地基—非线性结构相互作用空间效应的动力学行为研究
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摘要
中文摘要
本文应用非线性动力学理论研究了弹性地基-多自由度非线性结构相
互作用体系非线性耦合运动中的内共振问题，主要内容如下：
1．采用非对称结构的恢复力为双线性模型，地基土则简化为集中参
数模型，建立了无质量基础的地基－结构空间相互作用模型，利用能量
法得到了结构水平侧移和扭转位移相耦合的振动方程，为以下的分析建
立了基础。
2．用多尺度法对上面建立的方程进行分析，得到了体系发生内共振
的条件，通过算例讨论了水平刚度与扭转刚度成某一关系时的振动特性
与能量传递，并且利用数值分析方法得到了体系发生主共振时的数值解，
将其与摄动解进行了比较。
3．采用多尺度法进行研究，得到了体系在强激励下产生超谐和亚谐
共振的条件，分析了体系处于超谐和亚谐共振时的非线性动力学行为，
并且把多尺度法求得的摄动解同数值解进行了比较。
本文通过非线性动力学理论提示了 SSI 非线性耦合体系不同于线性耦
合体系的主共振以及超谐和亚谐共振现象，得到了一些有益的结论，为
结构抗震(振)分析与结构破坏预测提供了更为深刻的理论依据。
In this paper,the nonlinear dynamical theories are used to study the
internal resonance of the elastic foundation-multiple degree nonlinear
structure interaction system under the nonlinear coupling of horizontal
oscillation and torsion oscillation.The work of this paper are as follows:
1. A bilinear stiffness model is taken as a restoring force of the
unsymmetrical structure,and a lumped parameter model is used to simulate
the viscoelastic halfspace,then the massless foundation- soil-structure
interaction model considering spatial effect is established.The Lagrange
energy equation is used to build the nonlinear coupling equation of
horizontal motion and torsion motion and the following research is based on
the equation.
2. Perturbation analysis are used to analyze the above equation,the
situations of the internal resonance are obtained. Through the analyses of
numerical examples,the vibration characteristics and energy transfer
behavior are discussed while some relation lies in the horizontal stiffness
and the torsion stiffness.And the numerical analysis is carried out.The
perturbation solutions and numerical solutions are compared.
3. Perturbation analysis are used to analyze situations of superharmonic
resonance and subharmonic resonance under strong excitation.The nonlinear
dynamical behavior of superharmonic resonance and subharmonic resonance
is studied.The perturbation solutions and numerical solutions are compared.
Primary resonance and superharmonic resonance and subharmonic
resonance of nonlinear coupling system are studied,which are different from
linear coupling system.Some helpful conclusions are obtained, and these
can provide more profound theoretic reference for the earthquake(vibration)
resistant design and analysis.
本文应用非线性动力学理论研究了弹性地基-多自由度非线性结构相
互作用体系非线性耦合运动中的内共振问题，主要内容如下：
1．采用非对称结构的恢复力为双线性模型，地基土则简化为集中参
数模型，建立了无质量基础的地基－结构空间相互作用模型，利用能量
法得到了结构水平侧移和扭转位移相耦合的振动方程，为以下的分析建
立了基础。
2．用多尺度法对上面建立的方程进行分析，得到了体系发生内共振
的条件，通过算例讨论了水平刚度与扭转刚度成某一关系时的振动特性
与能量传递，并且利用数值分析方法得到了体系发生主共振时的数值解，
将其与摄动解进行了比较。
3．采用多尺度法进行研究，得到了体系在强激励下产生超谐和亚谐
共振的条件，分析了体系处于超谐和亚谐共振时的非线性动力学行为，
并且把多尺度法求得的摄动解同数值解进行了比较。
本文通过非线性动力学理论提示了 SSI 非线性耦合体系不同于线性耦
合体系的主共振以及超谐和亚谐共振现象，得到了一些有益的结论，为
结构抗震(振)分析与结构破坏预测提供了更为深刻的理论依据。
In this paper,the nonlinear dynamical theories are used to study the
internal resonance of the elastic foundation-multiple degree nonlinear
structure interaction system under the nonlinear coupling of horizontal
oscillation and torsion oscillation.The work of this paper are as follows:
1. A bilinear stiffness model is taken as a restoring force of the
unsymmetrical structure,and a lumped parameter model is used to simulate
the viscoelastic halfspace,then the massless foundation- soil-structure
interaction model considering spatial effect is established.The Lagrange
energy equation is used to build the nonlinear coupling equation of
horizontal motion and torsion motion and the following research is based on
the equation.
2. Perturbation analysis are used to analyze the above equation,the
situations of the internal resonance are obtained. Through the analyses of
numerical examples,the vibration characteristics and energy transfer
behavior are discussed while some relation lies in the horizontal stiffness
and the torsion stiffness.And the numerical analysis is carried out.The
perturbation solutions and numerical solutions are compared.
3. Perturbation analysis are used to analyze situations of superharmonic
resonance and subharmonic resonance under strong excitation.The nonlinear
dynamical behavior of superharmonic resonance and subharmonic resonance
is studied.The perturbation solutions and numerical solutions are compared.
Primary resonance and superharmonic resonance and subharmonic
resonance of nonlinear coupling system are studied,which are different from
linear coupling system.Some helpful conclusions are obtained, and these
can provide more profound theoretic reference for the earthquake(vibration)
resistant design and analysis.
引文
参考文献
[1]谷岩，地基－单自由度结构相互作用体系的非线性动力学行为研究：
[硕士学位论文],天津：天津大学,2000
[2]姜忻良，谷岩，地基-非线性结构相互作用体系的主共振与分叉，地
震工程与工程振动，2000，20(3)：22~27
[3]谷岩，姜忻良，地基-非线性结构相互作用体系的超谐和亚谐振动，
地震工程与工程振动，2001,21(1)：56~60
[4]姜忻良，郑刚，常好诵，结构－地基相互作用系统主共振及分叉研究，
振动工程学报，2001，14(2)：233~236
[5]常好诵，结构－地基相互作用系统主共振及分叉研究：[硕士学位论
文]，天津：天津大学，2000
[6]姜忻良，贾勇，结构-滑移基础-土非线性相互作用体系主共振与分叉，
工程力学，2002,19(5)：103~107
[7]钱家欢，殷宗泽，土工原理与计算，中国水利水电出版社，1996
[8]陈国兴等，土的动模量和阻尼比的经验估计，地震工程与工程振动，
1995,15(1)：73~83
[9]严人觉等，动力基础半空间理论概论，北京：中国建筑工业出版社，
1981：193~205
[10]陈文良,分析动力学,上海交通大学出版社,1991，39~51
[11]王海期，非线性振动，北京：高等教育出版社，1992.5
[12]陈予恕, 丰定国，非线性振动，天津：科学技术出版社，1983
[13]王洪礼，张琪昌，非线性动力学理论及应用，天津：天津科学技术
出版社，2002，191~198，262~266
[14]郑俊武，船舶内共振动力行为的研究：[硕士学位论文]，天津：天
津大学，1998
[15]骆兰月，结构内共振动力行为研究：[硕士学位论文]，天津：天津
大学，2003
[16]包世华，方鄂华，高层建筑结构设计(第二版)，北京：清华大学出
版社，1990
[17]车宏亚，钢筋混泥土结构原理，天津：天津大学出版社，1990
[18]李国强等著,建筑结构抗震设计,北京：中国建筑工业出版社,2002，
65~86
- 56 -
天津大学硕士学位论文 参考文献
[19]陈予恕，徐健学，线性动力学、非线性振动和稳定性，北京：科学
技术出版社，1995
[20]冯康等编，数值计算方法，北京：国防工业出版社，1978，345~385
[21]张宜华，精通 MATLAB 5，北京：清华大学出版社，1999
[22]苏金明，阮沈勇，MATLAB6.1 实用指南，北京：电子工业出版社，
2002
[23]叶宗泽，杨万禄，常微分方程组与运动稳定理论，天津：天津大学
出版社，1985，91~95
[24]李英杰，陈丹玲，牛顿法的一种新的迭代格式及程序实现，杭州电
子工业学报，2002，22(1)：62~66
[25]Ali Hasan Nayfeh, Dean T. Mook. Nonlinear oscillations, New York:
Wiley, c1979
[26]Derek P. Atherton, Stability of nonlinear systems, Chichester [Eng.];
New York: Research Studies Press,c1981
[27]F. PELLICANO and F. VESTRONI. Nonlinear dynamics and
bifurcations of an axially moving beams. Journal of vibration and Acoustics.
2000, 122: 21-30
[28]Tae-Woo Kim, Ji-Hwan Kim. Parametric instability of a cross-ply
laminated beam with viscoelastic properties under a periodic force.
Composite Structures.2001, 51: 205-209
[29]Y.-M. Wang. THE DYNAMIC ANALYSIS OF A BEAM-MASS
SYSTEM DUE TO THE OCCURRENCE OF TWO-COMPONENT
PARAMETRIC RESONANCE. Journal of Sound and Vibration 2002, 258(5):
951~967
[30]A. Maccari. Vibration control for the primary resonance of a cantilever
beam by a time delay state feedback. Journal of Sound and Vibration 2003,
259(2): 241~251
[31]Vincenzo Gattulli,Marco Lepidi. Nonlinear interactions in the planar
dynamics of cable-stayed beam. International Journal of Solids and
Structures.2003, 40: 4729-4748
[32]A. A. Popov. Parametric resonance in cylindrical shells: a case study in
the nonlinear vibration of structural shells. Engineering Structures. 2003,
25: 789–799
- 57 -
天津大学硕士学位论文 参考文献
[33]Brownjohn, J. M. W., Lee, J., Cheong, B. Dynamic performance of a
curved cable-stayed bridge. Engng. Structures. 1999, 21:1015-1027
[34]R. Heuer, F. Ziegler. Nonlinear resonance phenomena of panel-type
structures. Computers and Structures.1998, 67:65-70
[35] H.R.Oz,M.PAKDEM IRLI
,E.OZKAYAANDM.YILMAZ .NON-LINEAR VIBRATIONS
OF A SLIGHTLY CURVED BEAM RESTING ON A NON-LINEAR
ELASTIC FOUNDATION. Journal of Sound and Vibration. 1998,
212(2):295-309
[36]Sekhar Chandra Dutta,Rana Roy.A critical review on idealization and
modeling for interaction among soil-foundation-structure system.Computers
and structures,2002,80:1579-1594
[37]O.von Estorff,M.Firuziaan.Coupled BEM/FEM approach for nonlinear
soil/structure interaction. Engineering Analysis with Boundary
Elements,2000,24:715-725
[38]Bang-Fuh Chen. Dynamic responses of coastal structures during
earthquakes including sediment-sea-structure interaction. Soil Dynamics
and Earthquake Engineering.2000,20:445-467
[39]Y.X.Cai a,P.L.Gould b,C.S.Desai c. Nonlinear analysis of 3D seismic
interaction of soil-pile-structure systems and application. Engineering
Structures.2000,22:191-199
[40]T. Inabaa, H. Dohia, K. Okutaa, T. Satoa, H. Akagib.Nonlinear
response of surface soil and NTT building due to soil-structure interaction
during the 1995 Hyogo-ken Nanbu (Kobe) earthquake.Soil Dynamics and
Earthquake Engineering .2000,20:289-300
[41]T. Kobayashi a, K. Yoshikawa b, E. Takaoka b, M. Nakazawa c, Y.
Shikama c. Time history nonlinear earthquake response analysis considering
materials and geometrical nonlinearity. Nuclear Engineering and Design.
2000, 212:145-154
[1]谷岩，地基－单自由度结构相互作用体系的非线性动力学行为研究：
[硕士学位论文],天津：天津大学,2000
[2]姜忻良，谷岩，地基-非线性结构相互作用体系的主共振与分叉，地
震工程与工程振动，2000，20(3)：22~27
[3]谷岩，姜忻良，地基-非线性结构相互作用体系的超谐和亚谐振动，
地震工程与工程振动，2001,21(1)：56~60
[4]姜忻良，郑刚，常好诵，结构－地基相互作用系统主共振及分叉研究，
振动工程学报，2001，14(2)：233~236
[5]常好诵，结构－地基相互作用系统主共振及分叉研究：[硕士学位论
文]，天津：天津大学，2000
[6]姜忻良，贾勇，结构-滑移基础-土非线性相互作用体系主共振与分叉，
工程力学，2002,19(5)：103~107
[7]钱家欢，殷宗泽，土工原理与计算，中国水利水电出版社，1996
[8]陈国兴等，土的动模量和阻尼比的经验估计，地震工程与工程振动，
1995,15(1)：73~83
[9]严人觉等，动力基础半空间理论概论，北京：中国建筑工业出版社，
1981：193~205
[10]陈文良,分析动力学,上海交通大学出版社,1991，39~51
[11]王海期，非线性振动，北京：高等教育出版社，1992.5
[12]陈予恕, 丰定国，非线性振动，天津：科学技术出版社，1983
[13]王洪礼，张琪昌，非线性动力学理论及应用，天津：天津科学技术
出版社，2002，191~198，262~266
[14]郑俊武，船舶内共振动力行为的研究：[硕士学位论文]，天津：天
津大学，1998
[15]骆兰月，结构内共振动力行为研究：[硕士学位论文]，天津：天津
大学，2003
[16]包世华，方鄂华，高层建筑结构设计(第二版)，北京：清华大学出
版社，1990
[17]车宏亚，钢筋混泥土结构原理，天津：天津大学出版社，1990
[18]李国强等著,建筑结构抗震设计,北京：中国建筑工业出版社,2002，
65~86
- 56 -
天津大学硕士学位论文 参考文献
[19]陈予恕，徐健学，线性动力学、非线性振动和稳定性，北京：科学
技术出版社，1995
[20]冯康等编，数值计算方法，北京：国防工业出版社，1978，345~385
[21]张宜华，精通 MATLAB 5，北京：清华大学出版社，1999
[22]苏金明，阮沈勇，MATLAB6.1 实用指南，北京：电子工业出版社，
2002
[23]叶宗泽，杨万禄，常微分方程组与运动稳定理论，天津：天津大学
出版社，1985，91~95
[24]李英杰，陈丹玲，牛顿法的一种新的迭代格式及程序实现，杭州电
子工业学报，2002，22(1)：62~66
[25]Ali Hasan Nayfeh, Dean T. Mook. Nonlinear oscillations, New York:
Wiley, c1979
[26]Derek P. Atherton, Stability of nonlinear systems, Chichester [Eng.];
New York: Research Studies Press,c1981
[27]F. PELLICANO and F. VESTRONI. Nonlinear dynamics and
bifurcations of an axially moving beams. Journal of vibration and Acoustics.
2000, 122: 21-30
[28]Tae-Woo Kim, Ji-Hwan Kim. Parametric instability of a cross-ply
laminated beam with viscoelastic properties under a periodic force.
Composite Structures.2001, 51: 205-209
[29]Y.-M. Wang. THE DYNAMIC ANALYSIS OF A BEAM-MASS
SYSTEM DUE TO THE OCCURRENCE OF TWO-COMPONENT
PARAMETRIC RESONANCE. Journal of Sound and Vibration 2002, 258(5):
951~967
[30]A. Maccari. Vibration control for the primary resonance of a cantilever
beam by a time delay state feedback. Journal of Sound and Vibration 2003,
259(2): 241~251
[31]Vincenzo Gattulli,Marco Lepidi. Nonlinear interactions in the planar
dynamics of cable-stayed beam. International Journal of Solids and
Structures.2003, 40: 4729-4748
[32]A. A. Popov. Parametric resonance in cylindrical shells: a case study in
the nonlinear vibration of structural shells. Engineering Structures. 2003,
25: 789–799
- 57 -
天津大学硕士学位论文 参考文献
[33]Brownjohn, J. M. W., Lee, J., Cheong, B. Dynamic performance of a
curved cable-stayed bridge. Engng. Structures. 1999, 21:1015-1027
[34]R. Heuer, F. Ziegler. Nonlinear resonance phenomena of panel-type
structures. Computers and Structures.1998, 67:65-70
[35] H.R.Oz,M.PAKDEM IRLI
,E.OZKAYAANDM.YILMAZ .NON-LINEAR VIBRATIONS
OF A SLIGHTLY CURVED BEAM RESTING ON A NON-LINEAR
ELASTIC FOUNDATION. Journal of Sound and Vibration. 1998,
212(2):295-309
[36]Sekhar Chandra Dutta,Rana Roy.A critical review on idealization and
modeling for interaction among soil-foundation-structure system.Computers
and structures,2002,80:1579-1594
[37]O.von Estorff,M.Firuziaan.Coupled BEM/FEM approach for nonlinear
soil/structure interaction. Engineering Analysis with Boundary
Elements,2000,24:715-725
[38]Bang-Fuh Chen. Dynamic responses of coastal structures during
earthquakes including sediment-sea-structure interaction. Soil Dynamics
and Earthquake Engineering.2000,20:445-467
[39]Y.X.Cai a,P.L.Gould b,C.S.Desai c. Nonlinear analysis of 3D seismic
interaction of soil-pile-structure systems and application. Engineering
Structures.2000,22:191-199
[40]T. Inabaa, H. Dohia, K. Okutaa, T. Satoa, H. Akagib.Nonlinear
response of surface soil and NTT building due to soil-structure interaction
during the 1995 Hyogo-ken Nanbu (Kobe) earthquake.Soil Dynamics and
Earthquake Engineering .2000,20:289-300
[41]T. Kobayashi a, K. Yoshikawa b, E. Takaoka b, M. Nakazawa c, Y.
Shikama c. Time history nonlinear earthquake response analysis considering
materials and geometrical nonlinearity. Nuclear Engineering and Design.
2000, 212:145-154
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