Multi-body modelling of timing belt dynamics

Multi-body modelling of timing belt dynamics

M Callegari1 F Cannella1 and G Ferri21Department of Mechanics University of Ancona Ancona Italy2Dayco Europe SrL Via Gorizia Chieti Scalo (CH) Italy

Abstract Although timing belt drives have recently been increasingly used in mechanical design theirbehaviour is still considered to a large extent to be unpredictable especially under varying operativeconditions The acoustic emission of the transmissions above all has been thoroughly investigated in pastyears but noise still represents an unresolved problem for many applications and a concern for beltdesigners therefore the availability of good predictive models would be very useful for both design andapplication phases The present work describes a multi-body numerical model that has been developed forthe characterization of the dynamic behaviour of timing belt transmissions with the nal goal of assessingthe acoustic radiation of a given design Modelling and simulation have been performed by means ofcommercial software packages but more additional programming was required to obtain dynamic modelscapable of simulating the complex behaviour of toothed belt transmissions Several experimental tests havebeen performed to identify the many parameters that in uence system dynamics and to validate the resultingcomputer aided engineering (CAE) model

Keywords timing belts lumped-parameter dynamic models mechanical vibrations acoustic radiationmulti-body simulation tools automotive applications virtual prototyping

NOTATION

BEM boundary element methodCAE computer aided engineeringfn frequency of transverse vibration (Hz)FEM nite element methodFFT fast Fourier transformh tooth thickness on pitch circle (mm)l length of the belt span (m)m gear module (mm)MSS mechanical systems simulatorn sprocket angular velocity (rmin) order of vibra-

tion modeSLDV scanning laser doppler vibrometerT belt tension (N)vi impact velocity (mms)z number of teeth

r belt linear density (kgm)

1 INTRODUCTION

Nowadays regulations in the automotive eld are increas-ingly focused on user safety and machine design has tocomply with increasingly strict parameters in order to meetthe related requirements Moreover companies especially inthe automotive sector have to keep up with a shorter time-to-market in order to satisfy new customers and to winlarger market shares The advent of the new economy puts afurther stress on such trends and highlights the need forvirtual prototyping tools able to grant shorter developmenttimes and assess beforehand innovative solutions In the eld of power transmission for automotive applications forinstance much effort has been expended in past years onstudying the noise characteristics of the transmission itselfbut in many cases the acoustic emission still represents anunresolved problem and a concern for the designer This isemphasized even more by the fact that a great deal of efforthas recently been put into reducing noise from enginestyres covers etc in order to optimize vehicle cabinacoustics and therefore noise from other sources becomesmore remarkable and annoying to car passengers

Within this context the availability of good predictivemodels of the acoustic emission of timing belt transmissionswould be very useful for both design and application phasesThe present article describes a detailed numerical model thathas been developed for the characterization of the dynamic

The MS was received on 7 March 2002 and was accepted after revision forpublication on 9 October 2002

Corresponding author Department of Mechanics University of Anconavia Brecce Bianche 60131 Ancona Italy

63

K01402 IMechE 2003 Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics

behaviour of timing belt transmissions with the nal goal offorecasting the acoustic radiation of a given design andsuggesting possible design changes In research previouslydeveloped at the University of Ancona [1] synchronous beltnoise has been analysed by processing vibration data andthe method has been validated by comparing the results withexperimental values obtained from acoustic intensitymeasurements The work described here represents a furtherstep of the research that outlines a fully numerical procedurewhere the vibration data analysed through the acousticmodel are obtained from belt transmission design bymeans of a multi-body model of the mechanical equipment

11 Literature survey

Synchronous belts are very common for transmitting powerand motion in mechanical systems and they are oftenpreferred to more expensive and complicated drive compo-nents such as chains gears or linkages The exibility ofbelts allows them to absorb shock loads but on the otherhand the related compliance plays an important role withrespect to large amplitude vibrations and noise problems[2ndash5] which represent a key concern especially for appli-cations in the automotive industry Unfortunately the analy-sis of belt noise is a rather dif cult task and its correlationwith belt mechanical vibrations is an even more complicatedproblem in fact the running noise generated in a powertransmission belt can appear in a wide variety of types andmay be caused by several generating mechanisms as shownby Kubo et al [6] and commented upon in more detailbelow In addition things are made more dif cult by the factthat not all the noise detected near a belt drive arises fromthe belt itself for instance in the case of a belt operatingin the power transmission drive of an internal combustionengine bearings fans and other mechanical pieces ofequipment generate interfering noise that may be taken forbelt noise

A great amount of literature is available on the generalsubject of mechanical vibrations and many studies have beenpublished in past years on the dynamics of at belts orV-belts also with the development of powerful non-linearmodels [7] recently interesting models have been proposedin the related eld of chain drive dynamics [8] Few inves-tigations by contrast have been performed on the noisegenerated by timing belts and only some of these have beenbased on dynamic modelling of the transmission system [9]Among them several models in order to study longitudinalvibrations [10] or transversal vibrations [11 12] consider thetransmission line divided into belt spans each one treated likea string and apply to them the basic equations of the beltsOnly a few authors use lumped-parameter models such as theone proposed here Gerbert et al [13] developed a model forquasi-static loading Uchida et al [14] studied the loaddistribution for different engine speci cations Karolevand Gold [15] assessed the effects of variable torques andvalidated experimentally their models and Kagotani et al

[16] dealt with the speci c case of helical timing beltsRecently Johannesson and Distner [17] proposed a multi-body model quite similar to the one described here onlyneglecting tooth mass and without using linear dampers butconcentrating on dry friction history which greatly affectsbelt expected life Since multi-body models can be veryaccurate even if relatively simple this particular kind ofapproach was chosen for the present work [18]

The present authors claim that to their knowledge themodel described in this paper is the rst one of this kind tostudy the transversal vibrations of toothed belts by means ofnumerical analysis It must be noted too that very fewstudies focus their attention on the relationship betweennoise and vibration and even fewer experimental studies areavailable on the same subject [19] This is probably due tothe dif culty of measuring both noise and vibrations in thecase of belt transmission lines the vibration velocity in factcan hardly be measured at the required high number ofpoints while the belt is running by using conventionaltransducers The identi cation of belt noise on the otherhand is made dif cult by the presence of many disturbingnoise sources existing in the engine moreover a high spatialresolution of the measurement technique is needed in orderto collect useful and signi cant information The relatedwork [1] has characterized the vibroacoustic behaviour ofsynchronous belts by laser velocimetry and has been usefulin the phase of tuning the present multi-body model Thenoise information from both the experimental measurementsand the dynamic model has been processed by a boundaryelement modelling (BEM) tool to compare the results andgive a qualitative assessment of the performance of an actualdesign as explained in the following sections

12 Main causes of belt noise

Noise radiation of toothed belts is due to several differentcauses among which are [6]

(a) impact generated by collision of the belt tooth againstthe bottom land of the sprocket at the beginning ofmeshing

(b) impact generated by collision of the sprocket tooth tipagainst the bottom land of the timing belt at thebeginning of engagement

(c) collision between the anks of the two teeth at thebeginning of meshing

(d) transverse and torsional vibrations of the belt(e) vibrations of the pulleys(f) air ow between belt and pulley(g) frictional noise coming from contact between fabric

and steel

Among these causes two kinds of sound are far morepowerful than the others [20] the impact generated by theengagement between sprocket tooth tip and belt bottom landand the transverse vibrations generated by the belt whenresonant conditions are induced by meshing frequency

Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics K01402 IMechE 2003

64 M CALLEGARI F CANNELLA AND G FERRI

The impact velocity vi (mms) can be estimated by thefollowing approximate expression

vi ˆ p30

p iexcl hm

sup3 acutemn iexcl p

180p iexcl h

m

sup3 acute3mnz3

Dagger cent cent cent hellip1dagger

where n is the sprocket angular velocity (rmin) m is themodule (mm) z is the number of teeth and h is the estimatedtooth thickness (mm) on the pitch circle (approximately thelength of the tooth tip land of the wheel) Equation (1)shows that the impact velocity increases with belt speed andthe polygon effect of the wheel It is seldom affected by thenumber of teeth and is determined mainly by the module Inthe case of usual toothed belts the impact velocity is alsoaffected by transverse vibrations which in turn depend onthe exural rigidity and tension of the belt itself

As for the sound generated by transverse vibration of thebelt it must be pointed out that it is a forced vibration due tothe polygon effect of the wheel Therefore when the wheelapproaches an angular velocity yielding a meshing frequencythat coincides with the natural frequency of the transversevibration of the belt such vibration becomes very largeKubo et al [21] and then Tokoro et al [22] showed that thetransverse vibration of the belt becomes stronger along withincreasing belt tension and decreasing bending stiffnessTherefore if both belt tension and natural frequency arehigh the noise of transverse vibration of the belt becomesvery large and masks all other noises

The present research limited the investigation only to thenoise coming from the transverse vibrations of the belt spanand the dynamic model has been built accordingly Withinthis limitation all the most important causes of belt vibra-tions during operative conditions have been taken intoconsideration and the possible effects of their variation onsystem behaviour have been duly assessed

2 MODELLING AND SIMULATION

21 Multi-body model of the transmission line

A lumped-parameter model based on a discretization of thebelt has been de ned to emulate the dynamic behaviour ofthe transmission each tooth is represented by a rigid bodywith suitable mass and inertia (Fig 1) connected by springndashdamper constraints to the adjacent teeth and to the pulley

the wheels and sprocket by contrast have been modelled bymeans of rigid bodies In order to manage properly themeshing phase and the polygonal effect the geometry of theteeth (of both belt and wheels) has been duly reproducedOn the other hand to facilitate modelling and simulation aplane model has been developed and linear componentshave been chosen to model belt elasticity (linear springs)and internal dissipations (linear dampers) while Coulombdampers have been chosen to model contact frictionJohannesson and Distner [17 23] showed that linear springsand dampers can be properly used to model the character-istic response of the rubber tooth in the load ranges of a carengine since the major source of non-linearity is the varyingcontact geometry and not the rubber non-linearity itself

The described model is therefore composed of the follow-ing rigid bodies (Fig 2)

(a) a driving sprocket connected to the crankshaft respon-sible for motion generation

(b) one or more driven pulleys each one connected to acamshaft

(c) mechanical or hydraulic tensioners in order to providethe desired preload to the belt (idler pulleys can be usedinstead)

(d) belt teeth which can be joined at will to build up thetransmission line

The axial reaction was modelled by a spring and a damperit must be noted that the stiffness of this spring can be easilyobtained because it is mainly due to the properties of thecord which is generally made of glass bre or steel threadand therefore can be easily measured experimentally Thebending behaviour on the other hand is caused by both thecord and the rubber and was modelled through rotationalsprings and dampers

One of the most important geometric parameters of themodel is the relative position of the revolute joint connectingtwo adjacent teeth since the meshing impacts exciting thebelt and producing the transversal oscillations actually takeplace on the fulcrum itself Two contact forces acting on abelt tooth have been considered on the ank (with theconjugate ank of a pulley tooth) and on the tip (with thebottom land of a toothed pulley) Moreover the contactforces acting at the belt bottom land (with the sprockettooth tip) and on the back side (with an idler pulley or atensioner) have also been considered Therefore proper setsof constraints have been placed at these points to model the

Fig 1 Lumped-parameter model of two adjacent teeth

K01402 IMechE 2003 Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics

MULTI-BODY MODELLING OF TIMING BELT DYNAMICS 65

belt compliant behaviour during the meshing (mainly radialand axial deformations) of course such constraints had to beactivated only during contact phases by means of triggersthat were turned on according to the periodic motion of thetransmission The forces acting on each of these points wereevaluated a priori through a nite element method (FEM)simulation of teeth meshing under the same load conditionsas the belt under examination (Fig 3) the intensity the lineof action and the point of impingement of the internalreaction force proved to be parameters of great importance

The model can be driven by external kinematic ordynamic constraints which allow the user to specify eitherthe rotational velocity or the torque applied to the crankshaftand to the camshaft(s) it is also possible to de ne perturb-ing components (up to the fth harmonic) in order toreproduce realistic behaviour of the belt drive

Some of the described parameters were known a priorifrom manufacturerrsquos data obtained by experimental tests or

found from the reference literature Other parameters weretotally unknown though they were undoubtedly needed toimplement the model In this case their values were(roughly) identi ed by running several test case simulationsand by comparing the experimental results with the numer-ical data as shown below

As for the geometry of the tooth (pitch height width cordposition etc) this was imported from computer aideddesign (CAD) data provided by the belt manufacturerwhile the mass properties (eg mass position of centre ofmass inertia etc) were found accordingly by using thecapabilities of the computer aided tools mentioned Ofcourse all the data that were needed completely to de nethe transmission line (eg the number of teeth of the beltthe dimensions and the positions of the pulleys and of theidlers) were speci ed through an input interface and thenglobal congruence was checked (eg on the length of thebelt or on the feasibility of the path)

Fig 2 Model of a simple transmission line

Fig 3 FEM model of two adjacent teeth

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66 M CALLEGARI F CANNELLA AND G FERRI

22 Simulation tools

The model of the complete belt transmission was implemen-ted by means of a multi-body code such kinds of tool arelargely used in engineering to simulate the motion of complexmechanical systems the constitutive parts of which can beseen as a set of rigid or exible elements connected to eachother by several types of constraint ADAMS was chosen forthis project because no commercial package already providedall the necessary features for the implementation of thedescribed model so ease of extensions and open codestructure were considered a priority in the choice In parti-cular the version of ADAMS that was used was not able tomanage the contact between bodies and therefore it wasnecessary to describe this kind of interactionusing an externalFORTRAN subroutine able to control the occurrence of animpact between belt and pulleysand in this case to reproduceboth the normal reaction and the friction force that originatedfrom the contact itself The realization of this macrocode wasmade possible because ADAMS makes available at everysimulation step the values of the position velocity andacceleration of each part composing the model thereforesince the geometry of the system was known all the neededkinematic and dynamic data were obtained Such a feature ofthe tool was essential in order to know and therefore tocontrol the movements of the different parts of the model

At the end of dynamic simulation it was possible toobtain detailed information on system behaviour andalso to assess the values of some important data thatowing to the small size of the teeth or the impossibility ofapplying sensors could not be observed experimentally forinstance Fig 4 plots the time history of the contact forcebetween tooth and pulley during two complete revolutionsof the driving sprocket

The ultimate aim of the simulations was to assess thetransversal vibrations of the belt during its constant rotationin order to use them as an input for SYSNOISE a programbased on the BEM technique able to give a prevision of theacoustic power emitted by the belt Once again interfacesoftware was written to extract internal data that were notavailable to the user in the standard con guration of thesimulation tool In particular simulation results are normallyprovided by ADAMS according to a Lagrangian approachsince the time history of selected (travelling) points of thebelt is made available in the local reference frame Onthe other hand the assessment of acoustic radiation canonly be performed on the basis of data arranged in aEulerian fashion since the vibration of selected points thatare in xed positions must be evaluated in the globalreference frame (see Fig 5) Of course this step was crucialalso to validate the model and therefore to compare simula-tion results with the vibration measurements carried out onthe test rig set up in the laboratory as shown below

Since ADAMS libraries do not allow the user to access thiskind of data it was necessary to write an external FORTRANsubroutine once de ned the selected output points (in xedpositions of the free span relative to the global frame) and thedisplacements velocities and accelerations of the travellingteeth were stored for each selected position

The two simulation tools were interfaced by means of apurposely developed code written in a LabView environ-ment this program processed the output le created by themulti-body analysis in two steps so as to create a le in theformat required by the BEM package

1 In the rst step the program reads the output le generatedby ADAMS and builds a three-dimensional matrixcontaining the velocity time history of all the teeth

Fig 4 Time history of the contact force between tooth and pulley during two complete revolutions

K01402 IMechE 2003 Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics

MULTI-BODY MODELLING OF TIMING BELT DYNAMICS 67

2 In the second step the program displays the vibrations ofthe belt spans (at selected points) and performs the a fastFourier transform (FFT) of the raw data Once afrequency value for the analysis of acoustic radiationhas been selected (suitable values have been obtained byexperimental tests) the program writes two les (for boththe real and imaginary part of the signal) in a format thatcan be read by SYSNOISE Of course this output datamust be sorted in such a way that a correspondencebetween the target points in ADAMS and the nodes ofthe mesh in SYSNOISE is kept

3 VALIDATION OF THE MODELS

31 Description of the experimental set-up

In order to validate the model of the belt drive laboratorymeasurements were performed on the belt stand illustratedin Figs 6 and 7 The box-shaped structure housed an electric

motor a cylinder head of a 16-valve internal combustionengine immersed in an oil bath and a cooling system Themotor and cylinder head were mounted on two separatevertical steel beams xed to the stand base Tables 1 and 2show some characteristics of belts and transmission linesthat were taken into consideration in the experimentsaccording to the different tested layouts It has beenshown by Tokoro et al [24] that great dynamic tensilevariation occurs in synchronization with rotations of thecamshaft pulley and belt (mainly caused by the valve driveand to a minor extent by pulley eccentricity and belt cordnon-uniformity) Therefore in order to reproduce as well aspossible the torque oscillations caused by conventionalautomotive engines a real engine head dipped in lubricatingoil was used driven at the desired rotational velocity by aconventional three-phase four-pole induction motor with aspeed variator In this way the torque variations of thecamshaft pulley depended directly on the camshaftdynamics moreover the immersion in the oil bath alsomade it possible to reduce the noise emitted by the test rig

To set the initial tension of the toothed belt a tensioningidler was used and in order to reach the prede ned initial

Fig 5 Time history of the transversal vibration velocity of a point in the xed frame

Fig 6 Testing rig con gured in a typical layout with measure-ment points Fig 7 Testing rig with measurement system

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68 M CALLEGARI F CANNELLA AND G FERRI

tension T0 (N) the well-known formula of vibrating cordswas applied

fn ˆ n2l

Tr

s

hellip2dagger

where l is the length of the belt span (m) r is the belt lineardensity (kgm) n is the order of the mode and fn is thefrequency of the related transverse vibration (Hz)

All the tests were performed using idle pulleys mountedon eccentrics to preload the belt This choice was madebecause spring-based tensioningsystems would have compli-cated the model by introducing dif cult-to-characterizecompliance moreover this solution is the most commonlayout in modern industrial and automotive applications

32 Experiment sound power estimation andtransverse vibration measurement

Three different kinds of measurement were performed bymeans of the instrumentation sketched in Fig 8 andsummarized in Table 3

(a) total sound power radiation that is all the noisegenerated by all moving parts (belt pulleys idlers etc)

(b) vibrations of each component of the rig (belt pulleysidlers etc)

(c) vibrations of the belt only

Acoustic intensity measurements were performed by a differ-ent working group in the same Department of the Universityof Ancona [25] and some results are commented upon herefor convenience The same reference also provides details onthe measurement techniques used for the identi cation of themodels and an assessment of the related uncertainty

Some noise measurements were performed using acousticintensity techniques in order to collect sound power spectraunder operating conditions and to compare them with the

numerical predictions of the BEM model The sound inten-sity was measured by a two-microphoneprobe at a number ofpoints aligned on a grid built around the belt drive using nethreads The grid consisted of ve lateral surfaces of arectangular parallelepiped divided into square elements of10 mm side length such measurement surfaces togetherwith the steel plate lying beneath the belt drive formed aclose surface that made it possible to estimate the total soundpower emitted by the belt drive as the surface integral of theacoustic intensity measurements (according to ISO 9614-1)

Useful information was extracted by the power spectra ofthe signal a typical sound power spectrum is illustrated inFig 9 where the reference value used for the description ofthe acoustic power levels in dB was 10iexcl12W Several peaksappeared in the frequency range of interest resulting fromthe multiplicity of noise sources present in the test standGenerally these peaks coincided with the harmonics of therotating frequency with the meshing frequency and with thenatural frequencies of the belt strands

By varying some parameters of the considered transmis-sion it was shown that at meshing frequency the power levelincreases by 4ndash6 dB every time the speed doubles (inaccordance with theory since the sound power is directlyproportional to the square of vibration velocity) and that thepower radiation increases by more than 3 dB per 10 mmincrease in belt width

Table 1 Main parameters of the belts used for the tests

Pro letype

Pitch(mm)

Tooth height(mm)

Tooth width(mm)

Belt width(mm) Cord

Body andteeth Tooth

Parabolic 9525 35 3 25 Spirally woundglass-made

Chloroprenecompound

Nylon facing fabric

Table 2 Main parameters of the transmission lines for the testedlayouts

Layout 1 Layout 2 Layout 3

Number of belt teeth 132 162 217Number of camshaft

pulley teeth48 48 48

Number of crankshaftsprocket teeth

22 22 22

Number of idle toothed pulleys 0 1 1Number of idlers 1 2 1

Fig 8 Measurement chain

K01402 IMechE 2003 Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics

MULTI-BODY MODELLING OF TIMING BELT DYNAMICS 69

A boundary element analysis involving a vibrating struc-ture requires the de nition either numerical or experimen-tal of the vibration velocity values of the equipment undertest In the present case transverse vibrations were measuredusing a scanning laser Doppler vibrometer (SLDV) Such aninstrument allows a fast scanning of the test surfaces so thata great number of points were measured over each belt span(every 4 mm along the axial direction and every 8 mm alongthe transversal direction)

Attention was paid to the interference in the determina-tion of the normal component of the velocity This disturb-ing effect is caused by belt movement in the axial directionand can be reduced by keeping the incident laser beam asmuch as possible orthogonal to the running belt Thereforethe belt drive under test was arranged with vertical shaftaxes which made it possible to position the laser head faraway from the belt and then to restrict the vibrometerscanning angle On the other hand as a result of an increasein the distance of the measurement surface from the laserhead there was a reduction in the optical energy of thebackscattered laser beam appearing to the vibrometer opticsa fairly good compromise was found by positioning the laserhead at a distance of about 2 m from the belt span

As in the case of noise measurement several peaks werepresent in the measured spectra and they could be asso-ciated with harmonics of the fundamental frequency ofrotation and in particular with the meshing frequencyThe expected belt span resonance frequencies were foundin the spectra If each belt span is considered as a stringclamped at the edges such frequencies can be evaluated bysimply reusing equation (2) for each span

In general all the detected natural frequencies wereslightly lower than their theoretical values This is probablydue to the effect of the axial velocity which as shown inreference [26] makes the natural frequency of the axiallymoving belt span lower than that of the stationary belt Insome cases only a few peaks were clearly evident in thespectra it emerged that in this case the meshing frequencycoincided with one nth mode of one belt span Thisresonance phenomenon is expected to occur under precisecombinations of the test conditions and in this case thevibration velocity pro le over the belt span at meshingfrequency becomes a mode shape and the transverse vibra-tion amplitudes are stronger than under other resonanceconditions with the consequent in uence on noise radiation

Finally the results were processed through the programSYSNOISE The vibration velocities measured on the beltspans were the input data necessary to predict the acousticradiation In fact the boundary element code convertsexperimental data into velocity boundary conditions to beapplied for the acoustic eld computation It is noted thatboundary element codes are actually becoming a lsquostandardrsquotool in the eld of acoustic prediction because of theirnoticeable potential to import both data and geometryfrom other codes and to solve both interior and exterioracoustic problems In this way it was possible to assess thesound power radiation caused only by the transversal vibra-tions of the belt span without direct interference of impactnoise friction wheel vibrations and so on The same kind ofprocessing was then performed on the corresponding outputdata of dynamic simulation and the resulting numericalsound power radiation was compared with the experimentalone thus minimizing the effect of any interfering noise ofpulleys or rig vibration

A total of 18 different test cases were treated by varyingthe type of transmission line (layouts 1 2 and 3) the initialtension of the belt (150 and 350 N) and the angular velocityof the driving wheel (1000 1500 and 2000 rmin) Suchresults were also used in identifying the dynamic model andfor nal validation of the model itself as described below

Table 3 Measuring instruments

Instrumentation Function

Scanning laser Dopplervibrometer

Measurements of belt velocities

Spectrum analyser Check of signal qualityAcoustic probe Measurement of sound power radiation

Fig 9 Sound power spectrum of layout 3 (1500 rmin preload 350 N) lsquoLrsquo and lsquoArsquo are the linear and weightedsum of emitted sound power respectively

Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics K01402 IMechE 2003

70 M CALLEGARI F CANNELLA AND G FERRI

33 Comparison between model behaviour andexperimental results

More than 100 simulations were run in order to identify thevalues of the unknown parameters in uencing the acousticradiation and nally to validate the multi-body model Thevibration velocities computed by the mechanical systemssimulator (MSS) software for each belt span were the inputdata necessary for the BEM code to predict the acousticradiation the results were then compared with the experi-mental data measured on the test rig

Some parameters proved to be unimportant because achange in their values produced no signi cant variations inthe emitted sound power The radial damping for instanceis among the less in uential parameters as shown in Table 4in fact the contact forces are mainly exchanged between thewheel tooth tip and the belt bottom land and to a minorextent at the contact interface of the anks of the belt andwheel These results support the model because the numer-ical behaviour is similar to the experimental behaviourHowever when a parameter did not in uence the acousticradiation it was deleted from the model

It was decided to analyse the vibroacoustic behaviour ofthe belt only at meshing frequency and under resonantconditions for two basic reasons

1 It was shown in a previous study [1] that there is a strongrelation between the vibration of the belt spans at mesh-ing frequency and the total acoustic power emitted at thesame frequency

2 The test stand was suitably designed for the purpose ofintroducing disturbances in the form of camshaft pulleyrotational vibrations or eccentricities which can directly[11] or parametrically [3] give rise to resonancephenomena

Table 5 summarizes some results for the second and thirdlayouts With a decrease in the bending stiffness the noisedecreases accordingly until a lower limit is reached and itthen remains constant This behaviour of the modelcon rmed by experimental data can be explained verysimply One of the highest sound power radiations occursat the meshing frequency (the other ones being the reso-nance of the belt spans) Therefore if the variation in aparameter moves the resonance of the belt span away fromthe meshing frequency the contribution of this parameter tothe total sound power decreases over and over and the totalsound power of the belt remains more or less constant Thisphenomenon is also evident in the histogram of Fig 9 thesound power radiation has one of the highest peaks atmeshing frequency in this case at 540Hz Thus theweight of the parameter in the model can be somehowconsidered as the capability of changing the resonancefrequency of the belt

Table 6 shows that the acoustic radiation forecast by thecomputer aided tool is usually some 13 dB higher thanthe experimental data which is a signi cant differenceOne factor potentially in uencing the nal result isthe accuracy of the measurements In order to verify the

Table 4 Sensitivity of the model to changes in the value of radial damping

Preload(N)

Speed(rmin)

Bendingstiffness(Nmmdeg)

Bendingdamping(Nmmdegs)

Radialdamping(Nmms)

Backdamping(Nmms)

Point ofappliedforce(mm)

Angle ofappliedforce(rad)

Soundpower(dB)

Experimentalsoundpower (dB)

Layout 3 350 2000 173 01 20 059 144 0175 788 604Layout 3 350 2000 173 01 12 059 144 0175 790 604Layout 3 350 2000 173 01 00 059 144 0175 789 604

Table 5 Sensitivity of the model to changes in the value of some signi cant parameters

Preload(N)

Speed(rmin)

Bendingstiffness(Nmmdeg)

Bendingdamping(Nmmdegs)

Radialdamping(Nmms)

Backdamping(Nmms)

Point ofappliedforce(mm)

Angle ofappliedforce(rad)

Soundpower(dB)

Experimentalsoundpower(dB)

Layout 3 350 2000 173 01 10 049 144 0175 767 604Layout 3 350 2000 173 005 10 049 144 0175 740 604Layout 2 350 1500 100 07 30 080 144 0175 849 536Layout 2 350 1500 173 05 30 080 144 0175 810 536Layout 3 350 2000 173 01 20 059 144 0175 788 604Layout 3 350 2000 173 01 20 080 144 0175 800 604Layout 3 350 2000 173 01 12 059 134 0880 910 604Layout 3 350 2000 173 01 12 059 144 0175 790 604Layout 2 350 1500 100 07 30 080 144 0175 849 536Layout 2 350 1500 100 07 30 080 144 0087 824 536Layout 2 150 1500 50 01 10 049 144 0175 658 524Layout 2 150 1500 50 01 10 049 144 0087 649 524Layout 3 350 2000 1000 01 01 049 144 0175 770 604Layout 3 350 2000 173 01 05 049 144 0175 760 604Layout 3 350 2000 150 01 05 049 144 0175 761 604

K01402 IMechE 2003 Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics

MULTI-BODY MODELLING OF TIMING BELT DYNAMICS 71

sensitivity of the BEM model to the accuracy of velocitymeasurements an uncertainty analysis of the computa-tional values was conducted [25] In fact there were atleast two interfering inputs during the laser velocitymeasurements namely the generation of speckle noise(introducing a random error) and owing to the non-orthogonality of the measurement direction with respectto the surface the addition of part of the tangentialcomponent to the desired normal component in the results(generating a systematic error) As an outcome of thisanalysis it has been pointed out that for an uncertainty of20 per cent in the velocity measurements (which is areasonable value for the present application) the maximumdeviation in the acoustic power is around 2 dB a similarvalue could be expected for microphone-based acousticmeasurements

Another possible cause of the bias between the predictedacoustic emission and the measured values comes from thediscretization of the continuous mechanical system in thelumped-parameter dynamic model In fact numericalmodels usually appear lsquostifferrsquo than physical models asexplained for instance in references [27] and [28] Thiseffect is even more evident in the case of non-linearbehaviour of the real system which is usually linearizedin the lumped-parameter model This is actually thepresent case where the rubber material constitutingthe belt has a highly non-linear behaviour moreoverwhile meshing the belt undergoes a relaxation processthat progressively decreases the tension and therefore thestiffness [29]

One further issue potentially affecting the results of thisparticular analysis must be considered The experimentaldata made it possible to build up a spatial model since thebelt width was scanned in order to identify torsional vibra-tions The simulation on the other hand was based on aplane model so that in order to match the results thenumerical outputs of the two-dimensional multi-bodymodel were distributed along the belt width In this wayall the points of the belt spans were vibrating in coincidentphase which does not happen in reality because of thepresence of torsional oscillations that dephased the vibra-tions Figures 10 and 11 show the different shapes assumedby the belt in the three-dimensional experimental model andin the plane multi-body one

Figure 10 for instance shows the output of theSYSNOISE software for the third layout with a 350 Npreload at 2000 rmin allowing an easy comparison betweennumerical and experimental results the belt shape shows thedisplacements when excited at the rst resonance frequencywhile the displayed grey levels relate to the transversalvelocity In this case the expected (numerical) soundpower radiation was 770 dB the same value actuallymeasured on the test stand Figure 11 shows the samekind of data for the second layout with 150 N preload at1500 rmin the expected numerical sound power radiationwas 658 dB while a value of 504 dB was actuallymeasured in this case

Tab

le6

Com

pari

son

betw

een

sim

ulat

ions

ofla

yout

s2

and

3an

dex

peri

men

tal

valu

es

Prel

oad

(N)

Spee

d(r

min

)

Ben

ding

stif

fnes

s(N

mm

deg

)

Ben

ding

dam

ping

(Nm

md

egs

)

Rad

ial

dam

ping

(Nm

ms

)

Bac

kda

mpi

ng(N

mm

s)

Poin

tof

appl

ied

forc

e(m

m)

Ang

leof

appl

ied

forc

e(r

ad)

Soun

dpo

wer

(dB

)

Exp

erim

enta

lso

und

pow

er(d

B)

Dif

fere

nt(d

B)

Lay

out

215

015

005

00

103

049

144

017

565

853

4sect

312

4sect

3L

ayou

t3

150

2000

173

005

10

491

440

175

740

562

sect3

178

sect3

Lay

out

335

020

0017

30

051

049

144

017

574

060

4sect

313

6sect

3L

ayou

t3

350

1000

173

005

10

491

440

175

660

522

sect3

138

sect3

Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics K01402 IMechE 2003

72 M CALLEGARI F CANNELLA AND G FERRI

Finally several tests showed that the model is somehowcoherent because when the layout is changed but thesame values are kept for the parameters the gapsbetween the numerical and the experimental data remainsimilar (see Table 6) In the authorsrsquo opinion the bias of

about 12ndash13 dB shown by this complex model isacceptable and the results can still be considered signi -cant especially taking into consideration the coherentbehaviour shown when the transmission layout ischanged

Fig 10 Sample output of the SYSNOISE software (layout 3 2000 rmin 350 N preload)

Fig 11 Sample output of the SYSNOISE software (layout 2 1500 rmin 150 N preload)

K01402 IMechE 2003 Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics

MULTI-BODY MODELLING OF TIMING BELT DYNAMICS 73

4 CONCLUDING COMMENTS

This paper has described a detailed numerical model thathas been developed for the characterization of the dynamicbehaviour of timing belt transmissions with the nal goal offorecasting the acoustic radiation of a given design andsuggesting possible design changes Modelling and simula-tion of toothed belts have been performed by means ofcommercial software packages but additional programmingwas required to tune the models and to interface the toolsSeveral experimental tests have been performed for theidenti cation of the most important parameters and even-tually a correlation has been found between experimentaldata and the CAE model

To this end several relations describing the behaviour ofthe belt tooth and above all the beltpulley contact havebeen worked out Some design parameters were rather easyto identify through a literature survey or experimentalmeasurements but other dynamic properties proved to bemuch more dif cult to obtain even by means of experi-mental tests Therefore some parameters were determinedby means of a sensitivity analysis comparing the vibrationvelocities resulting from simulations with experimentaldata Only the key parameters (eg axial belt stiffnessbelt preload and velocity etc) were eventually retained inthe model It has been shown that belt vibrationsand emitted acoustic power follow the same trendswhen varying system parameters even if the simulatedacoustic radiation is still some 12 dB higher than actualradiation

In the authorsrsquo opinion such a difference between numer-ical results and experimental values may be at least partlydue to the following effects

1 The lumped-parameter numerical model discretizes thecontinuous mechanical system in a nite number of rigidbodies and (linear) springs and dampers so frequencyand amplitude of vibrations are affected

2 The numerical model which is a plane model can beconsidered to consist of masses vibrating in-phase rela-tive to the motion plane this causes constructive inter-ferences giving rise to maximum sound power emissionThe experimental model by contrast is a spatial modeland is therefore in uenced by several mode shapes sonot all the points are in-phase and not all the interfer-ences are constructive

3 The experimental measurements were affected by noisewhich could have caused a signal reduction and even-tually a lower measured sound power

In conclusion the complex multi-body model that has beendescribed is able to simulate very well the kinematics anddynamics of timing belt transmissions and provides a roughestimate of sound power emission Nevertheless when themost important belt parameters are changed the trend ofvariation in acoustic radiation is well reproduced and there-fore quantitative estimates of the nal result can also beprovided This tool is presently being used as CAD support

to help evaluate beforehand the effect of design choices andit is currently being inserted in the rmrsquos virtual prototypingchain

REFERENCES

1 Di Giacomo T Di Sante R Ferri G Rossi G LSantolini C and Tomasini E P Vibro-acoustic analysis ofsynchronous belts by laser Doppler vibrometry and boundaryelement techniques 4th International Conference on VibrationMeasurements by Laser Techniques Advances and Applica-tions 2000 Proc SPIE Vol 4072 pp 542ndash552

2 Naguleswaran S and Williams C J H Lateral vibrationsofband-saw blades pulley belts and the like Int J Mech Sci1968 10 239ndash250

3 Rhodes J E Jr Parametric self-excitation of a belt intotransverse vibration Trans ASME J Appl Mechanics 197020 1055ndash1060

4 Koyoama T Watanabe K Nagai K and Kagotani MA study on timing belts noise (how to reduce resonant noise)Trans ASME J Mech Des 1990 112 419ndash423

5 Pellicano F Catellani G and Fregolent A Parametricinstability of belts theory and experiments In Proceedings ofISMA Noise and Vibration Engineering Leuven 16ndash18September 2002 Vol IV pp 1497ndash1506

6 Kubo A Ando T Sato S Aida T and Hishiro T On therunning noise of a toothed belt drive Report 1 mechanism ofnoise generation Bull Jap Soc Mech Engrs 1971 14(75)991ndash997

7 Moon J and Wickert J A Non-linear vibration ofpower transmission belts J Sound and Vibr 1997 200(4)419ndash431

8 Troedsson I and Vedmar L A method to determine thedynamic load distribution in a chain drive Proc Instn MechEngrs Part C J Mechanical Engineering Science 2001215(C5) 569ndash579

9 Abrate A S Vibration of belts and belt drives Mechanismsand Mach Theory 1992 27 645ndash659

10 Mizuno N Inagaki H Nakakubo T and Nakada MAnalysis of synchronous belts vibration in automotivevalve train International Congress and ExpositionDetroit Michigan 29 Februaryndash4 March 1988 SAE paper880077

11 Gasper R G S and Hawker L E Resonance frequencyprediction of automotive serpentine belt drive systems bycomputer modeling ASME paper DE-18 1989 pp 13ndash16

12 Watanabe K Koyoama T Nagai K and Kagotani MA study on timing belts noise (theoretical analysis for forcedtransversal vibration of timing belt with parametric excitation)Trans ASME J Mech Des 1990 112 424ndash429

13 Gerbert G Jonsson H Persson U and Stensson G Loaddistribution in timing belts Trans ASME J Mech Des 1978100 208ndash215

14 Uchida T Hanada N and Yoshihisa T K Devlopment ofsimulation model for calculating loads of synchronous drivebelt SAE Spec Publs 1993 972 49ndash58

15 Karolev N A and Gold P W Load distributionof timing beltdrives transmitting variable torque Mech Mach Theory 199530(4) 553ndash567

Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics K01402 IMechE 2003

74 M CALLEGARI F CANNELLA AND G FERRI

16 Kagotani M Ueda H and Koyoama T Transmission errorin helical timing belt drives (case of a period of pulley pitch)Trans ASME J Mech Des 2001 123 104ndash110

17 Johannesson T and Distner M Dynamic loading of synchro-nous belts Trans ASME J Mech Des 2002 124 79ndash85

18 Callegari M and Cannella F Lumped-parameter model oftiming belt transmissions In Proceedings of 15th AIMETACongress of Theoretical and Applied Mechanics TaorminaItaly 26ndash29 September 2001

19 Fawcett J N and Burdess J S An experimental investigationon the vibration of toothed belts ASME paper 800-C2DET-941980

20 Hechler T and Anderson S Noise about synchronous beltsPT Des 2000 42(6) 13ndash14

21 Kubo A Ando T Sato S Aida T and Hishiro T On therunning noise of a toothed belt drive Report 2 in uence ofrunning condition and some noises reduction methods BullJap Soc Mech Engrs 1971 14(75) 998ndash1007

22 Tokoro H Nakamura M Sugiura N Tani HYamamoto K-I and Shuku T Analysis of high frequencynoise in engine timing belt Jap Soc Automot Engrs Rev1998 19 33ndash38

23 Johannesson T and Distner M Model for toothbelt mechanics In Proceedings of 4th World Congresson Gearing and Power Transmission 1999 Vol 2 pp1357ndash1369

24 Tokoro H Nakamura M Sugiura N Tani HYamamoto K-I and Shuku T Analysis of transverse vibra-tion in engine timing belt Jap Soc Automot Engrs Rev 199718 33ndash38

25 Di Sante R Revel G M and Rossi G L Measurementtechniques for the acoustic analysis of synchronous beltsMeasmt Sci and Technol 2000 11 1463ndash1472

26 Mote C D Jr On the non-linear oscillation of an axiallymoving string J Appl Mechanics 1966 33 463ndash464

27 Ewings D J Modal Testing Theory Practice and Applica-tions 2nd edition 2000 pp 416ndash446 (Research Studies PressBaldock UK)

28 Ginsberg J H Mechanical and Structural Vibrations Theoryand Applications 2001 (Wiley New York)

29 DalgarnoK W Day A J Childs T H C and MooreR BStiffness loss of synchronousbelts Composites Part B Engng1998 29(3) 217ndash222

K01402 IMechE 2003 Proc Instn Mech Engrs Vol 217 Part K J Multi-body Dynamics

MULTI-BODY MODELLING OF TIMING BELT DYNAMICS 75