A Wavelet Based Damage Identification for Ship

A Wavelet Based Damage Identification for Ship building Gantry Cranes

By Wang Haifeng

Supervised by Professor Mohammad Noori

May

2015

Abstract

A Wavelet Based Damage Identification for Shipbuilding Gantry Cranes Abstract With the rapid development of China’s economy, shipbuilding industry has experienced significant growth and advancement in the last decade. Shipbuilding gantry cranes are large, massive and complex structures that are used for lifting extremely heavy loads and are subject to harsh and corrosive working environment, which makes them susceptible to damage and deterioration. These damages accumulate gradually during the lifetime of these structures, and the damage accumulation may lead to catastrophic failures, collapse of these structures and/or fatal accidents. Hence, the development of a reliable and effective damage identification method can help provide viable damage information that can be used to prevent further damage accumulation and to repair the present damage. In this research a comprehensive overview of several existing damage identification methods is presented. Subsequently, considering the many useful features of wavelet analysis, as a time-frequency method, over other techniques, it is selected as a basis for the development signal analysis tool for damage identification. Given the type and the nature of the damage in gantry cranes, and considering several wavelets, proper and most suitable wavelet functions are considered and a specific wavelet based methodology is proposed. The proposed wavelet-based time-frequency technique is capable of detecting singularities in the mode shapes of a structure.

Finite element models of beam structures, single layer frames and scaled gantry

cranes are analyzed to obtain the response signals of the structure. From these mode shapes and the corresponding acceleration response data damages are successfully identified. By utilizing long-gage fiber optic strain sensors, a modified wavelet packet energy rate (MPWER) is introduced.

This approach is based on the fact that long-gage strain signal is sensitive to damage. Based on

the analytical results and the comparison of wavelet packet energy rate and envelope area of strain-time curvature, this MPWER is proposed as a viable damage identification technique. Computational results are utilized to identify the feasibility of MPWER. The results show that MPWER is capable of identifying the location as well as the severity of damage. Moreover, the results demonstrate that MPWER is robust to noise. Finally, experimental study is conducted on a simply supported steel beam. The strain response signals are used to calculate the MWPER. Based on MWPER, the location of damage is detected with a high degree of accuracy and the severity of the damage from the computational results closely match the experimental ones.

This comparison confirms the feasibility of MWPER as a viable technique for detecting the location

as well as the severity of the damage. The comprehensive study carried out in this research, consisting of analytical, computation and an experimental study verifies the viability of the proposed wavelet based method for damage identification and provide the basis for the construction of a comprehensive structural health monitoring tool for gantry cranes. Keywords: Damage Identification, Gantry Crane, Wavelet Transform, Time-Frequency Analysis, Long-gage strain sensing, Deflection Mode Shape I

Contents

Abstract! Chapter 1 Introduction ............................................................................................................................. 1! 1.1 Importance of structural damage identification and structural health monitoring ......................... 1! 1.2 Research background ..................................................................................................................... 2! 1.3 State of the art of damage identification methods ......................................................................... 3! 1.3.1 Natural frequency based methods ........................................................................................... 4! 1.3.2 Mode shape based methods .................................................................................................... 5! 1.3.3 Flexibility matrix based methods ............................................................................................ 5! 1.3.4 Stiffness matrix based methods .............................................................................................. 5! 1.3.5 Other methods ......................................................................................................................... 6! 1.4 Development of damage identification based on Wavelet Transform........................................... 6! 1.5 Introduction of distributed long-gage fiber optic sensors used in this research ............................ 9! 1.6 Research work and structure of this thesis ................................................................................... 10! Chapter 2 Theoretical basis of wavelet analysis and a proposed damage identification method .......... 13! 2.1 How Wavelet Transform is introduced. ....................................................................................... 13! 2.1.1 Applications of Fourier transform ........................................................................................ 13! 2.1.2 Applications of Wavelet Transform...................................................................................... 16! 2.2 Fourier transform ......................................................................................................................... 19! 2.2.1 Fourier series ......................................................................................................................... 20! 2.2.2 Continuous Fourier transform ............................................................................................... 21! 2.2.3 Comparison between Fourier transform and Fourier series .................................................. 22! 2.2.4 Discrete Fourier transform .................................................................................................... 22! 2.3 Short time Fourier transform ....................................................................................................... 23! 2.3.1 Heisenberg Uncertainty Principle ......................................................................................... 24! 2.3.2 Resolutions of the STFT ....................................................................................................... 25! 2.4 Continuous Wavelet Transform ................................................................................................... 26! 2.4.1 Conditions on mother wavelets ............................................................................................. 29! 2.4.2 Resolutions of Wavelet Transform ....................................................................................... 29! 2.5 Discrete Wavelet Transform ........................................................................................................ 30! 2.6 Wavelet packet transform ............................................................................................................ 31! 2.7 Wavelet properties ....................................................................................................................... 33! 2.8 Singularity detection principle of Wavelet Transform ................................................................ 34! 2.8.1 In displacement and acceleration data analysis .................................................................... 34! 2.8.2 In strain time-history signals ................................................................................................. 36! 2.9 Noise effects................................................................................................................................. 37! 2.9.1 Wavelet based denoising....................................................................................................... 37! 2.9.2 Simulation of noise ............................................................................................................... 38! 2.10 Conclusions ................................................................................................................................ 39! II

Contents

Chapter 3 Damage Identification based on modal displacement ........................................................... 40! 3.1 Wavelet selection ......................................................................................................................... 40! 3.1.1 Signal analysis ...................................................................................................................... 40! 3.1.2 Wavelet properties ................................................................................................................ 41! 3.2 Finite element analysis ................................................................................................................. 41! 3.2.1 Numerical Mmodel ............................................................................................................... 42! 3.2.2 Damage scenarios ................................................................................................................. 45! 3.3 Data processing ............................................................................................................................ 46! 3.3.1 Boundary effect problem ...................................................................................................... 46! 3.3.2 Comparison between deflection shape and displacement mode shape ................................. 48! 3.3.3 Comparison of different wavelets ......................................................................................... 50! 3.3.4 Selection of deflection mode shape level.............................................................................. 52! 3.3.5 Detecting damages by using CWT and selection of the best level for DWT ....................... 54! 3.2.6 Multiple damages locations and severities ........................................................................... 55! 3.2.7 Calculated deflection mode shape ........................................................................................ 56! 3.4 Damage identification process ..................................................................................................... 57! 3.5 Validation by scaled gantry crane model ..................................................................................... 58! 3.6 Conclusions .................................................................................................................................. 60! Chapter 4 Damage detection based on modified Wavelet Packet Energy Rate .................................... 61! 4.1 Wavelet selection ......................................................................................................................... 62! 4.1.1 Signal analysis ...................................................................................................................... 62! 4.1.2 Wavelet properties ................................................................................................................ 62! 4.2 Finite element analysis ................................................................................................................. 63! 4.2.1 Numerical model ................................................................................................................... 63! 4.2.2 Damage Scenarios ................................................................................................................. 65! 4.3 Data Processing ............................................................................................................................ 66! 4.3.1 Comparison of the proposed damage index and conventional indexes ................................ 66! 4.3.2 Comparison between hammer excitation and running car excitation ................................... 67! 4.3.3 Detectability of damages on simply supported beam ........................................................... 69! 4.3.4 Detectability of multiple damages on the main beam and damage on the column............... 70! 4.3.5 Verification of noise effect on the proposed method ............................................................ 71! 4.3.6 Effects of stiffness loss level................................................................................................. 72! 4.3.7 Effects of damage location.................................................................................................... 73! 4.3.8 Consideration of both damage location and damage level ................................................... 73! 4.4 Damage identification process ..................................................................................................... 74! 4.5 Validation by scaled gantry crane model ..................................................................................... 76! 4.6 Conclusions .................................................................................................................................. 77! Chapter 5 Experiment on a steel beam .................................................................................................. 78! III

Contents

5.1 Design of the experiment ............................................................................................................. 78! 5.2 Eexperiment process .................................................................................................................... 79! 5.3 Damage identification based on the data acquired by the experimental steel beam. ................... 81! Chapter 6 Conclusions ........................................................................................................................... 84! References .............................................................................................................................................. 86!

IV

Chapter 1 Introduction


1.1 Importance of structural damage identification and structural health monitoring With the development of modern technology, modern infrastructure and engineered systems are becoming larger and more complex. All these structures, such as aircraft, long span bridges, high-rise structures, large span steel grid-frame structures, large engineering structures, suffer from complex working conditions, heavy working loads and other effects that will cause damages on structures and accelerate the development of damages. With the accumulation of damages on structures, the safety of both the structure and human life are threatened. Although, over the past two decades, significant research has been carried out in the area of structural health monitoring to make structures work safely, highly reliable and practical damage identification methods are still lacking and many disasters occur due to lack of damage identification. All these structural failures and subsequent disasters cause great losses to both lives and property. On 1st February, 2003, the Space Shuttle Columbia disintegrated over Texas and Louisiana when it reentered Earth’s atmosphere. All its seven crew members died in that accident. This disaster was caused by a damage on its left wing, which was caused by a piece of foam insulation during the launch process. The damage led to a penetration of extremely hot atmospheric gases when it reentered Earth atmosphere. And then the internal wing structure was destroyed by temperature. On 23rd May, 2004, roof of Terminal 2E in Paris airport collapsed, killed 5 people and injured 3 people. The collapse of roof was caused by a penetration on the connection parts between concrete roof and metal support. (Figure 1.1)

Figure 1.1 Collapse of Terminal 2E

On 1st August 2007, Minneapolis I-35W bridge over the Mississippi River collapsed due to the undersized gusset plates, killing 13 people with 145 injured. The collapse was reported as an accident caused by damage (Figure 1.2).

1

A Thesis For the Academic Degree of Master of Science

Figure 1.2 Minneapolis I-35W bridge, before and after the collapse

These collapse accidents are only a few examples of numerous similar incidents that show how damages are often observed in engineering systems during their service life time, which could be caused by excessive response, accumulative crack growth, wear and tear of working parts and impact by other structures. And damages could lead to disasters. Even though the damage may not lead to catastrophic loss of a structure, many time the delay in early detection of damage may lead to unnecessary increase in the cost of repair that is required due to propagation of the extent of undetected damage. The earlier the damage can be identified, the less the cost would be. For structural damage identification, the most effective way to circumvent catastrophic failures is to develop a real time and online damage detection and monitoring. It is due to increasing necessity for a reliable, accurate, and effective online and real time monitoring of the health of infrastructure systems that a basic platform for structural health monitoring (SHM) is established. A fully functional SHM system could monitor the system performance, detect damages, evaluate the structural health condition and make decisions for maintenance. With an SHM system that helps evaluate the structure constantly, we can decide if it is necessary to have a maintenance procedure for inspection or replacement to ensure the structure’s safety and functionality. SHM has emerged as a reliable, economical and efficient approach to maintain high system performance and low maintenance cost [1]. Besides, the SHM is a viable and the most effective and reliable approach for assessing the life remaining life of a structure, assess it safety as well as developing a performance based design. 1.2 Research background With the rapid expansion of China’s foreign trade, the number of freight ports in China is rapidly increasing. While the development of foreign trade accelerates the development of ports, prosperous ports provide foreign trade a stable platform. Especially in the past decade, the construction of new ports in China has reached a peak and now is becoming the largest construction market in the world. In the meantime, the equipment used at ports in China have achieved great-leap-forward development, and the utilization of advanced technology and equipment in these ports has reached a highly advanced level in the world. Port facilities made in China are heavily used for international trade and business and have become a major hub for export and import in the world. Presently, there are a large number of gantry cranes in China, being used in shipyards, ports and factories. Shipbuilding gantry cranes are large, massive and complex structures that are used for lifting extremely heavy loads and are subject to harsh and corrosive working environment, which subject them to potential damage and deterioration. As a result, these severe conditions contribute to the possibility of damages, or even gantry crane accidents. The consequences can be catastrophic. In 2008, a gantry crane at Shanghai Zhenhua Heavy Industries collapsed because of burst in the box girder, killing 2 people. In the same year, a gantry crane at Yixian Road port, Shanghai, collapsed while workers were using it to load up cargos, falling on 2 cargo ships. These accidents could be due to three 2


reasons: 1. Accumulation of fatigue cracks; 2. Corrosion in various components of the structure, and 3. Structural collapse caused by local buckling failure. Many times all these three types of failure may occur simultaneously due to prolonged lack of structural damage inspection or effective damage prediction strategies and/or technology; hence resulting in the collapse. If structural damages and defects of gantry cranes are not evaluated properly and in a timely manner, it might make the repair or saving the structure impossible and lead to inevitable rapid progression of the damage and shortening the structure’s life time. On the other hand, too frequent routine maintenance, using traditional and inadequate inspection approaches, can be costly for they require extended periods of shutting down the operation or lowering the loading and operational capacity. To assure gantry cranes run safely and to prevent otherwise subsequent economic losses, it is necessary to do periodic maintenance and regular inspections. So far, various requirements have been proposed and various regulations have been developed, adopted and mandated by law. However, majority of the current inspection methods used in ports and shipyards, including large scale shipyards and modern ports, are still dominantly limited to traditional local damage identification methods. And most of these methods and procedures are limited to manual inspection which heavily rely on traditional methods where the key to maintenance is workers’ accumulated experience. Yet worse, some of these inspection and maintenance results are even recorded manually on paper forms. Given the advances in modern sensors and data processing techniques, a modern real-time and online inspection method to replace these traditional, unreliable and inefficient techniques and maintenance procedures is feasible and it is possible to develop practical, cost-effective and highly reliable structural health monitoring systems tailored for the detection of the specific types of damage incurred in gantry cranes, to assure a safe and enduring operation of these life line structures, a viable structural health monitoring system for gantry cranes will also result in increasing their life expectancy and significant financial benefits for shipbuilding industry. In this research, two damage identification methods that can detect the damages and defects in a gantry crane efficiently, which can be the basis for a health monitoring system, are established. 1.3 State of the art of damage identification methods Current damage detection techniques can be classified into two broad categories: local and global damage identification. Generally, the strategies employed for either local or global identification can be divided into four levels based on the nature and the scope of the necessary damage assessment and identification: Level 1: the presence of damage; Level 2: the location of the damage on the structure; Level 3: the severity identification of the damage; Level 4: the assessment of remaining life time of the structure [2]. Most techniques used for damage detection of gantry cranes, such as visual, magnetic field, eddy current, acoustic, X-ray inspection, etc., are local damage identification methods, which require a priori knowledge of the damage location and the accessibility of the part to be inspected. In many cases, the accessibility to various components of a shipyard gantry crane cannot be guaranteed. Thus, the application of local damage identification is limited to only some components of these structures. On the other hand, the global methods assess the structural health by examining changes in its structural dynamic characteristics. In most cases, when adequate information about the dynamic characteristics of a structure, i.e., the inertia, stiffness and damping, is available, a mathematical model of the system can be developed and the frequency response function (FRF) can be constructed. Changes in FRF and/or modal parameters will indicate a damage in the 3


structure. Thus, global damage identification is suitable for real-time assessment of structure health, while both methods should be used to complement each other and ensure the reliability and efficiency. Thus, depending on the availability of the structure’s dynamic characteristics, damage identification methods can also be classified into two categories: 1.model-based methods; 2. response data based (nonmodel-based) methods [3] [4] [65] [67]. For model-based methods, a numerical model with details of the structure is essential for damage identification. However, for response-based methods, experimental response data from structures are the only thing needed for damage identification. However, for response based methods, experimental response data from structures are the only thing needed for damage identification, where Artificial Neural Network (ANN), Support Vector Machine and other advanced algorithms are introduced [5] [65] [67]. A large volume of research work has been carried out and reported in the literature, and numerous damage identification methods have been proposed in the past several decades, such as frequency-based methods, modal shape-based methods, and mode shape curvature-based methods. A few of these methods are briefly introduced below: 1.3.1 Natural frequency based methods The basic feature of natural frequency-based methods is that they use the change in natural frequencies of a structure, or a component, as part of the damage index. Damages cause losses in both mass and stiffness. The structure’s natural frequency is a function of stiffness and mass of a structure and thus, it is is sensitive to the changes incurred in the mass or stiffness.

For instance, development of cracks in a simply supported

or cantilevered beam results in changes in the natural frequency. Consequently, any changes in the location and severity of the damage affects the eigenfrequencies. Frequency based methods are attractive because of the insensitivity or immunity to noise, since natural frequencies can be acquired by several accessible points and are usually less contaminated by experimental noise. Although the utilization of the change in frequencies has been extensively explored in the past decades, there are several limitations for the utilization of natural frequency based damage identification methods. First of all, the frequency changes caused by damage are so negligible that cannot be differentiated by the changes in the dynamic response caused by environmental and operational conditions. For this reason, the natural frequency based methods are usually verified in controlled laboratory environment rather than in field tests. Second, the modeling of structure and damage is another limitation. All natural frequency based methods are model-based while most beam-type structures’ damage identification methods are based on Euler-Bernoulli beam theory, which is well known for over-predicting the natural frequencies on short beams and high frequency bending modes. Moreover, the natural-frequency based methods can hardly identify the location of damage since different damages at different locations could cause the same change in the natural frequencies [6]. For instance, a damage at two symmetric locations of a symmetric structure will cause the same change in frequency. Lastly, in a multiple damage case, the location of damages will be extremely hard to identify. In summary, the natural frequency based damage identification methods can be applied to simple structures with single damages in a controlled environment. Due to these drawbacks, the applications for complex structures with multiple damages have been limited.

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1.3.2 Mode shape-based methods Mode shape-based methods have several merits over the natural frequency-based methods. First of all, mode shapes contain local information, which makes them sensitive to the location of damage.

They are

also capable of detecting multiple damages. In addition, mode shapes are less sensitive to environmental effects and the structure’s support conditions, compared with natural frequencies. Moreover, estimate and constructing and estimating the mode shapes requires using more information from multiple sensors installed across the structure.

This means the mode shapes contain more information about the structure.

Some of the drawbacks of mode shape-based methods are as follows. First, the measurement of mode shapes requires relatively a large number of sensors to be installed throughout the structure.

They also

depend on the accuracy of the measured data. Second, the measured mode shapes are more sensitive to contamination of signals, as compared with natural frequencies. In addition, the measurement of mode shapes requires a high degree of accuracy. Moreover, if a mode shape that has been affected more severely by a damage is overlooked or mis-calculated, the damage identification results will be inaccurate. 1.3.3 Flexibility matrix based methods Damages on a structure lead to a loss of mass and stiffness, leading to an increase of system flexibility. Based on the change of flexibility matrix, damages can be identified [7]. Besides the flexibility matrix contains local information. By comparing flexibility matrixes before and after the occurrence of damage, the element with the maximum value in the difference matrix indicates the location of damages. The biggest advantage of flexibility matrix based methods is their accuracy. The flexibility matrix is inversely proportional to the square of the natural frequencies [8], which means the lowest few natural frequencies contribute most to the assessment of flexibility matrix. With the increase of natural frequency level, their influence on the flexibility matrix becomes negligible. Hence, with the first several levels’ modal parameters, the flexibility matrix can be assessed with high accuracy. Pandey and Biswas [9] conducted valuable research on utilizing the changes in flexibility matrix. Wu and Law [10] [11] proposed a damage identification method based on the modal flexibility sensitivity. Other researchers did research based on the decomposition of flexibility matrix and have obtained good results [12] [13]. 1.3.4 Stiffness matrix based methods Damages on a structure lead to a loss of both mass and stiffness, leading to a loss of stiffness and mass matrices. Usually, the stiffness matrix can provide more information than the mass matrix. A large amount of work has been done based on the change of stiffness matrix. As discussed in the previous section, by comparing the stiffness matrices before and after the occurrence of damage, the maximum element in the difference stiffness matrix indicates the location of damage. While the fundamental concept and the analytical tools for damage identification using stiffness matrixbased methods are the same as that of the flexibility matrix-based methods, stiffness matrix-based methods suffer from their sensitivity to damage. The stiffness matrix is proportional to the square of the natural frequencies [8], hence the largest contribution to stiffness matrix is made by higher frequencies. However, the higher mode frequencies are hard to obtain, and sometimes these higher modes are coupled.

This leads

to an insensitivity to damage. Usually, stiffness matrix-based methods are capable of detecting large damages. 5


1.3.5 Other methods Energy-based damage identification methods are based on change of energy indexes. Due to the variety of energy indexes, many energy-based damage identification methods have been proposed. Damage identification methods based on change of energy indexes have several merits over other identification methods that utilize damage indexes. Lew [14] [15] proposed a damage detection method based on the Energy Transfer Ratio (ETR). Damage detection based on ETR does not need a finite element model and is more sensitive to damages than natural frequency.

Moreover, it does not consider noise or the higher modes.

Philip et al [16] proposed damage identification methods based on the strain energy. Since strain sensors provide more local information on the damages and are more sensitive to damages than other sensors. Hence, they can provide more accurate damage identification results. Other methodologies for damage detection have also been proposed, such as transfer function based techniques. The change in transfer function is only affected by damage type and location. Thus, methods based on transfer function changes are established. David et al. [17] combined the frequency response function and finite element model and proposed a damage identification scheme for truss structures [18]. Schulz, Mark, et al. [19] proposed a damage identification scheme based on the transmittance function (TF). TF is the ratio of the acceleration cross-spectrum and the acceleration auto-spectrum of any two points. Under a certain series of excitations that have the same root-mean-square value, TF is a function of frequency response function array and is independent of the input excitation. Thus, the change of TF represents damages on a structure. Damage identifications based on TF need neither numerical model of the structure nor prior knowledge of the structure and can be used for a real time monitoring system. 1.4 Development of damage identification based on Wavelet Transform As discussed earlier, global damage identifications suit gantry cranes better than local methods. Since vibration based damage identifications are capable of obtaining a large amount of information from the structure, vibration based global damage identification will be adopted in this research. The basic concept in vibration based global damage identification methods is the same: the differences or changes in the signals mean occurrence of damage. A large volume of nondestructive methods for damage detection weree discussed in the previous sections. Based on those discussions, most global damage identification methods rely on damage introduced changes in the dynamic properties of the identified structure. In these identification methods, system parameters such as frequencies, deflected mode shapes, flexibility matrix, etc. are used as part of the damage index. Among the discussed damage detection methods, some are directly based on Fourier transform (e.g. natural frequency-based damage identification methods).

Fourier transform provides information in the

frequency domain and is not capable of detecting when (or where) a particular damage occurs. To overcome this disadvantage, Dennis Gabor proposed the short-time Fourier transform (STFT). This well-known windowing technique separates the signals into a series of sections. Each time STFT is done within a small window that represents a section of the signal. Thus, information on both frequency-domain and time-domain can be kept. However, STFT has a disadvantage, the information about time and frequency (or space and frequency) are acquired with a limited precision. According to the Heisenberg uncertainty theory, this limitation is determined by the size of the time-frequency window. Once the window size is set, it is the same for all frequencies, so a higher resolution in time and frequency domain (or space and frequency domain) cannot be acquired simultaneously. 6


The Wavelet Transform (WT) is a relatively new signal processing tool to analyze data. Based on the theory of WT, it can be viewed as an extension of traditional STFT transform with adjustable window location and size. With its unique merit of examining signals with a "zoom lens having an adjustable focus", the original signal is separated in two different levels of details and approximations [20] [21] [22]. Therefore, transient information of the signal can be retained. WT has become a powerful tool for damage identification due to its capability of capturing the transient behavior of a signal. The Wavelet Transform has two parameters, just like the STFT. For time signals, the two parameters of the Wavelet Transform are time t and scale a. The scale a is related to frequency ω and thus stands for frequency domain. For spatially distributed signals, the location variable x is treated the same way as the time variable t in temporally distributed signals. Thus, results of Wavelet Transform can be used to analyze the location of damages same way as is used to determine the occurrence time of damages in temporally distributed signals. So far, few researchers have introduced wavelet analysis for the damage identification of crane structures. However, significant amount of work has been done for wavelet based damage identification in other structures, which can provide the necessary background for our proposed research. Hou and Noori [23] are the first researchers in the United States who introduced wavelet analysis for structural damage identification. They proposed a wavelet based approach and applied it on a MDOF system, where the occurrence of damage could be determined clearly in the details of the wavelet decomposition. They investigated the Effects of noise intensity and damage severity and presented it by a detectability map. Liew and Wang [24] used spatial wavelet coefficients in a numerical solution for the deflection of a simply supported Euler-Bernoulli beam under sinusoidal excitation and demonstrated its merits. Wang and Deng [25] utilized Haar Wavelet Transform in analyzing spatially distributed signals for damage detection. In their research, a cracked beam and plate were excited under different combinations of supporting conditions and excitation levels. Their research proved the feasibility of wavelet analysis in damage detection and showed the importance of the choice of mother wavelet. Hou and Hera [21] proposed a concept of pseudo-wavelet, with which the structural parameters are successfully estimated in both SDOF and MDOF systems. Then, they applied wavelet analysis on the ASCE structural health monitoring benchmark to detect and locate a structural damage. In their research, the effects of measurement noise and severity of damage were taken into consideration. Wavelet analysis illustrates promising results for the SHM, especially for its on-line application, under these adverse conditions. Quek et al. [26] compared Haar wavelet and Gabor wavelet on beams under different conditions and proved the feasibility of wavelet analysis. Their result showed that Haar wavelet is superior in terms of determining the location and extent of damage as compared with Gabor wavelet. Hong et al. [27] utilized the continuous Wavelet Transform (CWT) on mode shapes. Their selection of wavelet was Mexican hat wavelet, which helps estimate the Lipschitz exponent effectively for damage detection of a damaged beam. It was proved in their research that the selected wavelet should have at least two vanishing moments for crack detection in beams. Douka et al. [28] applied 1D symmetrical 4 Wavelet Transform on mode shapes for crack identification in a cantilever beam and plate. Sudden changes in the wavelet coefficients are used to determine positions of cracks. Via this method, the depth of the crack could be estimated from an intensity factor defined according to coefficients of the Wavelet Transform. Bayissa et al. [29] utilized the vibration responses in the time–frequency domain to generate energy density function. And based on the statistical moments, they proposed a new damage identification technique. They transformed wavelet coefficients into a new damage identification parameter (Zeroth-Order Moment, 7


ZOM) in space domain and used it on I-40 bridge to test it, which proved their method was more sensitive to damage than other methods. Gentile and Messina [30] did a comprehensive summary on mode shape based application of CWT for damage detection on a cracked beam. The investigation result showed that CWT with m vanishing moments could perform a good approximation for the m th derivatives of the spatial domain signal, when the scale parameter a is small, except for the discrepancies caused by boundary effects. At the finest scales, the Gaussian derivative wavelet family caused no interruption of the singularity in the signal. Hence, this approach demonstrated a particular advantage in damage detection and suggested to be an ideal choice for damage identification. The wavelets were shown to be capable of denoising the mode shape data without losing peaks that indicate damage locations by adopting a tradeoff between the finest and the larger scales. It was also shown that mode shape weighed through a window helps reduce the boundary effect caused by signal discontinuity effectively. Classical Gaussian wavelets suffer from processing a low number of sampling points. Hence, for application of this method, relatively high density of sensors was necessary. Yam and Yan [31] calculated energy variation of the structural vibration responses before and after the damage by utilizing WPT. Based on the calculated energy variation, they proposed damage feature proxy vectors that were used as the input for artificial neural network (ANN). By ANN, a mapped relationship of the damage feature proxy, damage location and damage severity was formed. The application on a composite structure showed great potential in online structural damage detection and SHM. Detecting the structural damages and extending the life of structures can be achieved by utilizing smart materials, intelligent systems and Structural Health Monitoring (SHM). In this study, structural damage detection is performed using Least Square Support Vector Machines (LS-SMVs) based on a new combinational kernel. Thin Plate Spline Littlewood-Paley Wavelet (TPSLPW) kernel function introduced in this paper is a novel combinational kernel function, which combines Thin Plate Spline Radial Basis Function (RBF) kernel with local characteristics and a modified Littlewood-Paley Wavelet kernel function with global characteristics. During the process of structural damage detection, a Social Harmony Search (SHS) algorithm optimizes the parameters of LS-SVM and the TPSLPW kernel. The results obtained by this method are compared with LS-SVM based on the other combinational and conventional kernels. These results show that the accuracy of damage detection based on LS-SVM with TPSLPW kernel is higher than the other methods based on the conventional kernels under the same conditions. In comparison with other combinational kernels, LSSVM with TPSLPW kernel possesses a better dissemination and learning ability by incorporating the advantages of RBF kernel and wavelet kernel functions. Based on the special properties of stationary Wavelet Transform (SWT), Zhong and Oyadiji [32] proposed a crack detection method for symmetric beam-like structures. With SWT of the acquired mode shape data, two sets of mode shape, which constituted two new signal series, were, respectively, obtained from the left half and reconstructed right half of displacement modal data of a damaged simply supported beam. These two newly formed signal series were compared and the difference of their detail coefficients was used for damage detection. By the utilization of symmetry, this method could detect damages without baseline modal parameters. Chang and Chen [33] presented a damage identification algorithm based on spatial Gabor Wavelet Transform for a beam with multiple cracks. Damage status (location and severity) were calculated by a traditional characteristic equation based optimization process with the prior knowledge of frequency and mode shapes acquired via system identification methods. The locations of damages were acquired first by mode shape data and the severity of damages were predicted by utilizing frequency information. The identification

8


results were sensitive to the crack depth, and the reliability of this method was good. The limitation of this algorithm was that cracks nearby boundaries could not be detected. Ren and Sun [34] proposed a damage identification algorithm that combined Wavelet Transform and information entropy. They investigated and compared Wavelet entropy, wavelet time entropy and relative wavelet entropy on results acquired via both numerical simulation and laboratory experiments. Damage identification results proved that their method was sensitive to damage. But this method was too sensitive to damage location, thus, damage could be detected only when a sensor was placed at the damaged location. Poudel et al. [35] proposed a new approach that utilized the difference between mode shapes in the damaged status and undamaged status. The difference function of mode shapes between two statuses was analyzed by Wavelet Transform and the high order mode shape change was obtained, indicating structural damage. They compared several wavelets to get the best result for an experimental study. However, the spatially intensive data were essential for this identification. Vibration mode shape based experimental methods are the most widely used experimental damage detection techniques in structures [22]. Both damage identification methods proposed in this research utilize vibration based experiments. The first damage identification method proposed in this paper is based on the mode shapes obtained by vibration experiment. The second method is based on strain time history data that is also obtained via vibration experiment. For the mode shape based damage identification method, the damage localization is based on the Wavelet Transform of the mode shapes. For the applications in this thesis, the deflection mode shapes are utilized, location x and scale a are used as parameters of the Wavelet Transform. For the strain energy based method, Wavelet Packet Transform (WPT) are used to separate the strain signal into different frequency bands. Then, time t and scale a are used as parameters of the Wavelet Transform. 1.5 Introduction of distributed long-gage fiber optic sensors used in this research

As discussed above, damage identification could be classified into global and local damage identifications. Global damage identification is more suitable for gantry cranes due to its merits over local damage identification. For instance global damage identification needs less accessibility of damaged locations. However, if the accessibility of damaged locations can be guaranteed, these two methods should be used complementary to each other to ensure the reliability and efficiency. Strain measurements have proved to be sensitive to damage. However, if traditional strain gauges are not installed exactly on the damaged locations, the damage detection result won’t be accurate. Thus, the idea of distributed sensing system was proposed to overcome the limitation of traditional strain gauge in obtaining the strain, where they are installed [36]. Distributed sensing system is different from multipoint sensing system. It is capable of capturing the information within certain range of the structure, leading to integrated information of the entire structure. However, this requirement of distributed sensing systems was unreachable until Horiguchi et al. [37] proposed the relationship between the strain and Brillouin frequency shift in optical fibers. Based on this relationship, a concept of distributed sensing system was proposed. Subsequently, Fiber Bragg grating (FBG) sensors [38] [64] [66] [68] [69], Brillouin optical domain reflectometer (BOTDR) sensors [39] [40] and Brillouin optical domain analysis (BOTDA) have been widely used in long-gage strain monitoring systems. FBG sensors are based on the principle that the wavelength of reflective signal from the grating changes when it has longitudinal deformation. By measuring the changes of wavelength, accurate deformation of the grating can be achieved. Based on wavelength-division multiplexing technology, several FBG sensors can be 9


combined in one Fiber to achieve multipoint sensing. For Long-gage FBG sensors [41], fiber Bragg grationgs are packaged in series to extend the effective sensing length. FBG sensors have their own merit: the signals used in FBG sensors are wavelength modulated signals, which means that they don’t need to suffer from the limitations of inaccurate phased measurement used in other kinds of fiber optic sensors. BOTDR and BOTDA sensors are based on Brillouin scattering [39] [40]. Once there is a change in strain or temperature on the fiber, the central frequency of Brillouin scattering light changes. Based on the relationship, Brillouin scattering light can be used to acquire changes in strain and temperature. For BOTDR and BOTDA sensors, any part of the fiber is both a sensing unit and a signal transmission unit, leading to their capability of spatial continuous measurement. BOTDR/BOTDA sensors and FBG sensors have different merits based on their sensing concept. FBG sensors have better accuracy in both dynamic and static sensing. However, theoretically, FBG sensors are point sensors. Their sensing length is limited. For BOTDR/BOTDA sensors, the accuracy and sampling rage is lower than FBG sensors. But they are more suitable for distributed sensing, especially in large scale structures. With the consideration of various aspects, both FBG and BOTDR/BOTDA will be used in this research in order to obtain a better sensing result. 1.6 Research work and structure of this article In this research, commonly used damage identification methods are investigated. Based on the investigation and properties of WT, two wavelet based damage identification schemes are established, where the data collected via accelerometers, displacement measurements, and fiber Bragg grating (FBG) sensors are utilized. Based on the proposed damage identification schemes, finite element analysis and experimental study of steel beams are used to validate the reliability of both schemes. With finite element models, properties of both identification schemes are examined, parameters and maps that will be used for damage identification are calculated, and noise effects are taken into consideration to examine the robustness of both schemes. Data gathered via experiments on the steel beam are used to examine the effectiveness of both schemes on real structures. Since there is no financial support from a research project, no experiment has been done on a real crane structures. But based on the survey did before this research, gantry cranes are smaller than civil structures such as long span bridges, high-rise frame structures and other reinforced concrete structures. Besides, the materials used in gantry cranes are also simpler than other structures, most of which is steel. So, experiments on simple beam structures and numerical models can be used to validate the methods used in this research. According to the survey, the most common damages found on gantry cranes are caused by corrosion. Thus, the form of researched damage is set to be corrosion, which leads to stiffness loss on the structure. Based on the development stage of damage, damages on structure could be classified into 3 stages: micro damage (stiffness loss under 0.1%), early damage (stiffness loss around 1%) and large damage (stiffness loss above 10%). In this research stiffness losses of 0.1%, 3.5%, 5%, 10%, etc. are introduced to the experimental structure to simulate all three stages. The excitations used in this research are considered to be applicable on gantry cranes. For example, the hammer excitation and running car excitation can be acquired by releasing load and moving the cargo on the main beam. The structure of this thesis is as follows: (1) Introducing the damage identification and research background: The necessity and feasibility of wavelet based damage identification on shipyard gantry cranes are analyzed. Several known model based 10


damage identification methods are briefly summarized. The advantages of damage identification based on Wavelet Transform and FBG sensors are introduced. (2) Introduction of wavelet theory and proposed damage identification methods: The development of signal processing is introduced. The theoretical background of both Fourier transform and Wavelet Transform are discussed with a brief comparison between short-time Fourier transform and Wavelet Transform. Two damage identification methods are established based on the properties of signals used in each of them. (3) Damage identification based on mode shape: The proposed mode shape based damage identification method is clearly analyzed by finite element analyses. Factors, such as the selection of wavelets, selection of mode shapes, supporting conditions, damage severity, etc. are taken into consideration to form a completed damage identification scheme. Then a scaled numerical model of a gantry crane is used to examine the efficiency of the proposed damage identification method. (4) Damage identification based on modified strain energy: The proposed energy based damage identification method is improved and incorporated in a complete damage identification scheme with a clear flow chart. Several factors are taken care of. Based on the properties of signals used in this section, the selection of wavelet and signal processing procedure are changed. Finally, the scaled numerical model of a gantry crane is used to validate the proposed damage identification procedure. (5) To better validate the damage identification based on long-gage string sensing system, a real steel beam is used to examine the completed scheme. In this chapter, details, such as the selection of excitation method, the supporting conditions and sampling rage are discussed. Then twenty four set of experimental testes are recorded to obtain the original data. Subsequently, the proposed damage identification method is utilized to process these data. Finally, the location and severity of damage are identified, showing the effectiveness and the robustness of the proposed methods. (6) Conclusions and recommendations for future work.

11


12

Chapter 2 Theoretical Basis

Chapter 2 Theoretical basis of wavelet analysis and introducing proposed damage identification schemes

2.1 How Wavelet Transform is introduced. The goal of signal processing is to find a simple and effective way to extract the desired information from the signals. Fourier transform has been the most widely used and most effective way for signal processing, ever since Fourier proposed “Théorie analytique de la chaleur” [42] . A signal can be treated as a function f of the time variable t . In the basic theory of Fourier transform, a signal is decomposed into various frequency components by representing the signal with basic sine and cosine functions: (2.1)

sin(nt ), cos(nt ) where n is a parameter of frequency ω = n / 2π . 2.1.1 Applications of Fourier transform

Limited by sensors’ performance and other undeniable factors, usually, the first problem in signal processing is denoising. The following function will be used as an example to show the basic denoising theory of Fourier transform: f (t ) = sin(t ) + 3cos(3t ) + 0.2sin(50t )

(2.2)

Since f (t ) is the sum value of three trigonometric functions, sin(t ), 3cos(3t ) and 0.2 sin(50t ) , it has three frequency components: ω1 = 1/ 2π , ω2 = 3 / 2π and ω3 = 50 / 2π . 4

Amplitude

2

0

%2

%4 0

2

4

6

Time (s)

Figure 2.1 Plot of the sample signal

Figure 2.1 shows the time-history data of signal f (t ) . As can be seen in the figure, the signal can be separated into 2 parts: the main trend and the wiggles. To remove the wiggles from the signal, Fourier transform can be introduced by setting the coefficient 0.2 in Eq.2.2 to zero. To achieve the goal of setting target coefficients to zero, three steps are needed. First, Fourier transform is used to represent the signal in the following form: 13


f (t ) = a0 + ∑ an sin(nt ) + bn cos(nt )

(2.3)

n

where n > 0 , a0 stands for the mean value of the signal, an and bn are Fourier transform coefficients that represent the frequency component of ω = n / 2π . Second, coefficients corresponding to the unwanted wiggles are set to be zero to eliminate the corresponding frequency component from the signal. Finally, the signal is represented by Eq. 2.4 with the remaining coefficients. Thus, the example signal f (t ) will be:

f (t ) = sin(t ) + 3cos(3t )

(2.4)

Figure 2.2 shows that the result of the data processing removed wiggles in the signal and the main trends are kept well. 4

Amplitude

2

0

%2

%4 0

2

4

6

Time (s)

Figure 2.2 Plot of trends of the sample signal with wiggles removed

For the utilization of Fourier transform in denoising, both high frequency components and low frequency components could be treated as the noised components. So the selection of coefficients to be eliminated must be made according to the actual situation. For example, during a phone call, both voice signal and noise signal are received. This kind of noise is usually caused by electric current and thus has a higher frequency than human voice. Hence, during the denoising process, coefficients that stand for high frequency components are set to zero. But in the situation that the received signal is a strain time history of a bridge and the impact of traveling cars on the bridge need to be studied, low frequency components need to be treated as noise. Both cars driving on the bridge and sunlight lead to strain changes. The frequency components caused by sunlight is lower than those of traveling cars since the cars cause impact loads on the bridge while sunlight only affects strain via thermal effect. Thus, low frequency components will be treated as noise and eliminated. Fourier transform can also be applied in data compression. The compression of an audio file (The mass, Era) will be discussed as an example in the following paragraphs. Figure 2.3 shows the raw signal of that audio file, where the horizontal axis is time and the vertical axis shows the output electric voltage on speakers. Usually the document size of raw file is extremely large, compared to the file usually used on digital devices. Take the sample file as an example, document size of the raw audio file is 39,064,020 bytes, while the compressed mp3 file we use is only 3,542,784 bytes, whose compression rate contains nearly everything people can hear. For raw audio files, they contain all the frequency bands that are allowed by the sampling rate, including those frequency bands that are beyond human hearing range (20~20000Hz). The purpose of audio file compression is to filter these unheard frequency components. So, the main objective of that compression is 14


discarding coefficients corresponding to those unheard components from the Fourier series that express the signal:

f (t ) = ∑ n an sin(nt ) + bn cos(nt )

(2.5)

Only those coefficients that are within the range of selected frequency range are remained. Subsequently, the audio file can be reconstructed from these remained coefficients, and the data file size will be much smaller than the raw file. 1

Amplitude

Amplitude

1

0

$1

0

$1 0

100

200

0

100

Time (s)

200

Time (s)

(a) Left channel

(b) Right channel Figure 2.3 Raw file of the sample file 1

Amplitude

Amplitude

1

0

$1

0

$1 0

100

200

0

100

Time (s)

200

Time (s)

(a) Left channel

(b) Right channel

Frequency

Frequency

Figure 2.4 Compressed sample file

44

45

46

47

48

49

50

51

52

Time (s) (a) Raw file

44

45

46

47

48

Time (s)

(b) Compressed file

Figure 2.5 Spectrogram of the raw file and compressed file

15

49

50

51

52


Figure 2.5 shows the spectrogram of both the raw file and the compressed file, where channels are shown. To show the spectrogram on the entire time line, the FFT size is set to be 8192, and it is clear that high frequency components are removed from the raw file in both channels. 2.1.2 Applications of Wavelet Transform When a signal is presented as a time-history graph, the information in time domain is exact, but there is no information in frequency domain. The contribution of Fourier transform is undeniable with its merits that help extract frequency domain information from a signal. However, Fourier transform suffers from the disadvantage of converting the whole signal from time domain into frequency domain, without any time domain information left. The reason for this disadvantage is that its building blocks are only sines and cosines, which are periodic functions that distribute on the whole time line. For stationary signals, there is no need for time domain information since the signal statistics of a stationary signal are constants. However, most signals that need to be analyzed are nonstationary signals. In case of nonstationary signals, the signal statistics are time-varying functions. So, for nonstationary signals, global characteristics of time domain or frequency domain are insufficient. The frequency domain information of a certain time section or point and the time domain information corresponding to a certain frequency band are needed for the analysis of nonstationary signals. Hence, STFT was developed for the analysis of nonstationary signals based on Gabor transform [43]. With fixed window sizes, the time resolution of STFT is limited. To overcome limitations of STFT, Wavelet Transform was introduced with a self-adapting window width. The wavelet method was first proposed by Haar. But it was not developed into an advanced signal processing tool until 1984, when geophysicist Morlet introduced it to analyze seismic survey data for the exploration of oil and mines. Usually, mines and oil reservoir in a certain layer of subsurface, and an exact map of these layers are necessary for the exploration of these resources. Since these layers are invisible and inaccessible, the mapping work of stratigraphic distribution are done by seismic methods. With geophones placed at equally spaced intervals along the seismic line, a twodimensional picture of the seismic line can be acquired by the records of earth moment from these geophones. And a three dimensional map could be established based on a certain number of two-dimensional pictures. In seismic surveys, geophones are distributed on the ground at first according to the requirements. Then a seismic wave is generated on a certain point in the ground by setting off a dynamite charge at that point. All the moments of earth caused by the seismic wave of each point are recorded by these geophones until they finish. Figure 2.6 shows a standard seismic trace, where displacement is plotted versus time.

Acceleration (m/s2)

2

0

'2 0

2

4

6

8

Time (s)

Figure 2.6 A standard seismic trace 16

10


For seismic surveys, oscillations mean reflections and scatterings from different layers, and the occurrence time instances of these oscillations show the length (thickness) of different layers, so both of them are very important. For the recorded data in Figure 2.6, the first peak means arrival of the direct wave, since it travels along the shortest path, the surface path. Usually, the first peak is predictable since the propagation velocity is known. For the subsequent oscillations, their propagation path are different since they are reflected by different layers. These subsequent waves are important because their paths contain the information of the stratigraphic distribution. As discussed earlier, the combination of propagation velocity and time information lead to the information of where the corresponding seismic wave is reflected. Each peak in the trace shows detail information of the layer. All the information extracted from the records of a geophone set that distributed along a line can be combined to draw an image of the slice beneath the line. Based on the theory of extracting information from a seismic trace, each trace needs to be analyzed properly and exactly to form an accurate seismic survey. Since Fourier transform provides only global frequency information of the entire time history, it is not suitable for seismic surveys. With the capability of extracting both time and frequency information, STFT is capable of processing seismic traces. But the fixed window width limits its ability of detecting short-duration and high-frequency bursts. As a result, Wavelet Transform becomes the best tool for the analysis of seismic traces, since it is capable of acquiring high time resolution at higher frequency bands and high frequency resolution at lower frequency bands with an adjustable time-frequency window. 2

Amplitude

1

0

(1

(2 0.0

0.2

0.4

0.6

0.8

1.0

Time (s)

Figure 2.7 Plot of a signal with isolated noise

To show the advantages of Wavelet Transform in analyzing nonstationary signals with localized features, another example is presented. Figure 2.7 shows a sound signal that has two wiggles representing noise features. As discussed before, the building blocks of Fourier transform, sines and consines, distribute on the entire time axis. Every coefficient of Fourier transform has an influence on the entire time line. So, Fourier transform is not suitable for denoising in this example. However, the building blocks of Wavelet Transform, called wavelets are more suitable for this type of signals. Wavelets are designed to analyze these localized signals. Compared with trigonometric functions, wavelets look like a single wave that travels on the entire time line. The graph for the mother wavelet of Daubechies (N = 2) is showed in Figure 2.8a. To make wavelets used in Wavelet Transform capable of representing signals within different frequency bands, the mother wavelet can be stretched or compressed by utilizing the scaling parameter a , which will be elaborated in the following sections. Except for the scaling parameter a , the translation parameter b is also used to translate the mother wavelet forward or backward in time, which helps matching signals on the entire time line. Figure 2.8(b) shows that the translated wavelet has the same shape of the mother wavelet, but the location and amplitudes are different from that of the mother wavelet. 17


Since Wavelet Transform uses scaling parameter a to match the signal at different frequency bands, it can be used the same way as Fourier transform in filtering and compressing signals. In the process of filtering or compressing, four steps are needed. First of all, the selection of mother wavelet is essential. Compared with Fourier transform, where the building blocks used is limited to sines and cosines, there are many kinds of mother wavelets for selection. The selection of mother wavelets has a great influence on the results of application, which will be discussed later based on the theory of Wavelet Transform. Second, the given signal needs to be decomposed into a combination of different wavelet coefficients, where the difference is introduced by the scaling parameter a and translation parameter b . Third, the coefficients corresponding to the unwanted components will be set to zero or other value that result in better filtering effects. For stationary signals, the translation parameter b is not needed, since unwanted components distributed on the entire time domain. However, for nonstationary signals (Figure 2.7), the translation parameter b will be needed to localize the noise signal, and corresponding coefficients will be set to zero or other values. Finally, the signal is reconstructed by the remaining coefficients and then noise effects are eliminated. 2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

'0.5

'0.5

'1.0

'1.0

'1.5

'1.5

'2.0

'2.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

(a) Mother wavelet

0.5

1.0

1.5

2.0

2.5

3.0

(b) Translation and scaling of mother wavelet

Figure 2.8 Plot of mother wavelet of Db2 and its translation and scaling

Except for applications involving the elimination of isolated noise, Wavelet Transform is capable of capturing changing frequency information in a signal. To elaborate the capability of Wavelet Transform, the softening signal in Figure 2.9 is used as an example, where the transient frequency at t = 0 s is 1 Hz and the transient frequency at t = 200 s is 0.5 Hz. With a changing frequency from 0.5 Hz to1 Hz, the power spectrum based on Fourier transform shows that the frequency components of the signal are within a range of 0.2~1.1 Hz (Figure 2.10). But the changing trend from 1 Hz to 0.5 Hz is not shown in the power spectrum, since all time-domain information are lost in Fourier transform. 1.0

f(t)

0.5

0.0

'0.5

'1.0 0

20

40

60

80

100

120

140

160

180

200

Time (s)

Figure 2.9 Sin wave with instantaneous frequency change

18

Chapter 2 Theoretical Basis 0

Power Spectrum Mangnitude (dB)

&2 &4 &6 &8 &10 &12 &14 &16 &18 0.0

0.5

1.0

1.5

2.0

2.5

Freqeuncy (Hz)

Figure 2.10 Power spectral density of the softening signal

While Fourier transom does not seem to be suitable in analyzing the frequency changing trend, Wavelet Transform is able to show the changing frequency components. Figure 2.11 shows the coefficients of a continuous Wavelet Transform by DB8. The horizontal axis represents time and the vertical axis shows the scale parameter a , the color map is set to be gray, which means the brighter a point is, the higher the value of corresponding coefficient will be. From the trend shown in Figure 2.11, it is easy to determine that the transient frequency of the sample signal changes from higher value to a lower value. Based on the relationship of a and frequency, the exact trend of frequency change can be determined.

Figure 2.11 CWT of the softening signal (DB8)

Because of stiffness loss, for engineering structures, the response signals usually share the softening trend as the sample signal. In this kind of cases, Wavelet Transform provides a new way to analyze the signal, with its merits of “zoom in” for transient features and “zoom out” for slow changes. So, by the introduction of Wavelet Transform, both frequency information and time information can be captured for signal analyzing. 2.2 Fourier transform In this section, Fourier transform and inverse Fourier transform are presented. Fourier transform was first proposed by Fourier [42], based on the concept that the certain functions can be extended to a linear combination of trigonometric functions. There are several different forms of Fourier transform, such as continuous transform, discrete Fourier transform, short time Fourier transform and fast Fourier transform.

19


2.2.1 Fourier series Fourier series can be used for the decomposition of a periodic function f ( x) with a period of T . The basic idea of Fourier series is to write the periodic function in the following form: ∞

f ( x) = a0 + ∑ [an sin(nx) + bn cos(nx) ]

(2.6)

n =1

which means coefficients a0 , an and bn need to be determined. The calculation steps for these coefficients are elaborated below. For trigonometric functions used in Fourier series, they satisfy the following relations:

1

$0,, n ≠ m

π

sin(nt )sin(mt )dt = % π∫π '1,, n = m ≥ 1

(2.7)

−

#0, other $ cos(nt ) cos(mt )dt = &1, n = m ≥ 1 ∫ π −π $2, n = m = 0 ' 1

π

1

π

(2.8)

(2.9)

sin(nt ) cos(mt )dt = 0 π∫π −

where n and m are integers, and this relationship is called orthogonality. If both sides of Eq. 2.3 are multiplied by cos(nx) / π and integrated over the interval −π

1

π

π ∫π −

f ( x) cos(nx)dx =

1

∞

$

π

a + ∑a π ∫ π &( 0

−

k

k =1

% sin(kx) + bk cos(kx) ' cos(nx)dx )

≤ x ≤π

:

(2.10)

Then according to orthogonality, Eq. 2.10 can be simplified to:

bn =

1

π

(2.11)

f ( x) cos(nx)dx, n > 0 π∫π −

Similarly, an and a0 can be expressed by

an = a0 =

1

π 1

π

(2.12)

∫ π f ( x) sin(nx)dx, n > 0 −

π

(2.13)

f ( x)dx π∫π −

Thus, most functions over the interval −π ≤ x ≤ π can be expressed by Fourier series. To make calculation suitable for any periodic functions f ( x) with a period of T over the entire time axis, Eq. 2.14 can be used for the conversion of intervals from −π ≤ x ≤ π to C ≤ x ≤ C + T by the change of variables:

2 C +T 2π t 2 C +T /2 2π t 1 f( )dt = ∫ f( )dt = ∫ C C − T /2 T T T T π

∫

C +π

C −π

f ( x)dx

(2.14)

which means any periodic function f ( x) with a period of T can be expressed by Fourier series. Euler equation provides a conversion relation between trigonometric function and complex exponential:

eit = cos(t ) + i sin(t )

(2.15)

where t is a real number and i = −1 . Sometimes, for the convenience of calculation specific cases, Fourier series is expressed by its complex form based on Eq. 2.6 and Eq. 2.15:

f (t ) =

−1

∞

n =−∞

n =1

∑ α neint + α 0 + ∑α neint 20

(2.16)


where α n =

1 2π

π

∫ π f (t )e

− int

−

dt .

2.2.2 Continuous Fourier transform The Fourier transform is based on the study of Fourier series. In this section, the deduction of inverse Fourier transform is presented. For Fourier series, the decomposed function defined over the interval C − T / 2 ≤ x ≤ C + T / 2 is combined by integral frequencies (Eq.2.6 and Eq. 2.16). If T is extended to ∞ , then Eq. 2.16 can be expressed as:

% ∞ ' 1 T /2 % ∞ 1 T /2 ( 2inπ x /T & −2 inπ t /T f ( x) = lim + ∑ ) ∫ f (t )e dt *e = lim + ∑ ∫ f (t )e 2inπ , − T /2 T →∞ . / n=−∞ - T 0 T →∞ / n=−∞ T −T /2 If we set

λn = 2nπ / T and Δλ = 2π / T ∞

f ( x) = lim

T →∞

Let Fl (λ ) =

1 2π

∫

T /2

−T /2

x − ) t /T

& dt , 0

(2.17)

, Eq. 2.17 can be written as:

& 1

∑ )+ 2π ∫

T /2

−T /2

n =−∞

' f (t )e 2λni x − )t dt *Δλ ,

(2.18)

f (t )eλi ( x −t ) dt , and Eq. 2.18 can be rewritten as:

f ( x) = lim

T →∞

∞

∑ F (λ )Δλ

(2.19)

l

n =−∞

As discussed before, n in Fourier series is an integer. Since T has been extended to ∞ , n should be extended to any real number. Therefore, Eq. 2.19 takes the following integral form: , where

Fl (λ ) can be presented by the integral form of f (t ) . Thus, f ( x) =

1 2π

∞

f ( x) = lim ∫ Fl (λ )d λ T →∞ −∞

∞

∫ ∫

∞

−∞ −∞

f (t )eλi ( x −t ) dtd λ

. Usually this formula is written as:

f ( x) =

1 2π

% 1 −∞ ' ) 2π

∫

∞

∫

∞

−∞

& f (t )e− λit dt ( e− λix d λ *

(2.20)!

Eq. 2.20 can be separated into 2 equations:

1 ∞ ˆ (2.21) f (λ )e− λix d λ ∫ −∞ 2π 1 ∞ (2.22) fˆ (λ ) = f (t )e− λit dt ∫ −∞ 2π Based on Eq. 2.21 and Eq. 2.22, a function f (t ) can be written in an integral form of fˆ (λ ) , which is expressed as an integral function of f (t ) . Subsequently, the definition of Fourier transform and its inversion f ( x) =

are formed: Fourier transform is defined by Eq. 2.22 and the inverse Fourier transform is defined by Eq. 2.21. Although the deduction process of both Fourier transform and its inversion are simply presented in this section, many assumptions that are necessary for the deduction process are not elaborated herein. Further information can be found in any classical signal processing textbook. The most important assumption is that the signal is a square integrable real function

f (t ) ∈ L2 ( R) . 21


2.2.3 Comparison between Fourier transform and Fourier series Compare Eq. 2.16 with Eq. 2.21. Since integration is a special case of summation, the complex form of ∞ f ( x) = 1/ 2π × ∫ fˆ (λ )e− λix d λ defined

the inverse Fourier transform

−∞

shares a lot of similarities with the complex form of the Fourier series over the interval −l ≤ x ≤ l . First of all, summation of

α n eint

over interval −∞ ≤ x ≤ ∞ ∞

f (t ) = ∑ n=−∞ α n eint , which is defined

is analogous to the integration of

fˆ (λ ) ,

Second, the variant λ in inverse Fourier transform corresponds to the integer n in Fourier series. Both of these stand for corresponding frequency components, for instance

fˆ (λ ) stands for the component at frequency λ ,

and the change of λ and n stand for the change of analyzed frequency components. Third, the interval

C − T / 2 ≤ x ≤ C + T / 2 and −∞ ≤ x ≤ ∞ are also analogous to each other. The difference between Fourier transform and Fourier series also lie in these aspects. For instance, integral frequency components are different from infinite frequency components. 2.2.4 Discrete Fourier transform Based on the discussion presented in section 2.2.2, the Fourier transform and its inversion are applicable to continuous or analog signals. However, almost all signals that we are dealing with in today’s information age are digital or discontinuous signals. Even those analog signals recorded by old equipment are digitalized for the convenience of storage and transmission. Digital signals have a common characteristics: the stored data are discrete (e.g. a song stored on a CD). In order to deal with digital signals, a new version of Fourier transform that is capable of analyzing discrete data was developed, which is called the discrete Fourier 2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

Amplitude

Amplitude

transform (DFT).

0.0 !0.5

0.0 !0.5

!1.0

!1.0

!1.5

!1.5

!2.0 !1.0

!0.8

!0.6

!0.4

!0.2

0.0

0.2

0.4

0.6

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(a) Continuous signal

(b) Discrete signal

Figure 2.12 A continuous signal and its discrete version

As shown in Figure 2.12, the discrete version of a continuous signal contains only part of the information contained in the original signal. Information lost during the process of digitization cannot be restored from the discrete signal. However, the basic goal of signal processing is to extract all information conveyed by and contained in signals. Based on these two objectives, the basic requirement for DFT is that no information contained in a data point is missed during the transform process. Assume that the only data we have is the discrete signal and no DFT is available for signal processing. What one will need to do first is to convert the discrete signal back to a continuous signal. There are several practical methods that help restore the continuous signal, such as least square method, Lagrange polynomial, 22


Newton interpolation, Newton iterative method, etc. But all these methods generate new data points that cannot be located on the original time axis. Although these restoration methods are not perfect, but the concept of converting the digital data back to a continuous function is the basis for DFT analysis. Since Fourier series and Fourier transforms share quite a lot of similarities and Fourier series has integral frequency components and a limited interval, Fourier series is used as a benchmark to find the best way of transforming Fourier transform into DFT. Based on the trapezoidal rule, a step size of h = 2π / n is set for the approximated integration. Then

2π 1 (2π ) −1 ∫ F (t )dt is expressed as: 0 2π

∫

2π

0

1 n−1 , where F (t )dt ≈ ∑ Y j n j =0

Y j = F (hj ) = F (2π j / n) and the period of F (t ) is 2π . Thus, the Fourier coefficient will be ak ≈

1 n−1 yiω jk , where yi = f (2π j / n) , ω = e 2π j / n . Since the only exact values available are those ∑ n j =0

provided by these discrete points, k < n is the necessary condition for extracting the exact and complete information from the discrete signals. Based on this necessary condition, DFT is defined as:

yˆ k ≈

1 n−1 −2π ikj / n ∑ yie , k < n n j =0

(2.23)

2.3 Short time Fourier transform As discussed in previous sections, Fourier transform suffers from its inherent drawbacks in processing nonstationary signals. Eq. 2.22 indicates that

fˆ (ω ) is the integration of f (t ) on time interval t ∈ (−∞, +∞)

and this infinite interval means that no time information, which are important for nonstationary signals, will be left intact during the process of Fourier transform. While most engineering signals are nonstationary, a new method that keeps the time information is needed. If the signal used for Fourier transform have nonzero values on part of the time axis, it can be determined that the result of Fourier transform stands for that portion of the time. In this manner, the time information is kept in a particular way. Based on this concept, if a signal is separated into a set of signals, for which only a certain part of the data is nonzero, the Fourier transforms set contains time information along the entire time axis. Based on this premise, the window function w(t) centered at time τ is introduced by Gabor [43] to the Fourier transform and thus the localized time-frequency atom φ is proposed φu ,ω (t ) = eiωt w(t − u ) . Subsequently, short time Fourier transform (STFT) is proposed as: +∞

STFT (τ ,f) = ∫ x(t ) w* (t − τ )e−i 2π ft dt

(2.24)

−∞

where * means complex conjugate and i = −1 . For STFT, the data on the selected section is assumed to be stationary, and thus the spectral coefficients for the corresponding time interval can be calculated. Although the basic mathematical concept behind STFT is relatively simple, this was the first general approach to obtain a time-frequency distribution based on Fourier transform. With its unique merits that allows STFT obtain both time and frequency information, it became quite a popular signal processing tool before Wavelet Transform was introduced. The applications of Fourier transform on a signal with a finite length is the same as having a corresponding rectangular window function on for an infinite function. Since Fourier transform is proposed for infinite signals, this kind of truncation introduced by windowing in time domain leads to energy leakage, which means part of energy that should concentrate on a certain frequency component is separated to nearby 23


frequency components, causing errors in frequency spectrum analysis. To solve the problem of energy leakage, two methods can be adopted: 1. Extend the length of the data analyzed by STFT. 2. Choose a better window function that attenuate the signal slowly to zero at the truncation point. Since longer data length means lower time resolution, the best way to reduce energy leakage is choosing a better window function. And thus, the selection of window function will have a large influence on the analysis results of STFT, and a good selection leads to good location in time and frequency domain at the same time. Over the years, extensive research has been carried out to develop strategies and practical techniques for the selection of most effective windows. A lot of researches have been done to find the best strategy for window function selection As a result, numerous window functions have been proposed for various application, including the rectangular window, Hanning window, Hamming window, Blackman windows, etc. [44]. A lot of researches have been done to find the best strategy for window function selection. Usually, for the selection of a window function, the side lobes need to be as small as possible and the band width of the main lobe needs to be as wide as possible. While STFT is widely used with its ability of analyzing signals in both time and frequency domain, it still suffer from drawbacks, most important of which is the fixed width of the window and lack of zooming capability: the shortcomings that are easily overcome by Wavelet Transform. 2.3.1 Heisenberg Uncertainty Principle In quantum mechanics, the Uncertainty Principle states that both the position and momentum of the particle cannot be determined at the same time. And the uncertainty of the position and momentum uncertainty follows the inequality ΔxΔpx

> h / 2 , where h is the reduced Planck constant.

Based on the same theory used in quantum mechanics, an uncertainty of time and frequency exists in STFT, which means time resolution and frequency resolution of STFT have an inverse relationship. It is easy to understand for signals, frequency resolution and time resolution are negatively correlated. This means in order to get a higher time resolution, the window length needs to be set smaller. But as discussed previously, a short window length leads to higher error in frequency resolution, and vice versa. Extensive research has been reported on how to get an exact definition of time resolution and frequency

f (t ) that belongs to L2 ( R) , its time resolution at time t = a

resolution for Fourier transform. For a function is defined as:

Δa f

∫ =

∞

−∞

2

(t − a ) 2 f (t ) dt

∫

∞

−∞

2

(2.25)

f (t ) dt

where the time resolution Δ a f is an index of its concentration at time t = a . Similarly, the frequency resolution of Fourier transform at frequency α is defined as:

Δα

∫ fˆ =

∞

−∞

2

(λ − α ) 2 fˆ (λ ) dt

∫

∞

−∞

For a function

2

fˆ (λ ) dt

(2.26)

f ∈ L2 ( R) , its frequency resolution Δα fˆ and time resolution Δ a f follow the following

rule:

1 Δ a f × Δα fˆ ≥ 4

24

(2.27)


This rule is called Heisenberg Inequality or Heisenberg Uncertainty Principle. According to Eq. 2.27, the time resolution Δ a f at time t and frequency resolution Δα fˆ at frequency α cannot be arbitrarily small at the sometime. 2.3.2 Resolutions of the STFT It is the window function w(t) that introduced time resolution into STFT, the determining factor of timefrequency resolution is the window function for STFT. For the localized time-frequency atom

φu ,ω (t ) = eiωt w(t − u) , the central time is u , and its time span at u is independent of u and frequency center ξ . Since w(t) is normalized as w = 1 and the central is set to be u , the time resolution of the window function can be rewritten:

σ

2 t

∫ =

∞

2

−∞

(t − u ) 2 w(t ) dt

∫

∞

2

−∞

∞

2

= ∫ t 2 w(t ) dt −∞

w(t ) dt

(2.28)

Similarly, with a center frequency ξ , the frequency span can be rewritten as:

σ ω2

∫ =

∞

−∞

2

(ω − ξ ) 2 wˆ (ω ) dt

∫

∞

−∞

2

wˆ (ω ) dt

∞

2

= ∫ ω 2 wˆ (ω ) dt −∞

(2.29)

ˆ (ω ) is its Fourier Transform. According to the uncertainty where w(t ) is the window function and w inequality, the time resolution and frequency resolution cannot have product that is smaller than 1 / 4 , or

σ tσ ω ≥ 1/ 2 . The time-frequency analysis of STFT follows the Heisenberg Uncertainty Principle. So, for STFT, with the definition of uncertainty at time

u and frequency ξ , the uncertainty in both time and frequency domain

can be covered by a rectangular window on the time-frequency plane, which is called the Heisenberg box:

[t − σ t , t + σ t ]× [ f − σ ω , f + σ ω ]

(2.30)

where σ t and σ ω stand for time resolution at time u and frequency at frequency ξ . Eq. 2.28 and 2.29 indicate that σ t and σ ω are independent of central time u and frequency center ξ . Hence, for a selected window function w(t) , the Heisenberg box is always the same, which means the corresponding time-frequency resolution is unchangeable.

25

A Thesis For the Academic Degree of Master of Science Figure 2.13 Concept Heisenberg box for STFT

Figure 2.13 shows that the Heisenberg box for STFT is independent from frequency and time. And the only thing that affects the Heisenberg box is the selection of window function w(t ) . Frequency is proportional to the number of cycles in a specific time interval. If we want to extract high frequency features form a signal by STFT, a window function that has narrower time span must be selected. On the contrary, if low frequency features is needed, a window function with wider time span is needed. With a selected window function, STFT does not fit well with signals that have both high and low frequency components simultaneously (Figure 2.13). For higher frequency components, the Heisenberg box covers more cycles than lower frequency components, leading to a waste of time information. And for lower frequency the time span is fixed, leading to the possibility of missing lower frequency information. As discussed earlier, signals used in engineering are usually nonstationary, which means both high frequency and low frequency components need to be extracted from the original data. The fixed resolutions limit the applications of STFT on nonstationary signals that contain both high frequency and low frequency components. To overcome this limitation, Wavelet Transform is proposed with self-adjustable window size that makes adaptive time-frequency analysis possible. Ferlez et al. [45] proved that Wavelet Transform is more suitable for the fault detection for gear-tooth box. Noori et al. [23] [46] applied Wavelet Transform in SHM and demonstrated good results. Wavelet transform will be elaborated in the following sections. 2.4 Continuous Wavelet Transform As discussed previously, Wavelet Transform provides a new way to analyze the signal, with its merits of “zoom in” for transient features and “zoom out” for slow changes. Thus, by the introduction of Wavelet Transform, both frequency and time information can be utilized for the analysis of signals. Based on the features that help extract the maximum information from signals, Wavelet Transform is deducted below Based on Heisenberg Uncertainty Principle, there is a limit in time-frequency analysis for signals. While the minimum size of Heisenberg box is set by Δa f

⋅ Δα fˆ ≥ 1/ 4 . The ultimate goal of signal time-frequency

processing is to reach the minimum size of Heisenberg box. For STFT, the fixed window size limits the frequency resolution over a certain band of frequencies, while the time resolution is also limited, which means that by STFT the Heisenberg Uncertainty Principle cannot be utilized to maximum. Since the window function is the limitation for STFT’s applications on time-frequency analysis, modifications must be made on the window function to improve the time-frequency analysis performance until it reaches the limit of Heisenberg Uncertainty Principle. Equation 2.24 shows that data was truncated by the window function. But each part of that integration has equal effect on the resulting integration. This means, if we consider w* (t − τ )e−i 2π ft as the entire part, the trigonometric functions are localized by the window function and then the result will remain the same since only the way of understanding the STFT is changed. If the basic blocks are localized functions that only last a certain length on the time axis, which means the building blocks are functions

f * (t − τ ) = w* (t − τ )e−i 2π ft

, the STFT can be rewritten as: +∞

STFT (τ ,f) = ∫ x(t ) f * (t − τ )dt −∞

(2.31)

Let us now return to the analysis of limitations of window function used in STFT. For Fourier transform, the time span is infinite, and thus all the frequency information obtained can be highly accurate, based on Heisenberg Uncertainty Principle. For STFT, the time span and frequency span are fixed because of the window function introduced and only the frequency component that fits the best time span and frequency span 26


of window function can be analyzed with the maximum utilization of Heisenberg uncertainty. Thus, if we can introduce a localized function with adjustable window that have higher frequency resolution at higher frequency bands and higher time resolution at lower frequency bands (where the time information needs to be exact), it will have the most valuable and accurate information based on the Heisenberg Uncertainty Principle. The proposed function that can fulfill all these requirements is called wavelet function. Therefore, the new analytical method, Wavelet Transform, was proposed. A large volume of research has been carried out in the development of Wavelet Transform and its application for signal processing. In this section, the detailed features of Wavelet Transform will be presented. More detailed overview and mathematical assumption behind the development of Wavelet Transform can be found in other research work [47] [48]. Similar to Fourier transform, the basic building blocks used in Wavelet Transform fits orthogonality, with the advantage of decomposing any function uniquely and thus allowing inverse transform. Just as windowed trigonometric function used in Fourier transform, the wavelet is a smooth and quick vanishing oscillating function. Wavelet functions have a smoothness and concentration in both time and frequency domain, which are quantified by vanishing moments. The definition of vanishing moments and its effects will be elaborated later in this chapter. Numerous types of wavelet functions have been proposed and applied in many diagnostic and signal processing related research. Some of these include, but are not limited to Haar, Daubechies (dbN), Morlet and Meryer. Based on properties of different wavelet functions, the result of Wavelet Transforms are different. And the selection will greatly affect the Wavelet Transform results. If needed, one can also develop and propose a wavelet function that suits the requirement of a specific signal to be analyzed. 1.0

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4

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(b) Db4 1.4

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0

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2

3

4

(d) Sym4

Figure 2.14 Sin wave and 3 mother wavelet functions (db4, Haar, Sym4)

The building blocks of transformation need to cover the entire time axis and all frequency components. For Fourier transform, trigonometric function is infinite and the factor λ allows Fourier transform to cover all frequency components. For STFT, the window function can be translated forward or backward by factor τ to cover the entire time axis. But for Wavelet Transform, utilization is localized, so it is impossible to cover the 27


entire time and frequency domain with only one wavelet function. Thus, a wavelet function set

Ψ a ,b (t ) is

generated by dilations and translations of the mother wavelet Ψ ( t ) :

1 t −b Ψ( ) a a

Ψ a ,b ( t ) =

(2.32)

where b ∈ R, a > 0 are the scale and translation parameters. The variable t is time, but in spatial distributed signal processing it can be location. As the scale parameter a increases, the wavelet becomes wider. This way, the wavelet function set can show the signal at different scales on the entire time/location axis by changing the parameter a and b . Just as Fourier transform converts signals into a coefficient function STFT (τ , f) with parameters τ and

f standing for time and frequency. Wavelet transform converts a function f ( t ) into a coefficient function of parameters a and b by the convolution with the generated wavelet function set Ψ a ,b (t ) .

C ( a, b ) =

1 a

∫

∞

−∞

# t −b $ f (t ) Ψ % & dt ' a (

(2.33)

Since continuous parameter variable that stands for all frequency components, this transformation is

C ( a, b ) can be used for extracting time and frequency features of the transformed signal. For example, a large amplitude in C ( a, b ) stands for large frequency component corresponding to parameters a and b at function f ( t ) . C ( a, b ) is normalized by the

called continuous Wavelet Transform (CWT). The result

root value of scale parameter a to make sure the integral energy given by wavelet and dilation are is

independent from each other. Eq. 2.33 shows the calculation of Wavelet Transform in time domain, while it can be achieved in frequency domain by utilizing Fourier transform. Once the signal is transformed by Fourier transform, the

ˆ f ) . Then equation 2.33 can be achieved in frequency domain: signal can be expressed as X( ∞ ˆ f )Ψ ˆ * (af )df W (a, t ) = a ∫ X(

(2.34)

−∞

ˆ * stands for the Fourier transform of the wavelet function set. where Ψ

If the mother wavelet used in CWT meet the admissibility condition, there is an inverse CWT that reconstructs the signal from the coefficient function C ( a, b ) , the inverse CWT is defined as:

f (t ) = The constant

1 Kψ

+∞+∞

da ∫ ∫ C ( a, b )ψ ( t ) db a a ,b

(2.35)

2

a −∞

Kψ is called admissibility constant and is defined as Kψ = ∫

∞

−∞

admissibility condition mentioned before is 0