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Composite Damage Detection Based on Redundant Second-Generation Wavelet Transform and Fractal Dimension Tomography Algorithm of Lamb Wave Xuefeng Chen, Xiang Li, Shibin Wang, Zhibo Yang, Binqiang Chen, and Zhengjia He
Abstract—In the purpose of achieving composite structure damage identification and localization of structural health monitoring, a denoising algorithm of redundant second-generation wavelet transform considering neighboring coefficients is selected as the best solution from 18 denoising schemes performed in this paper. Through introducing fractal dimension as a damage-sensitive feature and adopting the probabilistic reconstruction algorithm, the damage status on the composite panel could be identified and located as tomography maps. The practicability of the presented approach is validated in the experiment operating in the composite damage monitoring system. Index Terms—Composite, fractal dimension (FD), Lamb wave, redundant second-generation wavelet transform (RSGWT), structural health monitoring (SHM), tomography algorithm.
I. I NTRODUCTION
A
T PRESENT, composites have played important roles in areas of wind power plant, aeronautics, and astronautics. Along with its expanding applications, presences in essential parts, and higher proportion of structures, composites already become indicators for appraising advancement of structure design. It makes structural health monitoring (SHM) indispensable because of ensuring safety, implementing timely maintenance, avoiding disastrous events, and lowering costs [1]–[5]. Several research works to realize more effective detection for the healthy status of structures have been already dedicated into this field. Su et al. provide a comprehensive review on the Lamb-wave-based damage identification approaches for composite structures [2]. Yan and Gao enhance the ability of the continuous wavelet transform in feature extraction from vibration signals [3]. Therefore, composite damage monitoring rises as the top priority problem of SHM, and damage identification acts as the primary condition of evaluating and managing the structure damage.
Manuscript received June 21, 2012; revised August 17, 2012; accepted August 21, 2012. This work was supported by in part by the National Natural Science Foundation of China under Grant 51225501 and Grant 51175401, by the Fok Ying Tung Education Foundation under Grant 121052, and by the Program for Changjiang Scholars and Innovative Research Team in University. The Associate Editor coordinating the review process for this paper was Dr. Ruqiang Yan. The authors are with the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2012.2224277
Since composite laminates are complex in lay-up, which evidently have characters of layering and anisotropy, there seldom runs a reliable and mature damage monitoring method, particularly being short of measures for detecting large area and irregular structure. In existing SHM studies, the methods based on PZTs and active Lamb wave are still the most frequently employed effective method due to its sensitivity for small-size damages, such as crack or delamination. The Lamb wave is able to be propagated for a long distance without significant amplitude attenuation in plate structures, which quite fit for monitoring large-area structures such as blades of wind-driven generators, wings, and bodies of aircrafts. However, the phenomenon of dispersion and complicated transition, which the mode shape possesses, are hard to be analyzed and interpreted. In addition, anisotropy further aggravates this difficulty [2], [6]–[9]. In addition, the Lamb wave is unavoidably polluted from multi-interference sources and strong noise, which is in sore need of more precise and efficient advanced signal processing and feature extraction method to accurately identify damage information, which, in turn, offer more reasonable feature parameters [10], [11]. Obviously, it would be convenient and ideal to deeply investigate a quantitative imaging technique that could present an image of damage severity status in terms of location and size in detail as the composite structure being interrogated [12]. Aiming at combining Lamb waves and tomography algorithms to quantify damage in complex structures, several research works have been made [13]–[15]. These algorithms are mostly based on time of flight (ToF) measurements of a particular mode. However, the small errors in these measures have the risks to make the reconstruction unluckily corrupt. Moustafa and Salamone [16] present a new signal-processing technique that utilizes the fractal proprieties of guided ultrasonic waves for SHM in an isotropy aluminum plate structure. However, for the diagnosis of composite structures, the laminated plates with unisotropy properties have several kinds of damage manners. The signals detected for identification are possibly polluted by strong noises. They make the damage feature extraction hard to be obtained. Therefore, a composite SHM approach is proposed in this paper. Through implementing denoising approach based on redundant second-generation wavelet transform (RSGWT) considering neighboring coefficients, introducing the fractal dimension (FD) as a damage-sensitive feature and adopting the probabilistic reconstruction algorithm (PRA), the damage of composite could be identified and located precisely as
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tomography maps [17]–[19], which can overcome the drawbacks mentioned. To testify and choose the relative best denoising method for the signals propagating in the composite panel, 18 denoising solutions based on both second-generation wavelet transform (SGWT) and RSGWT were compared in simulated signals. The algorithm of RSGWT based on neighboring coefficients won as a result. Therefore, it was introduced to a practical experiment. The improvements are achieved for both denoising and reserving relatively more complete feature information for raw signals. The practicability is validated by the experimental program using smart piezoelectric sensors/actuators sparse array supported by composite damage monitoring system mainly based on National Instruments PCI eXtensions for Instrumentation (NI PXI) platform and advanced signal processing algorithms. The result indicates the availability of the approach presented in this paper.
Fig. 1.
Lifting scheme of SGWT. (a) Forward and (b) inverse transforms.
II. D ENOISING OF THE S IGNAL —C OMPARING SGWT- AND RSGWT-BASED M ETHODS For extracting the feeble signals from the polluted raw signals propagating in composite panel, the denoising methods based on SGWT and RSGWT are considered and discussed. Fig. 2. Redundant scheme of RSGWT. (a) Forward and (b) inverse transforms.
A. Review of SGWT Using the Lifting Scheme To construct biorthogonal wavelets in the spatial domain using SGWT, the lifting scheme of SGWT is a popular and flexible tool [20], [21]. The algorithm of the lifting scheme is fast, simple, efficient, and invertible. Three main steps constitute this scheme. 1) The split step: the original signal x = (xi )i∈Z is split into (0) even samples s(0) = (si )i∈Z and odd samples d(0) = (0) (di )i∈Z , i.e., (0)
si
= x2i
(0)
di
= x2i+1 .
(1)
2) The prediction step: To predict d(0) , operator P on s(0) is adopted. The prediction error d = (di )i∈Z is viewed as wavelet coefficients of x, i.e.,
N/2
di =
(0) di
−
(0)
pr si+r
(2)
r=−N/2+1
where pr values are coefficients of the prediction operator P , and N is the number of prediction coefficients. 3) The update step: An update of even samples s(0) is fulfilled by employing an update operator U on wavelet coefficients d and then adding its result to s(0) . The update sequence s = (si )i∈Z is considered as the approximation coefficients of x, i.e.,
When s comes as the input signal for the lifting scheme, the wavelet coefficients and the approximation coefficients at the lower resolution can be obtained. Through reversing the prediction and update operators, and changing each “+” into “−” and vice versa, the inverse lifting scheme can be directly implemented. Fig. 1 shows the forward and inverse transforms of the lifting scheme of SGWT. B. Review of RSGWT Using the Lifting Scheme For the translation invariance of the classical wavelet transform, a solution is provided by RSGWT, which discards the split step and keeps the original length’s information of lowand high-frequency signals. The forward and inverse transforms of the redundant lifting scheme of RSGWT are shown in Fig. 2 [21]. After the design of the customized wavelet, the prediction operator Popt and the update operator Uopt , which closely match the inspected signal, were obtained. As the split step is eliminated, the redundant prediction operator P [k] and the redundant update operator U [k] are computed by padding the prediction operator Popt and the update operator Uopt with zeros at the corresponding level k. The decomposition results of an approximation signal s (k) at level k with the redundant lifting scheme are expressed by following:
˜ /2 N
si =
(0) si
+
(k+1)
uj di+j−1
= s (k) − P (k) s (k) d (l+1) (l) (l) (l+1) =s + U d s
(3)
˜ /2+1 j=−N
where uj values are coefficients of the update operator U , ˜ is the number of update coefficients. and N
(k+1)
(4) (5)
where d and s (k+1) are wavelet coefficients and approximation coefficients at level k + 1.
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C. Denoising Algorithm of Soft and Hard Thresholds The new wavelet coefficients can be estimated by the hardand soft-thresholding functions as follows, where βk is the threshold value: ⎧ (k) (k) ⎨ d , ≥ βk d Hard(k) i i di = (6) (k) ⎩ 0, di < βk ⎧ (k) (k) (k) ⎨ d − β d , sgn d ≥ βk k Sof t(k) i i i di = (k) ⎩ 0, di < βk . (7) Compared with the hard-thresholding method, the softthresholding method has a stronger ability in smoothing the signal. However, the hard-thresholding method is used in the denoising procedure for retaining the impact components in the raw signal and guaranteeing that it would not be smoothed [22]. Generally, there are four schemes for determining threshold value βk according to the basic noise model [23]. 1) Rigrsure: a method of selection using the principle of Stein’s unbiased risk estimate in which the threshold is equal to 2 loge (N log2 N ). 2) √ Sqtwolog: a fixed form threshold that is equal to 2 log N . 3) Heursure: the synthesis of the former two with the result of optimal forecasting variable threshold. 4) Minimaxi: a stationary method based on the extremum principle.
where k is the scale level. L denotes the length of the raw signal. σk is the noise standard deviation at the k level, i.e., σk = median d(k) /0.6745 (11) where median (·) is the midvalue obtaining function. This denoising method of RSGWT based on neighboring coefficients can be summarized as following steps: 1) Decompose the raw signal by RSGWT. Wavelet coefficients at each scale level are then calculated. 2 2) Compute parameters Nk,i and βk at each scale level. 3) Implement thresholding for the wavelet coefficients, according to (8). 4) Obtain the reconstruction signal by adopting the inverse transform of RSGWT. E. Denoising Performance Indexes For evaluating denoising results, three indicators are introduced: signal-to-noise ratio (SNR), mean square error (MSE), and correlation coefficient (CC) [30]–[32], i.e., l l 2 2 xi (xi − x ˆi ) (12) SNR = 10 lg i=1
CC =
where (9) (10)
(13)
l ¯ (xi − x ¯x ˆi − x ˆ) i=1
l i=1
To denoise the composite structure damage signal, a proper algorithm of RSGWT based on neighboring coefficients is proposed to help in overwhelming the drawbacks of conventional wavelet threshold denoising [24]–[28]. Cai and Silverman presented a thresholding scheme through taking immediate neighboring coefficients into account. The idea of their thoughts is that, if the current wavelet coefficient contains useful features of the signal, its two neighboring coefficients also have them [21], [29]. Suppose, after RSGWT decomposition of the raw signal, (k) (k) (k) d j , d j−1 , d j+1 are jth, j − 1th, and j + 1th wavelet coefficient at level k, thresholding scheme for wavelet denoising are as follows:
(k) β2 2 di ≥ (βk )2 1 − N 2k , Nk,i (k) k,i (8) di = 2 0, Nk,i < (βk )2
i=1
l 1 (xi − x ˆ i )2 MSE = l i=1
D. Denoising Algorithm of RSGWT Based on Neighboring Coefficients
2 2 2 2 = dki−1 + dki + dki+1 Nk,i
βk = 2σk2 ln(L)
3
(xi − x ¯ )2
l
(14)
¯ˆ)2 (ˆ xi − x
i=1
where xi and x ˆi are the raw damage signal and the corresponding denoising damage signal at the i point in time, respectively. ¯ˆ are their mean values, and l is the length of the data. x ¯ and x III. F EATURE E XTRACTION AND L OCALIZATION BASED ON FD T OMOGRAPHY A LGORITHM A. DI Based on FD For researching damage detection on the composite panel, the FD of Lamb waves as a damage-sensitive feature is introduced. The FD is usually employed as the index of the complexity of natural objects. Among several presented algorithms, the box-counting algorithm is adopted as the frequently used method. In this approach, the curve is covered with square boxes with the same size, and the counting of boxes of a given size is implemented to inquire how many of them are needed to cover the curve fully, as shown in Fig. 3. As the size of the box approaches zero, the total amount covered by the boxes will converge to the measure of the curve as follows: log Nl D = lim − (15) l→0 log l
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posed DI to quantitatively evaluate the signal changes is defined as DI = 1 −
DD DB
(17)
where DD is the relative FD of the damaged signal, and DB is the FD for the baseline signal. For the nth actuator–sensor path, a high DI value implies a higher probability of the existence of damage in the monitoring area of this sensing path [12], [16]. B. Tomography Algorithm Based on PRA To combine Lamb waves and tomographic imaging algorithms, the PRA is introduced to provide a probabilistic-based imaging in platelike structures, such as the composite panel [7], [16], [21]. The algorithm is expressed as P (x, y) =
N i=1
pi (x, y) =
N i=1
Fi
β−M β−1
(18)
where P (x, y) is the flaw probability at position (x, y), and pi (x, y) is the estimation from the ith actuator–sensor path. N is the total number of actuator–sensor paths on the panel. The damage-sensitive feature Fi is DI according to (17). β is a scaling parameter controlling the size of the effective elliptical distribution area, and β > 1. Here, β is equal to 1.05 [17]. For the ith path, M is the ratio of the total distance measured from point (x, y) to actuator (xai , yai ) and to sensor (xsi , ysi ) to the distance between the transmitter and the receiver as follows: (x−xai )2 + (y−yai )2 + (x−xsi )2 + (y−ysi )2 . M= (xai −xsi )2 + (yai −ysi )2 (19)
Fig. 3. Two varying box sizes for counting over a signal.
where Nl is the counting of boxes of size l needed to cover the curve fully. However, in practice, the box-counting algorithm estimates D of the curve by counting the number of boxes needed to cover the curve for several box sizes and fitting a straight line to the log-log plot of Nl versus l, expressed as log(Nl ) = −D · log l + C
(16)
where C is a constant [16]. The key idea of this method is that, for every single actuator– sensor path in damage interrogation, the counting a signal changes reflects the panel property changes, which indicate the presence and the status of a flaw. Comparing the DD representing the FD of the damaged signal with the DB representing the FD the baseline signal acquired from the undamaged structure, a damage index (DI) that quantifies how the signal has changed is introduced as a feature. The pro-
Usually, actuator–sensor pair signals will be influenced by the flaw, while others without the flaw will not. As a result, in the defect distribution probability image, the pixel where the flaw is located will have larger probability than the others. Through adopting this 2-D probability density function, an image showing a distribution of the possibility of damage location could be obtained. The expectation-maximization algorithm can be used for a sharper estimation of the location. IV. E VALUATING D ENOISING M ETHODS FOR S IMULATED S IGNAL For the purpose of evaluating the nine various schemes respectively using the SGWT- and RSGWT-based signal denoising method, a simulated signal was adopted as a subject. As to SGWT, the prediction operator P is [0.0117, −0.0977, 0.5859, 0.5859, −0.0977, 0.0117], and the update operator U is [0.0059, −0.0488, 0.2930, 0.2930, −0.0488, 0.0059]. Because of the split step is discarded in RSGWT, the redundant prediction and update coefficients P (k) and U (k) are calculated
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Fig. 4. Denoising results of 18 methods for simulated signal. Red curve: Signal without noise; blue curve: signal with noise. x-axis: time (in microseconds); y-axis: amplitude (in millivolts).
by padding the original prediction and update coefficients with zeros at the corresponding level k. When the numbers of prediction and update coefficients vary, the interval and the smoothness will change, but the waveforms are similar. The numbers are properly chosen as 6 and are obtained by implementing interpolation subdivision method. Values are [0.0117, −0.0977, 0.5859, 0.5859, −0.0977, 0.0117] and [0.0059, −0.0488, 0.2930, 0.2930, −0.0488, 0.0059], respectively [20], [21]. The results are shown in Fig. 4. The denoising performance indexes MSE, SNR, and CC are calculated and reported in Table I. As shown in Table I, the denoising algorithm of RSGWT based on neighboring coefficients won in the competition result. Because the noise is possibly strong and always unknowable, thus, the best denoising solution here was adopted as our denoising approach in our experiments. In addition to this, RSGWT has all the SGWT merits, such as the construction of base wavelet that can be fulfilled in the time domain, faster calculating speed, less memory using, and the adaptability for the arbitrary length of the signal. In addition, it specially owns the advantage of the translation invariance, which allows the same length of decomposition signals and raw signals.
This advance can provide more abundant information from signals. The result in Table I also indicated that neighboringcoefficient-based denoising method was more effective than soft-threshold denoising method, which was better than hardthreshold method as well. In addition, the RSGWT-based signal denoising method had more remarkable performance than the SGWT-based method, and compared with Rigr and Heur, Sqtw and MM acted better. V. E XPERIMENTAL S YSTEM BASED ON L AMB WAVE Lamb wave, which has a high sensibility to damage and boundary on a propagation path, can travel over a long distance in composite without a high attenuation ratio. It allows a broad area to be inspected and fulfills the possibility of detecting internal damage, as well as on the surface over the entire thickness. With a rapid velocity, waves reflected from boundaries may easily conceal damage-scattered components in the signals. To ensure precision, the inspection area is required to be a relatively small unit on the large structure. Dispersive properties of multiple wave modes throughout the thickness of
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TABLE I D ENOISING P ERFORMANCE I NDEXES OF D ENOISING R ESULTS FOR THE S IMULATED S IGNAL
Fig. 5.
the medium are not identical, even for the same mode but in different frequency scopes [2], [7], [10]. The Lamb wave is the frequently used method when analyzing the plate structure. The applied excitation signal for actuating Lamb wave in this paper has been chosen as a wave packet (narrow frequency band sine modulated), generated by 2πfc t u(t) = A [H(t) − H(t − n/fc )] 1 − cos sin 2πfc t n (20) where A is the amplitude of the signal, n is the number of peaks, fc is the central frequency, and H(t) is the Heaviside step function. It is verified that there are two fundamental modes A0 and S0 propagating in the composite laminates structure [6], [33]. Their phase/group velocity is dependent on the algebraic product of the laminate thickness and the mode central frequency [1], [10]. It indicates the chosen central frequency for the product of frequency–thickness up to 1 MHz · mm could reasonably ensure that none of higher modes will need to be considered in the specimen, which can reduce the difficulties of the analysis. In this experiment, fc = 100 kHz and n = 5.
Excitation Signal. (a) Time and (b) frequency domains.
The experimental excitation signal in the time and frequency domains can be shown in Fig. 5. The validating experiment is supported by a composite damage monitoring system, as schematically shown in Fig. 6. It is established for emphasizing on the advanced signal processing algorithm and mainly constructed on the NI PXI platform, which allows further investigations of the composite damage mechanism and detection. A glass/epoxy laminated composite plate with dimensions 500 mm × 500 mm × 2.5 mm (±0.2) as experimental specimen is adopted in this paper. Its lay-up contains eight plies with a configuration of [0/ ± 45/90]S . The material density was 1981 g/cm3 . In the form of a mass of the load, artificial damage has a diameter of 8 mm, and the active sensor network is configured by the array of eight piezoelectric ceramic wafer (PZT) with a diameter of 8 mm and 0.5 mm in thickness, as shown as the layout coordinate (unit: in millimeters) in Fig. 6. The photo of the overall view of the experiment setup is presented in Fig. 7. Experiments were carried out using the composite panel to validate the proposed damage localization approach. The plate was instrumented with an array of eight circular PZT, as shown in Fig. 8.
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Fig. 6. Schematic of damage detection using the proposed monitoring system with active sensor network.
Fig. 7.
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Fig. 8. Sensor configuration on composite panel. (a) Composite panel specimen with attached sensors. (b) Sensors arrangement with coordinates (unit: millimeters) on panel.
Overall view of experiment setup.
VI. E XPERIMENTAL S IGNAL A NALYSIS A. Experimental Signals In the experiment, the amplitude of the excitation signal is 10 V, and the sample rate is 1 MHz. A raw baseline signal and a raw damage signal via actuator–sensor path P1–P4 as a sampling example are presented in Fig. 9. Considering the influence of the composite unisotropy property, the signal would be possibly feeble. The best denoising algorithm of RSGWT based on neighboring coefficients, which was discussed in Section IV was employed in the processing procedure. Their denoising results are shown in Fig. 10.
Fig. 9. Raw signals via actuator–sensor path P1–P4. (a) Raw baseline signal. (b) Raw damage signal.
B. Tomography Maps Based on PRA After PRA operations, the tomography maps of the raw signal are shown as the following single maps’ figures in Fig. 11. Their fusion map and thresholding fusion map are presented
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Fig. 10. Denoising signals via actuator–sensor path P1–P4. (a) Denoising baseline signal. (b) Denoising damage signal.
Fig. 12. Fusion tomography map of raw signal, based on PRA; damage location: (235, 294). (a) Fusion map of eight paths (coordinate unit: millimeters). (b) Thresholding fusion tomography map (coordinate unit: millimeters).
Fig. 13. Thresholding fusion tomography map of denoising signal, based on PRA; damage location: (232,290) (coordinate unit: millimeters). Fig. 11. Eight single tomography maps based on PRA.
in Fig. 12. The final thresholding fusion map of the denoising signal is shown in Fig. 13. Clearly, the center of the damage was located as the top value point from the fusion tomography map. It proved the ability of the proposed approach to identify the correct position of the
simulated damage (i.e., mass of load) on the composite panel with the anisotropy character. C. Error Analysis According to the real damage location (225, 285), the identification location result error based on the raw signal and the
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TABLE II IDENTIFICATION ACCURACY COMPARING BETWEEN THE R AW AND D ENOISING S IGNALS
location result error based on the denoising signal were computed through the following, with the result listed in Table II: xi − xo × 100% (21) xei = xo yi − yo yei = × 100%. (22) yo As comparing result shows between Figs. 12 and 13, the damage identification accuracy of the denoising signal has the better performance rather than the accuracy of the raw signal analysis. In addition, the DI calculation accuracy will get improved with the smaller size l in the FD approach, which means the identification error could be controlled in a small scale. The more proper configuration of the PZT sensor network can also promote the accuracy. VII. C ONCLUSION In this paper, after comparing 18 solutions based on both RSGWT and SGWT in simulated signals, the algorithm of RSGWT based on neighboring coefficients has been introduced for the denoising of composite panel signals in the SHM. Wavelet coefficients at each decomposition level have been dealt with neighboring coefficients thresholds, after which they have been reconstructed by the inverse transform of RSGWT. This approach could effectively identify and extract the feeble signal in composite structures, and improve three performance indexes SNR, MSE, and CC. It can also overcome the drawbacks of conventional wavelet denoising methods that lost information in transforms. After the preprocessing work by RSGWT, the evaluation and the extraction of composite damage features have been given based on the FD method. The localization of the damage in the composite panel could be identified and located as tomography maps using PRA. Finally, the proposed approach has been validated in the experiment, and the ability of the proposed approach has been proven to identify the correct position of the simulated damage (i.e., mass of load) on the composite panel with the anisotropy character. In future work, further study would be planted in improving the accuracy of the localization by investigating the mechanism of the anisotropy of the laminate composite panel and the tomography method and realizing classification of flaws, such as crack, delamination, and porosity. R EFERENCES [1] V. L. Saponara, D. A. Horsley, and W. Lestari, “Structure health monitoring of glass/epoxy composite plates using PZT and PMN-PT transducers,” J. Eng. Mater. Technol., vol. 133, pp. 011011-1–011011-8, Jan. 2011.
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Xuefeng Chen received the Ph.D. degree from Xi’an Jiaotong University, Xi’an, China, 2004. He is a Professor of mechanical engineering with Xi’an Jiaotong University. His current research interests include finite-element method, machinery condition monitoring, and prognostics. Dr. Chen was a recipient of the National Excellent Doctoral Dissertation of China in 2007 and the Second Awards of Technology Invention of China in 2009.
Xiang Li is currently working toward the Ph.D. degree in the Department of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China. His research interests include structural health monitoring and diagnosis, signal processing, sensors, and sensor networks, particularly composites.
Shibin Wang is currently working toward the Ph.D. degree in mechanical engineering in the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, China. His research interests include time-frequency analysis and signal processing in machine condition monitoring.
Zhibo Yang is currently working toward the Ph.D. degree in the Department of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China. His research interests include wavelet finite element method and signal processing in machine condition monitoring.
Binqiang Chen is currently working toward the Ph.D. degree in mechanical engineering at Xi’an Jiaotong University, Xi’an, China. His current research focuses on discrete wavelet theory and nonstationary signal processing techniques for machinery health monitoring.
Zhengjia He received the Ph.D. degree in information engineering from Kyushu Institute of Technology, Kitakyushu, Japan, in 1998. He is currently a Professor of mechanical engineering with Xi’an Jiaotong University, Xi’an, China. He has published eight books and more than 360 papers. His academic research areas are mechanical systems and signal processing, condition monitoring and fault diagnosis, dynamic analysis, and design of machinery. Dr. He was the recipient of six Awards of Science and Technology Progress including the Third Award of National Science and Technology Progress of China in 1999 and the Second Awards of Technology Invention of China in 2009.
