﻿ 大跨度斜拉桥非线性振动模型与理论研究进展
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 力学学报  2016, Vol. 48 Issue (3): 519-535  DOI: 10.6052/0459-1879-15-436 0

### 引用本文 [复制中英文]

[复制中文]
Kang Houjun, Guo Tieding, Zhao Yueyu. REVIEW ON NONLINEAR VIBRATION AND MODELING OF LARGE SPAN CABLE-STAYED BRIDGE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 519-535. DOI: 10.6052/0459-1879-15-436.
[复制英文]

### 文章历史

2015-12-03 收稿
2016-01-18 录用
2016-03-11 网络版发表.

1 斜拉索的非线性振动

 图1 斜拉索和悬索构形 Fig.1 Inclined cable and suspension cable
1.1 力学模型与数学模型

1.1.1 考虑弯曲刚度的斜拉索动力学模型

 $EA{\left[ {u' + y'v' + \frac{1}{2}\left( {{{v'}^2} + {{w'}^2}} \right)} \right]^\prime } + {P_u} - {\mu _u}\dot u = m\ddot u$ (1)
 $\begin{array}{l} {\left\{ {Hv' + EA\left( {y' + v'} \right)\left[ {u' + y'v' + \frac{1}{2}\left( {{{v'}^2} + {{w'}^2}} \right)} \right]} \right\}^\prime } + \\ {P_v} - {\mu _v}\dot v - EIv'''' = m\ddot v \end{array}$ (2)
 $\begin{array}{l} {\left\{ {Hw' + EAw'\left[ {u' + y'v' + {\textstyle{1 \over 2}}\left( {{{v'}^2} + {{w'}^2}} \right)} \right]} \right\}^\prime } + \\ {P_w} - {\mu _w}\dot w - EIw'''' = m\ddot w \end{array}$ (3)

1.1.2 考虑温度影响的斜拉索动力学模型

 $H = {H_0} - EA\alpha \Delta T$ (4)

 $Hv{\rm{''}} + EAe(t)\left( {y{\rm{''}} + v{\rm{''}}} \right) = m\ddot v$ (5)

 $e(t) = \frac{1}{L}\int_L {\left( {y'v' + {\textstyle{1 \over 2}}v{\rm{''}}} \right)} dx$ (6)
$L$为索长.

1.1.3 CFRP斜拉索动力学模型

 ${\left[ {{E_{11}}A(e + y{\rho _1} + z - y){T_{11}}} \right]^\prime } + {q_1} = m\ddot u + {c_1}\dot u$ (7)
 ${\left[ {{E_{11}}A(e + y{\rho _1} + z - y){T_{12}}} \right]^\prime } + {q_2} = m\ddot v + {c_2}\dot v$ (8)
 ${\left[ {{E_{11}}A(e + y{\rho _1} + z - y){T_{13}}} \right]^\prime } + {q_3} = m\ddot w + {c_3}\dot w$ (9)
 $\begin{array}{l} \{ {E_{11}}ye + [{G_{13}}{y^2} + {G_{12}}({z^2} - 2yz)]{\rho _{11}}\} ' - {E_{11}}ye{\rho _2} - \\ {E_{11}}ze{\rho _3} + {q_4} = J\ddot \varphi + {c_4}\dot \varphi \end{array}$ (10)

1.1.4 考虑支座运动的斜拉索动力学模型

 $\begin{array}{l} mv + {c_2}v + EA{\left( {\frac{{mg}}{{{T_x}}}} \right)^2}\int_0^L {\frac{v}{L}} ds = \\ \left[ {\frac{{{T_x}}}{{\sin \alpha }} + \frac{{EA}}{L}({v_B}\cos \alpha - {u_A}\sin \alpha )} \right]\frac{{{\partial ^2}v}}{{\partial {s^2}}} + {f_v} \end{array}$ (11)

1.1.5 斜拉索振动控制力学模型

 $\begin{array}{l} \ddot v + {c_v}\dot v - \frac{{v{\rm{''}}}}{{{\pi ^2}}} - \frac{\alpha }{{{\pi ^2}}}(y{\rm{''}} + v{\rm{''}})\int_0^1 {\left[ {y'v' + \frac{1}{2}({{v'}^2} + {{w'}^2})} \right]} dx = \\ - {c_d}\dot v(t - \tau )\delta (x - a) + f \end{array}$ (12)
 $\begin{array}{l} \ddot w + {c_w}\dot w - \frac{{w{\rm{''}}}}{{{\pi ^2}}} - \frac{\alpha }{{{\pi ^2}}}w{\rm{''}}\int_0^1 {\left[ {y'v' + \frac{1}{2}({{v'}^2} + {{w'}^2})} \right]} dx = \\ - {c_d}\dot w(t - \tau )\delta (x - a) \end{array}$ (13)

1.2 研究方法

1.2.1 多尺度法

 $\begin{array}{l} {q_i} = \varepsilon {q_{i1}}({T_0},{T_1},{T_2}) + {\varepsilon ^2}{q_{i2}}({T_0},{T_1},{T_2}) + \\ {\varepsilon ^3}{q_{i3}}({T_0},{T_1},{T_2}) + \cdots \end{array}$ (14)

 ${q_i} = \varepsilon {q_{i1}}({T_0},{T_2}) + {\varepsilon ^3}{q_{i3}}({T_0},{T_2}) + \cdots$ (15)

 $\begin{array}{l} {q_i} = \varepsilon {q_{i1}}({T_0},{T_2}) + {\varepsilon ^2}{q_{i2}}({T_0},{T_2}) + \\ {\varepsilon ^3}{q_{i3}}({T_0},{T_2}) + \cdots \end{array}$ (16)

 $\begin{array}{l} {q_i} = {q_{i1}}({T_0},{T_1},{T_2}) + {\varepsilon ^1}{q_{i2}}({T_0},{T_1},{T_2}) + \\ {\varepsilon ^2}{q_{i3}}({T_0},{T_1},{T_2}) + \cdots \end{array}$ (17)

1.2.2 实验研究

2 梁的非线性振动

2.1 弹性地基梁动力响应研究

 图2 弹性地基梁及地基模型 Fig.2 Beam on elastic foundation and its model

2.2 桩-土系统动力响应研究

 图3 桩-土相互作用体系 Fig.3 Soil-pile interaction system

3 斜拉桥整体模型的非线性振动

 图4 索-梁一般模型 Fig.4 Simple cable-stayed beam model
3.1 索-梁组合模型动力特性研究

 图5 索-梁模型的实测响应 Fig.5 Measured response of cable-beam

3.2 其他斜拉桥动力学模型及理论研究

 图6 斜拉桥的质量块模型 Fig.6 Multi-mass models of cable-stayed bridge
 图7 索-质量块参数振动模型 Fig.7 Cable-mass model for parametric vibration of cable-stayed bridge
 图8 索-质量块的参强激励模型 Fig.8 Cable-mass model for parametric and forced vibration of cable-stayed bridge
 图9 斜拉桥的多体模型 Fig.9 Multi-body model of cable-stayed bridge

3.3 斜拉桥非线性动力学的实验研究

 图10 斜拉桥实验模型 Fig.10 Bridge model

4 展望

(1)考虑边界约束和弹性支承斜拉索的非线性振动问题

(2)斜拉桥下部结构非线性振动

(3)斜拉桥大系统的非线性振动

(4)高效有限元法

(5)高维非线性动力学求解方法

(6)斜拉桥大系统的随机振动

(7)斜拉桥振动能量采集与控制