multibody analysis of the desmodromic …diem1.ing.unibo.it/mechmach/rivola/pub38.pdfmultibody...

MULTIBODY DYNAMICS 2007, ECCOMAS Thematic ConferenceC.L. Bottasso, P. Masarati, L. Trainelli (eds.)

Milano, Italy, 25–28 June 2007

MULTIBODY ANALYSIS OF THE DESMODROMIC VALVE TRAIN OFTHE DUCATI MOTOGP ENGINE

David Moreno?, Emiliano Mucchi†, Giorgio Dalpiaz†, and Alessandro Rivola‡

?Departamento de Ingenieria de Sistemas IndustrialesUniversidad Miguel Hernandez, Campus de Elche, Avda. de la Universidad s/n, 03202 Elche, Spain

e-mail: [email protected]

†Department of EngineeringUniversity of Ferrara, Via Saragat 1, 44100 Ferrara, Italy

e-mails: [email protected], [email protected]

‡DIEMUniversity of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy

e-mail: [email protected]

Keywords: Desmodromic valve train, Motorcycle, Experimental validation.

Abstract. This paper presents a preliminary study concerning a multibody model of the des-modromic valve train used in the Ducati MotoGP engines. The desmodromic mechanism hasa positive cam that causes the dynamic effects to be partly different from common valve trains,where the valve spring plays an important role. The presented model includes only one cam-valve mechanism. In a further step of the research, it will be possible to develop and expand themodel by introducing the other cam-valve mechanisms and other mechanical parts that com-pose the system in order to obtain a complete model of the valve train. In the first part of thiswork, the generation of the cam profiles is explained. The second part is focused on the descrip-tion of the multibody model employed for the dynamic simulations. Finally, the experimentalvalidation is presented and discussed. The comparison between the numerical results and theexperimental data is encouraging even if it shows that the effectiveness of the model is not com-pletely achieved. Therefore, it will be necessary to improve the model by including the presenceof other mechanical parts of the valvetrain, as well as other important dynamic effects as, forexample, the link flexibility.

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David Moreno, Emiliano Mucchi, Giorgio Dalpiaz and Alessandro Rivola

1 INTRODUCTION

When a mechanism operates at a high speed, its dynamic behaviour is deeply affected bylink elastic flexibility and mass distribution, as well as the effects of backlash and friction injoints. As a consequence, motion alterations may occur, causing mechanisms to fail in theproper execution of their tasks. High accelerations and dynamic stress levels may also produceearly fatigue failures, and high levels of vibration and noise may arise. In the particular fieldof valve trains for high-performance engines, these dynamic effects are particularly importantsince they may cause serious functional troubles, such as jump and bounce phenomena.

Thus, increasing attention is addressed to the elastodynamic analysis in order to predictthe dynamic behaviour, forces and impacts, and to identify the causes of failures and poorperformances. However, the papers on this topic generally concerns widely-used trains withclosing springs. In this case, the valve spring plays an important role in the system dynamicsand its accurate modelling is required. On the other hand, in the case of desmodromic valvetrains (mechanisms with positive-drive cams) the dynamic effects are partly different, as studiedin [1] and [2].

C o n j u g a t ec a m

N e g a t i v er o c k e r

P o s i t i v er o c k e r

A d j u s t e r

V a l v e - h e a d

V a l v es e a t

Figure 1: (a) General view of the desmodromic valve train of Ducati motorbike engines with four valves percylinder; (b) schematic of the cam mechanism driving a single valve.

Some of the authors have already developed non-linear lumped-parameter models of the des-modromic valve train, Ref. [1] and [2]. The aim of this paper is to present a different modellingapproach based on a multibody code, namely LMS Virtual.Lab Motion [3], integrated withuser defined procedures implemented in Matlab and FORTRAN.

An integrated multibody-FEM model, will make it possible to study the dynamic behaviourof the timing system, considering the elasticity of the bodies and evaluating the stress, strain andvibrational states of the components under different operating conditions in a more accurateway. Attention will be addressed to the benefits and drawbacks of multibody approach withrespect to lumped parameter models.

2 DESCRIPTION OF THE DESMODROMIC VALVE TRAIN

This work concerns the timing system of the fourcylinder ’L’ engine of Ducati racing mo-torbikes, having double overhead camshafts, desmodromic valve trains and four valves per

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cylinder. This system is partially shown in the general view of Fig. 1(a): two camshafts withfour conjugate cams each one are driven by a gear transmission; one camshaft drives the fourintake valves and the other one the four exhaust valves.

The schematic of the cam mechanism driving a single valve is shown in Fig. 1(b): the lobesof a conjugate cam are in contact with their respective rockers; these two rockers are then incontact with the adjuster located at the tip of the valve. Thus, it is possible to identify two partsof the mechanism, each one consisting of one cam lobe and its corresponding rocker. They givevalve acceleration in positive and negative directions respectively, where the positive directionis considered to be the direction of the opening valve. Here the terms ’positive’ and ’negative’cam disc/rocker are used; however, these links are commonly, but improperly, called ‘opening’and ‘closing’ cam disc/rocker, respectively.

With respect to the more widely-used trains with closing springs, the desmodromic trainsmake it possible to give higher valve accelerations, preventing the follower from jumping offthe cam, without employing a very stiff closing spring; on the other hand, the mechanical com-plexity of the desmodromic system is justified only in high-speed engines with single-cylinderheads, such as Ducati engines.

3 GENERATION OF THE DESMODROMIC CAM PROFILES

This section is intended to describe the cam synthesis process, i.e. starting from the valvedisplacement law (Fig. 2 left) and taking into account the geometry of the mechanism, the pro-files of both cams will be obtained (Fig. 2 right). It is worth noting that, in order to control thebackslash as a function of the camshaft angle, each subsystem (positive and negative) was cre-ated following a different motion law. Both are shown in Fig. 2; red for the positive subsystemand blue for the negative one.

0 100 200 300

0

0.2

0.4

0.6

0.8

1

Camshaft angle [deg]

Val

ve d

ispl

acem

ent [

norm

aliz

ed]

30

210

60

240

90

270

120

300

150

330

180 0

Figure 2: Cam generation process. Both cam profiles (positive and negative) are generated starting from thedisplacement laws shown in the graph on the left.

In order to tackle the generation problem, a specific multibody model is created. The camsynthesis is clearly a kinematic case. Therefore, the backslashes of the different links thatcompose the mechanism are not considered and a set of ideal constrains is used. The schematicof the model is depicted in Figure 3.

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Figure 3: Multibody model employed for the CAM generation.

As it is shown, the model inputs (depicted as joint drivers) are the imposed motion of thevalve and the camshaft angle. The kinematic chain that links these inputs with the output,i.e. the profile of the CAM, consist of four bodies and 5 kinematic constrains. Although thestructure of this chain is always the same, the bodies that compose it depend on the conjugateCAM to be generated. For each CAM, positive or negative, their corresponding rocker andadjuster are used.

The active adjuster is bracketed to the valve through a bracket joint that constrains all theDOFs between both bodies. The valve is also linked to ground by means of a translationaljoint, which is driven so that the valve follows the desired displacement. The motion is thentransmitted to the active rocker through a curve-to-curve joint. This kinematical joint defines abilateral contact constraint between two curves, one on each body, in order to keep the tangentvectors parallel at the contact point. Slip is also allowed.

The CAM body consists only of an axis system, which is used to reference the points thatcompose the profile of the CAM. A driven revolute joint is used in order to attach this bodyto ground. It is worth noting that the camshaft angle was properly synchronized with valvedisplacement.

Finally, in order to relate the motion of the rocker with the CAM body, a specific tool fromthe Virtual.Lab Engine Library is used. It is called CAM Generator and allows to create a CAMprofile by taking as input the motion of the follower and other parameters like the dwell radius

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and the number of points of the profile. The result is a file containing the profile points.By means of the described model, the positive and negative cams were generated. This pro-

files will be used in the following sections to carry out dynamic simulations of the desmodromicsystem.

4 DYNAMIC SIMULATION

4.1 General description of the dynamic model

This section describes the multibody model of the desmodromic valve train, which is beingdeveloped using LMS Virtual.Lab Motion. At this stage, only a single valve sub-system hasbeen modelled in order to simplify the estimation of the model parameters. Figure 4 shows theschematic of this model, composed of eight rigid bodies and twelve massless parts. The latter,also called dummy bodies, will be used in a forthcoming flexible model as an interface to allowmeshed parts to interact with the rest of the multibody model.

The transmission that drives the camshaft has been neglected in order to focus the modellingprocess on the valve mechanism. Because of this, the model input is the angular speed of thecamshaft, which is depicted in the schematic as Joint Driver. This input drives the revolute jointthat links the camshaft with the dummy camshaft. Unlike the rest of the dummy bodies, whichwill be ignored for now, this one has a key role in the rigid model. It is introduced to be ableto consider the camshaft bending compliance at the point where the CAM is placed. To doso, a linear bushing (i.e. a 6 DOF spring-damper) and a planar joint, both acting between theground and the dummy body, are employed to model the camshaft stiffness in the XY plane.The driven revolute joint, on the other hand, constrains the relative angular motion between thedummy camshaft and the camshaft. Otherwise, if both elements were applied on the camshaft,the revolute joint would cancel the effect of the bushing.

As shown in the schematic, both the positive and the negative rocker are related to thecamshaft by means of specific CAM contact elements taken from the Virtual.Lab Motion En-gine library. Therefore, the cam surface is modelled using a 5th order spline. For each cam inthe simulation, this spline is used to determine the penetration depth, velocity and sliding ve-locity between the cam and the follower. These parameters are then used to generate the forcesin the contact according to the following expressions:

FN(xp) = k1xp + k2x2p + cpxp (penetration) (1)

FN(xp) = k1xp + k2x2p + csxp (separation) (2)

FT = µFN (3)

Where xp is the penetration depth, k1 and k2 are the linear and quadratic stiffness coefficientsrespectively, cp and cs are the penetration and separation damping coefficients and µ is thefirction coefficient. Thus, the contact algorithm is capable of predicting valve float, since thecam and follower are allowed to separate and re-impact throughout the simulation. Moreover,the Coulomb friction is also taken into account by this contact element.

The rocker-adjuster interaction is slightly different. In this case, unlike the former, the curva-ture radius of the parts that take part in the contact does not vary with time. Therefore, a simplercontact element, i.e. the sphere to revolved surface contact, is used in order to take advantage ofthis fact. This element allows to properly model the interaction between the cylindrical surfaceof the rocker (sphere) and the planar face of the adjuster (revolved surface). The contact force is

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Figure 4: Scheme of the multibody model employed for the dynamic simulations.

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based on the depth of penetration and the relative velocity normal to the contact surface, whilefriction forces are calculated based upon the relative velocities tangential to the surface.

Both rockers are bracketed to the ground by means of the parallel combination of a linearbushing and a planar joint in order to take into account the compliance of the hubs in the XYplane. As it is shown in the schematic, between the rockers and the ground there are also bracketjoints marked with an asterisk. These constrains, which remove all the relative DOFs betweenparts when they are active, allow to study the behaviour of the system when the complianceof the hubs is neglected. The same constraint is added between the ground and the dummycamshaft.

The valve is divided into two different bodies, the valve stem and the valve-head. As it isrepresented in the schematic, they are connected by means of a spring-damper element and atranslational joint. Thus, it is possible to introduce in the model the axial stiffness of the valve.However, it is worth noting that this configuration will be modified for the elastic model, wherethe entire valve will be a flexible part. The connection among the valve stem and the two partsthat compose the adjuster is made through bracket joints. On the other hand, a nonlinear spring-damper element is introduced between the valve-head and the ground in order to model thecontact with the seat. Unlike the other spring-damper elements described above, this element isonly active when the valve-head is in contact with the seat.

4.2 Hertzian stiffness

As it will be shown in the last section, the initial configuration of the model did not workproperly for high speeds. As an attempt to avoid this problem, it was introduced the Hertzianstiffness for both contacts adjuster-rocker. This stiffness was modeled according to the follow-ing equation:

FHertzian = 0.733E

√1

C

[1− 1− ε2

1 + ε2tanh

(2.5

δ

νε

)]‖δ‖ 3

2 sign(δ) (4)

Where E is the effective Young’s modulus of the two surfaces involved in the contact, δ isthe penetration, ε is the restitution coefficient, C is the compound radius of curvature of bothsurfaces at the current point of contact and νε is the transition velocity.

4.3 Model limitations

The current multibody model present the following limitations with respect to the previousSimulink models developed by some of the authors:

• The Hertzian stiffness of both contacts CAM-rocker is not considered since the programdoes not allow to parameterize the stiffness as a function of the contact point. Because ofthis reason, a constant stiffness is used. At present it is being developed a subroutine toconsider the Hertzian stiffness of this contacts.

• The Squeeze effect is not considered either because the CAM Contact element does notallow to take into account any force when there is no contact. Also in this case a subrou-tine will be introduced to consider this effect.

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5 MODEL PARAMETERS

The values of inertial parameters are computed automatically by the program. However, therest of model parameters1 were preliminary evaluated on the basis of both literature data andprevious Simulink models and subsequently adjusted in order to fit the experimental results.The following tables (Tables 1-5) summarize all the used parameters.

Parameter ValueX Stiffness Coefficient 2e8 N/mY Stiffness Coefficient 2e8 N/mX Damping Coefficient 400 Kg/sY Damping Coefficient 400 Kg/s

Table 1: Linear bushing parameters: camshaft

Parameter ValueStiffness Coefficient 2.214 N/m (deactivated)Damping Coefficient 1560 Kg/s (deactivated)

Table 2: Valve seat spring-damper

Parameter ValueLinear Stiffness 2e7 N/mQuad Stiffness 0Penetration Damping 200 Kg/sSeparation Damping 200 Kg/sFollower Radius 20 mmTransition Velocity 0.001 m/sFriction Coefficient 0.136

Parameter ValueLinear Stiffness 2.5e7 N/mQuad Stiffness 0Penetration Damping 200 Kg/sSeparation Damping 200 Kg/sFollower Radius 27.5 mmTransition Velocity 0.001 m/sFriction Coefficient 0.136

Table 3: Contact parameters: positive & negative CAM-rocker contacts

Parameter ValueStiffness Coefficient 1e7 N/mDamping Coefficient 200 Kg/sTransition Velocity 0.001 m/sFriction Coefficient 0.164Restitution coefficient 1

Parameter ValueStiffness Coefficient 1e7 N/mDamping Coefficient 200 Kg/sTransition Velocity 0.001 m/sFriction Coefficient 0.164Restitution coefficient 1

Table 4: Contact parameters: positive & negative CAM-adjuster contacts

6 COMPARISON BETWEEN EXPERIMENTAL AND NUMERICAL RESULTS

6.1 Experimental set up

Experimental tests were carried out on a test bench developed at the laboratory of DucatiCorse. The experimental apparatus includes a test stand, two cylinder heads pertaining to the

1These parameters were obtained from the validation of a single cam-valve mechanism without considering thetransmission that drives it. For this reason, the values of model parameters are not definitively assessed.

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Parameter ValueX Stiffness Coefficient 4.1e8 N/mY Stiffness Coefficient 4.1e8 N/mX Damping Coefficient 1226 Kg/sY Damping Coefficient 1226 Kg/s

Parameter ValueX Stiffness Coefficient 2.8e8 N/mY Stiffness Coefficient 2.8e8 N/mX Damping Coefficient 138 Kg/sY Damping Coefficient 138 Kg/s

Table 5: Linear bushing parameters: positive & negative rockers

same bank, an electrically powered driveline to operate the camshafts, a lubrication circuit, andthe measurement instrumentation. In particular, only the components required for the opera-tion of the valve train were included into the system, that is, crankshaft, piston, cylinder, andconnecting rod were excluded. As a result, no gas forces, combustion, or spurious vibrationsoccurred in the considered system.

The system under test is clearly different from the real one: in particular, it is only a portionof the real valve train and it is worth noting that only the components required to operate thevalve train are included in the test bench. The system response is therefore dissimilar from theactual one, i.e. the response of the motorbike engine system in working conditions. However,such differences, as well as the inclusion (or exclusion) of the forces due to compressed gases,do not compromise the validity of the experimental data as a tool for model validation.

The small valve mass and its high velocity do not allow contact measurements; on the otherhand, the high-frequency range and temperature do not suggest the use of proximity transducers.In addition, the valve motion measurements may be affected by vibration of the cylinder head;it is therefore necessary to measure the relative motion between the valve-head and its seat. Forthese reasons, a specific high-speed differential laser vibrometer was used for this experimentalstudy: the Polytec High-Speed Vibrometer (HSV). Finally, the valve motion has to be referredto the camshaft position. To this purpose, the experimental apparatus includes an encoder whichis placed on the camshaft.

By means of the described test bench it possible to run either only one camshaft or both. Theexperimental results reported in this study are relative to tests performed on the timing systemof the horizontal bank; in particular, only the intake camshaft was driven and, consequently,only four cam-valve mechanisms were operated.

For the comparison of the results of the multibody model with the experimental measure-ments, the valve motion of the desmodromic mechanism closest to the driveline (named as”first” cam mechanism) has been employed in this work. In fact, since the multibody modelrefers to a single cam-valve mechanism, it seems reasonable to make use of the experimentalmotion of the ”first” cam mechanism for the comparison with the numerical results due to thefact that the motion of this cam-valve mechanism is less affected than the others by the torsionaland flexural behaviour the camshaft.

6.2 Results

Next, the comparison with experimental data is presented. However, it is worth noting thatdue to the differences that exist between the multibody model and the experimental one (i.e. inthe current multibody model there is only one valve mechanism and the transmission drivingthis mechanism is not taken into account), it is not possible to talk about a strict validation. Itcould be considered as a ’preliminary validation of the model’.

The following figures show the acceleration of the valve for different camshaft speeds. Thecurves in blue depict the experimental values while the red ones depict the numerical results.

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0 50 100 150 200 250 300 350−3

−2

−1

0

1

2

3Low regime


Val

ve a

ccel

erat

ion

[nor

mal

ised

]

0 50 100 150 200 250 300 350−3

−2

−1

0

1

2

3Medium regime


Val

ve a

ccel

erat

ion

[nor

mal

ised

]

0 50 100 150 200 250 300 350−3

−2

−1

0

1

2

3High regime


Val

ve a

ccel

erat

ion

[nor

mal

ised

]

Figure 5: Results without considering Hertzian stiffness.

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0 50 100 150 200 250 300 350−3

−2

−1

0

1

2

3Low regime


Val

ve a

ccel

erat

ion

[nor

mal

ised

]

0 50 100 150 200 250 300 350−3

−2

−1

0

1

2

3Medium regime


Val

ve a

ccel

erat

ion

[nor

mal

ised

]

0 50 100 150 200 250 300 350−3

−2

−1

0

1

2

3High regime


Val

ve a

ccel

erat

ion

[nor

mal

ised

]

Figure 6: Results considering Hertzian stiffness.

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The only difference between both figures is the consideration of the Hertzian stiffness. Theresults obtained by neglecting it are presented in Fig. 5, while it is taken into account in Fig. 6.Both experimental and simulated results are overlapped in order to evaluate the the effectivenessof the developed multibody model.

As shown in Fig. 5, the results are quite good for low speeds but they tend to get worseas the camshaft speed increases. In fact, for the low regime both curves, experimental andnumerical, are quite similar. However, although the response of the model at the medium regimeis acceptable, there are some peaks in the experimental signal that are not followed by thenumerical model. Finally, at the high regime both curves are quite dissimilar and the frequencyof the oscillations does not even match.

On the other hand, at Fig.6 the response of the model is different. In this case, at the low andmedium regimes, the presence of the Hertzian stiffness increases the frequency of the numer-ical response with respect to the experimental one and introduces several considerable peaks.However, at the high regime the numerical results are improved with respect to the former caseonly in the first phase of positive acceleration, but in general the frequency of the oscillations ishigher than in the experimental signal.

In the light of this results, it seems reasonable to suppose that taking into account the flex-ibility of the members of the mechanism could increase the effectiveness of the model aboveall for high speed regimes in which the elastodynamic behavior is more important. To this end,a flexible model is currently being developed in order to study its influence. Once this modelis validated, it will be possible in a further study to add the other cam-valve mechanisms, thetransmission driving the camshaft and the missing external forces in order to obtain a completesystem model.

7 CONCLUSIONS

In this paper, a simplified model of the desmodromic valve train of the Ducati MotoGPengine has been introduced. Instead of using the lumped parameter approach like the mod-els previously presented by some of the authors, a multibody model has been developed andpreliminary compared with experimental data.

The presented model is composed of eight rigid bodies which are related by different kindof joints and by two kinds of contact forces that act between the pairs rocker-cam and rocker-adjuster. Some parts of the real system have been neglected in order to simplify the modellingprocess of the valve mechanism. Particularly, there is only one valve mechanism and the drivingtransmission system is not taken into account.

Although the presented results show that the model still needs to be improved, they are veryencouraging considering the simplicity of this model. Taking into account the good behavior ofthe model at low regimes and its bad response at high speeds (Fig. 5), it is quite probably thatthe introduction of link flexibility will improve its response above all at high speeds.

Therefore, it will be necessary, in a further study, to improve the model by including impor-tant dynamic effects as the link flexibility as well as the presence of other mechanical parts ofthe valve train.

8 ACKNOWLEDGEMENTS

The work presented in this paper has been performed in the framework of the Marie CurieHost Fellowship EDSVS for early stage researchers. The author would like to thank the Eu-ropean Commission for the grant received. The authors wish to thank DucatiCorse for active

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co-operation during the course of this research. Part of this project has been carried out withinthe Laboratorio di Acustica e Vibrazioni (LAV) which is supported by Regione Emilia Romagna- Assessorato Attivit Produttive, Sviluppo Economico, Piano telematico - Fondi Obiettivo 2 (I).

REFERENCES

[1] G. Dalpiaz and A. Rivola. A Non-Linear Elastodynamic Model of a Desmodromic ValveTrain. Mechanism and Machine Theory, 35(11), 1551–1562, 2000.

[2] Rivola A.,Troncossi M., Dalpiaz G. and Carlini A. Elastodynamic analysis of the desmo-dromic valve train of a racing motorbike engine by means of a combined lumped/finite el-ement model. Mechanical Systems and Signal Processing, ISSN: 0888-3270, 21(2), 735–760, 2007.

[3] LMS International. LMS Virtual.Lab 6A, 2006.

[4] Carlini A., Rivola A., Dalpiaz G., Maggiore A. Valve Motion Measurements on MotorbikeCylinder Heads using High Speed Laser Vibrometer. Proceedings of the 5th InternationalConference on Vibration Measurements by Laser Techniques: Advances and Applications,Ancona (Italy), 564–574, 2002.

[5] Carlini Andrea. Studio di una Distribuzione Desmodromica: Progettazione Cinematica,Simulazione Elastodinamica e Verifica Sperimentale. Ph.D Thesis, Dottorato di Ricercain Meccanica Applicata, University of Bologna, 2003.

[6] Fogli Federico. Analisi Multibody di una distribuzione desmodromica in ambiente LMSVirtual.Lab. Degree Thesis, University of Ferrara, 2007.

[7] R. R. Craig and M. C. C. Bampton. Coupling of substructures for dynamics analyses.AIAA Journal, 6(7), 1313–1319, 1968.

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