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Y H ZWEIRI, J F WHIDBORNE AND L D SENEVIRATNE392
T r reciprocating inertia torque (N m)
torsional stiffness torque (N m)T Svelocity of the slider (m/s)U
V d displacement volume (m3)
W load on the slider (N/m)
x Cartesian coordinates in the x direction
piston displacement (m) y
h weighting coefficienti connecting rod angle (rad)
l piston pin offset (m)
Dt time step (s)
q crankshaft angular position (rad)
q1 dynamometer angular position (rad)
v dynamic viscosity of the oil (N s/m2)
x characteristic and inclination angle of the
upper of a tilted ring profile (rad)
oil density (kg/m3)z
� connecting rod angle when the piston is at
TDC (rad)
1 INTRODUCTION
The direct-injection (DI) diesel engine model has long
been established as an effective tool for studying en-
gine performance and contributing to evaluation and
new developments. Most of the work done in this area
has concentrated on steady state models for the pur-
pose of modifying engine design parameters in order
to minimize emissions and maximize power and fuel
economy of the engine. However, recent regulations
have imposed stringent emission and fuel economy
standards that cannot be addressed by a steady state
analysis of the engine. Simulation of transient engine
response is needed to predict performance and fuel
economy of diesel engines that frequently experience
rapid changes in speed and load. Hence, to contribute
towards solving this problem, the current research
work is conducted with the aim of developing a
non-linear dynamic model for direct-injection single-
cylinder diesel engines that can simulate the engine
performance under transient and steady state operat-
ing conditions.
Previous efforts in the area of engine dynamic mod-
elling can be grouped into two major categories:steady state non-linear and transient non-linear mod-
els. Examples of steady state non-linear models can be
found in references [1] t o [4], which simulate real
spark-ignition engines in order to estimate engine
torque and cylinder pressure.
Some examples of transient non-linear dynamic
models can be found in reference [5], where a model
composed of thermodynamic and dynamic constitutive
elements for a transient, multicylinder diesel engine
simulation is developed. This model utilizes a quasi-
steady thermodynamic process. A comparison of pre-
dicted and measured pressure traces during the
transient response was satisfactory overall, but also
indicated some limitations of the quasi-steady process
submodels, and so Filipi and Assanis [6] have ex-
tended the steady state diesel engine simulation to
include the prediction of instantaneous engine speed
and torque during transients.
Important aspects of engine dynamic operation are
the instantaneous torque and the cyclic nature of the
gas pressure force and the slider– crank kinematics.
Therefore, the objective of this work is to develop a
non-linear single-cylinder diesel engine model with full
transient capability explaining the relationship between
the net engine torque and the angular speed of the
crankshaft. Another aspect that plays an important
role in engine transient modelling is the evaluation of
frictional losses, especially piston assembly friction be-
cause it is a factor that strongly affects the economy,
performance and durability of the reciprocating inter-
nal combustion engines. The model takes this pointinto consideration in dealing with the detailed analysis
of engine friction components. Another advantage of
this model is that full dynamic dynamometer mod-
elling with step loading (to avoid the chatter effect) is
included. In addition, the piston pin offset has been
taken into consideration during the transient analysis,
and motoring analysis capability could also be imple-
mented. This paper presents the salient features of the
developed model, along with a brief description of a
SIMULINK [7] implementation. The dynamic engine
operation is illustrated by simulation results, and the
predicted engine response is validated through com-parison with measured data from two different en-
gines.
The paper is arranged as follows. Firstly, the engine
and dynamometer model, which is composed of the
engine and dynamometer dynamic model, the instanta-
neous single-cylinder engine torque model and the fric-
tion torque model, is formulated on a crank angle
basis. Next is a description of the implementation,
followed by some simulation results to show the model
behaviour and validation. Finally, there is a discussion
and some conclusions are drawn.
2 ENGINE MODELLING
2.1 Engine dynamic model
Figure 1 shows a model of the engine coupled to a
dynamometer. The following two equations describe
the dynamic system:
T i− %5
k =1
T f k −T r=J q 8 +T S+T D (1)
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DYNAMIC SIMULATION OF A SINGLE-CYLINDER DIESEL ENGINE 393
T D+T S=J 1q 8 1+ %
n
j =1
T L j (2)
The above equations simply state Newton’s second law
for a rotational body. The variables used for these and
all the other equations are defined in the Notation. The
indicated engine torque, T i, is generated by the conver-
sion of chemical to thermal to mechanical energy dur-
ing the combustion process. The reciprocating torque,
T r, is produced by the motion of the piston assemblyand the small end of the connecting rod. The recipro-
cating torque and the friction torque terms, k =1
5 T f k ,
are subtracted from the instantaneous indicated torque
value to produce the brake torque at the shaft. The
resistance torque, j =1
n T L j , which is the result of exter-
nal loading imposed on the engine by the dynamome-
ter, is in addition to the dynamometer inertia. Owing to
rapid changes in the cylinder pressure and consequent
changes in the forces acting on the crank during a cycle,
the instantaneous rotational speed of the crankshaft is
unsteady during any engine cycle, even if the mean
speed is constant. The torsional stiffness torque, T S,
and damping torque, T D, which depend on the stiffness
and damping in the coupling between the engine and
dynamometer, are given by the linear relationships
T S=S (q−q1) (3)
and
T D=D(q: −q: 1) (4)
2.2 Instantaneous engine torque model
Figure 2 shows the piston–crank mechanism with ap-
proximate kinetically equivalent point masses replacing
the connecting rod. The model includes the piston pinoffset. Important geometrical parameters are the
crankshaft angular position, q, the angle of the con-
necting rod, i, the crank radius, r, which is equal to
half of the stroke, the connecting rod length, L, the
piston pin offset, l, and the connecting rod angle when
the piston is at top dead centre (TDC), �.
The relationship between the indicated gas pressure,
Pi, and the indicated torque, T i, is deterministic and is
a function of engine geometry. This relationship is
expressed as
Fig. 2 Forces and acceleration of the piston–crank mecha-
nism
T i=( pi− patm)ArG (q) (5)
where
G (q)=sin(q+i)
cos i=sin q+'1−u
ucos q (6)
and
u=1−l+r sin(q−�)
L
n2(7)
From the piston–crank geometry, the piston displace-
ment, y, is given by
y=(r+L)2−l2− [L cos i+r cos(q−�)] (8)
where angles � and i can be expressed as
�=sin−1l
r+Land
i=sin−1l+r sin(q−�)
L(9)
2 .2 .1 Reciprocating torque, T r
This term is the torque produced by the motion of the
engine reciprocating components and is given as
T r=MrG (q) y =MrG (q)[G 1(q)q: 2+G 2(q)q
8 ] (10)
where geometrical functions G 1(q) and G 2(q) are
G 1(q)=r!
cos(q−�)
1+(r/L) cos(q−�)
u3/2
n
−'1−u
usin(q−�)
"(11)
G 2(q)=r
sin(q−�)+'1−u
ucos(q−�)
n(12)
Fig. 1 Engine and dynamometer model
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Y H ZWEIRI, J F WHIDBORNE AND L D SENEVIRATNE394
where M is the mass of the piston, rings, pin and small
end of the connecting rod, and y is the acceleration of
the reciprocating components. The connecting rod is
treated as an equivalent mass system, the first concentric
mass is assumed to be connected to the crankpin as a big
end while the second concentric mass is attached to the
piston assembly as a small end. The forces acting on the
connecting rod, the inertia forces and the bearing forces
act at the ends of the rod. It is assumed that the big endof the connecting rod may be placed at the crankpin
centre rather than at the correct point. Thus, there are
no transverse components of the force between the ends
of the rod to bend or shear the link, and therefore the
member is in axial tension or compression.
Implementation of the instantaneous torque model
obviously requires accurate masses of the reciprocating
components in addition to the detailed engine geometry.
The piston pin is slightly offset in order to reduce engine
noise and wear during the change in direction of the
normal force on the piston at the end of compression.
2.3 Friction torque model
2 .3 .1 Piston ring assembly friction torque, T f 1
The literature [8–10] suggests that the piston ring assem-
bly may be responsible for 50–75 per cent of the entire
engine friction. The components that contribute to
friction are: compression rings, oil control ring, piston
skirt and piston pin. The forces acting on the piston
assembly include static ring tension, the gas pressure
force and the inertia force. The piston assembly friction
is dominated by the ring friction components [11]. This
model takes into account only the hydrodynamic lubri-cation, since the friction torque is identically zero at the
top and the bottom dead centre position. The piston
assembly friction torque is expressed as
T f1=F f1rG (q) (13)
where
F f1=sgn( y; )% F f
RLi +F f
SL
n(14)
where sgn( y; ) is the signum function (i.e. the sign of
friction force is the same as the sign of piston velocity)
defined as
sgn( y; )=>1, y; \0
0, y; =0 (15)
−1, y; B0
The present approach is based on calculating the piston
assembly friction using a simplified model [12, 13] that
is based on hydrodynamic lubrication. The lubrication is
considered to be one-dimensional as both ring and bore
are assumed to be perfectly circular with the same centre,
in which case the clearance in the circumferential direc-
tion is constant, the ring is considered to be infinitely
long and there is no gap effect. In this case the Reynoldsequation becomes
(
(x
h3
v
( p
(x
=−6U
(h
(x+12
(h
(t(16)
The load equation is
W =& B
0
p dx (17)
and the friction force is
F f =& B
0
−
h
2
( p
(x+vU
h
dx (18)
By integrating the Reynolds equation twice with
boundary conditions x=0, p= p1(t) and x=B , p=
p2(t), the oil-film pressure is expressed as
p=6{U −2(hl −hm)/[tan x(Dt)]}vB hm
2 K
× 1
h2
−K +1
h2
2(K +2)−
1
K +2
n+ p1
+( p1− p2)(K +1)2(h2
2−1)
[(K +1)2−1]h2
2(19)
where
h2=h/hm and K =B tan x
hm
(20)
Finally, from equation (18) the friction force per circum-ferential unit length is
F f
l =
hm
2[ p1− p2(K +1)]+
1
2
W
l
tan x
+vUB
hmK ln(K +1) (21)
where W is obtained from equation (17).
2 .3 .2 Bearings friction torque, T f 2
Bearings friction contributions come from the journal
bearings and their associated seals. Journal bearings areusually designed to provide a minimum film thickness of
about 2 mm. The journal bearings operate under hydro-
dynamic lubrication, which means a large load can be
carried by the journal bearing with low energy losses
under normal operating conditions. Following work
done by Rezeka and Henein [14], the friction torque, T f2,
in the bearing is expressed as
T f2=hADb
2( pi− patm)
cos q q:
(22)
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DYNAMIC SIMULATION OF A SINGLE-CYLINDER DIESEL ENGINE 395
Fig. 3 Transient diesel engine model representation using SIMULINK
2 .3 .3 Val 6e train friction torque, T f 3
The valve train carries high loads over the entire speed
range of the engine. Loads acting on the valve train at
lower speeds are due primarily to the spring forces,
while at higher speeds the inertia forces of the compo-
nent masses dominate. From reference [15] (p. 738), the
friction torque, T 3, in the valve train is expressed as
T 3=169.8(1−0.00127q: )nivD iv
1.75V d
2d 2r
n(23)
2 .3 .4 Pumping losses torque, T f 4
The pumping work is the integral of the product of the
pressure and the volume over inlet and exhaust strokes.
The work measures two effects: the first is the restric-
tions outside the cylinder, in the inlet and exhaust
systems—the air filter and intake manifold on the inlet
side and the exhaust manifold, muffler and tail pipe on
the exhaust side; the second effect is the valve flow,
which corresponds mainly to pressure losses in the inlet
and exhaust valves. The pumping losses torque is the
summation of the two effects. From reference [15] (p.
728), it is expressed as
T 4=1.0618 V d
2.28
(nivncD iv
2 )1.28nq: 1.7 (24)
2 .3 .5 Pumps friction torque, T f 5
The pumps are employed to circulate the oil, water and
fuel. From reference [16] (p. 246), the pumps friction
torque is expressed as
T 5=6.79×10−6zV d(2r)2.5
'D
v
nq: 2.5 (25)
The formulation obtained by the three models pre-
sented in this section leads to a set of non-linear
differential equations. These can be numerically inte-
grated to obtain the simulated engine performance.
3 MODEL IMPLEMENTATION
The simulation is created using MATLAB/SIMULINK
[7, 17]. Figure 3 shows the structure of the single-cylin-
der diesel engine SIMULINK model. The main advan-
tage of SIMULINK is its capability to represent the
entire engine model by an assemblage of interconnected
blocks. Also, it has eight variable-step solvers and six
fixed-step solvers for the integration of differential
equations, and hence the most suitable integration
method can be chosen. Input design parameters are
passed on to the blocks from an input file, but all of theoperating parameters come from the block (functions)
for the other components of the system.
4 MODEL BEHAVIOUR AND VALIDATION
In order to validate the behaviour of the engine dy-
namic model with experimental results, simulations
have been performed for two single-cylinder diesel en-
gines labelled A and B. Some geometrical specifications
for engine A are shown in Table 1.
Table 1 Engine A geometrical specifications
130 mmBore, d 80 mmCrank radius, r
Connecting rod length, L 269.3 mm15Compression ratio, c
Piston pin offset, l 1.69 mmEngine moment of inertia, J 1.4 kg m2
Dynamometer moment of inertia, J 1 0.37 kg m2
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Y H ZWEIRI, J F WHIDBORNE AND L D SENEVIRATNE396
Fig. 4 Comparison between the predicted (—) and measured (· · ·) instantaneous speed of engine A with
dynamometer load (– – –)
Fig. 5 Fluctuation in the crankshaft angular acceleration of engine A during transient response
A comparison between predicted and measured val-
ues of the crankshaft instantaneous angular velocity
during engine transient behaviour is illustrated in Fig.
4. The measured values are taken from reference [6].
Almost no external load is imposed by the dynamome-
ter for the first two seconds, so the engine accelerates
from low idle speed and passes through the entire speed
range until it is at high idle speed. The acceleration
during the transient is shown in Fig. 5; the engine
accelerates because the net torque value is positive. As
shown in Fig. 4, between 2.4 and 3 s, the dynamometer
increases the external load in order to keep the engine
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DYNAMIC SIMULATION OF A SINGLE-CYLINDER DIESEL ENGINE 397
speed constant, hence acting as a cut-off of the fuel
pump. Finally, the external load is increased signifi-
cantly after 3 s in order to reduce the engine speed
while indicated torque remains at the same value. The
resulting net torque value is negative, so the engine
decelerates as shown in Figs 4 and 6.
The overall agreement between the measured and
predicted traces is excellent. The very small discrepan-
cies at a speed of 1900–2180 r/min are linked to using
dynamometer step loading rather than fuel pump cut-
off to avoid engine over-running, inaccuracies in the
values of the engine model parameters, changes in
Fig. 6 Fluctuation in the crankshaft angular deceleration of engine A during applied external loads
Fig. 7 Fluctuations in the indicated torque on the crankshaft of engine A during steady state low idle speed
over one engine cycle
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Y H ZWEIRI, J F WHIDBORNE AND L D SENEVIRATNE398
Fig. 8 Fluctuations in the instantaneous crankshaft angular speed of engine A during steady state low idle
speed (from t1 to t2 represents one engine cycle)
Fig. 9 Steady state high idle crankshaft speed of engine A
friction and oil viscosity during transient process and to
the fact that the engine angular velocity fluctuations are
subject to the effect not only of engine torque but also
of the reactive forces from the engine and dynamometer
mounting in a sharp transient operation.
The instantaneous torque produced by the engine at
low idle speed over one engine cycle (two revolutions) is
shown in Fig. 7. The maximum torque value represents
the maximum pressure in the cylinder during the combus-
tion stroke. As a consequence of the huge fluctuations
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DYNAMIC SIMULATION OF A SINGLE-CYLINDER DIESEL ENGINE 399
Fig. 10 Fluctuation in the indicated torque on the crankshaft of engine A during transient response
in engine torque during the cycle, the variations in the
instantaneous crankshaft rotational speed are obvious
between t1 and t2 in Fig. 8; the sudden drop in the speed
and its subsequent increase can be linked with the
negative and positive peaks of the engine torque. The
amplitudes of the cyclic speed fluctuations tend to
increase as the mean engine speed decreases owing to the
fact that at low engine speed the cycle time is long and
the engine deceleration at the end of the compression
stroke is dominant and vice versa, as shown in Figs 8 and
9. This is a very important criterion in making compro-
mises between the flywheel size, the engine speed of
response and the engine low idle speed limit. Owing to
the harmonic motion of the reciprocating assembly, the
relation between the phase of indicated torque and the
instantaneous engine acceleration is virtually identical, as
shown in Figs 10 and 5.
To test the dynamic model behaviour under steady
state with exerted external load, a simulation has been
performed for a low-speed single-cylinder diesel engine;
some geometrical specifications for engine B are shownin Table 2.
Figure 11 shows the predicted instantaneous angular
velocity during the starting process until steady state
angular velocity at rating torque. The starter-off speed
was about 40 rad/s. After the starter torque is turned off,
the angular velocity is accelerated by increasing input
torque to achieve rating indicated torque. The external
load torque (from the dynamometer) was increased
between 4 and 9 s in three steps (to avoid the chatter
effect) until it reached the rating load torque (maximum
engine output torque) and the steady state angular enginespeed was achieved. Figure 12 shows the comparison
between predicted and measured steady state angularengine velocity at rating torque and it is in excellentagreement.
Figure 13 shows the average values of the engine
friction components at steady state. The piston assemblyfriction torque is about 60 per cent, the crankshaft and
camshaft bearing friction torque is about 15 per cent andthe valves and pumping friction torque are nearly equal
at steady state and are about 10 per cent each. Finally,the pump losses are about 5 per cent. Figure 14 illustratesthe mean engine acceleration (averaged over each enginecycle) to describe the engine step acceleration decreases
during dynamometer loading steps and the zero averageacceleration at steady state.
5 DISCUSSION AND CONCLUSIONS
This paper presents a dynamic model for a single-
cylinder diesel engine that can simulate engine per-formance under both transient and steady state operating
Table 2 Engine B geometrical specifications
100 mmBore, d 62.5 mmCrank radius, r218.8 mmConnecting rod length, L18Compression ratio, c
Piston pin offset, l 1.75 mm1.7 kg m2Engine moment of inertia, J
Dynamometer moment of inertia, J 1 0.37 kg m2
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Y H ZWEIRI, J F WHIDBORNE AND L D SENEVIRATNE400
Fig. 11 Instantaneous speed of engine B from starting (—) with dynamometer load (– – –)
Fig. 12 Comparison between predicted (— ) and measured (– – –) steady state fluctuation speed of engine
B at rating torque
conditions. The model has been implemented in SIM-
ULINK. Validation has been performed for two types
of diesel engine, one for transient response and the
other for steady state. Predicted profiles of the instanta-
neous engine speeds through the transient and steady
state are in excellent agreement with measurements.
The model includes all the engine friction components,
namely the piston assembly, the crankshaft bearings,
the valve train, the pumping losses and the pumps.
Figures 4 and 12 illustrate the importance of this
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DYNAMIC SIMULATION OF A SINGLE-CYLINDER DIESEL ENGINE 401
friction during both transient response and at steady
state. The model also includes consideration of the
piston pin offset. A dynamic dynamometer model is
also included, which enables a variety of engine tests to
be carried out.
The model has been developed with the aim of
investigating different strategies for transient fuel con-
trol. The work presented here does not include any
modelling of the thermodynamic processes within the
engine. Work is ongoing in developing such models. In
Fig. 13 Average friction torque components for engine B at steady state: (a) piston assembly; (b)
crankshaft bearings (– · –·), pumping (—), valve train (······) and pumps (– – –)
Fig. 14 Mean acceleration of engine B from starting until steady state
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