Paper Number
Modeling of In-cylinder Pressure Oscillations under Knocking
Conditions: Introduction to Pressure Envelope Curve
G. Brecq1 and O. Le Corre2
1
Now at Gaz de France
Ecole des Mines de Nantes
2
Copyright © 2005 SAE International
ABSTRACT
High frequency pressure oscillations are generated
under knocking conditions within the combustion
chamber of Spark Ignition engines. Although acousticoscillation model can give the natural frequencies of
these oscillations, very few mathematical models are
today available, in scientific literature, to describe the
oscillation deadening effect. An analytical formulation of
the deadening has been highlighted. Analytical solution
has been established for future ECU implementation.
Coupling this new concept and an existing highfrequency model, an achieved model of the knocking
pressure high frequencies is compared to experimental
data. Good behavior is obtained on a natural gas fuelled
spark ignition engine.
BACKGROUND
Knock is an undesirable combustion mode occurring in
SI engines. It results in an abnormal auto-ignition of the
end gas ahead of the propagating flame front. This
phenomenon, characterized by the occurrence of
pressure oscillations within the combustion chamber
resulting in a metallic noise, can lead to irreversible
engine damage.
Because of increasing environmental concerns and
enlargement of fuel diversification, natural gas is more
and more considered as a valuable fuel for reciprocating
engines and especially SI engines. Because of the
absence of anti-knock-additive for natural gas (such as
lead tetraethyl for gasoline) and the relative high
variability of the natural gas composition (natural gas is a
crude hydrocarbon whose composition changes with
feedstock location), knocking conditions can easily occur
in gas engines. Consequently, although this
phenomenon has been extensively studied for more than
a century, renewed interest has occurred in the last 1015 years in order to detect and control its occurrence.
Either intrusive or non-intrusive techniques are used to
detect knock and preserve engines from damages. Nonintrusive sensors (accelerometer) are mainly used in
industrial applications, while intrusive techniques (mainly
piezoelectric sensors) support research developments.
Up to now, the intrusive in-cylinder pressure
measurement is likely the most investigated way to
detect knock because of the direct connection between
knock and pressure oscillations.
Signals generated by both accelerometer and pressure
sensor are generally filtered in order to provide highfrequency oscillations. This signal is then used to
calculate knock indices. Many kinds of indices exist.
Leppard [1] in 1982 was one of the first to propose an
indicator using filtered pressure signal. Later, extensions
of his work were reported in [2-4]. Gradient in-cylinder
pressure indices were developed by [3,5-7]. Lastly,
many authors [8-23] studied or used filtered pressure
indices, such as:
1. Maximum Amplitude of Pressure Oscillations
N
1
MAPO = ∑1 max pˆ
(1)
θ 0 ;θ 0 +ζ
N
2. Integral of Modulus of Pressure Gradient
ˆ
1
N θ 0 +ζ d p
IMPG = ∑1 ∫
dθ
(2)
θ0
N
dθ
3. Integral of Modulus of Pressure Oscillations
N θ 0 +ζ
1
IMPO = ∑1 ∫
pˆ dθ
(3)
θ0
N
A scheme of calculation methods is proposed for IMPO
and MAPO in Fig. 1.
Integration
Pass-band
Filter
Absolute
Value
Comparison
Present knock models are based on works done by
Draper [24] in 1935, who proposed a mathematical
description of the pressure oscillations on the basis of
acoustic wave theory (Eq. 4).
)
∂² p
)
= v ² ∆p
(4)
∂t ²
Analytical solution of equation 4 can be obtained under
disk geometry assumptions. Different oscillation modes
were established according to the literature [10, 13, 2526]. Nevertheless, this theory does not describe the time
deadening effect on the oscillations, that can be
observed in Fig. 2. This acoustic model is then unable to
determine the value of MAPO for modeled oscillations.
More recently, Trapy [27] attempted to account for knock
effects on wall heat losses. He introduced the
perturbation ∆h of the heat exchange coefficient due to
knock as a decreasing exponential function depending
on time (Eq. 5).
 −t 

(5)
∆h = ∆h0 exp
 1,7 10 −3 


Indeed, the wall transfers are strongly affected by the
presence of oscillations [28]. Equation 5 has no real
scientific
fundaments
based
on
experimental
observations but allows concluding that the oscillation
deadening is driven by an exponential function. After
testing, it appears too rough to reproduce the
phenomenon accurately.
The aim of this paper is to propose a global model for
the pressure oscillation signal under knocking
conditions. This model merges an acoustic model
(Draper [24] cf Appendix A) and a deadening model. We
introduce the definition of the Pressure Envelope Curve,
denoted by PEC, to reflect the oscillations deadening
phenomenon. Using the least square method,
investigations show good occurrence between
experiments and identified PEC.
Crank Angle CA
Absolute Filtered Pressure [bar]
Among all existing indices, MAPO is likely the most
employed, but none of them can be considered as a
universal indicator in terms of knock detection
performance. To achieve the definition of such an
indicator, oscillation modeling could be, for instance,
very helpful.
Filtered Pressure [bar]
Fig. 1: Determination of two knock indices from filtered pressure
Crank Angle CA
Fig. 2: Filtered pressure post-treatment
EXPERIMENTAL SET-UP
Main technical features of the engine used are
presented in Table 1. The engine is a natural gas fuelled
SI engine derived from a Diesel engine, with a bowl
piston head. It was adapted to spark ignition by reducing
the compression ratio and fixing a spark plug in the
injector location. The engine was connected to an
electrical dynamometer, which maintained the speed at
1500 rev/min. A view of the setup is shown in Fig.3.
Natural gas composition was analysed by a gas
chromatograph and it is given in Table 2 with the
maximum fluctuations encountered.
LINEAR RELATION FROM LITERATURE
In 1998, Diana et al. [29] plotted the mean values of
IMPO and MAPO versus IMPG. They noted a linear
dependency between IMPO and IMPG. This observation
can be expressed by the following equation obtained for
a given CA window ∆ζ and N consecutive cycles (used
in the arithmetic average):
1
N
Spark Ignition, Air Cooled
Bore
95. 3 mm
Stroke
88.8 mm
Compression ratio
12.37:1
Cooling system
Forced air circulation
No of cylinders
One
θ 0 + ∆ζ
∫θ
Max.
absolute
fluctuation
CH4
90.4%
± 1.8%
C2H6
6.6%
± 1.7%
C3H8
1.9%
± 0.7%
CxHy
0.7%
± 0.5%
N2
0.3%
± 0.4%
0
dpˆ
dθ = a (∆ζ )
dθ
θ 0 + ∆ζ
∫θ
pˆ dθ
(6)
0
CONCEPT OF THE PRESSURE ENVELOPE
CURVE
The system for the acquisition of in-cylinder pressure
was composed of:
Piezoelectric
cylinder
pressure
sensor
AVL QH32D, gain 25.28pC/bar, range 0-200 bar
pˆ dθ
0
Equation (6) is very difficult to solve because the sign of
filtered pressure p̂ never stops varying on the domain
[ θ 0 ; θ 0 + ∆ζ ].
Table 2. Mean natural gas composition
•
N θ 0 + ∆ζ
∑1 ∫θ
Equation 6 has been verified by plotting IMPG versus
IMPO with a constant windows of ∆ζ = 60 CA (see for
instance, Figure 4a) at different equivalence ratios
covering both lean burn and stoichiometric modes of
operation. Investigations have been also done to study
effects of throttling on this property. A linear tendency
can then be observed for cyclic values of these 2
indices. More interesting results are proposed in figure
4b that reflects the evolution of the gain factor a with ∆ζ
: beyond a certain value of ∆ζ, a (∆ζ) is either linear or
constant (depending of the cycle).
Table 1. Technical features of the engine
Volumetric
content
0
dpˆ
1
d θ = a ( ∆ζ )
dθ
N
Since our interest is focused on formulating an equation
for the pressure oscillations, one attends to confirm this
property for cyclic values of IMPO and IMPG, (Eq. 6):
Fig. 3: Experimental Setup
Type
N θ 0 + ∆ζ
∑1 ∫θ
–
•
Charge amplifier - AVL 3066A0
•
Shaft position encoder – AVL 364C, sampling 0.1CA
•
Piezo resistive pressure sensor fixed inside the inlet
manifold with its amplifier – range 0-2.5 bar
The pressure envelope curve ~
p is defined as the curve
passing by the “local” maximum of the pressure
oscillations, see figure 5. Obtaining most of their values
is quite easy by picking up the consecutive maximum
values of the absolute filtered pressure signal from the
end of window θ 0 + ∆ζ down to θ max and from the
beginning of the window θ 0 to θ max (the angle θ max is
defined as the angle where pˆ ≡ max pˆ ), see figure 5.
ζ
IMPGexp (60) [bar]
From this concept, it is possible to define an extension of
the cyclic IMPO and IMPG to the pressure envelope
curve, denoted by ~p , as:
θ 0 + ∆ζ
θ 0 + ∆ζ
θ0
θ0
∫
IMPO PEC =
~
p dθ and IMPG PEC =
∫
d~
p
dθ
dθ
The linear behavior between cyclic IMPO PEC and cyclic
IMPG PEC still exists, as it can been seen in figures 6.
This means:
IMPOexp (60) [bar CA]
θ 0 + ∆ζ
θ0
θ0
∫
~
p dθ = a (∆ζ )
∫
d~
p
dθ
dθ
(7)
Eq (7) depends on the width of the observation window
(denoted by ∆ζ ). Assuming that θ 0 is taken as the
reference with 0 CA.
limit
a [CA-1]
θ 0 + ∆ζ
Denoting by ζ the end angle of the observation window,
one obtains:
ζ0
ζ
∫
θ0
∆ζ,[CA]
Fig. 4: Tendency between IMPG vs IMPO
~p
ζ
d~
p
dθ = a(ζ ) ∫ d~
p dθ
dθ
θ
(8)
0
Equation 8 is valid for ζ high enough. As the pressure
envelope curve is first increasing and then decreasing
(maximum is obtained at θ max ), one splits the
observation window into 2 domains to obtain a constant
sign for the derivative of ~
p.
Each integral member is then as follow:
ζ0
∫
θ0
Local maxima
- - - Envelope
Filtered Pressure [bar]
x
ζ
ζ0
ζ
d~
p
d~
p
dθ +
dθ = a (ζ )( d~
p dθ + d~
p dθ )
dθ
dθ
ζ
θ
ζ
∫
∫
∫
0
0
0
(9)
By choosing appropriately ζ 0 (that is to say ζ 0 high
enough, greater than θ max , see Fig 4) to verify the
property on the 2 domains. On [ζ 0 ; ζ ] , ~
p is decreasing:
d~
p
d~
p
=−
dθ
dθ
∀θ > θ max
(10)
Eq (9) can be rewritten as:
ζ0
∫
θ0
θ max
∆ζ Crank Angle [CA]
Fig. 5: Pressure Envelope Curve a) Definition – b) Experiment
ζ ~
ζ0
ζ
d~
p
dp
dθ −
dθ = a(ζ )( d~
p dθ + d~
p dθ )
dθ
dθ
ζ
θ
ζ
∫
∫
∫
0
0
0
(11)
IMPGPEC (60) [bar]
ζ
d  d~
p 
d
dθ = −
dζ ζ dθ 
dζ
 0

∫
Methane Number 71
FAR 0.98
Spark Advance 11CA
WOT
ζ


a (ζ ) d~
p dθ 


ζ0


∫
(13)
∫ pdθ
(14)
Then:
d~
p
da
p (ζ ) −
= −a (ζ ) ~
dζ
dζ
ζ
~
ζ0
NB: One notices that for a constant, Eq. 14 gives the
Trapy’s formulation, seen previously.
IMPOPEC (60) [bar CA]
IMPGPEC (60) [bar]
By deriving a second time, Equation 7 becomes:
Methane Number 76
FAR 0.85
Spark Advance 14CA
WOT
θ +ζ
d²~
p
d~
p
da ~
d ²a 0 ~
p=−
p dθ
+ a (ζ )
+2
dζ ²
dζ
dζ
dζ ² ξ
∫
0
On [ζ 0 ; ζ ] , a(ζ ) is almost linear and independent of the
window length, see figure 7. Hence, it can be expressed
by a first order Taylor development:
a(ζ ) = a (ζ 0 ) + (ζ − ζ 0 ) a ' (ζ 0 ) + o(ζ ) ≈ αζ + β
IMPOPEC (60) [bar CA]
Fig. 6: Tendency between IMPGPEC and IMPOPEC under different
conditions
ζ
ζ0 min
ζ1
ζ, CA
Fig. 7: Ratio IMPGPEC and IMPOPEC versus the window angle
∫
Solving this equation, we finally obtain:
ζ

A
~
p (ζ ) = − A + (αζ + β )e −(αζ + β )² / 2α  e t ² / 2α dt + B 
α
ζ

 0

Itegration properties give then:
ζ0
ζ 0 d~
p

dθ = a (ζ ) d~
p dθ
θ dθ
θ0
0
ζ
ζ
 d~
p
dθ = −a (ζ ) d~
p dθ

d
θ
ζ 0
ζ0
(17)
Equation (17) is an Ordinary Differential Equation of the
second order with non constant coefficients. Two
possibilities exist to solve it. The first one is numerical,
by using MATLAB/SIMULINK for instance. The second
one is analytical. One presents the last one since it could
be clever to program the analytical solution in an ECU
(Electronic Control Unit). In appendix B, the
mathematical steps are detailed. Using Runge-Kutta 4th
order, analytic and numeric solutions are identical.
Errors are lower than 0.01%.
a CA-1
(ζ)
(16)
The differential equation follows by the filtered in-cylinder
pressure (under knocking conditions) is approximated
by:
~
p ' ' (ζ ) + (αζ + β ) ~
p ' (ζ ) + 2α ~
p (ζ ) ≈ 0
∫
(15)
∫
∫
(18)
(12)
∫
The second equation from (12) can be differentiated as
follow:
A and B are the two constants from integration.
We have, at the knock onset, two time conditions:
p (θ 0 ) = 0
~
~
 p ' (θ 0 ) = C
where C is the origin slope of the pressure envelope
curve. It depends on the running conditions (air fuel
ratio, spark advance, etc.) and must be characterized
experimentally.
α [CA −2 ]
The analytical solution of equation 17 can be written as:
a2 
a2
β2
1 − 2α 1 − 2α
β − 2α 
~
+ e
p (ζ ) = aC e
ψ (ζ ) − a e
α
β

ζ
where ψ (ζ ) = ∫
(α t + β ) 2
e 2α
dt and




(19)
a = αζ + β
0
Envelope Pressure [bar]
On the figure 8, an example of pressure envelope curve
example is plotted with arbitrary values for constants C,
α and β .
β [CA −1 ]
α = 0.01 [CA-2]
β = 0.1 [CA-1]
C = 0.25 [bar CA-1]
ζ, CA
C [bar CA −1 ]
Fig. 8: Numerical pressure envelope curve: an example
APPLICATION
In order to use these new concepts, it remains to identify
the two constants of the slope α and β , ie the engine
signature, and C the cyclic knock feature.
It can be done by using least square method. Taken 100
consecutive cycles, cycle to cycle dispersion exists
(engine conditions are: Methane Number 81, FAR 0.98,
Spark Advance 11CA, 90% WOT).
MAPO PEC [bar]
Then, in this set we have different cyclic MAPOexp, cyclic
IMPOexp and hence different cyclic MAPOPEC and cyclic
IMPOPEC. Two ways were studied. The first one
evaluated the best value for the engine signature. The
cyclic triplet ( α , β , C) is identified. LSM results are
plotted in Fig. 9a,b,c; comparison in terms of MAPO is
proposed in fig 9.d. The second one consists to keep the
signature parameters as constants.
Now, let us take 3 typical cycles, marked as 1, 14 and
13. These cycles have different calculated MAPO
values: 0.25, 0.72 and 1.05 respectively, i.e. low,
medium and high knock.
Fig. 9: Integration coefficients and MAPO validation
Absolute Filtered Pressure [bar]
Absolute Filtered Pressure [bar]
Cycle 1
α = 0.54 [CA-2]
β = 6.15 [CA-2]
C = 1.84 [bar CA-1]
Cycle 14
α = 0.16 [CA-2]
β = 1.2 [CA-2]
C = 1.01 [bar CA-1]
Crank Angle CA
Absolute Filtered Pressure [bar]
Absolute Filtered Pressure [bar]
Crank Angle CA
Crank Angle CA
Cycle 14
α = 0.08 [CA-2]
β = 0.6 [CA-2]
C = 0.58 [bar CA-1]
Crank Angle CA
Absolute Filtered Pressure [bar]
Absolute Filtered Pressure [bar]
Crank Angle CA
Cycle 13
α = 0.11 [CA-2]
β = 0.88 [CA-2]
C = 1.37 [bar CA-1]
Cycle 1
α = 0.08 [CA-2]
β = 0.6 [CA-2]
C = 0.22 [bar CA-1]
Cycle 13
α = 0.08 [CA-2]
β = 0.6 [CA-2]
C = 0.94 [bar CA-1]
Crank Angle CA
Fig. 10: Cyclic Absolute Filtered Pressure versus crank angle
On the left: best fitting
On the right: two integration factor kept constant
Absolute Filtered Pressure [bar]
Absolute Filtered Pressure [bar]
Cycle 1
α = 0.08 [CA-2]
β = 0.6 [CA-2]
C = 0.22 [bar CA-1]
Cycle 14
α = 0.08 [CA-2]
β = 0.6 [CA-2]
C = 0.58 [bar CA-1]
Crank Angle CA
Absolute Filtered Pressure [bar]
Absolute Filtered Pressure [bar]
Crank Angle CA
Crank Angle CA
Absolute Filtered Pressure [bar]
Absolute Filtered Pressure [bar]
Cycle 14
α = 0.08 [CA-2]
β = 0.6 [CA-2]
C = 0.58 [bar CA-1]
Crank Angle CA
Crank Angle CA
Cycle 13
α = 0.08 [CA-2]
β = 0.6 [CA-2]
C = 0.94 [bar CA-1]
Cycle 1
α = 0.08 [CA-2]
β = 0.6 [CA-2]
C = 0.22 [bar CA-1]
Cycle 13
α = 0.08 [CA-2]
β = 0.6 [CA-2]
C = 0.94 [bar CA-1]
Crank Angle CA
Fig. 11: Absolute Filtered Pressure versus crank angle
On the left : Real data (after pass-band filter)
On the right : Coupling envelope curve and Draper high frequency Model
To achieve this paper, we propose to compare model of
oscillations using:
•
Pressure envelope curve (obtained by equation 7
with only one integration coefficient adapted)
•
Disk geometry Model from Draper [24]
Since knock behavior is analyzed in terms of MAPO,
IMPO or IMPG in the literature, comparison between
experiment and model is qualitative. Same data (cycle 1,
14 and 13) are used. Results are shown in figure. 11.
CONCLUSIONS
1. The cyclic Integral of Modulus of Pressure
Oscillations in a SI engine occurring under knock
conditions has been analyzed.
2. A curve called “pressure envelope curve” describing
the deadening of the oscillation pressure signal
under knocking conditions has been introduced.
3. An analytical formulation has been found to drive the
deadening of the oscillations from two constants,
assimilated to the engine signature, and one cyclic
constant (to characterize the knock intensity).
4. Using Draper work [24], a complete model is
obtained to simulate pressure oscillations due to
knock.
5. Perspectives of this work deal with the modeling of
knock oscillations on both the combustion processes
and the wall heat exchange. More experimental data
would be required to extend this approach and set
correlation between the three constants and the
operating conditions (spark advance, load, air-fuel
ratio, EGR etc…).
6. Implementation of the model in an ECU is possible
by taking the engine signature as a known
parameter. The knock intensity would be evaluated
on the two or three first local peaks (describing the
pressure envelope curve) in order to evaluate the
cyclic constant C and hence to act on the
combustion.
BIBLIOGRAPHY
1. Leppard, W. R. Individual-cylinder knock occurence
and intensity in multicylinder engine, SAE Technical
Paper, N°820074, 1982.
2. Checkel, M. D. and Dale, J. D. Characterisation
knock detection from engine pressure records, SAE
paper N° 860028, 1986
3. Chun, K.M. and Kim, K.W. Measurement and
analysis of knock in a SI engine using the cylinder
pressure and block vibration signals, SAE paper
N°940146, 1994
4. Russ, S. A review of the effect of engine operating
conditions on borderline knock, SAE paper N°
960497, 1996
5. Checkel, M.D. and Dale, J.D. Testing a third
derivative knock indicator on a production engine,
SAE paper N° 861216, 1986
6. Ando, H., Takemura, J. and Koujina, E. A knock
anticipating strategy basing on the real-time
combustion mode analysis, SAE paper N°890882,
1989
7. Dhall, S.N. and Beans, E.W. Correlation of knock
with engine parameters for ammonia-nitrous oxide
mixtures, SAE paper N° 912360, 1991
8. Millo, F. and Ferraro, C.V. Knock in SI engines: a
comparison between different techniques for
detection and control. SAE paper N°982477, 1998
9. Thomas, J.R., Clarke, D.P., Collins, J.M., Sakonji,
T., Ikeda, K., Shoji, F. and Furushima, K. A test to
evaluate the influences of natural gas composition
and knock intensity, ASME ICE, 22, 1994
10. Lee, J.H., Hwang, S.H., Lim, J.S., Jeon, D.C. and
Cho, Y.S. A new knock-detection method using
cylinder pressure block vibration and sound pressure
signals from SI engine. SAE paper N° 981436, 1998
11. Westin, F., Grandin, B. and Angstrom, H.E. The
influence of residual gases on knock in turbocharged
SI engines, SAE paper N° 2000-01-2840, 2000
12. Attar, A.A. and Karim, G.A. Knock rating of
gaseous fuels, Fall Technical Conference of ASME,
Vol. 31-3, 1998
13. Brunt, M.F., Pond, C.R. and Biundo, J. Gasoline
engine knock analysis using cylinder pressure data,
SAE paper N° 980896, 1998
14. Ortmann, S., Rychetsky, M., Glenser, M., Groppo,
R., Tubetti, P. and Morra, G. Engine knock
estimation neural networks based on a real-world
database, SAE paper N° 980513, 1998
15. Schmillen, K.P., and Rechs, M. Different methods
of knock detection and knock control. SAE paper N°
910858, 1991
16. Ryan, T.W., Callahan, T.J. and King, S.R. Engine
knock rating of natural gases - methane number,
Journal of Engineering for Gas Turbines and Power,
115, 1993
17. Callahan, T.J., Ryan, T.W., Buckingham, J.P.,
Kakockzi, R.J. and Sorge, G. Engine knock rating
of natural gases - expanding the methane number
database, Proceedings of 18th annual fall Technical
conference of ASME Internal Combustion Engine
Division, Vol 27-4, 1996
18. Ferraro, C.V., Marzano, M. and Nuccio, P. Knocklimit measurement in high speed SI engines, SAE
paper N°850127, 1985
19. Najt, P.M. Evaluating thershold knock with a semiempirical model - initial results, SAE paper
N°872149, 1987
20. Huo, S.Y. and Kuo, T.W. A hydrocarbon
autoignition model for knocking combustion in SI
engine. SAE paper N°971672, 1997
21. Dubel, M., Schmillen, K. and Wackertapp, H.
Influence of gas composition on the knocking
behaviour of spark-ignited gas engines, International
Gas Research Conference Transactions, p 952-963,
1983
22. Abu-Qudais, M. Exhaust gas temperature for knock
detection and control in spark ignition engine.
Energy Conversion Management, Vol 37, pp 13831392, 1996
23. Goto, S. and Itoh, Y. A contribution of lean burn
high-output spark-ignited gas engines(experimental
study in lean gas engines). In Nippon Kikai Gakkai
Ronbunshu editor, Transactions of the Japan
Society of Mechancal Engineers, Volume B, pp
1055-1061, 1997
24. Draper, C.S. The physical Effect of Detonation in a
closed cylindrical chamber, Technical Report 493,
NACA, 1935
25. Blundson, C.A. and Dent, J.C. The simulation of
auto-ignition and knock in a spark ignition engine
with disk geometry. SAE Technical Paper 940524,
1994
26. Worret, R., Bernhardt, S., Schwarz, F. and
Spicher, U. Application of different cylinder pressure
based knock detection methods in spark ignition
engines, SAE technical paper 2002-01-1668, 2002
27. Trapy, J. Les Transferts Thermiques dans le Moteur
à Allumage Commandé : Conception et Mise au
Point d’un Modèle spécifique du Coefficient
d’Echange gaz-parois, Société Française de
Thermique, Journée d’Etude G.U.T., Mai 1985 (in
french)
28. Syrimis, M. and Assanis, D.N.The effect of the
location of knock initiation on heat flux into a SI
combustion chamber. SAE Technical Paper,
N°972935, 1997.
29. Diana, S., Gilio, V. , Iorio, B. and Police, G.
Evaluation of the effect of EGR on engine knock,
SAE Technical Paper, 982479, 1998
APPENDIX A : DRAPER ACOUSTIC MODEL
In 1935, Draper worked on the physical effect of
detonation in a closed cylindrical chamber. He proposed
to consider the time pressure dependence with the
following assumptions:
1/ in-cylinder geometry is cylindrical
2/ elastic surrounding that is to say a wave propagation
(cylindrical co-ordinates (r ,φ , z ) )
p 1 ∂² ~p ∂² ~p 1 ∂² ~p
∂² ~p 1 ∂~
+
+
+
=
∂r ² r ∂r r ² ∂φ ² ∂z² c² ∂t ²
where c is the speed of sound:
c=
γ p0
under ideal gas assumption.
ρ0
He determined three wave numbers, denoted by u, s
and g describing the in-cylinder vibration mode.
Time pressure dependence was obtained by using
Fourier decomposition:
~p( r ,φ , z , t ) = A
∑ u ,s ,g J s ( β u ,s r ) cos( sφ )
cos(
gπ
z ) cos( 2π f u ,s ,g t )
h
• Limit conditions leads to transcendent equation:
dJ s ( β u ,s R )
=0
dr
with Js designs first Bessel function of the s order and u
is the uth root.
• The wave oscillation frequency is calculated by:
f u ,s ,g = c
•
Au ,s ,g
β u2,s
+
g2
4π 2 4h 2
is the module of the oscillating mode u/s/g.
By assuming that g equals to 0, as proposed by
Blundson and Dent [25] near the top dead center, the
wave oscillation frequency is reduced to the main
modes:
u = 0,1,2,3

s = 0,1
g = 0

The piezo transducer sensor is located at ( rsens ,0, z sens ).
The pressure measured is then:
~p
sens ( θ ) =
1
3
∑ ∑ Au ,s
s =0u =0
J s ( β u ,s rsens ) cos( 2π f u ,s
θ
6N
)
where θ is the crankshaft angle.
Resonance
Oscillating
Mode
Authors
Draper Blundson Lee
[24]
[25]
[10]
Brunt
[13]
1/0
6.5
6.5
10.6
13.4*
14.7
-
7.4*
7.8
12.3*
12.9
15.5*
16.2
16.9*
17.8
6.6
6.5
10.9
11.8
13.7
15.0
16.3
2/0
0/1
3/0
*
Theory/
Meas.
Theory/
Meas.
Theory/
Meas.
Theory/
Meas.
7.4*
7.3
12.2*
12.8
16.8*
17.1
calculated by the equation above for f u , s , g
Engine
Draper
CFR
Bore [mm]
82.4
Spark position
Speed of sound 914
[m/s]
Blundson Lee
DOHC
75.0
950
75.5
950
Brunt
Ford
Zetec
84.8
center
950
α y1 z ' '+ (2α y1 '+ xy1 ) z ' = 0
APPENDIX B : PROOF OF PRESSURE
ENVELOPE CURVE
Using the change of variable as following u = z' , we
obtain:
Let us introduce new variables:
x = α ζ + β

~
 y ( x) = p (ζ )
(P1)
Equation (9) is then written as:
α y ' ' ( x) + xy ' ( x) + 2 y ( x) = 0
y ' x
u'
= −2 1 −
u
y1 α
∑ cn x
y ( x) =
(P2)
x²
u ( x) =
(P3)
Hence using equation P3 in equation P2, a recurrence
relation is deduced between cn and cn-2:
−1
cn =
c n−2
α (n − 1)
(P4)
(P12)
∞
(−1) p
∑ α p 1* 3 * ... * (2 p − 1) c0
∞
(−1) p
(P5a)
(P5b)
Final solution of P2 is linear combination :
(P6)
With
∞

(−1) p
x2 p
 y0 =
p

p = 0 α 1 * 3 * ... * ( 2 p − 1)

∞
(−1) p
y =
x 2 p +1
p
 1
α
2
*
4
*
...
*
(
2
)
p
p =0

∫
 −
A
y ( x) = − A +  Γ( x) + B  xe 2α

α
(P14)
Initial conditions are:
p =0
y = c0 y0 + c1 y1
(P13)
x²
p =0
∑ α p 2 * 4 * ... * (2 p) c1
x²

A
A
 z ( x) = − e 2α + Γ( x) + B
x
α


x t²
Γ( x) = e 2α dt

β

Complete solution is given by:
Then, we have two cases:
c 2 p +1 =
A 2α
e
x²
Coming back to the original variable z, we rewrite:
n
n =0
c2 p =
(P11)
After calculations, solution of equation (11) is:
By developing with infinite polynomial series as:
∞
(P10)
∑
(P7)
∑
p (0) = 0
 y( β ) = ~

dζ d~
p
D
 dy
=
(
)
(0) =
β
 dx
dx dζ
α

(P15)
Using the previous variables, equation (15) supplies two
conditions :

 y ( β ) = z ( β ) y1 ( β ) = 0 ⇒ z ( β ) = 0

D

 y ' ( β ) = z ( β ) y1' ( β ) + z ' ( β ) y1 ( β ) =
α

D

 ⇒ z ' ( β ) y1 ( β ) = α

(P16)
One notices that,
y1 = x e
−
x²
2α
(P8)
A= D
Let us introduce a function z as:
y = z y1
The second equation can be developed and gives the
first integration constant :
(P9)
One uses the so-called method “variation of the
constant”. Differential equation is then written as:
β
α
(P17)
First condition with equation (17) leads to the second
integration variable:
z( β ) = −
D
α
β²
e 2α + Dβ Γ( β ) + B = 0 ⇒ B =
D
α
β²
e 2α
(P18)
since Γ( β ) = 0 by definition.
Finally, we obtain:

~
 p (ζ ) = y ( x )
x²

−
 y ( x) = − A +  A Γ( x) + B  xe 2α

α


x = α ζ + β

β
A = D
α


x t²
Γ( x) = e 2α dt

β

(P19)
∫
NOMENCLATURE
MAPO : Maximum Amplitude of Pressure Oscillations
bar
IMPG : Integral of Modulus of Pressure Gradient
bar
IMPO : Integral of Modulus of Pressure Oscillations
bar.CA
PEC : Pressure Envelope Curve
θ:
Crank Angle
CA
θ 0 : Initial crank angle (Auto-Ignition)
CA
ζ:
Window crank angle
Filtered Pressure
CA
bar
Outer Layer Pressure
N:
Number of cycles tested
WOT : Wide Open Throttle
FAR : Fuel Air Ratio
a:
Linear function of the window
C:
Integral factor
α : Integral factor
β : Integral factor
v:
Wave propagation speed
bar
p̂ :
~
p :
CA-1
-1
bar.CA
CA-2
CA-1
m.s-