Extraction and imaging of aerodynamically generated sound field of rotor blades in the wind tunnel test

Mechanical Systems and Signal Processing 116 (2019) 1017–1028

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Extraction and imaging of aerodynamically generated sound field of rotor blades in the wind tunnel test Liang Yu a, Haijun Wu a, Jerome Antoni b, Weikang Jiang a,⇑ a b

Institute of Vibration, Shock and Noise, State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China Laboratoire Vibrations Acoustique (LVA), University of Lyon, INSA-Lyon Batiment St. Exupery 25 bis av. Jean Capelle, 69621 Villeurbanne Cedex, France

a r t i c l e

i n f o

Article history: Received 13 April 2018 Received in revised form 9 July 2018 Accepted 21 July 2018

Keywords: Moving beamforming Cyclostationary signal processing Ffowcs Williams-Hawkings (FW-H) equation Rotor blades signal extraction

a b s t r a c t The acoustic beamforming has been widely applied in the imaging of flow-induced aeroacoustic sound sources. However, the measured signals of the rotor blades often accompanied with the unwanted interference from other components of the experimental setup in the wind tunnel test (for example, the rotor shaft and the stand), which results in the undistinguished sources in the beamforming result. In this paper, the signals of rotor blades are defined first as the cyclostationary process based on the Ffowcs WilliamsHawkings (FW-H) equation in the form of Wold-Cramer decomposition, which connects the statistical definition of rotor blades signals with the wave propagation model of moving sources. Then the developed cyclostationary signal processing tools of a second order, specifically the reduced-rank cyclic Wiener filter, can be applied in the wind tunnel test of rotor blades. The rotor blades signals can be extracted from the noisy measurements with other interferences, which aide to purify the image results of beamforming in the final experiment of wind tunnel test. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction During the operation of any equipment with rotor blades (for example, helicopters [1,2] or wind turbines [3–5]), the undesirable aerodynamic noise is usually generated. The physical mechanism of the generating noise is due to the interaction between the rotor blades and the surrounding air. This strong noise not only affects the physical and mental health of the occupants near the noise sources, but also causes the side effects such as acoustic-structural coupling, acoustic fatigue and so on, which affects the safety of the equipment. Therefore, studying the physical mechanism of rotor blades aerodynamic noise generation and effectively reducing aerodynamic noise are important research topics in the development of modern equipment. The earlier aeroacoustic theory is based on the Lighthill’s analogy, which specifically addresses the problem of the sound generation by a region of high-speed turbulent flow instead in a stationary medium [6]. By introducing the generalized function theory, Ffowcs Williams-Hawkings generalized Lighthill’s acoustic analogy approach to tackle a wider range of the problem, such as very general types of surfaces and motions [7]. Thus, the Ffowcs Williams-Hawkings (FW-H) equation has been generally accepted as the foundation for modeling the aerodynamically generated sound of main rotor. Thickness noise is caused by the displacement of the fluid in the flow during the passage of the blade, which is modeled as the monopole term in the FW-H equation; the loading noise is due to the forces that the blades exert on the surrounding fluid during their ⇑ Corresponding author. E-mail address: [email protected] (W. Jiang). https://doi.org/10.1016/j.ymssp.2018.07.042 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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periodic motion (Blade-Vortex Interaction (BVI) noise and Broadband noise are two important types of loading noise), which is modeled as the dipole term in the FW-H equation; and the quadrupole term model the high-speed-impulsive (HSI) noise when the flow becomes transonic and shocks appear. With the FW-H equation, the computational fluid dynamics (CFD) [8,9] and its application to rotor blades aerodynamics have become an inevitable step for accurate noise prediction. The cause and effect are seen to be clear from a theoretical perspective. However, the aerodynamic noise of rotor blades in the experimental research of wind tunnel still encounter difficulties. The cause is usually required to be sought with the effect of the experimental research, which is known as the inverse problem. Moving Beamforming [10,11] is one of the most exploited acoustic visualization method by phased microphone array measurements for rotor blades noise, which is an extended technique to identify moving sources by combining the de-Dopplerization technique with conventional beamforming. However, one problem is the rotor blades noise measurement usually accompany with the aerodynamic noise of wind tunnel (the background noise or the interference of other facilities), and the extraction of rotor blades noise from the total measurement is needed in the experimental research of wind tunnel. In the conventional setting, the signals are supposed to be stationary and the extraneous noise is assumed to be uncorrelated with the signals. After the computation of the cross-spectral matrix (CSM), the uncorrelated noise will concentrate on the diagonal elements of the CSM. A common practice is thus to set the CSM diagonal to zero, which is known to improve the dynamic range of the source localization maps. However, diagonal removal technique also leads to under-estimated source levels. More advanced methods have been recently proposed to avoid such problems by preserving the source information that lies in the CSM diagonal. Diagonal Reconstruction [12,13] was proposed to remove as much noise as possible as long as the denoised CSM remains non-negative. Wavenumber decomposition to filter out high wavenumbers associated with turbulent boundary layer noise have been proposed in Ref.[14]. Robust Principal Component Analysis [15] was proposed to recover the acoustic signal in case of poor signal-to-noise ratio due to the boundary layer noise, which is based on a low rank and sparse decomposition of the CSM. Probabilistic Factor Analysis [16] was proposed to de-noise the CSM in a probabilistic framework, which is based on the decomposition of the CSM into a low-rank part and a residual diagonal part attached to the unwanted noise. All these methods can not be directly used to de-noise the rotor blades sound signals due to the aforementioned fundamental assumption, both the rotor blades and interference noise of other facilities may not be stationary. The cyclostationary signal processing methods [17,18] has been widely applied in the machine fault diagnosis, identification of machine systems and separation of mechanical sources. However, few researchers have addressed the problem of aerodynamic noise generated by the rotor blades with the cyclostationary signal processing tool. One fundamental problem is the lack of formal definition of cyclostationary noise based on the partial wave equation which has been generally accepted and used in the aerodynamic noise engineering. This also becomes the main knowledge gap to further apply the cyclostationary signal processing tools in the wind tunnel test of rotor blades [19]. In this paper, the cyclostationary signal is defined firstly from the Wold-Cramer decomposition of non-stationary signals by imposing the periodic Green’s function. Then the moving sources forward model is derived in the form of Wold-Cramer decomposition, which connects the signal processing definition of the cyclostationary signal with the wave propagation model. By this work, then the developed cyclostationary signal processing tools can be applied in the wind tunnel test of rotor blades. Finally, the results of moving Beamforming can be further purified with the proposed signal extraction method. In this paper, the propagation model of moving monopole source is first given in Section 2, and the rotor blades noise is mathematically formulated as a cyclostationary process based on a periodic Green’s function, and a formal definition of the cyclostationary process of rotor blades noise is described; a signal extraction problem of the rotor blades noise from noisy measurements is formulated in Section 3, and extraction and visualization of rotor blades signals is given in Section 4. The experimental validation is shown in Section 5: one typical background noise is generated from the rotor shaft and the stand that interact with the flow, which is the main interference of aero-acoustic measurements of the rotor blades. The beamforming results show that the rotor shaft noises are eliminated from the image after the signal extraction. 2. Cyclostationary process modeling of rotor blades noise 2.1. The Wold-Cramer decomposition of non-stationary signals The basic definition of Wold-Cramer decomposition [20] is briefly described in this subsection for the preparation of the derivation of the next. In the stationary case, Wold’s decomposition uniquely describes any stationary stochastic process YðtÞ as the output of a causal, linear, and time-invariant system hðtÞ excited by strict white noise XðtÞ:

Z

t

YðtÞ ¼ 1

hðt  sÞXðsÞds:

ð1Þ

Wold’s decomposition imposes no other restriction on XðtÞ than having a flat spectrum almost everywhere. The frequency counterpart of wold’s decomposition is known as Cramer’s decomposition,

Z

YðtÞ ¼

1

1

Hðf Þej2pft dXðf Þ;

ð2Þ

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where the transfer function Hðf Þ is the Fourier transform of hðsÞ (s is a dummy variable for time). dXðf Þ is the so called specR1 tral increments associated with the stochastic signal XðtÞ : XðtÞ ¼ 1 ej2pft dXðf Þ, which is interpreted as the stochastic Fourier coefficient of XðtÞ at frequency f. It is noted that Xðf Þ is the integrated spectrum of XðtÞ, and the power spectral density of XðtÞ can be defined by taking the averaged derivative of Xðf Þ [21]. Hðf Þej2pft dXðf Þ may be interpreted as the results of filtering YðtÞ with an infinitely narrow-band filter centred on frequency f. A natural solution for extending the Wold-Cramer decomposition to non-stationary processes is to make the filter hðsÞ time varying. Let us define hðt; sÞ the causal impulse response at time t of a system excited by an impulse at time t  s, then

Z

t

YðtÞ ¼

hðt; t  sÞXðsÞds;

1

ð3Þ

such a representation has been shown to hold true for any non-stationary process and, most importantly, to be unique under mild regularity conditions of the impulse response hðt; sÞ. The frequency counterpart is

Z

YðtÞ ¼

1

Hðt; f Þej2pft dXðf Þ;

ð4Þ

1

where Hðt; f Þ is the Fourier transform of the time-varying impulse response hðt; sÞ. Cyclostationary process are nonstationary process whose statistics are periodically varying, which is easily obtained from the proposed representation by imposing that hðt; sÞ be a periodic function of time hðt; sÞ ¼ hðt þ T; s þ TÞ. 2.2. Sound field propagation of a moving monopole source The acoustic field generated by a moving source in a uniform flow is governed by an inhomogeneous wave equation, which is based on the derivative of Ffowcs Williams et al. [22,23].

1 D2 pð~ x; tÞ  r2 pð~ x; tÞ ¼ qð~ x; tÞ; c2 Dt2

ð5Þ

where pð~ x; tÞ is the acoustic field distribution, qð~ x; tÞ is the sound source, ~ x is the vector of spatial coordinate (Cartesian system with the orthogonal directions referred by the indices ~ x1 ; ~ x2 ; ~ x3 , respectively), t is the time variable, and c is the speed of D is defined by sound. The convective derivative Dt

D @ ~ ¼ þU  r; Dt @t

ð6Þ

~ is a constant speed vector of the uniform flow and r the divergence operator. The sound propagation of rotor blades where U can be modeled as the sound propagation of moving point sources, which has been already applied in the imaging of the blades of helicopter and wind turbine [10,11]. The Green’s function Gð~ x; tj~ y; t s Þ for an impulsive point source located at point ~ y that emits a sound pulse at time ts is defined as the one that satisfies the wave equation of

1 D2 Gð~ x; tj~ y; t s Þ  r2 Gð~ x; tj~ y; t s Þ ¼ dð~ x ~ yÞdðt  ts Þ; c2 Dt2

ð7Þ

where d is the Dirac delta function. Solution of Eq. (7) for the impulsive point source in an unbounded region yields the freefield Green’s function,

Gð~ x; tj~ y; t s Þ ¼

~ ~ ~

s Þj dðt  ts  jxyUðtt Þ c : ~ x ~ y  Uðt  t s Þj 4pj~

ð8Þ

The Green’s function method allows for writing the solution of Eq. (5) as

pð~ x; tÞ ¼

Z

t

1

Z V

3 Gð~ x; tj~ y; ts Þqð~ y; ts Þd ~ yds;

and the spatial integration is over a volume V ¼ ten as the convolution form of [24]

pð~ x; tÞ ¼

Z

t

1

Z

ð9Þ R

3 Gð~ y; t  t s Þqð~ y; t s Þd ~ ydts :

V

3 y that enclose the sound source. The above equation Eq. (9) can be writd~

ð10Þ

V

It is assumed that the source qð~ x; tÞ ¼ qðtÞdð~ y ~ xs ðtÞÞ is a moving point source with time dependent position ~ xs ðtÞ, then the measured pressure at a fixed position ~ x from the sound source with the given positions ~ xs ðtÞ can be reformulated as,

pðtj~ xÞ ¼

Z

t

1

Gðt; t  ts j~ xs ðts ÞÞqðts j~ xs ðts ÞÞdt s :

ð11Þ

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Let us define Gðt; t s Þ the causal impulse response at time t of the wave propagation function excited by an impulse at time xÞ [20]. The t  ts , then the above representation is recognized as the Wold-Cramer decomposition to a random process of pðtj~ random process pðtj~ xÞ is non-stationary due to the convolution between the time varying impulse response function Gðt; t  t s j~ xs ðtÞÞ and the stationary source qðt s j~ xs ðt s ÞÞ (the sources are statistically stationary in time domain even though its positions are varying). It should be noted that the Wold-Cramer decomposition is unique under mild regularity conditions of the impulse response Gðt; t  ts Þ. The frequency counterpart of Eq. (11) is

pðtj~ xÞ ¼

Z

1 1

Gðt; f j~ xs ðt s ÞÞej2pft dqðf j~ xs ðt s ÞÞ;

ð12Þ

where Gðt; f Þ is the Fourier transform of the time-varying impulse response Gðt; ts Þ and dqðf Þ is the spectral increment associated with qðts Þ. When the trajectory of the moving source is a circle due to the circular motion as shown in Fig. 1, the random process pðtj~ xÞ can be further characterized as Cyclostationary process due to the Green’s function Gðt; ts Þ is a periodic P function of time Gðt; ts Þ ¼ Gðt þ T; t s þ TÞ ¼ k Gk ðsÞej2pkt=T [25]. Cyclostationary process pðtj~ xÞ is a non-stationary process whose statistics are periodically varying. It is noted that the Cyclostationarity of pðtj~ xÞ of Eq. (11) can be alternatively interpreted as the linear periodically time-variant (LPTV) transform of the stationary process qðt s j~ xs ðt s ÞÞ [26]. Thus, the rotor blades noise can be generally considered as the cyclostationary process, and the cyclostationarity of some specific rotor blades noise is explained as follows. 1. The cyclostationarity of BVI noise (a shed tip vortex subsequently interacting with a following blade) and broadband noise (produced by turbulence and vortices) can be simply modeled by introducing the periodicity (with random fluctuation) into the waveform generated by a model of single impact as in Ref. [27]. 2. The cyclostationarity of HSI noise is trivial since the impulsive noise of high intensity is associated with the blade pass frequency (BPF). 2.3. The formal definition of Cyclostationary process of rotor blades With above analysis, it is infers that the rotor blades noise can be generally modelled as the Cyclostationary process. The formal definition of cyclostationary process pðtj~ xÞ of rotor blades noise is given in this section and the position ~ x is omitted for simplicity. Formally, the stochastic process pðtÞ is said to be strict-sense stationary with cycle T if its joint probability density function f ðpðt1 Þ; pðt 2 Þ; . . . ; pðt n ÞÞ is periodic in t with period T, i.e.if

f ðpðt1 Þ; pðt2 Þ; . . . ; pðt n ÞÞ ¼ f ðpðt1 þ TÞ; pðt 2 þ TÞ; . . . ; pðtn þ TÞÞ:

ð13Þ

The most basic cyclostationary signal is the cyclostationarity at the first-order (first-order cyclostationarity: CS1), whose first order moment or expected value mp ðtÞ is periodic with period T:

mp ðtÞ ¼ EfpðtÞg ¼ mp ðt þ TÞ:

ð14Þ

Here the expected value Efg means ensemble average. CS1 sound signals are periodic waveform with possibly additive stationary random noise. A more general cyclostationary signal is that the cyclostationary at the second-order (a second-order cyclostationary:CS2), i.e. whose second-order moments are periodic. In particular, the autocorrelation function R2x ðt1 ; t2 Þ is a periodic function with period T:

R2p ðt 1 ; t2 Þ ¼ Efp ðt 1 Þpðt 2 Þg ¼ R2p ðt 1 þ T; t 2 þ TÞ:

Fig. 1. Propagation model of a moving source in a uniform flow.

ð15Þ

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CS2 sound signals are stochastic processes with periodic amplitude or/and frequency modulation, where  denotes the transpose for real vector (or matrix) and conjugate transpose for complex vector (or matrix) in this paper. One typical example of amplitude modulation is a rotor blades operating in a very uneven inflow (such as the rotor blades operating near a wall), the rotor blades pass through a turbulent boundary layer that extends over about one-fourth of the rotor disc plane [6]. The noise levels from a particular blade are low when it is in the free stream flow outside the boundary layer, and all the leading-edge noise is generated when the blade passes through the high levels of turbulence in the boundary layer. The signal from each blade is therefore strongly modulated. Signals that are CS1 and CS2 are referred to as wide-sense cyclostationary. Frequency modulation sound may be generated due to the modulation of the vibrations by the blade pass frequency. The frequency modulation sound signals usually look stationary (but sound cyclostationary), thus the hidden periodicity of sound is not obvious and demand specific signal processing tool to detect (e.g. spectral correlation [28]). One typical example of frequency modulation noise is the rotor blades noise in the scenario of without flow [17]. Finally, an n-th-order cyclostationary (CSn) signal is that whose n-th-order moments are periodic, In particular, if all moments up to infinity are periodic, then the signal is strictly cyclostationary. 3. Rotor blades noise extraction problem in wind tunnel test During the wind tunnel test of rotor blades, the target noise measurements usually accompany with the background noise and the flow-induced noise from other facilities. Especially the noise caused by the flow through the rotor shaft, which causes strong interference measurements. Therefore, the total measurement zðtÞ ¼ ðz1 ðtÞ; zm ðtÞ; . . . ; zM ðtÞÞT of M microphones are composed of the rotor blades noise pðtÞ ¼ ðp1 ðtÞ; pm ðtÞ; . . . ; pM ðtÞÞT , the rotor shaft noise

v ðtÞ ¼ ðv 1 ðtÞ; v m ðtÞ; . . . ; v M ðtÞÞT

T

and the background noise of wind tunnel nðtÞ ¼ ðn1 ðtÞ; nm ðtÞ; . . . ; nM ðtÞÞ as described in

zðtÞ ¼ pðtÞ þ v ðtÞ þ nðtÞ:

ð16Þ

pðtÞ is assumed to be cyclostationary (CS2 signal explicitly), v ðtÞ is usually non-stationary and not exhibit any periodicity, and nðtÞ is uncorrelated with the previous two noises. Thus, the problem to be tackled in the current paper is formulated to extract the pure rotor blades noise from the noisy measurements. In view of the cyclostationary modeling of rotor blades noise in the previous section, the cyclostationary property of the rotor blades noise can be exploited as the signal feature to aid the rotor blades noise extraction (Fig. 2). 4. Extraction and imaging of rotor blades CS2 signal The conventional approach to separate cyclostationary sources from noisy measurements is the cyclic Wiener filter [29,30]. It can be shown that a CS2 signal pðnÞ (the discretized version of pðtÞ) implies that

X a : X ak k EfzðxÞzH ðx  2pak Þg¼ Sz;z ðxÞdðx  2pak Þdxda ¼ Sp;p ðxÞdðx  2pak Þdxda – 0: ak 2A

ð17Þ

ak 2A

k It is noted that the spectral correlation density Sap;p ðxÞ (resp. Saz;zk ðxÞ) is the Fourier transform of the cyclic autocorrelation k ðsÞ (resp. Raz;zk ðsÞ), where ak is the cyclic frequency [17]. It implies that there exist correlations between different function Rap;p ^ðxÞ from the measurefrequency bins spaced apart by ak in an CS2 signal. The spectral redundancy can be used to estimate p ments zðx  2pak Þ by designing an optimal filter and shifting the signal zðxÞ at each cyclic frequency ak in the frequency domain. The filtration of the measurements at shifted cyclic frequencies is destructive for the noise and constructive for the desired sources due to the averaging effect. Thus, the estimation is represented as a regression,

^ m ð xÞ ¼ p

M X K X Wm j;k ðxÞzj ðx  2pak Þ; m ¼ 1; . . . ; M; k ¼ 1; . . . ; K; j¼1 k¼1

Fig. 2. Measured noise is mainly consisted of the rotor blades noise and flow-induced shaft noise.

ð18Þ

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^m ðxÞ is the CS2 signal on the m-th position of the array and W m where p j;k ðxÞ is the designed filter relating the m-th microphone at frequency x with the j-th microphone at frequency x  2pak . The measurements are first duplicated and shifted in the frequency domain by amounts corresponding to the ak , and second multiplied with the filter coefficients to obtain the CS2 signal. The frequency counterpart with reduced rank (with the assumption of limited number of cyclostationary sources) of Eq. (18) is formulated as follows. It is assumed that e 2 CMK is the extended observation vector of measurements p 2 CM for a given frequency, denoted as T

e ¼ ðzðx  2pa1 ÞT ; . . . ; zðx  2paK ÞT Þ , where a1 ; . . . ; aK are the cyclic frequencies of i-th cyclostationary source and K the number of cyclic frequencies. The frequency shift are implemented by zðnÞe2pjak n in time domain. Then Eq. (18) for the reduced-rank cyclic regression can be reformulated in a vector form. It aims at finding a filter W 2 CMMK such that the cyclo^ can be estimated as p ^ ¼ We, with stationary signal p 1

^  WeÞk2 g; W ¼ Argmin EfkSzz2 ðp

ð19Þ

under the constraint that rankfWg ¼ 1. It is important to note that the rank-one constraint is imposed to force the accuracy of only one cyclostationary source, this number should be changed correspondingly when more cyclostationary sources are observed. The solution of this optimization problem gives [31]: 1=2 H W ¼ S1=2 Sze S1 zz Utrun Utrun Szz zz ;

ð20Þ

where Utrun are the eigenvectors associated with the first eigenvalue of 1=2 1=2 Szz Sze S1 : ee Sez Szz

ð21Þ

The spectral matrices Szz 2 CMM ; See 2 CMKMK and Sze 2 CMMK are respectively defined as Szz ¼ EfzzH g; See ¼ EfeeH g, and Sze ¼ EfzeH g, which can be estimated by Welch’s method [32]. ^ is extracted, then it can be applied for the moving sources imaging. Assume the source Once the cyclostationary signal p plane is equally discretized into N grids at the known positions rn ; n ¼ 1; . . . ; N, and S incoherent broadband sound sources with coordinates rs ; s ¼ 1; . . . ; S are distributed on these grids (the position set frs g belongs to the set frn g). The radiated sound is measured by M microphones (N > M > S) with coordinates rm ; m ¼ 1; . . . ; M. Assumed that pm ðtÞ the cyclostationary ^ (the Inverse Fourier transform of the sound pressure of m-th microphone that is the Inverse Fourier transform of extracted p ^ ðxÞ with all the frequencies), then the de-dopplerized signal with respect to the n-th scanning point is m-th row of p

qnm ðt s Þ ¼ T 1 nm ðt; t s Þpm ðtÞ; where

T 1 nm ðt;

ð22Þ

sÞ is the inverse of the transfer function T nm defined as the following equation,

T nm ðt; ts Þ ¼

1 ; ~ ~ ~ 4pjx  xs ðt s Þ  Uðt  t s Þjj1  M so ðts Þj

~ ~ ð~ x~ xs ðt s ÞUðtt s ÞÞ ~ j~ x~ xs ðt s ÞUðtt s Þj

and M so ðts Þ ¼ ~v c U

ð23Þ

is the dimensionless speed in Mach number, which is the component of the source velocity ~ v

(in Mach) in the source-to-observer direction. The cross-spectral matrix with respect to the n-th scanning point is

^n ðf Þ ¼ 1 S L

L X ½Q nml ðf ÞQ nml ðf Þ;

ð24Þ

l¼1

where Q nml ðf Þ is the de-dopplerized signal qnm ðts Þ in the f-th frequency bin of l-th block. The idea behind each beamformer is to steer the microphone array to search for the source position, and the beamformer power output Bðrn ¼ rs Þ at position rn of measurement qnm ðt s Þ is defined as [10]

(

^n wn Bðrn ¼ rs Þ ¼ wHn S wn ¼ ð1; 1 . . . ; 1Þ; n ¼ 1; . . . ; N

ð25Þ

The weights wn 2 RM is a simple unit column vector since the de-dopplerization procedure has already considered both the

^n for the distance factor and the frequency shift in the microphone signals. It should be noted that only the spectral matrix S n-th scanning point is used in the moving beamforming, which is different with the conventional beamforming that the spectral matrix of all measurements is used. 5. Wind tunnel test experiment The experiment was conducted in a 5.5 m  4 m (metre) acoustic wind tunnel at the China Aerodynamic Research and Development Center as shown in Fig. 3(a). The acoustic wind tunnel is a single-return tunnel of low velocity and low turbulence level with both the open test section and closed test section. The dimension of the test section is 14 m (length)  5.5 m (width)  14 m (height), and the cross-section is rectangular. The wind speed ranges from 8 m/s to 100 m/s for open

L. Yu et al. / Mechanical Systems and Signal Processing 116 (2019) 1017–1028

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Fig. 3. (a) Equipment and setups for rotor blades tests in a wind tunnel; (b) the coordinates of the microphone array.

test section and from 8 m/s to 130 m/s for closed test section. The open test section is mainly used for acoustic tests with a background noise level of 75.6 dB (A) and the turbulence level in the central region of the open test section is e 6 0:2%. An anechoic chamber with a clearance size of 27 m (length)  26 m (width)  18 m (height) is constructed surround the open test section. The cutoff frequency of the anechoic chamber is 100 Hz. The reverberation time satisfy T 60 6 0:1 s (second) and the noise attenuation index of a wall is greater and equal than 50 dB (A). A dual-blade test piece with the diameter of 2 m is used in the test as shown in Fig. 4(a). The blade is with long axis of 0.03 m and short axis of 0.01 m. The cross section of the blade is oval and the blade is not twisting. The setup of the experiment is shown in Fig. 4(b). A microphone array with 139 microphones (G.R.A.S-40PH) is used and its coordinates are shown as in Fig. 3(b), and the microphones are connected with a PXI (PCI extensions for Instrumentation) dynamic data acquisition system of National Instruments (NI). The microphone array is positioned outside the flow and obliquely below the plane of the rotor blades in order to reduce the interference in the transmission path. The center coordinates of the array is ð2:6; 0:1; 5Þ m. The rotor blades is horizontally placed in the flow with the speed of 70 m/s and its rotation speed is 1200 rpm. The sampling frequency of microphones array measurement is 51.2 kHz, the duration of the measurement is 30 s. The time domain signals with different lengths of one microphone at ð3:16; 1:26; 5:02Þ m are shown in Fig. 5(a). The upper figure of Fig. 5(a) shows the signal within the record time of 1 s and the lower figure shows the record time of 0.1 s. The signals that are shown in different scales indicates the strong amplitude modulation phenomenon, and the amplitude modulation signals are corrupted with strong fluctuations due to the flow-induced shaft and stand noise. The power spectral

Fig. 4. (a) A two-blade rotor for wind tunnel tests and (b) a schematic diagram of rotor blades tests in a wind tunnel.

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Fig. 5. (a) Time domain pressure signal of one channel (at position ð3:16; 1:26; 5:02Þ m) of within 1 s (upper) and within 0.1 s (lower), and (b) the power spectral density of measured signal of one channel (the frequency resolution Df ¼ 20 Hz).

Fig. 6. Spectral correlation density of measured signal of one channel (at position ð3:16; 1:26; 5:02Þ m), it is a one dimensional representation by summing the spectral frequency with given cyclic frequency.

Fig. 7. Time domain measured pressure signal of one channel (at position ð3:16; 1:26; 5:02Þ m) of (a) within 1 s and (c) within 0.1 s; the corresponding extracted rotor blades signal of (b) within 1 s and (d) within 0.1 s.

L. Yu et al. / Mechanical Systems and Signal Processing 116 (2019) 1017–1028

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Fig. 8. Power spectral density of the measured signal, extracted cyclostationary signal and residual signal of (a) the first channel (at position ð3:16; 1:26; 5:02Þ m) , (b) the second channel (at position ð3:281:26  5:02Þ m). Power spectral density of the measured signal is denoted by blue circle, the extracted cyclostationary signal is denoted by red cross, the residual is denoted by black asterisk. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Imaging results of (a) direct beamforming without signal extraction at 1600 Hz, (b) beamforming after the rotor blades signals extraction at 1600 Hz, (c) direct beamforming without signal extraction at 2500 Hz, (d) beamforming after the rotor blades signals extraction at 2500 Hz.

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density (PSD) of the corresponding measured signal (one channel) is shown in Fig. 5(b) (the window length is 2560 samples, the overlap is 75% of the window length, the power spectral density is estimated with the Welch’s method [32]). It is seen that the measured signal have its power continuously distributed in the frequency domain, thus indicating that it is essentially random in nature (little information is conveyed by the frequency analysis). Thus the analysis of measured signal is resorted to the spectral correlation density [28], which is a waterfall of envelope spectra for all carrier frequencies of the signal. The spectral correlation density of the measured signal is shown in Fig. 6, and the cyclic frequency is revealed by summing the spectral frequency with given cyclic frequency, which results in a one dimensional representation of spectral correlation density. The cyclic frequency corresponding to the blade pass frequency of 40 Hz is revealed, and the harmonics of shaft frequency of the 20 Hz are also shown in the figure. The reduced rank cyclic Wiener filter is applied: the fundamental cyclic frequency is determined as 40 Hz by previous spectral correlation density analysis; the orders of cyclic frequency to use is 6 (the top 6 peaks of largest amplitude), and the number of cyclostationary sources is chosen to be 4 (this can be identified by the main singular value number of the spectral matrix). The exacted rotor blades signal is shown in Fig. 7. Fig. 7(a)(c) are the measured signals, and the extracted rotor blades signals are separately shown in Fig. 7(b)(d). It is evident that the fluctuation due to the rotor shaft disappeared and an amplitude modulation signals with a clean period are shown (the CS2 signals are extracted). It should be noted that the flow-induced noise from the stand and the rotor shaft are usually non-stationary, this is why the conventional winner filter can not be applied directly in the current setting of this paper. The PSD of the measured signals, which is composed of the extracted cyclostationary signals of rotor blades and the residual signals of the rotor shaft and stand (the results of the first two channels at positions ð3:16; 1:26; 5:02Þ m and ð3:281:26  5:02Þ m are given here) are shown in Fig. 8 (the window length is 256 samples, the overlap is 75% of the

Fig. 10. Imaging results of (a) direct beamforming without signal extraction at 3200 Hz, (b) beamforming after the rotor blades signals extraction at 3200 Hz, (c) direct beamforming without signal extraction at 4000 Hz, (d) beamforming after the rotor blades signals extraction at 4000 Hz.

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Fig. 11. Imaging results of (a) direct beamforming without signal extraction at 5000 Hz, (b) beamforming after the rotor blades signals extraction at 5000 Hz, (c) direct beamforming without signal extraction at 6300 Hz, (d) beamforming after the rotor blades signals extraction at 6300 Hz.

window length). It is generally known that the cyclostationary signals (non-stationary signals) contributed most of the peaks in the total measurement (the stationary noise contribute mainly the flat spectra), therefore, the extracted cyclostationary signals of rotor blades have recovered the most of peaks of total measurement. Fig. 9(a)(c), Fig. 10(a)(c) and Fig. 11(a)(c) respectively show the imaging results of direct moving beamforming without signal extraction at 1600 Hz, 2500 Hz, 3200 Hz, 4000 Hz, 5000 Hz and 6300 Hz (1/3 octave). It is evident that the rotor blades sources are corrupted by the rotor shaft noises (especially in the middle positions of two blades). Fig. 9(b)(d), Fig. 10(b)(d) and Fig. 11(b)(d) respectively show the imaging results of beamforming after the rotor blades signals extraction at 1600 Hz, 2500 Hz, 3200 Hz, 4000 Hz, 5000 Hz and 6300 Hz (1/3 octave). It is shown that the image of beamforming is purified and the rotor shaft sources (usually appear in the middle of two blades) are eliminated.

6. Conclusion The cyclostationary signal processing has been widely applied in the machine fault diagnosis, identification of machine system and separation of mechanical sources. However, few researchers have addressed the problem of aerodynamic noise generated by the rotor blades with the cyclostationary signal processing tool. One fundamental problem is the lacking of a formal definition of cyclostationary noise based on the FW-H equation which has been generally accepted in aeroacoustic. This also become the main theoretical gap to further apply the cyclostationary signal processing tools in the wind tunnel test of rotor blades. In this paper, the cyclostationary signal is defined firstly from the Wold-Cramer decomposition of nonstationary signals by imposing the periodic Green’s function of FW-H equation. Then the moving sources forward model is derived in the form of Wold-Cramer decomposition, which connects the signal professing definition of cyclostationary

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signal with the propagation model of FW-H equation. Specifically, for the acoustic imaging problem of rotor blades in the wind tunnel test, the rotor blades signals are corrupted by the rotor shaft noise which is clearly shown in the sound image of moving beamforming. By the work of current paper, the corresponding reduced-rank cyclic Wiener filter can be applied to extract the rotor blades signals from the total measurements in the wind tunnel test. An experiment is constructed to validate the proposed concept, the rotor blades signals can be extracted from the total measurements with the interferences of flow-induced rotor shaft and stand noise, and the beamforming result of the extracted rotor blades signals is further purified. Acknowledgements We would like to thank the continuous discussion with Mr. Jianzheng Gao, Mr. Dr. Pinxi Mo, Mr. Shichuan Huang (Shanghai Jiao Tong University) and Mr. Iurii Storozhenko (University of Calgary). The authors appreciate the Dr. Zhengwu Chen (China Aerodynamic Research and Development Center) for providing the experimental data. This work was supported by the National Natural Science Foundation of China (Grant No. 11574212, 11704248). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

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