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SPECIAL ISSUE PAPER 349

Multi-physics analysis of valve train systems: fromsystem level to microscale interactionsM Teodorescu1∗, M Kushwaha2, H Rahnejat3, and S J Rothberg3

1School of Engineering and Mathematical Sciences, City University, London, UK2Ford Motor Company, Ford Engineering Centre, Essex, UK3Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire, UK

The manuscript was received on 3 May 2006 and was accepted after revision for publication on 8 January 2007.

DOI: 10.1243/14644193JMBD77

Abstract: The paper highlights a holistic, integrated, and multi-disciplinary approach to designanalysis of valve train systems, referred to as multi-physics. The analysis comprises various formsof physical phenomena and their interactions, including large displacement inertial dynamics,small amplitude oscillations due to system compliances, tribology, contact mechanics, and dura-bility at the cam–tappet contact. Therefore, it also represents a multi-scale investigation, wherethe phenomena can be investigated at system level and referred back to underlying causes atsubsystem or component level, in other words, implications of an event at microcosm can beascertained on the overall system performance. This approach is often referred to in industryas down-cascading and up-cascading. The particular case reported here to outline the merits ofthis approach concerns a four-stroke single-cylinder engine. This promotes a system approachto engineering analysis for integrated noise, vibration, and harshness, durability and frictionalassessment (efficiency). Experimental validation is provided with a motored test rig, using laserdoppler vibrometry.

Keywords: valve train systems, noise, vibration, and harshness, frictional/tribological assess-ment, durability, multi-physics/multi-scale analysis

1 INTRODUCTION

The operation of a four-stroke internal combustion(IC) engine is heavily dependent on the behaviourof its valve train system. The concept is very sim-ple and has been used long before the IC was evenconceived, for example, in steam engines. Basically,a set of valves, which are closed during the high-pressure region of the thermodynamic cycle openprogressively during the gas exchange region of thecycle. However, the main design characteristics invalve train systems have evolved significantly sincethe early days. Given the low operational demand andover-design, most of the early engine components suf-fered malfunction because of gradual degradation in

∗Corresponding author: School of Engineering and Mathematical

Sciences, City University, Northampton Square, London EC1V 0HB,

UK. email: [email protected]

the form of rust, rather than through mechanical fail-ures. The development of later steam engines, andparticularly, the then newly introduced IC engines,brought additional challenges, including the need toreduce mechanical and frictional losses, durability,and more recently noise and vibration refinement.Although engines have reduced in size and weight (useof lighter durable materials) the operating speeds andloads have actually increased by at least an order ofmagnitude. The IC engines have been evolved in linewith the increasing demand put on efficiency withthe additional requirement of reduced manufactur-ing costs. In recent years, the new generations of valvetrains have become significantly lighter and subjectedto much higher forces. Therefore, for every mecha-nism, the opening and the closing events are carefullydesigned to synchronize their mechanical cycle withthe combustion cycle to achieve maximum efficiency.This has brought new challenges, which in the case ofvalve train systems translates into complex regimes of

JMBD77 © IMechE 2007 Proc. IMechE Vol. 221 Part K: J. Multi-body Dynamics

350 M Teodorescu, M Kushwaha, H Rahnejat, and S J Rothberg

lubrication in cam–follower conjunctions (reductionof lubricant film, thus increased chance of wear),stressing of load bearing surfaces (greater tendencyfor inelastic subsurface deformation) and vibrationbecause of component flexibility. Therefore, to bein line with these changes, a holistic and integratedapproach is desired, which necessitates breakdownof the traditionally isolated engineering disciplinesto promote the concept of first time right in prod-uct development in an ever increasing competitivemarket.

A typical valve train comprises a large number ofcontacting elements whose interactions are governedby a wide range of coupled phenomena. Consequently,to accurately predict their mechanical behaviour, adetailed transient dynamic model of the mechanismis an important prerequisite.

The ideal model should account for the transientinteractions between several physical phenomenaranging from system level interactions to those atmicroscale. This approach is called multi-physicsmulti-scale. The approach requires integrated solu-tion for all the interactive phenomena, such asLagrangian dynamics for rigid body motions (of theorder of few to several millimetres for valve motion),and Reynolds equation for lubricated conjunctions(for submicrometre thick films). For a complex sys-tem such as a valve train, this requires differen-tial equations of motion for several parts, constraintfunctions for their assembly, Reynolds and elasticityequations for all the contacts. The result is a sys-tem of differential–algebraic equations, which mustbe solved simultaneously both in time and spacedomains.

The most loaded contact in the valve train mech-anism is the cam–tappet conjunction. The slid-ing nature of this lubricated conjunction, togetherwith a highly transient loading regime, ren-ders this contact a major cause of valve trainmechanical inefficiency. Without the protectiveeffect of a lubricant film formed in the con-tact, the applied load together with vibrationscan cause significant frictional losses and inextreme cases wear and scuffing of contactingsurfaces. A multi-physics model is reported here,which incorporates all the aforementioned physicalphenomena.

The validity of a mathematical model can only beaccepted if at least some of its predictions are ver-ified against experimental measurements. In orderto understand the effect of certain physical phe-nomenon, a fundamental test (fully independent ofthe method of analysis) is required. The experiencegained, is then integrated back into the comprehen-sive framework of the whole system. The oppositeextreme is the use of a fully instrumented engine test[1, 2]. Although much closer to reality, the resulting

measurements should be very carefully analysed inorder to eliminate the influence of other additionalinteractions. Because of the combustion process, theresults are usually affected by noise, and great careshould be taken in applying corrective methods.Therefore, the optimal approach may be the use ofa test rig, in which an engine is motorized, ratherthan fired [3–6]. The advantage of this technique isthat most of the additional noise is eliminated and theresulting signal still preserves the interactions betweenthe key physical elements of the system.

2 THE MULTI-PHYSICS VALVE TRAIN MODEL

2.1 System dynamics

A constrained inertial dynamics model is required,which should incorporate the elastic behaviour ofthe various valve train elements. Each componentcontributes to the system dynamics through its stiff-ness and damping properties, and special care istaken while modelling the contacting/impacting con-ditions. The general framework is built on the methoddescribed by Teodorescu et al. [7]. Figure 1 is aschematic representation of the equivalent valve trainmechanism as a two-mass system model with threedegrees of freedom. Each lumped mass element(m1, m2) is considered to be rigid and represents themass of several components of the valve train sys-tem. Mass m1 represents the tappet, the push-rod,and the proportion of the rocker arm in translationon the push-rod side of the assembly. Mass m2 rep-resents the valve, effective mass of the valve spring(being one-third of its actual mass) and the contri-bution of the mass of the rocker arm on the valveside. The connecting springs and dampers are consid-ered to be massless. The contacts in the mechanismare shown in Fig. 1 by two horizontal parallel lines.The possible frictional losses are also introduced inthe model through inclusion of appropriate coeffi-cients of friction for the contacting solids in relativemotion.

The governing equations of motion are obtainedfor each element of the lumped masses, using theNewton–Euler method, and solved together in a timemarching scheme. The general format for equation ofmotion is

mizi + cei (zi − zi−1) + kei (zi − zi−1) − kei+1(zi+1 − zi)

− cei+1(zi+1 − zi) + Fi = 0 i = 1, 2 (1)

where, the friction between various components of thevalve train, as well as the possible additional forces are

Proc. IMechE Vol. 221 Part K: J. Multi-body Dynamics JMBD77 © IMechE 2007

Multi-physics analysis of valve train systems 351

Fig. 1 Dynamic model for the valve train system

given by the term Fi, and

⎧⎪⎪⎪⎨⎪⎪⎪⎩

kei = k′i · ki−1

k′i + ki−1

, cei = c ′1 · ci−1

c ′i + ci−1

contact

kei , cei = 0 loss of contact

i = 1, 2

ke3 = k2, ce3 = c3 (2)

The contact force between the cam and the tappet isrequired in equation (1) for the mass m1. This is effec-tively the integrated contact pressure distribution inthe cam–tappet conjunction, which means that forproper analysis transient consideration of this contactis necessary.

2.2 Cam–tappet conjunction

The conjunction between the cam and the tappetaccounts for the major portion of frictional lossesin the valve train system and should, therefore, beincluded in the system model. This is a finite lineconjunction, which can be represented at any instantof time by the contact of an equivalent roller neara semi-infinite elastic half-space. Not withstandingthe stress concentrations at the edges of the cam totappet contact in the direction of cam width, the pres-sure gradient along the length of the equivalent rollermay be considered small compared with that in thedirection of lubricant entrainment: ∂P/∂y � ∂P/∂x.Therefore, as a first approximation, an infinite linecontact solution may be undertaken, with the reper-cussion that the minimum film thickness is considered

to be in the vicinity of the exit boundary, and notthat in the exit region and to the sides of the contact,as would be under physical conditions. Furthermore,any side leakage of lubricant is considered to be neg-ligible. Thus, solution for this conjunction may beobtained using the following one-dimensional formof Reynolds’ equation

∂

∂x

(ρh3

η

∂P∂x

)= 12

[u

∂(ρh

)∂x

+ ∂(ρh

)∂t

](3)

where, the contact is considered as an infinite lineconfiguration with x denoting the direction of entrain-ing motion of the lubricant. Ideally, the cam–tappetcontact should be considered as a finite line con-tact and the complex tappet dynamics [8] taken intoconsideration.

The elastic film shape is given by

h = h0 + δ (4)

where, h0 is the instantaneous rigid separation, includ-ing the profile of the cam at any location in thex-direction against the flat tappet, and δ is the instan-taneous local deflection at any location in the con-tact, obtained by solution of the contact elasticityproblem as

δ(x) = 1πE ′

∫ x∗=xoutlet

x∗=xinlet

P (x∗) D (x, x∗) dx∗ (5)

where, E ′ = 2/(1 − ν21 /E1) + (1 − ν2

2 /E2) is the reducedYoung’s modulus of elasticity of contacting pairs.



Where, δ is obtained at any location in the contactbecause of all pressure elements within the instan-taneous contact pressure distribution and D is theinfluence coefficient matrix [9].

Now, clearly the extent of pressure distributionfrom inlet to outlet must be determined. The outletboundary condition is thus required, and Reynolds’boundary condition is implemented as

P|xinlet= 0 at x = −∞ (fully flooded), and P|xoutlet

= ∂P∂x

∣∣∣∣xoutlet

= 0 (6)

Now to solve Reynolds’ equation, one needs the rheo-logical state of the lubricant (η, ρ). Viscosity variationwith pressure is taken into account using Roelands’equation [10]

η = η0 exp[ln η0 + 9.67][(1 + 5.1 × 10−9Ph)Z − 1] (7)

where Z = α/(5.1 × 10−9[ln η0 + 9.67]).Density variation with pressure is given by Dowson

and Higginson [11]

ρ = ρ0[1 + 0.6P/(1 + 1.7P)] (8)

Now the lubricant reaction acting on mass m1, asalready described is found as

Fi =∫ xoutlet

xinlet

P dx for i = 1 (9)

Kinematics of the contact (u, ∂h/∂t) are essentialfor the solution of Reynolds’ equation. The speedof entraining motion of the lubricant is the averagevelocity of the contacting surfaces in the x-direction.Disregarding tappet spin [12]

u =[

Rc + 1ω2

∂2s∂t 2

]ω

2(10)

The elastic squeeze film velocity is ∂h/∂t = ∂h0/∂t +∂δ/∂t (equation (4)). This shows that the approach/separation of contacting surfaces takes place by a com-bination of rigid body motion (the first term, whennegative indicates approach of surfaces) and theirlocal rate of deformation (when positive indicates anemerging gap by local deformation of the surfaces).This is easily computed under transient conditions as(first-order approximation) ∂h/∂t = 1/t(hj − hj−1),where j denotes the time step and t the analysistime step size. Therefore, in transient analysis ∂h/∂trepresents the historical variation in local contact con-ditions at any location x, and can be termed as filmmemory.

2.3 Determination of friction

Friction in cam–tappet contact arises because ofviscous and boundary contributions. Where a coher-ent lubricant film is assured, the contribution tofriction is because of viscous action under sliding con-dition alone. However, thin films are encountered atinlet reversal positions (where the cessation of entrain-ing motion takes place, prior to and after the camnose). Therefore, interruptions in a coherent film mayoccur, depending on the composite roughness of thecontacting surfaces in relative motion. A parameter,γ = h/ψ (ratio of film thickness to the compositeroughness of the contiguous surfaces), is used to ascer-tain any boundary contributions, which occur becauseof asperity interactions. This occurs when γ < 3. Thecomposite surface roughness of cam and tappet sur-faces for the contact considered is measured to be0.2 μm.

The elemental areas in the contact region aregiven as

dA = B dx (11)

For each element, the asperity contact area andthe load carried by the asperities are computed,based upon the model proposed by Greenwood andTripp [13] as

dAa = π2(ζβψ)2F2(γ ) dA (12)

dFa = 0.75π(ζβψ)2

√γ

βE ′F5/2(γ ) dA (13)

The two statistical functions F2 and F5/2 are determinedusing the formulation proposed in [13] as

Fn(γ ) = 1√2π

∫∞

λ

(s − γ )ne−s2/2 ds (14)

In order to speed-up the numerical predictions, themethod proposed by Teodorescu et al. [14] for a linecontact is adopted here, and a fifth degree polyno-mial is fitted to each of the corresponding statisticalfunctions.

The total friction force, acting on an element dA iscomputed as the sum of the viscous and boundaryfriction components as

F =∫

(dFb + dFv) dx (15)

The boundary friction force results from the shearingof a very thin film (down to several layers of molecules),which prevails in the contact between the asperitytips during their contact. The friction force is com-puted as described in references [13] and [14]. Thenon-Newtonian shear stress τE is given as a function ofthe normal load components [15]. Consequently, the



boundary friction force, acting on an element can beexpressed as

dFb = τ0 dAa + m dFa (16)

The general form for the viscous friction force onlytakes into account the shear of the lubricant film,which is trapped between the sliding surfaces outsidethe asperity contact areas. Thus

dFv =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

ηuh

(dA − dAa) −→ ηuh

� τE ≈ 2 MPa(τE + 0.8

dF − dFa

dA

)(dA − dAa)

−→ ηuh

� τE ≈ 2 MPa

(17)

2.4 Subsurface stress field

Load bearing contacts represent the most critical com-ponents in many machines and mechanisms. An idealload bearing conjunction would possess sufficientcompliance to undergo local deformation under load.This enhances the formation of a protective lubri-cant film, while having an adequate hardness, to avoidexcessive wear because of surface asperity interactionsof the contiguous solids. Excessive contact pressurescan also cause inelastic deformation due to subsurfacestress field, exceeding the elastic limit, and causingfatigue spalls. Cam–tappet contact is the most loadedconjunction in IC engines, and any multi-physics anal-ysis should incorporate proper contact mechanics ofthis conjunction.

Harmonic decomposition techniques have alwayspresented an opportunity to be used in such appli-cations, although their primary use has been in thedetermination of global deformation of elastic solids,such as in bending and buckling of plates and shells[16]. The small dimensions of non-conforming con-tacts provide an interval of very short length. Thismeans that a pressure distribution, acting upon thecontact area, must be approximated by a harmonicseries of significant number of orders. An additionalconsideration is that the local nature of deformationis preserved below the natural modal behaviour ofthe contacting solids (see the Hertzian impact the-ory), thus restricting the upper limit of the harmonicseries. This combination of restrictions, together withthe averaging nature of the fast Fourier analysis haspresumably been the reason for the late take-up of thisapproach as an alternative to the traditional methodof predicting contact behaviour.

Figure 2 shows the contact conditions as well as thepressure distribution. This practical problem corre-sponds to the plane strain condition, as pointed out

Fig. 2 Transverse pressure distribution on the contact

by Hertz [17]. To achieve a realistic prediction, boththe contact load as well as the contact friction shouldbe considered. Since the pressure distribution over thecontact footprint is half an ellipse, it is also assumedthat shear also has a similar distribution.

Both the contact pressure and surface shear distri-butions can be decomposed into Fourier series overa chosen interval, larger than the footprint width inthis case. The size of this interval 2L should be largeenough to avoid excessive influence because of edgeeffects, thus⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩P = 0.5P0 +

∞∑k=1

Pk

(2πx

λk − ϕk

)

τ = 0.5τ0 +∞∑

k=1τk

(2πx

λk − ψk

)

(18)

where

αk = 2π/λk and λk = L/k

The stress field can be computed for each harmoniccomponent (wavelength) separately and the final pre-diction is obtained using the superposition princi-ple. Teodorescu et al. [18] have shown that the finalsubsurface stress tensor can be expressed as

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

kσxx sech αky= [Pk cos(αkx − ϕk)(αky − 1)

+τk sin(αkx − ψk)(αky − 2)](tanh αky − 1)

kσyy sech αky= [−Pk cos(αkx − ϕk)(αky + 1)

+τk sin(αkx − ψk)αky](tanh αky − 1)

kσxy sech αky= [−Pk sin(αkx − ϕk)αky

+τk cos(αkx − ψk)(αky − 1)](tanh αky − 1)

(19)

Note that most of the modern cam and tappets areprotected with wear resistant coatings. The general



method to deal with such cases takes into account theeffect of thin- or thick-layered bonded solids [19].

The pressure distribution is decomposed into anumber of harmonic components and a line of con-stant pressure. The effect of this latter term on thecontact stresses is taken into account here.

For any location x, y under the surface the stressescan be evaluated as [20]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0σxx = − P0

2π[2(θ1 − θ2) + (sin 2θ1 − sin 2θ2)]

+ τ0

2π

[4 ln

(r1

r2

)− (cos 2θ1 − cos 2θ2)

]

0σyy = − P0

2π[2(θ1 − θ2) − (sin 2θ1 − sin 2θ2)]

+ τ0

2π(cos 2θ1 − cos 2θ2)

0σxy = P0

2π(cos 2θ1 − cos 2θ2) − τ0

2π[2(θ1 − θ2)

+(sin 2θ1 − sin 2θ2)]

(20)

where

r1,2 = √(x ∓ L/2)2 + y2, tan θ1,2 = y/(x ∓ L/2)

3 EXPERIMENTAL SET-UP

In the current work, a 5-Hp single-cylinder Honda ICengine was powered by a variable speed electric motor.The goal was to validate the dynamics of the com-plete valve train model, as well as the cam profile, bymonitoring the movements of the tappet in the testrig. Since the dynamics of the valve train system is ofprimary interest, the piston and the rest of engine aux-iliaries were removed. Several openings were made inthe engine block to ensure access for measurementsto be carried out. Figure 3 depicts an electric motor (1)and the test engine (2).

Monitoring of the motion of valve train componentsat high acceleration can be achieved with a variety ofmethods. Norton et al. [21] have analysed the vibra-tion characteristics of a valve train system, providing acritique of the various available methods. The cylin-der head was mounted on a separate fixture, andone of the intake valves was instrumented. Teodor-escu et al. [14] installed a miniature accelerometeron the valve seat of a fired engine and monitoredthe acceleration over a large range of engine speeds.One of the common conclusions made is that ide-ally the method of measurement should not interferewith the valve train geometry. Therefore, for the cur-rent study, a non-contacting method was employed.A laser Doppler vibrometer (LDV) was used to mea-sure the tappet lift velocity. The method consists of a

Fig. 3 The experimental set-up

laser source (point 3 in Fig. 3), which transmits a sin-gle beam towards the tappet. The upper side of thetappet is covered with a special reflective paper toreduce the chance of beam scatter. Figure 4 shows theprinciple of operation of an LDV. To reduce the noiseand to minimize the measurement errors, the beamwas carefully centred orthogonal to the upper side ofthe tappet at all times. An attribute of cam–tappetcontact is that the cam is designed to approach thetappet with a predefined eccentricity. During the event

Fig. 4 Principle of use of the LDV



cycle, this misalignment generates a moment alongthe vertical axis of the tappet, and consequently, thetappet tends to spin. Teodorescu et al. [22] have inves-tigated the tappet spin phenomenon. This, togetherwith the vertical motion of the tappet can alter theLDV measurements and result in an error. However,this shortcoming can be eliminated if the laser beamis directed orthogonally to the reflective surface of thetarget.

4 RESULTS AND DISCUSSION

4.1 Experimental validation

A suitably designed mechanism should ensurerequired valve timing and sealing.This can be achievedthrough numerous methods. The most commonlyused method is to introduce a small clearance in themechanism, ensuring correct valve closure durationswith no additional applied force. A downside to thisapproach is that at the start of the event cycle, theclearance is suddenly taken-up by the cam lift, withall the mechanism components coming into contact,thereby, resulting in an undesired impact. Therefore,

Fig. 5 Comparison between the experimental andtheoretical tappet velocity

Fig. 6 Experimental and theoretical cam lifts

the engine life may be affected by the high impactforces. A simple method of overcoming this problemis to introduce a ramp at the beginning and at the endof the cam profile. The geometrical profile of such aramp has been investigated in some detail by Chen[23]. Because of the vertical orientation of the experi-mental set-up, the cam and tappet are likely to be incontact during the part of the cycle, when the valve isnot actuated. Figure 5(a) shows the measured tappetvelocity, together with the predictions provided by themodel. Two constant velocity ramps, one before theopening part of the event and the other after closurecan be observed. Since the mechanism is not loadedduring these ramps, the velocity remains constant andconsequently, there is no acceleration.Thus, the rampsare of little interest in the combined durability andnoise, vibration, and harshness assessment, and thusare not included in the model.

The experiment was repeated for several enginespeeds to verify consistent behaviour. Figure 5(b)shows the tappet velocity for three different enginespeeds, together with the corresponding model pre-dictions. The stretch of the curve at lower speeds isdue to a longer time required for the mechanism todescribe the cam profile.

Fig. 7 Lubricant entrainment velocity



Fig. 8 Transient pressure and shear distributions and film shape through the inlet reversal duringthe wind-up process

The predicted values for tappet velocity containlower noise levels than those obtained experimentally.This is partly random, due to measurement error, andpartly repetitive across the engine speed spectrum,due to surface irregularities, not accounted for in themodel. The simulation of the dynamic behaviour ofthe mechanism requires an accurate prediction of thetappet acceleration. However, by differentiating the

experimental signal, the noise levels are extensivelyamplified. The integration of the tappet velocity curvewith respect to time yields the tappet lift as shownin Fig. 6.

It should be noted that the integration process alsoacts as a smoothing algorithm. Therefore, the cumula-tive error over the whole cam event results in a smallskew of the curve at the end of the closure event.



A high-order polynomial cam is utilized in the math-ematical model. The polynomial function for the camwas achieved by employing curve fitting techniques tothe experimentally obtained cam lift data [2, 23].

4.2 Down-cascading from system level tomicroscale contact interactions

Now that good agreement is obtained between the pre-dictions and the experimental data, the model resultscan be used to study many important design and per-formance aspects, where measurements are difficultto undertake, such as from cam–tappet interface.

Figure 7 shows the lubricant entrainment veloc-ity for a cam cycle. At the beginning and at the endof the cam event, the cam–tappet contact is on thecam flanks, where the speed of entraining motion andinstantaneous radius of curvature of the cam attaintheir highest values. Consequently, the oil film is rel-atively thick and the regime of lubrication is usuallyhydrodynamic, with low friction, and thus not of greatinterest. However, around ±70◦ from the position ofcam nose the speed of entraining motion ceases andchanges direction. These positions are known as inletreversal regions because of a change in the directionof lubricant flow into the contact. At these locationsthe supply of fresh lubricant by wedge action ceasesand an oil film survives purely due to the squeezefilm action [8]. In the extreme cases of high load andlow engine speeds, a mixed regime of lubrication ismost likely in these locations, where boundary frictionwould contribute to the contact frictional behaviour.The cam nose region usually suffers the highest con-tact load, but the nearly constant speed of entrainingmotion ensures a continuous supply of lubricant, andthe regime of lubrication is predominantly piezo-viscous elastic (i.e. elastohydrodynamic lubrication(EHL)). It is interesting to ascertain whether theseexperiential findings are borne out by the predictionsof the multi-physics model.

Figure 8(a) shows a series of pressure distributionsand the corresponding lubricant film thickness, as thecam traverses through the wind-up region (inlet rever-sal point at around −70◦). Note that the inlet trail of thepressure distribution (on the right-hand side of thefigure) gradually changes direction (to the left-handside). The solitary pressure spike, typical of elasto-hydrodynamic (EHD) pressure distributions, migratesfrom the initial outlet position (on the left-hand side)to the final outlet location. In the process of migration,gradually a pressure spike emerges at the opposite endof the contact as the initial spike begins to dimin-ish in magnitude. This is an important phenomenon,as the pressure spike inhibits the flow of lubricantin the region directly beneath it, thus depleting the

film and increasing the chance of direct surface inter-actions (see later). Correspondingly, a cave (dimple)appears in the elastic film shape. This is because ofthe elastic squeeze effect. This cave moves within thecontact, which indicates the movement of a pressurewave through the contact at inlet reversal points. Thisphenomenon tends to generate a higher film thicknessduring these parts of the cycle, where the cessationof entraining motion would normally be expected tocause the depletion of the oil film. The elastic squeezefilm is dominated by the local distortion of surfaces,being a much quicker elastic phenomenon than theslow viscous response of the lubricant being pushedout of the contact by rigid approach, and is referredto above as film memory. It is an unintentional fortu-nate event that happens to be also engrained in nature,such as in natural mammalian joints [24]. The resultshere also corroborate the conclusions of Kushwahaand Rahnejat [25] that squeeze caving phenomenonis really a by-product of kinematics of the contact.

Figure 8(b) shows the corresponding shear stressdistribution for those cases shown in Fig. 8(a). Theseindicate the prevalence of mixed regime of lubricationthrough the wind-up inlet reversal event, being thecontribution due to boundary lubrication. It is alsoclear that the total shear behaviour puts the contactcondition firmly into non-Newtonian for the most ofthe contact as τ > τE. However, the viscous contribu-tion remains largely in the Newtonian regime, exceptin the vicinity of the migrating pressure spike in thelatter parts of the wind-up process. In fact, its contri-bution is almost insignificant prior to the inlet reversal,and during the lift event. As the cave depth initiallyincreases the shear stress is reduced and as the inlet iscompletely reversed (the culmination of the wind-upprocess), both the viscous and boundary contribu-tions increase accordingly. This shows that squeezecaving event is in fact quite a favourable tribologicalevent on its own accord, which has been somewhatunderrated.

The same conditions also occur at around +70◦,where the wind-down process (subsequent inlet

Fig. 9 Cyclic variations of coefficient of friction



Fig. 10 Subsurface stress fields



reversal) takes place, but lower pressure spikes andshallower dimple formation are noted.

As the regime of lubrication alters from the camflank-to-tappet contact through the inlet reversalpoints and onto the cam nose, as well as the rheo-logical behaviour of fluid from dominantly Newtonianto non-Newtonian, the coefficient of friction changesaccordingly. Figure 9 shows that the coefficient offriction falls rapidly on the flank contacts as hydrody-namic conditions are attained, with mainly Newtonianviscous friction. Between the reversal points and ontothe cam nose, the coefficient of friction hardly altersand has its maximum value, with the prevalent mixedregime of lubrication, dominated by boundary fric-tion. At and in the vicinity of inlet reversals due to lowspeed of entraining motion boundary contributiondominates, as already shown in Fig. 8(b), but the for-mation of the squeeze cave actually mitigates againstthe undesired effect of friction. This is a major funda-mental find, suggesting that application of soft coatinglayers in these regions of cam may lead to a reduc-tion in frictional losses. Investigations of this type, inmore detail, may pave the way to localized econom-ical application of appropriate coatings (modulus ofelasticity and layer thickness), rather than the currentexperimental ad hoc approach. A migrating pressurespike in a softer conjunction can lead to a greater depthof squeeze cave, a phenomenon already demonstratedin Fig. 8(a).

The pressure spike sets up localized subsurfacestress field of its own, superimposed upon the overallsubsurface stress tensor, increasing both the magni-tude of stresses and reducing the depth beneath thesurface that such stresses attain their highest val-ues, thus exists a greater chance for fatigue spalling.Figure 10(a) shows this asymmetry in the maximumshear stress field, indicated by the isocline at 120 MPa,tending towards the surface of the tappet (i.e. at y = 0).This absolute maximum island of shear stress followsthe Hertzian theory at 0.3 PH, where PH is the max-imum Hertzian pressure (the primary pressure peakin the EHL pressure distribution, at the top left-handpressure distribution of Fig. 8(a)). Under elastostaticHertzian condition, the contact semi-half-width in thex-direction would be 0.03 mm. The depth at which themaximum shear stress should occur would be 78 percent of this semi-half-width (or 0.0234 mm), which ispredicted to be 0.024 mm in the case of Fig. 10(a).This shows very good agreement with the classicaltheory. The shift along the x-axis is due to the pres-ence of the EHL pressure spike. The asymmetry dueto pressure spike is better observed in the subsurfacestress tensor component σxy . This is the orthogonalreversing shear stress, which is of a cyclic nature ascan be seen at the top left-hand inset in Fig. 10.Note that the isoclines in the vicinity of the pres-sure spike pack very closely and approach the surface.

This shows that sufficiently high pressures can lead tofatigue.

However, the contact is subjected not only to appliedpressures, but also to shear as already discussed. Thus,Fig. 10(a) is only informative. The correct predictionsfor the contact condition are those in Fig. 10(b), wherethe effect of shear leads to an increase in magnitude ofshear stresses near the surface as well as further asym-metry in the subsurface stress tensor. Further islandsof stress are formed due to the migration of the pres-sure pip, with its localized field propagating from x > 0to x < 0 with the squeeze cave (Figs 10(a) to (d)).

Figure 11 provides better visualization of the for-mation of the squeeze cave and its migration at theinlet reversal during the wind-up process. Figure 11(a)shows the depth of the dimple emerging during theonset of contact transition from cam flank (inletreversal). The step increase in the film thickness indi-cates the emergence of the dimple (Fig. 8(a)). Thedepth increases gradually (and also migrates along thex-direction (Fig. 11(c))) until the dominance of elasticsqueeze ceases and entraining motion of the lubricantcommences, indicated by the sharp drop in the dim-ple depth. In the corresponding Fig. 11(b), a negativevalue for the squeeze velocity indicates rigid approach.Thus, to the left of the indicated demarcation line (onthe cam flank just prior to the inlet reversal) a smallapproach of surfaces is noted. At the event, a sharprise in ∂h/∂t > 0 indicates the dominance of the elas-tic nature of squeeze velocity (i.e. |∂δ/∂t | � |∂h0/∂t | ∴∂h/∂t > 0), creating the squeeze cave. Thereupon,the effect of squeeze diminishes (∂h/∂t ≈ 0) and thedepth of the cave remains more or less unchanged

Fig. 11 Kinematics of squeeze cave phenomenon



(Fig. 11(a)), but is pushed along the contact (Fig. 11(c))with the reinstatement of entraining motion. This isthe sign of migration of the pressure spike across thecontact (Fig. 8(a)), with the eventual large approachof the surfaces (∂h/∂t > 0) to smoothen the cave andattain the familiar EHL pressure distribution shapedominated by entraining motion (Fig. 8(a), the insetat the bottom left-hand corner). The predictions hereshow that squeeze caving is essentially governed bycontact kinematics, as correctly observed by Kush-waha and Rahnejat [25] and need not be explainedin terms of a thermal wedge as has been invariablysuggested as a cause, which is in fact clearly an effect.

5 CONCLUSIONS

The multi-physics model provides an integrated tribo-dynamic analysis, coupled with the evaluation ofsubsurface stress field. The model results at the systemlevel agree with experimental measurements. Withdown-cascading, even to microscale not only addi-tional information of a practical nature is found, butalso some fundamental aspects of physics of motioncan be ascertained. Further physical phenomena suchas heat generation due to work done should be addedin future work, which would affect lubricant rheol-ogy, thus the contact conditions and consequently thesystem behaviour.

REFERENCES

1 Ball, W. F., Jackson, N. S., Pilley, A. D., and Porter, B. C.The friction of a 1.6 litre automotive engine – gasolineand diesel, SAE paper 860418, 1986.

2 Teodorescu, M., Taraza, D., Henein, N. A., and Bryzik,W. Experimental analysis of dynamics and frictionin valve train systems. Trans. SAE, J. Engine, 2002,1027–1038.

3 Crane, M. E. and Meyer, R. A process to predict fric-tion in an automotive valve train. SAE paper 901728,1990.

4 Pieprzak, J. M., Willermet, P. A., and Dailey, D. P.Experimental evaluation of tappet/bore and cam/tappetfriction for a direct acting bucket tappet valve train. SAEpaper 902086, 1990.

5 Van Helden, A. K., Van der Meer, R. J., Van Staaden, J.J., and Van Geldern, E. Dynamic friction in cam/tappetlubrication. SAE paper 850441, 1985.

6 Yutaro, W., Mitsushiro, S., Yoshito, E., Toshiro, H.,and Tatsumi, K. Studies on friction characteristics ofreciprocating engines. SAE paper 952471, 1995.

7 Teodorescu, M., Rahnejat, H., and Rothberg, S. Elasto-multi-body dynamics of valve train systems with springsurge effect and coil clash. In Multi-body dynamics:monitoring and simulation techniques III (Eds H. Rah-nejat and S. Rothberg), 2004, pp. 273–284 (ProfessionalEngineering Publishing, IMechE, London).

8 Kushwaha, M. and Rahnejat, H. Transient elastohydro-dynamic lubrication of finite line conjunction of camto follower concentrated contact. J. Phys. D, Appl. Phys.,2002, 35 (3), 2872–2890.

9 Hamrock, B. J. Fundamentals of fluid film lubrication,1994 (McGraw-Hill International Editions, New York).

10 Roelands, C. J. A. Correlation aspects of viscosity–temperature–pressure relationship of lubricating oils.PhD Thesis, Delft University of Technology, The Nether-lands, 1996.

11 Dowson, D. and Higginson, G. R. Elasto-hydrodynamiclubrication, 1966 (Pergamon Press Ltd, Oxford).

12 Teodorescu, M., Kushwaha, M., Rahnejat, H., andTaraza, D. Elastodynamic transient analysis of four-cylinder valve train system with camshaft flexibility. Proc.IMechE, Part K: J. Multi-body Dynamics, 219(K1), 2005,13–25.

13 Greenwood, J. A. and Tripp, J. H. The contact of twonominally flat rough surfaces. Proc. Instn Mech. Engrs,1970–71, 185(48/71), 625–633.

14 Teodorescu, M., Taraza, D., Henein, N. A., and Bryzik,W.Simplified elasto-hydrodynamic friction model of thecam – tappet contact. Trans. SAE, J. Engine., 2003, 1271–1283.

15 Evans, C. R. and Johnson, K. L. The rheological proper-ties of elastohydrodynamic lubricants. Proc. Instn Mech.Engrs, Part C: J. Mechanical Engineering Science, 1986,200(C5), 303–312.

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18 Teodorescu, M., Rahnejat, H., and Gohar, R. Har-monic analysis to determine contact characteristics ofconcentrated counterformal contacts. In Proceedingsof the 7th Biennial ASME Conference on EngineeringSystems Design and Analysis, Manchester, UK, 19–22July 2004.

19 Teodorescu, M. and Rahnejat, H. Mathematical modelfor elastic layered contacts with direct application on acam-tappet conjunction within a multi-physics frame-work. In Applied mathematical modelling (Elsevier Sci-ence).

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23 Chen, J. S. Dynamic analysis of a 3D finger followervalve train system coupled with flexible camshafts. SAEtechnical paper 2000.

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APPENDIX

Notations

A contact areaAa asperity contact areaB cam widthci generic damping coefficientE1, E2 elastic moduli for cam and follower

materialsE ′ reduced modulus of elasticity: E ′ =

2/{(1 − ν21)/E1 + (1 − ν2

2)/E2}F total friction forceFa force carried by asperitiesFb boundary friction forceFi applied forcesFn statistical functionsFv viscous friction forceh elastic film shapeh0 rigid separationk harmonic orderki generic stiffnessL Fourier decomposition intervalm pressure coefficient of the boundary

shear strengthm1 equivalent mass of push-rod and

tappetm2 equivalent mass of valve assemblyP pressurePH maximum Hertzian pressurePk amplitude of the kth harmonic of the

applied pressure k = 0 → Nr1, r2 radial extent of the subsurface com-

pressive stress fieldRc instantaneous radius of curvature of

cam

s cam liftt timeu speed of entraining motion of the

lubricantx coordinate along the contact surfacey coordinate into the depth of the con-

tacting solidszi rigid body motionsZ pressure viscosity index

α piezo-viscosity indexακ =2kπ/Lβ asperity radius of curvatureγ film thickness parameterδ local elastic deflectiont time step sizeζ surface density of asperity peaksη dynamic viscosity of the lubricantη0 atmospheric dynamic viscosity of the

lubricantθ1, θ2 circumferential extent of the sub-

surface compressive stress fieldλk =L/K wavelength for the kth har-

monicν1, ν2 Poisson’s ratio for cam and follower

materialsρ atmospheric bulk density of the lubri-

cantkσij kth component of the stress tensor,

where i, j ∈ {x, y}τk amplitude of the kth harmonic of the

applied shear stress k = 0 → NτE Eyring shear stress of the oilψ composite surface roughnessω camshaft angular velocity


multi-physics analysis of valve train systems: from system ......(ic) engine is heavily dependent on...

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