitation parameters on rattle

Fig. 15 shows the experimental results of Fig. 7 (curves in thin lines), on which resultsgiven by the model are superimposed (bold curves, each point corresponding to a

Fig. 13. Temporal signal obtained by calculation with a sinusoidal excitation at 30 Hz, 700 rad/s2 RMS. Wesuccessively visualize: accelerations of the input shaft and unloaded gear, (in m/s2), relative displacement betweengears (in mm), input shaft and unloaded gear speeds (in m/s), relative speed between gears (in m/s).

Fig. 14. Temporal signal measured on our test rig with a sinusoidal excitation at 30 Hz, 700 rad/s2 RMS. Wesuccessively visualize: accelerations of the input shaft and unloaded gear, (in m/s2), relative displacement betweengears (in mm), input shaft and unloaded gear speeds (in m/s), relative speed between gears (in m/s).

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Fig. 15. Comparison of measured (curves in thin lines) and calculated data (bold curves, each pointcorresponding to a computation configuration). Evolution of the RMS acceleration on the unloaded gear inrelation to the RMS acceleration imposed.

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computation configuration). The general assessment of this comparison is highly satisfac-tory. Except for an excitation at 30 Hz where calculation is approximately 25% higher, thevalues given by the model are in the dispersion interval of measurements on test rig. How-ever, it should be noted that the rattle threshold is not reproduced in the model, because ofthe uncontrolled initial conditions.

6.3. Geometrical parameters influence

The following figures show the predictions of the model concerning the influence of theunloaded gear inertia and the backlash.

Fig. 16 gives the evolution of the RMS acceleration of the unloaded gear for a constantinertia (initial inertia) and three different backlashes (60, 100 and 160 lm). The variationrange of backlash in our numerical simulation was wider so as to better release the generaltrend.

Model gives us the evolution of the RMS acceleration of the unloaded gear for a con-stant backlash (120 lm) and 3 different inertias: 0.00098, 0.00149 and 0.00225 kg m2

(Fig. 17). Used inertia values correspond to the ones used for measurements. Comparisonwith experiment can be done. For example, Fig. 18 compares experimental results (linecurves), and numerical result (points) in the case of a sinusoidal excitation at 30 Hz. Wehave checked that the numerical simulation gives us the same qualitative or quantitativeresults as those obtained by experimentation.

6.4. Utility and limitations of that model

Comparison of measures on our test rig and of numerical results obtained with aKelvin–Voigt model is satisfactory. Excitation parameters and geometrical parameters

0

20

40

60

80

100

120

0 200 400 600 800 1000 1200 1400

RMS input shaft acceleration (rad/s²)

RM

S U

nlo

aded

gea

r ac

cele

ratio

n (

m/s

²)

75 µm - 30 Hz83 µm - 30 Hz100 µm - 30 Hz75 µm - 60 Hz83 µm - 60 Hz100 µm - 60 Hz

Fig. 16. Influence of the backlash on rattle (constant inertia) – Numerical results. Evolution of the RMSunloaded gear acceleration according to the RMS acceleration imposed on the primary shaft, for sinusoidalexcitations at 30 Hz and 60 Hz.

Fig. 17. Influence of the unloaded gear inertia on rattle (constant backlash) – Evolution of the RMS unloadedgear acceleration according to the RMS acceleration imposed on the primary shaft, for sinusoidal excitations at30 Hz and 60 Hz.

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influences can be well found, qualitatively and quantitatively. The noted differences can beexplained by the approximation used in the model: the fact that the helix angle is not takeninto account (that partially explains axial impacts and unsteady behaviour), the fact thatdrag torques are simplified and the fact that the secondary shaft mean angular speed is nottaken into account.

A Kelvin–Voigt model is sufficient if we consider only one gearing but can not be usedin the case of a real gearbox.

Fig. 18. Comparison of measured and calculated data – Influence of the unloaded gear inertia on rattle: evolutionof the RMS acceleration of the unloaded gear for a constant backlash (120 lm) and 3 different inertias:0.00098 kg m2, 0.00149 kg m2 and 0.00225 kg m2.

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7. Conclusions

In this article, an experimental study of the rattle noise phenomenon is realized on asimplified gearbox and allows us to assess the validity of a simple model as Kelvin–Voigtapplied to rattle noise.

A test rig has been design to produce rattle phenomenon under a perfectly controlledexcitation and equipped to achieve acoustic and vibratory measurements.

One specification of our test rig is to impose to the gearbox input shaft an acyclism notonly sinusoidal but composed of several harmonics with relative amplitudes and phasesare adjustable at will, which allows a very precise measurement of the gearing dynamics.

We have been interested in the rattle threshold (i.e. excitation conditions imposed togearbox that cause rattle to occur) and in the influence of excitation parameters and geo-metrical gearbox parameters on rattle.

About the rattle threshold, we have observed that, for a constant gearbox configura-tion, the threshold is, at first, linked to the kinetic energy imposed to the input shaft ofthe gearbox. In other words, the spectral composition of the acyclism has little influence.The threshold increases when backlash increases and decreases when the unloaded gearincreases.

When rattle is triggered, its level increases when the excitation amplitude and frequencyincreased.

The introduction of a 4th order harmonic into the excitation gives similar evolutions,but increases dispersion and is important in auditory perception since it leads to a ‘‘jumpsphenomenon’’, i.e. fast variations of rattle level and/or rattle sonority.

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Otherwise, increasing the backlash or decreasing the unloaded gear increases rattlenoise level.

Overall the comparison of numerical results and experimental results has been satisfac-tory. The influence of different parameters can be investigated, qualitatively and quantita-tively, with a simple Kelvin–Voigt model.

Acknowledgements

This work is supported by RENAULT. The authors would like to thank the MechanicalDirection of Renault and more particularly the 66126 NVH GMP acoustics department.

References

[1] Johnson O, Hirami N. Diagnosis and objective evaluation of gear rattle. SAE no. 911082, 1991, p. 381–96.

[2] Lang CH, Lechner G. Klappern und Rasseln von Fahrzeuggetrieben – Simulation der Losteilschwing-ungen und Moglichkeiten zur Gerauschminimierung (Automotive gear rattle noise – simulation of theoscillations due to the loose of contact in gears and possibilities for noise minimization). VDI Berichte1995;1175:99–122.

[3] Dogan SN. Zur minimierung der Losteilgerausche von Fahrzeuggetrieben (To the minimization of the noisesdue to loose of contact in gears) – These. Institut fur Maschinenelemente, Stuttgart: Stuttgart; 1999.

[4] Couderc P. Comportement dynamique des chaınes de transmission automobiles (Torsional dynamicbehaviour of drive powertrains) – These. INSA de Lyon: Lyon; 1997.

[5] Develay C. Etude et reduction du bruit de graillonnement des boıtes de vitesses automobiles (Study andreduction of the automotive gear rattle noise) – These. INSA de Lyon: Lyon; 2000.

[6] Kim TC, Singh R. Dynamic interactions between loaded and unloaded gear pairs under rattle conditions.SAE no. 2001-01-1553, 2001.

[7] Fudula GJ, Engle C. A system approach to reducing gear rattle, SAE, 1987.[8] Sakai T, Doi Y, Yamamoto K. Theorical and experimental analysis of rattling noise of automotive gearbox.

SAE, 1981.[9] Fujimoto T, Chikatani Y, Kojima J. Reduction of idling rattle in manual transmission. SAE no. 870396,

1987.[10] Pfeiffer F, Prestl W. Decoupling measures for rattling noise in gearboxes. In: IMechE 1990, 1990.[11] Singh R, Xie H, Comparin RJ. Analysis of automotive neutral gear rattle. J. Sound Vib. 1989;131(2):177–96.[12] Kim TC, Singh R. Prediction of rattle noise from an unloaded gear pair. In: Inter Noise 2000. Nice: France;

2000.[13] Padmanabhan C, Singh R. Influence of clutch design on the reduction and perception of automotive

transmission rattle noise. In: Noise-con 93, Williamsburg, Virginia, 1993.[14] Padmanabhan C, Rook TE, Singh R. Modeling of automotive gear rattle phenomenon: state of the art. SAE

no. 951316, 1995b, p. 669–80.[15] Meisner S, Campbell B. Development of gear rattle analytical simulation methodology. SAE no. 9511317,

1995, p. 681–9.[16] Weidner G, Lechner G. Rattling Vibrations in automotive transmissions. In: JSME international conference

on motion and powertransmissions, 1991, Hiroshima, Japan.[17] Croker MD, Greer RJ. Transmission rattle analysis. In: IMechE 1990, 1990.[18] Soine DE, Evensen HA, Vankarsen CD. Threshold level as an index of squeak and rattle performance. SAE

no. 1999-01-1730, 1999.[19] Lee CH, Byrne KB. Impact statistics for a simple random rattling system. J. Sound Vib. 1987;119(3):

529–43.[20] Szadkowski A. Mathematical model and computer simulation of idle gear rattle. SAE no. 910641, 1991, p.

81–97.[21] Blomqvist J. Detection of impact sources using an impulse response recognition technique. Applications on

rattle sources in gearboxes 2000.

M. Barthod et al. / Applied Acoustics 68 (2007) 1003–1025 1025

[22] Wang H, Glover R. Analytical and experimental study on gear rattle in supercharger. SAE, 2005-01-2369,2005.

[23] Pfeiffer F. Theorie des Getrieberasselns. VDI Berichte 1988;679:45–65.[24] Pfeiffer F. Modelling problems of rattling in gear boxes. In: JSME international conference on motion and

powertransmissions, 1991, Hiroshima, Japan.[25] Chae C-K, Won K-M, Kang K-T. Measurement of transmission rattle sensitivity and calculation of driveline

torsional vibration for gear rattle analysis. SAE, 2005-01-1785, 2005.[26] Forcelli A, Grasso C, Pappalardo T. The transmission gear rattle noise: parametric sensitivity study. SAE,

2004-01-1225, 2004.[27] Dai L, Singh MC. Diagnosis of periodic and chaotic responses in vibratory systems. J Acoust. Soc. Am.

1997;102(6):3361–71.[28] Blazejczyk B, Kapitaniak T, Wojewoda J, Barron R. Experimental observation of intermittent chaos in a

mechanical system with impacts. J Sound Vib 1994;178(2):272–5.[29] Perret-Liaudet J, Sabot J. Sur quelques comportements dynamiques non-lineaires des engrenages (On some

non-linear dynamic behaviours of the gears). Mecanique industrielle et materiaux 1998;51(4):170–2.[30] Comparin RJ, Sing R. Non-linear frequency response characteristics of an impact pair. J Sound Vib

1989;134(2):259–90.[31] Comparin RJ, Singh R. An analytical study of automotive neutral gear rattle. J Mech Des 1990;112:237–45.[32] Padmanabhan C. Computational issues associated with gear rattle analysis. J Mech Des 1995;117:185–92.[33] Philippe-Damagnez C. Correlation between gearbox housing vibration and teeth impacts. In: European

automotive congress, 2001, Bratislava, Slovaquie, EAEC.[34] Forcelli A. The transmission gear rattle noise: parametric sensitivity study 2005.[35] Pfeiffer F. Rattling in gears – a review. VDI Berichte 1996;1260:719–37.[36] Azar RC, Crossley FRE. Digital simulation of impact phenomenon in spur gear systems. Trans ASME

1977:792–8.[37] Pfeiffer F, Prestl W. Hammering in gears. in 3eme Congres Mondial des Engrenages et deTransmissions.

1992. Paris, France.[38] Ligier JL, Baron E. Acyclisme et vibrations: applications aux moteurs thermiques et aux transmissions – 1.

Notions de base, 2. Analyses avancees et experimentales [Acyclism and vibrations: Applications to thethermal engines and the transmissions – 1. Basic notions, 2. – advanced analyze and experimentation],Technip; 2000.

[39] Barthod M, Hayne B, Tebec JL. Auditory perception of noise known as ‘‘rattle’’ in gearboxes. Acta Acustica2005;90(1):91–9.

[40] Ligier JL, Baron E. Acyclisme et vibrations: applications aux moteurs thermiques et aux transmissions – 1.Notions de base – 2. Analyse avancees et experimentales, Technip; 2000.

[41] Fremau N, Pilippe-Damagnez C. Numerical and experimental study of gear rattle. Document interneRenault, Direction de la Mecanique.

[42] Barthod M, Tebec JL, Pin JC. Study on the threshold of gearbox rattle noise in automotive. In: Congrescommun SFA/DAGA ’04, 2004, Strasbourg, France.

[43] Fujimoto T, Kizuka T. Predictive calculation of idling rattle in manual transmissions – Based onexperimental measurements of gear vibration occuring in backlashes. SAE, 2003-01-0678, 2003.

[44] Barthod M, Hayne B, Tebec JL, Pin JC. Study of the influence of acyclism characteristics on gearbox rattlenoise. in Inter Noise 2004, 2004, Prague, Czech republic.

[45] Wang Y, Zhongfang T. Sound radiated from the impact of two cylinders. J Sound Vib 1992;159(2):295–303.