基于折线包络的经验模态分解方法及其应用
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摘要
Hilbert-Huang变换(HHT)是一种新的非平稳信号处理方法,它由经验模态分解(EMD)与Hilbert谱分析(HSA)两部分组成。HHT方法自提出后已经在地震动工程特性的研究及结构系统识别等领域得到了应用。本文即围绕HHT方法及其应用展开研究工作。
作为对HHT方法的研究,针对传统的经验模态分解方法在处理长周期信号时所面临的问题,本文提出了一种分解效率更高的基于折线包络的EMD方法,利用具有明确解析表达式的函数验证了该方法所得结果的数学意义,并且讨论了该方法分解地震波所得本征模态分量(IMF)的波形与频谱特征。
为了深入验证该方法所得结果的物理意义,本文将所提出的基于折线包络的经验模态分解方法应用于分析线性时不变和线性时变单自由体系(SDOF)的动力响应,并对其动力特性进行识别。其研究结果表明:在规则输入下,基于折线包络EMD方法能够将体系动力反应中的稳态反应与瞬态反应部分分解出来:稳态反应部分能够提供有关输入频谱特性方面的信息;瞬态反应部分则能够提供有关体系动力特性方面的信息。这部分研究不仅深入验证了基于折线包络的经验模态分解方法应用于基本的结构动力学问题所得结果的物理意义,而且也为该方法在结构系统识别领域的深入应用提供一定的基础。
最后,作为此基于折线包络EMD方法在地震动工程特性研究中的应用,本文利用该方法分析了1999年台湾Chi-Chi地震中记录到的含速度脉冲的近断层地震记录,从中识别出了加速度脉冲,并利用反应谱讨论了加速度脉冲对近断层脉冲型地震动工程特性的贡献。
Hilbert-Huang transform (HHT) is a new processing method for non-stationary signal, which comprises Empirical Mode Decomposition (EMD) and Hilbert Spectral Analysis (HSA). After its introduction, HHT method has found its applications in the fields of the study of engineering properties of ground motion and structural system identification, ect. And it is just the method of HHT and its applications that the studies in this paper are centered around.
As the study of the HHT method itself, aiming at the problem confronted by the traditional EMD method when processing long-period signals, a more efficient folding-line-envelope (FLE) based EMD method is proposed in this paper. By the function with definite analytical expression, the mathematical significance of the results obtained by the proposed method is demonstrated, and the characteristics of waveforms and spectra of the intrinsic mode function (IMF) resulted from decomposing the seismic wave by the method are also discussed.
In order to validate in depth the physical significance of the results obtained by the proposed method, the FLE-based EMD is applied in analyzing the dynamic responses of linear time-invariant and linear time-variant single-degree-of-freedom (SDOF) systems, whose dynamic properties are identified meanwhile. The results show that, under the regular input, the FLE-based EMD can decompose the stationary part and the transient part from the system dynamic response, where the stationary part can provide the information on the spectral properties of input and the transient one can provide the information on the dynamic properties of system. This study not only demonstrates deeply the physical significance of the results obtained by applying the FLE-based EMD in the basic structural dynamic issues, but also offers to some extent the foundation for the in-depth application of this method in the field of structural system identification.
At last, as the application of the FLE-based EMD in the study of the engineering properties of ground motion, the method is used to analyze the near-fault seismograms recorded in the Chi-Chi, Taiwan, earthquake in 1999, which contain velocity pulses. And the acceleration pulses are identified, whose contributions to the engineering properties of near-fault pulse-like ground motions are discussed using the response spectra.
作为对HHT方法的研究,针对传统的经验模态分解方法在处理长周期信号时所面临的问题,本文提出了一种分解效率更高的基于折线包络的EMD方法,利用具有明确解析表达式的函数验证了该方法所得结果的数学意义,并且讨论了该方法分解地震波所得本征模态分量(IMF)的波形与频谱特征。
为了深入验证该方法所得结果的物理意义,本文将所提出的基于折线包络的经验模态分解方法应用于分析线性时不变和线性时变单自由体系(SDOF)的动力响应,并对其动力特性进行识别。其研究结果表明:在规则输入下,基于折线包络EMD方法能够将体系动力反应中的稳态反应与瞬态反应部分分解出来:稳态反应部分能够提供有关输入频谱特性方面的信息;瞬态反应部分则能够提供有关体系动力特性方面的信息。这部分研究不仅深入验证了基于折线包络的经验模态分解方法应用于基本的结构动力学问题所得结果的物理意义,而且也为该方法在结构系统识别领域的深入应用提供一定的基础。
最后,作为此基于折线包络EMD方法在地震动工程特性研究中的应用,本文利用该方法分析了1999年台湾Chi-Chi地震中记录到的含速度脉冲的近断层地震记录,从中识别出了加速度脉冲,并利用反应谱讨论了加速度脉冲对近断层脉冲型地震动工程特性的贡献。
Hilbert-Huang transform (HHT) is a new processing method for non-stationary signal, which comprises Empirical Mode Decomposition (EMD) and Hilbert Spectral Analysis (HSA). After its introduction, HHT method has found its applications in the fields of the study of engineering properties of ground motion and structural system identification, ect. And it is just the method of HHT and its applications that the studies in this paper are centered around.
As the study of the HHT method itself, aiming at the problem confronted by the traditional EMD method when processing long-period signals, a more efficient folding-line-envelope (FLE) based EMD method is proposed in this paper. By the function with definite analytical expression, the mathematical significance of the results obtained by the proposed method is demonstrated, and the characteristics of waveforms and spectra of the intrinsic mode function (IMF) resulted from decomposing the seismic wave by the method are also discussed.
In order to validate in depth the physical significance of the results obtained by the proposed method, the FLE-based EMD is applied in analyzing the dynamic responses of linear time-invariant and linear time-variant single-degree-of-freedom (SDOF) systems, whose dynamic properties are identified meanwhile. The results show that, under the regular input, the FLE-based EMD can decompose the stationary part and the transient part from the system dynamic response, where the stationary part can provide the information on the spectral properties of input and the transient one can provide the information on the dynamic properties of system. This study not only demonstrates deeply the physical significance of the results obtained by applying the FLE-based EMD in the basic structural dynamic issues, but also offers to some extent the foundation for the in-depth application of this method in the field of structural system identification.
At last, as the application of the FLE-based EMD in the study of the engineering properties of ground motion, the method is used to analyze the near-fault seismograms recorded in the Chi-Chi, Taiwan, earthquake in 1999, which contain velocity pulses. And the acceleration pulses are identified, whose contributions to the engineering properties of near-fault pulse-like ground motions are discussed using the response spectra.
引文
[1]胡聿贤(1988).地震工程学.地震出版社,北京
[2] Kostadinov MV and Yamazaki F. Detection of Soil Liquefaction from Strong Motion Records. Earthq Engrg Struct Dyn, 2001, 30: 173~193
[3] Wang JJ, Fan LC, Qian S, et al. Simulations of Non-stationary Frequency Content and Its Importance to Seismic Assessment of Structures. Earthq Engrg Struct Dyn, 2002, 31: 993~1005
[4] Iyama J and Kuwamura H. Application of Wavelets to Analysis and Simulation of Earthquake Motions. Earthq Engrg Struct Dyn, 1999, 28: 255~272
[5] Mukherjee S and Gupta VK. Wavelet-based Characterization of Design Ground Motions. Earthq Engrg Struct Dyn, 2002, 31: 1173~1190
[6] Liu SC. Evolutionary Power Spectral Density of Strong Earthquake. Bull Seism Soc Am, 1970, 60: 891~900
[7] Udwadia FE and Trifunac MD. Time and Amplitude Dependent Response of Structures. Earthq Engrg Struct Dyn, 1974, 2(4): 359-378
[8] Kitada Y. Identification of Nonlinear Structural Dynamic Systems Using Wavelets. J Engrg Mech, ASCE, 1998, 124(10): 1059-1066
[9] Huang NE, Shen Z, Long SR, et al. The Empirical Mode Decomposition and Hilbert Spectrum for Nonlinear and Nonstationary Time Series Analysis. Proc Roy Soc Lond, 1998, A(454): 903-995
[10] Huang NE, et al. A new spectral representation of earthquake data: Hilbert spectral analysis of station TCU129, Chi-Chi, Taiwan, 21, September 1999. Bull Soc Seism Am, 2001, 91(5): 1310-1338
[11] Zhang RR, Ma S, Safak E, et al. HHT Analysis of Earthquake Recordings. J Engrg Mech, ASCE, 2003, 129(8): 861-875
[12] Loh CH, Wu TC, and Huang NE. Application of the Empirical Mode Decomposition– Hilbert Spectrum Method to Identify Near-fault Ground- motion Characteristics and Structural Responses. Bull Soc Seism Am, 2001, 91(5): 1339-1357
[13] Yang JN, Lei Y, Pan SW, et al. System Identification of Linear Structures Based on Hilbert-Huang Spectral Analysis. Part 1: Normal Modes. Earthq Engrg Struct Dyn, 2003, 32: 1443-1467
[14] Yang JN, Lei Y, Pan SW, et al. System Identification of Linear Structures Based on Hilbert-Huang Spectral Analysis. Part 2: Complex Modes. Earthq Engrg Struct Dyn, 2003, 32: 1533-1554
[15] Salvino LW. Empirical Mode Analysis of Structural Response and Damping. Proc the 18th Inter Modal Analysis Conf, San Antonio, TX, 2000: 503-509
[16] Pines and Salvino LW. Health Monitoring of Structures Using Empirical Mode Decomposition and Phase Dereverberation. Proc the 2001 Mech and Materials Summer Conf. UCSD: San Diego, CA, 2001: 228-229
[17] Yang JN, Lei Y, Lin S, et al. Hilbert-Huang Based Approach for Structural Damage Detection. J Engrg Mech, ASCE, 2003, 130(1): 85-95
[18] Zhang YS, Liang JW and Hu YX. Hilbert spectrum and intrinsic oscillation mode of dynamic response of a bilinear SDOF system: influence of harmonic excitation amplitude. Earthq Engrg Engrg Vibr, 2005, 4(1): 17-26
[19]张郁山,梁建文,胡聿贤.双线性单自由度体系强迫动力反应的Hilbert谱与本征振动模态:输入简谐波频率的影响.地震工程与工程振动, 2005, 25(4): 1-10
[20]张郁山.希尔伯特—黄变换(HHT)与地震动时程的希尔伯特谱——方法与应用研究.中国地震局地球物理研究所博士研究生学位论文,北京, 2003
[21]胡聿贤,张郁山,梁建文.应用HHT方法从竖向地震动记录识别场地液化的物理过程.地震工程与工程振动, 2004, 24(3): 1-11
[22] Hu YX, Zhang YS, Liang JW, et al. Recording-based Identification of Site Liquefaction. Earthq Engrg Engrg Vibr, 2005, 4(2): 181-189
[23]胡聿贤,张郁山,梁建文.基于HHT方法的场地液化的识别.土木工程学报, 2006, 39(2): 66-72
[24] Huang NE, Shen Z, and Long RS. A New View of Nonlinear Water Waves– Hilbert Spectrum. Ann Rev Fluid Mech, 1999, 31: 417-457
[25]邓拥军,王伟,钱成春等. EMD方法及Hilbert变换中边界问题的处理.科学通报, 2001, 46(3): 257-263
[26]张郁山,梁建文,胡聿贤.应用自回归模型处理EMD方法中的边界问题.自然科学进展, 2003, 13(10): 1054-1059
[27] Cohen L. Time-Frequency Analysis: Theory and Applications. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1995
[28] Hann SJ. Hilbert Transforms in Signal Processing. Artech House: Boston, 1996
[29] Gabor D. Theory of Communication. Proc IEEE, 1946, 93: 429-457
[30] Bedrosian E. A Product Theorem for Hilbert Transform. Proc IEEE, 1963, 51: 868-869
[31] Boashash B. Estimating and Interpreting the Instantaneous Frequency of a Signal.Ⅰ. Fundamentals. Proc IEEE, 1992, 80: 520-538
[32] Feldman M. Investigation of the Natural Vibrations of Machine Elements Using the Hilbert Transform. Soviet Machine Science, 1985, 2: 44-47
[33] Feldman M. Non-linear System Vibration Analysis Using Hilbert Transform―Ⅰ: Free Vibration Analysis Method FREEVIB. Mechanical Systems and Signal Processing, 1994, 8(2): 119-127
[34] Feldman M. Nonlinear Free-vibration Identification via the Hilbert Transform. J Sound Vibr, 1997, 208(3): 475-489
[35] Aki K. Seismic displacements near a fault. J Geophys Res, 1968, 73(16): 5359-5376
[36] Bolt BA. The contribution of directivity focusing to earthquake intensities. Report No. 20, U.S. Army Corps of Engineers, Vicksburg, MS, 1983
[37] Singh JP. Earthquake ground motions: Implications for designing structures and reconciling structural damage. Earthquake Spectra, 1985, 1(2): 239-270
[38] Iwan WD. Drift spectrum: measure of demand for earthquake ground motions. J Struct Engng, ASCE, 1997, 123(4): 397-404
[39] Mavroedis GP, Dong G, and Papageorgiou AS. Near-fault ground motions, and the response of elastic and inelastic single-degree-of-freedom (SDOF) systems. Earthq Engng. Struct. Dyn., 2004, 33: 1023-1049
[40] Someville P. Development of an improved representation of near-fault ground motions. SMIP98 Seminar on Utilization of Strong-Motion Data (Sept. 15), Oakland, Calif., 1998
[41] Mavroedis GP and Papageorgiou AS. A mathematical representation of near-fault ground motions. Bull Seism Soc Am, 2003, 93: 1099-1131
[42] Bertero VV. Establishment of design earthquakes― Evaluation of present methods. Proc Int Symp on Earthq Struct Engrg, St. Louis, 1976, 1, 551-580
[43] Makris N and Black CJ. Evaluation of peak ground velocity as a“good”intensity measure for near-source ground motions. J Struct Engng, ASCE, 2004, 130(9): 1032-1044
[44] Jennings PC. Ground motion pulses and structural response. Earthquake Engineering Frontiers in the New Millennium, Spencer & Hu (eds), Swets & Zeitlinger, 2001
[45] Dai JW, Mai T, Lee GC, Qi XZ, et al.. Dynamic responses under the excitation pulse sequences. Earthq Engng Engng Vibr, 2004, 3(2):157-170
[46] Zhang YS, Hu YX, Zhao FX, Liang JW, and Yang CH. Identification of acceleration pulses in near-fault ground motions by the method of EMD. Earthq Engrg Engrg Vibr, 2005, 4(2): 201-212
[2] Kostadinov MV and Yamazaki F. Detection of Soil Liquefaction from Strong Motion Records. Earthq Engrg Struct Dyn, 2001, 30: 173~193
[3] Wang JJ, Fan LC, Qian S, et al. Simulations of Non-stationary Frequency Content and Its Importance to Seismic Assessment of Structures. Earthq Engrg Struct Dyn, 2002, 31: 993~1005
[4] Iyama J and Kuwamura H. Application of Wavelets to Analysis and Simulation of Earthquake Motions. Earthq Engrg Struct Dyn, 1999, 28: 255~272
[5] Mukherjee S and Gupta VK. Wavelet-based Characterization of Design Ground Motions. Earthq Engrg Struct Dyn, 2002, 31: 1173~1190
[6] Liu SC. Evolutionary Power Spectral Density of Strong Earthquake. Bull Seism Soc Am, 1970, 60: 891~900
[7] Udwadia FE and Trifunac MD. Time and Amplitude Dependent Response of Structures. Earthq Engrg Struct Dyn, 1974, 2(4): 359-378
[8] Kitada Y. Identification of Nonlinear Structural Dynamic Systems Using Wavelets. J Engrg Mech, ASCE, 1998, 124(10): 1059-1066
[9] Huang NE, Shen Z, Long SR, et al. The Empirical Mode Decomposition and Hilbert Spectrum for Nonlinear and Nonstationary Time Series Analysis. Proc Roy Soc Lond, 1998, A(454): 903-995
[10] Huang NE, et al. A new spectral representation of earthquake data: Hilbert spectral analysis of station TCU129, Chi-Chi, Taiwan, 21, September 1999. Bull Soc Seism Am, 2001, 91(5): 1310-1338
[11] Zhang RR, Ma S, Safak E, et al. HHT Analysis of Earthquake Recordings. J Engrg Mech, ASCE, 2003, 129(8): 861-875
[12] Loh CH, Wu TC, and Huang NE. Application of the Empirical Mode Decomposition– Hilbert Spectrum Method to Identify Near-fault Ground- motion Characteristics and Structural Responses. Bull Soc Seism Am, 2001, 91(5): 1339-1357
[13] Yang JN, Lei Y, Pan SW, et al. System Identification of Linear Structures Based on Hilbert-Huang Spectral Analysis. Part 1: Normal Modes. Earthq Engrg Struct Dyn, 2003, 32: 1443-1467
[14] Yang JN, Lei Y, Pan SW, et al. System Identification of Linear Structures Based on Hilbert-Huang Spectral Analysis. Part 2: Complex Modes. Earthq Engrg Struct Dyn, 2003, 32: 1533-1554
[15] Salvino LW. Empirical Mode Analysis of Structural Response and Damping. Proc the 18th Inter Modal Analysis Conf, San Antonio, TX, 2000: 503-509
[16] Pines and Salvino LW. Health Monitoring of Structures Using Empirical Mode Decomposition and Phase Dereverberation. Proc the 2001 Mech and Materials Summer Conf. UCSD: San Diego, CA, 2001: 228-229
[17] Yang JN, Lei Y, Lin S, et al. Hilbert-Huang Based Approach for Structural Damage Detection. J Engrg Mech, ASCE, 2003, 130(1): 85-95
[18] Zhang YS, Liang JW and Hu YX. Hilbert spectrum and intrinsic oscillation mode of dynamic response of a bilinear SDOF system: influence of harmonic excitation amplitude. Earthq Engrg Engrg Vibr, 2005, 4(1): 17-26
[19]张郁山,梁建文,胡聿贤.双线性单自由度体系强迫动力反应的Hilbert谱与本征振动模态:输入简谐波频率的影响.地震工程与工程振动, 2005, 25(4): 1-10
[20]张郁山.希尔伯特—黄变换(HHT)与地震动时程的希尔伯特谱——方法与应用研究.中国地震局地球物理研究所博士研究生学位论文,北京, 2003
[21]胡聿贤,张郁山,梁建文.应用HHT方法从竖向地震动记录识别场地液化的物理过程.地震工程与工程振动, 2004, 24(3): 1-11
[22] Hu YX, Zhang YS, Liang JW, et al. Recording-based Identification of Site Liquefaction. Earthq Engrg Engrg Vibr, 2005, 4(2): 181-189
[23]胡聿贤,张郁山,梁建文.基于HHT方法的场地液化的识别.土木工程学报, 2006, 39(2): 66-72
[24] Huang NE, Shen Z, and Long RS. A New View of Nonlinear Water Waves– Hilbert Spectrum. Ann Rev Fluid Mech, 1999, 31: 417-457
[25]邓拥军,王伟,钱成春等. EMD方法及Hilbert变换中边界问题的处理.科学通报, 2001, 46(3): 257-263
[26]张郁山,梁建文,胡聿贤.应用自回归模型处理EMD方法中的边界问题.自然科学进展, 2003, 13(10): 1054-1059
[27] Cohen L. Time-Frequency Analysis: Theory and Applications. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1995
[28] Hann SJ. Hilbert Transforms in Signal Processing. Artech House: Boston, 1996
[29] Gabor D. Theory of Communication. Proc IEEE, 1946, 93: 429-457
[30] Bedrosian E. A Product Theorem for Hilbert Transform. Proc IEEE, 1963, 51: 868-869
[31] Boashash B. Estimating and Interpreting the Instantaneous Frequency of a Signal.Ⅰ. Fundamentals. Proc IEEE, 1992, 80: 520-538
[32] Feldman M. Investigation of the Natural Vibrations of Machine Elements Using the Hilbert Transform. Soviet Machine Science, 1985, 2: 44-47
[33] Feldman M. Non-linear System Vibration Analysis Using Hilbert Transform―Ⅰ: Free Vibration Analysis Method FREEVIB. Mechanical Systems and Signal Processing, 1994, 8(2): 119-127
[34] Feldman M. Nonlinear Free-vibration Identification via the Hilbert Transform. J Sound Vibr, 1997, 208(3): 475-489
[35] Aki K. Seismic displacements near a fault. J Geophys Res, 1968, 73(16): 5359-5376
[36] Bolt BA. The contribution of directivity focusing to earthquake intensities. Report No. 20, U.S. Army Corps of Engineers, Vicksburg, MS, 1983
[37] Singh JP. Earthquake ground motions: Implications for designing structures and reconciling structural damage. Earthquake Spectra, 1985, 1(2): 239-270
[38] Iwan WD. Drift spectrum: measure of demand for earthquake ground motions. J Struct Engng, ASCE, 1997, 123(4): 397-404
[39] Mavroedis GP, Dong G, and Papageorgiou AS. Near-fault ground motions, and the response of elastic and inelastic single-degree-of-freedom (SDOF) systems. Earthq Engng. Struct. Dyn., 2004, 33: 1023-1049
[40] Someville P. Development of an improved representation of near-fault ground motions. SMIP98 Seminar on Utilization of Strong-Motion Data (Sept. 15), Oakland, Calif., 1998
[41] Mavroedis GP and Papageorgiou AS. A mathematical representation of near-fault ground motions. Bull Seism Soc Am, 2003, 93: 1099-1131
[42] Bertero VV. Establishment of design earthquakes― Evaluation of present methods. Proc Int Symp on Earthq Struct Engrg, St. Louis, 1976, 1, 551-580
[43] Makris N and Black CJ. Evaluation of peak ground velocity as a“good”intensity measure for near-source ground motions. J Struct Engng, ASCE, 2004, 130(9): 1032-1044
[44] Jennings PC. Ground motion pulses and structural response. Earthquake Engineering Frontiers in the New Millennium, Spencer & Hu (eds), Swets & Zeitlinger, 2001
[45] Dai JW, Mai T, Lee GC, Qi XZ, et al.. Dynamic responses under the excitation pulse sequences. Earthq Engng Engng Vibr, 2004, 3(2):157-170
[46] Zhang YS, Hu YX, Zhao FX, Liang JW, and Yang CH. Identification of acceleration pulses in near-fault ground motions by the method of EMD. Earthq Engrg Engrg Vibr, 2005, 4(2): 201-212