小波分析在结构损伤识别中的应用研究
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摘要
本文以美国ASCE健康监测的Benchmark模型为对象,利用小波变换分析方法探讨研究了目前结构在线实时损伤识别技术存在的若干基本问题,为结构在线实时损伤识别技术的实用化尝试提供了新的思路。主要工作如下:
首先,简要介绍了小波分析的基本理论,包括小波分析的发展、连续小波变换、离散小波变换以及小波变换的快速算法—Mallat算法。
其次,针对目前结构实时在线损伤识别研究存在的问题,在ASCE结构健康监测任务组编写的Benchmark模型动力响应计算程序的基础上,修改编制了适用于本文研究目的的数值模拟试验程序。1)修改程序使其可以计算在荷载作用过程中发生突然损伤体系的非线性动力响应,并通过对比分析数值模拟结果,发现王焕定教授提出的高阶单步β(Wang-β)法要明显优于目前最常用的Newmark-β法和Wilsion-θ法;2)改编程序以增加地震动输入下发生突然损伤benchmark模型的动力响应计算功能,使其可以考查不同荷载输入对小波变换损伤识别方法的影响;3)修正了Benchmark源程序中加噪声函数存在的无法准确控制噪声输入水平的问题。
大量的数值模拟试验结果表明,对响应加速度信号进行小波变换分析,通过细节信号“尖峰”在时间轴上的位置可以识别系统损伤的时刻,通过“尖峰”的高度可以识别结构损伤的位置。主要结论如下:
⑴在线的实时的损伤数值模拟中,结构响应计算应采用高精度的计算方法,如Wang-β法,以获取较高的频率信息。
⑵使用离散小波变换的方法进行损伤识别,选取ReverseBior小波效果较好。
⑶损伤程度越大,利用小波变换进行损伤识别所需的最低采样频率越小;损伤程度越小,所需的最低采样频率越大。在相同损伤程度下,风荷载作用所需的可识别最低频率,高于地震作用的可识别最低频率。推荐采样频率范围1000～2000Hz。
⑷“尖峰”的高度与荷载的强度、损伤程度以及测点(传感器的位置)距损伤位置的距离有关。随着荷载强度的增加,“尖峰”高度增加,随着荷载强度的减小,“尖峰”高度减小。随着损伤程度的增大,“尖峰”高度增大;反之,“尖峰”高度减小。随着测点距损伤位置距离的增加,“尖峰”高度减小;反之,“尖峰”高度增大。
⑸实测信号噪声的影响规律为:对于在低荷载强度激励下,微小损伤的识别要求传感器有非常高的精度。在测点离损伤位置很远的情况下,传感器自身的噪声可能将微小损伤信号的“尖峰”淹没;在较大损伤或较强的荷载激励下,噪声的影响则非常小。
Taking the ASCE health monitoring Benchmark model as the research object, several basic problems about the wavelet transform method used in real-time online damage identification for structures had been studied in this thesis. A tentative new approach is also developed for the application practice of the real-time online damage identification for structures. The main works are as follows:
Firstly, the wavelet analysis theory is introduced briefly, including the development history of wavelet analysis, continuous wavelet transform, discrete wavelet transform and the fast“Mallat algorithm”for wavelet analysis.
Secondly, for satisfying the research purpose of this thesis, a numerical simulation program has been developed to fix several critical problems in real-time online damage identification for structures. This program is modified and updated based on the simulation program developed by ASCE Task Group on Health Monitoring to simulate the dynamic response of Benchmark model. That is, the source program is modified and updated to be able to 1) calculate the nonlinear dynamic response of the Benchmark model with“sudden damage”during a certain loading excitation, in which numerical simulation results shows that the high order single-stepβ(Wang-β) method developed by professor Wang Huanding is obviously much accurate in calculation of the nonlinear response than most commonly used other two methods: Newmark-βmethod and Wilsion-θmethod; 2) calculate the dynamic response of the Benchmark model with“sudden-damage”characteristic under earthquake excitations to satisfy the purpose of examining the influence of different type of load excitation to the wavelet transform method used in structural damage identification; 3) rectify the error using of the noise adding function in the source program to provide accurate input noise level for examining the noise effects in wavelet analysis for damage identification.
Mounts of numerical simulation test results indicates that, through carrying on wavelet transform to the time history of the response acceleration, the accurate damage instant can be read from the appearing of the spike in the time axis as well as the damage location can be predicted from the height of related spikes.
Following conclusions are obtained through abovementioned studies:
(1) In numerical simulations of the real-time online damage identification for structures, high accuracy method such as Wang-βmethod should be used in order to obtain high frequency information from structural responses.
(2) The ReverseBior wavelet should be used as the mother wavelet since it’s much better than others when discrete wavelet transform is used for damage identification.
(3) Generally, the severer of the structural damage, the lowest identifiable sampling frequency will be relatively lower by using the wavelet transform. As well, the slighter of the structural damage, the lowest identifiable sampling frequency will be much higher. At the same damage level, the lowest identifiable sampling frequency under wind loading is relatively higher than that of under earthquake in damage identification simulation. Here, a reasonable sampling frequency range from 1000Hz to 2000Hz is suggested to be used in wavelet analysis.
(4) The height of the identified signal spike is strongly related with the load intensity, the severity of damage as well as the distance from the measuring point (sensor position) to the damage location. With the increase of the excitation intensity, the height of the spike increases; With the decrease of the load intensity, the height of the spike decreases; With the increase of severity of the structural damage, the height of the spike increases, Contrariwise, the spike height decreases; With the increase of the distance from the measuring point to the damage location, the height of the spike decreases, Contrariwise, the spike height increases.
(5) The noise effects of the measuring signal are: for identifying of the minus damage at a relatively lower excitation level, the sensors with very high precision are required. If the distance from the measuring point to the damage location is relatively large, the spike representing the minus damage might be submerged by the noise. If the damage is quite severe or the excitation level is quite large, the influence of the noise to the identified spike will be very slight.
首先,简要介绍了小波分析的基本理论,包括小波分析的发展、连续小波变换、离散小波变换以及小波变换的快速算法—Mallat算法。
其次,针对目前结构实时在线损伤识别研究存在的问题,在ASCE结构健康监测任务组编写的Benchmark模型动力响应计算程序的基础上,修改编制了适用于本文研究目的的数值模拟试验程序。1)修改程序使其可以计算在荷载作用过程中发生突然损伤体系的非线性动力响应,并通过对比分析数值模拟结果,发现王焕定教授提出的高阶单步β(Wang-β)法要明显优于目前最常用的Newmark-β法和Wilsion-θ法;2)改编程序以增加地震动输入下发生突然损伤benchmark模型的动力响应计算功能,使其可以考查不同荷载输入对小波变换损伤识别方法的影响;3)修正了Benchmark源程序中加噪声函数存在的无法准确控制噪声输入水平的问题。
大量的数值模拟试验结果表明,对响应加速度信号进行小波变换分析,通过细节信号“尖峰”在时间轴上的位置可以识别系统损伤的时刻,通过“尖峰”的高度可以识别结构损伤的位置。主要结论如下:
⑴在线的实时的损伤数值模拟中,结构响应计算应采用高精度的计算方法,如Wang-β法,以获取较高的频率信息。
⑵使用离散小波变换的方法进行损伤识别,选取ReverseBior小波效果较好。
⑶损伤程度越大,利用小波变换进行损伤识别所需的最低采样频率越小;损伤程度越小,所需的最低采样频率越大。在相同损伤程度下,风荷载作用所需的可识别最低频率,高于地震作用的可识别最低频率。推荐采样频率范围1000～2000Hz。
⑷“尖峰”的高度与荷载的强度、损伤程度以及测点(传感器的位置)距损伤位置的距离有关。随着荷载强度的增加,“尖峰”高度增加,随着荷载强度的减小,“尖峰”高度减小。随着损伤程度的增大,“尖峰”高度增大;反之,“尖峰”高度减小。随着测点距损伤位置距离的增加,“尖峰”高度减小;反之,“尖峰”高度增大。
⑸实测信号噪声的影响规律为:对于在低荷载强度激励下,微小损伤的识别要求传感器有非常高的精度。在测点离损伤位置很远的情况下,传感器自身的噪声可能将微小损伤信号的“尖峰”淹没;在较大损伤或较强的荷载激励下,噪声的影响则非常小。
Taking the ASCE health monitoring Benchmark model as the research object, several basic problems about the wavelet transform method used in real-time online damage identification for structures had been studied in this thesis. A tentative new approach is also developed for the application practice of the real-time online damage identification for structures. The main works are as follows:
Firstly, the wavelet analysis theory is introduced briefly, including the development history of wavelet analysis, continuous wavelet transform, discrete wavelet transform and the fast“Mallat algorithm”for wavelet analysis.
Secondly, for satisfying the research purpose of this thesis, a numerical simulation program has been developed to fix several critical problems in real-time online damage identification for structures. This program is modified and updated based on the simulation program developed by ASCE Task Group on Health Monitoring to simulate the dynamic response of Benchmark model. That is, the source program is modified and updated to be able to 1) calculate the nonlinear dynamic response of the Benchmark model with“sudden damage”during a certain loading excitation, in which numerical simulation results shows that the high order single-stepβ(Wang-β) method developed by professor Wang Huanding is obviously much accurate in calculation of the nonlinear response than most commonly used other two methods: Newmark-βmethod and Wilsion-θmethod; 2) calculate the dynamic response of the Benchmark model with“sudden-damage”characteristic under earthquake excitations to satisfy the purpose of examining the influence of different type of load excitation to the wavelet transform method used in structural damage identification; 3) rectify the error using of the noise adding function in the source program to provide accurate input noise level for examining the noise effects in wavelet analysis for damage identification.
Mounts of numerical simulation test results indicates that, through carrying on wavelet transform to the time history of the response acceleration, the accurate damage instant can be read from the appearing of the spike in the time axis as well as the damage location can be predicted from the height of related spikes.
Following conclusions are obtained through abovementioned studies:
(1) In numerical simulations of the real-time online damage identification for structures, high accuracy method such as Wang-βmethod should be used in order to obtain high frequency information from structural responses.
(2) The ReverseBior wavelet should be used as the mother wavelet since it’s much better than others when discrete wavelet transform is used for damage identification.
(3) Generally, the severer of the structural damage, the lowest identifiable sampling frequency will be relatively lower by using the wavelet transform. As well, the slighter of the structural damage, the lowest identifiable sampling frequency will be much higher. At the same damage level, the lowest identifiable sampling frequency under wind loading is relatively higher than that of under earthquake in damage identification simulation. Here, a reasonable sampling frequency range from 1000Hz to 2000Hz is suggested to be used in wavelet analysis.
(4) The height of the identified signal spike is strongly related with the load intensity, the severity of damage as well as the distance from the measuring point (sensor position) to the damage location. With the increase of the excitation intensity, the height of the spike increases; With the decrease of the load intensity, the height of the spike decreases; With the increase of severity of the structural damage, the height of the spike increases, Contrariwise, the spike height decreases; With the increase of the distance from the measuring point to the damage location, the height of the spike decreases, Contrariwise, the spike height increases.
(5) The noise effects of the measuring signal are: for identifying of the minus damage at a relatively lower excitation level, the sensors with very high precision are required. If the distance from the measuring point to the damage location is relatively large, the spike representing the minus damage might be submerged by the noise. If the damage is quite severe or the excitation level is quite large, the influence of the noise to the identified spike will be very slight.
引文
[1] 陈长征等. 结构损伤检测与智能诊断[M]. 北京:科学出版社,2001
[2] 飞思科技研发中心. 小波分析理论与 MATLAB7 实现[M]. 北京:电子工业出版社 2005
[3] 郭健,基于小波分析的结构损伤识别方法研究[D]. 杭州,浙江大学博士论文, 2004.
[4] 胡聿贤等,“地震工程学”,地震出版社,1989。
[5] 李宏男,孙鸿敏. 小波分析在土木工程领域中的应用[J]. 哈尔滨:地震工程与工程振动, 2003,19(2):16-22
[6] 李建平. 小波分析一与信号处理一理论、应用及软件实现[M]. 重庆:重庆出版社,1997
[7] 林宝龙. 小波变换在结构损伤识别中的应用研究[D]. 杭州,浙江大学硕士学位论文, 2006
[8] 欧进萍,重大工程的智能监测与健康诊断[J]. 工程力学,2002 增刊.
[9] 冉启文. 小波变换与分数傅里叶变换理论及应用[M]. 哈尔滨:哈尔滨工业大学出版社,2001
[10] 任伟新,韩建刚,孙增寿. 小波分析在土木工程结构中的应用[M]. 北京:中国铁道出版社,2006
[11] 孙鸿敏, 李宏男. 土木工程结构健康监测研究进展[J]. 防灾减灾工程学报, 2003,23(3):92-98。
[12] 闫桂荣. 基于广义柔度矩阵和小波分析的结构损伤识别方法[D]. 哈尔滨,哈尔滨工业大学博士学位论文,2006
[13] 叶蔚. 声发射源安全监测系统[J].武汉:江汉大学学报,第16卷第3期,1999:67-69
[14] 袁万城,崔飞,张启伟. 桥梁健康监测与状态评估的研究现状与发展[J]. 上海:同济大学学报,1999,27(2):184-188.
[15] Adriana Hera, Zhikun Hou. Application of wavelet analysis for structural health monitoring[J]. ASCE, Journal of Engineering Mechanics, 2004. 130(1): 96-104
[16] Al-khalidy A, Noori M, Hou Z, et al. Health monitoring systems of linear str-uctures using wavelet analysis[A]. Stanford: International Workshop on Structural Health Monitoring [C ]. 1997: 164-175.
[17] Al-khalidy A, Noori M, Hou Z, Carmona R, Yamamoto S, Masuda A, et al. A study of health monitoring systems of linear structures using wavelet analy-sis[J ] . A SMEP VP,1997,347:49一58.
[18] Amaravadi V, Rao V, Koval LR, et al. Structural health monitoring using wavelet transforms. Smart structures and integrated systems[A]. Proc
[19] Arata Masuda, Yuito Hashimoto and Akira Sone. Experimental Verification of Wavelet-Based Structural Damage Identification Method[C]. Proceedings of the 4th International Workshop on Structural Health Monitoring, Stanford University, Stanford, CA, 2003: 517-524
[20] D. E. Newland. Wavelet Analysis of Vibration. Part 1: Theory[J]. Journal of Vibration and Acoustics, 1994, 116: 409-416.
[21] G. W. Housner, L. A. Bbergman,T. K. Caughey et al. Structual control: Past, Present, Future[J]. ASCE, Journal of structural Engineering Mechanics, Vo1. 123, No. 9, 1997: 940～943.
[22] Gary G Yen, Kuo-Chang lin. Wavelet Packet Feature Extraction for Vibration Monitoring[J].IEEE Transactions on Industrial Electronics. 2000, 47(3): 650-667
[23] Han J G,Sun Z S,Ren W X. Wavelet based damage identification of beams [A]. In: Proceedings of the eighth international symposium on structural engineering for young experts[C].Beijing: sp, 2004: 56一363
[24] Hera, A., and Hou, Z. K. Wavelet-based approach for ASCE structural health monit- oring benchmark studies[C]. Proc., 3rd Int. Workshop on Structural Health Monitoring, Stanford Univ., Stanford, Calif. ~2001!.
[25] Hou, Z. K., and Noori, M. Application of wavelet analysis forstructural healthmonitoring.’’ Proc., 2nd Int. Workshop on StructuralHealth Monitoring, StanfordUniv., Stanford, Calif., 1999, 946–955.
[26] Hou, Z. K., Noori, M., and St. Amand, R. ~2000!. ‘‘Wavelet-based approach for structural damage detection.’’ J. Eng. Mech., 126~7!, 677–683.
[27] Hou, Z. K., and Hera, A. A system identification technique using pseudo-wavelets[J]. J. Intell. Mater. Syst. Struct. 2001, 12~10: 681–688.
[28] Hou, Z. K., and Hera, A. Progress of phase II study of the ASCE health monitoring benchmark data using wavelet approach[C]. Columbia Univ. New York: Proc., 15th ASCE Engineering Mechanics Conf., 2002. Control, Como, Italy, in press.
[29] Hou, Z. K., Yoshiyuki Suzuki, Y., Shimizu, H., and Hera, A. ~2002!. ‘‘Damage detection of a wooden house during shaking table testing using wavelet-based approach.’’ Proc., 3rd World Conf. on Structural
[30] Kim YY, Kim EH. A new damage deflection method based on a wavelet transf- orm[A]. In: Proceedings of the International Modal Analysis Conference [C]. Texas,SEM,2000,1207-1212
[31] Liew K M, Wang Q. Application of wavelet theory for crack identification Struct-ure[J]. Journal of Engineering Mechanics. 1998, 124(2): 152-157.
[32] Liew K M, Wang Q. Application of wavelet theory for crack identification Struct-ure[J]. Journal of Engineering Mechanics. 1998, 124(2): 152-157.
[33] Rytter A. Vibration based inspection of civil engineering structures: [D]. Aalborg, Denmark: Aalborg univ., 1993.
[34] Sohn et al. A review of structural health monitoring literature: 1996-2001[R].Los Alamos National Laboratory Report,LA-13976-MS,2002.
[35] Surace C, Ruotolo R. Crack detection of a beam using the wavelet transform [ C ]. Honolulu Hawaii: Proceedings the 12th International Modal Analysis Conference. 1994, 1141 ～ 1147.
[36] Wang Huanding, Zhang Yongshan and Wang Wei. A high order single step-β method for nonlinear structural dynamic analysis[J]. Journal of Harbin Institut e of Technology ( New Series) , Vol.10, No. 2 , 2003: 113-119.
[37] Wang Q, Wang D, Sun X. Crack detection of structure for plane problem with spatial wavelets[J].Acta MechS in,1999,15(1):39-51 SPIE[C]. Newport beach, 2001. 4327:258-269
[38] Wang WJ, McFadden P D. Application of orthogonal wavelets to early gear damage detection[J]. Mechanical Systems and Signal Processing, 1995, 9(5): 497-507.
[39] Y.J Wu and R. Du Feature Extraction And Assessment Using Wavelet Packets for Monitoring Machining Processes. Mechanical System and Signal Processi- ng[J].1996, 10(1): 29-53
[40] Y.J Yan and L.H. Yam. Online Detection of Crack Damage in Composite Plat es Using Embedded Piezoelectric Actuators/Sensors And Wavelet Analysis[J]. omposite Structures. 2002, 58:29-38
[41] Y.J Yan and L.H. Yam. Detection Of Delamination Damage in Composite Plate s Using Energy Spectrum of Structural Dynamics Responses Decomposed byW avelet Analysis[J]. Computers and Structures. 2004. 82: 347-358
[42] Z. Sun and C.C. Chang. Structural Damage Assessment Based on Wavelet Pa-c ket Transform. Journal ofStructural Engineering. 2002, 128(10):1354-1361
[2] 飞思科技研发中心. 小波分析理论与 MATLAB7 实现[M]. 北京:电子工业出版社 2005
[3] 郭健,基于小波分析的结构损伤识别方法研究[D]. 杭州,浙江大学博士论文, 2004.
[4] 胡聿贤等,“地震工程学”,地震出版社,1989。
[5] 李宏男,孙鸿敏. 小波分析在土木工程领域中的应用[J]. 哈尔滨:地震工程与工程振动, 2003,19(2):16-22
[6] 李建平. 小波分析一与信号处理一理论、应用及软件实现[M]. 重庆:重庆出版社,1997
[7] 林宝龙. 小波变换在结构损伤识别中的应用研究[D]. 杭州,浙江大学硕士学位论文, 2006
[8] 欧进萍,重大工程的智能监测与健康诊断[J]. 工程力学,2002 增刊.
[9] 冉启文. 小波变换与分数傅里叶变换理论及应用[M]. 哈尔滨:哈尔滨工业大学出版社,2001
[10] 任伟新,韩建刚,孙增寿. 小波分析在土木工程结构中的应用[M]. 北京:中国铁道出版社,2006
[11] 孙鸿敏, 李宏男. 土木工程结构健康监测研究进展[J]. 防灾减灾工程学报, 2003,23(3):92-98。
[12] 闫桂荣. 基于广义柔度矩阵和小波分析的结构损伤识别方法[D]. 哈尔滨,哈尔滨工业大学博士学位论文,2006
[13] 叶蔚. 声发射源安全监测系统[J].武汉:江汉大学学报,第16卷第3期,1999:67-69
[14] 袁万城,崔飞,张启伟. 桥梁健康监测与状态评估的研究现状与发展[J]. 上海:同济大学学报,1999,27(2):184-188.
[15] Adriana Hera, Zhikun Hou. Application of wavelet analysis for structural health monitoring[J]. ASCE, Journal of Engineering Mechanics, 2004. 130(1): 96-104
[16] Al-khalidy A, Noori M, Hou Z, et al. Health monitoring systems of linear str-uctures using wavelet analysis[A]. Stanford: International Workshop on Structural Health Monitoring [C ]. 1997: 164-175.
[17] Al-khalidy A, Noori M, Hou Z, Carmona R, Yamamoto S, Masuda A, et al. A study of health monitoring systems of linear structures using wavelet analy-sis[J ] . A SMEP VP,1997,347:49一58.
[18] Amaravadi V, Rao V, Koval LR, et al. Structural health monitoring using wavelet transforms. Smart structures and integrated systems[A]. Proc
[19] Arata Masuda, Yuito Hashimoto and Akira Sone. Experimental Verification of Wavelet-Based Structural Damage Identification Method[C]. Proceedings of the 4th International Workshop on Structural Health Monitoring, Stanford University, Stanford, CA, 2003: 517-524
[20] D. E. Newland. Wavelet Analysis of Vibration. Part 1: Theory[J]. Journal of Vibration and Acoustics, 1994, 116: 409-416.
[21] G. W. Housner, L. A. Bbergman,T. K. Caughey et al. Structual control: Past, Present, Future[J]. ASCE, Journal of structural Engineering Mechanics, Vo1. 123, No. 9, 1997: 940～943.
[22] Gary G Yen, Kuo-Chang lin. Wavelet Packet Feature Extraction for Vibration Monitoring[J].IEEE Transactions on Industrial Electronics. 2000, 47(3): 650-667
[23] Han J G,Sun Z S,Ren W X. Wavelet based damage identification of beams [A]. In: Proceedings of the eighth international symposium on structural engineering for young experts[C].Beijing: sp, 2004: 56一363
[24] Hera, A., and Hou, Z. K. Wavelet-based approach for ASCE structural health monit- oring benchmark studies[C]. Proc., 3rd Int. Workshop on Structural Health Monitoring, Stanford Univ., Stanford, Calif. ~2001!.
[25] Hou, Z. K., and Noori, M. Application of wavelet analysis forstructural healthmonitoring.’’ Proc., 2nd Int. Workshop on StructuralHealth Monitoring, StanfordUniv., Stanford, Calif., 1999, 946–955.
[26] Hou, Z. K., Noori, M., and St. Amand, R. ~2000!. ‘‘Wavelet-based approach for structural damage detection.’’ J. Eng. Mech., 126~7!, 677–683.
[27] Hou, Z. K., and Hera, A. A system identification technique using pseudo-wavelets[J]. J. Intell. Mater. Syst. Struct. 2001, 12~10: 681–688.
[28] Hou, Z. K., and Hera, A. Progress of phase II study of the ASCE health monitoring benchmark data using wavelet approach[C]. Columbia Univ. New York: Proc., 15th ASCE Engineering Mechanics Conf., 2002. Control, Como, Italy, in press.
[29] Hou, Z. K., Yoshiyuki Suzuki, Y., Shimizu, H., and Hera, A. ~2002!. ‘‘Damage detection of a wooden house during shaking table testing using wavelet-based approach.’’ Proc., 3rd World Conf. on Structural
[30] Kim YY, Kim EH. A new damage deflection method based on a wavelet transf- orm[A]. In: Proceedings of the International Modal Analysis Conference [C]. Texas,SEM,2000,1207-1212
[31] Liew K M, Wang Q. Application of wavelet theory for crack identification Struct-ure[J]. Journal of Engineering Mechanics. 1998, 124(2): 152-157.
[32] Liew K M, Wang Q. Application of wavelet theory for crack identification Struct-ure[J]. Journal of Engineering Mechanics. 1998, 124(2): 152-157.
[33] Rytter A. Vibration based inspection of civil engineering structures: [D]. Aalborg, Denmark: Aalborg univ., 1993.
[34] Sohn et al. A review of structural health monitoring literature: 1996-2001[R].Los Alamos National Laboratory Report,LA-13976-MS,2002.
[35] Surace C, Ruotolo R. Crack detection of a beam using the wavelet transform [ C ]. Honolulu Hawaii: Proceedings the 12th International Modal Analysis Conference. 1994, 1141 ～ 1147.
[36] Wang Huanding, Zhang Yongshan and Wang Wei. A high order single step-β method for nonlinear structural dynamic analysis[J]. Journal of Harbin Institut e of Technology ( New Series) , Vol.10, No. 2 , 2003: 113-119.
[37] Wang Q, Wang D, Sun X. Crack detection of structure for plane problem with spatial wavelets[J].Acta MechS in,1999,15(1):39-51 SPIE[C]. Newport beach, 2001. 4327:258-269
[38] Wang WJ, McFadden P D. Application of orthogonal wavelets to early gear damage detection[J]. Mechanical Systems and Signal Processing, 1995, 9(5): 497-507.
[39] Y.J Wu and R. Du Feature Extraction And Assessment Using Wavelet Packets for Monitoring Machining Processes. Mechanical System and Signal Processi- ng[J].1996, 10(1): 29-53
[40] Y.J Yan and L.H. Yam. Online Detection of Crack Damage in Composite Plat es Using Embedded Piezoelectric Actuators/Sensors And Wavelet Analysis[J]. omposite Structures. 2002, 58:29-38
[41] Y.J Yan and L.H. Yam. Detection Of Delamination Damage in Composite Plate s Using Energy Spectrum of Structural Dynamics Responses Decomposed byW avelet Analysis[J]. Computers and Structures. 2004. 82: 347-358
[42] Z. Sun and C.C. Chang. Structural Damage Assessment Based on Wavelet Pa-c ket Transform. Journal ofStructural Engineering. 2002, 128(10):1354-1361