﻿ 基于线性递推滤波算法的地震反应谱计算方法
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 大地测量与地球动力学  2021, Vol. 41 Issue (11): 1200-1206  DOI: 10.14075/j.jgg.2021.11.018

### 引用本文

XU Yucong. The Seismic Response Spectrum Calculation Method Based on the Linear Recursive Filtering Method[J]. Journal of Geodesy and Geodynamics, 2021, 41(11): 1200-1206.

### 第一作者简介

XU Yucong, engineer, majors in analysis of reservoir earthquake and dam strong earthquake data, E-mail: xuyc1988@foxmail.com.

### 文章历史

1. 长江三峡勘测研究院有限公司(武汉), 武汉市创业街99号，430074

1 单自由度震动体系原理

 $\ddot{u}+2\xi \omega \dot{u}+{{\omega }^{2}}u=-{{\ddot{u}}_{g}}\left( \text{t} \right)$ (1)

$f\text{=}-{{\ddot{u}}_{\text{g}}}\left( \text{t} \right)$$\ddot{u}$$\dot{u}$u分别为震动体系相对于地面的加速度、速度和位移反应；${{\ddot{u}}_{\text{g}}}\left( \text{t} \right)$为地面加速度；ξω分别为系统的阻尼比和固有频率。

 $W\left( t \right)\text{=1+}{{w}_{1}}{\mathit{\Gamma}} \text{+}{{w}_{2}}{{{\mathit{{\mathit{\Gamma}}}} }^{2}}\text{+}{{w}_{3}}{{{\mathit{\Gamma}} }^{3}}$ (2)

u$\dot{u}$$\ddot{u}f进行级数展开：  u={{u}_{n}}+{{{\mathit{\Lambda}} }_{1}}{{\dot{u}}_{n}}\tau +{{{\mathit{\Lambda}} }_{2}}{{\ddot{u}}_{n}}{{\tau }^{2}}+{{{\mathit{\Lambda}} }_{3}}\frac{{{{\ddot{u}}}_{\text{n+1}}}-{{{\ddot{u}}}_{n}}}{\Delta t}{{\tau }^{3}} (3)  \dot{u}={{\dot{u}}_{n}}+{{{\mathit{\Lambda}} }_{4}}{{\ddot{u}}_{n}}\tau +{{{\mathit{\Lambda}} }_{5}}\frac{{{{\ddot{u}}}_{\text{n+1}}}-{{{\ddot{u}}}_{n}}}{\Delta t}{{\tau }^{2}} (4)  \ddot{u}={{\ddot{u}}_{n}}+{{{\mathit{\Lambda}} }_{6}}\frac{{{{\ddot{u}}}_{\text{n+1}}}-{{{\ddot{u}}}_{n}}}{\Delta t}\tau (5)  f={{f}_{n}}+{{\dot{f}}_{n}}\tau (6) 式中，Λ1Λ2Λ3Λ4Λ5Λ6为一系列自由参数。 将式(1)代入式(2)可得：  \int_{0}^{\Delta t}{W\text{(}\ddot{u}+2\xi \omega \dot{u}+{{\omega }^{2}}u-}f\text{)d}t=0 (7) u\dot{u}$$\ddot{u}$f的渐近展开式代入式(7)可得：

 \begin{align} & {{{\ddot{u}}}_{n}}+{{{\mathit{\Lambda}} }_{6}}{{W}_{1}}\text{(}{{{\ddot{u}}}_{\text{n+1}}}-{{{\ddot{u}}}_{n}}\text{)}+2\xi \omega \text{(}{{{\dot{u}}}_{n}}+{{{\mathit{\Lambda}} }_{4}}{{W}_{1}}{{{\ddot{u}}}_{n}}\Delta t+ \\ & \ \ \ {{{\mathit{\Lambda}} }_{5}}{{W}_{2}}\text{(}{{{\ddot{u}}}_{\text{n+1}}}-{{{\ddot{u}}}_{n}}\text{)}\Delta t\text{)}+{{\omega }^{2}}\text{(}{{{\dot{u}}}_{n}}+{{{\mathit{\Lambda}} }_{1}}{{W}_{1}}{{{\dot{u}}}_{n}}\Delta t+ \\ & {{{\mathit{\Lambda}} }_{2}}{{W}_{2}}{{{\ddot{u}}}_{n}}\Delta t+{{{\mathit{\Lambda}} }_{3}}{{W}_{3}}\text{(}{{{\ddot{u}}}_{\text{n+1}}}-{{{\ddot{u}}}_{n}}\text{)}\Delta {{t}^{2}}\text{)}={{f}_{n}}+{{W}_{1}}{{{\dot{f}}}_{n}}\Delta t \\ \end{align} (8)

 ${{u}_{n+1}}={{u}_{n}}+{{{\mathit{\lambda}} }_{1}}{{\dot{u}}_{n}}\Delta t+{{\lambda }_{2}}{{\ddot{u}}_{n}}\Delta {{t}^{2}}+{{\lambda }_{3}}\text{(}{{\ddot{u}}_{\text{n+1}}}-{{\ddot{u}}_{n}}\text{)}\Delta {{t}^{2}}$ (9)
 ${{\dot{u}}_{n=1}}={{\dot{u}}_{n}}+{{\lambda }_{4}}{{\ddot{u}}_{n}}\Delta t+{{\lambda }_{5}}\text{(}{{\ddot{u}}_{\text{n+1}}}-{{\ddot{u}}_{n}}\text{)}\Delta t$ (10)

 ${{\mathit{\boldsymbol{d}}}_{n+1}}={\mathit{\boldsymbol{A}}}{{\mathit{\boldsymbol{d}}}_{n}}+{{\mathit{\boldsymbol{L}}}_{n}}$ (11)

 $\mathit{\boldsymbol{A}} = \left[ \begin{array}{l} 1 - {\lambda _3}{\kern 1pt} {A_{{\rm{31}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _1} - {\lambda _3}{\kern 1pt} {A_{32}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _2} - {\lambda _3}{\kern 1pt} {A_{33}}{\kern 1pt} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _5}{\kern 1pt} {A_{{\rm{31}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1 - {\lambda _5}{\kern 1pt} {A_{{\rm{32}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _4} - {\lambda _5}{\kern 1pt} {A_{33}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {A_{{\rm{31}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {A_{32}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {A_{{\rm{33}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \end{array} \right],$
 ${{\mathit{\boldsymbol{L}}}_n} = \frac{1}{D}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( \begin{array}{l} {\lambda _3}\Delta {t^2}\left[ {\left( {1 - {W_1}} \right)} \right]{f_n} + {W_1}{f_{n + 1}}\\ {\lambda _5}\Delta {t^2}\left[ {\left( {1 - {W_1}} \right)} \right]{f_n} + {W_1}{f_{n + 1}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Delta {t^2}\left[ {\left( {1 - {W_1}} \right)} \right]{f_n} + {W_1}{f_{n + 1}} \end{array} \right)$

ALn中变量可整理为：

 ${A_{31}}{\kern 1pt} {\kern 1pt} = {\Omega ^2}/D, {A_{32}}{\kern 1pt} {\kern 1pt} = \left( {2\xi \Omega + {{\mathit{\Lambda}} _1}{W_1}{\Omega ^2}} \right)/D$
 ${A_{{\rm{33}}}} = 1 - {\rm{(}}1 + 2{{\mathit{\Lambda}} _4}{W_1}\xi \Omega + {{\mathit{\Lambda}} _2}{W_2}{\Omega ^2}{\rm{)}}/D$
 $D = {{\mathit{\Lambda}} _6}{W_1} + 2{{\mathit{\Lambda}} _5}{W_2}\xi \Omega + {{\mathit{\Lambda}} _3}{W_3}{\Omega ^2}, \Omega = \omega {\kern 1pt} \Delta t$
2 计算原理

Z变换域中输入、输出和传递函数的形式为：

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Y\left( Z \right) = \\ \frac{{{B_{\rm{1}}} + {B_{\rm{2}}}{z^{ - {\rm{1}}}} + {B_{\rm{3}}}{z^{ - {\rm{2}}}} + {B_{\rm{4}}}{z^{ - {\rm{3}}}} + \cdots + {B_{{n_b} + {\rm{1}}}}{z^{ - {n_b}}}}}{{{\rm{1}} + {A_{\rm{1}}}{z^{ - {\rm{1}}}} + {A_{\rm{2}}}{z^{ - {\rm{2}}}} + {A_{\rm{3}}}{z^{ - {\rm{3}}}} + \cdots + {A_{{n_a} + {\rm{1}}}}{z^{ - {n_a}}}}} \end{array}$ (12)

 $\begin{array}{l} y\left( m \right) = {B_{\rm{1}}}x\left( m \right) + {B_{\rm{2}}}x\left( {m - {\rm{1}}} \right) + \cdots + \\ \;\;\;\;\;{B_{{n_b} + {\rm{1}}}}x\left( {m - {n_b}} \right) - {A_{\rm{1}}}y\left( {m - {\rm{1}}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; \cdots + {B_{{n_a} + {\rm{1}}}}y\left( {m - {n_a}} \right) \end{array}$ (13)

 ${A_1} = \mathit{\boldsymbol{I}} :\mathit{\boldsymbol{A}}, {A_2} = \frac{1}{2}\left( {A_1^2{\kern 1pt} {\kern 1pt} - \mathit{\boldsymbol{A}}:\mathit{\boldsymbol{A}}} \right), {A_3} = {\rm{det}}\mathit{\boldsymbol{A}}$
 ${B_1} = \Delta {t^2}{\lambda _3}{W_1}/D$
 $\begin{array}{l} {B_2} = \Delta {t^2}\left[ {{\lambda _3}{\rm{(}}1 - {W_1}{\rm{)}} - {\rm{(}}{A_{22}}{\kern 1pt} + {A_{{\rm{33}}}}{\kern 1pt} {\rm{)}}{\kern 1pt} {\lambda _3}{W_1}{\kern 1pt} + } \right.\\ {\kern 1pt} \left. {\;\;\;\;\;\;\;{A_{12}}{\kern 1pt} {\kern 1pt} {\lambda _5}{W_1}{\kern 1pt} + {A_{13}}{\kern 1pt} {\kern 1pt} {W_1}} \right]/D \end{array}$
 $\begin{array}{l} {B_3} = \Delta {t^2}\lambda \left[ { - {\rm{(}}{A_{22}} + {A_{33}}{\rm{)(}}1 - {W_1}{\rm{)}}{\lambda _3} + } \right.{\kern 1pt} \\ \;\;\;\;\;\;\;{A_{12}}{\kern 1pt} {\kern 1pt} {\lambda _5}{\rm{(}}1 - {W_1}{\rm{)}} + {A_{13}}{\rm{(}}1 - {W_1}{\rm{)}} + \\ \;\;\;\;\;\;\;{\rm{(}}{A_{22}}{\kern 1pt} {A_{{\rm{33}}}} - {A_{{\rm{23}}}}{A_{{\rm{32}}}}{\kern 1pt} {\rm{)}}{\kern 1pt} {\lambda _3}{W_1} - {\rm{(}}{A_{12}}{\kern 1pt} {A_{{\rm{33}}}} - \\ \left. {\;\;\;\;\;\;\;{A_{{\rm{13}}}}{A_{{\rm{32}}}}{\kern 1pt} {\rm{)}}{\kern 1pt} {\lambda _5}{W_1}{\rm{ + (}}{A_{12}}{\kern 1pt} {A_{{\rm{23}}}} - {A_{{\rm{13}}}}{A_{{\rm{22}}}}{\kern 1pt} {\rm{)}}{\kern 1pt} {W_1}} \right]/D \end{array}$
 $\begin{array}{l} {B_4} = \Delta {t^2}\lambda \left[ {{\rm{(}}{A_{22}}{\kern 1pt} {A_{{\rm{33}}}} - {A_{{\rm{23}}}}{A_{{\rm{32}}}}{\rm{)(}}1 - {W_1}{\rm{)}}{\lambda _3} - } \right.\\ \;\;\;\;\;\;\;{\rm{(}}{A_{12}}{\kern 1pt} {A_{{\rm{33}}}} - {A_{{\rm{13}}}}{A_{{\rm{32}}}}{\kern 1pt} {\rm{)}}{\kern 1pt} {\lambda _5} - {\rm{(}}1 - {W_1}{\rm{)}} + \\ \;\;\;\;\;\;\;\left. {{\rm{(}}{A_{12}}{\kern 1pt} {A_{{\rm{23}}}} - {A_{{\rm{13}}}}{A_{{\rm{22}}}}{\kern 1pt} {\rm{)}}{\kern 1pt} {\rm{(}}1 - {W_1}{\rm{)}}} \right]/D \end{array}$
 $y{\rm{(}}m{\rm{)}} = {B_1}x{\rm{(}}m{\rm{)}} + {z_1}{\rm{(}}m - 1{\rm{)}}$ (14)
 ${z_1}{\rm{(}}m{\rm{)}} = {B_2}x{\rm{(}}m{\rm{)}} + {z_2}{\rm{(}}m - 1{\rm{)}} - {A_1}y{\rm{(}}m{\rm{)}}$ (15)
 ${z_2}{\rm{(}}m{\rm{)}} = {B_3}x{\rm{(}}m{\rm{)}} + {z_3}{\rm{(}}m - 1{\rm{)}} - {A_2}y{\rm{(}}m{\rm{)}}$ (16)
 ${z_3}\left( m \right) = {B_4}x\left( m \right) - {A_3}y\left( m \right)$ (17)

3 合成数据分析计算 3.1 计算精度分析

 $\begin{array}{l} f\left( t \right) = {\rm{sin}}(2{\rm{ \mathsf{ π} }} \times {f_1} \times t) + {\rm{sin}}(2{\rm{ \mathsf{ π} }} \times {f_2} \times t), \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t \in \left[ {8, 15} \right] \end{array}$ (18)
 $f\left( t \right) = 0, t \in \left[ {0, 8} \right)\;或\; t \in\; \left( {15, 30} \right)$ (19)

 图 1 位移反应谱计算结果及误差 Fig. 1 The calculation results and errors of the displacement response spectrum

 图 2 速度反应谱计算结果及误差 Fig. 2 The calculation results and errors of the velocity response spectrum

 图 3 加速度反应谱计算结果及误差 Fig. 3 The calculation results and errors of the acceleration response spectrum

3.2 计算效率分析

 图 4 反应谱计算耗时对比 Fig. 4 Comparison of the response spectrum calculation time

4 实际数据分析计算

 图 5 地震动加速度时程 Fig. 5 The ground motion acceleration

 图 6 不同台站速度反应谱计算结果 Fig. 6 The calculation results of velocity response spectrum of different stations

 图 7 不同台站加速度反应谱计算结果 Fig. 7 The calculation results of acceleration response spectrum of different stations
5 结语

1) LRFM方法计算合成数据得到的反应谱精度和稳定性显著优于Duhamel逐步积分法，略优于Newmark-β方法。与ASM方法相比，LRFM方法计算位移、速度和加速度反应谱的误差均小于2%，满足精度要求。

2) LRFM方法的计算效率与Duhamel逐步积分法相当，且显著优于目前普遍采用的Newmark-β方法，因此对于快速处理强震及大地震产生的数量多且持续时间长的强震动记录具有重要意义。

3) 对于实际强震动记录数据的处理效果，LRFM方法与ASM方法计算的速度反应谱结果一致。

4) 与ASM方法相比，LRFM方法计算得到实际强震动加速度反应谱结果产生的误差主要集中在周期0~1.0 s之间，且随着阻尼的增大误差逐渐增大，但计算结果总体满足精度要求。

5) 针对不同卓越周期和频谱范围的实际强震动记录，采用LRFM方法均可得到可靠的计算结果，稳定性和适用性较好。

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The Seismic Response Spectrum Calculation Method Based on the Linear Recursive Filtering Method
XU Yucong1
1. Three Gorges Geotechnical Consultants Co Ltd, 99 Chuangye Street, Wuhan 430074, China
Abstract: In order to improve the calculation accuracy and efficiency of the seismic response spectrum, we introduce the linear recursive filtering method(LRFM) to calculate the seismic response spectrum. We derive the general expression of the response spectrum, which is based on the single degree of freedom dynamic system equation. To verify the calculation accuracy and efficiency of this method, we use the synthetic sinusoidal simple harmonic wave as the system input to compare the results of response spectrum which are calculated by the LRFM, the Duhamel step-by-step integration method, the Newmark-β method and the analytical solution method (ASM), and verify the error comparing with the ASM. We prove that the calculation method of the LRFM has good stability and high efficiency. In order to further prove the effectiveness of this method in calculating the actual data, we select the acceleration data of strong earthquakes with different prominent periods and spectrum from the ESM strong earthquake database, and the response spectrum of the LRFM and ASM is compared and analyzed under different damping conditions. The results show that the LRFM is consistent with the ASM in calculating the velocity response spectrum. The calculation result of the acceleration response spectrum has some errors with the increase of the damping, but the overall calculation accuracy could meet the requirements. The LRFM proposed in this paper could quickly and efficiently calculate the response spectrum information, so it has great significance for the rapid assessment of seismic intensity characteristics of the site and the stress condition of engineering structure.
Key words: recursive filtering; single degree of freedom; Newmark-β method; seismic response spectrum