Blind Deconvolution De-Noising for Helicopter ... - Semantic Scholar
Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007
WeC13.6
Blind Deconvolution De-noising for Helicopter Vibration Data Bin Zhang, Taimoor Khawaja, Romano Patrick and George Vachtsevanos
Abstract— Critical aircraft assets are required to be available when needed, while exhibiting attributes of reliability, robustness and high confidence under a variety of flight regimes, and maintained on the basis of their current condition rather than on the basis of scheduled maintenance practices. New and innovative technologies must be developed and implemented to address these concerns. Condition Based Maintenance (CBM) requires that the health of critical components/systems be monitored and diagnositics/prognostic strategies be developed to detect and identify incipient failures and predict the failing component’s remaining useful life (RUL). Typically, vibration and other key indicators on-board on aircraft are severely corrupted by noise thus curtailing our ability to accurately diagnose and predict failures. This paper introduces a novel blind deconvolution de-noising scheme that employs vibration model in the frequency domain and attempts to arrive at the true vibration signal through an iterative optimization process. Performance indexes are defined and data from a helicopter are used to demonstrate the effectiveness of the proposed scheme.
I. INTRODUCTION Epicyclic, or planetary, gear trains are widely used in the main transmission of many systems, such as the helicopter and other aircraft. This kind of gear system consists of an inner sun gear, two or more rotating planet gears, a stationary outer ring gear and a planetary carrier. The planet gears, which surround the sun gear, are riding on the planetary carrier and also rotating inside the outer ring gear. In the operation, torque is transmitted through the sun gear to the planets and planetary carrier. The carrier, in turn, transmits torque to the main rotor shaft and blades [1]–[5]. The gearbox is a critical component which directly impacts the safety and performance of the aircraft. The UH-60A Blackhawk main transmission employs a five-planet-gear epicyclic gear system [2]. Recently, a crack in the planetary carrier was found as shown in Figure 1. This resulted in flight restrictions on many helicopters. Manual inspection of all transmissions is not only costly, but also time prohibitive. A CBM based on vibration signal analysis is a cost-effective solution [12]. To measure the vibration signals, the transducer is mounted at a fixed point on the frame of the gearbox. If a fault occurs on the gear system, it is expected that the vibration signal will exhibit a characteristic signature which reveals the severity and location of the fault. In fault diagnosis and prognosis, gear health is typically described in terms of feature or condition indicator (CI) values, which are extracted from the vibration signals [2]. The success of diagnosis and prediction of the remaining The authors are with the Intelligent Control Systems Laboratory, School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA. [email protected]
1-4244-0989-6/07/$25.00 ©2007 IEEE.
Fig. 1. The crack of planetary gear carrier plate of the UH-60A helicopter
useful life of a gear system highly depend on the quality of these features. However, during the course of a flight, the vibration signals derived from accelerometers often have multiple excitation sources and, in most cases, information about the noise characteristics is not available. The measured output signal is, therefore, the combined effect from these different sources. Noise signals will degrade the quality of the features and, consequently, the performance of diagnosis and prognosis, such as the detection threshold, the accuracy of remaining useful life of components, etc. For example, when a fault is at its early stage, the indication in the vibration signals and the features is often masked by noises. It is important, therefore, to develop an efficient and reliable de-noising algorithm to remove noise and make the fault characteristics perceptible in the extracted features. The widely used de-noising technique in planetary gears is Time Synchronous Averaging (TSA) [2]–[5], [12]–[14]. However, in the frequency domain, it is possible that some noise components are enhanced by TSA processing. Blind source separation (BSS) algorithms are alternative de-noising techniques, which are successfully used in speech processing. BSS techniques are proposed to extract individual but physically different excitation sources from the combined output measurement [8], [9], [11]. In many mechanical systems, due to the noisy nature, the complex environment and the large number of noise sources, the application of BSS algorithms is severely hindered [10]. A practical solution is to focus on the main vibration source that contributes mostly to the vibration while it treats all other sources as a combined noise. Then, the objective is reduced to separating the vibration source from noise, which, in this sense, is the cumulative contribution from many different sources. The vibration signal collected from the transducer on the frame of the gearbox is amplitude modulated and corrupted by additive noise. Blind deconvolution has shown
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Fig. 2.
Blind deconvolution de-noising scheme
very promising results for similarly formulated problems in image and speech processing [6], [7] but it is rarely applied to vibration signals. In this paper, a blind deconvolution scheme, which consists of vibration modeling, nonlinear projection, and cost function optimization, is employed to the de-noising of the vibration signals from a UH-60A helicopter gearbox, whose planetary carrier has a seeded crack fault growing with the evolving operation of the system. The analysis of the system, the theoretical basis of the blind deconvolution algorithm, and the proposed performance indexes are addressed in detail. Experimental results show the effectiveness of the proposed de-noising scheme.
Fig. 3.
The configuration of epicyclic gear system
II. D E -N OISING S CHEME A RCHITECTURE The architecture of the proposed blind deconvolution denoising scheme is shown in Figure 2. In this scheme, nonlinear projection, which is based on vibration modeling in the frequency domain, and cost function minimization are critical components, which will be described in later sections. The proposed de-noising approach transforms the measured vibration signal s(t) to S(f ). With an inverse filter ¯ ) being convoluted with the composite signal S(f ), an Z(f ¯ ) is obtained. Then, B(f ¯ ) estimated vibration signal B(f ¯ passes through a nonlinear projection, which maps B(f ) to a subspace which contains only known characteristics ¯ ) of the vibration signal to yield Bnl (f ). By adjusting Z(f ¯ iteratively to minimize the difference between B(f ) and Bnl (f ) and when the difference between them reaches a ¯ ) → B(f ) can be regarded minimal value, the signal B(f ¯ ) as the de-noised vibration signal. At the same time, Z(f converges to Z(f ). III. SYSTEM DESCRIPTION The vibration signals are derived from the main transmission gearbox of the UH-60A Blackhawk Helicopter. The gearbox is an epicyclic gear system with five planet gears, whose configuration is shown in Figure 3. The following analysis assumes an ideal system. The transducer is mounted at a fixed point on the annulus gear at the position θ = 0, the tooth meshing vibration signal is amplitude modulated as the planetary carrier rotates. It
is natural to assume that all mesh vibrations from different planetary gears are of the same amplitude but of different phases in the ideal case. If there is only one planetary gear, then the vibration observed by the transducer should have the largest amplitude at θ = 0, 2π, · · ·. Similarly, the vibration should have the smallest amplitude at θ = π, 3π, · · ·. Suppose Np = 5 planetary gears are evenly spaced and the planetary carrier has a rotation frequency of f s . In this case, the planetary gear p has a phase of: ¶ µ p−1 (1) θp = 2π f s t + Np For planetary gear p, the amplitude modulating signal can be written in the time domain as: ap (t) =
N X
αn cos(nθp )
(2)
n=−N
where N is the number of sidebands about the harmonics under consideration and αn is the amplitude of component of modulating signal at frequency nf s . The annulus gear has Nt = 228 teeth. Suppose that the meshing vibration signal of a single planet gear and its harmonics has amplitude βm , the vibration signal generated from this planet gear can be written in form of:
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bp (t) =
M X
m=1
βm sin(mNt θp )
(3)
WeC13.6 rotation frequency f s ) that are multiples of the number of planetary gears. 2) The phase of each sideband depends on the geometrical angular position of the planet and is given by θp (mNt + n) for sideband (m, n). 3) The amplitude of the modulating signal a(t) decreases monotonically on either side of the maximum until it reaches the minimum. Note that this assumption is somewhat restrictive.
where M is the number of harmonics under consideration. The vibration signal observed by the transducer is given as the product of the meshing vibration signal and the amplitude modulating signal. Then, the observed vibration signal of the planetary gear p is ap (t)bp (t) and is given as: yp (t) =
M N 1 X X αn βm sin((mNt + n)θp ) 2 m=1
(4)
n=−N
Since the planetary gears are evenly spaced, the phase angel of the sidebands will be evenly spaced along 2π [1]. For sideband mNt + n, if it is not a multiple of Np , when the vibrations generated from different planetary gears are combined, these sidebands add destructively and become very small. On the other hand, if mNt + n is a multiple of Np , the sidebands add constructively and are reinforced. This process finally generates asymmetrical sidebands. A partial frequency spectra of the sidebands about the first order harmonic that illustrates this asymmetry is shown in Figure 4, where Np = 5 and Nt = 228. Note that the peak of the spectrum does not appear at n = 0 (order 228) but at n = −3 (order 225), n = 2 (order 230), etc. The largest spectral amplitude appears at the frequency closest to the gear meshing frequency.
IV. BLIND DECONVOLUTION DE-NOISING From the vibration analysis of the gearbox, we know that the vibration signals collected from the transducer are amplitude modulated [1], [3]. Multiple sources of noise may further corrupt the signal. A simplified model for such a complex signal may be defined as : s(t) = a(t)b(t) + n(t)
where s(t) is the measured vibration signal, b(t) is the noisefree un-modulated vibration signal, a(t) is the modulating signal and n(t) is the cumulative additive noise. Note that the modulating signal a(t) is itself affected by noise in the system. Let a ˆ(t) denotes the ideal modulating signal and na (t) denotes the noise introduced in this signal. Consequently, a(t) can be represented as:
0.4
a(t) = a ˆ(t) + na (t)
Amplitude
0.3
0.2
s(t)= (ˆ a(t) + na (t))b(t) + n(t) =a ˆ(t)b(t) + n ˆ (t)
0
220
225
230
235
240
Order of fs
Fig. 4.
Vibration spectrum of combined signal with Nt =228 and Np =5
The vibration signal observed by the transducer, with the noise component excluded, is the superposition of vibrations from each individual gear and has the form of: y(t) =
(9)
Thus, Equation (8) can be rewritten as follows:
0.1
−0.1 215
(8)
Np M N 1XX X αn βm sin((mNt + n)θp ) 2 p=1 m=1
(5)
where n ˆ (t) = na (t)b(t) + n(t) contains the total additive noise in the system. On the other hand, the factor a ˆ(t) describes the multiplicative noise in the system. The goal for a de-noising scheme, such as the one described here, is to recover the unknown vibration signal b(t) from the observed signal s(t) given partial information about the noise sources and characteristics of the vibration signal. A typical approach would be to find the inverse of a ˆ(t),
n=−N
According to earlier research findings [1]–[3], only terms at frequencies that are multiples of the number of planetary gears survive. Then, the Fourier transform of the vibration data can be written as: Y (f ) = fnl (γm,n (mNt + n)f s )
(6)
where γm,n is the magnitude of the spectral amplitude at (mNt + n)f s and fnl is the nonlinear projection: ½ 1 if mNt + n is a multiple of Np fnl = (7) 0 otherwise From the above analysis of the system vibration behavior, the following assumptions are made and used in the blinddeconvolution scheme: 1) The sidebands of the vibration spectra exist only at frequencies (with a resolution of the planetary carrier
(10)
such that
zˆ(t) = 1/ˆ a(t)
(11)
b(t)= (s(t) − n ˆ (t)) · zˆ(t) = s(t)ˆ z (t) − n ˆ (t)ˆ z (t)
(12)
Note however that little can be assumed about n ˆ (t). We propose an iterative de-noising scheme that starts with z¯(t), an initial estimate of the inverse of modulating signal, which demodulates the observed signal s(t) to give a rough noisefree vibration signal ¯b(t) as below: ¯b(t) = s(t) · z¯(t)
(13)
If some knowledge about how a system is influenced by a modulating signal a ˆ(t) and a reasonable understanding of the true vibration signal for the system under consideration are available, then the ideal characteristics of the vibration signal can be obtained by projecting this estimated signal ¯b(t) into
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WeC13.6 a subspace with only the known ideal characteristics of the vibration signal to yield a refined signal bnl (t). Since this non-linear projection, as the subscript signifies, removes all uncharacterisitic components that exist in the rough estimate ¯b(t) according to our understanding of the vibration signal, it is necessary to stress the importance of a good understanding of the underlying process. An iterative scheme then refines these results by minimizing the error between the two signals ¯b(t) and bnl (t), i.e. min ke(t)k = min k¯b(t) − bnl (t)k
(14)
Much has been published about the spectral characteristics of vibration signals for rotating equipment [1]–[3]. For this reason, the measured noisy vibration will be investigated in the frequency domain. The convolution theorem states that the product of two signals in the time domain equals their convolution in the frequency domain. Thus, model (10) can be written in the frequency domain as: ˆ ) ∗ B(f ) + N ˆ (f ) S(f ) = A(f
(15)
ˆ ), B(f ) with ∗ being the convolution operator and S(f ), A(f ˆ and N (f ) are the Fourier transforms of s(t), a ˆ(t), b(t) and n ˆ (t), respectively. Then, the goal in the frequency domain is to recover B(f ). Writing Equation (13) in the frequency domain, we have ¯ ) = S(f ) ∗ Z(f ¯ ) B(f
frequencies is used to define critical frequencies. All these windows form the support Dsup . In Equation (19), assumptions 4) and 5) are used to arrive at the second term to avoid an all-zero inverse filter Z(f ), which leads to the trivial solution for error minimization. Moreover, an iterative optimization routine is required to implement this scheme. The iterative conjugate gradient method has been related to address the optimization problem. This method has faster convergence rate in general as compared to the steepest descent method [7]. V. EXPERIMENTAL RESULTS The planetary carrier has a seeded fault with an initial length of 1.344 inches. The crack grows with the evolving operation of the gearbox. The gearbox is installed on a testbed and operates Ground-Air-Ground (GAG) cycle at different torque levels. This way, the vibration data under the operational condition of different torque levels and different crack length can be acquired. The load profile in the first 320 GAG cycles is shown in Figure 5(a). From the 321st GAG cycle on, the torque profile is shown in Figure 5(b). In this experiment, the vibration data at 100% torque in the first 320 GAG cycles and that at 93% torque in the later GAG cycles are considered together. Therefore, torque levels at 20%, 40%, 100% are investigated.
(16)
¯ ) and Z(f ¯ ) being the Fourier transforms of ¯b(t) with B(f ¯ ) through the nonlinear and z¯(t), respectively. Passing B(f projection, it yields Bnl (f ). Then, in the frequency domain, ¯ ): we will minimize the difference between Bnl (f ) and B(f ¯ ) − Bnl (f )k min kE(f )k = min kB(f
(17)
¯ ) to minimize Equation The iterative process refines Z(f ¯ ) converges to (17). When it reaches the minimal value, Z(f ¯ Z(f ). Then, with this Z(f ) replacing Z(f ) in Equation (16), a good estimate of B(f ) is obtained as: B(f ) = S(f ) ∗ Z(f )
(a) GAG cycle 1-320
(18)
Lastly, the estimate is transformed back into the time domain to recover the noise-free vibration signal b(t). To solve this problem, the following assumptions are made (continued from the previous section): 4) P Z(f ) exists and is absolutely summable, i.e. |Z(f )| < ∞. 5) Since the modulating signal a ˆ(t) is always positive, its inverse should be positive too. Hence, the Fourier transform of its inverse Z(f ) contains a dc component. With the above two assumptions being taken into consideration, the cost function is defined as: X X J= [B(f ) − Bnl (f )]2 + ( Z(f ) − 1)2 (19) f ∈Dsup
where Dsup is the frequency range that contains the main vibration information. Because of the periodic fade of signal spectra between harmonics, a window centered at harmonic
(b) GAG cycle 321-1000 Fig. 5.
The load profile of the gearbox
The signal-to-noise ratio (SNR) before and after the denoising is investigated and the results at 40% and 100% torque levels are shown in Figures 6(a) and 6(b), respectively. The improvement of SNR is remarkable. Although the blind deconvolution routine shows the improvement of SNR, it is desirable that it can improve the
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WeC13.6 quality of the features or condition indicators. The accuracy and precision of mapping between the propagation of features and actual crack growth along the operation is closely related to the performance of diagnosis and prognosis, and the performance of CBM as well. To evaluate the quality of the features, two performance indexes are developed.
where n is the number of entities in feature vector x and x ˜ is the smoothed feature vector. Obviously, the value of correlation coefficient should be near 1 for a good feature. On the other hand, PMD for a good feature should have a very small value close to 0. TABLE I T HE GROUND TRUTH DATA OF CRACK LENGTH ( INCHES )
−0.5
SNR of de−noised data −1
GAG 1 36 100 230 400 550 650 714 750 988 Crack 1.34 2.00 2.50 3.02 3.54 4.07 4.52 6.15 6.78 7.63
Signal−to−Noise Ratio
−1.5 −2 −2.5 −3
The ground truth data of crack length at some GAG cycles are available and tabulated in Table I. With these data, the crack length for GAG cycles from 1 to 1000 can be obtained by interpolating, which results in a crack length growth curve shown in Figure 7.
−3.5 −4
SNR of TSA data
−4.5 −5
0
200
400
600
800
1000
GAG
(a) Signal-to-noise ratio at 40% torque 8
Crack Length (inches)
7
0
SNR of de−noised data
Signal−to−Noise Ratio
−1
6
5
4
3
−2 2
1
−3
0
200
400
600
800
1000
GAG
−4
Fig. 7.
The growth of crack length
−5
SNR of TSA data −6
0
200
400
600
800
1000
GAG
(b) Signal-to-noise ratio at 100% torque Fig. 6.
The signal-to-noise ratio
The first performance index is accuracy measure, which is defined as the linear correlation between the feature value curve and the crack length growth curve on the GAG cycle axis. Suppose x is the feature vector and y is the crack growth curve with x ¯ and y¯ being their means, respectively. The correlation coefficient between them is: s ss2xy (20) CCR(x, y) = ssxx ssyy P P where ss (xi − x ¯)(yi − y¯), ssxx = (xi − x ¯)2 and xy = P 2 ssyy = (yi − y¯) , respectively. Because of the changes of operating conditions and other disturbances, the feature values along GAG cycle are often very noisy. Hence, in applications, the feature values are often smoothed through a low passing filtering operation. In this case, the linear correlation is calculated based on the smoothed feature curve. The second performance index is precision measure, which is a normalized measure of the signal dispersion. It is termed as percent mean deviation (PMD) and is defined as: Pn xi −˜xi P M D(x, x ˜) =
i=1
n
x ˜i
× 100
(21)
The frequency spectra in Dsup are divided into two categories: Regular Meshing Components (RMC), which is defined as the dominant and other apparent frequency components that appear at frequencies that are multiple of Np , and Non Regular Meshing Components (NonRMC) that appear at frequencies that are not multiple of Np . The relative size of the NonRMC sidebands to RMC sidebands in Dsup or part of Dsup may indicate the fault of the planetary carrier. As the crack grows, the operating condition deviates from the ideal operating condition. In this case, the constructive superposition of vibration from different planetary gears is attenuated while the destructive superposition is reinforced. This phenomenon results in smaller RMC sidebands while larger NonRMC sidebands. This indicates that the ratio between RMC and NonRMC sidebands can be used as feature. Based on this idea, some features are successfully extracted in the frequency domain. For instance, in previous research, Sideband Ratio (SBR) and relSX extracted from the TSA data have shown very promising results in diagnosis and prognosis. For this reason, these two features from the TSA data and the blind deconvolution de-noised data will be used to demonstrate the effectiveness of the proposed de-noising routine. For each dominant and apparent frequency component, a group of frequency spectra is defined. Since the vibration signal is collected from a five-planetary-gear gearbox system, the group size is defined as 5 from previous system description. Then, Dsup is defined by the number of groups on each side of the dominant component.
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WeC13.6 SBR: Let Dsup is defined by m (the index of harmonics) and X (the number of groups on each side of the dominant components), SBR is defined as the ratio between the energy of RMC and NonRMC, which is formulated as: Pm PX g=−X N onRM C h=1 (22) SBR(m, X) = Pm PX g=−X N onRM C + RM C h=1 relSX: RelSX is defined as: PX
avg(N onRM C) g=−X N onRM C+RM C
relSX(m, X) =
2X + 1
(23)
3.5
feature from de−noised data
Feature value
3
smoothed de−noised feature
2.5
TABLE III T HE PERCCENT MEAN DEVIATION OF THE FEATURES Torque SBR relSX
The research reported in this paper was partially supported by DARPA and Northrop Grumman under the DARPA Prognosis program contract no. HR0011-04-C-003. We gratefully acknowledge the support and assistance received from our sponsors.
smoothed TSA feature features from TSA data 0
200
400
600
800
1000
GAG
(a) SBR(6,2) at 40% torque
R EFERENCES
0.21
Feature from de−noised data 0.2
Smoothed de−noised feature
Feature value
0.19 0.18 0.17 0.16 0.15
Feature from TSA data
0.14
Smoothed TSA Feature
0.13
0
200
400
600
800
1000
GAG
Fig. 8.
(b) relSX(6,3) at 100% torque Feature values vs. GAG before and after de-noising
The results of SBR(6,2) at 40% torque and relSX(6,3) at 100% torque are shown in Figures 8(a) and 8(b), respectively. The correlation coefficient and PMD for the two features at different torque levels are summarized in Tables II and III, respectively. The improvements of the feature qualities are remarkable. In the tables, D-N simply means de-noised. TABLE II T HE CORRELATION COEFFICIENTS OF THE FEATURES Torque SBR relSX
100% TSA D-N 5.65% 2.90% 3.05% 0.80%
Acknowledgement
1
0.12
40% TSA D-N 4.85% 3.31% 1.19% 0.93%
the gearbox of a helicopter testbed. The de-noising scheme uses a recursive filtering technique to minimize a given cost function to improve the SNR of the vibration signals. Then, the qualities of the features extracted from the de-noised data and those from TSA data are compared in terms of accuracy and precision indexes. Experimental results about SNR and performance indexes demonstrate the efficiency of the blind deconvolution de-noising routine.
2
1.5
0.5
20% TSA D-N 1.12% 1.84% 1.05% 0.96%
20% TSA D-N 0.951 0.985 0.981 0.992
40% TSA D-N 0.987 0.993 0.993 0.993
100% TSA D-N 0.972 0.998 0.966 0.979
VI. C ONCLUSIONS The quality of the features is closely related to the success of fault diagnosis and prognosis. A blind deconvolution denoising scheme is developed for the vibration signals from
[1] McFadden, P.D. and J.D. Smith, An Explanation for the Asymmetry of the Modulation Sidebands about the Tooth Meshing Frequency in Epicyclic Gear Vibration, Proc. Institution of Mechanical Engineers, Part C: Mechanical Engineering Science, 199(1), 1985, pp. 65-70. [2] J. Keller and P. Grabill, Vibration Monitoring of a UH-60A Main Transmission Planetary Carrier Fault, the American Helicopter Society 59th Annual Forum, Phoenix, Arizona, 2003. [3] Patrick Romano, Relating the modulation sidebands in the vibration spectrum of planetary gears to planet shifting or misalignment, Pending Review, 2005. [4] B. Wu, A. Saxena, T. Khawaja, R. Patrick, G. Vachtsevanos, and R. Sparis, An approach to fault diagnosis of helicopter planetary gears, IEEE AUTOTESTCON, San Antonio, Texas, Sept. 21, 2004. [5] G. R. Ayers and J. C. Dainty, Iterative blind deconvolution method and its applications, Optics Section, Blackett Laboratory, Imperial College, London SW7 2BZ, UK, 1998. [6] D. Kundur and D. Hatzinakos, Blind Image Deconvolution, IEEE Signal Processing Magazine, vol. 13, no. 3, pp. 43-64, May 1996. [7] R. Prost and R. Goutte, Discrete constrained iterative deconvolution with optimized rate of convergence, Signal Process., vol. 7, pp. 209230, Dec. 1984. [8] J. Antoni, Blind separation of vibration components: Principles and demonstrations, Mechanical Systems and Signal Processing, vol. 19, pp.1166-1180, 2005. [9] A. K. Nandi, D. Mampel, and B. Roscher, Blind deconvolution of ultrasonic signals in nondestructive testing applications, IEEE. Trans. Signal Processing, vol. 45, no. 5, May, 1997. [10] R. Peled, S. Braun, M. Zacksenhouse, A blind deconvolution separation of multiple sources, with application to bearing diagnostics, Mechanical Systems and Signal Process., vol.19, pp. 1181-1195, 2005. [11] G. Gelle, M. Colas, and G.Delaunay, Blind sources separation applied to rotating machines monitoring by acoustical and vibrations analysis, Mechanical Systems and signal Processing, 14(3) (2000), 427-442. [12] A. Saxena, B. Wu, G. Vachtsevanos, A methodology for analyzing vibration data from planetary gear system using complex Morelt wavelets, American Control Conference 2005, vol. 7, pp. 4730-4735. [13] M. Lebold M. Lebold, K. McClintic, R. Campbell, C. Byington, and K. Maynard, Review of vibration analysis methods for gearbox diagnostics and prognostics, Proc. 5th Meeting of the Society for Machineary Failure Prevention Technology, VA, 2000, pp. 623-634 [14] B. Wu, A. Saxena, R. Romano, G. Vachtesevanos, Vibration monitoring for fault diagnosis of helicopter planetary gears, 16th IFAC.
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Blind Deconvolution De-noising for Helicopter Vibration Data Bin Zhang, Taimoor Khawaja, Romano Patrick and George Vachtsevanos
Abstract— Critical aircraft assets are required to be available when needed, while exhibiting attributes of reliability, robustness and high confidence under a variety of flight regimes, and maintained on the basis of their current condition rather than on the basis of scheduled maintenance practices. New and innovative technologies must be developed and implemented to address these concerns. Condition Based Maintenance (CBM) requires that the health of critical components/systems be monitored and diagnositics/prognostic strategies be developed to detect and identify incipient failures and predict the failing component’s remaining useful life (RUL). Typically, vibration and other key indicators on-board on aircraft are severely corrupted by noise thus curtailing our ability to accurately diagnose and predict failures. This paper introduces a novel blind deconvolution de-noising scheme that employs vibration model in the frequency domain and attempts to arrive at the true vibration signal through an iterative optimization process. Performance indexes are defined and data from a helicopter are used to demonstrate the effectiveness of the proposed scheme.
I. INTRODUCTION Epicyclic, or planetary, gear trains are widely used in the main transmission of many systems, such as the helicopter and other aircraft. This kind of gear system consists of an inner sun gear, two or more rotating planet gears, a stationary outer ring gear and a planetary carrier. The planet gears, which surround the sun gear, are riding on the planetary carrier and also rotating inside the outer ring gear. In the operation, torque is transmitted through the sun gear to the planets and planetary carrier. The carrier, in turn, transmits torque to the main rotor shaft and blades [1]–[5]. The gearbox is a critical component which directly impacts the safety and performance of the aircraft. The UH-60A Blackhawk main transmission employs a five-planet-gear epicyclic gear system [2]. Recently, a crack in the planetary carrier was found as shown in Figure 1. This resulted in flight restrictions on many helicopters. Manual inspection of all transmissions is not only costly, but also time prohibitive. A CBM based on vibration signal analysis is a cost-effective solution [12]. To measure the vibration signals, the transducer is mounted at a fixed point on the frame of the gearbox. If a fault occurs on the gear system, it is expected that the vibration signal will exhibit a characteristic signature which reveals the severity and location of the fault. In fault diagnosis and prognosis, gear health is typically described in terms of feature or condition indicator (CI) values, which are extracted from the vibration signals [2]. The success of diagnosis and prediction of the remaining The authors are with the Intelligent Control Systems Laboratory, School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA. [email protected]
1-4244-0989-6/07/$25.00 ©2007 IEEE.
Fig. 1. The crack of planetary gear carrier plate of the UH-60A helicopter
useful life of a gear system highly depend on the quality of these features. However, during the course of a flight, the vibration signals derived from accelerometers often have multiple excitation sources and, in most cases, information about the noise characteristics is not available. The measured output signal is, therefore, the combined effect from these different sources. Noise signals will degrade the quality of the features and, consequently, the performance of diagnosis and prognosis, such as the detection threshold, the accuracy of remaining useful life of components, etc. For example, when a fault is at its early stage, the indication in the vibration signals and the features is often masked by noises. It is important, therefore, to develop an efficient and reliable de-noising algorithm to remove noise and make the fault characteristics perceptible in the extracted features. The widely used de-noising technique in planetary gears is Time Synchronous Averaging (TSA) [2]–[5], [12]–[14]. However, in the frequency domain, it is possible that some noise components are enhanced by TSA processing. Blind source separation (BSS) algorithms are alternative de-noising techniques, which are successfully used in speech processing. BSS techniques are proposed to extract individual but physically different excitation sources from the combined output measurement [8], [9], [11]. In many mechanical systems, due to the noisy nature, the complex environment and the large number of noise sources, the application of BSS algorithms is severely hindered [10]. A practical solution is to focus on the main vibration source that contributes mostly to the vibration while it treats all other sources as a combined noise. Then, the objective is reduced to separating the vibration source from noise, which, in this sense, is the cumulative contribution from many different sources. The vibration signal collected from the transducer on the frame of the gearbox is amplitude modulated and corrupted by additive noise. Blind deconvolution has shown
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WeC13.6
Fig. 2.
Blind deconvolution de-noising scheme
very promising results for similarly formulated problems in image and speech processing [6], [7] but it is rarely applied to vibration signals. In this paper, a blind deconvolution scheme, which consists of vibration modeling, nonlinear projection, and cost function optimization, is employed to the de-noising of the vibration signals from a UH-60A helicopter gearbox, whose planetary carrier has a seeded crack fault growing with the evolving operation of the system. The analysis of the system, the theoretical basis of the blind deconvolution algorithm, and the proposed performance indexes are addressed in detail. Experimental results show the effectiveness of the proposed de-noising scheme.
Fig. 3.
The configuration of epicyclic gear system
II. D E -N OISING S CHEME A RCHITECTURE The architecture of the proposed blind deconvolution denoising scheme is shown in Figure 2. In this scheme, nonlinear projection, which is based on vibration modeling in the frequency domain, and cost function minimization are critical components, which will be described in later sections. The proposed de-noising approach transforms the measured vibration signal s(t) to S(f ). With an inverse filter ¯ ) being convoluted with the composite signal S(f ), an Z(f ¯ ) is obtained. Then, B(f ¯ ) estimated vibration signal B(f ¯ passes through a nonlinear projection, which maps B(f ) to a subspace which contains only known characteristics ¯ ) of the vibration signal to yield Bnl (f ). By adjusting Z(f ¯ iteratively to minimize the difference between B(f ) and Bnl (f ) and when the difference between them reaches a ¯ ) → B(f ) can be regarded minimal value, the signal B(f ¯ ) as the de-noised vibration signal. At the same time, Z(f converges to Z(f ). III. SYSTEM DESCRIPTION The vibration signals are derived from the main transmission gearbox of the UH-60A Blackhawk Helicopter. The gearbox is an epicyclic gear system with five planet gears, whose configuration is shown in Figure 3. The following analysis assumes an ideal system. The transducer is mounted at a fixed point on the annulus gear at the position θ = 0, the tooth meshing vibration signal is amplitude modulated as the planetary carrier rotates. It
is natural to assume that all mesh vibrations from different planetary gears are of the same amplitude but of different phases in the ideal case. If there is only one planetary gear, then the vibration observed by the transducer should have the largest amplitude at θ = 0, 2π, · · ·. Similarly, the vibration should have the smallest amplitude at θ = π, 3π, · · ·. Suppose Np = 5 planetary gears are evenly spaced and the planetary carrier has a rotation frequency of f s . In this case, the planetary gear p has a phase of: ¶ µ p−1 (1) θp = 2π f s t + Np For planetary gear p, the amplitude modulating signal can be written in the time domain as: ap (t) =
N X
αn cos(nθp )
(2)
n=−N
where N is the number of sidebands about the harmonics under consideration and αn is the amplitude of component of modulating signal at frequency nf s . The annulus gear has Nt = 228 teeth. Suppose that the meshing vibration signal of a single planet gear and its harmonics has amplitude βm , the vibration signal generated from this planet gear can be written in form of:
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bp (t) =
M X
m=1
βm sin(mNt θp )
(3)
WeC13.6 rotation frequency f s ) that are multiples of the number of planetary gears. 2) The phase of each sideband depends on the geometrical angular position of the planet and is given by θp (mNt + n) for sideband (m, n). 3) The amplitude of the modulating signal a(t) decreases monotonically on either side of the maximum until it reaches the minimum. Note that this assumption is somewhat restrictive.
where M is the number of harmonics under consideration. The vibration signal observed by the transducer is given as the product of the meshing vibration signal and the amplitude modulating signal. Then, the observed vibration signal of the planetary gear p is ap (t)bp (t) and is given as: yp (t) =
M N 1 X X αn βm sin((mNt + n)θp ) 2 m=1
(4)
n=−N
Since the planetary gears are evenly spaced, the phase angel of the sidebands will be evenly spaced along 2π [1]. For sideband mNt + n, if it is not a multiple of Np , when the vibrations generated from different planetary gears are combined, these sidebands add destructively and become very small. On the other hand, if mNt + n is a multiple of Np , the sidebands add constructively and are reinforced. This process finally generates asymmetrical sidebands. A partial frequency spectra of the sidebands about the first order harmonic that illustrates this asymmetry is shown in Figure 4, where Np = 5 and Nt = 228. Note that the peak of the spectrum does not appear at n = 0 (order 228) but at n = −3 (order 225), n = 2 (order 230), etc. The largest spectral amplitude appears at the frequency closest to the gear meshing frequency.
IV. BLIND DECONVOLUTION DE-NOISING From the vibration analysis of the gearbox, we know that the vibration signals collected from the transducer are amplitude modulated [1], [3]. Multiple sources of noise may further corrupt the signal. A simplified model for such a complex signal may be defined as : s(t) = a(t)b(t) + n(t)
where s(t) is the measured vibration signal, b(t) is the noisefree un-modulated vibration signal, a(t) is the modulating signal and n(t) is the cumulative additive noise. Note that the modulating signal a(t) is itself affected by noise in the system. Let a ˆ(t) denotes the ideal modulating signal and na (t) denotes the noise introduced in this signal. Consequently, a(t) can be represented as:
0.4
a(t) = a ˆ(t) + na (t)
Amplitude
0.3
0.2
s(t)= (ˆ a(t) + na (t))b(t) + n(t) =a ˆ(t)b(t) + n ˆ (t)
0
220
225
230
235
240
Order of fs
Fig. 4.
Vibration spectrum of combined signal with Nt =228 and Np =5
The vibration signal observed by the transducer, with the noise component excluded, is the superposition of vibrations from each individual gear and has the form of: y(t) =
(9)
Thus, Equation (8) can be rewritten as follows:
0.1
−0.1 215
(8)
Np M N 1XX X αn βm sin((mNt + n)θp ) 2 p=1 m=1
(5)
where n ˆ (t) = na (t)b(t) + n(t) contains the total additive noise in the system. On the other hand, the factor a ˆ(t) describes the multiplicative noise in the system. The goal for a de-noising scheme, such as the one described here, is to recover the unknown vibration signal b(t) from the observed signal s(t) given partial information about the noise sources and characteristics of the vibration signal. A typical approach would be to find the inverse of a ˆ(t),
n=−N
According to earlier research findings [1]–[3], only terms at frequencies that are multiples of the number of planetary gears survive. Then, the Fourier transform of the vibration data can be written as: Y (f ) = fnl (γm,n (mNt + n)f s )
(6)
where γm,n is the magnitude of the spectral amplitude at (mNt + n)f s and fnl is the nonlinear projection: ½ 1 if mNt + n is a multiple of Np fnl = (7) 0 otherwise From the above analysis of the system vibration behavior, the following assumptions are made and used in the blinddeconvolution scheme: 1) The sidebands of the vibration spectra exist only at frequencies (with a resolution of the planetary carrier
(10)
such that
zˆ(t) = 1/ˆ a(t)
(11)
b(t)= (s(t) − n ˆ (t)) · zˆ(t) = s(t)ˆ z (t) − n ˆ (t)ˆ z (t)
(12)
Note however that little can be assumed about n ˆ (t). We propose an iterative de-noising scheme that starts with z¯(t), an initial estimate of the inverse of modulating signal, which demodulates the observed signal s(t) to give a rough noisefree vibration signal ¯b(t) as below: ¯b(t) = s(t) · z¯(t)
(13)
If some knowledge about how a system is influenced by a modulating signal a ˆ(t) and a reasonable understanding of the true vibration signal for the system under consideration are available, then the ideal characteristics of the vibration signal can be obtained by projecting this estimated signal ¯b(t) into
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WeC13.6 a subspace with only the known ideal characteristics of the vibration signal to yield a refined signal bnl (t). Since this non-linear projection, as the subscript signifies, removes all uncharacterisitic components that exist in the rough estimate ¯b(t) according to our understanding of the vibration signal, it is necessary to stress the importance of a good understanding of the underlying process. An iterative scheme then refines these results by minimizing the error between the two signals ¯b(t) and bnl (t), i.e. min ke(t)k = min k¯b(t) − bnl (t)k
(14)
Much has been published about the spectral characteristics of vibration signals for rotating equipment [1]–[3]. For this reason, the measured noisy vibration will be investigated in the frequency domain. The convolution theorem states that the product of two signals in the time domain equals their convolution in the frequency domain. Thus, model (10) can be written in the frequency domain as: ˆ ) ∗ B(f ) + N ˆ (f ) S(f ) = A(f
(15)
ˆ ), B(f ) with ∗ being the convolution operator and S(f ), A(f ˆ and N (f ) are the Fourier transforms of s(t), a ˆ(t), b(t) and n ˆ (t), respectively. Then, the goal in the frequency domain is to recover B(f ). Writing Equation (13) in the frequency domain, we have ¯ ) = S(f ) ∗ Z(f ¯ ) B(f
frequencies is used to define critical frequencies. All these windows form the support Dsup . In Equation (19), assumptions 4) and 5) are used to arrive at the second term to avoid an all-zero inverse filter Z(f ), which leads to the trivial solution for error minimization. Moreover, an iterative optimization routine is required to implement this scheme. The iterative conjugate gradient method has been related to address the optimization problem. This method has faster convergence rate in general as compared to the steepest descent method [7]. V. EXPERIMENTAL RESULTS The planetary carrier has a seeded fault with an initial length of 1.344 inches. The crack grows with the evolving operation of the gearbox. The gearbox is installed on a testbed and operates Ground-Air-Ground (GAG) cycle at different torque levels. This way, the vibration data under the operational condition of different torque levels and different crack length can be acquired. The load profile in the first 320 GAG cycles is shown in Figure 5(a). From the 321st GAG cycle on, the torque profile is shown in Figure 5(b). In this experiment, the vibration data at 100% torque in the first 320 GAG cycles and that at 93% torque in the later GAG cycles are considered together. Therefore, torque levels at 20%, 40%, 100% are investigated.
(16)
¯ ) and Z(f ¯ ) being the Fourier transforms of ¯b(t) with B(f ¯ ) through the nonlinear and z¯(t), respectively. Passing B(f projection, it yields Bnl (f ). Then, in the frequency domain, ¯ ): we will minimize the difference between Bnl (f ) and B(f ¯ ) − Bnl (f )k min kE(f )k = min kB(f
(17)
¯ ) to minimize Equation The iterative process refines Z(f ¯ ) converges to (17). When it reaches the minimal value, Z(f ¯ Z(f ). Then, with this Z(f ) replacing Z(f ) in Equation (16), a good estimate of B(f ) is obtained as: B(f ) = S(f ) ∗ Z(f )
(a) GAG cycle 1-320
(18)
Lastly, the estimate is transformed back into the time domain to recover the noise-free vibration signal b(t). To solve this problem, the following assumptions are made (continued from the previous section): 4) P Z(f ) exists and is absolutely summable, i.e. |Z(f )| < ∞. 5) Since the modulating signal a ˆ(t) is always positive, its inverse should be positive too. Hence, the Fourier transform of its inverse Z(f ) contains a dc component. With the above two assumptions being taken into consideration, the cost function is defined as: X X J= [B(f ) − Bnl (f )]2 + ( Z(f ) − 1)2 (19) f ∈Dsup
where Dsup is the frequency range that contains the main vibration information. Because of the periodic fade of signal spectra between harmonics, a window centered at harmonic
(b) GAG cycle 321-1000 Fig. 5.
The load profile of the gearbox
The signal-to-noise ratio (SNR) before and after the denoising is investigated and the results at 40% and 100% torque levels are shown in Figures 6(a) and 6(b), respectively. The improvement of SNR is remarkable. Although the blind deconvolution routine shows the improvement of SNR, it is desirable that it can improve the
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WeC13.6 quality of the features or condition indicators. The accuracy and precision of mapping between the propagation of features and actual crack growth along the operation is closely related to the performance of diagnosis and prognosis, and the performance of CBM as well. To evaluate the quality of the features, two performance indexes are developed.
where n is the number of entities in feature vector x and x ˜ is the smoothed feature vector. Obviously, the value of correlation coefficient should be near 1 for a good feature. On the other hand, PMD for a good feature should have a very small value close to 0. TABLE I T HE GROUND TRUTH DATA OF CRACK LENGTH ( INCHES )
−0.5
SNR of de−noised data −1
GAG 1 36 100 230 400 550 650 714 750 988 Crack 1.34 2.00 2.50 3.02 3.54 4.07 4.52 6.15 6.78 7.63
Signal−to−Noise Ratio
−1.5 −2 −2.5 −3
The ground truth data of crack length at some GAG cycles are available and tabulated in Table I. With these data, the crack length for GAG cycles from 1 to 1000 can be obtained by interpolating, which results in a crack length growth curve shown in Figure 7.
−3.5 −4
SNR of TSA data
−4.5 −5
0
200
400
600
800
1000
GAG
(a) Signal-to-noise ratio at 40% torque 8
Crack Length (inches)
7
0
SNR of de−noised data
Signal−to−Noise Ratio
−1
6
5
4
3
−2 2
1
−3
0
200
400
600
800
1000
GAG
−4
Fig. 7.
The growth of crack length
−5
SNR of TSA data −6
0
200
400
600
800
1000
GAG
(b) Signal-to-noise ratio at 100% torque Fig. 6.
The signal-to-noise ratio
The first performance index is accuracy measure, which is defined as the linear correlation between the feature value curve and the crack length growth curve on the GAG cycle axis. Suppose x is the feature vector and y is the crack growth curve with x ¯ and y¯ being their means, respectively. The correlation coefficient between them is: s ss2xy (20) CCR(x, y) = ssxx ssyy P P where ss (xi − x ¯)(yi − y¯), ssxx = (xi − x ¯)2 and xy = P 2 ssyy = (yi − y¯) , respectively. Because of the changes of operating conditions and other disturbances, the feature values along GAG cycle are often very noisy. Hence, in applications, the feature values are often smoothed through a low passing filtering operation. In this case, the linear correlation is calculated based on the smoothed feature curve. The second performance index is precision measure, which is a normalized measure of the signal dispersion. It is termed as percent mean deviation (PMD) and is defined as: Pn xi −˜xi P M D(x, x ˜) =
i=1
n
x ˜i
× 100
(21)
The frequency spectra in Dsup are divided into two categories: Regular Meshing Components (RMC), which is defined as the dominant and other apparent frequency components that appear at frequencies that are multiple of Np , and Non Regular Meshing Components (NonRMC) that appear at frequencies that are not multiple of Np . The relative size of the NonRMC sidebands to RMC sidebands in Dsup or part of Dsup may indicate the fault of the planetary carrier. As the crack grows, the operating condition deviates from the ideal operating condition. In this case, the constructive superposition of vibration from different planetary gears is attenuated while the destructive superposition is reinforced. This phenomenon results in smaller RMC sidebands while larger NonRMC sidebands. This indicates that the ratio between RMC and NonRMC sidebands can be used as feature. Based on this idea, some features are successfully extracted in the frequency domain. For instance, in previous research, Sideband Ratio (SBR) and relSX extracted from the TSA data have shown very promising results in diagnosis and prognosis. For this reason, these two features from the TSA data and the blind deconvolution de-noised data will be used to demonstrate the effectiveness of the proposed de-noising routine. For each dominant and apparent frequency component, a group of frequency spectra is defined. Since the vibration signal is collected from a five-planetary-gear gearbox system, the group size is defined as 5 from previous system description. Then, Dsup is defined by the number of groups on each side of the dominant component.
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WeC13.6 SBR: Let Dsup is defined by m (the index of harmonics) and X (the number of groups on each side of the dominant components), SBR is defined as the ratio between the energy of RMC and NonRMC, which is formulated as: Pm PX g=−X N onRM C h=1 (22) SBR(m, X) = Pm PX g=−X N onRM C + RM C h=1 relSX: RelSX is defined as: PX
avg(N onRM C) g=−X N onRM C+RM C
relSX(m, X) =
2X + 1
(23)
3.5
feature from de−noised data
Feature value
3
smoothed de−noised feature
2.5
TABLE III T HE PERCCENT MEAN DEVIATION OF THE FEATURES Torque SBR relSX
The research reported in this paper was partially supported by DARPA and Northrop Grumman under the DARPA Prognosis program contract no. HR0011-04-C-003. We gratefully acknowledge the support and assistance received from our sponsors.
smoothed TSA feature features from TSA data 0
200
400
600
800
1000
GAG
(a) SBR(6,2) at 40% torque
R EFERENCES
0.21
Feature from de−noised data 0.2
Smoothed de−noised feature
Feature value
0.19 0.18 0.17 0.16 0.15
Feature from TSA data
0.14
Smoothed TSA Feature
0.13
0
200
400
600
800
1000
GAG
Fig. 8.
(b) relSX(6,3) at 100% torque Feature values vs. GAG before and after de-noising
The results of SBR(6,2) at 40% torque and relSX(6,3) at 100% torque are shown in Figures 8(a) and 8(b), respectively. The correlation coefficient and PMD for the two features at different torque levels are summarized in Tables II and III, respectively. The improvements of the feature qualities are remarkable. In the tables, D-N simply means de-noised. TABLE II T HE CORRELATION COEFFICIENTS OF THE FEATURES Torque SBR relSX
100% TSA D-N 5.65% 2.90% 3.05% 0.80%
Acknowledgement
1
0.12
40% TSA D-N 4.85% 3.31% 1.19% 0.93%
the gearbox of a helicopter testbed. The de-noising scheme uses a recursive filtering technique to minimize a given cost function to improve the SNR of the vibration signals. Then, the qualities of the features extracted from the de-noised data and those from TSA data are compared in terms of accuracy and precision indexes. Experimental results about SNR and performance indexes demonstrate the efficiency of the blind deconvolution de-noising routine.
2
1.5
0.5
20% TSA D-N 1.12% 1.84% 1.05% 0.96%
20% TSA D-N 0.951 0.985 0.981 0.992
40% TSA D-N 0.987 0.993 0.993 0.993
100% TSA D-N 0.972 0.998 0.966 0.979
VI. C ONCLUSIONS The quality of the features is closely related to the success of fault diagnosis and prognosis. A blind deconvolution denoising scheme is developed for the vibration signals from
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