小波分析具有良好的時域及頻域的解析能力，結構的裂縫或損傷會造成結構振動訊號上的微小擾動，靠著小波分析可以擷取出這個微小的擾動，所以本研究利用小波來作結構的破損偵測。
　　本研究一共研究了小波在樑、板、旋轉葉片等三種結構破損探測上的應用。發現這方法不論是在樑、板、旋轉葉片等三種結構中，小波分析對損傷的程度都有很高的靈敏度，即使損傷程度很小的損傷也能發現損傷的位置，同時也發現除非損傷位置很接近邊界不然都可將損傷的位置偵測出來，同時這方法可同時探測出多個損傷的位置，這也是這方法的一大優點。
　　在樑結構中，在利用小波分析推出多個裂縫的位置後，將樑的自然頻率代入樑的特徵方程式中可推出裂縫的深度，靠著這個方法可同時將多個裂縫的深度和位置同時估計出來，但有多少個裂縫就需要有多少個樑的自然振動頻率帶入特徵方程式中才可以推出所有裂縫的深度。
　　在板結構中，本研究必須同時作小波在x方向上的分析和在y方向的分析，由這兩種小波係數的圖互相做個比較就能找出靠近邊界損傷的位置。
　　在旋轉葉片結構中，發現小波係數分佈圖在較高轉速時在裂縫處的尖峰會較不明顯，這也是說這方法在很高轉速時會失效。

　　Damages on a structure lead to structure response perturbations at damage position. Although such local perturbations may not be apparent from the total response data, they are visible from wavelet analysis. This paper presents a technique for structure damage detection based on spatial wavelet analysis. The main purpose of this paper is to predict the positions of multi-damages based on spatial wavelet analysis.
　　First, the mode shapes of damaged structures are obtained. Then the mode shapes are analyzed by wavelet transformation to get the positions of the damages by showing a peak in the distributions of the wavelet coefficients. The structures studied here are the beam, plate and rotating blade.
　　When the positions of the cracks have been known from the plot of wavelet coefficients, the natural frequencies can be used to predict the depth of the cracks through the characteristic equation in the structure of beam. If the number of cracks is n, the first n natural frequencies are used to predict the depth of the cracks. It is observed that the positions and depths of the cracks can be predicted with acceptable precision even though there are many cracks in the beam.

1. A. D. Dimarogonas. Vibration Engineering. St. Paul, Minnesota, West Publishers, 1976.
2. T. Chondros. Thesis, Dynamic Response of Cracked Beams. M. Sc. Thesis, University of Patras, Greece, 1977.
3. H. J. Petroski. Simple Static and Dynamic Models for the Cracked Elastic Beam. International Journal of Fracture 1981, 17, R71-R76.
4. W. M. Ostachowicz and M. Krawkczuk. Analysis of the Effect of Cracks on the Natural Frequencies of a Cantilever Beam. Journal of Sound and Vibration 1991, 150, 191-201.
5. E. I. Shifrin and R. Ruotolo. Natural Frequencies of a Beam with an arbitrary number of Cracks. Journal of Sound and Vibration 1999, 222(3), 409-423.
6. P. Gudmundson. Eigenfrequency Changes of Structures due to Cracks, Notches or other Geometrical Changes. J. Mech Phys Solids 1982, 30, 339-353.
7. P. Gudmundson. The Dynamic Behaviour of Slender Structures with Cross-Sectional Cracks. J. Mech Phys Solids 1983, 31, 329-345.
8. G. Gounaris, A. D. Dimarogonas. A Finite Element of a Cracked Prismatic Beam for Structural Analysis. Computers & Structures 1988, 28, 309-313.
9. MHH. Hen, J. E. Taylor. An Identification Problem for Vibrating Cracked Beams. Journal of Sound and Vibration 1990, 138, 457-494.
10. MHH. Hen, Y. C. Chu. Vibrations of Beams with a Fatigue Crack. Computers & Structures 1991, 45, 457-484.
11. M. L. Kikids and C. A. Papadopoulos. Slenderness Ratio Effect on Cracked Beam. Journal of Sound and Vibration 1992, 155, 1-11.
12. ONL. Abraham, J. A. Brandon. The Modeling of the Opening and Closure of a Crack. J Vibr Acoustics-Trans ASME 1995, 117, 370-377.
13. A. D. Dimarogonas and G. Massouros. Torsional vibration of a shaft with a circumferential crack. Engineering Fracture Mechanics 1981, 15, 439-444.
14. C. A. Papadopoulos. Torsional Vibrations of Rotors with Transverse Surface Cracks. 1993 713-718.
15. P. Cawley and RD. Adams. The Location of Defects in Structures from Measurements of Natural Frequencies. J Strain Analysis 1979, 14, 309-313.
16. F. D. Ju and M. E. Mirovich. Experimental Diagnosis of Fracture Damage in Structure by the Modal Frequency Method. ASME Journal of Vibration and Acoustics, Stress and Reliability in Design 1988, 110, 456-463.
17. P. F. Rizos, N. Aspragathos and A. D. Dimarogonas. Identification of Crack Location and Magnitude in a Cantilever Beam from the Vibration Modes. Journal of Sound and Vibration 1990, 138, 381-388.
18. GL. Qian, SN. Gu and JS. Jiang. The Dynamic Behavior and Crack Detection of a Beam for Structural Analysis. Computers & Structures 1990, 138, 233-243.
19. R. Y. Liang, F. K. Choy and J. Hu. Detection of Cracks in Beam Structures Using Measurements of Natural Frequencies. Journal of the Franklin Institute 1991, 328(4), 505-518.
20. A. K. Pandey and M. Biswas. Damage Detection in Structure Using Changes in Flexibility. Journal of Sound and Vibration 1994, 169, 3-17.
21. H. T. Banks, D. J. Inman, D. J. Leo and Y. Wang. An Experimentally Validated Damage Detection Theory in Smart Structure. Journal of Sound and Vibration 1996, 191, 859-880.
22. Y. Hirano, and K. Okazaki. Vibration of Cracked Rectangular Plates. Bulletin of the JSME 1980, 23(179), 732-740.
23. G. L. Qian, S. N Gu and J. S. Jiang. A Finite Element Model of Cracked Plates and Application to Vibration Problems. Computers and Structures 1991, 39(5), 483-487.
24. P. Cornwell, S. W. Doebling and C. R. Farrar. Application of the Strain Energy Damage Detection Method to Plate-Like Structures. Journal of Sound and Vibration 1999, 224(2), 359-374.
25. J. Morlet, G. Arens, I. Fourgeau, and D. Giard. Wave Propagation and Sampling Theory. Geophysics 1982, 47, 203-236.
26. J. Morlet. Sampling Theory and Wave propagation. NATO ASI Series, Issues in Acoustic Signal / Image Processing and Recognition 1983, Vol. I, pp. 233-261.
27. Y. Meyer. Ondettes et Functions Splines, Lectures given at the University of Torino, Italy, 1986.
28. Y. Meyer. Ondettes, Functions Splines et Analyses Graduees, Seminaire EDP, Ecole Polytechnique, Paris, France, 1986.
29. S. G. Mallat. A Theory for Multiresolution Signal Decomposition, The Wave Representation. Comm. Pure Appl. Math 1988, 41, 674-693.
30. I. Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, 1992.
31. C. Surace and R. Ruotolo. Crack Detection of a Beam Using the Wavelet Transform. Proceedings of the 12th International Modal Analysis Conference 1994, Honolulu, 1141-1147.
32. W. J. Wang and P.D. McFadden. Application of Orthogonal Wavelets to Early Gear Damage Detection. Mechanical Systems and Signal Processing 1996, 9, 497-508.
33. S. T. Quek, Q. Wang, K. H. Ong and L. Zhang. Practical Issues in the Detection of Damage in Beams Using Wavelets. Smart Materials and Structures 2001, 10, 1009-1017.
34. D. U. Sung, C. G. Kim and C. S. Hong. Monitoring of Impact Damages in Composite Laminates Using Wavelet Transform. Composites 2002, part B 33, 35-43.
35. Y. J. Yan and L. H. Yam. Online Detection of Crack Damage in Composite Plates Using Embedded Piezoelectric Actuators/Sensors and Wavelet Analysis. Composite Structures 2002, 58, 29-38.
36. J. Hu and R. Y. Liang. An Integrated Approach to Detection of Cracks Using Vibration Characteristics. Journal of the Franklin Institute 1993, 330, 841-853.
37. R. Ruotolo and C. Surace. Damage Assessment of Multiple Cracked Beams, Numerical Result and Experimental Validation. Journal of Sound and Vibration 1997, 206(4), 359-374.
38. Rudolph Szilard, Theory and analysis of plates. New Jersey, Prentice-Hall, 1973.
39. B. A. H. Abbas. Dynamic analysis of thick rotating blades with flexible roots. Aeronaut. Jnl 1985, 10-16.