向心式摩擦阻尼器的理论分析及应用研究
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摘要
摘 要
向心式摩擦阻尼器（EDR）具有一些不同于常见阻尼器的特点：采用不同的
参数配置，可以提供多种滞回模型。其中三角形滞回模型和双旗帜形滞回模型具
有自回位功能，即完全卸载后，阻尼器不存在残余变形；另外具有三角形滞回性
能的单元实际上是一种变刚度单元，可通过被动方式在一定程度上实现的半主动
控制。本文从被动耗能减震的角度对向心式摩擦阻尼器的三种滞回模型（三角形、
双旗帜形和矩形滞回模型）的减震性能进行研究，包括以下几个方面的内容：
详细介绍向心式摩擦阻尼器提供三角形、双旗帜形和矩形滞回模型时的参数
配置情况；通过阻尼器内部元件静力平衡分析，推导了三角形和双旗帜形滞回模
型中加载刚度、卸载刚度与阻尼器构造参数之间的对应关系，给出了加载刚度和
卸载刚度的计算公式，为阻尼器的设计和制造提供依据。并基于这些公式，研究
了 5 个构造参数对加载刚度的影响，讨论了如何提高向心式摩擦阻尼器出力吨
位；进行了拟静力试验研究，实现了上述三种滞回模型，并对刚度公式进行了初
步的验证；
对具有矩形、三角形滞回性能的单自由度体系的自由振动以及具有矩形、三
角形和双旗帜形滞回性能的单自由度系统的稳态响应进行了研究和讨论，推导了
三种滞回模型的等效参数的计算公式，这三方面的研究结果定性的描述了三种滞
回模型的特性及其减震性能；
为计算包含滞回单元结构体系的地震响应，本文较为详细地研究了当非线性
单元的滞回模型为分段线性函数时，求解此类系统动力响应的四种非线性算法，
包括 Newton-Raphson 法、虚拟荷载迭代法、近似寻找拐点的非迭代方法以及精
确寻找拐点的非迭代方法。对四种算法的计算精度、效率进行了比较，对各算法
在编程中应该注意的问题进行了说明，给出了实现三种滞回模型的程序代码。
通过数值模拟计算，基于非线性反应谱，研究了三角形滞回模型的三个刚度、
双旗帜形滞回模型的起滑力、矩形滞回模型的起滑力和过渡刚度这 6 个参数对包
含它们的结构系统的动力响应的影响。研究结果表明：滞回参数对减震效果的影
响与线性结构周期 T 在线性反应谱中对应的位置有关，当 T 位于结构某项反应谱
的上升段时，增大滞回单元等效刚度可以减小结构的该反应，反之，则将增大结
构该项反应。
从特定地震激励下的减震效果以及地震强度变化时减震效果的变化情况这
两个方面对三角形、双旗帜形和矩形滞回单元的减震性能进行了比较，结果表明
矩形滞回单元具有很好的减震效果，但其减震效果随地震强度的增加而减弱；三
角形和双旗帜形滞回单元的减震效果随地震强度的变化很小。因而，在不能准确
－ I －
北京工业大学工学硕士学位论文
预料地震强度时，本文认为三角形和双旗帜形滞回单元较矩形滞回单元更具优
势。
最后，本文对三角形滞回单元在相邻结构减震控制中的应用进行了研究：推
导了两个单自由度互联结构体系在简谐力作用下的稳态响应解，并对连接单元的
刚度、阻尼参数以及相邻结构的周期比对互联体系稳态响应的影响进行了讨论，
并在此基础上通过数值模拟,讨论了三角形滞回单元在相邻结构减震控制中的应
用范围。结果表明：当地震能量比较集中时，不宜采用互联结构形式进行减震。
Energy Dissipating Restraint (EDR) has some specialty which differs from normal
energy dissipating device. This device is able to provide several forms of hysteretic
characteristics. The triangular and double flag hysteretic modes have the capacity of
self-centering, that is to say, after totally unloaded to naught, the residual deformation
of the device tends zero. In the addition, the element with triangular hysteretic
behavior enables to provide AVS control by passive method. Some researches on
seismic resistant performance of EDR with rectangular, triangular and double flag
hysteretic behaviors are conducted in this paper. These related researches include:
Introducing in detail how to yield the triangular, double flag, rectangular hysteretic
modes; according static equilibrium of interior element of EDR, deducing some
formulation to calculate the loading-stiffness and unloading-stiffness based on the
constitution models. These formulations provide help for designing and fabricating
EDR. On the basis of these formulations, the influence of model parameters on
loading-stiffness and unloading-stiffness is studied; furthermore, the approach for
improving the max restore force of EDR is discussed. Quasi-static tests on an EDR
are conducted in this dissertation, and all the three above hysteresis loops are
observed. The calculated stiffness is preliminary proved by the experimental results.
The free vibration of single-degree-of-freedom (SDOF) system with rectangular,
triangular hysteretic behaviors and the steady-state response of SDOF system with
three above hysteretic behaviors are studded, respectively. And equivalent parameters
of three hysteretic modes are derived. These works provide some insight to these three
hysteretic modes and their seismic resistant performance.
To calculate the earthquake response of structures equipped hysteretic elements whose
hysteretic modes are piece-wise, the algorithms conducting nonlinear dynamic
analysis are investigated, which include: Newton-raphson method, pseudo load
method, non-iterate approach in search of stiffness turn point by approximate or
accurate solution. The calculating precision and efficiency of these four methods are
compared. Besides, this paper emphasizes some problems which should be kept in
mind when programming in accordance of these algorithms, and show the method of
how to calculate the restore force for three hysteretic elements.
Based on nonlinear response spectra analysis, parametric studies are preformed. The
impact on structural earthquake response of six parameters are investigated, which
include loading-stiffness, unloading-stiffness, transition-stiffness of triangular
hysteretic element, slip-force of double flag hysteretic element, and slip-force,
transition-stiffness of rectangular hysteretic element. The results indicate that the
parameter influence of the hysteretic modes on structural response is related to what
position of linear structural period (T) is located on response spectra. If T closes to the
－ III －
北京工业大学工学硕士学位论文
period zone in which the gradient of certain response spectrum curve is positive, to
increase the equivalent stiffness of hysteretic element enables to decrease this
response, and otherwise, it will result in the response increasing.
The comparisons of seismic resistant performance of the three hysteretic elements are
conducted in two cases: one is that the intensity of earthquake is known, and the other
is that the intensity is unpredictable. The results show that: in the first case rectangular
element can get well control effect, but as the peak ground acceleration (PGA)
increasing, the control effect decrease. But for triangular and double flag hysteretic
element, the control effect has little change as PGA altering. So, when the intensity of
earthquake can’t exactly forecast, the triangular and double flag element should be
adopted.
At the last part, this paper investigates the application of triangular
向心式摩擦阻尼器（EDR）具有一些不同于常见阻尼器的特点：采用不同的
参数配置，可以提供多种滞回模型。其中三角形滞回模型和双旗帜形滞回模型具
有自回位功能，即完全卸载后，阻尼器不存在残余变形；另外具有三角形滞回性
能的单元实际上是一种变刚度单元，可通过被动方式在一定程度上实现的半主动
控制。本文从被动耗能减震的角度对向心式摩擦阻尼器的三种滞回模型（三角形、
双旗帜形和矩形滞回模型）的减震性能进行研究，包括以下几个方面的内容：
详细介绍向心式摩擦阻尼器提供三角形、双旗帜形和矩形滞回模型时的参数
配置情况；通过阻尼器内部元件静力平衡分析，推导了三角形和双旗帜形滞回模
型中加载刚度、卸载刚度与阻尼器构造参数之间的对应关系，给出了加载刚度和
卸载刚度的计算公式，为阻尼器的设计和制造提供依据。并基于这些公式，研究
了 5 个构造参数对加载刚度的影响，讨论了如何提高向心式摩擦阻尼器出力吨
位；进行了拟静力试验研究，实现了上述三种滞回模型，并对刚度公式进行了初
步的验证；
对具有矩形、三角形滞回性能的单自由度体系的自由振动以及具有矩形、三
角形和双旗帜形滞回性能的单自由度系统的稳态响应进行了研究和讨论，推导了
三种滞回模型的等效参数的计算公式，这三方面的研究结果定性的描述了三种滞
回模型的特性及其减震性能；
为计算包含滞回单元结构体系的地震响应，本文较为详细地研究了当非线性
单元的滞回模型为分段线性函数时，求解此类系统动力响应的四种非线性算法，
包括 Newton-Raphson 法、虚拟荷载迭代法、近似寻找拐点的非迭代方法以及精
确寻找拐点的非迭代方法。对四种算法的计算精度、效率进行了比较，对各算法
在编程中应该注意的问题进行了说明，给出了实现三种滞回模型的程序代码。
通过数值模拟计算，基于非线性反应谱，研究了三角形滞回模型的三个刚度、
双旗帜形滞回模型的起滑力、矩形滞回模型的起滑力和过渡刚度这 6 个参数对包
含它们的结构系统的动力响应的影响。研究结果表明：滞回参数对减震效果的影
响与线性结构周期 T 在线性反应谱中对应的位置有关，当 T 位于结构某项反应谱
的上升段时，增大滞回单元等效刚度可以减小结构的该反应，反之，则将增大结
构该项反应。
从特定地震激励下的减震效果以及地震强度变化时减震效果的变化情况这
两个方面对三角形、双旗帜形和矩形滞回单元的减震性能进行了比较，结果表明
矩形滞回单元具有很好的减震效果，但其减震效果随地震强度的增加而减弱；三
角形和双旗帜形滞回单元的减震效果随地震强度的变化很小。因而，在不能准确
－ I －
北京工业大学工学硕士学位论文
预料地震强度时，本文认为三角形和双旗帜形滞回单元较矩形滞回单元更具优
势。
最后，本文对三角形滞回单元在相邻结构减震控制中的应用进行了研究：推
导了两个单自由度互联结构体系在简谐力作用下的稳态响应解，并对连接单元的
刚度、阻尼参数以及相邻结构的周期比对互联体系稳态响应的影响进行了讨论，
并在此基础上通过数值模拟,讨论了三角形滞回单元在相邻结构减震控制中的应
用范围。结果表明：当地震能量比较集中时，不宜采用互联结构形式进行减震。
Energy Dissipating Restraint (EDR) has some specialty which differs from normal
energy dissipating device. This device is able to provide several forms of hysteretic
characteristics. The triangular and double flag hysteretic modes have the capacity of
self-centering, that is to say, after totally unloaded to naught, the residual deformation
of the device tends zero. In the addition, the element with triangular hysteretic
behavior enables to provide AVS control by passive method. Some researches on
seismic resistant performance of EDR with rectangular, triangular and double flag
hysteretic behaviors are conducted in this paper. These related researches include:
Introducing in detail how to yield the triangular, double flag, rectangular hysteretic
modes; according static equilibrium of interior element of EDR, deducing some
formulation to calculate the loading-stiffness and unloading-stiffness based on the
constitution models. These formulations provide help for designing and fabricating
EDR. On the basis of these formulations, the influence of model parameters on
loading-stiffness and unloading-stiffness is studied; furthermore, the approach for
improving the max restore force of EDR is discussed. Quasi-static tests on an EDR
are conducted in this dissertation, and all the three above hysteresis loops are
observed. The calculated stiffness is preliminary proved by the experimental results.
The free vibration of single-degree-of-freedom (SDOF) system with rectangular,
triangular hysteretic behaviors and the steady-state response of SDOF system with
three above hysteretic behaviors are studded, respectively. And equivalent parameters
of three hysteretic modes are derived. These works provide some insight to these three
hysteretic modes and their seismic resistant performance.
To calculate the earthquake response of structures equipped hysteretic elements whose
hysteretic modes are piece-wise, the algorithms conducting nonlinear dynamic
analysis are investigated, which include: Newton-raphson method, pseudo load
method, non-iterate approach in search of stiffness turn point by approximate or
accurate solution. The calculating precision and efficiency of these four methods are
compared. Besides, this paper emphasizes some problems which should be kept in
mind when programming in accordance of these algorithms, and show the method of
how to calculate the restore force for three hysteretic elements.
Based on nonlinear response spectra analysis, parametric studies are preformed. The
impact on structural earthquake response of six parameters are investigated, which
include loading-stiffness, unloading-stiffness, transition-stiffness of triangular
hysteretic element, slip-force of double flag hysteretic element, and slip-force,
transition-stiffness of rectangular hysteretic element. The results indicate that the
parameter influence of the hysteretic modes on structural response is related to what
position of linear structural period (T) is located on response spectra. If T closes to the
－ III －
北京工业大学工学硕士学位论文
period zone in which the gradient of certain response spectrum curve is positive, to
increase the equivalent stiffness of hysteretic element enables to decrease this
response, and otherwise, it will result in the response increasing.
The comparisons of seismic resistant performance of the three hysteretic elements are
conducted in two cases: one is that the intensity of earthquake is known, and the other
is that the intensity is unpredictable. The results show that: in the first case rectangular
element can get well control effect, but as the peak ground acceleration (PGA)
increasing, the control effect decrease. But for triangular and double flag hysteretic
element, the control effect has little change as PGA altering. So, when the intensity of
earthquake can’t exactly forecast, the triangular and double flag element should be
adopted.
At the last part, this paper investigates the application of triangular
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structures. SPIE.vol.2193. 1994:213~224
49. Alessandra Zambrano, Inaudi. Jose A, Kelly. J M. accuracy of the modal strain
energy method. SPIE,vol.2193:284~295
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50. Inaudi. Jose A, George. Leitmann, Kelly. J M. Single-Degree-Of-Freedom
Nonlinear Homogeneous Systems. Journal of Engineering Mechanics.
1994,120(7):1543~1562
51. Inaudi. Jose A. Kelly. J M. Linear Hysteretic Damping and the Hilbert Transform.
Journal of Engineering Mechanics. 1995,121(5):626~632
52. Inaudi. Jose A, Nicos. Makris. Time-Domain Analysis of Linear Hysteretic
Damping. Earthquake Engineering and Structural Dynamics. 1996,25:529~545
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57. Beucke.Karl.E, Kelly.James.M. Equivalent Linearizations for Practical Hysteretic
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61. 王建平，邹银生. 结构非线性动力分析中恢复力拐点的非迭代处理方法. 第
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62. 傅铁铭，熊建国. Wilson-θ法中非线性拐点的精确处理方法. 青年力学协会
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63. Roberto.villaverde, Russell.c.lamb. Scheme to improve numerical analysis of
hysteretic dynamic systems. Journal of structural engineering.
1989,115(1):228~233
64. 肖明葵，刘 纲，白绍良. 滞回恢复力模型中求折点的一种方法. 重庆大学学
报（自然科学版）. 2002，25（1）：13~16
65. 黄宗明， 孙勇. 决定单自由度体系弹塑性地震反应的结构参数分析. 重庆建
筑大学学报. 1996, 18(3):42~48
66. Stavros A. Anagnostopoulos, Konstantinos V. Spiliopoulos. An Investigation of
earthquake induced pounding between adjacent buildings. Earthquake
Engineering & Structural Dynamics. 1992.21(4):289~302
67. Kobori T, Yamada T, Takenaka Y, Maeda Y, Nishimura I. Effect of Dynamic
Tuned Connector On Reduction Of Seismic Response: Application To Adjacent
Office Buildings. In: Proceedings of the Ninth World Conference on Earthquake
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Linked By Fluid Dampers. Structural Engineering And Mechanics, 2003,15(5):
513~534
70. 闫维明，房屋建筑减震控制理论和试验研究．博士后研究工作报告．中国地
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72. 闫维明，陆瀛祺，彭凌云. 两结构高效阻尼控制体系非线性地震反应分析. 土
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the Seismic Restraint of Piping.ATC-29,4:seminar technical papers:393~404
48. Inaudi. Jose A. Nims. D K. Kelly. J M. linear-friction dissipators for truss
structures. SPIE.vol.2193. 1994:213~224
49. Alessandra Zambrano, Inaudi. Jose A, Kelly. J M. accuracy of the modal strain
energy method. SPIE,vol.2193:284~295
- 116 -
参考文献
50. Inaudi. Jose A, George. Leitmann, Kelly. J M. Single-Degree-Of-Freedom
Nonlinear Homogeneous Systems. Journal of Engineering Mechanics.
1994,120(7):1543~1562
51. Inaudi. Jose A. Kelly. J M. Linear Hysteretic Damping and the Hilbert Transform.
Journal of Engineering Mechanics. 1995,121(5):626~632
52. Inaudi. Jose A, Nicos. Makris. Time-Domain Analysis of Linear Hysteretic
Damping. Earthquake Engineering and Structural Dynamics. 1996,25:529~545
53. 周锡元，闫维明，杨润林，建筑结构的隔震、减振和振动控制，建筑结构学
报，2002，23(2):2~13
54. 李惠，彭君义，周锡元，张平，向心式变摩擦阻尼器控制结构地震反应分析，
地震工程与工程振动，2001，21(4):66~73.
55. Robert D.H. Soong T.T., Seismic Design With Supplemental Energy Dissipation
Devices, EERI Publication No.MNO-8,2001
56. A. Filiatrault. Seismic design spectra for friction-damped structures. Journal
of Structural Engineering, 1990,116(5):1334~1355
57. Beucke.Karl.E, Kelly.James.M. Equivalent Linearizations for Practical Hysteretic
Systems. International Journal of Non-Linear Mechanics. 1985,20(4): 211~238
58. 《地震工程概论》编写组，地震工程概论，科学出版社,1977:41~45
59. Wilson.E.L. Three-Dimensional Static And Dynamic Analysis Of Structures
(Third Edition), ComputersAnd Structures,Inc. Berkeley, California, Usa, 2002
60. 朱镜清. 非线性动力分析的拐点处理方法. 地震工程与工程振动. 1982第3期
61. 王建平，邹银生. 结构非线性动力分析中恢复力拐点的非迭代处理方法. 第
三届全国地震工程会议论文集.1990：1072~1078
62. 傅铁铭，熊建国. Wilson-θ法中非线性拐点的精确处理方法. 青年力学协会
第三届学术年会论文集.1989：301~314
63. Roberto.villaverde, Russell.c.lamb. Scheme to improve numerical analysis of
hysteretic dynamic systems. Journal of structural engineering.
1989,115(1):228~233
64. 肖明葵，刘 纲，白绍良. 滞回恢复力模型中求折点的一种方法. 重庆大学学
报（自然科学版）. 2002，25（1）：13~16
65. 黄宗明， 孙勇. 决定单自由度体系弹塑性地震反应的结构参数分析. 重庆建
筑大学学报. 1996, 18(3):42~48
66. Stavros A. Anagnostopoulos, Konstantinos V. Spiliopoulos. An Investigation of
earthquake induced pounding between adjacent buildings. Earthquake
Engineering & Structural Dynamics. 1992.21(4):289~302
67. Kobori T, Yamada T, Takenaka Y, Maeda Y, Nishimura I. Effect of Dynamic
Tuned Connector On Reduction Of Seismic Response: Application To Adjacent
Office Buildings. In: Proceedings of the Ninth World Conference on Earthquake
- 117 -
北京工业大学工学硕士学位论文
Engineering, vol. V, Tokyo–Kyoto, Japan, 1988：773–778.
68. Xu Y. L.， Zhan S.，Ko J. M., Zhang W. S., Experimental Investigation of
Adjacent Buildings Connected By Fluid Damper. Earthquake Engineering and
Structural Dynamics. 1999，28(6)：609~631
69. Xu Y. L., Yang Z., Lu X. L., Inelastic Seismic Response Of Adjacent Buildings
Linked By Fluid Dampers. Structural Engineering And Mechanics, 2003,15(5):
513~534
70. 闫维明，房屋建筑减震控制理论和试验研究．博士后研究工作报告．中国地
震局工程力学研究所．1998
71. 闫维明，陆赢祺，彭凌云. 两结构高效阻尼控制体系试验研究. 地震工程与
工程振动. 2003,23(3):160~169
72. 闫维明，陆瀛祺，彭凌云. 两结构高效阻尼控制体系非线性地震反应分析. 土
木工程学报. 2003,36(5)：58~63