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This paper
presents a simple mathematical
model which describe the
evolution of unsteady mechanical
characteristics of a diesel
engineduring its thermodynamic
cycle. The approach is based on
knowledge of cylinder pressure.
The inertia effects of the masses
in movement are also considered.
Knowledge of the unsteady
mechanical characteristics at the
shaft end is possible thanks to
integration of an unsteady
friction model worked out by
Rezeika and Henein[1]. The
results obtained are satisfactory.
Keywords :
Modelling –
thermal engines - friction.
The knowledge of the
torque and instantaneous rotation
velocity of the thermal
combustion engines is capital.
These sizes result directly from
the operating mode of the engine.
Consequently, the control of the
level of performances of these
engines and the optimization of
their energy and ecological
characteristics pass by the
control of the torque and
instantaneous rotation velocity
at any moment.
Currently, the experimental ways
giving access to the dynamic
characteristics of the engine,
are summarized in:
- the use of an instantaneous
torquemeter. This means have a
lot of advantages but requires
the knowledge of the
instantaneous resistive torque.
In addition, the good sensors are
expensive and require a high
technical level.
- the instantaneous rotation
velocity measurement. The engines
can be equipped with specific
wheel for this measurement. It is
a possible way but the
calculation of the instantaneous
torque from the velocity is a
complex problem [1][2].
- the use of a cylinder pressure
sensor. The variation of the
cylinder pressure allows to
calculate the indicated torque.
Except the delicacy of this
measurement, this technique
becomes expensive if one
associates the whole of the
cylinders sensors in the case of
a use of series.
Our work develops a mathematical
model which predicts the
instantaneous variation of
rotation velocity and the torque
of the engine from the cylinder
pressure measured. We consider
the case of one-cylinder engine.
The multicylinder case (N
cylinders) is a simple extension
which consists in considering N
combustion which occurs by cycle.
To approach from the reality of
operation engines, it is
necessary to take account of
frictions which occur there. To
this effect, a model of powerful
friction was selected, which
allows to evaluate the torque in
the end of shaft.
Thus, using the model that we
developed and within sight of the
current performances of wheel of
data processing, an
unquestionable improvement in
modelling works of the
performances of the engine in
transitory mode could be carried
out by freeing the traditional
quasi-steady approach. This
approach proves to be
insufficient in particular for
the fast engines.
The development of our model is
based on the kinematics and
dynamic study of the mobile
attachment of the piston, rod and
the crank, figure (1). The
results will be discussed.
| 2.
Mathematical model of the
torque of gases |
In order to apprehend this
model, it is necessary to
establish the dynamic balance of
the mobile attachment. For that,
it is interesting to insulate and
study each one of its components.
Within sight
of the complexity of the movement
of the rod, we substitutes one
system dynamically equivalent,
made up by two specific masses
m A and m B concentrated
at points A and B, figure (2). These
two masses must satisfy the three
conditions:
One
is thus in the presence of
three equations to determine two
unknown factors m A and
mB In practice, we
gives up the third condition
and we evaluates the efforts
developed in each end by
affecting each one of them in
proper fictitious mass.
Consequently, we considering
deduces
from the two first conditions the
masses m A and m B
like :
the third
condition means that the moment
of inertia around the centre of
inertia G remained unchanged.
The assessment of the forces
applied to the crank is
represented by the equation (1)
the piston constitutes
the mobile wall of the combustion
chamber. Its rectilinear
motion transmits to the driving
attachment the energy provided
by combustion gases. On figure
(3) are represented the forces
applied to the piston due to the
pressure of gases
the forces
reactive and applied in A, the
inertia and that of gravity
 The contact
segment – shirt generates
obviously a force of friction The general
balance-sheet of these actions
leads to the balance of the
piston:
| 2.3.
Balance of the crank |
figure(4)
shows the various forces applied
to the crank. This one is
subjected to the point B with
the weight of the big end
the reactive forces of
connection
 , and
the centrifugal inertia
 In
addition, the crank constitutes
the seat of manifestations of
the engine torque
 and
a resisting torque
 and
of drive of the auxiliary bodies
of the engine (injection
pump, alternator, water pump...).The equilibrium conditions of the
crankare translated by the system
of equations (3) and (4)
the kinematics study
of the crank-connecting rod
system has as an aim the
determination of the analytical
expression speeds and
instantaneous accelerations of
each component. Taking into
account this study and
equations (1) to (4) one leads to
a system of eight equations. A
suitablemathematical treatment
solves this system and led to
the algebraic model of the
instantaneous torque of gases
according to the angle q of the
crankshaft:
This model integrates
the geometrical characteristics
of the engine, the masses
moving, the pressure of gases as
well as the forces of friction
indicated by the term
 .
| 3. Expressions
of the elementary couples
of friction |
friction piston -shirt
was the several research task
object [ 3][4][5 ]. Most
famous are those of Rezeika and
Henein[ 3 ]. These researchers
establishedfor friction piston-shirt
two correlative models. The
firstcorresponds the mode of
hydrodynamic lubrication and the
second with the mode of mixed
lubrication. whenthe piston is
in the vicinity of the PMH in
phase of compression-combustion,
of significant efforts appear
following the high pressures of
gases and the mode of
hydrodynamic lubrication
degenerates towards a mixed mode.
The model [ 3 ] was adopted to
express the elementary torque of
friction. Its formulation is as
follows:
Friction Piston –shirt
the torque of frictions
piston-shirt transmitted to the
crankshaft are expressed by:
Hydrodynamic lubrication
Mixed
lubrification
Friction skirt-shirt
the torque of friction
skirt-shirt transmitted to the
crankshaft is expressed by:
Friction
of the crankshaft bearing
the torque
of friction of the crankshaft
bearing is described by:
the
model of adopted friction also
holds account couples necessary
to the drive of the auxiliary
bodies as well as the system of
distribution. One thushas the
expressions:
Losses due to
the
distribution
the input torque of the
distribution is defined by:
the
auxiliary bodies
input torque of the
auxiliary bodies (oil pump, water
pump, injection pump,
alternator...) is considered
constant during the cycle and is
expressed like
(11)
|
the
table (1)
gives a numerical values of the
parameters of adjustment which
intervene in the equations (6) to
(11)
Table 1:
parameters of the torque of
friction
| 4.
instantaneous Speed of
the wheel |
the
determination of instantaneous
rotation speed is obtained by
application to the driving wheel
of the fundamental law of
dynamics:
After
discretization of the equation (12)
and conversion angular velocity,
we obtain:
| 5. Results
and discussions |
For the needs
for simulation, we use the data
of a diesel engine at 4 times,
with direct injection with cooled
overfeeding. This engine has
the following characteristics:
Boring: 120 mm
Race: 145 mm
Volumetric Report/ratio of
Compression: 17
l =0.25,
 =3.2
kg,
 =7
kg, L=0.28m
Fig. 5 :
experimental statements of the
pressure rolls engine considered.(N=1000
rpm;Cr=1.85 daN).
The experimental
statement of the cylinder
pressure on the test bench of
this engine was carried out to
1000 (tr/min) under a weak
average resistive torque posted
by the brake of 1,85 (daNm).
Figure (5) shows the evolution
of this pressure on a cycle.
The frequency of selected
acquisition is 7 kHz. Figure (6)
illustrates the evolution of the
instantaneous torque by
considering the case of only one
cylinder without taking account
of the effects of inertia of the
mobile attachment. This
torque is characterized by an
appreciably constant evolution
during the admission and the
exhaust whereas it reaches the
maximum in the zone of
combustion. The negative values
of the torque during the phase
of compression require the
restitution of the energy stored
in the wheel of inertia, body
essential to the correct
operation of the engine. In
addition, figure (7) shows the
variation of the instantaneous
rotation speed on the level of
the wheel. It is minimal at
the end of the compression where
the speed of the piston cancels
and reaches the maximum at the
end of the relaxation. The
calculated mean velocity being of
1000 (tr/min) corresponds well to
the in experiments measured value.
The taking into account
of the inertia of the attachment
shows that its effect is very
distinguished during the cycle
Indeed, inertia confers has
better cyclic regularity have it
is shown one the figure (8).
The instantaneous rotation speed
shown in figure(9), also sees
in the presence of the effect of
inertia its form harmonize
itself during the phases of
admission and exhaust. It is
characterized by extremism with
each time the torque is cancelled.
figure (10) represents the
evolution of the total friction
torque according to the model
of Rezeika and Henein. The
representation of the effective
engine torque, indicated torque
and of the total friction torque
is shown on the figure (11).
On this figure can see that the
report/ratio of the effective
torque on the indicated torque is
almost constant, which
indicates that the instantaneous
mechanical output oscillates on
a cycle around the average value
of 93%.
This study shows the
possibility of estimating into
instantaneous the engine torque
indicated, the effective engine
torque and the total friction
torque using the cylinder
pressure measurement. The
developed model is simple, purely
algebraic and thus does not
suffer from any problem of
instability. The model of adopted
engine friction is local and
instantaneous. It as well takes
account of the losses with the
level of the elements of
attachment, like on the level of
the auxiliary bodies and the
elements of order. the average
sizes calculated of the torque
and instantaneous speed
correspond to those measured what
leaves predict the validity of
this model. In all cases the
tendencies obtained are in
perfect agreement with those of
work [2] and [5].
 |
 |
| Fig. 6 :
instantaneous
Engine torque without
taking account of the
effects of inertia |
Fig. 9
Instantaneous rotation
speed of the wheel with
without taking account of
the effects of inertia. |
 |
 |
| Fig. 7 :
Instantaneous rotation
speed of the
wheel |
Fig. 10 :
Instantaneous total
friction torque
according to Rezeika and
Henain. |
 |
 |
| Fig . 8
Instantaneous Engine
torque with taking
account of the effects of
inertia. |
Fig. 11:
Effective torque,
indicated torque and
total friction torque. |
[1] S. Ginoux, J.C.
Champoussin : Engine Torque
Determination by Crank angle
Mesurements State of the Art ,
Future Prospects. SAE, Technical
Paper Série 970532, pp.1-5, 1997.
[2] J. Williams, M.C.
Witter : Individual Cylinder IMEP
Estimation Using Crankshaft
Angular Velocity Measurements.
SAE- n°0990-01-2001.
[3] S.F
Rezeka, N.A. Henein : A new
approach to evaluate
instantaneous frictionengine. S.A.E
Paper, n°840179, 1984
[4] S.L.
Marek, N.A.Henein : Transient
Engine and Piston Friction During
Starting. S.A.E, Paper, n°
922195, pp.1863-1869, 1998.
[5] M.
Benhassaine : Etude expérimentale
et modélisation des frottements
locaux instantanés, piston-chemise
en moteur diesel. LTE, ECL, Lyon,
France, 1993.
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