An integrated approach to helicopter planetary gear fault diagnosis ...
AN INTEGRATED APPROACH TO HELICOPTER PLANETARY GEAR FAULT DIAGNOSIS AND FAILURE PROGNOSIS Romano Patrick†, Marcos E. Orchard†, Bin Zhang†, Michael D. Koelemay*, Gregory J. Kacprzynski*, Aldo A. Ferri‡, George J. Vachtsevanos† †
School of Electrical and Computer Engineering. Georgia Institute of Technology. Atlanta, GA 30332-0250. Phone: (404) 894-6252
*Impact Technologies, LLC. 200 Canal View Blvd. Rochester, NY 14623. Phone: (585) 424-1990
‡
School of Mechanical Engineering. Georgia Institute of Technology. Atlanta, GA 30332-0405. Phone: (404) 894-7403
This paper introduces the design of an integrated framework for on-board fault diagnosis and failure prognosis, and describes briefly its main modules. The framework offers the ability to detect with minimum false alarms incipient failure of a critical component of a helicopter transmission and predict its time to failure. The enabling technologies, illustrated in Figure 1, include (1) statistical tools for sensor data validation, (2) denoising of data using blind deconvolution, (3) model-based techniques for extracting an optimum condition-indicator vector, and (4) a combination of measurements and Bayesian estimation algorithms to detect and identify a fault and predict RUL with specified confidence.
Abstract - This paper introduces the design of an integrated framework for on-board fault diagnosis and failure prognosis of a helicopter transmission component, and describes briefly its main modules. It suggests means to (1) validate statistically and pre-process sensor data (vibration), (2) integrate modelbased diagnosis and prognosis, (3) extract useful features or condition indicators from data de-noised by blind deconvolution, and (4) combine Bayesian estimation algorithms and measurements to detect and identify the fault and predict remaining useful life with specified confidence and minimum false alarms.
INTRODUCTION Data acquisition
Recent events in aircraft failures accompanied by increasing cost for maintenance, repair and overhaul have highlighted the need for on-board health monitoring and off-board logistics support that will improve the safety, availability and reliability of such critical assets while minimizing maintenance costs. Available on-board health monitoring systems (HUMS,VMEP) for military helicopters are collecting baseline and fault/failure data and perform basic diagnostic functions aimed at providing useful information to the maintainer. The military sector is recognizing the need for onboard implementation of an integrated diagnostic/prognostic architecture that will provide the operator, the maintainer and, eventually, the system designer with accurate information regarding the current health status of the vehicle as well as an estimate of the remaining useful life (RUL) of failing components.
1-4244-1239-0/07/$25.00 ©2007
Sensor data validation
Data preprocessing
Prognosis of RUL Fault diagnosis Feature extraction Enhancement (de-noising)
Model of damage progression (damage vs. time) Diagnosis model (data / features vs. instant damage) Legend Process flow Supporting block Support+adaptation
Figure 1. Diagnosis and prognosis framework In May of 2002 the U.S. Army grounded almost a thousand of its helicopters after an unexpected and unexplained mechanical fault had been detected in a UH-60A “Black Hawk” [1]. The planetary gear carrier plate of the main rotor transmission of the helicopter had developed a crack, which had gone undetected by the gearbox warning instrumentation system. Similar cracks were found later in other helicopters. The carrier
547
plate is a critical component of the transmission, aiding to transmit mechanical power from the engines to the main rotor blades of the helicopter. There is concern that this fault may lead to the loss of aircraft and the lives of people on board.
The diagnostic model The model in support of the diagnosis task simulates changes in vibration data caused by the presence of the crack in the carrier plate, as illustrated in Figure 3. The crack reduces the stiffness of the plate, leading to non-uniform deformation and a change in the angular spacing of the planet gears. This change in spacing affects the vibration signature of the transmission. The appropriate design and selection of features can pick up the vibration changes and hence provide a means to assess the length of a crack. Details about this model can be found in [5].
The carrier plate crack, illustrated in Figure 2, developed on the root of one of the five planet gear mounting posts of the planetary carrier plate, which is a region of high stress [2]. It is highly desirable to detect the crack in the helicopter transmission through vibration data. Vibration sensors and instrumentation are already available in many of these aircraft, and data acquisition is feasible even while in flight. [3] and [4], among others, discuss vibration-based features and early detection techniques for the carrier plate crack.
Engine torque Crack length Vibration signature of test gearbox
Finite element model of carrier plate
Deformation (angular shift of planet gears)
Frequency response analysis
Estimate of frequency response coefficients
Transmission vibration model Field-like noise and processing Simulated vibration signal
Figure 3. Vibration model for diagnosis
Figure 2. Crack on a planetary carrier plate
The simulated vibration data provided by the model are used to determine what features are the most effective at identifying the crack length and to characterize feature values corresponding to various crack lengths. The model also supports the de-noising task (see de-noising section).
SENSOR DATA VALIDATION The ability to distinguish sensor faults from symptoms of mechanical component faults improves the integrity and robustness of the system by reducing false alarms. A suite of algorithms called FirstCheckTM, developed by Impact Technologies, is used for this purpose. The tool uses time and frequency domain methods to detect various potential sensor faults including faulty connections, loose accelerometer mounts, and damaged accelerometer elements.
The prognostic model
The algorithms include checking for bias, A/D clipping, weak signal and/or no power, kurtosis, dynamic range, random energy anomalies, filtered RMS, and frequency amplitude ratio. While generic, FirstCheck must still be configured for a given application to determine acceptable thresholds for sensor fault diagnosis. Any data that is diagnosed as a sensor fault is not used for any further processing.
The crack growth can be characterized using linear elastic fracture mechanics (LEFM), since the crack grows by the fatigue caused by cyclic loads acting on the operating transmission. Adequate models are variants of the famous Paris Law of crack growth, which can be stated as
The model in support of prognosis determines the growth rate of the carrier plate crack given a specific usage pattern of the transmission. This usage pattern, known as the loading profile, is an input parameter to the model simulations.
da = C (∆K ) m , dN
(1)
where da/dN is the increase in the crack length a per each N cycles of applied load; C and m are material constants determined empirically; the term ∆K, referred to as the stress intensity range, describes the change in stresses at the crack tip and varies with load and geometry. The architecture described in this paper used a variant of Paris Law considering the growth retardation
FAULT MODELS The health monitoring architecture is resorting to models for characterizing the physics-of-failure mechanisms of the transmission because the carrier-plate crack problem is so new that data indicative of the problem are very scarce. Different models are used for diagnosis and prognosis.
548
effect caused by plasticity-induced crack closure. The model is described in more detail in [5].
vibration signals offered by the diagnostic vibration model as a noiseless reference.
The performance of the model was assessed using data from a seeded-fault, crack growth experiment with an actual helicopter transmission. The model used values of ∆K corresponding to different crack lengths and found through finite element analysis as illustrated in Figure 4. Results of this test are discussed in a later section. Crack geometry
Crack tip stress table
Finite element model
Part geometry
Crack Length 1.5 2 2.5 3
Material properties Load / forces Fracture-related material properties
Kmax 27.92 25.68 21.23 17.82
Figure 6. Structure of de-noising scheme An accelerometer mounted on the gearbox frame collects vibration signals and the Time Synchronous Averaging (TSA) signal s(t) is calculated. The blind deconvolution de-noising algorithm is carried out in the frequency domain. The de-noising algorithm is applied to S( f ) which outputs the de-noised vibration data in the frequency domain B( f ). If the time domain signal is required, B( f ) can be inverse-Fourier transformed to obtain b(t). From B( f ) and b(t), features can be extracted and fused to be used subsequently for fault diagnosis and failure prognosis. With the main objective being RUL prediction, the failure prognosis algorithm also provides an estimate of crack length as a function of time [6]. Both the estimated crack length and loading profile serve as inputs to the diagnostic vibration model, which generates a simulated vibration signal m(t). The frequency spectrum of m(t) is normalized to obtain the weighting factor vector W( f ), which is used in a nonlinear projection of the de-noising algorithm.
Crack growth equation
da = C (∆K ) m dN
Crack length
Load
Loading profile
Kmin 30.29 27.25 21.52 19.47
Cycles
Cycles
Crack progression curve
Figure 4. Crack growth model The prognostic model provides a deterministic estimate of the crack-length progression curve. This estimate is dependent upon the initial crack length, which is determined by the diagnosis task. Bayesian considerations are used to manage and interpret the inherent uncertainty in the initial crack length (see corresponding section). The crack progression curve is used to estimate the number of load cycles remaining before a “failure” condition (i.e., a critical crack length) is reached, as illustrated in Figure 5. Loading profile Load
Model-based prognosis block Fracture mechanics crack growth model
Cycles
Expected load vs. time Real-time crack length estimate
Diagnosis (real time)
With uncertainty
Particle filtering • Uncertainty management • Real-time state PDF estimation (crack length) • Real-time model parameter updates
The blind deconvolution algorithm is shown in Figure 7. A nonlinear projection, which is based on frequency-domain vibration analysis, and a cost function minimization are critical components. Initially, an inverse filter Z '( f ) must be defined. This inverse filter is an initial estimate of the modulating signal in the frequency domain, and converges through an optimization routine that recovers the vibration signal from the measured noisy data S( f ). The inverse filter Z '( f ) is convolved with S( f ) to obtain a rough estimate of the vibration signal B'( f ). The signal B'( f ) passes through the nonlinear projection, which maps B'( f ) to a subspace that contains only known characteristics of the vibration signal, to yield Bnl( f ). The difference between B'( f ) and Bnl( f ) is denoted as E( f ). By adjusting Z '( f ) iteratively to minimize E( f ), and when E( f ) reaches a minimal value, the signal B'( f )ÆB( f ) can be regarded as
RUL estimate with confidence bounds
Figure 5. Model-based prognosis architecture
DATA DE-NOISING The de-noising architecture Once the sensor data have been validated, the de-noising algorithm, illustrated in Figure 6, is applied to the vibration signals collected from the gearbox. The de-noising scheme makes use of the frequency-domain characterizations of the
549
the de-noised vibration signal. At the same time, Z '( f ) converges to Z( f ). Through an inverse Fourier transform, the de-noised vibration signal in the time domain can be obtained as well.
provides a solid theoretical framework to handle model nonlinearities and non-Gaussian noise. A nonlinear dynamic state-space model may be used to represent the behavior of a faulted system and predict the evolution of the state probabilitydensity-function (PDF) in real time.
Diagnosis, prognosis and particle filter From a Bayesian standpoint, and especially in the case of particle-filtering algorithms, real-time fault diagnosis and prognosis rely upon estimating the current value of a fault dimension, as well as other important parameters, taking into account the information provided by a set of measurements. This approach involves a prediction step, based on a nonlinear process model (model-based consideration), and an update step, which includes new measurements into the a priori state estimate (data-driven consideration) [7].
Figure 7. Blind deconvolution de-noising scheme
Blind deconvolution de-noising A vibration signal model may be defined as s(t) = a(t)b(t) + n(t), (2) where s(t) is the measured vibration signal, b(t) the noise-free un-modulated vibration signal, a(t) the modulating signal, and n(t) the additive noise. This model can be written in the frequency domain as (3) S ( f ) = A( f ) ∗ B( f ) + N ( f ) , with ∗ being the convolution operator, and S( f ), A( f ), B( f ) and N( f ) the Fourier transforms of s(t), a(t), b(t) and n(t), respectively. Then, the goal is to recover B( f ).
Particle filtering approximates the state PDF by using samples or "particles" having associated discrete probability masses (“weights”), as N
p ( xt | y1:t ) ≈ ∑ w t ( x0:i t ) ⋅ δ ( x0:t − x0:i t ) ,
(7)
i =1
where x0:t is the state trajectory and y1:t are the measurements up to time t. The simplest implementation of this algorithm, the sequential importance resampling (SIR) particle filter [7], updates the weights using the likelihood of yt as (8) wt = wt −1 ⋅ p( yt | xt ) .
Starting from an initial guess of Z '( f ), we have (4) B '( f ) = S ( f ) ∗ Z '( f ) . Passing B'( f ) through the nonlinear projection yields Bnl( f ). Then, the difference between Bnl( f ) and B( f ) will be minimized with a cost function (5) J = ∑ [ B '( f ) − Bnl ( f )]2 + (∑ Z '( f ) − 1) 2 ,
Particle filtering for diagnosis Fault detection and identification involves the use of a feature vector (observations) to determine the operating conditions (state) of a system and the causes for deviations from desired behavioral patterns. The carrier plate crack is detected by using a particle-filter-based module built upon the nonlinear dynamic state model
f ∈Dsup
where Dsup is the frequency range that contains the main vibration information. The second term of J is used to a avoid an all-zero inverse filter Z '( f ). The iterative process refines Z '( f ) to minimize J. When it reaches the minimal value, Z '( f ) converges to Z( f ). Then, a good estimate for B( f ) is obtained as: (6) B( f ) = S ( f ) ∗ Z ( f )
⎛ ⎡ xd ,1 (t ) ⎤ ⎞ ⎡ xd ,1 (t + 1) ⎤ ⎢ x (t + 1) ⎥ = f b ⎜⎜ ⎢ x (t ) ⎥ + n(t ) ⎟⎟ ⎣ d ,2 ⎦ ⎝ ⎣ d ,2 ⎦ ⎠ xc (t + 1) = xc (t ) + β ⋅ xc (t ) ⋅ xd ,2 (t ) + ω (t )
BAYESIAN FRAMEWORK
, (9)
y (t ) = xc (t ) + v(t )
The use of Bayesian estimation algorithms in diagnosis and prognosis is appropriate because they are well suited to solve the problem of realtime state estimation, since they incorporate process data into an a priori state estimate by considering the likelihood of sequential observations. The algorithm of particle filtering
⎧[1 0]T , if x − [1 0]T ≤ x − [ 0 1]T ⎪ fb ( x) = ⎨ ⎪⎩[ 0 1]T , else ⎡⎣ xd ,1 (0) xd ,2 (0)
550
xc (0) ⎤⎦ = [1 0 0.25] T
T
∆K corresponding to different crack lengths are estimated off-line with finite element analysis. The nonlinear mapping h(•) is approximated from data collected in a previous experiment that was conducted at the same facility.
where xd,1(t) and xd,2(t) are Boolean states associated with the presence of a particular operational condition (normal operation vs. cracked plate); xc(t) is a continuous-valued state associated with the crack length; β is a timevarying model parameter dependent on the loading profile; n(t) is uniform white noise with zero-mean and independently and identically distributed; and ω(t), v(t) are non-Gaussian distributions that characterize the process and feature noise signals, respectively. The initial condition for the crack length, xc(0), is known from the setup of the seeded fault test.
Finally, the probability of failure at any future time instant, namely the PDF of the RUL, is estimated by combining both the weights of predicted trajectories and the specifications for a hazard zone through the application of the Law of Total Probabilities [6]. Standard procedures can be used to obtain 95% confidence intervals and RUL expectations from the computed PDF of the RUL.
This framework allows computing the probability of detection in an on-line fashion by comparing the state PDF estimate with statistical (historical) information about the normal operating conditions (“baseline”) of the process. In addition, PDF estimates for the continuous-valued system states allow swift transitioning to failure prognostic modules [6].
DIAGNOSIS AND PROGNOSIS GUI A graphical user interface (GUI) has been designed to display the results of the integrated algorithms for diagnosing and prognosticating the carrier plate crack, as illustrated in Figure 8. The GUI conveys system-health information in realtime, including sensor data validation, extracted features for different sensors and engine torque levels, and 95% confidence intervals for the crack length and time-to-failure (TTF) of three length thresholds used in lieu of hazard zones. The GUI allows an operator to specify the threshold values and the probabilities of false alarms and detection.
Particle filtering for prognosis Prognosis can be accomplished by generating long-term predictions that describe the evolution in time of a fault indicator. Since a prognostic scheme intends to project the current condition of the indicator in the absence of future measurements, it entails large-grain uncertainty.
On-line Diagnosis and Prognosis GUI Feature Values
In a particle-filtering-based approach, long-term predictions for the fault indicator are based on both an accurate estimate of the current state PDF and a model describing the fault progression. First, the state value associated with each particle is considered as an initial condition, and then the evolution of each particle is computed by successively taking the expectation of the model. For the carrier plate problem, a crack-growth state model based on Paris Law has been implemented to estimate on-line parameters and states as
Current State PDF
Detection Parameters
Baseline PDF (Healthy conditions) Earliest Detection Results
Estimates of Crack Length Progression
Fault Identification Mapping of Feature Values vs. Crack Length
User choices for display (on-line selectable)
⎧ L(t + 1) = L(t ) + C ⋅ α (t ) ⋅ ( ∆K (t ) )m + ω 1 (t ) ⎪⎪ , (10) ⎨α (t + 1) = α (t ) + ω 2 (t ) ⎪ ⎪⎩∆K (t ) = f ( Load(t ), L(t ) ) Feature(t ) = h( L(t )) + v (t )
Fault Detection Indicators corresponding to different features and operating conditions
Prognosis results (in remaining cycles of life) for different thresholds
Figure 8. Graphical user interface As mentioned earlier, the performance of the integrated architecture was assessed with data from a seeded fault test. Early detection, and precision and accuracy of the prognosis results were evaluated. The baseline PDF for each feature was constructed considering vibration data from healthy systems (no crack). The loading profile of the system was a controlled variable. The GUI was run after the seeded fault test was completed, but real-time operation was simulated.
where L(t) is the estimated crack length at load cycle t; α (t) is an unknown time-varying model parameter to be estimated (unitary initial condition); C, m, and ∆K are the parameters of Paris Law described earlier; and ω1(t), ω2(t) and v(t) are non-Gaussian white noises. The values of
551
The robustness of the GUI was also evaluated against missing data points (e.g., problems in data acquisition).
ACKNOWLEDGMENTS This work has been partially supported by the DARPA Structural Integrity Prognosis Program (SIPS) under the direction of Dr. Leo Christodoulou. We gratefully acknowledge this support and as well as valuable guidance from other SIPS team members including Steve Engel (Northrop Grumman), Mark Davis (Sikorsky Aircraft) and Bill Hardman (NAVAIR).
In general, model parameters can be determined in either of an off-line or on-line fashion. The offline approach determines the parameter values before the health monitoring architecture is run onboard an aircraft. The pre-determined parameters in the present implementation include maps relating vibration feature values to crack lengths (diagnosis) and values of ∆K for different amounts of damage (prognosis). These values are stored in a database to be accessed as needed by the monitoring architecture. Predetermination of the parameters is necessary because some simulations take hours to derive results, which precludes real-time operation.
REFERENCES [1] Strass, M. (2002). Transmission Crack Grounds Two-Thirds of Army Black Hawk Fleet. Defense Daily. 214(25). [2] Sahrmann, G. J. (2004). Determination of the crack propagation life of a planetary gear carrier. 60th Annual Forum Proceedings, Baltimore, MD, Jun. 7-10 2004, American Helicopter Society. [3] Keller, J. and P. Grabill (2003). Vibration Monitoring of a UH-60A Main Transmission Planetary Carrier Fault. The American Helicopter Society 59th Annual Forum, Phoenix, AZ, May 6-8, 2003. [4] Wu, B., A. Saxena, T. S. Khawaja, R. Patrick, G. Vachtsevanos and P. Sparis (2004). An approach to fault diagnosis of helicopter planetary gears, IEEE. [5] Patrick, R. (2007). A Model Based Framework for Fault Diagnosis and Prognosis of Dynamical Systems with an Application to Helicopter Transmissions. Ph.D. Thesis. Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA. [6] Orchard, M. and G. J. Vachtsevanos (2007). “A Particle Filtering-based Framework for Real-time Fault Diagnosis and Failure Prognosis in a Turbine Engine,” 15th Mediterranean Conference on Control and Automation MED’07, Athens, Greece. [7] Arulampalam, M. S., S. Maskell, N. Gordon, and T. Clapp (2002). “A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking,” IEEE Transactions on Signal Processing, 50(2)
The on-line approach uses information from a helicopter gearbox in operation. For the present architecture, on-line analysis and parameter determination is done by the routines for denoising and for diagnostic/prognostic particle filtering. The present architecture showed that all necessary computations can be performed in real time with a regular microprocessor. Results proved the efficacy of the algorithms. The crack was detected after just 13 load cycles out of a total run time of more than 1000. Post-test ground-truth crack lengths were always within the 95% confidence intervals of the prediction. Results also demonstrated that it would be feasible to implement the architecture on-board a helicopter. In addition, the combination of modelbased and data-driven techniques provided two important advantages. First, the prediction results were very robust, since the effect of outliers in the data was mitigated by the model. Second, the architecture allows on-line adaptation whenever the system undergoes environment changes, e.g., changes in the loading profile.
CONCLUSIONS This paper introduces the design of an integrated framework for on-board fault diagnosis and failure prognosis of a helicopter transmission component. Essential modules of the architecture have shown to perform satisfactorily in a “blind” test with a seeded crack fault on a gearbox test cell. The integrated algorithms detect with minimum false alarms the crack in the transmission component and predict its time-to-failure.
552
School of Electrical and Computer Engineering. Georgia Institute of Technology. Atlanta, GA 30332-0250. Phone: (404) 894-6252
*Impact Technologies, LLC. 200 Canal View Blvd. Rochester, NY 14623. Phone: (585) 424-1990
‡
School of Mechanical Engineering. Georgia Institute of Technology. Atlanta, GA 30332-0405. Phone: (404) 894-7403
This paper introduces the design of an integrated framework for on-board fault diagnosis and failure prognosis, and describes briefly its main modules. The framework offers the ability to detect with minimum false alarms incipient failure of a critical component of a helicopter transmission and predict its time to failure. The enabling technologies, illustrated in Figure 1, include (1) statistical tools for sensor data validation, (2) denoising of data using blind deconvolution, (3) model-based techniques for extracting an optimum condition-indicator vector, and (4) a combination of measurements and Bayesian estimation algorithms to detect and identify a fault and predict RUL with specified confidence.
Abstract - This paper introduces the design of an integrated framework for on-board fault diagnosis and failure prognosis of a helicopter transmission component, and describes briefly its main modules. It suggests means to (1) validate statistically and pre-process sensor data (vibration), (2) integrate modelbased diagnosis and prognosis, (3) extract useful features or condition indicators from data de-noised by blind deconvolution, and (4) combine Bayesian estimation algorithms and measurements to detect and identify the fault and predict remaining useful life with specified confidence and minimum false alarms.
INTRODUCTION Data acquisition
Recent events in aircraft failures accompanied by increasing cost for maintenance, repair and overhaul have highlighted the need for on-board health monitoring and off-board logistics support that will improve the safety, availability and reliability of such critical assets while minimizing maintenance costs. Available on-board health monitoring systems (HUMS,VMEP) for military helicopters are collecting baseline and fault/failure data and perform basic diagnostic functions aimed at providing useful information to the maintainer. The military sector is recognizing the need for onboard implementation of an integrated diagnostic/prognostic architecture that will provide the operator, the maintainer and, eventually, the system designer with accurate information regarding the current health status of the vehicle as well as an estimate of the remaining useful life (RUL) of failing components.
1-4244-1239-0/07/$25.00 ©2007
Sensor data validation
Data preprocessing
Prognosis of RUL Fault diagnosis Feature extraction Enhancement (de-noising)
Model of damage progression (damage vs. time) Diagnosis model (data / features vs. instant damage) Legend Process flow Supporting block Support+adaptation
Figure 1. Diagnosis and prognosis framework In May of 2002 the U.S. Army grounded almost a thousand of its helicopters after an unexpected and unexplained mechanical fault had been detected in a UH-60A “Black Hawk” [1]. The planetary gear carrier plate of the main rotor transmission of the helicopter had developed a crack, which had gone undetected by the gearbox warning instrumentation system. Similar cracks were found later in other helicopters. The carrier
547
plate is a critical component of the transmission, aiding to transmit mechanical power from the engines to the main rotor blades of the helicopter. There is concern that this fault may lead to the loss of aircraft and the lives of people on board.
The diagnostic model The model in support of the diagnosis task simulates changes in vibration data caused by the presence of the crack in the carrier plate, as illustrated in Figure 3. The crack reduces the stiffness of the plate, leading to non-uniform deformation and a change in the angular spacing of the planet gears. This change in spacing affects the vibration signature of the transmission. The appropriate design and selection of features can pick up the vibration changes and hence provide a means to assess the length of a crack. Details about this model can be found in [5].
The carrier plate crack, illustrated in Figure 2, developed on the root of one of the five planet gear mounting posts of the planetary carrier plate, which is a region of high stress [2]. It is highly desirable to detect the crack in the helicopter transmission through vibration data. Vibration sensors and instrumentation are already available in many of these aircraft, and data acquisition is feasible even while in flight. [3] and [4], among others, discuss vibration-based features and early detection techniques for the carrier plate crack.
Engine torque Crack length Vibration signature of test gearbox
Finite element model of carrier plate
Deformation (angular shift of planet gears)
Frequency response analysis
Estimate of frequency response coefficients
Transmission vibration model Field-like noise and processing Simulated vibration signal
Figure 3. Vibration model for diagnosis
Figure 2. Crack on a planetary carrier plate
The simulated vibration data provided by the model are used to determine what features are the most effective at identifying the crack length and to characterize feature values corresponding to various crack lengths. The model also supports the de-noising task (see de-noising section).
SENSOR DATA VALIDATION The ability to distinguish sensor faults from symptoms of mechanical component faults improves the integrity and robustness of the system by reducing false alarms. A suite of algorithms called FirstCheckTM, developed by Impact Technologies, is used for this purpose. The tool uses time and frequency domain methods to detect various potential sensor faults including faulty connections, loose accelerometer mounts, and damaged accelerometer elements.
The prognostic model
The algorithms include checking for bias, A/D clipping, weak signal and/or no power, kurtosis, dynamic range, random energy anomalies, filtered RMS, and frequency amplitude ratio. While generic, FirstCheck must still be configured for a given application to determine acceptable thresholds for sensor fault diagnosis. Any data that is diagnosed as a sensor fault is not used for any further processing.
The crack growth can be characterized using linear elastic fracture mechanics (LEFM), since the crack grows by the fatigue caused by cyclic loads acting on the operating transmission. Adequate models are variants of the famous Paris Law of crack growth, which can be stated as
The model in support of prognosis determines the growth rate of the carrier plate crack given a specific usage pattern of the transmission. This usage pattern, known as the loading profile, is an input parameter to the model simulations.
da = C (∆K ) m , dN
(1)
where da/dN is the increase in the crack length a per each N cycles of applied load; C and m are material constants determined empirically; the term ∆K, referred to as the stress intensity range, describes the change in stresses at the crack tip and varies with load and geometry. The architecture described in this paper used a variant of Paris Law considering the growth retardation
FAULT MODELS The health monitoring architecture is resorting to models for characterizing the physics-of-failure mechanisms of the transmission because the carrier-plate crack problem is so new that data indicative of the problem are very scarce. Different models are used for diagnosis and prognosis.
548
effect caused by plasticity-induced crack closure. The model is described in more detail in [5].
vibration signals offered by the diagnostic vibration model as a noiseless reference.
The performance of the model was assessed using data from a seeded-fault, crack growth experiment with an actual helicopter transmission. The model used values of ∆K corresponding to different crack lengths and found through finite element analysis as illustrated in Figure 4. Results of this test are discussed in a later section. Crack geometry
Crack tip stress table
Finite element model
Part geometry
Crack Length 1.5 2 2.5 3
Material properties Load / forces Fracture-related material properties
Kmax 27.92 25.68 21.23 17.82
Figure 6. Structure of de-noising scheme An accelerometer mounted on the gearbox frame collects vibration signals and the Time Synchronous Averaging (TSA) signal s(t) is calculated. The blind deconvolution de-noising algorithm is carried out in the frequency domain. The de-noising algorithm is applied to S( f ) which outputs the de-noised vibration data in the frequency domain B( f ). If the time domain signal is required, B( f ) can be inverse-Fourier transformed to obtain b(t). From B( f ) and b(t), features can be extracted and fused to be used subsequently for fault diagnosis and failure prognosis. With the main objective being RUL prediction, the failure prognosis algorithm also provides an estimate of crack length as a function of time [6]. Both the estimated crack length and loading profile serve as inputs to the diagnostic vibration model, which generates a simulated vibration signal m(t). The frequency spectrum of m(t) is normalized to obtain the weighting factor vector W( f ), which is used in a nonlinear projection of the de-noising algorithm.
Crack growth equation
da = C (∆K ) m dN
Crack length
Load
Loading profile
Kmin 30.29 27.25 21.52 19.47
Cycles
Cycles
Crack progression curve
Figure 4. Crack growth model The prognostic model provides a deterministic estimate of the crack-length progression curve. This estimate is dependent upon the initial crack length, which is determined by the diagnosis task. Bayesian considerations are used to manage and interpret the inherent uncertainty in the initial crack length (see corresponding section). The crack progression curve is used to estimate the number of load cycles remaining before a “failure” condition (i.e., a critical crack length) is reached, as illustrated in Figure 5. Loading profile Load
Model-based prognosis block Fracture mechanics crack growth model
Cycles
Expected load vs. time Real-time crack length estimate
Diagnosis (real time)
With uncertainty
Particle filtering • Uncertainty management • Real-time state PDF estimation (crack length) • Real-time model parameter updates
The blind deconvolution algorithm is shown in Figure 7. A nonlinear projection, which is based on frequency-domain vibration analysis, and a cost function minimization are critical components. Initially, an inverse filter Z '( f ) must be defined. This inverse filter is an initial estimate of the modulating signal in the frequency domain, and converges through an optimization routine that recovers the vibration signal from the measured noisy data S( f ). The inverse filter Z '( f ) is convolved with S( f ) to obtain a rough estimate of the vibration signal B'( f ). The signal B'( f ) passes through the nonlinear projection, which maps B'( f ) to a subspace that contains only known characteristics of the vibration signal, to yield Bnl( f ). The difference between B'( f ) and Bnl( f ) is denoted as E( f ). By adjusting Z '( f ) iteratively to minimize E( f ), and when E( f ) reaches a minimal value, the signal B'( f )ÆB( f ) can be regarded as
RUL estimate with confidence bounds
Figure 5. Model-based prognosis architecture
DATA DE-NOISING The de-noising architecture Once the sensor data have been validated, the de-noising algorithm, illustrated in Figure 6, is applied to the vibration signals collected from the gearbox. The de-noising scheme makes use of the frequency-domain characterizations of the
549
the de-noised vibration signal. At the same time, Z '( f ) converges to Z( f ). Through an inverse Fourier transform, the de-noised vibration signal in the time domain can be obtained as well.
provides a solid theoretical framework to handle model nonlinearities and non-Gaussian noise. A nonlinear dynamic state-space model may be used to represent the behavior of a faulted system and predict the evolution of the state probabilitydensity-function (PDF) in real time.
Diagnosis, prognosis and particle filter From a Bayesian standpoint, and especially in the case of particle-filtering algorithms, real-time fault diagnosis and prognosis rely upon estimating the current value of a fault dimension, as well as other important parameters, taking into account the information provided by a set of measurements. This approach involves a prediction step, based on a nonlinear process model (model-based consideration), and an update step, which includes new measurements into the a priori state estimate (data-driven consideration) [7].
Figure 7. Blind deconvolution de-noising scheme
Blind deconvolution de-noising A vibration signal model may be defined as s(t) = a(t)b(t) + n(t), (2) where s(t) is the measured vibration signal, b(t) the noise-free un-modulated vibration signal, a(t) the modulating signal, and n(t) the additive noise. This model can be written in the frequency domain as (3) S ( f ) = A( f ) ∗ B( f ) + N ( f ) , with ∗ being the convolution operator, and S( f ), A( f ), B( f ) and N( f ) the Fourier transforms of s(t), a(t), b(t) and n(t), respectively. Then, the goal is to recover B( f ).
Particle filtering approximates the state PDF by using samples or "particles" having associated discrete probability masses (“weights”), as N
p ( xt | y1:t ) ≈ ∑ w t ( x0:i t ) ⋅ δ ( x0:t − x0:i t ) ,
(7)
i =1
where x0:t is the state trajectory and y1:t are the measurements up to time t. The simplest implementation of this algorithm, the sequential importance resampling (SIR) particle filter [7], updates the weights using the likelihood of yt as (8) wt = wt −1 ⋅ p( yt | xt ) .
Starting from an initial guess of Z '( f ), we have (4) B '( f ) = S ( f ) ∗ Z '( f ) . Passing B'( f ) through the nonlinear projection yields Bnl( f ). Then, the difference between Bnl( f ) and B( f ) will be minimized with a cost function (5) J = ∑ [ B '( f ) − Bnl ( f )]2 + (∑ Z '( f ) − 1) 2 ,
Particle filtering for diagnosis Fault detection and identification involves the use of a feature vector (observations) to determine the operating conditions (state) of a system and the causes for deviations from desired behavioral patterns. The carrier plate crack is detected by using a particle-filter-based module built upon the nonlinear dynamic state model
f ∈Dsup
where Dsup is the frequency range that contains the main vibration information. The second term of J is used to a avoid an all-zero inverse filter Z '( f ). The iterative process refines Z '( f ) to minimize J. When it reaches the minimal value, Z '( f ) converges to Z( f ). Then, a good estimate for B( f ) is obtained as: (6) B( f ) = S ( f ) ∗ Z ( f )
⎛ ⎡ xd ,1 (t ) ⎤ ⎞ ⎡ xd ,1 (t + 1) ⎤ ⎢ x (t + 1) ⎥ = f b ⎜⎜ ⎢ x (t ) ⎥ + n(t ) ⎟⎟ ⎣ d ,2 ⎦ ⎝ ⎣ d ,2 ⎦ ⎠ xc (t + 1) = xc (t ) + β ⋅ xc (t ) ⋅ xd ,2 (t ) + ω (t )
BAYESIAN FRAMEWORK
, (9)
y (t ) = xc (t ) + v(t )
The use of Bayesian estimation algorithms in diagnosis and prognosis is appropriate because they are well suited to solve the problem of realtime state estimation, since they incorporate process data into an a priori state estimate by considering the likelihood of sequential observations. The algorithm of particle filtering
⎧[1 0]T , if x − [1 0]T ≤ x − [ 0 1]T ⎪ fb ( x) = ⎨ ⎪⎩[ 0 1]T , else ⎡⎣ xd ,1 (0) xd ,2 (0)
550
xc (0) ⎤⎦ = [1 0 0.25] T
T
∆K corresponding to different crack lengths are estimated off-line with finite element analysis. The nonlinear mapping h(•) is approximated from data collected in a previous experiment that was conducted at the same facility.
where xd,1(t) and xd,2(t) are Boolean states associated with the presence of a particular operational condition (normal operation vs. cracked plate); xc(t) is a continuous-valued state associated with the crack length; β is a timevarying model parameter dependent on the loading profile; n(t) is uniform white noise with zero-mean and independently and identically distributed; and ω(t), v(t) are non-Gaussian distributions that characterize the process and feature noise signals, respectively. The initial condition for the crack length, xc(0), is known from the setup of the seeded fault test.
Finally, the probability of failure at any future time instant, namely the PDF of the RUL, is estimated by combining both the weights of predicted trajectories and the specifications for a hazard zone through the application of the Law of Total Probabilities [6]. Standard procedures can be used to obtain 95% confidence intervals and RUL expectations from the computed PDF of the RUL.
This framework allows computing the probability of detection in an on-line fashion by comparing the state PDF estimate with statistical (historical) information about the normal operating conditions (“baseline”) of the process. In addition, PDF estimates for the continuous-valued system states allow swift transitioning to failure prognostic modules [6].
DIAGNOSIS AND PROGNOSIS GUI A graphical user interface (GUI) has been designed to display the results of the integrated algorithms for diagnosing and prognosticating the carrier plate crack, as illustrated in Figure 8. The GUI conveys system-health information in realtime, including sensor data validation, extracted features for different sensors and engine torque levels, and 95% confidence intervals for the crack length and time-to-failure (TTF) of three length thresholds used in lieu of hazard zones. The GUI allows an operator to specify the threshold values and the probabilities of false alarms and detection.
Particle filtering for prognosis Prognosis can be accomplished by generating long-term predictions that describe the evolution in time of a fault indicator. Since a prognostic scheme intends to project the current condition of the indicator in the absence of future measurements, it entails large-grain uncertainty.
On-line Diagnosis and Prognosis GUI Feature Values
In a particle-filtering-based approach, long-term predictions for the fault indicator are based on both an accurate estimate of the current state PDF and a model describing the fault progression. First, the state value associated with each particle is considered as an initial condition, and then the evolution of each particle is computed by successively taking the expectation of the model. For the carrier plate problem, a crack-growth state model based on Paris Law has been implemented to estimate on-line parameters and states as
Current State PDF
Detection Parameters
Baseline PDF (Healthy conditions) Earliest Detection Results
Estimates of Crack Length Progression
Fault Identification Mapping of Feature Values vs. Crack Length
User choices for display (on-line selectable)
⎧ L(t + 1) = L(t ) + C ⋅ α (t ) ⋅ ( ∆K (t ) )m + ω 1 (t ) ⎪⎪ , (10) ⎨α (t + 1) = α (t ) + ω 2 (t ) ⎪ ⎪⎩∆K (t ) = f ( Load(t ), L(t ) ) Feature(t ) = h( L(t )) + v (t )
Fault Detection Indicators corresponding to different features and operating conditions
Prognosis results (in remaining cycles of life) for different thresholds
Figure 8. Graphical user interface As mentioned earlier, the performance of the integrated architecture was assessed with data from a seeded fault test. Early detection, and precision and accuracy of the prognosis results were evaluated. The baseline PDF for each feature was constructed considering vibration data from healthy systems (no crack). The loading profile of the system was a controlled variable. The GUI was run after the seeded fault test was completed, but real-time operation was simulated.
where L(t) is the estimated crack length at load cycle t; α (t) is an unknown time-varying model parameter to be estimated (unitary initial condition); C, m, and ∆K are the parameters of Paris Law described earlier; and ω1(t), ω2(t) and v(t) are non-Gaussian white noises. The values of
551
The robustness of the GUI was also evaluated against missing data points (e.g., problems in data acquisition).
ACKNOWLEDGMENTS This work has been partially supported by the DARPA Structural Integrity Prognosis Program (SIPS) under the direction of Dr. Leo Christodoulou. We gratefully acknowledge this support and as well as valuable guidance from other SIPS team members including Steve Engel (Northrop Grumman), Mark Davis (Sikorsky Aircraft) and Bill Hardman (NAVAIR).
In general, model parameters can be determined in either of an off-line or on-line fashion. The offline approach determines the parameter values before the health monitoring architecture is run onboard an aircraft. The pre-determined parameters in the present implementation include maps relating vibration feature values to crack lengths (diagnosis) and values of ∆K for different amounts of damage (prognosis). These values are stored in a database to be accessed as needed by the monitoring architecture. Predetermination of the parameters is necessary because some simulations take hours to derive results, which precludes real-time operation.
REFERENCES [1] Strass, M. (2002). Transmission Crack Grounds Two-Thirds of Army Black Hawk Fleet. Defense Daily. 214(25). [2] Sahrmann, G. J. (2004). Determination of the crack propagation life of a planetary gear carrier. 60th Annual Forum Proceedings, Baltimore, MD, Jun. 7-10 2004, American Helicopter Society. [3] Keller, J. and P. Grabill (2003). Vibration Monitoring of a UH-60A Main Transmission Planetary Carrier Fault. The American Helicopter Society 59th Annual Forum, Phoenix, AZ, May 6-8, 2003. [4] Wu, B., A. Saxena, T. S. Khawaja, R. Patrick, G. Vachtsevanos and P. Sparis (2004). An approach to fault diagnosis of helicopter planetary gears, IEEE. [5] Patrick, R. (2007). A Model Based Framework for Fault Diagnosis and Prognosis of Dynamical Systems with an Application to Helicopter Transmissions. Ph.D. Thesis. Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA. [6] Orchard, M. and G. J. Vachtsevanos (2007). “A Particle Filtering-based Framework for Real-time Fault Diagnosis and Failure Prognosis in a Turbine Engine,” 15th Mediterranean Conference on Control and Automation MED’07, Athens, Greece. [7] Arulampalam, M. S., S. Maskell, N. Gordon, and T. Clapp (2002). “A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking,” IEEE Transactions on Signal Processing, 50(2)
The on-line approach uses information from a helicopter gearbox in operation. For the present architecture, on-line analysis and parameter determination is done by the routines for denoising and for diagnostic/prognostic particle filtering. The present architecture showed that all necessary computations can be performed in real time with a regular microprocessor. Results proved the efficacy of the algorithms. The crack was detected after just 13 load cycles out of a total run time of more than 1000. Post-test ground-truth crack lengths were always within the 95% confidence intervals of the prediction. Results also demonstrated that it would be feasible to implement the architecture on-board a helicopter. In addition, the combination of modelbased and data-driven techniques provided two important advantages. First, the prediction results were very robust, since the effect of outliers in the data was mitigated by the model. Second, the architecture allows on-line adaptation whenever the system undergoes environment changes, e.g., changes in the loading profile.
CONCLUSIONS This paper introduces the design of an integrated framework for on-board fault diagnosis and failure prognosis of a helicopter transmission component. Essential modules of the architecture have shown to perform satisfactorily in a “blind” test with a seeded crack fault on a gearbox test cell. The integrated algorithms detect with minimum false alarms the crack in the transmission component and predict its time-to-failure.
552
