Study of Timing Belt Tensioner Behaviour 52153

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STUDY OF TIMING BELT TENSIONER

BEHAVIOUR

Laszlo LOVAS

Daniel PLAY

Janos MARIALIGETI

Abstract: Toothed belt transmissions are widely used in

modern engineering, from inkjet printer transmissions

through motorbike primary and secondary transmissions

to airplane luggage elevator transmissions. They are

silent, do not need excessive maintenance, and allow cost

reduction. Similarly, internal combustion engine timing

drive transmissions are often realized by timing belt.

In engine timing drive transmissions, driving and driven

pulleys generally do not allow adjusting belt tension.

Thus, a belt tensioner is used to furnish and maintain

preload force. It also compensates belt length variation

under load. The tensioner consists of a short arm fixed by

pivot on the engine block on one end and has an idler pulley on the other end. A torsional stiffness and a

torsional damping element also join the arm and the

engine block.

Numerical model and simulation software were developed

to modelize the timing drive transmission. For a given

belt transmission layout, various stiffness values were

tested numerically. Simulation input data come from

measurements on real engines. Viscous and Coulomb-

type damping cases were tested. System response to

various parameter values is studied and discussed.

Keywords: toothed belt transmission, tensioner, viscous

damping, torsional stiffness

1. INTRODUCTION

Toothed belt transmissions are used in a wide spectrum of power transmissions, from head moving mechanism of inkjet printers through timing transmission of internalcombustion engines and wheel transmission of motorbikes to luggage elevator transmission of airplanes.Although these transmissions are widely used, influenceof many parameters concerning the transmission is notentirely known and some phenomena have missingsatisfactory theoretical explanation.For example, in internal combustion engines, timing belttransmissions have tensioner mechanism. There is no well

defined scientific method for determination of stiffnessand damping parameters of tensioner mechanism. Theseparameters are chosen upon technical experience, and bytry-and-error methods.Toothed belt transmission modelization is studied atINSA-Lyon since the 80s. Theoretical background andalgorithms were elaborated by Dancé [5] and Monternot

[6], under direction of Professor Daniel Play [3], [4]. Asimulation software (MDSTD – Mechanical DynamicalSimulation of Timing Drives) was elaborated incooperation with BUTE Dept. of Vehicle Parts andDrives, allowing numerical modelization of a giventoothed belt transmission. This paper deals with effect of tensioner mechanism damping and stiffness in steadystate conditions, with given torque transmissionparameters.

2. THEORETICAL BASES OF THE

NUMERICAL SIMULATION SOFTWARE

The transmission is described with a two dimensionalmodel. It is assumed that longitudinal and transversal beltspan oscillations are independent in order to simplify thenumerical simulation. Transversal belt span oscillationsare also not considered in the model.The belt and bearings of the pulleys are modelized withelastic and damping elements. Pulleys are assumed to beideally rigid comparing to the other stiffnesses. Pulleyaxes are included in the calculation of their mass andinertia.Bearing elements of the pulleys are characterized withhorizontal and vertical stiffnesses.For a toothed belt element, belt stiffness and tooth

stiffness are defined separately. Belt longitudinal andbending stiffness values are given for one pitch. Toothstiffness is given for one tooth. Numerical stiffness valuescome either from literature [1], [6] or from measurements[7]. Values of belt and tooth stiffness depend on workingconditions. Effective belt stiffness is a function of thefollowing parameters: preload force, belt velocity, massof one belt pitch, belt temperature, past life of the belt.Tooth stiffness is a function of all mentioned parametersfor the belt, and of the fact, that toothed belt pitch changesunder load while pulley pitch remains the same. Pitch of the pulley has usually its nominal value, while pitch of theunloaded belt is generally smaller than nominal. Tooth

stiffness is also influenced by tooth-pulley contact force.

3. MODEL OF THE STUDIED

TRANSMISSION

A PSA DV4 engine timing transmission is modelized(Fig. 1) for numerical simulations. The belt connects sixpulleys being the crankshaft pulley (1), idler pulley (2),camshaft pulley (3), injection pump pulley (4), tensionerpulley (5) and water pump pulley (6). The drivingelement of the transmission is the crankshaft pulley, withan angular velocity applied on it. The angular velocityvariation of the internal combustion engine crankshaft is

modelized with a harmonic motion, with high mean valueand small amplitude:

rad/s)4cos(0215,02,471)( t t crank ⋅+=ω (1)



474

The resistant torque of the water pump is assumed to be alinear function of the water pump pulley angular velocity[7]. The idler pulley constant resisting torque comes onlyfrom internal friction of the rolling bearing. The tensionerpulley resisting torque comes also from rolling bearinginternal friction. Resisting torque of the injection pumppresents oscillations. These oscillations come from the

working of the piston pump feeding the four cylinders.This torque is modelized with the following harmonicequation:

Nm)6cos(68)( t M injinjinj ⋅⋅+= ω ω (2)

Fig. 1. Timing belt transmission of the DV4 engine

Resisting torque of the camshaft also varies. The camshaftturns with the half of the crankshaft angular velocity andmoves two valves in each cylinder. For a four cylinderengine, it has to open and close two valves against theclosing springs and the pressure inside the cylinder duringeach 90° turning. This torque is also modelized with a

harmonic equation:

Nm)4sin(206)( t M camcamcam ⋅⋅+= ω ω (3)

4. MODEL OF THE TENSIONER

MECHANISM

The tensioner mechanism is composed of the followingparts: tensioner arm, tensioner pulley, torsional spring,torsional viscous damper and torsional friction damper(Fig. 2). One end of the tensioner arm is fixed on theengine block with a pivot “A”. At the other end of the

tensioner arm is fixed the tensioner pulley. The tensionerpulley can rotate free on the tensioner arm around thepivot “B”. The tensioner arm can rotate on the engineblock within a defined angular domain. This working

domain is delimited with bumpers, not modelized yet.Torsional spring and torsional damper elements linkengine block and tensioner arm. Usually, arm length issmaller than pulley diameter, and all the tensionermechanism is mounted inside the pulley, including aspring and a damper (Fig. 3).

Fig. 2. Model of belt tensioner

In tensioner design, torsional stiffness is usuallyconsidered as main parameter, because it is easy tomeasure and to study its influence [2]. Dry friction onpivot “A“ of the arm is used as damping, and pivotparameters have to be also studied.

Fig. 3. Belt tensioner

The force acting on the pivot “A” is the parallelcomponent of the arm force coming from the pulleybearing force being the resultant F res of the tensionerpulley forces. Tangential component of the resultant forceF res is neglected. A constant friction coefficient is

considered as a first approximation. The equation for thetensioner arm torque equilibrium is:

( )F resF r signk s M θ θ µ θ θ θ −⋅⋅⋅⋅++= sin)( && (4)

s k

F res

r A

µ

θ F

θ

B

1

2

3

4

5

6

0 33 66 mm

AB

F res



475

where s – torsional stiffness,k – equivalent viscous damping,θ – angular position of the tensioner arm,F res – resultant force on pulley bearing,θ F – angular position of the resultant force,r – radius of the pivot “A”,

µ – friction coefficient on the pivot “A”.

5. STUDY OF STIFFNESS EFFECT

WITHOUT DAMPING

Fig. 4. Belt force variation when tensioner stiffness

changes

Three stiffness cases were studied without any torsional

damping. Stiffness variation was one magnitude in eachcase, thus: s1=5Nm/rad (thin line curve on Figs. 4-5 and10-13), s2=50 Nm/rad (medium line curve on Figs. 4-5and 10-13), s3=500 Nm/rad (thick line curve on Figs. 4-5and 10-13).Studied quantities were belt force F belt in the crank pulleytight span, dynamic transmission error ∆tr in the camshaftpulley.From figure 4, it can be seen that torsional stiffnessvariation has an effect on maximum belt force. Whenstiffness increases, maximum belt force decreases, andintermediate force peaks between maximum force peaks

increase. Thus, to decrease belt span force, increasingtorsional stiffness is not the best way.Dynamical transmission error on camshaft pulleydecreases with increasing stiffness (Fig. 5). The decreaseis more important than in the case of belt span force. Itreaches 50% in peak to peak amplitude. However,stiffness value in the most efficient case is so high, that itcan not be used in current automobile applications.

6. STUDY OF DAMPING EFFECT

Four damping cases were studied with original torsional

stiffness value of s=5 Nm/rad :- without damping- only pivot dry friction damping, r=8 mm, µ=0,3, - only viscous damping, k=0,458 Nms/rad,

- dry friction and viscous damping together, withthe previously mentioned values.

Fig. 5. Dynamical transmission error variation when

tensioner stiffness changes

From the computed output values, the following werestudied:- angular displacement of the tensioner arm,- angular velocity of the tensioner arm,- belt tension force in the tight span of the crankshaftpulley,- transmission error angle of the camshaft pulley.

Fig. 6. Belt force in the tight span of crank pulley

Simulation results have shown, that computing with orwithout dry friction gives neglectible difference in theoutput values. Thus, only two cases are represented in thefigures: tensioner without damping (thin line curve onFigs. 6-9) and tensioner with viscous damping (thick linecurve on Figs. 6-9).

Viscous damping decreases the force variation amplitude

in the tight belt span of the crankshaft pulley (Fig. 6).Moreover, it is turned out, that presence of viscousdamping decreases also the transmission error angle onthe camshaft pulley by the half (Fig. 7.)

1300

1100

900

700

500

300

0 0,002 0,004 0,006 0,008 0,01 0,012Time [s]

s1=5 Nm/rads2=50 Nm/rads3=500 Nm/rad

Fbelt [N]

0 0,002 0,004 0,006 0,008 0,01 0,012Time [s]

11

8

5

2

0

-3

-5

∆tr [10-3

rad]s1=5 Nm/rads2=50 Nm/rads3=500 Nm/rad

0 0,002 0,004 0,006 0,008 0,01 0,012Time [s]

1300

1100

900

700

500

300

No dampingViscousdamping

Fbelt [N]



476

Fig. 7. Dynamical transmission error of the camshaft

pulley

Fig. 8. Tensioner arm angular displacement

Fig. 9. Tensioner arm angular velocity

Differences between effects of the two types of damping

can be explained by displacements and angular velocitiesof the tensioner arm. When the engine is working,tensioner arm oscillates in a small angular domain (Fig. 8)but with high frequency around the pivot “A” (Fig. 2). As

pivot radius is small, friction torque remains small, anddoes not have efficient damping effect. Meanwhile,tensioner arm angular velocity is high (Fig. 9). In suchconditions, viscous damping becomes very efficient.

7. STUDY OF STIFFNESS EFFECT IN

PRESENCE OF DAMPING

The three stiffness values mentioned in paragraph 5 werestudied in presence of a viscous torsional damping

k=0,458 Nms/rad . Studied output values were the same asin the previous paragraph.

Fig. 10. Tensioner arm angular displacement with

changing torsional stiffness, damped case

Fig .11. Tensioner arm angular velocity with changing

torsional stiffness, damped case

Tensioner arm angular position mean value changes dueto stiffness variation, while similar amplitudes and shapesof oscillations are seen (Fig. 10). Tensioner arm angularvelocity position remains invariable, while oscillationamplitude slightly increases with increasing stiffness (Fig.

11). Thus, stiffness variation has small effect on thesequantities.Influence of torsional stiffness in the damped tensionercase is small (Fig. 12): large torsional stiffness variations

0

-1

0 0,005 0,01Time [s]

s1=5 Nm/rads

2=50 Nm/rad

s3=500 Nm/rad

θ & [rad/s]

0 0,005 0,01Time [s]

49

48

47

s1=5 Nm/rads2=50 Nm/rad

s3=500 Nm/rad

θ [°]0 0,002 0,004 0,006 0,008 0,01 0,012Time [s]

11

8

5

2

0

-3No dampingViscousdamping

∆tr [10-3

rad]

0 0,002 0,004 0,006 0,008 0,01 0,012Time [s]

51

50

49

48

47

46

45

44


θ [°]

0 0,002 0,004 0,006 0,008 0,01 0,012Time [s]

10

8

6

4

2

0

-2

-4

-6

-8


θ & [rad/s]



477

give small belt span tension variations. Note that forcemaximum values are smaller than those obtained in thenot damped case (Fig. 12). Thus, belt span tension reactsmore to the damping than to the stiffness of the tensioner.When damping exists, shape of the camshaft pulleydynamic transmission error remains the same withtensioner stiffness variations (Fig. 13). These variations

change mainly the mean value of transmission error.Increasing stiffness decreases the mean value.

Fig. 12. Belt span tension variation with changing


Fig. 13. Dynamic transmission error with changing


8. DISCUSSION

When designing a belt tensioner, manufacturers giveparameters measured in static state such as torsionalstiffness and dry friction coefficient. However, a working

state of the tensioner can not be characterised with thevalues measured in static state. In real workingconditions, equivalent stiffness and equivalent damping of all parts of the tensioner and of the lubricant used into the

rolling bearing of the tensioner pulley are the values feltby the belt transmission.It was pointed out that modifying tensioner torsionalstiffness within reasonable limits has small effect on beltspan force and camshaft dynamical transmission error.Thus, use of small or high stiffness gives approximatelysimilar effect.

Similarly, dry friction damping on the tensioner arm pivothas negligible effect from practical point of view, as pivotdimensions are small and displacements also.In such conditions, use of viscous damping is moreefficient for decreasing belt span force and dynamicaltransmission error in the same time. Mounting a viscousdamper in the tensioner is recommended upon thesimulation results. Viscous damping coefficient valuemust be chosen so that equivalent viscous damping of thebelt transmission could be optimal. The optimal viscousdamping value is function of the crankshaft angularvelocity. Usually it decreases when crankshaft angularvelocity increases. Thus, a technical solution assuming

such variation must be chosen.Note that viscous dampers were used in tensioners in thefirst automotive applications. Then, viscous dampers werechanged for dry friction dampers, as in that time, theywere voluminous and of complicated structure. Today’stechnology allows small size and simpler structure inviscous damper realisation.Based upon the numerical simulation, the followingprinciples can be mentioned:• Small torsional stiffness can be chosen. Its aim is to

maintain belt preload force and to give the tensionerinitial angular position. Stiffness value has noinfluence when engine runs.

• Damping effect of tensioner arm pivot dry friction canbe neglected with the compact geometries of actualtensioners.

• Application of a viscous damper seems to be areasonable solution for belt tensioners. Dampingcoefficient value should change depending oncrankshaft angular velocity. Torsional stiffness anddamping values must be harmonized and should beverified by measurements and simulations.

9. CONCLUSION

Toothed belt transmission tensioners with high torsionalstiffness values and small dry friction dampers can beimproved. High belt span force peaks are not damped andimportant transmission error is not decreased by dryfriction damping of tensioner arm pivot only. However,use of a viscous torsional damper gives a solution to thesedifficulties: both decreases belt force peaks andtransmission error.

REFERENCES

Conference articles

[1] MANIN L., PLAY D.: Experimental Validation of a Dynamic Numerical Model for Timing Belt

Drives. ASME 2000 IDETC/CIE Conference,Baltimore, 10-13 Sept. 2000.

6

5

4

3

2

1

0

-1

0 0,002 0,004 0,006 0,008 0,01 0,012Time [s]


∆tr [10-3

rad]

900

800

700

600

500

400

300

200

0 0,004 0,008 0,012Time [s]


Fbelt [N]



478

[2] MANIN L., PLAY D., POTINET L.,SOLEILHAC P.: Analysis and prediction of a

pulley tensioner effect on the behaviour of an

automotive timing belt drive. Proceedings of theJSME International Conference on Motion andPower Transmissions, Fukuoka, 2001, pp 785-790.

[3] PLAY D., FRITSCH N., HUOT S., AYAX E.: Numerical simulations of timing belt camshaft

layout: Local and global behaviour . ASME DETC2003 Conference, paper no. PTG 48008, Chicago,2003.

[4] PLAY D., MONTERNOT C.: Dynamic behaviour

simulation and load distribution for timing belt

drives. Proceedings of the 4th CMET Congress,Paris, Mars 1999, pp 1303-1314.

PhD thesis

[5] DANCÉ, J.-M.: Comportement statique et

cinématique des transmissions par courroies

synchrones. Modélisation et détermination des

paramètres prépondérants. PhD thesis, INSA-

Lyon, 239 p., 1992.[6] MONTERNOT C.: Comportement dynamique des

transmissions de puissance par courroie dentée. PhD thesis, INSA-Lyon, 278 p., 1998.

Research reports

[7] PLAY D., MANIN L.: Rapport de recherche pour

PSA. Research report, 1994.[8] PLAY D., LOVAS L.: Research report for SKF

France. Research report, July 2004.

CORRESPONDENCE

Laszlo LOVAS, Assoc. Prof., PhD.Budapest University of Technology andEconomicsFaculty of TransportationDept. of Vehicle Parts and DrivesBertalan Lajos u. 2.1111 Budapest, [email protected]

Daniel PLAY, Prof., Ph.D.Institut National des Sciences Appliquéesde LyonDept. of Industrial Engineering20, av. Albert Einstein69621 Lyon Cedex, [email protected]

Janos MARIALIGETI, Prof., PhD.Budapest University of Technology andEconomicsFaculty of TransportationDept. of Vehicle Parts and DrivesBertalan Lajos u. 2.1111 Budapest, [email protected]

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Study of Timing Belt Tensioner Behaviour 52153