Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

57, 1, pp. 273-286, Warsaw 2019
DOI: 10.15632/jtam-pl.57.1.273

ANALYSIS OF DYNAMICS AND FRICTION RESISTANCE IN

THE CAM-TAPPET SYSTEM

Krzysztof Siczek

Lodz University of Technology, Department of Vehicles and Fundamentals of Machine Design, Łódź, Poland

e-mail: krzysztof.siczek@p.lodz.pl

Andrzej Stefański

Lodz University of Technology, Division of Dynamics, Łódź, Poland

e-mail: andrzej.stefanski@p.lodz.pl

In this paper, the influence of friction resistance and mutual contact interaction on dy-
namical properties of the cam-tappet mechanism is analyzed. A dynamical model of the
cam-tappet contact has been developed. Chosen results of numerical simulations of this
model are presented for cases with and without oil lubrication in contact zones. Various
phenomena accompanying the cam-tappet contact dynamics have been observed, e.g., chan-
ge of the global direction of tappet rotation, local oscillation of these revolutions, changes of
friction in function of frequency of the camshaft longitudinal vibrations. We confirm, that
the growing amplitude of camshaft longitudinal vibration causes an increase in the tappet
rotational speed, whereas its reducing to small values leads to stopping the tappet rotation.

Keywords: cam-tappet mechanism, friction, lubrication

1. Introduction

Many machines like combustion engines employ mechanisms forced by a roller or plane tappet
driven by cam devices. A characteristic feature observed in such mechanisms is the occurrence
of concentrated line contact between the touching surfaces. There exist conditions of elasto-
hydrodynamic lubrication (EHL) allowing variations of the friction type from the mixed to
the boundary one. The contact zone is loaded by the force changing both, in value and its
acting direction. Themain component of the sliding velocity which results from themechanism
operational cycle also varies. During the time when both surfaces remain in contact, additional
relative displacements, small in amplitude,may occur in peripheral, axial and normal directions
of the cam motion. They may be a result of torsional vibrations of the camshaft. The axial
displacements may result from bending vibrations and from forced displacement within the
axial clearance. Such displacements being normal to the contacting surfaces may arise from
changes of the loading force and bending vibrations of the camshaft. Obviously, they change the
course of contact loading.

The amount of friction between two lubricated sliding surfaces depends primarily on the
contacting materials, load, lubricant formulations and the lubrication regimes. Under boundary
and mixed lubrication conditions, where some asperities of both surfaces touch each other,
friction can be controlled by lubricant formulations and appropriate surface engineering.
Most studies on cam-follower contacts are addressed to friction and wear measurements for

investigating the influence of lubricant additives and surface coatings, finish and texture, see
Willermet et al. (1991), Soejima et al. (1999), Lindhom and Svahn (2006), Kano (2006), Lewis
and Dwyer-Joyce (2002). The friction force (or torque) is often calculated by subtracting the



274 K. Siczek, A. Stefański

contributions of other components and inertia actions from the measured values, as in Kano
(2006), Baş et al. (2003).
The action of friction forces between the cam and tappet or the valve stem can cause tappet

or valve rotations. Their effects can becomepositive, as they decrease friction resistance between
contacting surfaces. The rotations are forced when the symmetry plane of the cam is displaced
relative to the tappet or valve axis. Such behavior was described in Jelenschi et al. (2011). For a
particular application, a specific amountof valve rotation is required tomaintain even the sealing
level at the valve seat. Excessive amount of valve rotation will result in increased wear of the
contacting surfaces, limiting the engine lifetime. In the analysis described inRefalo et al. (2010),
a minimum engine speed of 3500 rpm was established as a target for valve rotation to begin.
Over a typical driving cycle, the valves can rotate at least once. The maximum value of valve
rotation can reach 15 rpm for any engine speed.Typical factors influencingvalve rotation include
oil temperature, assembly variation, machining variability, valve deposits, engine vibrations and
engine temperature.
Hyundai inKim et al. (2004) are describedwhat influencedand contributed tovalve rotation.

It was explained that during opening the valve also rotated along with the valve spring. Next,
during the closing phase, the valve moved back to its original position. As the speed of the
motion increased, there was a greater tendency for the valve to slide at both – the maximum
lift and the closing events. This slidingmotion provides a rotational net movement over a given
lift event. All thismotion is balanced by friction between each of the contact interfaces: between
the cylinder head, valve spring, valve, retainer, lock, drive mechanism, etc.
In the research described in Hiruma and Furuhama (1978) it was observed that such a

valve started to rotate after reaching the crankshaft speed of 3000rpm.While speeding up the
crankshaft, the valve rotation increased rapidly. Before reaching the level of 3000 rpm at the
crankshaft, the valve did not rotate or it rotated in varying directions. Also in Beddoes (1992)
similar behavior of random nature and different directions of valve rotation was reported.
Themain aim of this paper is to check if the occurrence of camshaft longitudinal vibrations

can decrease friction between the cam and tappet of the valve tip. Also analysis of dynamics of
cam-tappet contact is discussed.
The paper is organized as follows. In Section 2, profiles of the cam mechanism are briefly

described and illustrated. In Section 3, the dynamical model of the cam-tappet contact is in-
troduced. Results of numerical simulations of the cam-tappet dynamics in the case with and
without oil lubrication in the contact zones are demonstrated in Section 4. Finally, Section 5
contains discussion of the obtained results and conclusions.

2. Valve lift profiles in the cam mechanism

Avery aggressive cam profile is the bestwhen usedwith bucket tappets, as stated inBlair et al.
(2005). The base circle radius Rb of the cam equals 0.02m and the valve (and bucket tappet by
definition) stroke is 8.3mm. The bucket tappet is assumed flat. The bucket has the minimum
possible diameter (dt = 0.03m) in order to keep the declared width of the cam in full contact
conditions with the flat tappet surface throughout the working cycle. The design of cam profiles
was presented in Rothbart (2004). They can have a polynomial form

h = hmax
[

1+C2
(θc−θc(hmax)

β

)2
+Cp

(θc−θc(hmax)
β

)p
+Cq

(θc−θc(hmax)
β

)q

+Cr
(θc−θc(hmax)

β

)r
+Cs

(θc−θc(hmax)
β

)s
(2.1)

where p = 22, q = 42, r = 62, s = 82 is a constant power and C2 = −pqrs[(p− 2)(q − 2)(r −
2)(s−2)]−1, Cp =2qrs[(p−2)(q−2)(r−p)(s−p)]−1, Cq =−2prs[(q−2)(q−p)(r−q)(s−q)]−1,



Analysis of dynamics and friction resistance in the cam-tappet system 275

Cr =2pqs[(r−2)(r−p)(r−q)(s−r)]−1, Cs =−2pqr[(s−2)(s−p)(s−q)(s−r)]−1 are constant
factors, β = π/3 – angle of rise, hmax – cam stroke.

Fig. 1. (a) Diagram of thevalve lift h against the crankshaft angle θc = CA (crankshaft angle degree).
(b) Diagram of the normal force Nc−t between the cam and tappet against the crankshaft

angle θc = CA. (c) The instantaneous radius of curvature Rc of the cam profile against the crankshaft
angle θc = CA

The shape of the valve lift h against the crankshaft angle φ = CA is presented in Fig. 1a.
The typical trace of the normal force Nc−t between the cam and tappet against φ = CA has
been assumed to be similar as that in Taraza et al. (1999) for the crankshaft speed 1320rpm,
and is presented in Fig. 1b. The instantaneous radius of curvature Rc of the cam profile, shown
in Fig. 1c, can be calculated from the formula

Rc = Rb+h+
a

ω2c
(2.2)

where Rb is the base radius of cam, ωc – angular velocity of the crankshaft, a = d
2l/dt2, l(t) –

varying displacement (Fig. 2b).

3. Modelling of the cam-tappet contact

Mathematical models allowing prediction of lubricant film-thickness and Hertzian pressures at
the cam/tappet contact were presented in Gecim (1992). Themodel can predict the changes in
the cam/tappet interface friction due to changing operating conditions. Also amodel of tappet
spin allowing for slip at the cam/tappet interface has been included.Modelling the tappet spin
allows one to see the effects of the tappet crown radius and cam-taper angle on the interface
frictional loss. It is found that tappet rotation is affected by design and operating conditions,
and depends primarily on camshaft speed.
The use of advanced mathematical models to quantify power loss at cam/tappet contact,

tappet/bore contact and camshaft bearingswas presented inCalabretta et al. (2010). Calculated
andmeasured frictiondata for thevalve train of ahigh speedpassenger car enginewere compared
with those obtained from tests on a motored cylinder head test rig. The system friction was
measured and calculated across the operating speed rangewith different oil supply temperature.
The camshaftmodel considering both camshaft angular vibration and bendingvibrationwas

presented in Guo et al. (2015). Each follower element was treated as a multi-mass system. The
lumped masses were connected by spring elements and damping elements. The contact force
model at the cam-tappet interfaces was developed based on the elasto-hydrodynamic lubrica-
tion theory of finite line conjunction. It was that bending vibration of the camshaft wasmainly
in the normal direction at the cam-tappet interfaces. Bending vibration was mainly influenced
by overlapping of the inlet and exhaust cam functions of each cylinder. The angular vibra-
tion of the camshaft mainly focused at the fundamental frequency and the harmonic frequency
corresponding to the cylinder number.



276 K. Siczek, A. Stefański

3.1. Description of the model

The present dynamicmodel of the tappet-cam assembly is shown in Fig. 2a for the camshaft
oscillating in the z direction and rotating about such an axiswith a constant angular velocity ωc.
The tappet can rotate about its axis with the angle ε. It is assumed that friction torque MTt−p
between the bucket tappet and the valve stem of diameter d or between the tappet and pushrod
of cross-section diameter dp can be estimated as follows

MTt−p ≈
1

3
µt−pNc−tdp (3.1)

whereµt−p is the friction coefficient between the tappet andpushrod,whichmayvary depending
on the amount of oil, sliding velocity of the pushrod against the tappet, and Nc−t represents
the load of contact zone equal to the normal force between the cam and tappet (Fig. 2b). The
friction between the tappet and pushrod is complex and of amixed type, and sometimes even of
the boundary type, as wear debris and pollution can cumulate in the contact zone. The friction
coefficient can also vary with the loading in the contact zone and vibration amplitude. In order
to ignore the effect of variation of the friction coefficient µt−p on motion of the tappet, only
a constant value of that coefficient has been considered. For simplicity of calculations, it has
been assumed a constant value of the friction coefficient µt−p equal to the averaged one 0.2, and
diameter dp = 0.006m. The friction torque MTc−t between the cam and tappet is calculated
from the equation

MTc−t = µc−tNc−tAc1 sin(2πfc1t) (3.2)

Fig. 2. (a) Dynamic model of the tappet-cam assembly for the camshaft oscillating in the z direction
and rotating about such an axis. (b) The scheme of the cam-tappet contact; 1 – contact surface between
the tappet and pushrod, 2 – contact surface between the tappet and its guide, 3 – plane parallel to the
tappet frontal face and tangent to the cam surface, 4 – cam surface, 5 – camshaft, 6 – camshaft bearing,

7 – tappet guide

Longitudinal oscillations of the camshaft along the z axis have the amplitude Ac1 and fre-
quency fc1. The friction coefficient µc−t between the cam and tappet is complex and varies as
described further. The friction coefficient µt−g between the tappet and guide varies depending
on the amount of oil, sliding velocity of the tappet against its guide and load ib the contact zone.
For simplicity of calculations, it has been assumed a constant value of friction coefficient µt−g
equal to the averaged one, being 0.15. The friction torque MTt−g between the tappet and its
guide is calculated from

MTt−g =
1

2
µt−gFc−tdt =

1

2
µt−gµc−tNc−tdt (3.3)



Analysis of dynamics and friction resistance in the cam-tappet system 277

where diameter of the tappet equals dt = 0.03m. The balance of torques acting on the tappet
is described by a dynamical differential

Iyε̈+MTt−g +MTt−p = MTc−t (3.4)

where Iy is the mass moment of inertia of the tappet with respect to the axis Y (see Fig. 2b).
Substituting Eqs. (3.1)-(3.3) into Eq. (3.4), we have

Iyε̈+
1

2
µt−gµc−tNc−tdt+

1

3
µt−pNc−tdp =µc−tNc−tAc1 sin(ωc1t) (3.5)

where ωc1 =2πfc1t.

The initial conditions correspond to the situation of the rest, i.e., ε(t =0)=0, ε̇(t =0)=0.

3.2. Lubrication in the contact zone between the cam and tappet

When the camshaft rotates and undergoes axial oscillations, the oil flow in the contact zone
between the cam and tappet becomes very complex. The cammotion against the tappet caused
only by camshaft rotation allows the occurrence of an oil film over the contact zone between
the tappet and cam. This film is characterized by varying thickness and dominant fluidmotion
in the periphera θ direction. The characteristic central thickness of such an oil film is equal
to h0. Estimation of thickness h0 is described in Section 3.3. The cammotion against the tappet
caused only by camshaft longitudinal oscillations also results in varying thickness of the oil film
in thementioned contact zone. However, the dominant fluidmotion in the contact zone is in the
axial z direction. Averaged oil thickness of this oil film is equal to hv. Estimation of thickness hv
is described in Section 3.4. If camshaft motions occur both due to rotation and longitudinal
oscillations, the resulted oil film thickness h is higher than both h0 and hv. In real, the existence
of such two camshaft motions influences the oil thickness h in a very complex manner but, for
simplicity, it can be treated as a superposition. Therefore, it has been assumed that the oil
film thickness h ≈ h0+hv, where h0 is the central oil film thickness due to camshaft rotation,
hv – averaged oil film thickness due to camshaft longitudinal oscillation. When both rotation
and axis oscillation of the camshaft occurs, the resulted normal force Nresc−t can be estimated
from the equation given in Section 3.3 and corresponding to the force Nt−c. Also the resulted
friction force Fresc−t in this zone can be obtained from the equations given in Section 3.3 and
corresponding to the force Fc−t. In both these cases, the thickness h0 is substituted by h. Then
the friction coefficient µresc−tµc−t between the tappet and pushrod is estimated from the equation

µresc−t =
Nresc−t
Tresc−t

(3.6)

3.3. Elasto-hydrodynamic lubrication and the contact force model due to rotation

Highpressuredevelopedbetween the camprofile and the tappet requires conditions of elasto-
-hydrodynamic (EHD) lubrication in the contact zone. At high pressure, the viscosity of the oil
increases exponentially with pressure, and an oil film can be maintained between the cam and
the tappet, as described in Teodorescu et al. (2003). Theminimumoil thickness exists at the oil
exit, and the oil film can be assumed to stay nearly parallel along the lubricated zone. So the
oil film thickness can be estimated at the centre point of the lubricated zone. Calculation of the
central oil film thickness can be carried out from the equation

h∗0 =1.67G
∗0.421U∗0.541W∗0.059exp(−96.775w∗s) (3.7)



278 K. Siczek, A. Stefański

developed byRahnejat, and described inTeodorescu et al. (2005, 2007), Kushwaha et al. (2000),
for finite line concentrated contact conjunction for combined entraining and squeeze filmactions.
The dimensionless parameters in Eq. (3.7) are described by the following formulas

h∗0 =
h0
RC

G∗ = α1EC U
∗ = ueη0[ECRC]

−1

W∗ = Nc−t[ECRCLC]
−1 w∗s =

ḣ0
ue

(3.8)

whereh0 is the central oil filmthickness,α1 –pressureviscosity coefficient, η0 –dynamicviscosity
at the inlet of contact, ue – oil entraining velocity, EC – effective elasticmodulus,Nc−t – normal
load responsible for local deformation of the cam/tappet contact, RC – instantaneous radius of
curvature, LC =0.014m – cam width. The symbol w

∗

s represents the squeeze-roll ratio, and its
range of applicability is given, by Rahnejat, between 0 and 0.005, and in the present analysis it
is assumed to be constant and equal to 0.005. The effective modulus can be calculated from the
equation

E−1C =
1

2
[(1−ν21)E

−1
1 +(1−ν

2
2)E

−1
2 ] (3.9)

whereE1 =210GPa is the elasticmodulus of the cammaterial, E2 =210GPa – elasticmodulus
of the tappetmaterial, ν1 =0.3 – Poisson’s ratio of the cammaterial, ν2 =0.3 – Poisson’s ratio
of the tappet material.
The oil entraining velocity ue is calculated according to the formula

ue =
1

2
ωc(Rb+h(θ)+2a(θ)) (3.10)

The value of Nc−t can be approximated using the following equation presented in Guo et al.
(2011)

Nc−t = Kc−t(h(θ)−yC −yT)+Cc−t(v(θ)− ẏC − ẏT) (3.11)

where yC is the displacement of camshaft bending motion, yT – displacement of the tappet,
Kc−t =1.434·108N/m,Cc−t =115.292Ns/m–contact stiffness anddampingcoefficient between
the cam and tappet, respectively, as given in Guo et al. (2011). Values of yC and yT are not
known, so their sum can be estimated in the followingmanner. Let us assume that the course of
the cam force F against time t is the same as the course of Nc−t. Hence, we have the relationship

Kc−t(yC +yT)≫ Cc−t(ẏC + ẏT) (3.12)

so, fromEq. (3.3), thevalue of [yC+yT ](θ) canbe estimated. It is representedby theapproximate
formula

[yC +yT ](θ)≈ [Kc−th(θ)+Cc−tv(θ)−Nc−t(θ)]
1

Kc−t
(3.13)

Then the sum of velocities [ẏC + ẏT ](θ) can be estimated from the equation

[ẏC + ẏT ](θ)= ωC
∂[yC +yT ](θ)

∂θ
(3.14)

and finally the corrected form of Nc−t can be calculated from equation (3.11). The friction for-
ce Fc−t between the cam and tappet is due to two different mechanisms, the asperity contact
(boundary part Tb) and the shear of lubricant (hydrodynamic part Tv), as described inTeodore-
scu et al. (2003, 2005), Yang et al. (1996). The asperity interactionmodel is based on the theory



Analysis of dynamics and friction resistance in the cam-tappet system 279

developed by Greenwood and Tripp (1971). The boundary friction Tb was determined by Guo
et al. (2011)

Tb = τ0Aa+mPa (3.15)

where τ0 =2.0MPa is theEyring shear stress, described inRothbart (2004), m =0.17 –pressure
coefficient of theboundaryshear strength, described inGuo et al. (2011). ConsideringaGaussian
distribution of the asperities heights andfixed asperity radius of curvature, the area Aa occupied
by the asperity peaks and the load Pa carried by the asperities are calculated as in Guo et al.
(2011)

Aa = π
2(ζβσR)

2AF2(λ) Pa =
8
√
2

15
π(ζβσR)

2
√

σR
β
ECAF5/2(λ) (3.16)

where ζ is the asperity density, β – radius of curvature, σR = 0.4µm – composite surface
roughness parameter, A – Hertzian contact area, λ = h0/σR – constant. It has been assumed
that (ζβσR) = 0.055 and σR/β = 0.001, as in Guo et al. (2011). The Hertzian formula for the
contact of two cylinders can be used to calculate the contact area, see Patir and Cheng (1979)

ECA =

√

8

π
ECRCLCNc−t (3.17)

Two statistical functions F2(λ) and F5/2(λ) are defined by the equation

Fn(λ)=
1

2π

∞
∫

h0/σ

(

s−
h0
σR

)n
exp
(

−
1

2
s2
)

ds (3.18)

They can be approximated by the following formulas

F2(λ)=−0.0018λ5 +0.0281λ4 −0.1728λ3 +0.5258λ2 −0.8043λ+0.5003

F5/2(λ)=−0.0046λ
5 +0.0574λ4 −0.2958λ3 +0.7844λ2 −0.0776λ+0.6167

(3.19)

The viscous friction is given by

Tv = τ(A−Aa) (3.20)

where τ is the shear stress of the lubricant. Depending on the oil film thickness, the lubricant
may behave as aNewtonian or non-Newtonian oil film, as described in Teodorescu et al. (2003).
The behavior can be estimated by the Eyring shear stress τ0. If the shear stress is lower than
the Eyring shear stress τ0 then Newtonian behavior occurs, otherwise non-Newtonian behavior
takes place. So the shear stress can be expressed by the equation presented inGuo et al. (2011)

τ =







ηuS
h0

for τ ¬ τ0

τ0+γSp
∗ for τ > τ0

(3.21)

where η is the oil viscosity, η = η0exp(α1p
∗), η0 = 0.0057Pa/s, α1 = 1.8 · 10−8m2/N, uS –

sliding velocity, γS = 0.08 – rate of change of shear stress with pressure, and p
∗ – pressure on

the oil film described by the (Moraru, 2005)

p∗ =
Nc−t−Pa
A−Aa

(3.22)

The sliding velocity uS between the cam and tappet is calculated from the equation (Guo et al.,
2011)

uS = ωc(Rb+h(θ)) (3.23)

The total friction force Fc−t is given by the sum (Guo et al., (2011)

Fc−t = Tb+Tv (3.24)



280 K. Siczek, A. Stefański

3.4. Elasto-hydrodynamic lubrication and the contact force model due to axis oscillations

The contact between the tappet and cam is considered as the case of two parallel plates
sliding relative to each other. The loading force Nc−t is balanced by the load Pa1 carried by the
asperities and by the hydrodynamic force Pv as well as by the squeeze force PS described as

Nc−t = Pa1+Pv +Ps (3.25)

The load Pa1 carried by the asperities can be estimated from the following equation

Pa1 = A
(hv0−hv

c

)1/m
(3.26)

similarly to Yang et al. (1996). It should be remembered that A = 2bLC, where b is the Hertz
half-width of contact between the cam and tappet. The hydrodynamic force Pv, the squeeze
force PS and the velocity w are defined by formulas presented in Siczek (2016)

Pv =
6ηALcKpψv

h2v
Ps =

ηb2Aw

hv3
(3.27)

and

w = ḣv =−
[

Nc−t−A
(hv0−hv

c

)1/m
−
6ηALcKpψv

h2v

] h3v
ηAb2

(3.28)

whereKp =0.0265,ψ =0.06are coefficients characterizing hydrodynamic impacts in the contact
zone and hv0 is the oil film thickness, as described in Siczek (2016). The initial value of velocity
v =0 and the film thickness hv is equal to hv0. The initial film thickness hv0 can be estimated
assuming that, in the initial conditions, the load Pa1 carried by the asperities is equal to Pa, see
(3.16)2. After obtaining the oil film thickness hv, the force Nc−t results from equations (3.25)-
-(3.27) and the friction force Tc−t in the axial z direction from (3.15)-(3.24) by substituting force
Fc−t by Tc−t, thickness h0 by hv and velocity us by v2 given by the equation

v2 =2Ac1πf cos(2πfc1t) (3.29)

Then the friction coefficient µc−t between the tappet andpushrod is estimated fromthe equation

µc−t =
Fc−t
Nc−t

(3.30)

4. Results

The first set of numerically simulated results have been obtained for the case of a constant
(dry or mixed) friction coefficient µc−t occurring in contact between the cam and tappet, and
µt−p between the tappet and the pushrod, which are equal to 0.3. In Fig. 3, an exemplary
time-diagram of the Fc−t force obtained for the camshaft rotational speed ns = 1320rpm, the
camshaft longitudinal oscillations of the amplitude Acl =0.0001mand the frequency fcl =1Hz,
is demonstrated. It is a single segment of the force sequence for a time period of 0.1s. The entire
sequence includes a time period of 60s. Subsequent time-diagrams of the tappet rotation angle ε
for various values of the frequency fcl are shown in Figs. 4a-d. The case of tappet rotation,
corresponding to Fig. 3, is shown in Fig. 4a. An increase in such an angle with time in almost
step-like manner can bee observed.
If the frequency fcl is significantly elevated, up to 10Hz, we can observe almost a linear

increase in the angle ε (see Fig. 4b). Cyclic variations of the direction of tappet rotation occur,



Analysis of dynamics and friction resistance in the cam-tappet system 281

Fig. 3. Time-diagrams of the force Fc−t between the cam and the tappet for the frequency of camshaft
longitudinal oscillations fcl =1Hz. Values of remaining parameters: µc−t = µt−p =0.3, ns =1320rpm,

Acl =0.0001m

Fig. 4. Time-diagrams of the tappet rotation angle ε for various frequencies of the camshaft longitudinal
oscillations fcl: 1Hz (a), 10Hz (b), 20Hz (c), 100Hz (d). Values of remaining parameters:

µc−t = µt−p =0.3, ns =1320rpm, Acl =0.0001m

but the observed oscillations have a very small amplitude. Further growth of the camshaft
vibration frequency (fcl = 20Hz) causes reversal of the direction of tappet rotation. However,
as shown in Fig. 4c, cyclic variations of this direction have a quite large amplitude. The same
direction of tappet rotation and its increase with time t has been obtained for the frequency
fcl =100Hz, as it is depicted in Fig. 4d. However, in this case, variations of the tappet rotation
direction are very small.



282 K. Siczek, A. Stefański

In the next two diagrams (Figs. 5a and 5b) the influence of the amplitude change on tappet
rotation is demonstrated. In the case when the amplitude is doubled (Acl =0.0002m) and the
frequency fcl = 10Hz, the obtained time history of the tappet rotation angle ε is shown in
Fig. 5a. It can be seen that an increase in the angle ε with time occurs in the same direction as
in the case of Acl equal to 0.0001, but 4.44 times greater than the last one. On the other hand,
reducing the amplitude to a half of the initial value (Acl = 0.00005m and fcl = 10Hz) causes
stopping of the tappet rotation (see Fig. 5b).

Fig. 5. Time-diagrams of the tappet rotation angle ε for two different amplitudes of the camshaft
longitudinal oscillations Acl: 0.0002m (a), 0.00005m (b). Values of remaining parameters:

µc−t = µt−p =0.3, ns =1320rpm, fcl =10Hz

The results of numerical research corresponding to the case presented in Fig. 4b
(Acl = 0.0001m, fcl = 10Hz) but performed for two times larger rotational speed of the cam-
shaft, i.e., ns = 2640rpm, are illustrated in Fig. 6. Comparison of Figs. 3b and 6 shows that
such doubling of the speed ns does not influence significantly the rotational dynamics of the
tappet. The increase in the angle ε in time still has almost a linear character, and the speed of
the tappet rotation is about 15% higher only.

Fig. 6. Time-diagrams of the tappet rotation angle ε for the camshaft rotational speed ns =2640rpm.
Values of remaining parameters: µc−t = µt−p =0.3, fcl =10Hz, Acl =0.0001m

The second group of the results have been obtained for the case (f) when the oil is present
in the contact zones. Variations of mixed or hydrodynamic friction coefficients cause changes
in the loading and sliding velocity of the contacting surfaces. They result in varying friction
forces Fc−t (Eq. (3.24)) occurring in contact between the cam and tappet, and Ft−p (Eq. (3.1))
between the tappet and the push rod which have been calculated from the Reynolds equations.



Analysis of dynamics and friction resistance in the cam-tappet system 283

The time-diagram of the friction force Fc−t between the cam and tappet corresponding to the
graph shown in Fig. 4a (ns =1320rpm, fcl =10Hz) is depicted in Fig. 7a. The oil lubrication
causes values of the friction forces to be a few times lower (2-4) than in the case of constant
friction coefficients µc−t and µt−p. Time histories of the tappet rotation angle ε are shown in
Fig. 7b. It is clearly visible that, similarly to the case presented in Fig. 4a, an increase in the
angle ε has almost a step-like character.

Fig. 7. Time-diagram of the force Fc−t between the cam and the tappet (a) and the tappet rotation
angle ε (b) for the case of oil lubrication. Values of others parameters: fcl =1Hz, ns =1320rpm,

Acl =0.0001m

In the case when the camshaft longitudinal oscillations frequency fcl is equal to 100Hz,
the time history of the friction force Tc−t between the cam and tappet is presented in Fig. 8a.
Such values of the friction force are much lower, even two orders in magnitude lower than in
the case of frequency fcl = 1Hz (see Fig. 7a). It is due to the dominant role of hydrodynamic
lubrication in both contact zones, between the cam and tappet and between the tappet and
pushrod, respectively. Such small values of friction forces also result in lack of tappet rotation,
as it is shown in Fig. 8b.

Fig. 8. Time-diagram of the force Fc−t between the cam and the tappet (a) and the tappet rotation
angle ε (b) for the case with oil lubrication. Values of others parameters: fcl =100Hz, ns =1320rpm,

Acl =0.0001m



284 K. Siczek, A. Stefański

5. Summary and conclusions

We have carried out the analysis of friction resistance between the cam and its tappet for dif-
ferent values of the amplitude and frequency of camshaft longitudinal oscillations and camshaft
rotational speeds.Dependingon theseparameters or presence (or not) of oil lubrication, different
time diagrams of the tappet rotation angle ε are calculated and shown.
In the case of a constant value of the friction coefficient between the cam and tappet and

between the tappet and pushrod, the same camshaft rotational speed and the same values of the
oscillation amplitudes result with different shapes of time histories depending on the camshaft
longitudinal oscillation frequency. Some results are illustrated in Figs. 4a-d. They prove that
the increase in this frequency leads to a significant reduction of the average angular velocity
of tappet rotation and reversal of its direction of rotation. This velocity is represented by an
average slope of the time histories in Figs. 4a-d – its value is reduced from about 50rad/s for
fcl = 1Hz (Fig. 4a) to about −1rad/s (opposite direction) for fcl = 100Hz (Fig. 4d). Other
interesting phenomena detected during the analysis are:

• changes in the global direction of the tappet rotary motion (compare Figs. 4a,b with
Figs. 4c,d),

• cyclic variations in the direction of rotation with a relatively large amplitude (Fig. 4c),
which tend to zero with an increase in the frequency fcl, then the tappet rotates slowly
with an almost constant speed (Fig. 4d),

• step-like shape of the time-course of tappet rotation for small values of the frequency
fcl = 1Hz – one can observe revolutions interrupted with stopping periods (vertical sec-
tions of the diagram in Fig. 4a) what indicates the stick-slip character of the cam-tappet
dynamical contact (for am increasing frequency, the angular velocity of the tappet is sta-
bilized – see Fig. 4b).

Analyzing the influence of camshaft rotational speed and the amplitude of longitudinal oscilla-
tions, one can draw the following conclusions:

• growing amplitude of camshaft longitudinal vibration causes a considerable increase in the
tappet rotational speed (Fig. 5a), whereas its reduction to small values leads to stopping
of the tappet rotation (Fig. 5b),

• even a large increase in the camshaft rotational speed does not change significantly the
velocity of tappet revolutions (Fig. 6).

When the oil lubrication is applied to the contact zones between the cam and tappet and
between the tappet and pushrod, rather a large reduction of the friction force and tappet ro-
tational velocity takes place when compared to contact without any lubrication. This fact can
be treated as an obvious and expected effect. The comparisonwith corresponding cases without
lubrication (see Figs. 3 and 4a) indicates that after oil application the friction force declined
several times (Fig. 7a) and tappet rotations slowed at least 150 times (Fig. 7b).
However, our results demonstrate that further large reduction of friction can be obtained for

a high frequency of the camshaft longitudinal vibration. Comparison of the cases presented in
Figs. 7a and8adisplays that ahundredfold increase in the frequency causes almost ahundredfold
decrease in the friction force. This fact can be explained by the hydrodynamic lubrication effect
(squeeze effect) described in Guran et al. (1996). Hence, lack of tappet revolutions, depicted in
Fig. 8b, is not surprising due to the friction force which is too weak to maintain the rotations.
We can conclude that thepresentedphenomena, i.e., changes in the global direction of tappet

rotation, its local oscillations, the friction force dependence with the frequency and the other
factsmentionedaboveare effects ofmutual interaction of camshaft rotationswith its longitudinal
vibrations and possible different configuration of the phase synchrony between them. However,
an accurate identification and description of themechanism governing such phenomena requires



Analysis of dynamics and friction resistance in the cam-tappet system 285

moredetailed bifurcation analysis in the systemparameter space.This is our task for the nearest
future and the results will be reported soon.

Acknowledgment

This work has been supported by Polish National Centre of Science (NCN) under Project No.

DEC-2012/06/A/ST8/00356.

References

1. Baş H., Bıyıklıoǧlu A., Cuvalci H., 2003, A new test apparatus for the tribological behavior
of cammechanisms,Experimental Techniques, September/October, 28-32

2. Beddoes G.N., 1992, Valve materials and design, Ironmaking and Steelmaking, 19, 290-296

3. BlairG.,McCartanC.D., HermannH., 2005,Making the Cam. Valvetrain Design Part Two.
Special Investigation. Race Engine Technology, Issue 9, ISSN 1740-6803

4. Calabretta M., Cacciatore D., Carden P., 2010, Valvetrain friction – modeling, analysis
and measurement of a high performance engine valvetrain system, SAE International Journal of
Engines, 3, 2, 72-84, DOI: 10.4271/2010-01-1492

5. Gecim B.A., 1992, Tribological study for a low-friction cam/tappet system including tappet spin,
Tribology Transactions, 35, 2, 225-234

6. Greenwood J.A., Tripp J.H., 1971, The contact of two nominally flat rough surfaces, Proce-
edings of the Institution of Mechanical Engineers, 185, 625-633

7. GuoJ., ZhangW., ZhangX., 2015,Modeling andanalysis of the transientvibrationof camshaft
in multi-cylinder diesel engine,Advances in Mechanical Engineering, 7, 11, 1-14

8. GuoJ., ZhangW.,ZouD., 2011, Investigationofdynamic characteristicsof a valve train system,
Mechanism and Machine Theory, 46, 1950-1969

9. Guran A., Pfeiffer F., Popp K., 1996, Dynamics with friction: modeling, analysis and expe-
riment: (Part I),World Scientific, 23.08.1996 – 264

10. Han Y., Chan C., Wang Z., Shi F., Wang J., Wang N., Wang Q.J., 2015, Effects of shaft
axialmotion andmisalignment on the lubrication performance of journal bearings via a fastmixed
EHL computing technology,Tribology Transactions, 58, 2, 247-259

11. Hiruma M., Furuhama S., 1978, A study on valve recession caused by non-leaded gasoline –
measurement bymeans of R.I.,Bulletin of JSME, 21, 147-160

12. Jelenschi L., CofaruC., SanduG., AleontemM., 2011, State of the art of engine valve and
tappet rotation,Bulletin of the Transilvania University of Braşov, Series I, 4, 53, 2

13. Kano M., 2006, Super low friction of DLC applied to engine cam follower lubricated with ester-
containing oil,Tribology International, 39, 1682-1685

14. KimD.,LeeJ.,LeeJ., 2004,Effectofvalvetraindesignparametersonvalve rotationperformance
of a diesel engine, SAE Technical Paper, F2004V281

15. Kushwaha M., Rahnejat H., Jin Z.M., 2000, Valve-train dynamics: a simplified tribo-
elastomulti-body analysis,Proceedings of the Institution of Mechanical Engineers, Part K: Journal
of Multi-Body Dynamics, 214, 5-110

16. Lewis R., Dwyer-Joyce R.S., 2002, Wear of diesel engine inlet valves and seat inserts, Pro-
ceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering,
216, 205-216

17. LindhomP., SvahnF., 2006, Study of thickness of sputtered-carbon coating for low friction valve
lifters,Wear, 261, 241-250



286 K. Siczek, A. Stefański

18. Meng F., Wang Q.J., Hua D. Liu S., 2010, A simple method to calculate contact factor used
in average flowmodel, Journal of Tribology, 132, 2, 024505

19. Moraru L.E., 2005, Numerical predictions andmeasurements in the lubrication of aeronautical,
engine and transmission components, Doctoral Thesis, The University of Toledo

20. Patir N., Cheng H.S., 1978, An average flow model for determine effects of three dimensional
roughness on partial hydrodynamic lubrication, Journal of Lubrication Technology, 100, 1, 12-17

21. Patir N., Cheng H.S., 1979, Effect of surface roughness orientation on the central film thickness
in EHD contacts,Proceedings of the 5th Leeds-Lyon Symposium on Tribology, 15-21

22. Refalo K., Bowyer S., Smith M., Hijawi M., 2010, Optimizing valve rotational speed using
Taguchi techniques, SAE Technical Paper, 2010-01-1096

23. Rothbart H.A., 2004,Cam Design Handbook, McGrawHill, NewYork

24. Siczek K., 2016,Tribological Processes in the Valve Train Systems with Lightweight Valves. New
Research and Modelling, 1st Ed., Elsevier

25. Soejima M., Ejima Y., Wakuri Y., Kitahara T., 1999, Improvement of lubrication for cam
and follower,Tribology Transactions, 42, 4, 755-762

26. Taraza D., Henain N.A., Teodorescu M., Ceausu R., Bryzic W., 1999, Dynamics and
friction of valve trains,Arc Annual Meeting, University of Michigan

27. Teodorescu M., Kushwaha M., Rahnejat H., Taraza D., 2005, Elastodynamic transient
analysis of a four-cylinder valve train systemwith camshaft flexibility,Proceedings of the Institution
of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 219, 13-25

28. TeodorescuM., TarazaD.,NaeimA.H., 2003, Simplified elasto-hydrodynamic frictionmodel
of the cam-tappet contact, SAE Paper, 2003-01-0985

29. Teodorescu M., Theodossiades S., Rahnejat H., 2007, Multi-physics approach to design
analysis of powertrain sub-systems, 12th IFToMM World Congress, Besançon, France

30. Willermet P.A., Pieprzak J.M., Dailey D.P., Carter R.O., Lindsay N.E., Haack L.P.
and deVries J.E., 1991, The composition of surface layers formed in a lubricated cam/tappet
contact,Transaction of the ASME, Journal of Tribology, 113, 38-47

31. Wu C.W., Zheng L.Q., 1989, An average Reynolds equation for partial film lubrication with a
contact factor, Journal of Tribology, 111, 188-191

32. Yang L.S., Akemi I., Hideo N., 1996 A valve train friction and lubrication analysis model and
its application in a cam/tappet wear study, SAE Paper, 962030

Manuscript received April 7, 2017; accepted for print November 13, 2018