1-s2.0-S0003682X06000909-main

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

multi harmonic excitation imposed to the input shaft of the gearbox. Second, dierent gearbox con-gurations were used to characterize the rattle threshold and the rattle noise evolution, in relation to

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

* Corresponding author. Tel./fax: +33 1 44246229.E-mail address: [email protected] (M. Barthod).

Applied Acoustics 68 (2007) 9821002

www.elsevier.com/locate/apacoust0003-682X/$ - see front matter 2006 Elsevier Ltd. All rights reserved.excitation parameters and mechanical gearbox parameters. Third, a simplied model of the rattlephenomenon is drawn, aiming to determine the most signicant parameters to the rattle noise. 2006 Elsevier Ltd. All rights reserved.

Keywords: Gear noise; Teeth impacts; Gearbox; Experiment

1. Introduction

1.1. Context of the projecta Laboratoire de Mecanique Vibratoire et dAcoustique, Ecole Nationale Superieure dArts et Metiers 151,

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

Received 16 January 2006; accepted 26 April 2006Available online 17 July 2006

Abstract

This paper deals with the rattle noise, caused by the uctuation 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 simplied gearbox was designed in order to recreate the rattle noise phenomenon for aExperimental study of dynamic and noise producedby a gearing excited by a multi-harmonic excitationdoi:10.1016/j.apacoust.2006.04.012

been masked. This is the case of the rattle noise, caused by uctuations 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 aect the mechanical behavior of the gearing and do not lead to breakage. Rattlenoise is considered as particularly annoying and has a negative inuence on vehicle inte-rior sound quality.

1.2. Rattle phenomenon

Rattle is an impulsive phenomenon that occurs on unloaded gears which does nottransmit any power. These unloaded gears, free in rotation, can knock each other undersome operating conditions and thus cause rattle noise. Fig. 1 illustrates in a simplied

M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 983way the backlash crossing phenomenon. In the literature [1], several theoretical modelare described, with dierent kind of impacts (elastic or inelastic). Some authors [2,3] havetaken into account axial impact due to the axial play of the gear on the shaft. Theseimpacts will be also highlighted in our study in Section 5.4.

The acyclic excitation on the input shaft of the gearbox is function of the engine tech-nology (four or six cylinders, in line or in V), of the design of the driveline (design of clutchand drive shafts), and function of vehicle running conditions (load conditions and enginespeed). In the case of a four-stroke and four-cylinder engine, since there are two explosionsper revolution, the spectrum of the angular acceleration is in theory composed of enginespeed harmonics H2n.

1.3. Literature review

There are lots of studies on gearbox rattle noise; they deal with many dierent points ofview. The associated literature is considerable. The rattle noise can be studied with a glo-bal point of view or with a more local phenomenon point of view. In these two cases, thereare experimental and numerical studies; the considered excitation being more or lesssimplied.

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 toprovide the inuence of the driveline design choices on rattle noise. The dynamic of thegearing is not well dened, so we will not detail more this kind of study.Fig. 1. Backlash crossing phenomenon.

984 M. Barthod et al. / Applied Acoustics 68 (2007) 9821002Experimentally, the objective is to study the torsional dynamic behaviour of the drive-line (i.e. to measure torsional eigen mode of the kinematic driveline) which corresponds toan excitation amplication and so lead to rattle noise amplication [4,5,1].

In numerical models [69], nonlinearities in gearing stiness, clutch hysteresis and widthof backlash are taken into account. Generally, a rattle noise reduction is due to a clutchdesign optimization [1014].

However, it seems to be necessary to work on both the entire driveline design and thegearbox design [10]. Thus, some studies are focused either on a gearbox or only on onegear pair.

With regard to the prediction of the gearing dynamics with backlash crossing phenom-enon, most of studies deal with simplied models with only one degree of freedom (intranslation, on the line of action).

Some models take into account the gearing backlash and use a mean gearing stiness,constant in time with meshing [1518].

Other models take into account nonlinearities due to backlash and to gearing stinessvariations [19,20,6]. Dogan [3] and Lang and Lechner [2] proposed a model of teethimpacts where axial impacts are distinguished from backlash impacts in a gearbox.

Pfeier presents three methods making it possible to know the dynamics of one or sev-eral gears excited in rattle [21]: the patching method, the point mapping method andthe stochastic method which is a probabilistic approach.

Some studies [2224] deal with jumping phenomenon and with branching link to thenonlinearities of the mechanical system. Other studies deal with problems due to numer-ical resolution of the dynamic equations [25,26].

But in all these models, the excitation is supposed to be sinusoidal and the mechanicalsystem under study is very simplied compared to a real gearbox.

In experimentation, in most case, the incoming acyclic excitation on an isolated gearboxis 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 [4,5,27,28]. In that case, the excitation imposed tothe gearbox input shaft is sinusoidal.

Other studies [2,3,29] use a synchronous tree-phase motor to impose an acyclism com-posed of several harmonics, representative of a four or six cylinder engine. But the inu-ence of the harmonic composition of the excitation is not quantied. In theses studies, onlythe global sound pressure level or the root mean squared casing acceleration are measuredto quantify the rattle noise, the gearing dynamic is not precisely measured.

Otherwise, some authors use more simplied experimental test rigs which give access tothe gearing dynamics, but the system is still very simplied compared to a real gearbox.Pfeier [21,30] works on one gear with only one tooth, excited with an eccentric device.Crocker and Greer [31] and Weidner and Lechner [16] have measured the restitution coef-cient of a tooth impact while Azar and Crossley [32] has studied the contact between twoteeth.

1.4. Objectives

On a vehicle, the gearbox is more or less sensitive to rattle phenomenon. This sensi-tivity is function of three parameters: the excitation (angular acceleration) imposed to the

input shaft of the gearbox (i.e. the acyclism) [30,33], the dynamic response of the internal

M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 985gearbox architecture and then, the vibration transfer to the casing. Here, we are interestedon the inuence of the two rst parameters on rattle noise.

(1) Until now and in most studies, the torsional excitation is very often simplied: onlythe 2nd order harmonic (H2) of the engine speed is considered. A measure on a realvehicle [34] 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 [35] thattaking into account a multi-harmonic excitation has a main inuence on the sonorityof rattle noise produced.

(2) To measure the dynamic response of the internal gearbox architecture, parametersthat have to be taken into account are that ones: gear inertia and backlashes(between unloaded and gearing gears), position of the unloaded gears (on primaryor on secondary shaft), gear reductions and drag torques.

Gearbox rattle sensitivity can be obtained analytically by modelling the transmission.However it is very dicult to consider all the inuencing components such as gears, syn-chronizers, bearings, case, oil and nonlinear properties such as meshing stiness, viscosity.Integrating all plays and backlashes is very dicult. Even though analytic models havebeen fully developed, validity of results has not been checked. Our contribution is anexperimental investigation on the inuence of unloaded gears inertia and backlash.

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

2. Used modelization

The objective is to assess the validity of a very simplied model of the rattle phenom-enon to estimate the sensitivity of a simplied gearbox. We have to show the limitations ofsuch a model and underline the parameters the most important on rattle noise.

For Pfeier [30], rattle in a real gearbox is a cascade process. Such phenomenon is dif-cult to analyse since interaction between gears have to be taken into account for the res-olution of dynamical equations.

The model used is a KelvinVoigt model (Fig. 2), usually used to analyse the backlashcrossing phenomenon between gears [36,37]. It is a simple model with two degrees of free-dom (Fig. 2). It is made of a driving gear and an unloaded gear, whose motion are linkedduring contact phases, or are independent during free-ight phases, when the unloadedgear moves within backlash.

Some modications have been made (i.e. works of Azar and Crossley [32]) in order toavoid numerical discontinuity problems. We have introduced a nonlinear parameter in the

expression of the 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 radiuses of the driving andunloaded gears are respectively R1 and R2 (in m), their inertia around their rotation axesbeing I1 and I2 (in kg m

2), j is the backlash, k is the contact stiness and c is the contactdamping.

The hypotheses of the model are:

Fig. 2. Diagram of the model used.

986 M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 on neutral, no average torque is transmitted, there is only oscillating torque; gears are spur toothed; drag torques are assumed to be constant (Cdrag); the imposed torsional excitation (Cexcitation) is not inuenced by the dynamics of the sys-tem (driving gearunloaded gear).

Modelling is carried out under MATLAB version 6, we have chosen to use the New-mark method to solve the dynamic equations. Values of the parameters used in our modelare estimated by simple mechanical calculation, or experimentally measured or are derivedfrom comparison between numerical results and experimental measures.

3. Description of the experimental setup

3.1. Design specications and realization 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 congurations. We also need a good access of gears

to study their dynamics: sound pressure level, impact amplitude and relative motion ofgears have to be measured, which requires instrumentation on the gears.

Oil in the gearbox is not negligible since it inuences drag torques applied to unloadedgears.

Meisner and Campbell [20] and Weidner and Lechner [16] have measured drag torquesinside a gearbox. Inuence of temperature, viscosity and quantity of oil (linked to theunloaded gear splashing) [31] and so the inuence of the gearbox orientation [3] have beenstudied. Overall, the rattle noise sound pressure level decreased when the drag torqueapplied to the unloaded gears increased. In our study, the input shaft does not rotate,we chose to work on neutral and do not take into account the oil inuence: contactsare 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 impactsbetween gear teeth: all other noise sources should be excluded. Our tests are carried ona simplied gearbox whose gear forks and synchronization mechanisms have beenremoved, only one pair of cylindrical gears with helical teeth remains, and there is nooil (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 shaft

M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 987Fig. 3. Simplied gearbox opened gearbox case.

of the gearbox. An electrodynamic translation exciter is used, tied to the gearbox with acrank 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, similar inamplitude and frequency with those measured on a vehicle. It is necessary to ensure anexcitation from 0 to 1200 rad/s2 root mean squared (RMS), on a frequency range goingfrom 30 to 180 Hz. This range correspond to engine speeds from 900 to 1800 rpm, whererattle noise is signicant, engine noise being too weak to cover it. So, the whole excita-tion mechanism has to be carefully designed. In particular, transmission system has to bewithout backlash and rigid enough to avoid resonances in our frequency range of interest.

988 M. Barthod et al. / Applied Acoustics 68 (2007) 98210023.1.1. Instrumentation

The input shaft and the unloaded gear are equipped with an accelerometer and a non-contacting displacement sensor (eddy current). An accelerometer is attached to thegearbox case. A sound level meter near the gearbox is used to compare the sound pressuresignal of rattle noise from dierent excitation congurations.

All the excitation parameters (frequency, harmonics amplitudes and phases) can be sep-arately adjusted and allow continuous sweep. For example, an excitation device enables usto carry out progressive continuous sweeps of the global excitation amplitude imposed tothe input shaft, whatever the composition of this excitation. That allow us to observe theevolution of rattle in relation to the acyclic excitation amplitude for a given engine 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 amplitudes (in % relatively to the H2 amplitude), their phases (u4and u6) are in relation toH2. In the case of a composite excitation, the frequency known asthe excitation frequency corresponds to the frequency of the 2nd order harmonic (H2).

3.2. Gearbox congurations used

Six dierent congurations of gearbox were used (Table 1). Congurations 13 (namedwith modied inertia) are obtained from the same unloaded gear which successivelyundergoes an increase of inertia (by addition of a disc on a side) then a reduction of inertia(by machining). The modications of inertia are about 50% of initial inertia. Congura-tions 46 (named with modied backlash) are obtained with three dierent unloadedgears. Precise measurements of the backlash were taken using position sensors. The back-lashes were 75, 83 and 100 lm.

Table 1Gearbox congurations used

Number of conguration Corresponding value of inertia and backlash

1 Conguration {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 inertia

6 Unloaded gear with maximum size backlash, initial inertia

M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 9894. Study of the rattle noise threshold

Rattle threshold is dened as the angular acceleration amplitude imposed to the inputshaft of the gearbox (in rad/s2 RMS) from which the rattle phenomenon occurs and ismaintained (i.e. stable in time; that denition of threshold allows us to avoid the inuenceof the relative position of gears before the rst impact).

Threshold is studied in the literature in two manners.The simplest way is to dene rattle threshold as the possibility of contact loss between

two pieces [17,38,16].Rattle threshold index is dened either with clutch parameters [15], or by comparison of

the gears acceleration to drag torque applied to the unloaded gear [11]. That denition stillbeing theoretic and supposes that the acceleration imposed on input shaft is sinusoidal.

In experimentations, rattle threshold is detected by listening, or by measuring vibra-tions of the gearbox case [5,27], or by measuring root mean square angular accelerationof the unloaded gears, or by visualizing a contact loss between teeth [31,1].

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 eared or that is annoying [31,27,13]. Even so, the acyclism amplitude is not necessarycorrelated to the subjective perception of rattle noise [1].

Such conclusions are in agreement with our results, but works presented in the litera-ture still consider a sinusoidal excitation.

4.1. Rattle threshold in relation to excitation parameters

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

We have previously showed [39] 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 excitationcongurations.We have used the experiment designmethod in order to estimate the inuenceof the amplitude and phase of the 2nd, 4th and 6th order harmonics on the rattle threshold.

Thus, we have proved 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 hasnally little inuence on the occurrence of rattle.

This result can be explained by the fact that the 4th and 6th order harmonics have littleinuence on the kinetic energy. For example, when the amplitude of the H4 harmonic goesfrom 20% to 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, 45

and 60 Hz, and for the three dierent unloaded gears inertia (in that case, the backlash is

the more signicant the corresponding acceleration must be. For condential reasons,

990 M. Barthod et al. / Applied Acoustics 68 (2007) 9821002numerical values are not given.

5. Study of the rattle noise evolution

Let us suppose rattle started, we are interested in the evolution of rattle, in function ofthe frequency and the amplitude of the acceleration imposed to the input shaft of the gear-box, then in function of gearbox geometrical parameters.

5.1. Transfer path between unloaded gear and gearbox case

For each experimentation, we simultaneously record the rattle noise signal (with amicrophone near the gearbox), the gearbox case vibration, the unloaded gear accelerationand the input shaft acceleration.

We have previously proved [35] that, for an excitation with constant spectral and tem-poral parameters and during a progressive sweep of the root mean squared excitationamplitude, the root mean squared (RMS) acceleration of the unloaded gear is well corre-lated by a linear relationship with the acoustic pressure of the rattle noise produced. Inother words, the rattle noise sound pressure can be rather simply estimated from theRMS value of the impacts on the unloaded gear, and vice versa, as shown in Fig. 4. Thatresult can also be found in Fujimoto and Kizuka works [40], or in Pfeier [30] and Swa-dowski [15] works: a good estimation of the rattle noise sound pressure level is given by themean impulse force measured on the unloaded gears of the gearbox.

Curves of Fig. 4 are obtained by post processing of the data recorded during progres-constant and equal to 120 m). We show that the evolution of the threshold with theunloaded gear inertia is linear. The weaker the inertia is, the more signicant the acceler-ation necessary to start the rattle phenomenon is.

Threshold is obtained for a constant kinetic energy imposed on the input shaft of thegearbox. With regard to two gears with inertia noted I+ and I in the following equations:

1

2I _h2I

1

2I _h2I 1

so : Ih2I Ih2I 2with _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 prove that relationship on kinetic energy (butfor condential reasons, the numerical values are not given). This conrms the fact thatthe rattle threshold (for one gearing) is obtained for a constant kinetic energy introducedinto the system, the value of this energy depending on the gearbox architecture.

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

The more signicant the backlash is, the higher the rattle threshold is. To start rattle, itis necessary 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 thussive and continuous sweeps of the excitation amplitude for sinusoidal excitation at 30, 45

M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 991or 60 Hz. For each sweep (0 to more than 1000 rad/s2 RMS), we calculate the RMS valuesof 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 too evolutions to build Fig. 4.

5.2. Inuence of excitation parameters on rattle

Let us consider a gearbox (or only a gearing) excited by a sinusoidal angular excitation.Literature gives the evolution of the produced rattle noise in relation to excitation fre-

quency and excitation amplitude: the sensitivity curves.

Fig. 4. Evolution of the RMS acoustic pressure of the rattle noise (in Pa) in relation to the RMS unloaded gearacceleration (in m/s2).Overall, the sound pressure level increased with the engine speed, corresponding to exci-tation frequency, and with the acyclism level, corresponding to excitation amplitude.

Weidner and Lechner [16] studies a ball which moves in a U-shaped part. He gives (byexperimentation and mathematical model) the evolution of the sound pressure level inrelation to the excitation amplitude (sinusoidal).

With regard to only one gearing, Pfeier [10] links the rattle noise level to the geomet-rical parameters of the gearing and to excitation parameters.

Chae et al. [41] works on real gearboxes excited by a sinusoidal acyclism. He proves thatgearbox sensitivity to rattle is function of the gearbox design, even if the global shape is thesame for all gearboxes.

Dogan [3] and Forcelli et al. [42] measure the rattle noise level produced on real gear-boxes under sinusoidal or multi-harmonic acyclism. But harmonics inuence is notobserved in details.

5.2.1. Rattle produced in the case of a sinusoidal excitation

Results obtained for a sinusoidal excitation are presented in Fig. 5. We give the evolu-tion of the RMS acceleration of the unloaded gear (impact due to backlash crossing ofteeth, expressed in m/s2) in relation to the RMS acceleration imposed to the input shaft(expressed in rad/s2). The dierent curves correspond to several progressive sweeps in

992 M. Barthod et al. / Applied Acoustics 68 (2007) 9821002RMS excitation amplitude (increasing and decreasing) for sinusoidal excitations at 30, 45and 60 Hz.

The dispersion of measurements is weak enough (about 15% for an excitation at800 rad/s2 RMS) to clearly release the inuence 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 proportionally withthe RMS excitation amplitude. Besides, for the same excitation amplitude, the higher is theexcitation frequency, the higher is the rattle noise. It means that an excitation level whichis not critical (i.e. giving a low rattle noise) to weak driving regime can become criticalif the engine speed increases.

Fig. 5. Evolution of the RMS acceleration measured on the unloaded gear in relation to the RMS accelerationimposed to the input shaft. Sinusoidal excitations at 30, 45 and 60 Hz.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 inuence.

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. 6), 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 sweeps (H2 at 30 or 60 Hz,with various percentage of harmonicH4 and with a phase = 0). Every curve is indicated bythe corresponding excitation imposed to input shaft and is noted: frequency of harmonicH2 relative amplitude of H4 (in % with regard to that of H2) phase of H4 (with regardto H2). Curves in dotted line remind for comparison the average measures obtained forsinusoidal excitations at 30 and at 60 Hz.

M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 993We can see that the introduction of harmonics H4, do not modify the global evolutionof curves 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. 6. 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.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.

Remark. From a perceptive point of view, we have already proved the great inuence ofspectral and temporal parameters of the acyclism imposed to the gearbox on the producedrattle noise (level and sonority) [35,43].

5.3. Inuence of geometrical gearbox parameters on rattle

With regard to one gearing, Pfeier [10], links the rattle noise level to unloaded gearinertia and to gears radius. Weidner and Lechner [16] and Lang and Lechner [2] show thatamplitude of the impact on unloaded gear (and so rattle noise level) increases when thebacklash increases, even if the temporal shape of the impact signal changes.

With a three-degrees-of-freedom model, Wang and Glover [18] proved that the rattlenoise level increases when backlash increases and decreases when the unloaded gear inertiaincreases. Theses results are in agreement with our experimental results. Wang also showsthat there is an interaction between these parameters.

5.3.1. Unloaded gear inertia inuence

We compare gearbox congurations no. 1, 2 and 3 (initial, increased and decreased iner-tia). Fig. 7 gives for these three congurations, the evolution of the RMS unloaded gear

Fig. 7. Inuence of the unloaded gear inertia for an excitation at 45 Hz.

994 M. Barthod et al. / Applied Acoustics 68 (2007) 9821002acceleration 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 proves that for the same excitation amplitude, a reduction in the unloadedgear inertia gives more signicant impact RMS amplitude. In other words, decreasing theunloaded gear inertia tends to increase the sensitivity of the gearing to acyclism.

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. 8).

For the same excitation amplitude, the impact speed of the unloaded gear is smaller inthe case of the initial inertia conguration than in the case of the decreased inertiaFig. 8. Temporal signal (accelerations, speeds and displacements) initial inertia and decreased inertia.

conguration (whatever the unloaded gear is or not axially maintained i.e. Section 5.4) thisexplains a louder noise produced [44].

5.3.2. Backlash inuence

We now compare gearbox congurations no. 4, 5 and 6.Fig. 9 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 data wereobtained for excitations at 45 and 60 Hz.

It appears that the wider the backlash is, the higher the RMS value of the impact is(even if the eect is weak). This is explained by the fact that an increase in the backlashgives a longer free ight phase, and, actually, a more signicant speed dierence betweengears, just before the impact.

5.4. Description of an unsteady behaviour

Rattle phenomenon is not always a stable phenomenon. With regard to a ball movingin a U-shaped part with a sinusoidal displacement, Weidner and Lechner [16] distinguishesdierent kind of relative movement (periodic, chaotic).

Pfeier [19] uses a model of one gearing under a sinusoidal excitation and gives dia-

M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 995grams of the unloaded gear position inside the backlash with impact phase. He underlinesperiodic, quasi-periodic or chaotic dynamic and bifurcations [10].

Dai and Singh [24] dene the periodicity ratio (based on the number of points thatoverlap on Poincare diagram) to distinguish periodic and chaotic dynamic comportmentof the system. Blazejczyk et al. [45] dene intermittency as a chaotic dynamic comportmentcharacterized by a periodic comportment interrupted by short chaotic phases. That com-portment can occur in a gearbox and explains the irregular characteristic of the rattle noisesometimes perceived.Fig. 9. Backlash inuence for an excitation at 30 Hz.

996 M. Barthod et al. / Applied Acoustics 68 (2007) 9821002Fig. 10. Visualization of a usual and then an unsteady behaviour.Actually, for a strong level of excitation imposed on the input shaft, a very irregularrattle noise is sometimes obtained. The passage to a behaviour named unsteady seemsto be random and can be observed by listening to a rattle noise. We notice an increase ofthe noise level and dierent rhythms of impacts, on the unloaded gear acceleration signal,or on the casing vibration even if the excitation parameters are constant. Fig. 10 gives tem-poral signals of the unloaded gear acceleration measured for a steady then anunsteady 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 beexplained 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 conrmed 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 conguration in order to optimizerepeatability of measurements: the axial play has been suppressed.

6. Comparison of the results experimental and numerical results

6.1. Comparison of temporal shape

Figs. 11 and 12 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. Wesuccessively visualize: accelerations of the input shaft and unloaded gear (in m/s2), relative

M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 997displacement 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. Inuence of excitation parameters on rattle

Fig. 13 shows the experimental results of Fig. 5 (curves in thin lines), on which resultsgiven by the model are superimposed (bold curves, each point corresponding to a compu-tation conguration). The general assessment of this comparison is highly satisfactory.

Fig. 11. Temporal shape calculated for H2 at 30 Hz, excitation at 700 rad/s2 RMS.

Fig. 12. Temporal shape measured for H2 at 30 Hz, excitation at 700 rad/s2 RMS.

Except for an excitation at 30 Hz where calculation is approximately 25% higher, the val-

Fig. 13. Inuence of excitation parameters on rattle in a simplied gearbox comparison of measured andcalculated data.

998 M. Barthod et al. / Applied Acoustics 68 (2007) 9821002ues given by the model are in the dispersion interval of measurements on test rig. However,it should be noted that the rattle threshold is not reproduced in the model, because of theuncontrolled initial conditions.

6.3. Geometrical parameters inuence

The following gures show the predictions of the model concerning the inuence of theunloaded gear inertia and the backlash.

Fig. 14 gives the evolution of the RMS acceleration of the unloaded gear for a constantinertia (initial inertia) and three dierent backlashes (60, 100 and 160 lm). The variationFig. 14. Inuence of the backlash on rattle (constant inertia) numerical results.

range 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 three dierent inertias: 0.00098, 0.00149 and 0.00225 kg m2

(Fig. 15). Used inertia values correspond to the ones used for measurements. Comparisonwith experiment can be done. For example, Fig. 16 compares experimental results (linecurves), and numerical result (points) in the case of a sinusoidal excitation at 30 Hz. We

Fig. 15. Inuence of the unloaded gear inertia on rattle (constant backlash) sinusoidal excitation at 30 Hz.

M. Barthod et al. / Applied Acoustics 68 (2007) 9821002 999Fig. 16. Comparison of measured and calculated data - Inuence of the unloaded gear inertia on rattle: evolutionof the RMS acceleration of the unloaded gear for a constant backslash (120 lm) and 3 dierent inertias:

0.00098 kg.m2, 0.00149 kg.m2 and 0.00225 kg.m2.

1000 M. Barthod et al. / Applied Acoustics 68 (2007) 9821002have 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 a Kel-vinVoigt model is satisfactory. Excitation parameters and geometrical parameters inu-ences can be well found, qualitatively and quantitatively. The noted dierences 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 simplied and the fact that the secondary shaft mean angular speed is nottaken into account.

A KelvinVoigt model is sucient if we consider only one gearing but can not be usedin the case of a real gearbox.

7. Conclusions

In this article, an experimental study of the rattle noise phenomenon is realized on asimplied gearbox and allows us to assess the validity of a simple model as KelvinVoigtapplied 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 specication 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 get interested in the rattle threshold (i.e. excitation conditions imposed to gearboxthat cause rattle to occur) and in the inuence of excitation parameters and geometricalgearbox parameters on rattle.

About rattle threshold, we have proved that, for a constant gearbox conguration,threshold is, at rst, linked to the kinetic energy imposed to the input shaft of the gearbox.

In other words, the spectral composition of the acyclism has little inuence. Thresholdincreases when backlash increases and decreases when the unloaded gear increases.

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 jumpsphenomenon, i.e. fast variations of rattle level and/or rattle sonority.

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

Comparison of numerical results and experimental results is overall satisfactory. Theinuence of dierent parameters can be well found, qualitatively and quantitatively, witha simple KelvinVoigt model.

Acknowledgements

This work is supported by RENAULT. The authors would like to thank the Mechanical

Direction of Renault and more particularly the 66126 NVH GMP acoustics department.

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1002 M. Barthod et al. / Applied Acoustics 68 (2007) 9821002

Experimental study of dynamic and noise produced by a gearing excited by a multi-harmonic excitationIntroductionContext of the projectRattle phenomenonLiterature reviewObjectives

Used modelizationDescription of the experimental setupDesign specifications and realization of the test rigInstrumentation

Gearbox configurations used

Study of the rattle noise thresholdRattle threshold in relation to excitation parametersRattle threshold in relation to gearbox parameters

Study of the rattle noise evolutionTransfer path between unloaded gear and gearbox caseInfluence of excitation parameters on rattleRattle produced in the case of a sinusoidal excitationRattle produced in the case of a multi-harmonic excitation

Influence of geometrical gearbox parameters on rattleUnloaded gear inertia influenceBacklash influence

Description of an unsteady behaviour

Comparison of the results - experimental and numerical resultsComparison of temporal shapeInfluence of excitation parameters on rattleGeometrical parameters influenceUtility and limitations of that model

ConclusionsAcknowledgementsReferences