2002-01-0237

FEM Approximation of Internal Combustion Chambers forKnock Investigations

Sönke Carstens-Behrens, Mark Urlaub, and Johann F. BöhmeRuhr Universität Bochum

Jürgen Förster and Franz RaichleRobert Bosch GmbH

Copyright © 2002 Society of Automotive Engineers, Inc.

ABSTRACT

The resonances of SI engine combustion chambers areslightly excited during normal combustion but stronglyexcited by knock. In order to avoid knocking combustionsextensive knowledge about knock and its effects isnecessary. In this paper the combustion chamber of aserial production engine is modeled by finite elements.Modal analyses are performed in order to gaininformation about the resonances, their frequencies, andtheir frequency and amplitude modulations. Simulationresults are compared to measured data using a high-resolution time-frequency method. Furthermore, aconnection between knock origin and the excitation of theresonances is postulated applying transient analyses.

INTRODUCTION

Although the appearance of knocking combustions andtheir problems concerning pollutant production, engineefficiency, and engine damages have been investigatedfor decades [1,2], there are still many aspects that arenot completely understood and more knowledge wouldbe helpful. This holds for the primary effect, the chemicalprocesses of the combustion, as well as for thesecondary effect, the excitation of the pressureresonances of the combustion chamber, as well as forthe tertiary effect, the transfer of the resonances throughthe engine housing as structure-borne sound. This workmainly addresses the secondary effect, the excitation ofthe pressure resonances by knock. Since measuringpressure in a combustion chamber is relatively feasiblecompared to e.g. optical measurements of the flame frontpropagation, pressure signals have become the mostimportant information source for knock investigations. It iswell known that the pressure resonances are standingwaves in the combustion chamber exhibiting pressurenodes and antinodes. Therefore the pressure amplitudefor each resonance depends on the considered location.Since pressure can not be measured at many locationsinside an operating cylinder, the resonances must be

modeled. Draper [1] estimated the combustion chamberresonances of an aircraft engine by solving the waveequation for a right cylindrical resonant cavity. Withdecreasing combustion chamber volume thisapproximation is too simple. Hickling et al. [2] reported onthe additional influence of the cylinder head that causes amode splitting, i.e. a resonance may appear twice,rotated by a certain angle and with different frequency.They also mentioned the application of the finite elementmethod (FEM) for modeling the combustion chamber.Scholl et al. [3] used this approach to investigate thecombustion chamber resonances of a two- and a four-valve cylinder. They considered a crank angle intervalfrom 10° to 95° degrees and compared the simulationresults to measured data using the spectrogram as time-frequency representation. This way they proved that FEMis a useful tool in order to model more realisticcombustion chamber geometries. They demonstrated themode splitting and mentioned the problem of observingpressure resonances because sensors might be locatedat pressure nodes.

This work extends the investigations of Scholl et al. withrespect to following aspects:

• The considered crank angle interval covers the wholerange from 0° to 180°.

• Compared to the spectrogram a time-frequencyrepresentation with better resolution is applied sothat simulated and measured data can be comparedmore accurately.

• In addition to the frequency modulation an amplitudemodulation is observed and described.

• A sensor characteristic is presented to predict thesignal obtained by a pressure sensor depending onits position.

• For the first time transient simulations wereperformed in order to simulate the explosivecharacter of knock. That way a possible influence ofknock origin on the excitation of the resonances canbe studied.

MODELING

The investigations base on a spark ignition engine withfour valves, a central spark plug, and a capacity of circa0.42l per cylinder. The combustion chamber of thisengine was modeled using the commercial FEM packageANSYS. The element FLUID30 was used to model thecombustion chamber gas as fluid with zero viscosity. Thechamber walls were considered to be absolutely rigid,reflections at the chamber walls are lossless. The speedof sound was set constant to 1000m/s for all simulations.For comparison with measured data the speed of soundcan be corrected afterwards. The crank angle intervalcovered the whole range from 0=α ° to 180=α °. Thiscorresponds to a piston position range from 0=z mm to

7.82=z mm. Simulations were performed with a pistonposition step size of 2.0=∆z mm. Actually the pistonmoves continuously but this quasi-static approximation isjustified because the speed of sound is much faster thanthe piston velocity, [3]. Simulation results will also confirmthis assumption. For meshing, the combustion chambervolume was divided into two volumes: the cylinder headvolume, constant for all simulations, and the pistonvolume that depends on piston position. The cylinderhead volume was meshed in the same way for allsimulations while the piston volume mesh changeddepending on piston position. For all simulations themean node distance was approximately 3 mm. This wasthe finest mesh that could be simulated by the hardwareat bottom dead center. Using modal analyses afrequency range from 1 kHz to 27 kHz was searched forresonances at each piston position. Figure 1 shows thecombustion chamber model at a piston position of 10mm. Furthermore, three nodes are marked that will serveas virtual pressure sensors in order to compare theirsignals with those of real measurements. Sensor 1 couldbe mounted in a central spark plug. Its position is slightlydecentralized from the cylinder axis. Sensor 2 and 3could be mounted in the cylinder head.

METAMORPHOSIS

Figure 2 shows the shape of the first ten resonancesfound at a piston position of 10 mm. Bright regions meanhigh pressure, dark regions mean low pressure. Noticethat the figures only show a time instant of eachoscillation, at a later instant corresponding to a phaseshift of 180°, high and low pressure would be exchanged.Only the mean pressure indicated by mean gray does notchange because it marks pressure nodes. Res. 1 and 2demonstrate the mode splitting reported by Hickling et al.They have a very similar shape but they are rotated byan angle of 90°. Because of the influence of the cylinderhead they have different frequencies: 6.3kHz and7.5kHz, respectively. Res. 3 and 4, 6 and 7, and 8 and 9are also subjected to mode splitting, but their frequenciesare closer together. For each piston position or crankangle, respectively, the shape of the resonances is fixed.But for different piston position the shape of mostresonances changes, see Figure 3. While the shape ofthe first and third resonance do almost not change, theother resonances run through a metamorphosis. Forexample, Res. 2 and Res. 4 seem to exchange theirpressure distribution between piston position 60 mm and70 mm. But the piston position step size is too large inthis figure in order to understand the metamorphosis.Therefore, Figure 4 shows the piston position range ofinterest. Here it is visible how the antinodes tip over. Themetamorphosis of the other resonances can beexplained by the same way. Because of themetamorphoses it was necessary to have a smallsimulation step of 0.2 mm between two piston positions.In the considered frequency range between 15 and 70resonances were found depending on piston position.This means that more than 10000 resonances had to beclassified to the 70 possible resonances. This enormouswork was done using a pattern recognition algorithm thattook into account both the pressure distribution and thefrequencies of the resonances.

FREQUENCY MODULATION

Draper has already noticed that the frequencies of theresonances are modulated. There are two reasons forthe modulation: (a) The speed of sound of the gas

Figure 1. FEM model of the combustion chamber at apiston position of 10 mm and the positions of threevirtual pressure sensors.

Figure 2. Pressure distribution of the first tenresonances at a piston position of 10 mm.

depends on the absolute temperature. Since thetemperature decreases due to the expansion of the gaswhile the piston moves down, the speed of sound andthe resonance frequencies decrease, too. (b) Due topiston motion the combustion chamber volumeincreases, i.e. the corresponding wavelengths of mostresonances increase, so that again the frequenciesdecrease. In real combustion chambers both effectssuperimpose. Figure 5 shows the instantaneousfrequencies of the resonances estimated by FEMsimulations depending on crank angle and pistonposition, respectively. Obviously, all resonances exceptthe first are frequency modulated – the frequenciesdecrease. Actually, the frequency of the first resonanceslightly increases. Since for all simulations the speed ofsound were constant, the frequency modulations werecaused only by change of the chamber geometry. Thefrequency modulations correspond well to the shape ofthe resonances. If they contain axial oscillationcomponents, there is a strong dependency on the pistonposition because piston position directly influences thewavelength of the standing wave. If they do not containaxial components, there is only a weak dependency, ifany, except for small values of piston position, i.e. near

top dead center. In that case the influence of the cylinderhead causes a frequency modulation. Since thefrequency modulation depends on the shape of theresonances it corresponds to the metamorphosis. E.g.Res. 4 contains no axial components until a pistonposition of about 40 mm, see Figure 3. Therefore, it ishardly frequency modulated in this interval. Between 40mm and 68 mm the metamorphosis yields an axialcomponent, see Figure 3 and 4, corresponding to astrong frequency modulation, see Figure 5. From 68 mmto bottom dead center there is again no axial componentand therefore almost no frequency modulation. Anotherinteresting point is that the frequencies of the first andsecond (from 68 mm to bottom dead center the fourth)resonances move closer together while the piston movesdown. For large values of piston position the influence ofthe cylinder head decreases and the frequenciesconverge toward the frequency of the first circumferentialresonance of a right cylinder. Similarly it can be shownthat the frequency modulations can be approximatedusing the resonances of a right cylinder. E.g. for thefourth resonance between 40 mm and 68 mm thecombustion chamber height equals approximately half ofthe wavelength. Therefore, the frequency modulation canbe very well described by

where c is the speed of sound, z is the piston position,and a takes into consideration the effective cylinder headheight.

,)(2

)(za

czf

+≈

Figure 3. Pressure distribution of the first five resonances depending on piston position.

Figure 4. Metamorphosis of Res. 2 and Res. 4.

As mentioned before, the speed of sound also influencesthe resonance frequencies. Since the distribution ofpressure nodes and antinodes are determined only bythe combustion chamber geometry, the frequencymodulation of all resonances due to varying speed ofsound is the same. If the speed of sound differs from thebasic speed c0 by a factor cz, the frequencies of allresonances change by this factor. It is very difficult toestimate the speed of sound in a real combustionchamber using physical relations like the gas laws andadiabatic expansion of the gas because the necessaryassumptions do not hold. Therefore, in this work thespeed of sound was estimated using measured pressuredata. From a high-resolution time-frequencyrepresentation the instantaneous frequency of the first

resonance was estimate as )(1̂ αf , where α is crank

angle. Then the speed of sound can be estimated by

where )(1 αf is the instantaneous frequency estimated

by FEM and 0c is the speed of sound for the FEM

simulations. The frequency of the n-th resonance canthen be estimated by

Figure 6 shows the Wigner-Ville spectra of pressuresignals measured at 5500 rpm by two pressure sensorsat approximately sensor position 1 and 2, see Figure 1.The Wigner-Ville spectrum provides a high-resolutiontime-frequency representation of multi-componentsignals, [4,5]. Dark regions indicate strong components.In order to stress weak components a logarithmic gray

scale was used. The spectra are superimposed by theinstantaneous frequencies estimated by FEM simulationsand corrected as described above. Obviously, the signalscontain different components. This phenomenon will bediscussed in the next section. Each component of thespectra corresponds to at least one FEM frequency. Thespectrum of Sensor 1 mainly contains one componentthat corresponds to the fifth resonance until 50° crankangle. Res. 1 is observed weakly and Res. 4 between80° and 130° crank angle. Sensor 2 observes Res. 2,Res. 3 or 4, and Res. 6 or 7. The assignment of thefourth component of the spectrum is difficult becausethere are four resonances close together in the time-frequency domain.

AMPLITUDE MODULATION

The amplitudes of the oscillations in a real combustionchamber decrease because of heat losses, friction andgas expansion. These effects influence all resonances inthe same way. But there is another aspect that causesan additional amplitude modulation due to pressuresensor position and resonance metamorphosis. Asmentioned before the Wigner-Ville spectra of Figure 6contain different components. Since the sensorsobserved the same combustions in the same combustionchamber, this can be only explained by their positions.Sensor 1 was mounted in the central spark plug. Mostresonances have a pressure node on the cylinder axis,see Figure 2. Near top dead center only Res. 5 has apressure antinode and should be well observable bySensor 1. That is exactly what the spectrum of thissensor shows. In addition, Res. 1 is visible because thesensor position is slightly decentralized from the cylinderaxis and Res. 1 is usually excited very strongly. Between80° and 130° there is a component in the spectrum thatcorresponds to Res. 4 comparing the frequencies. This

01

1

)(

)(ˆ)(ˆ c

f

fc

ααα =

)()(

)(ˆ)(

)(ˆ)(ˆ

1

1

0

αααααα nnn f

f

ff

c

cf ==

Figure 5. Instantaneous frequencies estimated byFEM simulations depending on crank angle (left) andon piston position (right).

Figure 6. Estimated Wigner-Ville spectra of measuredsignals of two pressure sensors super-imposed byselected and corrected instantaneous frequenciesestimated by FEM simulations.

fits well to the metamorphosis of this resonance, seeFigure 3. Near top dead center and near bottom deadcenter the resonance has a pressure node at the sparkplug but meanwhile there is a pressure antinode.Therefore from Sensor 1’s point of view Res. 4 is stronglyamplitude modulated. The invisibility of most otherresonances can be regarded as a kind of amplitudemodulation, too. Figure 7 shows the amplitudemodulation of Res. 1, 2, and 4 estimated by evaluatingthe FEM simulation results at the virtual pressurepositions 1, 2, and 3, see Figure 1. Res.1 is observedbest by Sensor 3. Its amplitude decreases but thisamplitude modulation is caused by numerical effects.Sensor 1 observes this resonance with smaller amplitudeand Sensor 2 does not notice this resonance at all. Res.2 is visible only for Sensor 2 until 120° crank angle. Thendue to the metamorphosis of this resonance, all threesensors observe the resonance. The amplitudemodulation of Res. 4 is more complex. The three sensorsobserve this resonance at top dead center. But Sensor 1and Sensor 2 run through a pressure node at about 40°crank angle and 80° crank angle, respectively. Fromabout 120° crank angle this resonance is not visible forSensor 3 any more.

SENSOR CHARACTERISTIC

The previous two sections have shown that thefrequency modulation and the amplitude modulation ofthe resonances can be estimated using FEM simulations.In return the information can be used to predict thecharacteristic of a signal and its components measuredby a pressure sensor at a certain position in thecombustion chamber. The intensity of excitation of theresonances is random and can not be predicted, ofcourse. The frequency modulation due to varying speedof sound and the amplitude modulation due to heatlosses and friction are neither predicted here. But the

frequency modulation and amplitude modulation due tometamorphosis contains a lot of information with respectto assess a pressure sensor position. The Wigner-Villespectrum of such a predicted signal, called sensorcharacteristic, yields a simple tool to compare severalpressure sensor positions. Figure 8. shows thecharacteristic of the three virtual sensors. Thecharacteristic of Sensor 1 and 2 are quite similar to theWigner-Ville spectra of the measured data in Figure 6.

INFLUENCE OF KNOCK ORIGIN

It is well known that the excitation of the resonances israndom and hardly correlated from combustion tocombustion. The cause is the random character of thecombustion process and of knock. In this section we willshow that especially the origin of knock may have stronginfluence on the excitation.

MODEL: In order to simulate the explosive character ofknock transient simulations were performed. Thecombustion chamber model equals that of the modalanalysis at a piston position of 8.1=z mmcorresponding to 15° crank angle. This piston positionwas chosen because knock occurs near this point. Thespeed of sound again was set to 10000 =c m/s. A

certain set of nodes got an initial pressure levelsimulating the pressure step caused by knock. With thisinitial condition the pressure distribution at time ∆=t ,then ∆= 2t and so on was calculated. Altogether

4000=N steps were simulated at a step size ofsµ5.0=∆ . This means that a time interval of

2=∆N ms was considered. During this time interval thecrank angle would rotate to 17° at 1000 rpm whichmeans that the piston would move down to 3.2=z mm.But for simulation simplicity the piston motion wasneglected. This approximation is justified because thepiston velocity at 15° crank angle and 1000 rpm is about

Figure 7. Amplitude modulation of Res. 1, 2, and 4depending on the virtual pressure sensor position.

Figure 8. Sensor characteristic of Sensor 1, 2, and 3.

0.5m/s and therefore much slower as speed of sound(1000m/s). This means that even the lowest resonancefrequency is high enough so that the piston hardly movesduring an oscillation period. Furthermore, the modalanalyses together with measured data have shown thatresonances containing also axial components are stableeven at high engine speeds and high piston velocities.Therefore, only the excitation of the resonances iscrucial.

Theoretically the time signals of all nodes could beevaluated. But the amount of information is infeasible. Sothis investigations limit to the signals of the three virtualpressure sensors marked in Figure 1.

RESULTS: Figure 9 demonstrates three different kinds ofexcitation. The top line shows the combustion chamberwith the initial pressure impulse marked as dark gray.The first example is an annular excitation. The end gasthat is assumed to burn by knock is considered as a ringabove the piston at the cylinder wall. The secondexample is a half ring above the piston on the outlet sideof the combustion chamber. The third example considersa small excitation region at one side of the cylinder.

Figure 10 shows the propagating pressure wave for thefirst example at several time instants. Here, dark graygenerally means high pressure but each image is scaledrelatively between highest and lowest pressure values.Therefore, it is difficult to compare the amplitudes of

Figure 9. Transient analysis for different kinds of excitation (top), according time signals (middle) and accordingperiodograms (bottom). Solid lines mark Sensor 1, dotted lines Sensor 2, and dash-dotted lines Sensor 3.

different images. The annular excitation causes anannular pressure wave front propagating towards thecylinder axis yielding a pressure peak when they meetthere. Figure 9, middle, shows the time signals of thevirtual pressure sensors in the interval from 0 to 0.4ms.Depending on the excitation location and sensor positionthere is a delay between excitation instant and the timewhen the sensors get signals. This delay corresponds tothe distance between excitation and sensor and thespeed of sound. The delay can be observed best in thethird example. After the sensors have got the first signalsthey observe some kinds of oscillation. In order toinvestigate the oscillations the signals were transferred infrequency domain by estimating their periodograms, seeFigure 9, bottom. Table 1 contains the frequencies of theresonances at piston position 8.1=z mm in order to beable to assign the periodogram peaks to the combustionchamber resonances. The strongest component in thefirst example that is observed by Sensor 1 has afrequency of about 14 kHz. This corresponds well to Res.5. Comparing to Figure 2 this resonance has a pressureantinode on the cylinder axis and that is why Sensor 1can observe the resonance well. Figure 10 demonstratesthe excitation of Res. 5 too. Sensor 2 samples thisresonance also with high amplitude, because its positionis near a pressure antinode too. Sensor 3 observes thisresonance worse due to its position. The secondresonance is received well by Sensor 2 at about 8.5 kHz.Both other sensors receive this resonance too, but theiramplitudes are very low. In the second example Res. 2 isthe strongest component observed best by Sensor 2.Res. 5 again was excited strongly well received bySensor 1 and 2, similar to the first example. In the thirdexample Res. 1 and most likely Res. 4, 7 and 10 arestrongly excited. While it is easy to identify Res. 1, it ismore difficult for the other resonances because there aremore resonance with similar frequencies. But theexcitation in the third example takes place in pressureantinodes of the mentioned resonances. Otherresonances like Res. 3, 6, and 9 have a pressure node atthe location of excitation. So they are not or only weaklyexcited. Although Res. 1 is strongly excited according toSensor 3, this resonance is not visible in the periodogramof Sensor 2 at all because Sensor 2 is perfectly located in

a pressure node of Res. 1. The excitations in theseexamples are quite simple and arbitrary but they showthat a random knock origin may cause the randomexcitation of the resonances. If so, the periodogram ofthe sensor signals can be regarded as a kind of fingerprint of the kind of excitation. Vice versa, it could bepossible to determine the knock origin from theperiodogram.

CONCLUSION

The FEM approach has been successfully applied tomodel combustion chamber resonances. The frequencyand amplitude modulations of the resonancescorresponding to their metamorphosis were estimatedand a sensor characteristic to assess pressure sensorpositions was proposed. Simulation results fit well resultsof measured data using Wigner-Ville spectra as high-resolution time-frequency representation. Applyingtransient analyses it was demonstrated that the randomexcitation of the resonances is very likely caused by therandom origin of knock.

REFERENCES

1. Draper, C. S., “Pressure Wave AccompanyingDetonation in the Internal Combustion Engine”,Journal of the Aeronautic Sciences, Vol. 5, No. 6, pp.219-226, 1938.

2. Hickling, R., Feldmaier, D. A., Chen, F. H. K., Morel,J. S., “Cavity Resonances in Engine CombustionChambers and Some Applications”, J. Acoust. Soc.Am., 73 (4), pp. 1170-1178, 1983.

3. Scholl, D., Davis, C. Russ, S. Barash, T., “ TheVolume Acoustic Modes of Spark-Ignited InternalCombustion Chambers”, SAE Paper 980893, 1998.

4. Martin, W., Flandrin, P., “Wigner-Ville-SpectralAnalysis of Nonstationary Processes”, IEEE Trans.on ASSP, 33 (6), pp. 1461-1470, 1985.

5. Böhme, J. F., König, D., “Statistical Processing ofCar Engine Signals for Combustion Diagnosis”, Proc.IEEE 7th Workshop on Statistical Signal and ArrayProcessing, pp. 369-374, invited paper, 1994.

Res. Freq. [kHz] Res. Freq. [kHz]1 6.31 9 19.802 8.63 10 19.973 11.70 11 20.264 11.94 12 23.895 14.04 13 23.946 15.99 14 24.007 16.03 15 24.138 18.69 16 24.92

Table 1. Resonance frequencies at piston position8.1=z mm.

Figure 10. Instantaneous pressure distribution at timeinstants ∆∆∆= 140,...,20,10,0t in case of the

annular excitation.