RECONSTRUCTION OF DIESEL ENGINE CYLINDER PRESSURE USING A TIME DOMAIN SMOOTHING TECHNIQUE

Mechanical Systems and Signal Processing (1999) 13(5), 709}722 Article No. mssp.1999.1229, available online at http://www.idealibrary.com on

RECONSTRUCTION OF DIESEL ENGINE CYLINDER PRESSURE USING A TIME DOMAIN SMOOTHING TECHNIQUE Y. GAO

AND

R. B. RANDALL

School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney 2052, Australia (Received 3 March 1998, accepted 26 April 1999) The time waveform of engine cylinder pressures is the most useful parameter in evaluating cylinder working procedures, such as compression, combustion and expansion in a diesel engine cylinder. Unfortunately, direct measurement of cylinder pressure signals is impractical because installing pressure transducers is di$cult as well as uneconomical for general use. Therefore, indirect derivation or reconstruction of such signals from externally measured engine vibration is desired. Di!erent procedures have been developed to reconstruct cylinder pressures from externally measured vibration responses. In this paper, the theories of existing procedures are reviewed. Also the Laplace transform (LT) is used to design a time-domain smoothing technique. The properties of this technique were investigated theoretically. Then it was applied to reconstruct cylinder pressures from response vibrations measured externally on a single-cylinder diesel engine, and the results obtained corresponded well with the measured pressure waveforms.  1999 Academic Press

1. A REVIEW OF EXISTING PROCEDURES

1.1. THEORY Mathematically, the response vibration, y(t), of a linear time-invariant system, to a single source, x(t), is the convolution of the source, x(t), with the impulse response, h(t), of the path. In discrete digital signal processing, this relationship is often expressed as y(n)"h(n)*x(n)

(1.1)

where n"t/Dt and Dt is the sampling interval. Z-transforming both sides of equation (1.1) produces >(z)"H(z)X(z).

(1.2)

If the Z-transform is evaluated along the unit circle in the Z-plane, e.g. z"eHSLDR, equation (1.2) becomes >(u)"H(u)X(u)

(1.3)

where >(u), H(u) and X(u) are the discrete Fourier transforms (FT) of y(n), h(n) and x(n), respectively. H(u) is also called the frequency response function (FRF) of the system under consideration. Sometimes, the Z-transform is evaluated along a spiral rather than the unit circle, e.g. z"eQLDR (s"p #ju; p is a constant). In this case, the response and source are related by   0888}3270/99/050709#14 $30.00/0

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Y. GAO AND R. B. RANDALL

the transfer function (TF), H(s), of the system: >(s)"H(s)X(s)

(1.4)

where >(s) H(s) and X(s) are the discrete Laplace transforms (LT) of y(n), h(n) and x(n), respectively. The complex cepstrum of a discrete digital sequence, l(n), is de"ned as [1] lL (n)OZ\[log(Z[l(n)])].

(1.5)

Applying the operation of equation (1.5) to both sides of equation (1.1) produces yL (n)"hL (n)#xL (n)

(1.6)

where yL (n), hL (n) and xL (n) are the respective complex cepstra of y(n), h(n) and x(n). It can be seen that the convolution relationship in equation (1.1) is transformed to multiplication in equations (1.2}1.4). Any third quantity can be derived in each of the equations (1.2}1.4) if the other two are known somehow. Suppose that >(u), and H(u) in equation (1.3) have been obtained in some way and the third quantity X(u) is derived as X(u)">(u)/H(u).

(1.7)

This equation shows the basic idea for reconstruction of sources, and its practical use and limitation are further discussed in Section 1.2. In equation (1.6), the path and source cepstra become additive, and this makes the derivation of the source from the response very simple, e.g. xL (n)"yL (n)!hL (n).

(1.8)

This equation forms the base of cepstral procedures for reconstruction of sources from responses. 1.2. SPECTRAL PROCEDURES The operation represented in equation (1.7) is usually called inverse "ltering and the reciprocal of the FRF, B(u)"1/H(u)

(1.9)

is the so-called inverse "lter. Generally, this inverse "lter is not realisable because the FRF, H(u), has non-minimum-phase zeros and they are inverted to unstable poles in the inverse "lter, B(u). However, the inverse "ltration in equation (1.7) still holds even if the inverse "lter, B(u), itself is unstable. This is because, in the ideal case, all the poles and zeros (minimum and non-minimum phase) in the denominator, H(u), will always get cancelled with their correspondence in the numerator, >(u) [2]. Thus, the ideal inverse "ltering procedure in equation (1.7) can still be applied to reconstruct the source, X(u), in some situations if a perfect record of H(u) is obtained. For example, the ideal inverse "ltering procedure was applied to reconstruct the cylinder pressure waveform of a single-cylinder diesel engine in references [3, 4] (also see later). The ideal inverse "ltering procedure is not always applicable, and this is attributed to the variation of the FRF used in inverse "ltering. Firstly, it is impossible to obtain a perfect FRF. Usually, an estimated version of the FRF is used in the inverse "ltering procedure. The estimation accuracy is determined by many factors, such as valid measurements of response and source. Secondly, the FRFs of all machines with the same speci"cations deviate among themselves greatly and this deviation limits the application of the FRF

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measured on one machine to the others of the same class. Lyon tested 46 diesel engines with the same speci"cations on a production line under the same conditions [5] and he found that the standard deviation in the magnitude of the FRF is about 8 dB in the higher frequencies and of the order of 10 rad in phase. Finally, the FRF changes with operating conditions (with speed, load and temperature of an engine) [3, 6]. In order to cope with the above FRF variations, the ideal inverse "ltering procedure is modi"ed. In reference [3], FRFs obtained under di!erent operating conditions are averaged to get a version of 1 0 HM (u)" H (u) (1.10) G R G with equal weighting on di!erent operation conditions. It was found that the averaged FRF could be used to recover the source (cylinder pressure) in a wider range of operating conditions than any individual one. Kim and Lyon [7}9] proposed a process to reduce the e!ect on the recovery of the source waveform of the FRF variations using cepstral smoothing, as illustrated in Fig. 1. In this process, smoothing is achieved through lowpass liftering the corresponding complex cepstra (in other words, windowing the complex cepstra around zero quefrency). The liftering operation (multiplication in the quefrency domain) is the same as a running average of the log spectrum (convolution in the frequency domain), which smooths the log spectrum. Also, the shorter the lifter, the smoother the log spectrum. The smoothed magnitude and phase of the FRF are estimated using the smoothed magnitudes and phases of the response and source which could be obtained previously under a particular operating condition or on a given machine of the same class. Then, the thus estimated FRF can be used for the general purpose of inverse "ltering for the same machine under di!erent operating conditions or for the same class of machines. However, why cepstrum smoothing is robust is still not explained satisfactorily. It has been shown that combination of the above process with an adaptive recovery process can further improve the accuracy of the recovered source [7] and this is beyond the scope of the current discussion.

Figure 1. Source waveform recovery using cepstral smoothing.

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In the case where the source has minimum-phase properties, the response and path can be separated into minimum-phase and allpass components, respectively [2, 6, 10]. The minimum-phase components are used to recover the source waveform with wrong timing, and then the recovered source waveform is retimed using the allpass components [6, 10]. This procedure overcomes the problem of unstable inverse "ltering caused by incomplete cancellation of non-minimum-phase zeros in equation (1.7), and it is very useful in the case where the source has impact signatures, such as valve impact forces in reciprocating machines. However, this procedure cannot be used to recover sources such as diesel engine cylinder pressures that have non-minimum-phase properties [3, 4]. 1.3. CEPSTRAL PROCEDURES In reference [4], equation (1.8) was used to recover the source, the cylinder pressure waveform of a single-cylinder engine. The path complex cepstrum was estimated under a given operating condition and then applied to recover the sources under the other operating conditions. It was shown that the cepstral procedure produced better results than the spectral procedure of direct use of equation (1.7). Furthermore, the cepstral procedure could produce satisfactory results around the Top Dead Centre (TDC) in the case where pressure and acceleration were not well correlated even where the direct use of the FRF as an inverse "lter failed [4]. Note that the linear-phase delays of the response and path are taken out in the calculation of their respective complex cepstra and the delay of the source recovered from equation (1.8) needs to be calculated and inserted using the following equation: r "r !r , (1.11) V W F where r , r and r are the respective delays (in number of samples) of the source, response V W F and path [4]. Theoretically, equations (1.7) and (1.8) represent the same linkage between the source and the response in two di!erent domains. Both of them should produce the same results when the same response and path are used.

2. TIME-DOMAIN SMOOTHING

2.1. INVERSE FILTERING ALONG A LINE PARALLEL TO THE FREQUENCY AXIS IN THE S-PLANE The discussion in Section 1.2 shows that the ideal inverse "ltering procedure is not always applicable because of variations in the FRF. In order to reduce the undesirable e!ects on the recovered sources (engine pressure waveforms) of the path variations, the following inverse "ltering procedure is proposed in the current investigation. Firstly, the TF is estimated by >(s) HI (s)" X(u)

(2.1)

where >(s) is the response spectrum evaluated along a transform line (s"p ; p '0) and   X(u) is the FT of the source (cylinder pressure signal in our case). Both the response and the source are measured under a particular condition. Secondly, if > (s) is the response  spectrum which is measured under di!erent conditions and evaluated along the same transform line in the S-plane, the source spectrum is recovered, e.g. > (s) XI (u)"  . HI (s)

(2.2)

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Finally, inverse Fourier transforming the recovered source spectrum produces the discrete source time waveform, e.g. x(n)"F\[XI (u)].

(2.3)

This procedure can reduce the undesirable e!ect of the TF variations to a negligible extent, producing a smooth path spectrum. Mathematically, evaluating the path spectrum along a line parallel to the frequency axis in the S-plane is equivalent to multiplying the time signal in question with an exponential window. The multiplication operation in the time domain is the same as a running and smoothing average of the FT spectrum in the frequency domain, hence the term, &time domain smoothing (TDS)' 2.2. INVESTIGATION OF THE TDS PROCEDURE 2.2.1. Pole/zero model of transfer functions For an N degree-of-freedom vibrating system its TF can be expressed as a ratio of a factorised numerator to a factorised denominator in the complex Laplace domain [11, 15]: “+ (s!s ) XI (2.4) H(s)"B I  “, (s!s ) I NI where s and s are the zeros and poles, respectively, and M represents the number of XI NI zeros. In general, the poles appear in complex conjugate pairs while the zeros might appear in complex conjugate pairs and/or on the real damping axis in the S-plane [11]. At the driving point, the poles and zeros alternate along the frequency axis in the S- plane while the number of zeros might be further reduced in transfer measurements [11, 15]. In practice, a continuous vibrating system has an in"nite number of modes and the corresponding frequency range is also in"nite. It is impossible to represent the system TF with a complete model, so a truncated model is required to approximate the TF in a given frequency range, e.g. “+R

(s!s ) I XI #R(s), M (2N (2.5) “,R (s!s ) R R I NI where M and 2N are the numbers of zeros and poles in the truncated model, respectively, G G and R(s) is a polynomial of order p used to compensate for the contributions of truncated modes [11]. Equation (2.5) can be further factorised as H(s)"B

“,R>N

(s!s ) I XI . (2.6) “,R (s!s ) I NI In equations (2.4)}(2.6), B (i"1, 2, 3) are the respective scaling factors. G It can be seen that the total number of zeros in equation (2.6) is 2N #p rather than M . R R There are 2N #p!M extra zeros introduced. These zeros are not associated with the true R R physical model within the frequency band of interest, but are necessary to compensate for the contributions of truncated modes. We call such kinds of zeros &phantom zeros'. The distributions of &phantom zeros' were throughly investigated in reference [11]. It is found that &phantom zeros' are typically arranged in pairs around the frequency axis, so that the compensation for the magnitude is enhanced, while no (or very small) compensation is made for the phase. Under such a distribution, half of the &phantom zeros' have maximumphase properties even when the system is minimum phase. H (s)"B G 

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Y. GAO AND R. B. RANDALL

Figure 2. Poles and zeros of a TF in the S-plane: zeros; ¹ reverberation time.

, poles;

, minimum-phase zeros; 䊉, non-minimum phase

The number of &phantom zeros' required to compensate for the contributions of truncated modes depends on the separation of the source and response points. For a driving point measurement, poles and zeros appear alternately and the e!ect of truncation is very small, whereas at the other extreme, with no actual zeros, the number of &phantom zeros' becomes 2N #p. R Tohyama and Lyon studied the distributions of poles and zeros of a TF [12}14]. In their model (as shown in Fig. 2), the numbers of zeros and poles are equal. When the source and response points are at the same position, the poles and zeros alternate along the pole line in the S-plane. With the separation of the source and response points, the poles remain on the pole line but the zeros move. When the response point is in the reverberant "eld, half of the zeros remain on the pole line and the other half drift away from the pole line to form a &cloud of zeros' around the pole line. In the case of real-mode shape functions, the &cloud of zeros' are symmetrically distributed about the pole line [14], meaning that a quarter of the zeros are located on each side of the pole line. Such pairs of zeros cannot be observed from the TF evaluated along the frequency axis and their contributions to the TF are the same as the &phantom zero' pairs [11]. 2.2.2. Contributions of uncancelled poles and zeros It is the variations of the FRF that limit the application of the ideal inverse "ltering procedure [equation (1.7)]. These variations of the FRF could give rise to incomplete cancellation of the zeros (minimum and non-minimum phase) and poles in the denominator, H(u), with their correspondence in the numerator, >(u), in equation (1.7). The incomplete cancellation of non-minimum-phase zeros in the denominator, H(u), results in unstable inverse "ltering, and the existence of any &residual' pole or zero of H(u) in the inverse "ltering operation may produce an incorrect recovery of the source. It is not thought that every residual pole or zero (if any) has the same e!ect on the inverse "ltering operation and this needs to be further investigated. Figure 3 shows how a residual pole can result in one pole and one zero in the recovered source. Here H(s) is the variant version of H(s). The variation (say caused by di!erent operating conditions) is represented by one pole which is shifted higher in frequency. After inverse "ltering, all the other poles and zeros in H(s) are cancelled with their corresponding ones in H(s), but the uncancelled pole in H(s), becomes a zero and its correspondence in H(s) remains as a pole in the recovered source X(s). In Fig. 4, the contributions of such a pole and zero are investigated. In the case of small damping, the pole and the zero are very close to and just above the frequency axis, as shown

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Figure 3. Generation of a pole-zero pair because of the shift of one pole position in the response spectrum, >(s): , pole; , zero.

Figure 4. Contributions of an uncancelled pole (p"!5, p "50). Note that the phase curves have been shifted  down for clarity and their #at parts correspond to zero radians.

in Fig. 4. When the TF is evaluated along the frequency axis (Fig. 4a), a very deep valley and a very strong peak are produced by the zero and the pole in the recovered source spectrum, respectively. Also, the phase jumps from zero to nearly n and then down to zero. It can be expected that such a residual pole will cause the recovered time waveform to oscillate at the pole frequency. Figure 4(b) illustrates the case where the TF is evaluated along a line parallel to the frequency axis in the S-plane. The zero phasor (from the zero to the test frequency on the line) and the pole phasor (from the pole to the test frequency on the line) are nearly the same in length and their phases are also close in angle. Also the two phasors

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Figure 5. Contributions of an uncancelled non-minimum-phase zero (p"5, p "50). Note that the phase  curves have been shifted down for clarity and their #at parts correspond to zero radians.

rotate anticlockwise as the test frequency passes them in the increasing direction. Therefore, the zero phasor nearly cancels the pole phasor in the recovered source spectrum, meaning that such a residual pole would not give rise to serious problems in inverse "ltering. The above explanation also applies for the case of residual zeros except that the zero in H(s) becomes a pole and the zero in H(s) remains a zero in the recovered X(s). In Fig. 5 the case is studied where a non-minimum-phase zero in >(s) is shifted higher than its correspondence in H(s). This generates one non-minimum-phase zero and one unstable pole in the recovered X(s). When the TF is evaluated along the frequency axis [Fig. 5(a)], the recovered source has both non-minimum-phase zero and unstable pole behaviours, meaning that the inverse "ltering operation is not stable. However, when the TF is evaluated along a transform line parallel to the frequency axis, as shown in Fig. 5(b), the zero and pole phasors rotate anti-clockwise as the test frequency increases. This implies that the recovered source has minimum-phase behaviour. If the transform line is far enough from the pole and zero, their contribution becomes negligible because of mutual cancellation of the pole and zero phasors [Fig. 5(b)] and large damping properties of the pole and zero. The following comments can be made from the above discussions. Any uncancelled pole or zero (minimum or non-minimum phase) close to the frequency axis could result in incorrect recovery of the source. Evaluating the TF along the frequency axis, has a much bigger chance to generate this undesirable e!ect as shown in Figs 4(a) and 5(a) than along a transform line (s"p ; p '0) because all the poles and most of the zeros are very close to   and cluster around the frequency axis. Evaluating the TF along a transform line to the right of the frequency axis in the S-plane can greatly reduce these undesirable e!ects on the recovered source, producing a better quality of source time waveform when reconstructed by the TDS procedure than by the direct inverse "ltering.

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2.2.3. An explanation for the robustness of the CS procedure Now a satisfactory explanation can be made regarding to the robustness of the cepstral smoothing procedure [7]. The variations of a TF result in incomplete cancellation of its poles and zeros with their correspondence in the response spectrum. Any uncancelled pole or zero generates one zero and one pole in the recovered source spectrum. In the cepstrum domain, sources with impulsive signatures concentrate around the zero quefrency and so do those uncancelled zeros and poles far away from the frequency axis in the S-plane. The contribution of these poles and zeros is negligible because of mutual phasor cancellation [Figs 4(b) and 5(b)] and large damping. However, the higher quefrency range is occupied by those uncancelled poles and zeros close to the frequency axis and having small damping. Lowpass liftering the complex cepstra of the TF and the response vibration in question reduces the undesirable contribution of the latter residual poles/zeros, thus making the inverse "ltering procedure robust.

3. A PRACTICAL APPLICATION AND COMPARISON

Experiments were done on a single-cylinder, four-stroke diesel engine [3, 4]. Cylinder combustion pressures and external response vibrations (accelerations) were measured simultaneously at di!erent engine speeds and loads. Fig. 6 illustrates the respective TFs (2400 rpm and full load) estimated using the direct, TDS and CS procedures. From this "gure the following observations can be made: E =indow e+ect. This e!ect is represented by the di!erence between the TFs in the initial frequency range. For the solid (without smoothing) and dashed (with CS) lines, a rectangular window with half hanning tapers [3] was applied in the time domain in order to separate adjacent events, while for the dotted lines, an exponential window was applied in the time domain in order to shift the transform line to the right of the frequency axis in the S-plane and to separate adjacent events. It can be seen that the CS procedure produced an optimum TF with smooth variation in both magnitude and phase. In the initial frequency range, however, the TDS procedure generated a valley (dotted line) at the position of the second peak of the solid line. In the remaining frequency range, the dotted lines had nearly the same smoothness and general slopes as the dashed lines, but with a shift. In fact, the deviation in the initial frequency range can be mutually cancelled in the inverse "ltering procedure. E Magnitude shift [Fig. 6(a)]. The mean value of a log TF represents its scaling factor. This factor scales the time signal in question in the whole time range. Exponentially windowing the time signal reduces its general level and thus its &valid scaling factor'. So, its log TF is shifted down. In the cepstrum domain, however, the scaling factor is represented by the zero quefrency component and windowing a cepstrum around zero quefrency will not change its scaling factor. E Phase shift [Fig. 6(b)]. As has been pointed out above, the TDS procedure generated the deviation of replacing one peak with one valley in the initial frequency range. This deviation introduced a phase jump of 2n around the peak. For the remaining frequency range, the two phase curves obtained using the two di!erent smoothing procedures are nearly parallel to each other with a phase di!erence of about 2n [above 300 Hz in Fig. 6(b)]. Figure 7 shows how the recovered pressures tend to approach the measured ones with increased damping (further shifting the transform lines to the right of the frequency axis in the S-plane). Here the actual position of a transform line is indicated by a damping index (DI) for convenience of digital signal processing. This index is de"ned in such a way that the

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Figure 6. Comparison of time domain smoothing with cepstrum smoothing for the test condition of 2400 rpm and full load: (a) magnitude and (b) phase.

index equals seven where the exponential window is decayed by 60 dB at the right-hand end of the record. The pressures were recovered from the response vibrations measured at 3600 rpm and full load using the TFs estimated under the operating condition of 2400 rpm and full load. The recovered pressure spectrum in Fig. 7(a) was obtained using the direct inverse "ltering procedure [equation (1.7)], which is the case without smoothing (or damping index equals zero). The residual poles/zeros can be clearly viewed. The pressure spectra in Figs 7(b) and (c) were recovered using the TDS procedure but with di!erent damping indices [equation (2.6)]. The damping indices were three and seven for Figs 7(b) and (c), respectively. It can be seen that the undesirable contributions from the residual poles/zeros were reduced with the increase of the damping index. Figures 7(e)}(g) are the pressure time waveforms corresponding to the spectra in Figs 7(a)}(c). The recovered pressure time waveform (solid line) tends to approach the measured one (dashed line) as the damping index increases from zero to seven [Figs 7(e)}(g)]. Also the corresponding results

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Figure 7. Cylinder combustion pressures reconstructed from the response vibration measured at 3600 rpm and full load, solid lines are the reconstructed pressures and dashed lines are the measured ones. (a)}(d) Pressure spectra and (e)}(h) pressure time waveforms.

obtained using the CS procedure are given in Fig. 7(d) (spectrum) and (h) (time waveform) for comparison. In Fig. 8 are compared the pressure waveforms reconstructed using the transfer path estimated under the condition of 2400 rpm and full load and the response vibration

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Figure 8. Comparison of Pressure Waveforms Reconstructed Using di!erent procedures. Dashed lines are measured pressure waveforms and solid lines reconstructed ones. (a) Direct inverse "ltering, (b) Cepstrum domain inverse "ltering, (c) time domain smoothing and (d) cepstrum smoothing.

measured at the same engine speed but di!erent engine load (50%). The solid lines are the recovered pressure waveforms and the dashed lines are the measured ones. The results in Figs 8(a) and (b) were obtained using the inverse "ltering procedures represented by equations (1.7) and (1.8), respectively. These results illustrated that the two inverse "ltering procedures produced the same quality of the recovered pressure waveforms. Mathematically, the two inverse "ltering procedures are based on the same principle and, therefore, the same e!ectiveness is expected of them. However, equation (1.7) derives the source from the path by means of division, while in equation (1.8) the derivation of the source is implemented by subtraction, which leads to more robust and stable discrete digital manipulation. In Figs 8(c) and (d) are shown the recovered pressure waveforms with smoothing operations. The recovered pressure waveform in Fig. 8(c) was obtained using the TDS

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procedure discussed in Section 2. The recovered pressure waveform in Figs 8(d) was obtained applying cepstral smoothing to the complex cepstrum. It can be seen that the pressure waveforms recovered with smoothing operations match the measured ones better than those recovered without smoothing operations. Applying an exponential window to the transient vibrations has two functions. The "rst is to shift the transform line to the right of the frequency axis in the S-plane and the second to separate the adjacent transient events. The shorter the e!ective window length, the further the transform line is shifted to the right of the frequency axis in the S-plane and the better the separation of adjacent transient events. However, valid vibration information might be windowed out if the e!ective window length is too short. Therefore, a proper e!ective window length has to be determined. From the current investigation, it seems a good choice that the exponential window length decays by 60 dB at the end of the transient vibration pulse caused by cylinder pressures (over 1503 of crank angle in the current case). In practice, the window position can be determined either by using a tacho signal from the engine shaft rotation (if such a tacho signal is available) or by eye inspection of the time waveforms of the transient vibrations caused by cylinder pressures. It should be pointed out that any d.c. o!set of the acceleration signal needs to be removed before application of an exponential window. This will ensure that the leading edge of the exponential window does not produce a spurious peak. We have successfully reconstructed the cylinder pressure waveforms from the response vibrations measured on a six-cylinder engine where the crank angle between the adjacent combustion pulses is one}sixth of that of the single cylinder engine discussed in this paper (i.e. 1203 of crank angle) [16].

4. DISCUSSION AND CONCLUSIONS

Residual non-minimum-phase zeros cause unstable inverse "ltering while residual minimum-phase zeros and poles can produce incorrect time waveforms. The undesirable contributions from these residual poles and zeros can be made negligible by evaluating the FT of a time signal along a straight line located to the right of the frequency axis in the S-plane (p'0). This leads to the development of the time domain smoothing (TDS) procedure in this paper. The implementation of the TDS procedure is very simple and easy. It needs only one windowing and two Fourier transform operations, which is the same amount of calculation as required for a direct inverse "ltering procedure. For the cepstral smoothing (CS) procedure, however, the following extra operations are required: two FTs, log transform, exponential transform and phase unwrapping. Also, care must be taken when unwrapping the phase. Furthermore, the same quality of the recovered pressure waveforms was obtained from the TDS and CS procedures. The transfer path cannot be assumed to be invariant with operating condition, especially with engine speed. Usually, a correct pressure waveform could be recovered in the case where the FRF and the response vibration are measured at the same engine speed, while the recovery of a pressure waveform might fail in the case where the FRF and the response vibration are measured at di!erent engine speeds.

ACKNOWLEDGEMENTS

We are grateful to the Lombardini engine company and Professors G. Chiatti and A. Sestieri of the University of Rome for assistance with the measurements.

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REFERENCES 1. A. V. OPPENHEIM and R. W. SCHAFER 1989 Discrete ¹ime Signal Processing. Englewood Cli!s, NJ: Prentice-Hall. 2. M. TOHYAMA, R. H. LYON and T. KOIKE 1992 Journal of Acoustical Society of America 91, 2805}2812. Pulse waveform recovery in a reverberant condition. 3. R. B. RANDALL, Y. REN and H. NGU 1996 Proceedings of 21st International Seminar on Modal Analysis, K. ;. ¸euven, Belgium, 847}856. Diesel engine cylinder pressure reconstruction. 4. G. D. CASSINI, W. D'AMBROGIO and A. SESTIERI 1996 Proceedings of 21st International Seminar on Modal Anlaysis, K. ;. ¸euven, Belgium, 835}846. Frequency domain vs. cepstrum technique for machine diagnostics and input waveform reconstruction. 5. R. H. LYON 1988 Sound and