基于HHT的全非平稳地震动降维建模

Dimension-reduction modeling of fully non-stationary ground motion based on HHT

摘要

地震动的瞬时频率可能会引发工程结构的瞬时共振，进而加剧结构的动力响应，严重威胁结构安全。因此，合理考虑地震动的时-频全非平稳特性对于工程结构抗震设计具有重要意义。希尔伯特-黄变换(Hilbert-Huang transform，HHT)在描述信号的时-频全非平稳特性方面具有显著优势，本文将其引入到谱表示-降维模拟方法中，以更好地刻画地震动过程在时间和强度上的非平稳特性。首先，基于能量平衡的理念，利用已有的演变功率谱模型构建地震动的Hilbert时频谱模型，并对其关键参数进行了统计建模。然后，在非平稳地震动随机过程的源谱表达式中引入Hilbert谱，并将随机变量定义为随机正交函数的约束形式，实现了仅用一个基本随机变量便可精确模拟全非平稳地震动过程。数值算例表明，运用本文方法生成的地震动代表性样本能够较为精细地刻画地震动的时-频非平稳特征，其反应谱和幅值谱与实测强震记录拟合较好，验证了该方法的有效性与工程适用性。

Abstract

Instantaneous frequency of seismic motion can induce instantaneous resonance in engineering structures, thereby leading to more severe dynamic responses and posing significant threats to structural safety. Therefore, it is crucial for seismic design of engineering structures to appropriately consider the fully non-stationary characteristics of seismic motion in both time and frequency. The Hilbert-Huang transform (HHT) is notably effective in describing the fully non-stationary nature of signals and is thus incorporated into the spectral representation method based on dimension-reduction method to better characterize the non-stationary characteristics of earthquake motion processes in terms of time and intensity. Initially, a Hilbert spectral model of seismic motion is constructed using the existing evolutionary power spectral density model based on the concept of energy balance, and its key parameters are statistically modeled. Subsequently, the Hilbert spectrum is incorporated into the original spectrum representation of non-stationary seismic random processes, where the random variables being defined in a constrained form as random orthogonal functions. This enables the precise simulation of the fully non-stationary seismic process using just one elementary random variable. Numerical examples demonstrate that the representative samples of ground motions generated using the method presented in this paper can meticulously capture the time-frequency non-stationary characteristics of earthquakes. The response spectrum and amplitude spectrum of these samples match well with the strong motion records, thereby validating the effectiveness and engineering applicability of the proposed method.

关键词

全非平稳地震动 / 希尔伯特-黄变换 / 参数识别 / 演变功率谱 / 降维模拟

Key words

fully non-stationary ground motion / Hilbert-Huang transform / parameter identification / evolutionary power spectrum / dimension-reduction simulation

参考文献

[1] 王宏伟, 任叶飞, 温瑞智 . 一种随机有限断层的三维地震动模拟方法: 鲁甸地震为例[J]. 地震工程与工程振动, 2021, 41(2): 181-191.

[2] WANG Hongwei, REN Yefei, WEN Ruizhi . A modified stochastic finite-fault method for simulating ground motions in three dimensions: A case study of Ludian earthquake[J]. Earthquake Engineering and Engineering Dynamics, 2021, 41(2): 181-191. (in Chinese)

[3] 李英民, 刘立平 . 工程结构的设计地震动[M]. 北京: 科学出版社, 2011: 22-30.

[4] LI Yingmin, LIU Liping . Design ground motion for engineering structures[M]. Beijing: Science Press, 2011: 22-30. (in Chinese)

[5] YEH C H, WEN Y K . Modeling of nonstationary ground motion and analysis of inelastic structural response[J]. Structural Safety, 1990, 8(1/2/3/4): 281-298.

[6] 王君杰, 周晶 . 地震动频谱非平稳性对结构非线性反应的影响[J]. 地震工程与工程振动, 1997, 17(2): 16-20.

[7] WANG Junjie, ZHOU Jing . Effects of frequency nonstationarity of seismic ground motion on inelastic response of structures[J]. Earthquake Engineering and Engineering Dynamics, 1997, 17(2): 16-20. (in Chinese)

[8] 张翠然, 陈厚群, 涂劲 . 频率非平稳对大岗山拱坝非线性响应的影响[J]. 水力发电学报, 2012, 31(1): 77-81.

[9] ZHANG Cuiran, CHEN Houqun, TU Jin . Influence of frequency non-stationary earthquake accelerograms on the nonlinear responses of Dagangshan arch dam[J]. Journal of Hydroelectric Engineering, 2012, 31(1): 77-81. (in Chinese)

[10] 陈辉国, 李英民 . 基于非均匀调制模型的完全非平稳多点地震动特性分析与模拟[J]. 西安建筑科技大学学报(自然科学版), 2015, 47(4): 523-530.

[11] CHEN Huiguo, LI Yingmin . Characteristics analysis and simulation of non-stationary and spatially variable ground motion based on a non-uniform modulation model[J]. Journal of Xi’an University of Architecture & Technology (Natural Science Edition), 2015, 47(4): 523-530. (in Chinese)

[12] 赵凤新, 胡聿贤 . 地震动非平稳性与幅值谱和相位差谱的关系[J]. 地震工程与工程振动, 1994, 14(2): 1-6.

[13] ZHAO Fengxin, HU Yuxian . On the relationship of earthquake ground motion’s non-stationarity with its amplitude spectrum and phase difference spectrum[J]. Earthquake Engineering and Engineering Dynamics, 1994, 14(2): 1-6. (in Chinese)

[14] HUANG N E, SHEN Z, LONG S R, et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis[J]. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 1998, 454(1971): 903-995.

[15] 吴琛, 周瑞忠 . Hilbert-Huang变换在提取地震信号动力特性中的应用[J]. 地震工程与工程振动, 2006, 26(5): 41-46.

[16] WU Chen, ZHOU Ruizhong . Application of Hilbert-Huang transform in extracting dynamic properties of seismic signals[J]. Earthquake Engineering and Engineering Dynamics, 2006, 26(5): 41-46. (in Chinese)

[17] 吴昊, 张洵安 . 基于HHT方法的非平稳人工地震动模拟[J]. 地震工程与工程振动, 2011, 31(6): 30-37.

[18] WU Hao, ZHANG Xun’an . Simulation of non-stationary artificial seismic ground motions based on HHT theory[J]. Earthquake Engineering and Engineering Dynamics, 2011, 31(6): 30-37. (in Chinese)

[19] WEN Y K, GU P . Description and simulation of nonstationary processes based on Hilbert spectra[J]. Journal of Engineering Mechanics, 2004, 130(8): 942-951.

[20] 黄天立, 尚旭强, 程顺, 等 . 基于正交Hilbert时变功率谱的多分量完全非平稳地震动模拟及验证[J]. 中南大学学报(自然科学版), 2021, 52(12): 4336-4348.

[21] HUANG Tianli, SHANG Xuqiang, CHENG Shun, et al. Simulation and verification of multivariate fully non-stationary earthquake ground motions using orthogonal Hilbert spectra[J]. Journal of Central South University (Science and Technology), 2021, 52(12): 4336-4348. (in Chinese)

[22] LI Y, CONTE J P, BARBATO M . Influence of time-varying frequency content in earthquake ground motions on seismic response of linear elastic systems[J]. Earthquake Engineering & Structural Dynamics, 2016, 45(8): 1271-1291.

[23] LIU Z X, LIU Z J, RUAN X X, et al. Spectral representation-based dimension reduction for simulating multivariate non-stationary ground motions[J]. Soil Dynamics and Earthquake Engineering, 2018, 114: 313-325.

[24] 姜云木, 阮鑫鑫, 刘章军 . 主余震型地震动过程的降维模拟[J]. 振动与冲击, 2021, 40(24): 282-292.

[25] JIANG Yunmu, RUAN Xinxin, LIU Zhangjun . Dimension-reduction simulation of main aftershock type ground motion process[J]. Journal of Vibration and Shock, 2021, 40(24): 282-292. (in Chinese)

[26] LI J, CHEN J B . Stochastic dynamics of structures[M]. Singapore: John Wiley & Sons, 2009.

[27] LI J, CHEN J B . The principle of preservation of probability and the generalized density evolution equation[J]. Structural Safety, 2008, 30(1): 65-77.

[28] 郭锋, 吴东明, 许国富, 等 . 中外抗震设计规范场地分类对应关系[J]. 土木工程与管理学报, 2011, 28(2): 63-66.

[29] GUO Feng, WU Dongming, XU Guofu, et al. Site classification corresponding relationship between Chinese and the overseas seismic design codes[J]. Journal of Civil Engineering and Management, 2011, 28(2): 63-66. (in Chinese)

[30] 丁艳琼, 李杰 . 工程随机地震动物理模型的参数识别与统计建模[J]. 中国科学(技术科学), 2018, 48(12): 1422-1432.

[31] DING Yanqiong, LI Jie . Parameter identification and statistical modelling of physical stochastic model of seismic ground motion for engineering purposes[J]. Scientia Sinica Technologica, 2018, 48(12): 1422-1432. (in Chinese)

[32] 陈清军, 李英成 . 基于演变谱和正交化HHT法的类谐和长周期地震动合成[J]. 湖南大学学报(自然科学版), 2012, 39(11): 20-27.

[33] CHEN Qingjun, LI Yingcheng . Simulation of harmonic-like long-period ground motion based on evolutionary spectra and orthogonal HHT method[J]. Journal of Hunan University (Natural Sciences), 2012, 39(11): 20-27. (in Chinese)

[34] 刘章军, 刘子心 . 基于规范反应谱的全非平稳地震动过程模拟[J]. 振动工程学报, 2017, 30(3): 457-465.

[35] LIU Zhangjun, LIU Zixin . Simulation of fully non-stationary ground motion based on seismic design response spectrum[J]. Journal of Vibration Engineering, 2017, 30(3): 457-465. (in Chinese)

[36] DEODATIS G, SHINOZUKA M . Simulation of seismic ground motion using stochastic waves[J]. Journal of Engineering Mechanics, 1989, 115(12): 2723-2737.

[37] 刘章军, 刘增辉, 刘威 . 全非平稳地震动过程的概率模型及反应谱拟合[J]. 振动与冲击, 2017, 36(2): 32-38.

[38] LIU Zhangjun, LIU Zenghui, LIU Wei . Probability model of fully non-stationary ground motion with the target response spectrum compatible[J]. Journal of Vibration and Shock, 2017, 36(2): 32-38. (in Chinese)

[39] CLOUGH R W, PENZIEN J . Dynamic of structures[M]. New York: McGraw-Hill Inc, 1993.

[40] LEGATES D R, MCCABE G J Jr . Evaluating the use of “goodness-of-fit” Measures in hydrologic and hydroclimatic model validation[J]. Water Resources Research, 1999, 35(1): 233-241.

[41] SCHWARZ G . Estimating the dimension of a model[J]. The Annals of Statistics, 1978, 6(2): 461-464.

[42] 刘章军, 曾波, 吴林强 . 非平稳地震动过程模拟的谱表示-随机函数方法[J]. 振动工程学报, 2015, 28(3): 411-417.

[43] LIU Zhangjun, ZENG Bo, WU Linqiang . Simulation of non-stationary ground motion by spectral representation and random functions[J]. Journal of Vibration Engineering, 2015, 28(3): 411-417. (in Chinese)

[44] LIU Z J, LIU W, PENG Y B . Random function based spectral representation of stationary and non-stationary stochastic processes[J]. Probabilistic Engineering Mechanics, 2016, 45: 115-126.

[45] LIU Z J, LIU Z X, PENG Y B . Dimension reduction of Karhunen-Loeve expansion for simulation of stochastic processes[J]. Journal of Sound and Vibration, 2017, 408: 168-189.

基金资助

国家自然科学基金项目(52478557)

国家自然科学基金项目(52108444)

国家自然科学基金项目(51978543)

国家自然科学基金项目(51778343)

地震科技星火计划项目(XH23065YA)

湖北省高等学校优秀中青年科技创新团队计划项目(T2020010)