Stochastic Modeling and Reconstruction Method for Multi-Directional Long-Period Ground Motions
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摘要: 长周期地震动中显著的低频成分对复杂、大型柔性结构的动力响应有着不可忽视的影响。因此，合理地随机建模长周期地震动是进行大型柔性结构抗震分析与设计的关键。然而，现有的模型未能充分考虑长周期地震动的多维性，并且缺乏与规范一致的推荐参数值。为了应对这一挑战，本研究提出了一种方法，基于实测地震动数据，采用经验模态分解（EMD）技术，将长周期地震动分解为高频和低频成分，以便独立研究其频谱特性。这一方法为理解长周期地震动复杂的频谱特性奠定了基础。接着，本文初步识别并统计分析了高低频成分的演化功率谱密度（EPSD）函数参数，进一步揭示了长周期地震动的频谱特性及其变化规律。在此基础上，利用本征正交分解（POD）方法，分别模拟了不同方向的高频和低频成分样本，并重构了三维长周期地震动过程。数值算例与验证结果显示，所生成的代表性样本能够真实反映长周期、三维地震动的动力学特征，具有较高的工程准确性与实用性。本研究提出的长周期地震动随机模型不仅能够有效捕捉地震动的低频特征，初步探索了对长周期地震动工程特征随机建模，还为后续的精细化研究提供了监视的理论依据和技术支撑。Abstract: The significant low-frequency components in long-period ground motions have a non-negligible impact on the dynamic response of complex, large-scale flexible structures. Therefore, a reasonable stochastic modeling of long-period ground motions is a key foundation for the seismic analysis and design of large flexible structures. To address this issue, this study proposes an exploratory approach, based on measured seismic data, employing the Empirical Mode Decomposition (EMD) method to decompose long-period ground motions into high-frequency and low-frequency components, thereby separating and studying their spectral composition, which lays the foundation for revealing the complex spectral characteristics of long-period ground motions. Subsequently, a preliminary identification and statistical analysis of the evolution of the power spectral density (EPSD) function parameters for the high- and low-frequency components are conducted, further revealing the spectral characteristics and their variations of long-period ground motions. Based on this, the Proper Orthogonal Decomposition (POD) method is used to simulate the samples of high- and low-frequency components in different directions, and to reconstruct the three-dimensional long-period ground motion process. Numerical examples and validation results show that the generated representative samples can accurately reflect the dynamic characteristics of long-period, three-dimensional ground motions, with high engineering accuracy and practicality. The proposed stochastic model for long-period ground motions not only effectively captures the low-frequency characteristics of ground motions but also realizes the preliminary exploration of stochastic modeling for the engineering features of long-period ground motions, providing a reliable theoretical basis and technical support for future refined research.
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图 2 TCU029站记录的Chi-Chi地震时程 Fig.2 The time history of Chi-Chi earthquake measured by station TCU029
图 3 Chi-Chi地震动x方向的IMFs及相应的地震影响系数
Figure 3. The IMFs of Chi-Chi earthquake in x-direction and corresponding seismic influence coefficient
图 6 三维长周期地震动过程的代表性样本
Figure 6. Representative samples of simulated multi-direction long-period ground motion process
图 8 模拟地震影响系数均值与场地类型2实测地震影响系数均值的对比（样本数：100）
Figure 8. Comparisons between the mean value of simulated seismic influence coefficient and the mean value of recorded seismic influence coefficient in site 2 （the number of samples: 100）
表 1 各场地条件对应的长周期地震动
Table 1. The number of selected measured far-filed long-period ground motion records correspondence between site classes and $ {V}_{\text{S30}} $
实测记录 场地分类 1 2 3 $ {V}_{\text{S30}} $/(m·s−1) >450 300～450 <300 数量/条 93 72 130 
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表 2 长周期地震动的推荐EPSD参数
Table 2. The recommended EPSD parameters of long-period ground motions
场地类型 频率成分 方向 参数 t1/s (t1−t2)/s $ \alpha $ $ {\omega }_{\text{g}}\,\,/(\text{rad}\cdot {\text{s}}^{-1}) $ $ {\xi }_{\text{g}} $ $ D/\mathrm{s} $ $ {\omega }_{0}\,\,/(\text{rad}\cdot {\text{s}}^{-1}) $ 1 高频 x 26.57 66.98 0.51 28.68 0.74 0.16 3.87 y 25.73 65.61 0.51 27.79 0.80 0.14 3.30 z 21.61 63.10 0.45 57.71 0.77 0.13 3.34 低频 x 29.41 72.46 0.17 0.67 0.30 0.20 3.31 y 30.28 71.67 0.13 0.61 0.31 0.27 3.66 z 29.84 76.30 0.41 0.72 0.35 0.18 2.79 2 高频 x 26.86 67.73 0.20 20.52 0.85 0.08 2.03 y 26.29 67.79 0.37 24.10 0.82 0.09 1.89 z 22.89 71.26 0.41 43.60 0.93 0.007 1.47 低频 x 30.51 78.90 0.21 0.52 0.38 0.17 2.57 y 31.41 80.48 0.24 0.49 0.39 0.17 2.74 z 37.29 94.97 0.23 0.24 0.38 0.14 2.16 3 高频 x 22.95 63.00 0.46 10.86 0.99 0.04 2.03 y 23.06 62.60 0.46 15.51 0.78 0.03 0.74 z 19.26 58.76 0.40 72.04 0.83 0.05 1.97 低频 x 25.99 78.04 0.40 0.85 0.42 0.22 3.09 y 25.28 77.34 0.41 0.83 0.39 0.26 3.56 z 30.91 89.33 0.41 1.64 0.83 0.1 0.76
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表 3 x方向高频和低频分量幅值参数的平均比值
Table 3. Average ratio of amplitude parameter between high and low frequency components in x-direction
场地类型 参数 $ a_{\max ,1}^{}/a_{\max ,2}^{} $ $ {\bar{r}}_{1}/{\bar{r}}_{2} $ 1 1.2242 1.1888 2 1.9203 1.2727 3 1.3303 1.0816
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表 4 不同方向高频与低频分量幅值参数的平均比值
Table 4. Average ratio of amplitude parameter of high and low frequency components between different directions
参数 场地类型 频率分量 方向 x/y x/z amax 1 1 1.03 1.76 2 0.99 1.63 2 1 0.99 1.93 2 0.94 1.81 3 1 0.97 1.66 2 1.01 2.10 $ \bar{r} $ 1 1 0.96 0.91 2 0.99 1.01 2 1 1.00 0.89 2 1.00 1.04 3 1 0.98 0.93 2 0.98 1.06
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表 5 模拟参数
Table 5. The parameters for simulation parameters for POD
参数 取值 上限截止频率 /(rad·s−1) $ {\omega }_{\text{u}}=240 $ 下限截止频率 /(rad·s−1) $ {\omega }_{\text{l}}=0.628 $ 频率步长/( rad·s−1) $ \Delta \omega =\text{0.14}9 $ 频率点数 $ N=1600 $ 模拟持时/s $ T=120 $ 时间步长/s $ \Delta t=\text{0.02} $ 总时间点 $ {N}_{t}=5000 $ 样本数 100
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