华南地震

分数阶导数模型可以全面地反映粘弹性阻尼器的力学性能和变化机制，但其中的Riemann-Liouville积分复杂，给减震结构的动力响应求解带来一定的困难。传统分析方法仅考虑结构基频的影响，通过等效刚度和等效阻尼简化结构响应计算，计算结果精度有限。针对含分数阶导数粘弹性阻尼器的减震结构，通过引入Caputo分数阶导数及相关高阶预估-校正技术，提出一种高精度的动力响应直接数值解法。通过多层结构算例，检验本文算法的有效性，对比考察了传统等效方法在不同简谐激励及El Centro地震波作用下的相关计算偏差。

The fractional derivative model can fully represent the mechanical properties and the changing mechanism of viscoelastic dampers，however，the Riemann-Liouville integral involved is complex，which brings difficulties to the solution of dynamic structural responses. The conventional method often takes the fundamental natural frequency of the structure into account to obtain the equivalent stiffness and damping provided by the damper，which can simplify the solution while provides results with limited accuracy. In this paper，a high-precision numerical method to directly solve dynamic responses is proposed by introducing the Caputo fractional derivative and a corresponding high-order predictor-corrector algorithm. The effectiveness of the proposed method is verified via a numerical example of a multi-story structure，and comparisons with the conventional equivalent method are carried out to identify the relative errors of dynamic responses under different harmonic excitations and the EI Centro seismic wave.

引言
1 分数阶导数本构模型
2 恢复力的等效刚度与等效阻尼模型
3 多层减震结构运动方程
4 运动方程的直接数值求解
5 多层减震结构算例
6 结论

图1 n层结构模型图<br/>Fig.1 Model diagram of an n-story structure

图1 n层结构模型图
Fig.1 Model diagram of an n-story structure

表1 简谐激励(ωg=2.5 rad/s)作用下的结构各层最大响应结果<br/>Table 1 Maximum response results of each floor of the structure under harmonic excitation(ωg=2.5 rad/s)

表1 简谐激励(ωg=2.5 rad/s)作用下的结构各层最大响应结果
Table 1 Maximum response results of each floor of the structure under harmonic excitation(ωg=2.5 rad/s)

图2 简谐激励(ωg=2.5 rad/s)作用下第1层和第10层结构响应相平面<br/>Fig.2 Phase plane of structural responses at the first and tenth stories under harmonic excitation(ωg=2.5 rad/s)

图2 简谐激励(ωg=2.5 rad/s)作用下第1层和第10层结构响应相平面
Fig.2 Phase plane of structural responses at the first and tenth stories under harmonic excitation(ωg=2.5 rad/s)

表2 简谐激励(ωg=15 rad/s)作用下的结构各层最大响应结果<br/>Table 2 Maximum response results of each floor of the structure under harmonic excitation(ωg=15 rad/s)

表2 简谐激励(ωg=15 rad/s)作用下的结构各层最大响应结果
Table 2 Maximum response results of each floor of the structure under harmonic excitation(ωg=15 rad/s)

图3 简谐激励(ωg=15 rad/s)作用下第1层和第10层结构响应相平面<br/>Fig.3 Phase plane of structural responses at the first and tenth stories under harmonic excitation(ωg=15 rad/s)

图3 简谐激励(ωg=15 rad/s)作用下第1层和第10层结构响应相平面
Fig.3 Phase plane of structural responses at the first and tenth stories under harmonic excitation(ωg=15 rad/s)

表3 El Centro地震波作用下的结构各层最大响应结果<br/>Table 3 Maximum response results of each floor of the structure under the El Centro seismic wave

表3 El Centro地震波作用下的结构各层最大响应结果
Table 3 Maximum response results of each floor of the structure under the El Centro seismic wave

图4 El Centro地震波作用下结构第1层响应时程<br/>Fig.4 Response time history of the first floor of the structure under the El Centro seismic wave

图4 El Centro地震波作用下结构第1层响应时程
Fig.4 Response time history of the first floor of the structure under the El Centro seismic wave

图5 El Centro地震波作用下结构第10层响应时程<br/>Fig.5 Response time history of the tenth floor of the structure under the El Centro seismic wave

图5 El Centro地震波作用下结构第10层响应时程
Fig.5 Response time history of the tenth floor of the structure under the El Centro seismic wave

图6 El Centro地震波作用下第1层和第10层结构响应相平面<br/>Fig.6 Phase plane of structural responses at the first and tenth stories under the El Centro seismic wave

图6 El Centro地震波作用下第1层和第10层结构响应相平面
Fig.6 Phase plane of structural responses at the first and tenth stories under the El Centro seismic wave

[1] 王烨华，周云，丁鲲. 粘弹性阻尼减震结构研究与应用的新进展[J]. 防灾减灾工程学报，2006，26(1)：109-121.
[2] Pan P，Ye L P，Shi W，et al. Engineering practice of seismic isolation and energy dissipation structures in China [J]. Science China Technological Sciences，2012，55(11)：3036-3046.
[3] Chang K C，Lai M L，Soong T T，et al. Seismic behavior and design guidelines for steel frame structures with added viscoelastic damper [R]. Buffalo，NY：National Center for Earthquake Engineering Research，1993.
[4] 邹向阳，欧进萍. 粘弹性耗能器的性能与结构减振试验研究[J]. 振动工程学报，1999，12(2)：237-243.
[5] 魏春彤，赵森林，裴星洙. 谐振作用下粘弹性阻尼器影响因素的数值分析[J]. 江苏科技大学学报(自然科学版)，2015，29(4)：392-398.
[6] 周云.粘弹性阻尼减震结构设计[M]. 武汉：武汉理工大学出版社，2006.
[7] Ou J P，Long X，Li Q S. Seismic response analysis of structures with velocity-dependent dampers[J]. Journal of Constructional Steel Research，2006，63(5)：628-638.
[8] 黄兴淮，徐赵东，贺泽峰，等. 黏弹性材料等效标准固体模型的时域延拓方法[J]. 东南大学学报(自然科学版)，2019，49(3)：440-445.
[9] Tsai C S. Temperature effect of viscoelastic dampers during earthquakes[J]. Journal of Structural Engineering，ASCE，1994，120(2)：394-409.
[10] 徐业守，徐赵东，葛腾，等. 黏弹性材料等效分数阶微观结构标准线性固体模型[J]. 力学学报，2017，49(5)：1059-1069.
[11] 张晓棣，陈文. 三种分形和分数阶导数阻尼振动模型的比较研究[J]. 固体力学学报，2009，30(5)：496-503.
[12] Fu Y M，Kasai K. Comparative study of frames using viscoelastic and viscous dampers[J]. Journal of Structural Engineering，ASCE，1998，124(5)：513-522.
[13] 林孔容. 关于分数阶导数的几种不同定义的分析与比较[J]. 闽江学院学报，2003，24(5)：3-6.
[14] 李亚杰，吴志强，章国齐. 基于Caputo导数的分数阶非线性振动系统响应计算[J]. 计算力学学报，2018，35(4)：466-472.
[15] Zeng F H，Turner I，Burrage K. A stable fast time-stepping method for fractional integral and derivative operators[J]. Journal of Scientific Computing，2018，77(1)：283-307.
[16] Galeone L，Garrappa R. Fractional Adams-Moulton methods [J]. Mathematics and Computers in Simulation，2008，79(4)：1358-1367.
[17] Tzan S R，Pantelides C P.Hybrid structural control using viscoelastic dampers and active control systems[J]. Eart-hquake Engineering & Structural Dynamics，1994，23(12)：1369-1388.

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引言

1 分数阶导数本构模型

2 恢复力的等效刚度与等效阻尼模型

3 多层减震结构运动方程

4 运动方程的直接数值求解

5 多层减震结构算例

6 结论

期刊信息