Liquid piston engines by Gupta, Aman Narayan, Sunny Sharma, Shubham

94 Liquid Piston Engines

The main component of friction force acts between the cam and the fol-
lower. The loading on cam is due to spring force and the inertia load. The
contact may be sliding or rolling. These contacts are of mixed lubrication or
boundary lubrication regime due to high load which acts on a small area.

The losses in pump are due to intermeshing friction, bearing friction,
internal fluid friction. The torque is proportional to the third power of the
gear diameter and speed.

Poor lubrication has minimum effects on the friction as sliding velocity
is low. The drag force at the mid-stroke, is the major contributor to the
friction power.

4.3 Effects of Varying Speeds and Loads

In boundary regime friction depends on speed. In hydrodynamic one,
lubrication these force increases with speed. The pumping torque is depen-
dent on square of speed.

The boundary lubrication regime dominates in the valve train where
the load falls with increase in speed. So as the speed falls, the boundary
lubrication friction also increases.

As the load or fueling changes gas loading, wall temperature, clearances
in the cylinder also rise. Higher loading causes a rise in friction in expan-
sion stroke and the compression stroke near the TDC. The coolant controls
the lubricant temperature with higher values at higher load. The higher
temperatures cause a fall in viscosity and the viscous force. The clearances
due to thermal expansion increase the friction force.

During the mid-stroke higher temperatures effects are predominant due
to higher velocity of skirt. Forces in bearings, torque and friction in acces-
sories increase with loading.

Richardson found that the piston ring friction during motoring condi-
tion was lower compared to firing. He found that cylinder friction during
firing case was 0–20% higher compared to a motored.

4.4 Friction Reduction

Design measures can be adopted to reduce friction in engine components
depending upon lubrication regimes. Various factors that must be taken
into account for reduction of friction are as follows:

Durability and reliability
Minimize the load on interface.

Lubrication Dynamics 95

For boundary lubrication regime reducing the normal load
acting on the contact is effective.
Reduction of friction coefficient
For hydrodynamic lubrication regime reducing of lubricated
area or the viscosity
Use of multi grade oils with higher viscosity indexes
For hydrodynamic lubrication regime an increase in the
clearance may reduce friction. To increase the oil film thick-
ness profile of skirt is important.
For mixed or boundary lubrication type of regime can be
changed by changing clearance or profile
Use of special coatings on the rings and the skirt can help to
reduce friction.

4.5 Piston-Assembly Dynamics

From the multi-body dynamics modeling effects, the piston assembly
includes study of the piston skirt, piston rings, the piston pin and the
connecting rod as defined by SAE J2612.

The skirt must have lower distortion, friction and wear, noise, higher
cooling and lubrication. Piston rings act as a seal towards gas formed pre-
venting transfer of heat.

Piston slap is due to impacting of skirt with liner seen most prevalent
during warm-up. This causes vibrations of surface which become domi-
nant speed increases. Design of piston skirt, oil film thickness formed, and
slap kinetic energy must be optimized to reduce skirt noise.

During an engine cycle, piston moves laterally form thrust force to anti-
thrust side, due to skirt to – bore gap. The reaction force due to connecting
rod resists the combined gas and inertia forces. A small tilting of piston pin
is also seen due to the moments acting on it. These motions cause a change
in lubrication from mixed at the dead center positions to hydrodynamic
one at the mid-stroke.

Research in piston dynamics during 1960s began without taking into
account lubrication to the more complex multi-body dynamics.

These motions not only affect the piston slap noise, but also ring opera-
tion and wear. These need to be reduced to control blow-by and oil con-
sumption. The piston motions are also dependent on bore polishing.

The piston-assembly friction can account for up to 40–55% of the total
engine losses, with the contribute of skirt, rings and connecting rod about
15–20%, 15–20%, and the connecting rod 10–15%.

96 Liquid Piston Engines

4.6 Reynolds Equation for Lubrication Oil Pressure

Tribology of lubricating oil plays an important role in mechanical losses
in skirt assembly of piston in internal combustion engines. About 3–5%
of total energy in the engine is dissipated from piston skirt assembly [1].
Figure 4.1 shows a typical breakdown of mechanical losses for a typical
diesel engine, wherein it is clear that piston assembly accounts for about
20–30% [24].

Nature of lubrication is hydrodynamic at mid strokes, where sufficient
oil film thickness separates skirt from cylinder wall. At dead center posi-
tions, oil film thickness is much thinner and surface asperities on skirt and
liner make contact.

In 1886 Osborne Reynolds proved that hydrodynamic pressure gener-
ated can separate two sliding surfaces. For an incompressible fluid with
constant density the Reynolds equation gets modified as:

h3 P h3 P

x 12 x z 12 z

1 (U2 U1)h (V2 V1 ) 1 (W2 W1)h (4.1)
2 x 2 z

In this relationship, the left hand side term is called pressure term, whereas
the right hand side terms are called source terms. The terms of U & U

xz

Mechanical
losses

15%

30% 25% Brake
power
Exhaust losses
losses

Cylinder 30%
cooling
losses

Figure 4.1 Break up of total dissipation of fuel energy.

Lubrication Dynamics 97

in the above relationship are known as stretching action and h, h
are
xz

known as Wedging action. The velocity difference term (V1–V2) is known
as squeezing action as shown in Figure 4.2.

Assuming that lubrication oil used is Newtonian fluid, flow is incom-

pressible, value of viscosity is constant and neglecting inertial effects, slip,

angle of inclination, pressure gradient and stretching action Reynolds

equation can be simplified as:

h3 p h3 p 6 (U2 U1)h 12 h (4.2)
xx zz xz

In order to estimate the oil pressure distribution, the above given
Reynolds equation needs to be solved. One way to do this is to neglect the
pressure distribution in one direction and consider it in other direction as
shown in Figure 4.3. The numerical method adopted uses non-dimensional

UW Stretching action V1 – V2 = h
, t

xz

h3 P +x h3 P = 1 (U2 – U1)h + (V1 – V2) + 1 (W2 – W1)h
x 12 x 12 z 2 x 2 x

Wedge action Squeeze action (bearing
(inclinded surfaces) surfaces move perpendicular
to each other)
h, h
xz

Figure 4.2 Interpretation of Reynolds equation.

P
P

X
X

Z Z

Dimension in x-dir is much larger Dimension in z-dir is much larger
than dimension in z-dir than dimension in x-dir

Figure 4.3 Variation of pressure along various directions.

98 Liquid Piston Engines

analysis of space coordinates and oil pressure developed as given by equa-
tion 4.3.

Non-dimensionalization can be obtained as follows:

xx
X

yy
Y

hh
C

p pC3
6 UX2

t tU (4.3)
C

Solution of Reynolds equation depends upon geometry of surfaces. Two
most common approximations are short and long surfaces. Substituting
these non-dimensional values the new equation gets modified as:

h3 p X2 h3 p Ch (4.4)
xx Z2 z z Xx

In order to solve this equation, finite element analysis method can
be used for which the mating surface needs to be analyzed into num-
ber of nodes as shown in Figure 4.4. A mesh was made so that nodes on
lubrication zone of skirt correlates with nodes used in finite element analy-
sis to analyze the pressure distribution.

Pi,j+1

Z Pi–1,j Pi,j Pi+1,j
Pi,j–1

X

Figure 4.4 Nodal representation of surface.

Lubrication Dynamics 99

Various gradient terms of Reynolds relationship can be solved using
Taylors approximation which yields following results:

h3 p h p3 h p3 i 1, j (hi3, j 0.5 hi3, j 0.5 ) pi , j (4.5)
x i, j 0.5 i, j 1 i, j 0.5 (4.6)
(4.7)
x2

h3 p h p3 h p3
z i, j 0.5 i, j 1 i, j 0.5 i 1, j (hi3, j 0.5 hi3, j 0.5 ) pi , j
z2

h hi 1, j hi 1,j
x 2x

Substituting these relationships and rearranging them we have:

Pi,j = Ai,j Pi,j+1 + Bi,j Pi,j 1 + CI,j Pi+1,j + Di,j pi 1,j + Ei,j (4.8)

As most of values of nodal pressure (Pi,j) are unknown hence iterative
loop must be employed to get values of fluid pressure. The process of itera-
tions must be repeated till convergence is satisfied. i.e.

pn m pn m (4.9)

i 1 j 1 i, j iteration k i 1 j 1 i, j iteration k 1

pn m

i 1 j 1 i, j iteration k

A Matlab code has been developed to analyze the lubrication behavior
of oil between piston skirt and liner considering its motion analogous to
motion of a journal in bearing. The oil film thickness (h) at a given crank
angle (θ) can be calculated from values of nominal clearance (c) and piston
eccentricity (e) as:

h = c(1 + e cos ) (4.10)

During analysis the piston is assumed to be as a short bearing and circum-
ferential pressure gradient is neglected as compared to axial pressure. Using
these assumptions the Reynolds equation discussed above gets modified as:

h3 p 6 h (4.11)
zz x

100 Liquid Piston Engines

Using boundary conditions of P , z= L 0, the closed form of
2

pressure distribution p can be expressed as:

p 3 x2 L2 sin (4.12)
c2 4 (1 cos )3

The variations in density of lubricant can be expressed in terms of
generated oil pressure (P) and density at mean liner temperature ( 0)
as [29]:

01 0.6 10 9 P (4.13)
1 1.7 10 9 P

The variations of oil viscosity can be expressed in terms of relationship
as given by Reoland [30]:

0e (4.14)

Where

1 [log 0 9.67] 138 s0 PZ 1
P 0 138 1.988 108
1

S0 and Z are constants which depend upon temperature and pressure, θ is
temperature, θ0 is bulk oil temperature.

Hydrodynamic oil pressure distribution was analyzed on piston skirt
plane for each 90° crank angle rotation at speed of 2000 RPM and nominal
skirt-liner gap (0.05 mm) as shown in Figures 4.5–4.12.

During intake stroke the peak value of pressures are close to top part
of skirt showing gradual slope. At 90° crank angle the piston is at mid
stroke and peak value of oil pressure is witnessed almost at center of skirt.
At 180° crank angle towards the end of intake stroke peak pressures are
again observed at skirt midpoint with slopes slightly towards the right side.
During mid compression stroke the peak hydrodynamic pressure starts
to shift towards bottom of skirt at 270° crank angle. At 360° crank angle
towards end of compression stroke the pressures shift towards bottom
of skirt. During middle of power stroke at 450° crank angle oil pressures

Lubrication Dynamics 101

Oil pressure-Pa 5

4 60 80
40
3 00 20
Skirt length-mm
2

1

0
80

60
40

Skirt width-mm 20

Figure 4.5 Oil pressure distribution (90° crank angle).

Oil pressure-Pa 5

4 60 80
40
3 00 20
Skirt length-mm
2

1

0
80

60
40

Skirt width-mm 20

Figure 4.6 Oil pressure distribution (180° crank angle).

Oil pressure-Pa 5

4 60 80
40
3 00 20
Skirt length-mm
2

1

0
80

60
40

Skirt width-mm 20

Figure 4.7 Oil pressure distribution (270° crank angle).

rise from mid part of skirt towards bottom part. At 540° crank angle same
trends in oil pressures were observed. During exhaust stroke peak pres-
sures are generated throughout skirt length having slopes shifting towards
bottom part of skirt.

102 Liquid Piston Engines

Oil pressure-Pa 5
4
3 20 40 60 80
2 Skirt length-mm
1 00
0
80 60 40 20

Skirt width-mm

Figure 4.8 Oil pressure distribution (360° crank angle).

5

Oil pressure-Pa 4

3

2

1

0 60 80
80 40
20 20
60 Skirt length-mm
40 00

Skirt width-mm

Figure 4.9 Oil pressure distribution (450° crank angle).

Oil pressure-Pa 5
4
3 60 80
2 20 40
1 00
0 Skirt length-mm
80

60
40
20

Skirt width-mm

Figure 4.10 Oil pressure distribution (540° crank angle).

4.7 Introduction

This part describes a mathematical model to simulate the piston second-
ary motion which is the cause of piston slap noise. Piston slap is caused
when forces in connecting rod change direction. The piston impact against

Lubrication Dynamics 103

Oil pressure-Pa 5

4 60 80
40
3 00 20
Skirt length-mm
2

1
0
80

60
40

Skirt width-mm 20

Figure 4.11 Oil pressure distribution (630° crank angle).

Oil pressure-Pa 5
4
3 20 40 60 80
2 Skirt length-mm
1 00
0
80 60 40 20

Skirt width-mm

Figure 4.12 Oil pressure distribution (720° crank angle).

cylinder liner is a major source of noise and cause wear of liner. Major
factors that affect piston slap are [26]:

a. Cylinder Bore Temperature
b. Lubrication Oil Film Thickness
c. Oil Viscosity
d. Engine speed
e. Skirt Profile
f. Skirt Roughness
g. Skirt Waviness
h. Skirt Size
i. Wrist pin offset
j. Piston-Liner gap

Motion of Crankshaft picks up lubrication oil from sump. This oil is then
transported along cylinder bore due to motion of piston, piston rings and
gravity. Oil is consumed either inside combustion chamber or it returns
to sump or is consumed by blow by gases. Piston slap takes place due to

104 Liquid Piston Engines

Mixed Hydrodynamic

Coefficient of friction Boundary
0.1

Piston

0.01 Valve Bearings
train

0.001

Duty parameter

Figure 4.13 Stribeck lubrication curve.

changes in direction of piston side forces and occurs mainly near top dead
center (TDC) position.

According to Stibeck curve, the lubrication can be classified into three
major types: boundary, hydrodynamic and hydrostatic. In the boundary
lubrication zone, the asperities in mating parts come into contact whereas
in hydrodynamic zone there is no direct contact and the film of lubricant
separates the mating surfaces. The function of piston rings of skirt assembly
is to seal pressure in combustion chamber and prevent leakage of oil from
crankcase into combustion chamber. Type of lubrication of oil changes
with operational conditions of engine. As piston reaches dead center posi-
tions, the speed of piston approaches zero and hence boundary lubrication
dominates. At mid strokes, where piston speed is at its maximum value, the
type of lubrication changes to a hydrodynamic one.

4.8 Background

Increasing demand for noise, vibration and harness comfort levels have led
to detailed study of piston dynamics motion as skirt-piston contact plays an
important roles in frictional losses in engine [41]. Piston secondary motion
is a key concept to understand slapping motion phenomenon. Various forces
as well as moments are responsible for lateral displacement motion of piston
as well as its rotatory motion about pin axis as depicted in Figure 4.14.

Piston impacts occur on either side of liner which are identified as thrust
side (TS) and anti thrust side (ATS). These contacting motions cause vibra-
tions in liner which are transmitted from engine surface. In general three
major approaches have been identified to study slapping motion. The first
one includes study of piston secondary motion without taking into consid-
eration oil lubrication effects and piston rotatory effects. This approach is

Lubrication Dynamics 105

Thrust R Anti
side et thrust
side
L Rotation Translation
eb

Lateral
motion

Figure 4.14 Piston secondary motion.

known as static method. In another method piston side force is found using
solution of lubrication equations as represented by following equation [42].

Oil film thickness is third parameter to study slapping motion. Piston-
Liner contact occurs when this thickness is minimum towards both thrust
as well as anti thrust side [43]. There are two most common methods of
studying slapping motion. The first one involves calculation of maximum
energy transfer to cylinder wall using force variations and oil film thick-
ness results. The other one involves calculation of rate of minimum oil film
thickness. As the slope of this parameter changes, the squeeze action is
initiated indicating the instance of piston slap. Another method includes
study of initiation of squeezing action and occurrence of minimum oil
thickness. This method is known as angular duration method. Up to 16
instances of slapping motion can be identified in a engine cycle, however
only 6–10 are practically observed [44]. In order to validate positions and
instances of slap, data was measured using accelerometers using various
engine operating conditions. Previous works include that of Richmond to
capture vibration data [25]. However this data included mixture of piston
slap as well as combustion noise since both occur in vicinity of TDC posi-
tion. Pruvost used spectro filters to separate the two noise sources [26].
Liu and Randall used blind source separation methods to achieve effective
separation [27]. Chen has used concept of pseudo angular acceleration to
study phase and frequency variations of slapping noise [28].

4.9 Occurrence of Piston Slap Events

Figure 4.14 shows the free body diagram of piston in which various forces
acting on body include frictional force between liner and skirt (Ff), gas

106 Liquid Piston Engines Fg

My”

b a
Fh Mx” J ” Ff

dCOG FL

dP

Figure 4.15 Piston free body.

force (Fg) and oil reaction force (Fh) acting on piston of mass m and pin of
mass mp. Piston of moment of inertia J tilts by an angle having pin offset
dp and center of gravity offset dCOG. Connecting rod exerts a lateral force Ft
on bore wall given by following equation:

Ft = mpX p Fhyd = FL sin [FG mpYP Ff] tan (4.15)

Hydrodynamic reaction force is given by solution of Reynolds equation
[29]. Frictional forces can be neglected since they contribute less [30].

Applying moment balance equation we have:

(J + M[dCOG2 (a b)2])β + Mx (a b) My dCOG (4.16)
Tf + TG + Fhyh

Changes in direction of lateral forces is an important way to diagnose
piston slaps. This occurs when Tan ϕ = 0 which is instances of TDC(ϕ = 0)
and BDC(ϕ = k , k 0, 1, 2 …) positions.

Exact values of ϕ can be determined using following relation in terms of
connecting rod length (l), crank radius (r), Crank case offset (C) as:

sin 1 rcos C (4.17)
l

When gas force is greater than inertial force, slap occurs at thrust side
of liner and vice versa.

Lubrication Dynamics 107

Dimensionless force 14
12

10 G1

G2

8

6 G3

4

2

0 +sy py 90 180
–2
–180 –90 0
Crank angle ( )

Figure 4.16 Piston force distribution.

Figure 4.16 shows graphical representation of balance between various
forces acting on piston. The coincidence points between dimensionless
gas forces ( G) and total inertial forces ( sy + py) indicates the instances
of slap. As evident from figure number of instances of slap increase with
increase of speed.

Another approach takes into account elastic deformations of piston and
oil film thickness variations to find the lateral forces [29]. Thrust side con-
tact is indicated by negative values of force whereas anti thrust values are
indicated by positive values. Hence if lateral force changes into value from
negative to positive, slap is considered at anti thrust side and vice versa.

Diagnose of slap can be done by consideration of minimum oil film
thickness both towards thrust as well as Anti thrust side [30]. Tilting
motion of piston causes squeeze action of oil film which indicates piston-
liner contact. Next method includes consideration of squeeze velocity. In
Reynolds equation the term indicates squeeze velocity of lubricant. Hence
change in its sign indicates piston slap instance. The product of oil film
force and film displacement yields the energy transferred to liner. When
lubricant squeezes, the energy is transferred to cylinder liner. Due to rise
in lubrication pressure the squeezing action slows down. The position
of maximum energy transfer is assumed to be that of slapping motion.
Figure 4.17 shows plots of steady state and transient lateral forces acting
on liner for engine running at 2000 RPM.

Whenever the side thrust force changes its direction crossing the zero
mark, a slap event is expected to occur as depicted by circles in the above
figure. Negative values of force depicts contact with thrust side whereas

108 Liquid Piston Engines

1 ×105

Lateral force (N) Steady state
Transient

0

–1
0 90 180 270 360 450 540 630 720
Crank angle

Figure 4.17 Piston side thrust force (3000 RPM).

Table 4.1 Summary of slap events (Lateral force method).

Slap Steady state Transient state

1 70° 60°
2 170° 240°
3 240° 340°
4 320° 390°
5 340° 470°
6 400° 550°
7 470° 720°
8 560°
9 720° –
–

positive values indicate contact with anti thrust side. Hence when the value
of these lateral forces change from negative to positive one, slap is expected
to occur at ATS and vice versa. Table 4.1 gives summary of slap events as
predicted by the above mentioned methods.

Lateral forces are expected to reach their maximum value in vicinity
of TDC position when contributions due to gas force is at peak. However
there may be some deviations due to crank shaft offset.

The concept of oil film thickness has been examined next in figure assum-
ing that skirt-liner gap is fully flooded with lubricant. Four events were
identified both at thrust as well as anti thrust side. Locations of instances
of slapping motion was found to be [220°, 300°, 420°, 630°] for anti thrust
side and [230°, 420°, 540°, 620°] for thrust side. During the intake stroke the
film thickness falls and then again rises during compression stroke reaching
its maximum value. This indicates lesser piston secondary motion. During

Lubrication Dynamics 109

1 10–4Oil film thickness (mm)

Thrust side
Anti thrust side

0.8

0.6

0.4

0.2

0
0 120 240 360 480 600 720
Crank angle

Figure 4.18 Oil film thickness behavior at 2000 RPM.

1 10–3 Thrust side
0.8 Anti thrust side

Energy (J) 0.6

0.4

0.2

0 600 720
0 120 240 360 480
Crank angle

Figure 4.19 Transferred energy behavior at 2000 RPM.

expansion stroke film thickness again drops to minimum value which indi-
cates development of full hydrodynamic lubrication. During expansion stoke
the film thickness rises again reaching maximum value at 720 ° crank angle.

Locations of instances of slapping motion was on basis of maximum
energy transfer method was found to be [120°, 240°, 370°, 500°, 720°] for
anti thrust side and for [130°, 200°, 480°, 630°, 710°] thrust side as shown
in Figure 4.19. The variations in values of energy are due to fluctuations in
in cylinder pressure values.

Locations of instances of slapping motion was on basis of analysis of
squeezing velocity was found to be [220°, 270°, 420°, 630°] for anti thrust
side and for [230°, 420°, 540°, 620°] thrust side.

The above mentioned location of events was validated by measuring
engine block accelerations both towards thrust as well as anti thrust side
and by doing time frequency analysis of filtering signals in 450–3000 Hz
range under full load conditions. The events having high energy are seen
having wider frequency ranges.

110 Liquid Piston Engines

Squeeze velocity (m/s) 2.5 10–7 Thrust side
2 Anti thrust side

1.5

1

0.5

0
0 120 240 360 480 600 720
Crank angle

Figure 4.20 Squeeze velocity of lubricant at 2000 RPM.

10000 10–2
14

12

Frequency (Hz) 5000 10

8

0 6
–5000
S9

S1 S2 S3 S4 S5 S6 S7 S8 4

2

0 120 240 360 480 600 720
Crank angle

Figure 4.21 Time frequency analysis of filtered acceleration signals (Thrust side).

It may be noted that 9–10 events of slapping motion were recorded both
at thrust as well as anti thrust side with highest energy level taking place
during intake stroke (event S2). There were other traces of energy levels
visible which maybe contributed to other cyclic events occurring during
operation of engine.

4.10 Literature Review

In engine the small gap between piston skirt and the cylinder liner wall
allows a small movement in lateral direction as well as rotational motion

Lubrication Dynamics 111

10000 10–12 4

12

Frequency [Hz] 5000 10

8

06

S9
S1 S2 S3 S4 S5 S6 S7 S8 4

–5000 2

0 120 240 360 480 600 720
Crank angle

Figure 4.22 Time frequency analysis of filtered acceleration signals (Anti thrust side).

of piston pin about its axis in addition to reciprocating motion of piston
during primary mode of engine operation. Although piston is held in its
cylinder bore by piston rings, there is a small gap between ring and grooves
and hence piston is free to move within this gap [31]. The existence of
piston-liner gap puts a limit on amplitude of piston motion but not on
degree of freedom in lateral as well as reciprocating motion [32].

The piston secondary motion is crucial to understand frictional forces
in piston assembly which contribute to 30–40% of total mechanical losses
and hence a major cause of inefficiency of engine [33, 34]. The piston trans-
lates from one side to liner to other side due to changing in direction of
side thrust force due to motion of connecting rod [35]. But inertial imbal-
ance as well as piston offset and reciprocating motion guides the piston to
move in a direction [34, 36], but piston collides with cylinder walls and
generates piston slap noise and engine vibrations.

Flores et al. [32] presented a computational methodology for slider
crank mechanism dynamics. The existence of translational clearance in
the slider crank mechanism makes the system highly nonlinear and the
dynamic nature of slider crank trends to be chaotic with gap increase. The
coefficient of restitution also plays an important role in dynamics of pis-
ton slap. As the coefficient of restitution decreases the motion between the
piston skirt and liner transforms from bouncing chaotic to transient and
periodic one [37]. This has been confirmed by the study of Farahanchi
and Shaw [38] who showed that sliding motion stops as the coefficient of
restitution approaches unity.

The primary motion of piston in the model has been studied by Mcfadden
and Turnbull [39] which is determined by combustion gas pressure.

112 Liquid Piston Engines

The secondary motion of piston is suppressed due to hydrodynamic
action of oil which plays a role of damper between skirt and liner. A two
degree of freedom system developed by Geng and Chen shows a corre-
lation between piston slap and induced vibrations of engine block. The
model was used to simulate piston head motion inside cylinder. The slap
induced vibration experiment is carried out and results verify the model.
However this model was limited to piston lateral motion.

Several simulations have been carried out to numerically simulate the
two dimensional model of piston slap [40]. Various parameters considered
include center of gravity offset [41], skirt profile [42], effects of variable
inertial force [43], effect of frictional force [44] and effects of lubricating
oil [45]. Another model has been developed which verifies the indirect
measurement of piston secondary motion by mounting accelerometers on
block surface to measure vibration response and to predict the piston sec-
ondary motion from impact force of slapping motion [46].

Several simulations have been carried out to numerically simulate the
two dimensional model of piston slap [47]. Various parameters consid-
ered include center of gravity offset [17], skirt profile [18,19], effects of
variable inertial force [20, 21], effect of frictional force [22] and effects of
lubricating oil [23]. Another model has been developed which verifies the
indirect measurement of piston secondary motion by mounting acceler-
ometers on block surface to measure vibration response and to predict
the piston secondary motion from impact force of slapping motion [5].
There can be several points of contact between liner and skirt as seen from
Figure 4.23.

Corners 1 or 2 or both of these can come into contact with liner when
skirt rotates counter clockwise. Similarly corners 3 or 4 or both can touch
liner as skirt moves in clockwise direction. The skirt comes in contact
with liner when its lateral displacement is greater as compared to skirt-
liner gap.

12 12

43 4 3
(I) (II) (III) (IV) (V) (VI)

Figure 4.23 Modes of contact during piston slap.

Lubrication Dynamics 113

Various conditions for impacts to occur are enlisted below:

Corner 1 in contact, Xc Xp Xc , 0
2 0
0
Corner 2 in contact, Xc Xp Xc ,
2 .
0
Corner 3 in contact, 0 Xp Xc ,
2 max

Corner 4 in contact, 0 Xp Xc , max
2

Corner 1, 2 in contact, Xc Xp Xc ,
2

Corner 3, 4 in contact, 0 Xp Xc , (4.18)
2

Subsequently various modes of piston secondary motion may be classi-
fied as in figure given below:

These modes can be expressed as:

a. Rattling motion-During this motion skirt rotates in clock
wise direction before ignition TDC position and turns its
direction after TDC position, rotating in clock wise direc-
tion with its top part striking anti thrust side of skirt as
shown in left part of the above figure. Amplitude of this
motion increased with increase of speed and load values.

Rattling Croaking Clatter noise

Th ATh Th ATh Th ATh

Figure 4.24 Modes of slapping motion.

114 Liquid Piston Engines

Thrust side Fp Anti-thrust Thrust side Fp Anti-thrust
Fi side Fi side

FSi s FSp Mp Mi FSp s FSi Mp Mi
Ms
Ms

Maximum diameter portion Center of rotation

Rattling potion Croacking motion

Figure 4.25 Force analysis during various modes of piston motion.

During this motion the inertial force component (FSi) of
side thrust force acts towards thrust side of liner as shown
in force analysis figure shown below, whereas in cylinder gas
component of side thrust force (FSp) acts towards anti thrust
side of liner.
b. Croaking motion-During this motion the top part of skirt
strikes thrust side of skirt. The inertial force component
(FSi) of side thrust force acts towards anti thrust side of liner,
whereas in cylinder gas component of side thrust force (FSp)
acts towards thrust side of liner. This mode of motion was
found to be least affected by engine speed or load conditions.
c. Clatter motion-Bottom part of skirt strikes thrust side of
liner which is typical during low piston speeds.

4.11 Piston Motion Simulation Using COMSOL

Most prominent instances of piston slap take place in vicinity of dead
center positions A multi body dynamics model of piston using COMSOL
Multi physics 7 was next used for analysis of dynamic model of piston. The
model used here includes the following:

a. Distortion of liner under thermal and assembly loads
b. Thermal expansion of the piston
c. Frictional forces and oil film action

Figures 4.26–4.30 shows quarter profiles of piston skirt analyzed using
FEA for testing cases enlisted in Table 4.2.

Lubrication Dynamics 115

Case 1

Figure 4.26 FEA Model of piston skirt (Case 1).

Case 2

Figure 4.27 FEA Model of piston skirt (Case 2).
Case 3

Figure 4.28 FEA Model of piston skirt (Case 3).

Results have been obtained for surface velocity of skirt at both thrust as
well as anti thrust sides in time as well as frequency domains.

As evident from these figures surface velocity of skirt is expected
to decrease due to higher hydrodynamic action of oil becoming more
dominant.

116 Liquid Piston Engines

Case 4

Figure 4.29 FEA Model of piston skirt (Case 4).
Case 5

Figure 4.30 FEA Model of piston skirt (Case 5).

Table 4.2 Engine parameters. Value

Parameter 68 mm
34 mm
S 121 mm
rp 62.65 mm
l 0.03 Pa-s
L 2000 RPM, 3000 RPM
μ 179 g
84 g
mpiston 100 g
mpin 363 g
msl 31.3250 mm
mt 36.9 mm
bc 6.6 10 8 kg-m2
ap 0 mm
Ipiston 0 mm
Cp
Cg

Lubrication Dynamics 117

4.12 Force Analysis

The dynamic model of piston secondary motion as simulated using param-
eters enlisted in Table 4.2. When gas acts on piston the hydro dynamic oil
film force Fh creates a moment Mh about piston pin.

The displacement of piston along liner (Z) can be expressed as:

Z rp cos θ (l2 Bs2) (4.19)

This equation may be differentiated to get values of piston velocity as
plotted in Figure 4.34.

Piston velocity (m/s) 3
Case 1

2 Case 2
1
0
–1
–2
–3
–4
0 120 240 360 480 600 720

Crank angle

Figure 4.31 Velocity of piston skirt (2000 RPM).

3 Case 4Piston velocity (m/s)
Case 3

2 Case 5
1
0
–1
–2
–3
–4

0 120 240 360 480 600 720
Crank angle

Figure 4.32 Velocity of piston skirt (3000 RPM).

118 Liquid Piston Engines

y FIC Fg

x FIC bc ap L
Fh MIC
Ff Mh + Mf

Cg FL
Figure 4.33 Piston skirt force balance. Cp

Piston velocity-m/s 15 2000RPM
10 3000RPM

5 100 200 300 400 500 600 700 800
0 Crank angle
–5
–10
–15 0

Figure 4.34 Piston velocity.

The inertial force acting along X axis (FIC) may be expressed as:

FIC (mpiston mpin msl )(rpw cos )

(rpw Bs cos )2 rpw2(rpcos 2 cos Bs ) (4.20)

l2 Bs2 l2 Bs2

Gas force acting on piston top (Fg) may be expressed in terms of in
cylinder pressure (Pg) and piston diameter (D) as:

Fg Pg D2 (4.21)
4

Lubrication Dynamics 119Inertial forces -N

40
3000RPM

20 2000RPM

0

–20

–40
0 100 200 300 400 500 600 700 800
Crank angle

Figure 4.35 Variation of Inertial force along X axis.

Inertial forces along Y axis FIC may be expressed as:

FIC (mpiston mpin msl ) de2t bc de2t de2b (4.22)
dt 2 L dt2 dt 2

As piston moves along liner the frictional force between liner and skirt
causes a shear force (τ) in oil film which can be expressed as [31]:

U
(4.23)

h

The hydrodynamic friction force Ff and its moment Mf about the wrist-
pin can be calculated based on the above shear stress and these can be
defined as follows:

Ff R (x,q)dx d (4.24)

M f R (x, )(R cos Cp )dx d (4.25)

The oil film force Fh and its moment about wrist pin Mh due to the non-
linear pressure distribution, can be calculated by the following integra-
tions [23]:

Fh R p(x, ) cos dx d (4.26)

Mh R p(x, )(ap x)cos dx d (4.27)

120 Liquid Piston Engines

The rotatory moment about wrist pin MIC can be calculated as:

M IC Ipiston de2t deb2 (4.28)
L dt2 dt 2

Further various force and moment balance equations for the system may
be expressed in form of:

Fg +FIC + Ff + FL cos φ 0 (4.29)
Fh + F IC +Fr + FL sin φ 0 (4.30)
Mh + MIC + FIC(ap bc)+FgCp FICCg + Mf 0 (4.31)
Which may be consolidated in matrix form as:

mpis 1 bc mpin 1 ap mpis bc mpin ap
L L L L

mpis 1 ap (bc ap) Ipiston mpis ap (bc ap) Ipiston
L L L L

et Fh Fr (FIC Fg Ff )Tan (4.32)
eb Mh M f FgCP FICCg FIC (ap bc )

These equations of motion are nonlinear and stiff differential equations

which can be solved using time step Runger Kutta method using initial

conditions of et 0, eb 0. The time period of simulation was taken as
2 engine cycle with time step of 0.34s.

4.13 Effects of Various Skirt Design Parameters

a) Effects of piston pin offset
The piston pin may be offset towards either thrust side or towards the anti
thrust side of liner with 0–2 mm amplitude as shown in Figure 4.36. The piston
pin offset distance inclines towards anti thrust side when Cp is negative and
towards thrust side when Cp is positive. Piston secondary motion equations
are defined by piston eccentricities normal to axis of liner. Figures 4.36, 4.37

Lubrication Dynamics 121

Cp Cp

TS ATS TS ATS

Figure 4.36 Variations of piston pin offset.

Top eccentricity [microns] 0.06 +1 mm
0.05 0 mm
0.04 –1 mm 720
0.03
0.02 120 240 360 480 600
0.01 Crank angle

0
–0.01
–0.02

0

Figure 4.37 Variations of Top eccentricities with piston pin offset.

shows the comparisons of dynamic features of piston secondary motion
using three different offset distances of 1 mm, 0 mm, 1 mm. The value
of side thrust force F = (Fg + FIC) Tan φ falls when φ inclines towards thrust
side. i.e. φ = φʹ and vice versa. Hence there is a tradeoff between two posi-
tions since in order to reduce slapping motion the piston pin is offset towards
thrust side whereas in order to reduce wear it must be offset towards anti
thrust side. The amplitude of moment due to gas pressure M increases when
offset is towards anti thrust side hence preventing tilt of skirt.

b) Effect of skirt-liner gap
Piston skirt presses against liner walls at dead center positions which
causes slapping motion of piston due to changing inertial forces. The

122 Liquid Piston Engines

0.02

Bottom eccentricity (microns) 0.01

0

–0.01 +1 mm
–0.02 0 mm
–0.03 –1 mm

–0.04

–0.05

–0.06 120 240 360 480 600 720
0

Crank angle

Figure 4.38 Variations of Bottom eccentricities with piston pin offset.

Top eccentricity (microns) 0.6
+1 mm
0 mm

0.4 –1 mm

0.2

0

–0.2 120 240 360 480 600 720
0 Crank angle

Figure 4.39 Variations of Top velocities with piston pin offset.

Bottom velocity [m/s] 0.4
+1 mm

0 mm
0.2 –1 mm

0

0.2

–0.4 120 240 360 480 600 720
0 Crank angle

Figure 4.40 Variations of Bottom velocities with piston pin offset.

Lubrication Dynamics 123

5 10–5

Tilt angle (degree) 0

–1 mm
–5 0 mm

+1 mm

–10

–15 120 240 360 480 600 720
0

Figure 4.41 Variations of piston Tilt angles with piston pin offset.

10–6
3

2

Tilt velocity (degree/s) 1

0

–1 +1 mm
0 mm
–2 M –1 mm

–3 ATS 240 360 480
–4 Ts Crank angle

–5

–6 600 720
0 120

Figure 4.42 Variations of piston Tilting velocities with piston pin offset.

impact velocity of skirt with liner causes vibrations and hence is impor-
tant source of noise. Larger skirt-liner gap causes larger impact velocities
which leads to a larger impact forces whereas smaller gap causes asperity
contact between mating surfaces. Nominal value of clearance is taken as
0.03–0.04 mm.

Larger values of clearances increase the piston lateral displacements,
however main effecting factor of slapping motion is lateral velocity of pis-
ton. The energy of impacts due to piston secondary motion (EI) consists of
contributions due to both rotational as well lateral components. i.e.

E 1 MV 2 1 J 2 (4.33)
22

124 Liquid Piston Engines

The results show that maximum values of velocity of skirt is 0.071 m/s,
0.064 m/s and 0.062 m/s for clearance values of 0.04 mm, 0.05 mm and
0.06 mm respectively and these occur in vicinity of TDC position. From the
plots of velocity of skirt it is clear top edge is moving away from anti thrust
side of liner and the tilting angle increases with increase in piston-liner
gap. Hence slap force caused by lateral component of velocity increases
with skirt-liner gap. The reason for this is that load bearing capacity of the
oil film increases with a decrease of skirt-liner gap which prevents piston
from striking the liner but this in turn increases the wear of skirt.

Apart from lateral velocity, rotational component also plays a vital role,
hence rotational velocity of skirt must be compared. Figure 4.45 shows
results for comparisons of rotational velocity of skirt. The results show that
rotational velocity falls with decrease of skirt-liner gap. This is due to the
fact that rotational moment about piston pin increases with fall of clear-
ance values which prevents further rotation of skirt about piston pin.

0.05

Top eccentricity [microns] 0.04
0.06 mm

0.03 0.05 mm

0.04 mm
0.02

0.01

0

–0.01 120 240 360 480 600 720
0 Crank angle

Figure 4.43 Variations of Top eccentricities with skirt-liner gap.

Bottom eccentricity [microns] 0.01

0

–0.01 0.06 mm
–0.02 0.05 mm
–0.03 0.04 mm

–0.04

–0.05 120 240 360 480 600 720
0 Crank angle

Figure 4.44 Variations of Bottom eccentricities with skirt-liner gap.

Lubrication Dynamics 125

0.25

0.2

Top velocity (m/s) 0.15 0.06 mm

0.1 0.05 mm
0.04 mm

0.05

0

–0.05

–0.1

–0.15 120 240 360 480 600 720
0

Figure 4.45 Variations of Top velocities with skirt-liner gap.

Bottom velocity (m/s) 0.15 0.06 mm
0.1 0.05 mm
0.04 mm 720
0.05
0 120 240 360 480 600

–0.05
–0.1

–0.15
–0.2

–0.25
0

Figure 4.46 Variations of Bottom eccentricities with skirt-liner gap.

2 10–5

0

Tilt angle (degree) –2 0.06 mm

0.05 mm
–4 0.04 mm

–6

–8

–10 120 240 360 480 600 720
0

Figure 4.47 Variations of Tilting angle with skirt-liner gap.

126 Liquid Piston Engines

3 10–6

Tilt angle (degree) 2

T3 M ATS

1

0

–1 0.04 mm
0.05 mm

–2 0.06 mm

–3

–4

–5 720
0 120 240 360 480 600
Crank angle

Figure 4.48 Variations of Tilting velocities with skirt-liner gap.

0.05

Top eccentricity (microns) 0.04

62.67 mm

0.03 62.65 mm
62.6 mm

0.02

0.01

0

–0.01 120 240 360 480 600 720
0

Figure 4.49 Variations of piston Top eccentricities with skirt length.

c) Effects of variations in skirt length
Effects of length of skirt has been investigated next. As length of skirt
increases, the load compacting area falls which in turn leads to fall in oil
pressure. This causes top velocity to fall and bottom velocity to increase.
The concentration of oil will move down towards bottom of skirt which
leads to higher oil film force Fh and its associated moment Mh. This pre-
vents further rotation of skirt and hence rotational velocity of skirt falls
with increase of skirt length.

d) Effects of engine speed
With an increase in engine speed, combustion features as well as in
cylinder pressure trace changes due to increased fuel mass injected. This
leads to increased thermal deformations due to higher temperatures.

Lubrication Dynamics 127

0.01

Bottom eccentricity (microns) 0

–0.01 62.67 mm
–0.02 62.65 mm
–0.03 62.6 mm

–0.04

–0.05 120 240 360 480 600 720
0

Figure 4.50 Variations of piston Bottom eccentricities with skirt length.

4 10–6Tilt velocity [degree/s]

2 720

0

–2 62.67 mm
62.65 mm

–4 62.6 mm

–6
0 120 240 360 480 600

Figure 4.51 Variations of piston Tilt velocities with skirt length.

Figures 4.32, 4.33 shows the variations of axial velocity of piston and piston
lateral thrust forces. As speed of piston increases, lubrication type changes
to hydrodynamic one as higher fluid pressure will develop in oil film.
Figure shows variations of piston lateral forces and velocity for the engine
under consideration. It can be seen that lateral forces on skirt increase with
an increase in engine load. It can be observed that piston speed is zero at
dead center position.

Figures 4.52 and 4.53 shows effects of higher speeds (2000 RPM, 3000
RPM). These results show that at higher speeds secondary motion of piston
decreases due to higher hydrodynamic action of lubrication oil induced at
higher speeds.

Effects of wrist pin and piston mass was further analyzed.
Figures 4.54–4.55 shows the effects of lighter piston skirt (2/3 rd of orig-
inal mass) and heavier wrist pin mass (double original mass) on piston

128 Liquid Piston Engines

0.04

Top eccentricity (microns) 0.03 3000 RPM
2000 RPM

0.02

0.01

0

–0.01 120 240 360 480 600 720
0 Crank angle

Figure 4.52 Effect of engine speed on Top eccentricities.

Bottom eccentricity (microns) 0.01

0

–0.01 2000 RPM
–0.02 3000 RPM

–0.03

–0.04 120 240 360 480 600 720
0 Crank angle

Figure 4.53 Effect of engine speed on Bottom eccentricities.

secondary motion. These plots show that these parameters have little
affects on piston secondary motion as a slight increase was observed in
skirt eccentricities with a decrease in skirt and pin mass.

e) Effects of engine load
As engine load increases in order to maintain constant speed, the amount
of charge to be brought into cylinder must increase. Hence peak in cylinder
reached inside cylinder increases. As evident from figure, the axial velocity
of piston has no significant increase with increase in load values.

Other Factors Effecting Piston Secondary Motion

f) Effects of Inertia of Connecting Rod
This factor has pronounced effect on piston lubrication as well as second-
ary motion especially at high engine speeds.

Lubrication Dynamics 129

0.05

Top eccentricity (microns) 0.04 Lighter skirt
Normal skirt

0.03

0.02

0.01

0

–0.01 120 240 360 480 600 720
0 Crank angle

Figure 4.54 Effect of skirt weight on Top eccentricities.

0.01

Bottom eccentricity (microns) 0 Normal skirt
–0.01 Lighter skirt

–0.02

–0.03

–0.04

–0.05 120 240 360 480 600 720
0 Crank angle

Figure 4.55 Effect of skirt weight on Bottom eccentricities.

0.05

Top eccentricity (microns) 0.04
Normal pin mass

0.03 Heavy pin mass

0.02

0.01

0

–0.01 120 240 360 480 600 720
0 Crank angle

Figure 4.56 Effect of pin mass on Top eccentricities.

130 Liquid Piston Engines

g) Effects of lubrication oil supply
The oil supply has major effects on friction between liner and piston skirt
[27]. The presence of oil film between skirt and liner reduces frictional
force hence reducing asperity contact. Presence of oil film does not affect
the occurrence of piston slap motion but cushions and reduces piston tiling
and bouncing across liner.

h) Effects of surface finish
Skirt is machined so piston rings can fit in its grooves [26]. These grooves
behave as reservoirs of oil for lubrication. The grooves in liner are also
honed at an angle to horizon with shallow angles of honing allowing the
flow of oil laterally. A smooth piston should need lesser oil for supporting
hydro dynamic lubrication as compared to a rough one.

I) Shape of skirt
Changing flatness of skirt changes boundary and hydro dynamic friction
of piston. A flat skirt profile shows a larger wetted area and thicker oil film
thickness hence decreasing the boundary contact. The sharp skirt profile
experiences boundary friction than a flat profile.

J) Size of skirt
This is also an important design parameter for piston assembly. Material
selection forms an important aspect of skirt design. Smaller skirts must dis-
tribute load over a smaller area, hence they tend to have more of boundary
lubrication. Skirt ovality can be increased to distribute load over a larger area.

In addition to above stated factors, the wear of skirt as well liner during
engine operation must also be taken into consideration which sometimes

0.01

Bottom eccentricity (microns) 0

–0.01 Heavy pin mass
–0.02 Normal pin mass

–0.03

–0.04

–0.05 120 240 360 480 600 720
0 Crank angle

Figure 4.57 Effect of pin mass on Bottom eccentricities.

Lubrication Dynamics 131

leads to gas blow hence increasing engine emissions. Wear factor can be
calculated by taking integral of contact pressure and piston velocity over
stroke distance.

4.14 Numerical Model of Slapping Motion

The motion of piston skirt can be considered as a model with three degree
of freedom system as depicted in Figure 4.58. The skirt can be considered
as having dual degree of freedom (Xp, ) with mass (mp) equal to 0.363 kg
and moment of inertia (Ip) equal to 7.8540 10−9 kg-m2. The engine block
can be considered as a lumped system having single degree of freedom (Xb)
with mass (mb) equal to 48.5 kg. The nominal clearance between skirt and
liner was taken as 0.05 mm.

Motion of skirt can be represented mathematically in matrix form of
equation 4.34.

Mp 0 0 Xp Cp Cp 0 Xp
0 mb 0 Xb Cp Cb Cp 0 Xb
00 Ip q 0 0C

Kp Kp 0 Xp Fx (4.34)
Kp Kb Kp 0 Xb 0
0 K MZ
0

xc/2 xc/2 x2

kp x1 kp kb
mb
kb cp I mp cp
ccb ccb
kcp kcp
y
x ccp yp ccp
D
l
mr

rc mc

c

Figure 4.58 Model of Piston secondary motion.

132 Liquid Piston Engines

4.15 Piston Side Thrust Force

The existence of clearance between skirt and cylinder liner allows the pis-
ton to move and rotate freely within the confined region resulting in piston
secondary motion and slap. The main driving force is the side thrust force
imparted to skirt by connecting rod as shown in Figure 4.59.

The frictional forces between piston skirt and cylinder liner as well as
between rings and liner act vertically along Y axis. The center of mass of
piston assembly is at horizontal offset of LX and at vertical offset distance of
Ly from connecting rod position.

The force exerted by connecting rod on piston pin can be vertically
decomposed along X as well as Y axis. As the angle of connecting rod
changes as piston moves from bottom dead center (BDC) to top dead cen-
ter (TDC) there will be lateral force pushing the piston on to the cylinder
liner. The piston side thrust force takes into consideration both inertial
forces as well as gas forces as developed by Guzzomi and is given by fol-
lowing equation in terms of crank radius-connecting rod length ratio (K):

Fx = [Fg mp r 2(cos(θ)+K cos(2θ)]) (4.35)

Dp

Piston Ffr
grooves

FIX Lx
Piston skirt

FIY Ly Frodx Ff
Frody

Figure 4.59 Free body Piston diagram.

Lubrication Dynamics 133

Where Sin2
l2 (r sin )2

4.16 Frictional Forces

The piston ring friction force is predominant in the total engine mechani-
cal loss and is the highest single contributing factor. According to Zweiri,
the piston ring pack friction force can be expressed as the product of the
elastic ring tension and the coefficient of friction. As the piston speed
increases, the friction coefficient decreases gradually until minimum at
mid stroke as hydrodynamic lubrication region is achieved. The frictional
forces between liner and skirt (Ff) and piston rings and liner (Ffr) can be
written in terms of sliding velocity of piston (V), nominal clearance (h),
lubricating oil viscosity ( ), number of piston rings (n) and shear area of
contact (As) as:

Ff = μV (4.36)
Ffr = μV (4.37)

Where As1 is shear contact area between liner and skirt and As2 is the shear
contact area between liner and rings.

4.17 Determination of System Mobility

Mechanical mobility (M) can be defined as the ratio of resulting velocity of
structure to input force causing excitation. This parameter can be used to find
the dynamic mass, stiffness and damping constant of skirt-liner system. In
frequency domain the mechanical mobility M(J ) can be expressed as: [26]

M(J ) V(J ) (4.38)
F(J )

M(J ) J ((K M 2 ) JC ) (4.39)
M 2(K JC )

Where F(J ) is exciting force in frequency domain and V(Jω) is a frequency
domain velocity response function.

134 Liquid Piston Engines

Lateral forces -N 3000 Case 2
2500 Case 1
2000
1500 –180 0 180 360
1000

500
0

–500
–1000

–360

Crank angle

Figure 4.60 Piston side Thrust force (2000 RPM).

The measurement of mechanical mobility was carried out to compute

the dynamic parameters of skirt-liner system. The response of cylinder

block was captured using accelerometer mounted on the top of engine

block normal to axis of piston motion, whereas the response of piston skirt

was analyzed using COMSOL software.

Frequency domain plots of mobility have shown that frequency range

below first anti resonance frequency a K is dominated by dynamic
m
mass of system. Hence the point mobility equation can be approximated as:

M(J ) J (4.40)
ma

Above this anti resonance frequency, response of system is spring domi-
nated and hence the point mobility may be expressed as [26]:

M(J ) Ja (4.41)
K

Figures 4.60, 4.61 shows the plots of piston side thrust forces at 2000 and
3000 RPM conditions. As seen from these plots, side thrust force changes
its direction four times in a complete engine cycle indicating instances of
piston slapping contact as depicted by circles.

COMSOL 7 software was then used to simulate piston secondary motion
and hence compute the piston velocity as shown in Figures 4.62, 4.63.

As it is seen from these plots, the velocity of piston approaches zero
values near top dead center position. Major variations in velocity profile

Lubrication Dynamics 135

Lateral forces -N 3000 Case 3
2500
2000 Case 4
1500 Case 5
1000
–180 0 180 360
500
0

–500
–1000

–360

Crank angle

Figure 4.61 Piston side Thrust force (3000 RPM).

Piston velocity -m/s 8 Case 1
6 Case 2
4
2 –180 0 180 360
0
–2
–4
–6
–8
–10
–360

Crank angle

Figure 4.62 Piston velocity (3000 RPM).

Piston velocity -m/s 8 Case 5
6 Case 3
4 Case 4
2
0 –180 0 180 360
–2
–4
–6
–8
–10
–360

Crank angle

Figure 4.63 Piston velocity (2000 RPM).

136 Liquid Piston Engines

were observed during exhaust stroke after 630° crank angle position due to
variations in inertial forces. Further using equation 5, mechanical mobil-
ity of skirt was computed for the given testing condition as depicted in
Figures 4.64, 4.65.

Variations in values of mobility shows same trends hence confirming
that mobility is least affected by change in the engine operational condi-
tions. First anti resonant frequency for skirt was found to be close to 60 Hz
range. Block velocities of engine were simulated using numerical integra-
tion of accelerometer data as shown in Figures 4.66, 4.67.

The values of block velocities were used to plot graphs of block mobility
as seen in Figures 4.68, 4.69. At lower speeds gas force acting on skirt is a
major factor affecting mechanical mobility, hence despite almost same val-
ues of first anti resonant frequency, the values of block mobility were found
to be lower as compared with piston mobility.

Using the concept of anti-resonant frequency as discussed in previous
section, various dynamic parameters of liner-piston were computed for
given test conditions. The results can be seen in Table 4.1.

Piston mobility -m/N-s 1st anti resonant Case 1
frequency Case 2
10–4

10–6

10–8 102 103
101 Frequency-Hz

Figure 4.64 Piston mobility (3000 RPM).

Piston mobility -m/N-s 1st anti resonant
10–4 frequency

10–6 Case 3
Case 4

Case 5

10–8 102 103
101 Frequency-Hz

Figure 4.65 Piston mobility (2000 RPM).

Lubrication Dynamics 137

Block velocity -m/s Case 1
Case 2
100

0

–100 –180 0 180 360
–360 Crank angle 360

Figure 4.66 Block velocity (2000 RPM).
100
Block velocity -m/s
0 Case 3
Case 4 0 180
–100 Case 5 Crank angle
–360
–180

Figure 4.67 Block velocity (3000 RPM).

Block mobility -m/N-s 1st anti resonant frequency
10–5

Case 1
Case 2

10–10 102 103
101 Frequency-Hz

Figure 4.68 Block mobility (2000 RPM).

Block mobility -m/N-s 10–5

1st anti resonant Case 3
10–10 frequency Case 4
Case 5

101 102 103
Frequency-Hz

Figure 4.69 Block mobility (3000 RPM).

138 Liquid Piston Engines

Table 4.1 Dynamic parameters of system.

Case Value Parameter
1
Liner Parameter Piston Parameter
2 a 67 Hz
3 a 65 Hz
4 [M(J )] a = 10 7 m/N-s [M(J )] a = 2.5 10 5 m/N-s
5 c 42884 (kg/s) c 109330 (kg/s)
k 4.2 109 (kg/s2)
m 23754 (kg) k 1.63 107 (kg/s2)

a 67 Hz m 63 (kg)
[M(J )] a = 10 7 m/N-s
c 42884 (kg/s) a 65 Hz
k 4.2 109 (kg/s2) [M(J )] a = 3.98 10 5 m/N-s
m 23754 (kg) c 109330 (kg/s)

a 63 Hz k 1 107 (kg/s2)
[M(J )] a = 1.99 10 7 m/N-s
c 69669 (kg/ s2) m 39 (kg)
k = 1.98 109 (kg/s2)
m = 11937 (kg) a 100 Hz 10 5 m/N-s
[M(J )] a = 3.1623
a = 63 Hz
[M(J )] a = 2.5 10 7 m/N-s c 172750 (kg/s)
c = 69669 (kg/ s2)
k = 1.5 109 (kg/s2) k = 1.98 107 (kg/s2)
m = 10105 (kg)
m = 50 (kg)
a = 63 Hz
[M(J )] a = 2.5 10 7 m/N-s a 100 Hz
c = 69669 (kg/s2) [M(J )] a = 2.5 10 5 m/N-s
k = 1.63 109 (kg/s2) c = 172750 (kg/s)
m = 10105 (kg)
k = 2.5 107 (kg/s2)

m = 63 (kg)

a = 100 Hz
[M(J )] a = 1.9 10 5 m/N-s
c = 172750 (kg/s)

k = 3.3 107 (kg/s2)

m = 83 (kg)

The rotational motion of skirt about pin axis was simulated by solving
dynamic equations of motion and compared with results obtained from
COMSOL as seen in Figures 4.70–4.74. Both the trends showed a good
correlation. As seen from figures, the angle of tilt of skirt changes at both
dead centers due to changing position of connecting rod. Piston was found

Lubrication Dynamics 139

1
Experimental

0.8 Simulated

Angle, deg 0.6

0.4

0.2 Suction Compression Exhaust
0 Power 540
0 720
180 360
Crank angle, degrees

Figure 4.70 Piston tilting motion (2000 RPM-80% load)

0.8

Experimental

0.6 Simulated

Angle, deg 0.4

0.2

0 Suction Compression Exhaust
–0.2 Power

0 180 360 540 720
Crank angle, degrees

Figure 4.71 Piston tilting motion (2000 RPM-100% Load).

0.6
Experimental

0.4 Simulated

Angle, deg 0.2

0

–0.2 Suction Compression Exhaust
–0.4 Power

0 180 360 540 720
Crank angle, degrees

Figure 4.72 Piston tiling motion (3000 RPM-Motored).

0.4

Experimental

0.2 Simulated

Angle, deg 0

–0.2

–0.4 Suction Compression Exhaust
–0.6 Power

0 180 360 540 720
Crank angle, degrees

Figure 4.73 Piston tiling motion (3000 RPM-80% Load).

140 Liquid Piston Engines

0.2 Experimental
0 Simulated

Angle, deg –0.2 Suction Compression Exhaust
–0.4 Power 540
–0.6 720
–0.8 0 180 360
Crank angle, degrees

Figure 4.74 Piston tiling motion (3000 RPM-100% Load).

Block vibration amplitude-mm 5 10–3

4

3
Experimental

2 Simulated

1

0 90 180 270 360 450 540 630 720
Crank angle

Figure 4.75 Block vibrations (2000 RPM-80%Load).

to slide for a some crank angle duration before reaching TDC along the
cylinder liner. Also piston tilting angle decreases with increase in load.

During motion of skirt from TDC to BDC, the piston tilts towards coun-
ter clock wise direction corner 1 of skirt touches liner at 135° crank angle
during suction stroke. The piston skirt tilts towards clockwise direction
during compression stroke during motion from BDC to TDC hence corner
1 touches skirt at around 240° crank angle position. During power stroke,
again counter clockwise tilt is observed as corner 2 touches liner at around
410° crank angle position. During exhaust stroke, skirt rotates in counter
clock wise direction as corner 4 touches liner at 650° crank angle position.

Figures 4.75–4.79 shows the simulated and measured vibratory response
of cylinder block as captured by accelerometer. The trend of the simulated
vibration response of the cylinder block shows a good agreement with
the measured vibration response. The plots show harmonic peaks that are
related to fundamental firing frequency of engine. In these figures, the
impact of the piston on the cylinder wall results in a sudden increase of
the vibration amplitude and this is clearly marked in the diagram where a
few impacts occurs. The induced vibration amplitude of the cylinder block

Lubrication Dynamics 141

5 10–3

Block vibration amplitude-mm 4
3 Experimental

Simulated

2
1

0 90 180 270 360 450 540 630 720
Crank angle

Figure 4.76 Block vibrations (2000 RPM-100%Load).

5 10–3
Block vibration amplitude-mm
4
3

Experimental
2 Simulated
1

0 90 180 270 360 450 540 630 720
Crank angle

Figure 4.77 Block vibrations (3000 RPM-motored).

5 10–3
Block vibration amplitude-mm
4

3
Experimental

2 Simulated
1

0 90 180 270 360 450 540 630 720
Crank angle

Figure 4.78 Block vibrations (3000 RPM-80%Load).

measured experimentally is slightly higher than the predicted induced
vibration amplitude due to severe oscillations taking place during expan-
sion stroke owing to high combustion gas pressure. Major variations were
found at 540° crank angle due to piston impacts with liner. These vibrations

142 Liquid Piston Engines Block vibration amplitude-mm

Lateral amplitude - microns 5 10–3

4

3
2 Experimental

Simulated
1

0 90 180 270 360 450 540 630 720
Crank angle

Figure 4.79 Block vibrations (3000 RPM-100%Load).

20

15

10
Case 2
Case 1

5

0

–5
0 90 180 270 360 450 540 630 720
Crank angle

Figure 4.80 Piston lateral motion (2000 RPM).

gradually decay and are induced upon next instance of impact after some
crank angle rotation.

The effects of load and speed variations on piston lateral motion were
next investigated. The datum for lateral motion of skirt was taken as its
lower edge. The range of amplitude of this motion is within 0–0.05 mm
which depicts nominal value of skirt-liner gap. As the speed of engine
increases, the side thrust force which is dependent upon speed, also
increases. An increase in the magnitude of this side thrust force acting on
the piston, results in the piston bouncing off the liner more frequently for
longer durations and the sliding duration of the piston skirt along liner
falls. At low engine speeds, the vibration response of the cylinder block
induced by the slapping contact of the piston has a longer duration till
decay as compared with higher engine speeds.

As evident from these figures, with an increase in engine speed, the
secondary motion of piston became more dominant. From these plots it is
clear that piston rotates in counter clockwise direction till 270° crank angle

Lubrication Dynamics 143

Lateral amplitude - microns 6 Case 3
5 Case 4
4 Case 5
3
2 90 180 270 360 450 540 630 720
1 Crank angle
0
–0.01

0

Figure 4.81 Piston lateral motion (3000 RPM).

position. During the course of this motion corner a comes in contact near
BDC position. During the motion of skirt from BDC to TDC position,
piston skirt changes direction of rotation to clockwise sense and corner b
comes in contact with skirt near 360° crank angle position.

An increase in engine speed causes increase in the side thrust force. This
further results in a decrease in sliding motion of skirt along liner as piston
bounces off frequently for longer time durations. This result is similar to
one discussed in [20]. At a lower values of engine speeds, the vibration
response of liner induced by the slapping motion takes longer time to decay.

4.18 Conclusion

Piston slapping motion is a major cause of noise and vibrations in engines.
In order to understand this motion, a numerical model was presented in
the present work. Various dynamic parameters of system were calculated
using concept of mobility which were later used to simulate the lateral
motion of piston as well as resulting engine block vibrations. The values of
first anti resonant frequencies of both skirt and liner were found to be near
60 Hz range and it remains unaffected by variations in the engine opera-
tional conditions. Several peaks were found in the simulated block vibra-
tions which were related to firing frequency of engine. COMSOL software
was further used to analyze the tilting motion of piston which showed a
good match with that simulated by solving dynamic equations of motion.

Amplitude of piston secondary motion was found to be maximum value
in middle of intake stroke, when lubrication action of oil is minimum.
Piston was also found to slide along liner a few crank angle degree before
TDC position. Sliding motion is less dominant during power and exhaust