﻿ 漂浮体系斜拉桥黏滞阻尼器参数的简化计算
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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (5): 574-579, 638.  DOI: 10.11908/j.issn.0253-374x.2018.05.002 0

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XU Yan, TONG Chuan, LI Jianzhong. Simplified Calculation of Viscous Damper Parameter for Floating-system Cable-stayed Bridge[J]. Journal of Tongji University (Natural Science), 2018, 46(5): 574-579, 638. DOI: 10.11908/j.issn.0253-374x.2018.05.002

文章历史

Simplified Calculation of Viscous Damper Parameter for Floating-system Cable-stayed Bridge
XU Yan , TONG Chuan , LI Jianzhong
State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China
Abstract: In the light of velocity pulse characteristics of near-fault ground motion and the dominated vibration mode to the dynamic response of floating-system cable-stayed bridge, a three-mass simplified dynamic model was derived and the differential dynamic equation was established. Based on the method of equivalent damping ratio and equivalent linearization of nonlinear viscous dampers, design formulas for determining the damping coefficient of nonlinear viscous dampers of the bridge subjected to near-fault ground motion were deduced. Finally, the accuracy of the proposed design formulas for nonlinear viscous dampers in predicting the damping ratio of the bridge was verified by a real cable-stayed bridge.
Key words: cable-stayed bridge    viscous damper    three-mass simplified dynamic model    velocity pulse    equivalent damping ratio

1 三质点简化动力模型运动微分方程的建立与求解

 图 1 三质点简化动力模型 Fig.1 Three-mass simplified dynamic model

 $\mathit{\boldsymbol{M\ddot u}}\left( t \right) + \mathit{\boldsymbol{C\dot u}}\left( t \right) + \mathit{\boldsymbol{Ku}}\left( t \right) = \mathit{\boldsymbol{p}}\left( t \right) = - \mathit{\boldsymbol{MI}}{{\ddot u}_g}\left( t \right)$ (1)

 $\begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{m_1}}&0&0\\ 0&{{m_2}}&0\\ 0&0&{{m_3}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{{\ddot u}_1}}\\ {{{\ddot u}_2}}\\ {{{\ddot u}_3}} \end{array}} \right\} + \left[ {\begin{array}{*{20}{c}} {{c_{11}}}&{{c_{12}}}&{{c_{13}}}\\ {{c_{21}}}&{{c_{22}}}&{{c_{23}}}\\ {{c_{31}}}&{{c_{32}}}&{{c_{33}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{{\dot u}_1}}\\ {{{\dot u}_2}}\\ {{{\dot u}_3}} \end{array}} \right\} + \\ \;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} {{k_{11}}}&{{k_{12}}}&0\\ {{k_{21}}}&{{k_{22}}}&{{k_{23}}}\\ 0&{{k_{32}}}&{{k_{33}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{u_1}}\\ {{u_2}}\\ {{u_3}} \end{array}} \right\} = - \left[ {\begin{array}{*{20}{c}} {{m_1}}&0&0\\ 0&{{m_2}}&0\\ 0&0&{{m_3}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} 1\\ 1\\ 1 \end{array}} \right\}{{\ddot u}_g} \end{array}$ (2)

 $\left| {\mathit{\boldsymbol{K}} - \omega _n^2\mathit{\boldsymbol{M}}} \right| = 0,n = 1,2,3$ (3)

 $\omega _1^2 = A - \frac{C}{A} + B$ (4)
 $\omega _2^2 = \frac{{C - {A^2} + 2AB}}{{2A}} - \frac{{\sqrt 3 \left( {A + \frac{C}{A}} \right){\rm{i}}}}{2}$ (5)
 $\omega _3^2 = \frac{{C - {A^2} + 2AB}}{{2A}} + \frac{{\sqrt 3 \left( {A + \frac{C}{A}} \right){\rm{i}}}}{2}$ (6)

 $\begin{array}{l} A = \left\{ {{{\left[ {{C^3} + {{\left( {E - \frac{{{F^3}}}{{27m_1^3m_2^3m_3^3}} + D} \right)}^2}} \right]}^{\frac{1}{2}}} - E - D + } \right.\\ \;\;\;\;\;\;\;{\left. {\frac{{{F^3}}}{{27m_1^3m_2^3m_3^3}}} \right\}^{\frac{1}{3}}} \end{array}$
 $B = \frac{F}{{3{m_1}{m_2}{m_3}}}$
 $C = \frac{G}{{3{m_1}{m_2}{m_3}}} - \frac{{{F^2}}}{{9m_1^2m_2^2m_3^2}}$
 $D = \frac{{FG}}{{6m_1^2m_2^2m_3^2}}$
 $E = \frac{{{k_{11}}{k_{23}}{k_{32}} - {k_{11}}{k_{22}}{k_{33}} + {k_{12}}{k_{21}}{k_{33}}}}{{2{m_1}{m_2}{m_3}}}$
 $F = {k_{11}}{m_2}{m_3} + {k_{22}}{m_1}{m_3} + {k_{33}}{m_1}{m_2}$
 $\begin{array}{l} G = {k_{11}}{k_{22}}{m_3} - {k_{12}}{k_{21}}{m_3} + {k_{11}}{k_{33}}{m_2} + {k_{22}}{k_{33}}{m_1} - \\ \;\;\;\;\;\;{k_{23}}{k_{32}}{m_1} \end{array}$

 ${\mathit{\boldsymbol{\varphi }}_n} = \left( {\begin{array}{*{20}{c}} {{\varphi _{1n}}}\\ {{\varphi _{2n}}}\\ {{\varphi _{3n}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1\\ {\frac{{{k_{11}} - {m_1}\omega _n^2}}{{ - {k_{12}}}}}\\ {\left( {\frac{{{k_{11}} - {m_1}\omega _n^2}}{{ - {k_{12}}}}} \right)\frac{{{k_{32}}}}{{{k_{33}} - {m_3}\omega _n^2}}} \end{array}} \right),n = 1,2,3$ (7)

 ${\mathit{\boldsymbol{u}}_n}\left( t \right) = {\mathit{\boldsymbol{\varphi }}_n}{Y_n}\left( t \right),n = 1,2,3$ (8)

 $\begin{array}{l} \mathit{\boldsymbol{u}}\left( t \right) = {\mathit{\boldsymbol{\varphi }}_1}{Y_1}\left( t \right) + {\mathit{\boldsymbol{\varphi }}_2}{Y_2}\left( t \right) + {\mathit{\boldsymbol{\varphi }}_3}{Y_3}\left( t \right) = \\ \;\;\;\;\;\;\;\;\;\;\sum\limits_{n = 1}^3 {{\mathit{\boldsymbol{\varphi }}_n}{Y_n}\left( t \right)} \end{array}$ (9)

 ${M_n}{{\ddot Y}_n}\left( t \right) + {C_n}{{\dot Y}_n}\left( t \right) + {K_n}{Y_n}\left( t \right) = {P_n}\left( t \right)$ (10)

 ${{\ddot Y}_n}\left( t \right) + 2{\xi _n}{\omega _n}{{\dot Y}_n}\left( t \right) + \omega _n^2{Y_n}\left( t \right) = \frac{{{P_n}\left( t \right)}}{{{M_n}}}$ (11)

 ${{\dot u}_g}\left( t \right) = V\sin \left( {{\omega _{\rm{p}}}t} \right),0 \le t \le T$ (12)
 ${{\ddot u}_g}\left( t \right) = {\omega _{\rm{p}}}V\cos \left( {{\omega _{\rm{p}}}t} \right),0 \le t \le T$ (13)
 图 2 近断层地震动速度和加速度脉冲模拟 Fig.2 Velocity and acceleration pulse simulation of near-fault ground motion

 ${Y_n}\left( t \right) = \\\left\{ \begin{array}{l} \left[ {{A_n}\cos \left( {{\omega _n}t} \right) + {B_n}\sin \left( {{\omega _n}t} \right)} \right]\exp \left( { - {\xi _n}{\omega _n}t} \right) + {G_n}\cos \left( {{\omega _{\rm{p}}}t} \right) + {D_n}\sin \left( {{\omega _{\rm{p}}}t} \right),\left( {0,T} \right)\\ \left[ {{u_{T,n}}\cos \left( {{\omega _n}t} \right) + \left( {\frac{{{{\dot u}_{T,n}} + {u_{T,n}}{\xi _n}{\omega _n}}}{{{\omega _n}}}} \right)\sin \left( {{\omega _n}t} \right)} \right]\exp \left( { - {\xi _n}{\omega _n}t} \right),\left( {T, + \infty } \right) \end{array} \right.$ (14)

 ${A_n} = {u_{{\rm{st}},n}}\frac{{ - 1 + \beta _n^2}}{{{{\left( {1 - \beta _n^2} \right)}^2} + {{\left( {2{\xi _n}{\beta _n}} \right)}^2}}}$
 ${B_n} = {u_{{\rm{st}},n}}\frac{{ - {\xi _n}\left( {1 + \beta _n^2} \right)}}{{{{\left( {1 - \beta _n^2} \right)}^2} + {{\left( {2{\xi _n}{\beta _n}} \right)}^2}}}$
 ${G_n} = {u_{{\rm{st}},n}}\frac{{1 - \beta _n^2}}{{{{\left( {1 - \beta _n^2} \right)}^2} + {{\left( {2{\xi _n}{\beta _n}} \right)}^2}}}$
 ${D_n} = {u_{{\rm{st}},n}}\frac{{2{\xi _n}{\beta _n}}}{{{{\left( {1 - \beta _n^2} \right)}^2} + {{\left( {2{\xi _n}{\beta _n}} \right)}^2}}}$

 $\mathit{\boldsymbol{u}}\left( t \right) = \left( {\begin{array}{*{20}{c}} {{u_1}\left( t \right)}\\ {{u_2}\left( t \right)}\\ {{u_3}\left( t \right)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{\varphi _{11}}}&{{\varphi _{12}}}&{{\varphi _{13}}}\\ {{\varphi _{21}}}&{{\varphi _{22}}}&{{\varphi _{23}}}\\ {{\varphi _{31}}}&{{\varphi _{32}}}&{{\varphi _{33}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{Y_1}\left( t \right)}\\ {{Y_2}\left( t \right)}\\ {{Y_3}\left( t \right)} \end{array}} \right)$ (15)

 ${u_1}\left( t \right) = {\varphi _{11}}{Y_1}\left( t \right) + {\varphi _{12}}{Y_2}\left( t \right) + {\varphi _{13}}{Y_3}\left( t \right)$ (16)

 $\Delta u\left( t \right) = \sum\limits_{n = 1}^3 {\left( {{\varphi _{1n}} - {\varphi _{3n}}} \right){Y_n}\left( t \right)}$ (17)
2 线性黏滞阻尼器参数的简化计算

FEMA[5]提供了较为实用和方便的公式来计算建筑结构中附加阻尼器所提供的附加阻尼比，如下所示：

 ${\xi _{\rm{e}}} = {\xi _{\rm{0}}} + {\xi _{\rm{d}}}$ (18)

 $\mathit{\boldsymbol{C}} = a\mathit{\boldsymbol{M}} + b\mathit{\boldsymbol{K}}$ (19)

 $\left\{ \begin{array}{l} a = \frac{{2{\omega _1}{\omega _2}\left( {{\xi _1}{\omega _2} - {\xi _2}{\omega _1}} \right)}}{{\omega _2^2 - \omega _1^2}}\\ b = \frac{{2\left( {{\xi _2}{\omega _2} - {\xi _1}{\omega _1}} \right)}}{{\omega _2^2 - \omega _1^2}} \end{array} \right.$ (20)

 $\left\{ \begin{array}{l} a = \frac{{2{\omega _1}{\omega _2}\xi }}{{{\omega _1} + {\omega _2}}}\\ b = \frac{{2\xi }}{{{\omega _1} + {\omega _2}}} \end{array} \right.$ (21)

 $\mathit{\boldsymbol{C}} = \left( {\begin{array}{*{20}{c}} {\frac{{2\xi \left( {{k_{11}} + {m_1}{\omega _1}{\omega _2}} \right)}}{{{\omega _1} + {\omega _2}}}}&{\frac{{2\xi {k_{12}}}}{{{\omega _1} + {\omega _2}}}}&0\\ {\frac{{2\xi {k_{21}}}}{{{\omega _1} + {\omega _2}}}}&{\frac{{2\xi \left( {{k_{22}} + {m_2}{\omega _1}{\omega _2}} \right)}}{{{\omega _1} + {\omega _2}}}}&{\frac{{2\xi {k_{23}}}}{{{\omega _1} + {\omega _2}}}}\\ 0&{\frac{{2\xi {k_{32}}}}{{{\omega _1} + {\omega _2}}}}&{\frac{{2\xi \left( {{k_{33}} + {m_3}{\omega _1}{\omega _2}} \right)}}{{{\omega _1} + {\omega _2}}}} \end{array}} \right)$ (22)

 ${\mathit{\boldsymbol{C}}_{\rm{L}}} = \mathit{\boldsymbol{C}} + \left( {\begin{array}{*{20}{c}} {{c_{{\rm{dl}}}}}&0&{ - {c_{{\rm{dl}}}}}\\ 0&0&0\\ { - {c_{{\rm{dl}}}}}&0&{{c_{{\rm{dl}}}}} \end{array}} \right) = \\\left( {\begin{array}{*{20}{c}} {\frac{{2\xi \left( {{k_{11}} + {m_1}{\omega _1}{\omega _2}} \right)}}{{{\omega _1} + {\omega _2}}} + {c_{{\rm{dl}}}}}&{\frac{{2\xi {k_{12}}}}{{{\omega _1} + {\omega _2}}}}&{-c_{{\rm{dl}}}}\\ {\frac{{2\xi {k_{21}}}}{{{\omega _1} + {\omega _2}}}}&{\frac{{2\xi \left( {{k_{22}} + {m_2}{\omega _1}{\omega _2}} \right)}}{{{\omega _1} + {\omega _2}}}}&{\frac{{2\xi {k_{23}}}}{{{\omega _1} + {\omega _2}}}}\\ 0&{\frac{{2\xi {k_{32}}}}{{{\omega _1} + {\omega _2}}}}&{\frac{{2\xi \left( {{k_{33}} + {m_3}{\omega _1}{\omega _2}} \right)}}{{{\omega _1} + {\omega _2}}} + {c_{{\rm{dl}}}}} \end{array}} \right)$ (23)

 ${\mathit{\boldsymbol{\varphi }}_1} = \left( {\begin{array}{*{20}{c}} 1\\ {{\varphi _{21}}}\\ {{\varphi _{31}}} \end{array}} \right)$ (24)

 ${M_1} = {m_1} + {m_2}\varphi _{21}^2 + {m_3}\varphi _{31}^2$ (25)
 ${C_1} = {\left( {1 - {\varphi _{31}}} \right)^2}{c_{{\rm{dl}}}} + 2\xi \frac{{{k_{11}} + {k_{33}}\varphi _{31}^2 + {\varphi _{21}}\left( {2{k_{21}} + 2{k_{32}}{\varphi _{31}} + {k_{22}}{\varphi _{21}}} \right) + \left( {{m_1} + {m_2}\varphi _{21}^2 + {m_3}\varphi _{31}^2} \right){\omega _1}{\omega _2}}}{{{\omega _1} + {\omega _2}}}$ (26)

 $\begin{array}{l} {\xi _1} = \frac{{{C_1}}}{{2{M_1}{\omega _1}}} = \\ \frac{{2\left( {{\omega _1} + {\omega _2}} \right){{\left( {1 - {\varphi _{31}}} \right)}^2}{c_{{\rm{dl}}}} + 2\xi \left[ {{k_{11}} + {k_{33}}\varphi _{31}^2 + {\varphi _{21}}\left( {2{k_{21}} + 2{k_{32}}{\varphi _{31}} + {k_{22}}{\varphi _{21}}} \right) + \left( {{m_1} + {m_2}\varphi _{21}^2 + {m_3}\varphi _{31}^2} \right){\omega _1}{\omega _2}} \right]}}{{2\left( {{\omega _1} + {\omega _2}} \right)\left( {{m_1} + {m_2}\varphi _{21}^2 + {m_3}\varphi _{31}^2} \right)}} \end{array}$ (27)

 $\begin{array}{l} {c_{{\rm{dl}}}} = \\ \frac{{2{\xi _{\rm{e}}}\left( {{\omega _1} + {\omega _2}} \right)\left( {{m_1} + {m_2}\varphi _{21}^2 + {m_3}\varphi _{31}^2} \right) - 2\xi \left[ {{k_{11}} + {k_{33}}\varphi _{31}^2 + {\varphi _{21}}\left( {2{k_{21}} + 2{k_{32}}{\varphi _{31}} + {k_{22}}{\varphi _{21}}} \right) + \left( {{m_1} + {m_2}\varphi _{21}^2 + {m_3}\varphi _{31}^2} \right){\omega _1}{\omega _2}} \right]}}{{\left( {{\omega _1} + {\omega _2}} \right){{\left( {1 - {\varphi _{31}}} \right)}^2}}} \end{array}$ (28)

3 非线性黏滞阻尼器参数的简化计算

 ${f_{\rm{D}}} = c{\mathop{\rm sgn}} \left( {\dot u} \right){\left| {\dot u} \right|^\alpha }$ (29)

 ${c_{\rm{d}}} = \frac{{{c_{{\rm{dl}}}}{\rm{ \mathsf{ π} }}}}{{u_0^{\alpha - 1}{\omega ^{\alpha - 1}}}}\frac{{{\rm{\Gamma }}\left( {\alpha + 2} \right)}}{{{2^{\alpha + 2}}{{\rm{\Gamma }}^2}\left( {\alpha /2 + 1} \right)}} = \frac{{{c_{{\rm{dl}}}}{\rm{ \mathsf{ π} }}}}{{u_0^{\alpha - 1}{\omega ^{\alpha - 1}}}}\lambda$ (30)
 $\lambda = \frac{{{\rm{\Gamma }}\left( {\alpha + 2} \right)}}{{{2^{\alpha + 2}}{{\rm{\Gamma }}^2}\left( {\alpha /2 + 1} \right)}}$

4 实例验证

 图 3 某斜拉桥有限元模型 Fig.3 Finite element model of a cable-stayed bridge

4种模型在相同速度脉冲型地震动作用下的塔梁相对位移时程如表 1所示.

10%、20%、30%，速度指数取0.4时，模型1中非线性黏滞阻尼器阻尼系数和模型2中线性阻尼器阻尼系数如表 2所示.表 2中，目标等效阻尼比E为3%即相当于未设置附加黏滞阻尼器的情况.通过非线性时程分析、线性时程分析和数值计算得到不同等效阻尼比时4个模型的塔梁相对位移时程曲线，如图 4所示.

 图 4 塔梁相对位移时程 Fig.4 Relative displacement time histories

5 结论

(1) 三质点简化动力模型符合漂浮体系斜拉桥的纵桥向受力特点，在初步设计阶段可以替代全桥进行纵桥向动力特性和动力反应计算.

(2) 本文提出的黏滞阻尼器阻尼系数简化计算公式可避免斜拉桥初步设计阶段为选定阻尼器参数而进行的大量有限元计算，适用于以速度脉冲为主导的近断层地震作用下漂浮体系斜拉桥黏滞阻尼器参数简化计算.

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