A real-time plane-wave decomposition algorithm for characterizing perforated liners damping at multiple mode frequencies Dan Zhaoa) Acoustics Laboratory, Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
(Received 2 May 2010; revised 9 December 2010; accepted 11 December 2010) Perforated liners with a narrow frequency range are widely used as acoustic dampers to stabilize combustion systems. When the frequency of unstable modes present in the combustion system is within the effective frequency range, the liners can efficiently dissipate acoustic waves. The fraction of the incident waves being absorbed (known as power absorption coefficient) is generally used to characterize the liners damping. To estimate it, plane waves either side of the liners need to be decomposed and characterized. For this, a real-time algorithm is developed. Emphasis is being placed on its ability to online decompose plane waves at multiple mode frequencies. The performance of the algorithm is evaluated first in a numerical model with two unstable modes. It is then experimentally implemented in an acoustically driven pipe system with a lined section attached. The acoustic damping of perforated liners is continuously characterized in real-time. Comparison is then made between the results from the algorithm and those from the short-time fast Fourier transform (FFT)-based techniques, which are typically used in industry. It was found that the real-time algorithm allows faster tracking of the liners damping, even when the forcing frequency C 2011 Acoustical Society of America. [DOI: 10.1121/1.3533724] was suddenly changed. V
I. INTRODUCTION
To suppress combustion instabilities,1 perforated linalong the bounding walls of a combustor are widely ers used as acoustic dampers; they dissipate acoustic waves, preventing further perturbations in heat release.5 The liners are typically made of metal, are arranged in layers, and have tiny perforated holes in them, as shown in Fig. 1. Their main damping mechanism involves vortex shedding generated over the rims of the perforated holes. They tend to be effective over a narrow frequency range. If the frequency of the unstable mode present is within the effective frequency range, the liners can efficiently dissipate acoustic waves. The acoustic damping of perforated liners is generally characterized by power absorption coefficient, which describes the fraction of incident waves being absorbed. To estimate it, acoustic plane waves either side of the liners need to be decomposed into incident and reflected waves and characterized in terms of frequencies, amplitudes and phases. This is typically done by implementing the fast Fourier transform (FFT)-based plane-wave decomposition techniques.6,7 Seybert and Ross6 proposed a two-microphone technique for plane-wave decomposition, basing on autoand cross-spectral density analysis. Their technique was demonstrated to be an accurate and reliable method to decompose plane waves. However, it required long duration of data acquisition and involved singular frequencies.8–10 2–4
a)
Author to whom correspondence should be addressed. Present address: Aerospace Engineering Division, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore. Electronic mail:
[email protected]
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J. Acoust. Soc. Am. 129 (3), March 2011
Pages: 1184–1192
A similar method was developed by Chung and Blaser.7 They used a transfer function technique to decompose a broadband stationary random acoustic wave. While most plane-wave decomposition techniques are carried out in the frequency domain, Moeck et al.11 recently proposed a time-domain multiple microphone method (MMM). It decomposed plane waves in real-time by solving a matrix equation involving the phase differences between pressure measurements. They argued that MMM could reject flow noise and remove singular frequencies. However, acquiring multiple pressure measurements is difficult to achieve in a real combustion system due to the space limit and high temperature environment. Furthermore, FFT was also carried out to estimate the plane wave frequencies and amplitudes. FFT-based techniques6,7,11 are widely applied for characterizing perforated liners damping in industry.12 However, such techniques are mainly used for off-line signal processing. To achieve online characterizing and decomposing plane waves, short-time fast Fourier transform (ST-FFT) is needed. ST-FFT has a frequency resolution Fs/Ns and involves Ns log2 Ns multiplications. Here, Fs denotes the sampling frequency and Ns is the sample length, which is generally chosen to be a power of 2. To improve the frequency resolution and to distinguish the spectral response of two or more modes, a larger Ns and an appropriate data window13 have to be chosen. This means that a larger memory and a longer duration are needed, which could result in a serious challenge for a microprocessor to accomplish Ns log2 Ns multiplications within each sample interval. Moreover, many measured processes have slowly time-varying spectra in practice that may be considered constant only for short record lengths.
0001-4966/2011/129(3)/1184/9/$30.00
C 2011 Acoustical Society of America V
Author's complimentary copy
PACS number(s): 43.20.Ye, 43.20.Mv, 43.55.Ev, 43.58.Bh [AJZ]
pffiffiffiffiffiffiffi mean flow speed, and j ¼ 1 ; Am,n and Bm,n are the amplitude components of pm,n(t). These amplitude components can be estimated by calculating the Fourier coefficients of pm,n(t) (in complex notation) as, ð 1 t pm;n ðtÞejXn v dt Tn tTn ^ n Þejki;n xm þ RðX ^ n Þejkr;n xm : ¼ IðX
p^m;n ðXn Þ ¼
1 ¼ ðBm;n jAm;n Þ 2
(2)
FIG. 1. Schematic of a typical liner.
II. THEORY OF THE REAL-TIME PLANE-WAVE DECOMPOSITION ALGORITHM
Figure 2 is a diagram showing a rigid pipe with plane waves propagating along the x axis. Two pressure measurements, p1(t) and p2(t) consisting of N sinusoidal modes (not necessarily harmonic) are acquired at x1 and x2, respectively, as given by, pm ðtÞ ¼
N X n¼1
¼
N X
pm;n ðtÞ ¼
N X
Evaluating the complex coefficients p^1;n and p^2;n and solving for the complex amplitudes of the incident and reflected ^ n Þ, yields ^ n Þ and RðX waves, IðX ^ nÞ ¼ IðX
p^1;n expðjkr;n x2 Þ p^2;n expðjkr;n x1 Þ ; exp½jðkr;n x2 ki;n x1 Þ exp½jðkr;n x1 ki;n x2 Þ (3a)
^ nÞ ¼ RðX
p^2;n expðjki;n x1 Þ p^1;n expðjki;n x2 Þ : exp½jðkr;n x2 ki;n x1 Þ exp½jðkr;n x1 ki;n x2 Þ (3b)
If the mode frequency Xn was known, then the amplitude components Am,n and Bm,n could be determined using ^ n Þ using Eqs. (3a) and (3b) ^ n Þ and RðX Eq. (2), and so IðX with the assumption of no singularity. (For completeness, singularity analysis is carried out and compared with results given by Seybert,10 as shown in Appendix A.) However, the frequency Xn is generally not known, an estimate of it, denoted by xm,n (with corresponding period sm,n) is used to perform a ST integration transform to pm,n(t), as shown as
½In ðxm ; tÞ þ Rn ðxm ; tÞ
n¼1
½Am;n sin Xn t þ Bm;n cos Xn t ;
(1)
ðt
(4)
where cm,n and sm,n are the integration components given by
n¼1
where pm,n(t) denotes the nth mode (with corresponding frequency Xn) of the pressure measurement, pm(t) at xm, In ðxm ; tÞ ^ n ÞejðXn tki;n xm Þ and Rn ðxm ; tÞ ¼ RðX ^ n ÞejðXn tþkr;n xm Þ denote ¼ IðX the incident and reflected waves at Xn, respectively, c þ uÞ and kr;n ¼ Xn =ð c uÞare the incident and ki;n ¼ Xn =ð reflected wave numbers, c is the speed of sound, u is the
1
pm;n ðtÞejxm;n t dt sm;n tsm;n 1 ¼ ðcm;n jsm;n Þ ; 2
q^m ðxm;n Þ ¼
cm;n ¼
2
ðt
2
ðt
pm;n ðtÞ cos xm;n t dt sm;n tsm;n 1 1 p_ m;n ðtÞ cos xm;n t jttsm;n ¼ pm;n ðtÞ sin xm;n t þ p xm;n ð X2 t þ n pm;n ðtÞ cos xm;n t dt; (5a) pxm;n tsm;n
sm;n ¼
pm;n ðtÞsinxm;n tdt sm;n tsm;n 1 1 pm;n ðtÞcosxm;n t p_ m;n ðtÞsinxm;n t jttsm;n ¼ p xm;n 2 ðt X þ n pm;n ðtÞsinxm;n tdt: (5b) pxm;n tsm;n
FIG. 2. (Color online) Schematic of a cylindrical pipe with plane waves propagating. J. Acoust. Soc. Am., Vol. 129, No. 3, March 2011
Overdot denotes time derivative. Note that the lower limit of the integral in Eq. (4) involves the estimated, time-dependent Dan Zhao: Real-time calculating acoustic liners damping
1185
Author's complimentary copy
In this work, a real-time algorithm for plane-wave decomposition is developed to estimate perforated liners damping at multiple mode frequencies. This is described in Sec. II. The algorithm is not only easy to implement but also requires much less computational power to achieve online estimation via using a shorter sample length. The performance of the algorithm is evaluated first in a numerical model with two unstable modes, as described in Sec. III. The results from the algorithm are then compared with those from ST-FFT-based techniques. Finally, in Sec. IV, it is experimentally demonstrated in an acoustically driven pipe system that the acoustic damping of a perforated liner at the dominant mode frequency (with the largest amplitude) can be rapidly and continuously estimated, even when loudspeaker forcing frequency was suddenly changed.
cm;n ¼
xm;n ½p_ m;n ðtÞ p_ m;n ðt sm;n Þ cos xm;n t pðx2m;n X2n Þ þ
sm;n ¼
x2m;n ½pm;n ðtÞ pm;n ðt sm;n Þ sin xm;n t pðx2m;n X2n Þ
tanðxm;n t þ wm;n Þ ¼ ;
(6a)
xm;n ½p_ m;n ðtÞ p_ m;n ðt sm;n Þ sin xm;n t pðx2m;n X2n Þ
x2m;n ½pm;n ðtÞ pm;n ðt sm;n Þ cos xm;n t pðx2m;n X2n Þ
:
(6b)
If xm,n converges to Xn (i.e., jxm,n – Xn j< r, r ¼ Xn/100), it can be seen from Eqs. (2) and (4), that the amplitude components of cm,n and sm,n will converge to Bm,n and Am,n, respectively. Therefore, an approximation to the signal of pm,n(t) ¼ Am,n sin Xn t þ Bm,n cos Xn t is psm;n ðtÞ ¼ sm;n sin xm;n t þ cm;n cos xm;n t ;
(7)
where psm;n ðtÞ is the reconstructed signal and its time derivative p_ sm;n ðtÞ is p_ sm;n ðtÞ ¼ xm;n ½sm;n cos xm;n t cm;n sin xm;n t :
(8)
Substituting Eqs. (6a) and (6b) into Eq. (8) yields p_ sm;n ðtÞ ¼
x3m;n ½pm;n ðtÞ pm;n ðt sm;n Þ pðX2n x2m;n Þ
:
(9)
Manipulating Eq. (9) gives Xn in terms of xm,n, X2n ¼ x2m;n þ c
x3m;n ½pm;n ðtÞ pm;n ðt sm;n Þ : pp_ sm;n ðtÞ
(10)
It can be seen that the right hand side of Eq. (10) consists of two terms. The first term indicates the current estimate of the frequency and the second one is a correction term. Equation (10) can be iteratively used to determine the mode frequency, Xn. However, the correction term is not always meaningful, especially around the singularity values at which p_ sm;n ðtÞ ¼ 0. Therefore, Eq. (10) is used to update frequency only if the correction term is meaningful. Moreover, a relaxation coefficient c (0 < c 1) is applied in Eq. (10) to update the frequency in part and ensure the smoother convergence, which is guaranteed and detailed information can be found in Appendix B. For characterizing single mode oscillation, setting c ¼ 1 will result in the estimated frequency to the correct value after one time step. However, such rapid correction (convergence) to the frequency is not feasible in practice, due to the presence 1186
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of other modes and background noise. Therefore, we “filter” the frequency correction term by setting c < 1, so that each update is only partial.15,16 Generally, the converging time is of the order of a single period of the mode and the multiplications involved are OðN 2 Þ, where N denotes the sample length within the mode period. Compared with the sample length Ns used in ST-FFT,13,14,17 N is much less and so the computational power required by the algorithm. The predicted phase wm,n corresponding to the reconstructed signal psm;n ðtÞ can be estimated as, sm;n sinðxm;n tÞ þ cm;n cosðxm;n tÞ ; sm;n cosðxm;n tÞ cm;n sinðxm;n tÞ
(11) where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coswm;n ¼ sm;n = s2m;n þc2m;n and sin wm;n ¼ cm;n = s2m;n þc2m;n : The foregoing has been shown the amplitude and frequency of the nth mode of pm(t) can be iteratively deduced using Eqs. (5) and (10), if pm,n(t) were known. P However, pm,n(t) was usually unknown and only pm ðtÞ ¼ Nn¼1 pm;n ðtÞ are measurable. However, if pm(t) consists of a dominant mode pm,d(t) (with largest amplitude) such that pm(t) pm,d(t), application of the algorithm on pm(t) means that its dominant mode is characterized. Once the characteristics of the dominant mode have been identified, the dominant mode is subtracted from the pressure signal pm(t). The remaining signal, pm ðtÞ psm;d (reconstructed signal) is then used to characterize the next most dominant mode. This identification and subtraction procedure can be repeated several times so that many modes are characterized in terms of frequency, amplitude, and phase.15,18 To illustrate the algorithm’s implementation, its framework for characterizing the dominant mode pm,d(t) of pm(t) is now clearly described: (1) Estimate the dominant mode frequency, xm,d (with corresponding period sm,d) and initialize relaxation factor c, time t, and integration period T ¼ sm,d. (2) Calculate the amplitude components using Eq. (4), i.e., Ðt 2 jxm;d t p ðtÞe dt ¼ cm;d jsm;d and calculate sm;d tT m p_ sm;d ðtÞ using p_ sm;d ðtÞ ¼ xm;d ½sm;d cosxm;d tcm;d sinxm;d t. (3) Update the estimated frequency using x2m;d ðt þ DtÞ ¼ x2m;d ðtÞ þ c
x3m;d ½pm ðtÞpm ðtsm;d Þ pp_ sm;d ðtÞ
:
(4) Go back to step 2. In most cases, the assumption of pm(t) pm,d(t) is a reasonable one. However, there are some cases, where instead pm(t) consists of several modes with comparable amplitudes. To characterize such measured signal, the ST integral transform to pm(t) can still be performed, ð 1 t pm ðtÞejXn t dt T tT ð ðt 1 t pm ðtÞ cos Xn t dt j pm ðvÞ sin Xn t dt ¼ T tT tT 1 (12) ¼ ðbm;n jam;n Þ: 2
q^m ðXn Þ ¼
Dan Zhao: Real-time calculating acoustic liners damping
Author's complimentary copy
period sm,n. This indicates that the width of the function (window), ejxm;n t j ttsm;n ; in Eq. (4) is time-varying, unlike the ST-FFT which involves using a window function, but its width is fixed.13 If the complex exponential function was thought to be an altered mother wavelet, then the integral in Eq. (4) was a modified version of wavelet transform.14 Manipulating and simplifying Eqs. (5a) and (5b) gives cm,n and sm,n as
ð 2 t pm ðtÞ sin Xn t dt am;n ¼ T tT ð N X 2 t ½Am;i sin Xi t þ Bm;i cos Xi t sin Xn t dt ¼ T tT i¼1 ð ð Am;n t Bm;n t cos 2Xn t dt þ sin 2Xn t dt ¼ Am;n T tT T tT ð N X Am;i t ½cosðXi Xn Þt cosðXi þ Xn Þtdt þ T tT i6¼n ð N X Bm;i t þ ½sinðXi þ Xn Þt sinðXi Xn Þtdt: T tT i6¼n (13) With the relationships between frequency and period applied, manipulating Eq. (13) and integrating terms give the amplitude component am,n as Tn ½Am;n sin 2Xn t þ Bm;n cos 2Xn tjttT 4pT N X Ti Tn ½Bm;i cosðXi Xn Þt þ 2pTðT n Ti Þ i6¼n
am;n ¼ Am;n
þ Am;i sinðXi Xn ÞtjttT
N X
Ti Tn 2pTðTn þ Ti Þ i6¼n
½Bm;i cosðXi þ Xn Þt þ Am;i sinðXi þ Xn ÞtjttT : (14) Equation (14) indicates that am,n converges to Am,n (i.e., am,n Am,n), if T Tn, which is equivalent to T/Tn ¼ a and
a 1. Here, a is defined as an integration period coefficient. A similar analysis shows that bm,n converges to Bm,n, when T ¼ aTn and a 1. This provides guidance for determining the modes amplitude components. It is obvious that the bigger a is, the more accurate and slower the algorithm identified the modes amplitude components. To achieve faster convergence and low steady state error in practice,15–19 a is empirically set to 3 to characterize measured signals consisting of multiple modes with comparable amplitudes. However, in the case of pm(t) pm,d(t), setting a ¼ 1 is enough. In general, skillful selection of a proper a can improve the algorithm’s performance.15–19 With the amplitude components estimated (by choosing a proper a), the modal frequency can then be determined using Eq. (10), as before. The procedure for characterizing the measured signal consisting of multiple modes with comparable amplitudes is similar to that with a dominant mode. For convenience, a flow chart figure is included in Appendix C to illustrate the working procedure for characterizing multiple modes. Interested readers can refer to the work by Zhao15 for more detailed information. It is worthy noting that the real-time algorithm is presented in analog form. However, the numerical implementation is carried out digitally.15 Since the analog to digital conversion is well-known, the details are omitted. When the mode frequencies xm,n are estimated, the modulus of the difference between each other is checked. If the mean value of the predicted jx1,n x2,nj is negligible, P n ¼ 12 2m¼1 xm;n ; is used as an approximation frequencies, x n and the amplitude comof Xn. Once the mode frequency x ponents cm,n and sm,n, are estimated, the incident and ^x ^x n Þ, can be determined using n Þand Rð reflected waves, Ið Eqs. (3a) and (3b). The phase difference hn between the incident and reflected waves is calculated by ^x ^x n Þ R^ ðx n Þg==fIð n Þ R^ ðx n Þgg ; where hn ¼ tan1 f
