MODELLING AND ANALYSIS OF THE IN-PLANE DYNAMICS OF CABLE-STAYED BRIDGES CONSIDERING THE PYLON-CABLE-BEAM COUPLING EFFECT
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摘要: 为研究塔−梁模态间耦合作用对斜拉桥整体模态及1:1内共振影响, 考虑由斜拉索初始垂度及横向振动大位移引起的几何非线性, 建立固结体系斜拉桥塔−索−梁耦合面内整体动力学模型. 其中桥塔和主梁被简化为参数质量体系, 基于塔−索、索−梁及塔−梁动态平衡关系推导得到斜拉桥整体模型的动力平衡方程. 通过有限差分法对参数体系的微分方程进行代数转换, 运用模态拖拽法和分离变量法得到结构的运动方程和模态函数, 观察到局部−整体模态间、主导−非主导模态间的模态翻转现象, 并采用Runge-Kutta积分方法对无阻尼的系统方程进行数值仿真, 系统分析了塔−梁耦合作用对1:1内共振的影响, 结果表明: 在文章塔−索−梁耦合模型中, 塔−梁耦合作用对结构的低阶对称及高阶整体模态没有影响作用, 塔或梁的振动模态完全主导了该阶结构整体模态; 塔−梁耦合作用对结构的低阶反对称整体模态具有显著影响作用, 此时结构整体模态呈现为塔和梁振动模态共同参与的混合整体模态. 拉索的振动模态与此两种整体模态1:1耦合都将产生剧烈内共振, 各构件的模态局部化程度系数是量化内共振参与度的重要因素. 在完全整体模态内共振中, 非主导模态不参与结构内共振, 能量转换仅发生在主导的构件模态与拉索的振动模态间; 在混合整体模态内共振中, 非主导模态将参与内共振能量转换并改变拉索振动响应的动力特性, 其影响效应随整体模态阶次与构件参数变化而变化.Abstract: To study the pylon-beam modal coupling effect on the global modes and 1:1 internal resonance of rigid-frame cable-stayed bridges, a novel in-plane dynamic model, considering the geometric nonlinearity caused by cables’ initial sag and lateral large displacement, is established in this paper. The bridge pylon and beam in the model are modelled as integrated systems consisting of discretized parametric segments. The dynamic equations of the whole structure are derived based on the dynamic connections between the pylon-cable, cable-beam, and pylon-beam nodes. Using the finite difference method, the dynamic balance differential equations of the parametric system are transformed algebraically. The whole bridge governing equations and modal functions are obtained through the modal drag method and variable separation method. Based on the simulation of the undamped equations via Runge-Kutta method of 4 ~ 5-order, the modal veering phenomena between local-global modes and dominant-nondominant modes are observed. The simulation results shows that pylon-beam modal interaction has no significant effect on the low-order symmetric and high-order global modes of the structure, for which completely dominated by the local modes of the pylon or beam is complete global mode. Moreover, it is obtained that the local modes of the pylon or beam have a significant effect on the low-order anti-symmetric global modes, for which jointly dominated by both pylon and beam modes is hybrid global modes. It is verified that the coupling 1:1 internal resonance is excited when the local modal frequency of the cable meets both types of global modes. The localized factor of each component in the global modes plays an important factor in measuring the degree of internal resonance participation. In a complete global mode internal resonance, non-dominant modes do not participate in the internal resonance of the structure, and energy conversion occurs only between the dominant component mode and the cable local mode. In a hybrid global mode internal resonance, non-dominant modes participate in internal resonance and would change the dynamic characteristics of the cable vibration response. The influence varies with the order of global modes and cable mechanical parameters.
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引 言
斜拉桥整体刚度低, 受载荷作用产生全桥振动时, 将同时存在以拉索振动模态为主导的结构局部模态和以主塔或主梁振动模态为主导的结构整体模态[1]. 研究表明, 随着拉索参数变化, 当结构局部模态频率靠近某一阶结构整体模态频率时, 系统将出现明显的局部−整体模态耦合共振, 此时系统能量在拉索与其他构件间周期性转换, 进而产生复杂的非线性内共振 [2-5]. 已有研究表明斜拉桥普遍存在此类特征的全桥共振[6], 有的甚至已经严重影响到桥梁结构的安全性能与使用寿命[7], 引起了国内外学者的广泛关注.
在部分非线性共振研究中, 通过把塔、索和梁等构件等效为质量块[8], 或将主梁对拉索的激励作用等效为简谐载荷[9-12], 认为拉索自身模态与激励作用模态间相互独立, 彼此间无影响作用. 然而在斜拉桥这类塔−索−梁耦合结构的内共振中, 对拉索模态的激励作用主要来源于主梁或主塔的振动模态, 由于索−梁端、索−塔端存在动态协调关系, 拉索的振动模态与主梁或主塔振动模态间存在相互耦合的关系, 两者模态间实时变化且相互影响, 其内共振机理非常复杂[13]. 因此, 为了能更贴近工程实际结构, Fujino等[14-15]和Gattulli等[16-17]从试验和理论角度出发, 研究并建立了考虑局部−整体模态间相互作用的索−梁耦合动力学模型, 提出了振型局部化系数的计算方法, 并观察到了由于整体模态和局部模态线性耦合的模态失真现象. 康厚军等从连续介质力学角度出发, 建立了索−梁[18-19]、索−拱[20-21]、索−塔[22]、索−曲线梁[23]等动力学模型, 基于弹性体与柔性体间的非线性关系, 得到了各自约化动力学控制方程, 并就其非线性动力学行为进行了系列的理论与试验研究.
关于斜拉桥内共振的现有研究绝大多数以索−梁耦合结构为主[1-19,23-24], 其通过假设拉索上端固结而忽略了主塔振动模态对结构整体模态的耦合作用, 此时与拉索振动模态产生耦合关系的整体模态完全以主梁模态为主导. 然而在目前实桥或试验观测到, 全桥共振时斜拉索、主梁和主塔均发生了大幅振动, 且后两者模态的参与将改变斜拉索的某些非线性动力特性[5,13,25]. 极少数考虑了主塔激励的研究或将其视为理想激励[26-28], 或通过模态截断仅考虑了主塔的一阶模态 [29], 这些研究模型及结果显然不能全面揭示斜拉桥的动力学行为. 因此, 考虑塔−梁模态耦合作用建立的塔−索−梁耦合结构更契合工程实际结构体系, 其研究结果才能更准确反映斜拉桥内共振机理.
本文基于参数质量离散方法, 建立了考虑塔−梁模态耦合作用的塔−索−梁耦合动力学模型, 通过有限差分法将参数体系的动力平衡微分方程转换为代数方程, 采用模态拖拽法得到了结构面内运动方程和模态函数, 并与数值模拟结果进行了对比验证. 在此基础上, 采用4 ~ 5阶Runge-Kutta方法编写了振动方程的数值仿真程序, 重点分析和讨论了塔−梁间模态耦合作用对结构整体模态和结构1:1内共振的影响.
1. 斜拉桥面内动力学模型
忽略主塔和主梁轴向振动位移, 基于参数质量体系离散方法[2,30-31], 将斜拉桥主塔及主梁分别按照db, dp等间距划分为参数质量体系, 以此建立固结体系斜拉桥的面内离散动力学模型如图1所示.
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图 1 离散化的斜拉桥面内整体动力学模型Figure 1. The in-plane full bridge dynamic model of a cable-stayed bridge本文约定下标“p, c, b”分别代表塔、索和梁的相关参数, 其中r和j分别表示对塔段和梁段的计数(r∈[1, R], j∈[1, J]), i表示拉索的计数(i∈[1, I]), 定义Ci#和Bj#分别表示小里程至大里程方向的第i根拉索和第j个梁段, Pr#表示竖直由上至下方向的第r个塔段, 而Pri#和Bji#对应拉索锚固处的塔段及梁段; θci表示拉索与主梁大里程方向夹角. 需要说明的是, 为方便计算, 将P0#的质量归于P1#, 其余边界质点无位移, 故忽略其质量对结构整体运动的影响. 对Ci#及其连接的Pri#和Bji#开展受力分析如图2所示.
图中 zpr, xci和xbj分别表示Pr#, Ci#和Bj#在各自轴向的坐标; mpr和mbj分别表示Pr#和Bj#的质量, 另有mci表示Ci#的单位长度质量; wci , vci 和uci分别简写自wci(xci), vci(xci,t)和uci(xci,t), 其中wci为拉索的静平衡线形, vci为拉索的横向振动位移, uci为拉索的轴向振动位移; s0ci为Ci#静平衡长度, sci为Ci#振动弧长; vpr和vbj分别简写自vpr(zpr,t), vbj(xbj,t), 分别表示Pr#的横向振动位移及Bj#的竖向振动位移. βb(j-1,j)表示Bj-1#和Bj#面内运动的转角, Fb(j-1,j)与Nb(j-1,j)分别表示为Bj-1#与Bj#间的剪力与轴力, 同理如βp(r-1,r), Fp(r-1,r), Np(r-1,r)之于Pr#. 试验表明[32], 张紧弦的低阶模态是自由振动时的主导模态. 若仅考虑拉索的低阶模态, 则其轴向惯性效应可以忽略(准静态近似), 此时拉索的轴向动力学将通过静态缩聚的形式被其横向动力学所控制[33]. 在此基础上, 假设拉索质量沿弦向均匀分布, 建立拉索面内静平衡方程及静平衡附近的横向非线性振动方程分别如式(1)和式(2)所示
$$ {H_{ci}}\frac{{{{\rm{d}}^2}{w_{ci}}}}{{{\rm{d}}{x_{ci}}^2}} + {m_{ci}}g \cdot \cos {\theta _{ci}} = 0 $$ (1) $$\begin{split} & \frac{\partial\left(T_{c i}+\tau_{c i}\right)\left[\dfrac{\partial\left(w_{c i}+v_{c i}\right)}{\partial s_{c i}}\right]}{\partial s_{c i}} \cdot {\rm{d}} s_{c i}+m_{c i} {\rm{d}} x_{c i} g \cos \theta_{c i}= \\ &\qquad m_{c i} {\rm{d}} x_{c i} \frac{\partial^2 v_{c i}}{\partial t^2}+c_{c i} {\rm{d}} x_{c i} \frac{\partial v_{c i}}{\partial t} \end{split} $$ (2) 式中, g表示重力加速度, 取9.806 m/s2; Tci和τci分别表示Ci#在切向的初始索力与索力动增量, 其与弦向的初始索力(Hci)与索力动增量间(hci)存在近似关系[7,19]
$$ \frac{{{T_{ci}}}}{{{H_{ci}}}} = \frac{{{\tau _{ci}}}}{{{h_{ci}}}} = \frac{{{\rm{d}}{s_{ci}}}}{{{\rm{d}}{x_{ci}}}} \approx \frac{{{\rm{d}}{s_{0ci}}}}{{{\rm{d}}{x_{ci}}}} $$ (3) 拉索横向运动的单位动应变表达式为
$$ {\bar \varepsilon _{ci}} = \frac{{{\rm{d}}{s_{ci}} - {\rm{d}}{s_{0ci}}}}{{{\rm{d}}{s_{0ci}}}} \approx \frac{{{\rm{d}}{v_{ci}}}}{{{\rm{d}}{s_{0ci}}}} \cdot \frac{{{\rm{d}}{w_{ci}}}}{{{\rm{d}}{s_{0ci}}}} + \frac{1}{2}{\left( {\frac{{{\rm{d}}{v_{ci}}}}{{{\rm{d}}{s_{0ci}}}}} \right)^2} $$ (4) 对式(4)沿xci方向积分, 代入式(3)并简化后, 可以得到拉索弦向的索力动增量表达式为
$$ {h_{ci}} = {\tau _{ci}} \cdot \frac{{{\rm{d}}{x_{ci}}}}{{{\rm{d}}{s_{ci}}}} \approx \frac{{{E_{ci}}{A_{ci}}}}{{{L_{ci}}}} \cdot \left( {{U_{ci}} + \int_0^{{l_{ci}}} {{{\bar \varepsilon }_{ci}} \cdot {\rm{d}}{x_{ci}}} } \right) $$ (5) $$ {L_{ci}} = \int_0^{{l_{ci}}} {\frac{{{\rm{d}}{s_{ci}}}}{{{\rm{d}}{x_{ci}}}}} \cdot {\left( {\frac{{{\rm{d}}{s_{0ci}}}}{{{\rm{d}}{x_{ci}}}}} \right)^2}{\rm{d}}{x_{ci}} \approx {l_{ci}} \cdot \left[ {1 + 8 \cdot {{\left( {\frac{{{f_{ci}}}}{{{l_{ci}}}}} \right)}^2}} \right] $$ (6) $$ {f_{ci}} = \frac{{{m_{ci}}g{l_{ci}}^2}}{{8{H_{ci}}}} \cdot \cos {\theta _{ci}} $$ (7) 式中, lci表示Ci#上下端锚固点的弦向距离, Lci表示Ci#静平衡长度; Uci表示Ci#轴向振动伸长量. fci为Ci#的垂度, 表示静平衡状态下的索中点与弦向中点的横向距离[19,24,31]. Ci#轴向振动伸长量由其上下边界处连接的Pri#和Bji#运动分量构成
$$ {U_{ci}} \approx {u_{ci}}\left( {0,t} \right) - {u_{ci}}\left( {{l_{ci}},t} \right) $$ (8) 为了使对称拉索的两端边界条件符号一致, 认为塔段的右侧位移、梁段的向上位移、拉索沿轴线顺时针横向振动位移为正. 基于此, 拉索振动时边界条件为
$$\qquad\quad {u_{ci}}\left( {0,t} \right) = - {v_{p{r_i}}}\left( {{z_{p{r_i}}},t} \right)\cos {\theta _{ci}} $$ (9) $$\qquad\quad {v_{ci}}\left( {0,t} \right) = {v_{p{r_i}}}\left( {{z_{p{r_i}}},t} \right)\sin {\theta _{ci}} $$ (10) $$\qquad\quad {u_{ci}}\left( {{l_{ci}},t} \right) = - {v_{b{j_i}}}\left( {{x_{b{j_i}}},t} \right)\sin {\theta _{ci}} $$ (11) $$\qquad\quad {v_{ci}}\left( {{l_{ci}},t} \right) = - {v_{b{j_i}}}\left( {{x_{b{j_i}}},t} \right)\cos {\theta _{ci}} $$ (12) 根据牛顿定律和图2的受力分析, 并考虑到狄拉克函数性质(δ), 分别建立任意Pr#横向及Bj#竖向动力平衡方程
$$\begin{split} & - {m_{pr}}{{\ddot v}_{pr}} - {c_{pr}}{{\dot v}_{pr}}- {F_{p\left( {r - 1,r} \right)}} \cdot \cos {\beta _{p\left( {r - 1,r} \right)}} + \\ &\qquad {F_{p(r,r + 1)}} \cdot \cos {\beta _{p(r,r + 1)}}-{N_{p\left( {r - 1,r} \right)}} \cdot \sin {\beta _{p\left( {r - 1,r} \right)}} + \\ &\qquad {N_{p(r,r + 1)}} \cdot \sin {\beta _{p(r,r + 1)}}+ {\eta _{pk}} \cdot \delta \left( {r - {r_k}} \right) - \\ &\qquad \left( {{H_{ci}} + {h_{ci}}} \right)\cos {\theta _{ci}} \cdot \delta \left( {r - {r_i}} \right) = 0 \end{split} $$ (13) $$ \begin{split} & - {m_{bj}}{{\ddot v}_{bj}} - {c_{bj}}{{\dot v}_{bj}}-{F_{b\left( {j - 1,j} \right)}} \cdot \cos {\beta _{b(j - 1,j)}} + \\ &\qquad {F_{b(j,j + 1)}} \cdot \cos {\beta _{b(j,j + 1)}}-{N_{b(j - 1,j)}} \cdot \sin {\beta _{b(j - 1,j)}} + \\ &\qquad {N_{b(j,j + 1)}} \cdot \sin {\beta _{b(j,j + 1)}}- {m_{bj}}g + {\eta _{bk}} \cdot \delta \left( {r - {r_k}} \right)+ \\ &\qquad \left( {{H_{ci}} + {h_{ci}}} \right)\sin {\theta _{ci}} \cdot \delta \left( {j - {j_i}} \right) = 0 \end{split} $$ (14) 式中, 参数上标“·”表示对时间t求偏导; ηpk和ηbk分别表征固结体系斜拉桥中主塔及主梁运动通过索−梁固结点相互影响作用, 下标“k”表示与该点相关参数, 下同. 对于任一塔段而言, Pr#左右侧轴力表达式为
$$ {N_{p\left( {r - 1,r} \right)}} = \sum\limits_{r = 1}^{r - 1} {{m_{pr}}g} + \sum\limits_{i = 1}^{i - 1} {\left( {{H_{ci}} + {h_{ci}}} \right)\cos {\theta _{ci}}} $$ (15) $$ {N_{p\left( {r,r + 1} \right)}} = {N_{p\left( {r - 1,r} \right)}} + {m_{pr}}g + \left( {{H_{ci}} + {h_{ci}}} \right)\cos {\theta _{ci}} \cdot \delta \left( {r - {r_i}} \right) $$ (16) 对于任一梁段而言, Bj#左右侧轴力表达式为
$$ {N_{b\left( {j - 1,j} \right)}} = \sum\limits_{i = 1}^{i - 1} {\left( {{H_{ci}} + {h_{ci}}} \right)\sin {\theta _{ci}}} $$ (17) $$ {N_{b\left( {j,j + 1} \right)}} = {N_{b\left( {j - 1,j} \right)}} + \left( {{H_{ci}} + {h_{ci}}} \right)\sin {\theta _{ci}} \cdot \delta \left( {j - j_i} \right) $$ (18) 任意相邻的塔段或梁段间存在平衡关系
$$ {M_{pr}} = {M_{pr + 1}} - {F_{p\left( {r,r + 1} \right)}} \cdot {d_p} = - {E_{pr}}{I_{pr}}{v''_{pr}}({z_{pr}},t) $$ (19) $$ {M_{bj}} = {M_{bj + 1}} - {F_{b\left( {j,j + 1} \right)}} \cdot {d_b} = - {E_{bj}}{I_{bj}}{v''_{bj}}({x_{bj}},t) $$ (20) 式中, 参数上标“'”表示对轴向坐标求偏导; Mpr和Mbj分别表示Pr#和Bj#处的弯矩; Epr和Ebj分别表示Pr#和Bj#的弹性模量; Ipr表示Pr#的面内横向弯曲惯性矩; Ibj表示Bj#的面内竖向弯曲惯性矩. 假设质量体系分布较密, 相邻梁段间的相对位移较小, 采用差分法将对位移的偏微分多项式转换为差分代数方程 [2,31]
$$ {v''_{pr}}({z_{pr}},t) \approx \left( {\frac{{{v_{pr + 1}} - {v_{pr}}}}{{{d_p}}} - \frac{{{v_{pr}} - {v_{pr - 1}}}}{{{d_p}}}} \right)\Bigr/{d_p} $$ (21) $$ {v''_{bj}}({x_{bj}},t) \approx \left( {\frac{{{v_{bj + 1}} - {v_{bj}}}}{{{d_b}}} - \frac{{{v_{bj}} - {v_{bj - 1}}}}{{{d_b}}}} \right)\Bigr/{d_b} $$ (22) $$ \sin {\beta _{p(r - 1,r)}} \approx \tan {\beta _{p(r - 1,r)}} \approx \frac{{{v_{pr}} - {v_{pr - 1}}}}{{{d_p}}} $$ (23) $$ \sin {\beta _{b(j - 1,j)}} \approx \tan {\beta _{b(j - 1,j)}} \approx \frac{{{v_{bj}} - {v_{bj - 1}}}}{{{d_b}}} $$ (24) 对于塔−梁固结斜拉桥, 索−梁固结点处离散段同时为桥塔和主梁的一部分, 塔与梁振动模态间存在相互影响, 且通过该点进行传递. 截取索−梁节点受力示意图如图3所示.
如图3所示, 根据离散模型的假设, 该点处横向及纵向振动位移为零, 而其水平、竖向的剪力与轴力表达式间分别存在相互作用
$$ {N_{p({r_k} - 1,{r_k})}} + {F_{b\left( {{j_k} - 1,{j_k}} \right)}} - {F_{b\left( {{j_k},{j_k} + 1} \right)}} + {m_{p{r_k}}} \cdot g = {N_{p({r_k},{r_k} + 1)}} $$ (25) $$ {N_{b({j_k} - 1,{j_k})}} + {F_{p\left( {{r_k} - 1,{r_k}} \right)}} - {F_{p\left( {{r_k},{r_k} + 1} \right)}} = {N_{d({j_k},{j_k} + 1)}} $$ (26) 图1中, 主塔上端自由、下端固结, 主梁左右侧均为简支端, 其边界条件分别为
$$\qquad\quad {F_{p\left( {e - ,1} \right)}} = 0,\quad {M_{pe - }} = 0 $$ (27) $$\qquad\quad {v_{pe + }} = 0,\quad {\beta _{p\left( {R,e + } \right)}} = 0 $$ (28) $$\qquad\quad {v_{be - }} = 0,\quad {v_{be + }} = 0 $$ (29) $$\qquad\quad {M_{be - }} = 0,\quad {M_{be + }} = 0 $$ (30) 式中, e+, e−分别表示大小里程边界. 考虑边界条件式(27) ~ 式(30)后, 整理式(15) ~ 式(26)并分别代入式(13)和式(14), 可以得到面内的Pr#横向及Bj#竖向运动方程.
2. 结构整体模态定义与模态分析
2.1 运动方程与模态函数
为便于找到系统的内共振形式, 采用模态拖拽法定义第n阶斜拉桥塔、索、梁自由振动位移表达式[31,34]
$$ \begin{split} & {\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{v}}_{pr}}} \\ {{{\boldsymbol{v}}_{ci}}} \\ {{{\boldsymbol{v}}_{bj}}} \end{array}} \right]^{\left( n \right)}} = \sum\limits_{n = 1}^N {{{\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{\varphi}} _{pr}}{{\left( {{{\boldsymbol{z}}_{pr}}} \right)}^{\left( n \right)}}} \\ {{{\boldsymbol{f}}_{pi}}\left( {{{\boldsymbol{x}}_{ci}}} \right)} \\ {\boldsymbol{0}} \end{array}} \right]}^{\text{T}}} \cdot {{\boldsymbol{q}}_{npr}}} + \\ &\qquad \sum\limits_{n = 1}^N {{{\left[ {\begin{array}{*{20}{c}} {\boldsymbol{0}} \\ {{{\boldsymbol{\varphi}} _{ci}}{{\left( {{{\boldsymbol{x}}_{ci}}} \right)}^{\left( n \right)}}} \\ {\boldsymbol{0}} \end{array}} \right]}^{\text{T}}} \cdot {{\boldsymbol{q}}_{nci}}} + \sum\limits_{n = 1}^N {{{\left[ {\begin{array}{*{20}{c}} {\boldsymbol{0}} \\ {{{\boldsymbol{f}}_{bi}}\left( {{{\boldsymbol{x}}_{ci}}} \right)} \\ {{{\boldsymbol{\varphi}} _{bj}}{{\left( {{{\boldsymbol{x}}_{bj}}} \right)}^{\left( n \right)}}} \end{array}} \right]}^{\text{T}}} \cdot {{\boldsymbol{q}}_{nbj}}} \end{split} $$ (31) 式中, 加粗符号表示矩阵或向量(下同), 基本形式列于附录A. 式(31)是一个含有多维向量(塔/梁)和连续函数(索)的混合表达式, 其中, 元素为φpr(n), φbj(n)的向量分别表示塔、梁在第n阶面内振动模态下的振动模态振型向量; qnpr简写自qpr(t)(n), 表示与时间相关的主塔第n阶振动模态形状变化因子, 同理如qnci与qnbj; 元素为fpi(xci), fbi(xci)表示与边界条件式(10)和式(12)关联的模态拖拽函数向量, 定义其基本形式为[34]
$$\qquad\quad {f_{pi}}\left( {{x_{ci}}} \right) = \left( {1 - \frac{{{x_{ci}}}}{{{l_{ci}}}}} \right) \cdot \sin {\theta _{ci}} $$ (32) $$\qquad\quad {f_{bi}}\left( {{x_{ci}}} \right) = - \frac{{{x_{ci}}}}{{{l_{ci}}}} \cdot \cos {\theta _{ci}} $$ (33) 对于图1所示的主塔与主梁离散参数质量体系, Pr#和Bj#的形状变化因子已包括了各自各阶振动模态下的振型函数. 因此, Pr#和Bj#的振动位移表达式可以简写为
$$ {v_{pr}}\left( {{z_{pr}},t} \right) = \sum\limits_{n = 1}^N {{\varphi _{pr}}^{\left( n \right)} \cdot {q_{npr}}} = {A_{pr}}^{\left( n \right)} \cdot {q_{pr}} $$ (34) $$ {v_{bj}}\left( {{x_{bj}},t} \right) = \sum\limits_{n = 1}^N {{\varphi _{bj}}^{\left( n \right)} \cdot {q_{nbj}}} = {A_{bj}}^{\left( n \right)} \cdot {q_{bj}} $$ (35) 式中, Apr(n)(zpr)表示Pr#在结构第n阶振动模态下的振幅, 同理如Aci(n)(xci)和Abj(n)(xbj). 为简化表达, 选取三角函数作为式(31)中拉索振动方程的振型基函数[19,26-27]
$$ {\varphi _{ci}}{\left( {{x_{ci}}} \right)^{\left( n \right)}} = {A_{ci}}^{\left( n \right)}\sin \frac{{n\text{π} {x_{ci}}}}{{{l_{ci}}}} $$ (36) 基于此, 可以得到Ci#横向振动方程为
$$ {v_{ci}}\left( {{x_{ci}},t} \right){\text{ = }}{f_{pi}} \cdot {v_{p{r_i}}} + \sum\limits_{n = 1}^N {{\varphi _{ci}}{{\left( {{x_{ci}}} \right)}^{\left( n \right)}} \cdot {q_{nci}}} + {f_{bi}} \cdot {v_{b{j_i}}} $$ (37) 将上式连同式(34)和式(35)分别代入整合后的式(2)、式(13)和式(14), 使用Galerkin方法进行模态截断可得图1所示固结斜拉桥的面内整体运动方程. 考虑到张紧弦的低阶模态是其自由振动时的主导模态, 为简化计算, 在此仅考虑了拉索一阶振动模态, 并将qnci简写为qci, 得到结构方程如下所示
$$ \begin{split} & {{\ddot {\boldsymbol{q}}}_{pr}} + {{\boldsymbol{\varOmega }}_{pr}} \cdot {{\boldsymbol{q}}_{pr}} + {{\boldsymbol{\varPi}} _{pr,1}} \cdot {{\dot {\boldsymbol{q}}}_{pr}} + {{\boldsymbol{\varPi}} _{pr,2}} \cdot {{\boldsymbol{q}}_{p{r_i}}}^2 + \\ &\qquad {{\boldsymbol{\varPi}} _{pr,3}} \cdot {{\boldsymbol{q}}_{ci}} + {{\boldsymbol{\varPi}} _{pr,4}} \cdot {{\boldsymbol{q}}_{ci}}^2 + {{\boldsymbol{\varPi}} _{pr,5}} \cdot {{\boldsymbol{q}}_{b{j_i}}}+ \\ &\qquad {{\boldsymbol{\varPi}} _{pr,6}} \cdot {{\boldsymbol{q}}_{b{j_i}}}^2 + {{\boldsymbol{\varPi}} _{pr,7}} \cdot {{\boldsymbol{q}}_{p{r_i}}}{{\boldsymbol{q}}_{b{j_i}}} + {{\boldsymbol{\eta}} _{pk}} = {\boldsymbol{0}} \end{split} $$ (38) $$ \begin{split} & {{\ddot {\boldsymbol{q}}}_{ci}} + {{\boldsymbol{\varOmega}} _{ci}} \cdot {{\boldsymbol{q}}_{ci}} + {{\boldsymbol{\varPi}} _{ci,1}} \cdot {{\ddot {\boldsymbol{q}}}_{pr}} + {{\boldsymbol{\varPi}} _{ci,2}} \cdot {{\ddot {\boldsymbol{q}}}_{bj}}+ \\ &\qquad {{\boldsymbol{\varPi}} _{ci,3}} \cdot {{\dot {\boldsymbol{q}}}_{ci}} + {{\boldsymbol{\varPi}} _{ci,4}} \cdot {{\dot {\boldsymbol{q}}}_{pr}} + {{\boldsymbol{\varPi}} _{ci,5}} \cdot {{\dot {\boldsymbol{q}}}_{bj}}+ \\ &\qquad {{\boldsymbol{\varPi}} _{ci,6}} \cdot {{\boldsymbol{q}}_{ci}}^2 + {{\boldsymbol{\varPi}} _{ci,7}} \cdot {{\boldsymbol{q}}_{ci}}^3 + {{\boldsymbol{\varPi}} _{ci,8}} \cdot {{\boldsymbol{q}}_{pr}}+ \\ &\qquad {{\boldsymbol{\varPi}} _{ci,9}} \cdot {{\boldsymbol{q}}_{pr}}^2 + {{\boldsymbol{\varPi}} _{ci,10}} \cdot {{\boldsymbol{q}}_{bj}} + {{\boldsymbol{\varPi}} _{ci,11}} \cdot {{\boldsymbol{q}}_{bj}}^2+ \\ &\qquad {{\boldsymbol{\varPi}} _{ci,12}} \cdot {{\boldsymbol{q}}_{ci}}{{\boldsymbol{q}}_{pr}} + {{\boldsymbol{\varPi}} _{ci,13}} \cdot {{\boldsymbol{q}}_{ci}}{{\boldsymbol{q}}_{bj}} + {{\boldsymbol{\varPi}} _{ci,14}} \cdot {{\boldsymbol{q}}_{pr}}{{\boldsymbol{q}}_{bj}}+ \\ &\qquad {{\boldsymbol{\varPi}} _{ci,15}} \cdot {{\boldsymbol{q}}_{ci}}{{\boldsymbol{q}}_{pr}}^2 + {{\boldsymbol{\varPi}} _{ci,16}} \cdot {{\boldsymbol{q}}_{ci}}{{\boldsymbol{q}}_{bj}}^2+ \\ &\qquad {{\boldsymbol{\varPi}} _{ci,17}} \cdot {{\boldsymbol{q}}_{ci}}{{\boldsymbol{q}}_{pr}}{{\boldsymbol{q}}_{bj}} = {\boldsymbol{0}} \end{split} $$ (39) $$ \begin{split} & {{\ddot {\boldsymbol{q}}}_{bj}} + {{\boldsymbol{\varOmega}} _{bj}} \cdot {{\boldsymbol{q}}_{bj}} + {{\boldsymbol{\varPi}} _{bj,1}} \cdot {{\dot {\boldsymbol{q}}}_{bj}} + {{\boldsymbol{\varPi}} _{bj,2}} \cdot {{\boldsymbol{q}}_{b{j_i}}}^2 + {{\boldsymbol{\varPi}} _{bj,3}} \cdot {{\boldsymbol{q}}_{ci}}+ \\ &\qquad {{\boldsymbol{\varPi}} _{bj,4}} \cdot {{\boldsymbol{q}}_{ci}}^2 + {{\boldsymbol{\varPi}} _{bj,5}} \cdot {{\boldsymbol{q}}_{p{r_i}}} + {{\boldsymbol{\varPi}} _{bj,6}} \cdot {{\boldsymbol{q}}_{p{r_i}}}^2+ \\ &\qquad {{\boldsymbol{\varPi}} _{bj,7}} \cdot {{\boldsymbol{q}}_{p{r_i}}}{{\boldsymbol{q}}_{b{j_i}}} + {{\boldsymbol{\eta}} _{bk}} = {\boldsymbol{0}} \end{split} $$ (40) 式中, 对角矩阵Ωpr, Ωnci和Ωbj是主塔、拉索与主梁振动模态的特征对角矩阵, 其主元参数量纲与频率相同. 其中, Ωpr和Ωbj表达式为
$$\qquad\quad {{\boldsymbol{\varOmega}} _{pr}} = {{\boldsymbol{S}}_{pr}} + {{\boldsymbol{D}}_{pr}} + {{\boldsymbol{G}}_{pr}} + {{\boldsymbol{C}}_{pr}} $$ (41) $$ \quad\qquad {{\boldsymbol{\varOmega}} _{bj}} = {{\boldsymbol{S}}_{bj}} + {{\boldsymbol{D}}_{bj}} + {{\boldsymbol{C}}_{bj}} $$ (42) 式中, 矩阵S, D, C, G分别表征了剪力效应、轴力效应、拉索的弹性支承效应与重力效应对Pr#或Bj#振动模态的影响, 各参数表达式详见附录B. 从式(38) ~ 式(40)可以看出, Pr#, Ci#和Bj#的运动方程存在耦合项, 表明离散的塔段或梁段运动并不彼此独立, 其通过剪力与轴力效应相互影响, 共同参与结构的整体运动. 此外, 线性化后的式(38) ~ 式(40)实质是关于结构固有振动模态频率ωN的K阶齐次超越方程
$$ {{\boldsymbol{E}}_N} \cdot {{\boldsymbol{Q}}_N}{\text{ = }}{\boldsymbol{0}} $$ (43) 式中, 下标“N” 的参数表示结构面内固有振动模态参数; EN表示结构整体运动的特征矩阵, QN表示结构形状变化因子向量, 其表达式如下所示
$$ {{\boldsymbol{E}}_N} = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{\varOmega}} _{pr}}}&{{{\boldsymbol{\varPi }}_{pr,3}}}&{{{\boldsymbol{\varPi }} _{pr,5}}} \\ {{{\boldsymbol{\varPi }} _{ci,8}}}&{{{\boldsymbol{\varOmega}} _{ci}}}&{{{\boldsymbol{\varPi }} _{ci,10}}} \\ {{{\boldsymbol{\varPi }} _{bj,5}}}&{{{\boldsymbol{\varPi }} _{bj,3}}}&{{{\boldsymbol{\varOmega}} _{bj}}} \end{array}} \right] $$ (44) $$ {{\boldsymbol{Q}}_N} = {\left[ {{{\boldsymbol{q}}^{\rm{T}}_{pr}},{{\boldsymbol{q}}^{\rm{T}}_{ci}},{{\boldsymbol{q}}^{\rm{T}}_{bj}}} \right]} $$ (45) 式(44)中, 除塔、索和梁特征矩阵外, 存在Πbj,3等耦合项系数, 表征了塔、索、梁等构件振动模态在结构振动模态下的耦合作用. 式(43)存在非零解的前提须EN行列式为零. 因此, 对EN进行特征值求解可得结构的振动模态参数
$$ \left[ {{{ {\boldsymbol{A}}}_N}^{\left( K \right)},{{\boldsymbol{\omega}} _N}^{\left( K \right)}} \right] = {\rm{eig}}{\text{ }}{{\boldsymbol{E}}_N} $$ (46) 式中, ωN(K)表示结构前K阶固有振动模态频率. 其中第n阶频率ωN(n)对应的特征向量AN(n) 表示该阶振动模态下的塔、索和梁构件振动模态的振幅向量. 将结构第n阶固有振动模态频率ωN(n)和对应的特征向量AN(n)代入式(34)、式(35)和式(37)后, 可以得到塔、索、梁的振动位移表达式. 由于本文仅考虑了拉索的一阶频率, 则式(46)的K个解中包含了C1# ~ C4#的1阶局部模态频率, 其余为结构的整体振动模态频率, 此时
$$ K = R + 4 + J $$ (47) 若将主梁和拉索无限细分, 且使用拉索的前N阶模态进行截断, 则式(46)有N个解, 此时K = N, 按照ωN(k)数值大小升序排列即为结构的第1阶至第N阶固有振动模态频率.
2.2 基于FEM的斜拉桥整体模态验证
参考中国西北地区一座斜拉桥设计参数, 建立一座单塔四索斜拉桥的动力学简化模型, 构件参数如表1和表2所示.
表 1 拉索参数Table 1. Parameters of cablesNo. mci/(kg·m−1) Eci/Pa Aci/cm2 Hci/N lci/m θci/(°) $ {z}_{p{r}_{i}} $/m $ {x}_{b{j}_{i}} $/m Ωci/Hz C1# 68.1 2.0 × 1011 66.8 8.0 × 105 42.52 0.56 28 8 1.42 C2# 60.1 2.0 × 1011 27.5 6.2 × 105 28.84 0.98 24 24 1.87 C3# 60.1 2.0 × 1011 27.5 6.2 × 105 28.84 2.16 24 56 1.87 C4# 68.1 2.0 × 1011 66.8 8.0 × 105 42.52 2.42 28 72 1.42 表 2 主塔及主梁参数Table 2. Parameters of the pylon and beamComponent μm/(kg·m−1) E/Pa I/m4 L/m Lk/m pylon 8142.97 3.45 × 1010 0.21 40 32 beam 12819.29 3.45 × 1010 0.41 80 40 为辨别结构模态, 引入模态局部化程度系数Λc(n), Λp(n)和Λb(n) [1,24-25]以定义构件的振动模态在第n阶整体模态下的模态参与度
$$ \varLambda _p^{\left( n \right)} = \frac{{\displaystyle\sum\nolimits_{r = 1}^R {{ {\boldsymbol{A}}}_{pr}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{pr}} \cdot { {\boldsymbol{A}}}{{_{pr}^{\left( n \right)}}^{\text{T}}}} }}{{\displaystyle\sum\nolimits_{i = 1}^I {{ {\boldsymbol{A}}}_{ci}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{ci}} \cdot { {\boldsymbol{A}}}{{_{ci}^{\left( n \right)}}^{\text{T}}}} + \displaystyle\sum\nolimits_{r = 1}^R {{ {\boldsymbol{A}}}_{pr}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{pr}} \cdot { {\boldsymbol{A}}}{{_{pr}^{\left( n \right)}}^{\text{T}}}} + \displaystyle\sum\nolimits_{j = 1}^J {{ {\boldsymbol{A}}}_{bj}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{bj}} \cdot { {\boldsymbol{A}}}{{_{bj}^{\left( n \right)}}^{\text{T}}}} }} $$ (48) $$ \varLambda _c^{\left( n \right)} = \frac{{\displaystyle\sum\nolimits_{c = 1}^I {{ {\boldsymbol{A}}}_{ci}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{ci}} \cdot { {\boldsymbol{A}}}{{_{ci}^{\left( n \right)}}^{\text{T}}}} }}{{\displaystyle\sum\nolimits_{i = 1}^I {{ {\boldsymbol{A}}}_{ci}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{ci}} \cdot { {\boldsymbol{A}}}{{_{ci}^{\left( n \right)}}^{\text{T}}}} + \displaystyle\sum\nolimits_{r = 1}^R {{ {\boldsymbol{A}}}_{pr}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{pr}} \cdot { {\boldsymbol{A}}}{{_{pr}^{\left( n \right)}}^{\text{T}}}} + \sum\nolimits_{j = 1}^J {{ {\boldsymbol{A}}}_{bj}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{bj}} \cdot { {\boldsymbol{A}}}{{_{bj}^{\left( n \right)}}^{\text{T}}}} }} $$ (49) $$ \varLambda _b^{\left( n \right)} = \frac{{\displaystyle\sum\nolimits_{j = 1}^J {{ {\boldsymbol{A}}}_{bj}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{bj}} \cdot { {\boldsymbol{A}}}{{_{bj}^{\left( n \right)}}^{\text{T}}}} }}{{\displaystyle\sum\nolimits_{i = 1}^I {{ {\boldsymbol{A}}}_{ci}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{ci}} \cdot { {\boldsymbol{A}}}{{_{ci}^{\left( n \right)}}^{\text{T}}}} + \displaystyle\sum\nolimits_{r = 1}^R {{ {\boldsymbol{A}}}_{pr}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{pr}} \cdot { {\boldsymbol{A}}}{{_{pr}^{\left( n \right)}}^{\text{T}}}} + \sum\nolimits_{j = 1}^J {{ {\boldsymbol{A}}}_{bj}^{\left( n \right)} \cdot {{\boldsymbol{m}}_{bj}} \cdot { {\boldsymbol{A}}}{{_{bj}^{\left( n \right)}}^{\text{T}}}} }} $$ (50) 为辨别结构的整体模态与局部模态, 定义若Λp(n) + Λb(n) > 80%, 此时结构的整体运动以主塔或主梁振动模态为主导, 表明该阶为结构的整体模态(简写为“Gnth”); 若Λp(n) + Λb(n) < 10%, 此时结构的整体运动以拉索的振动模态为主导, 表明该阶为结构的局部模态; 其余数值为结构的混合模态. 为验证式(46)的模态参数计算方法, 建立了OECS (one-element cable system)有限元模型[25], 采用Lanczos方法获取有限元模型的整体模态参数. 由于本文仅考虑了拉索的低阶振动模态(高阶整体模态未反映拉索振动模态的耦合作用), 且为了对应到OECS模型获取的整体模态参数, 可对式(44)中结构特征矩阵进行简化
$$ {{\boldsymbol{E}}_N}^{\text{*}} = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{\varOmega}} _{pr}}}&{{{\boldsymbol{\varPi}} _{pr,5}}} \\ {{{\boldsymbol{\varPi}}_{bj,5}}}&{{{\boldsymbol{\varOmega}} _{bj}}} \end{array}} \right] $$ (51) 基于上式获取的结构模态参数仅包含结构整体模态的特征值与特征向量. 将本文方法(EMP)与有限元方法(FEM)得到的G1st ~ G5th整体模态振型及频率汇总如表3.
表 3 两种方法得到的结构体系整体模态振型Table 3. Global modal shapes obtained via these two methodsFEM 
EMP 
表3中, 两种方法得到的结构整体模态振型几乎一致, 表4中模态频率误差随离散间距d取值减小而减小, 这与式(21) ~ 式(24)中有限差分法应用假设相关. 采用本文方法求解复杂塔−索−梁结构体系模态参数, 在误差可接受范围内确定离散间距d及结构特征矩阵(式(44))元素的取值, 借助MATLAB, Excel等工具即可高效率获取结构体系固有振型及频率, 无需进行大量细化的有限元建模分析. 本文后续研究为避免较大计算量, 离散间距取值为4 m.
表 4 两种方法得到的结构体系整体模态频率Table 4. Global modal frequencies obtained via these two methodsMode no. FEM/Hz EMP/Hz Error/% d = 1 m d = 4 m d = 1 m d = 4 m G1st 1.17 1.13 1.12 −2.8 −4.2 G2nd 2.07 2.06 2.03 −0.6 −1.9 G3rd 2.66 2.66 2.52 −0.1 −5.2 G4th 3.75 3.76 3.61 −0.3 −3.6 G5th 4.55 4.51 4.38 −0.8 −3.7 2.3 主塔振动模态对整体模态的影响
表3显示, 在本文塔−索−梁耦合模型中, 结构的整体模态不再完全以主塔或主梁的振动模态为主, 而是存在由两者共同参与的混合整体模态, 如G1st, G4th等. 为开展进一步研究, 建立了3个动力学模型, 如表5所示.
表 5 3种索−支撑简化动力学模型Table 5. Three dynamic models reduced from the cable-support structuresSketch 
Instruction The upper end of the cable is anchored on the pylon while lower end is anchored on the beam. The upper end of the cable is consolidation while the lower end is anchored on the beam. The upper end of the cable is anchored on the pylon while the lower end is consolidation. pylon-cable-beam modal coupling cable-beam modal coupling pylon-cable modal coupling Abbreviation DM1# DM2# DM3# 表5中DM2#被广泛采用于索−梁模态耦合的相关非线性振动研究[2,15-18,26-27,31]. 采用本文方法对表5中的3个模型进行特征值求解, 剥离其整体模态频率汇总如图4所示.
图4中Bnth表示以梁第n阶局部模态为主的整体模态; Pnth表示以塔第n阶局部模态为主的整体模态. 上图表明, DM1#的整体模态由塔和梁的振动模态组成. 在数值大于10 Hz的高阶模态中, DM1#整体模态频率数值上分别与DM2#中梁的振动模态频率和DM3#中塔的振动模态频率相等, 表明塔−梁振动模态间的耦合作用无法影响结构的高阶整体模态频率; 而在低阶整体模态中, 三者模型的模态频率数值上存在一定差异. 为进一步研究, 取图4中阴影部分的G1st ~ G9th阶模态参数如图5所示.
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图 5 G1st ~ G9th结构的整体模态频率变化曲线Figure 5. The trends of the structural global modal frequencies of G1st ~ G9th图5中在G1st, G3rd, G4th, G5th, G7th和G8th阶等低阶反对称整体模态中, DM1#与其他两个模型得到的模态频率存在较大误差, 且局部化程度系数显示主塔振动模态参与了以主梁振动模态为主导的整体模态运动, 使得该阶整体模态由完全以主梁模态为主导的完全整体模态转为塔−梁振动模态耦合的混合整体模态. 而对于低阶的对称整体模态(如G2nd, G6th)、高阶整体模态(G9th及以上), 塔−梁模态耦合作用对结构整体模态没有明显影响, 此时塔或梁的振动模态完全主导了该阶整体模态. 这表明, 在考虑了主塔振动模态以后, 斜拉桥等塔−索−梁耦合结构的整体模态可以进一步被细分为完全整体模态和混合整体模态, 其细分流程如图6所示.
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图 6 斜拉桥面内固有振动模态的细分流程Figure 6. The reduced process of the in-plane natural vibration modes of cable-stayed bridges上述研究结果表明, 主塔振动模态参与度决定了结构整体模态的动力特性, 在斜拉桥这类塔−索−梁耦合模型中, 考虑主塔振动模态对研究因局部−整体模态耦合激励产生的内共振现象具有重要意义.
3. 斜拉桥1:1内共振分析
3.1 模态翻转与1:1内共振动力特性
为排除起振阈值影响以更清晰观测内共振响应变化规律[6], 本文以“1:1”内共振为主要研究对象. 在考虑拉索前2阶振动模态的基础上, 从纯数值角度变化斜拉索索力与垂度参数, 汇总结构第5阶固有振动频率(N5th)至结构第9阶固有振动频率(N9th)变化规律如图7所示.
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图 7 结构N5th ~ N9th整体模态频率随C1#的垂度与索力变化曲线Figure 7. The trends of the global modal frequencies from N5th to N9th with the sag and force of C1#图7中, 随着C1#垂度及索力参数变化, 当结构局部模态频率临近某一阶结构整体模态频率时, 两者模态振型发生了快速且连续的交换(Veering现象[1]). 此时局部−整体模态间发生耦合, 结构能量在拉索与系统内其他构件间进行周期性地转换, 容易激励产生1:1内共振. 研究表明, 内共振完全由系统非线性特性决定[35]. 为进一步研究这种现象并验证本文结构的非线性动力特性, 引入系数σci表示Ci#拉索局部频率(ωci)增量, ${{\varTheta }_{ci}}^{\left(n\right)}$表示Ci#拉索局部模态频率与结构第n阶整体模态频率(ωG(n))的靠近程度[24,31]
$$ {\omega _{ci}} = \sqrt {{\varGamma _{ci,1}}} + {\sigma _{ci}} $$ (52) $$ {\varTheta _{ci}}^{\left( n \right)}{\text{ = }}\left\{ \begin{aligned} & \frac{{\left| {{\omega _{ci}} - {\omega _G}^{(n)}} \right|}}{{{\omega _G}^{(n)}}},\;\;n = 1 \\ & \min \left( {\frac{{\left| {{\omega _{ci}} - {\omega _G}^{(n)}} \right|}}{{{\omega _G}^{(n)}}},\frac{{\left| {{\omega _{ci}} - {\omega _G}^{(n - 1)}} \right|}}{{{\omega _G}^{(n - 1)}}}} \right),n \ne 1 \end{aligned} \right. $$ (53) 采用4 ~ 5阶Runge-Kutta积分方法对式(38) ~ 式(40)进行数值仿真. 为清晰展示系统构件在达到稳态后一个振动周期内的动力特性, 提高仿真效率, 本文的数值仿真过程设定固定步长为0.001 s, 迭代时间为100 s. 在塔顶(P1#)、梁小里程端点(B1#)附近同时设置了0.02 m的初始位移, 改变σci至分别与第G6th与G8th模态频率满足“1:1”比例关系. 由图5可知, G6th为以主梁模态为主导的整体模态, 而G8th为以主塔模态为主导的混合模态. 对获取的拉索振动响应曲线进行快速傅里叶分析(FFT), 汇总拉索中点振动响应及对应频谱图如图8所示.
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图 8 局部−整体模态耦合激励产生“1:1”内共振现象Figure 8. The phenomenon of the 1:1 internal resonance excited by the local-global modal interactions图8(a)与图8(b)显示, C1#分别与G6th和G8th整体模态“1:1”耦合并产生了“拍”特性明显的内共振. 这表明, 拉索局部模态与HGM和EGM (定义见图6)两者“1:1”耦合都可以激励拉索产生“1:1”内共振. 而这样“拍”特性明显的内共振现象同样验证了本文推导方程及数值仿真方法可以有效进行非线性动力响应分析.
此外, 对比图8(a)与图8(b)后可得, 在同等的初始条件下, 由两种不同主导模态激励产生的内共振响应的“拍”频及幅值不同, 表明拉索的内共振动力特性随主导模态变化而变化. 实际上在塔−索−梁耦合模型中, 除局部−整体模态存在交换现象外, 还存在主导模态间的交换现象. 以表5中DM1#和DM2#为例, 变化主塔单位长度质量而使主梁构造参数保持不变, 计算DM#中G7th模态参数如图9所示.
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图 9 结构G7th模态参数随主塔单位长度质量变化规律Figure 9. The trend of the G7th -order modal property varied with the mass per unit length of the pylon上图显示, 随着塔单位长度质量变化至6.72 × 103 kg/m附近, DM1#模型的G7th主导模态发生了快速且连续的交换, 由以主梁模态为主导的HGM变化为以主塔模态为主导的HGM, 各构件的模态局部化程度系数也随之发生变化. 此时, DM1#模型中主导模态的变化将进一步影响拉索内共振响应的动力特性, 其变化规律与图8中一致, 在此不过多赘述.
3.2 HGM耦合内共振的参与度分析
不同于以往EGM耦合内共振的相关研究, 在塔−索−梁耦合系统中, 当某阶HGM与某根拉索的局部模态“1:1”耦合时, 主塔和主梁的振动模态将同时参与“1:1”内共振. 为研究以主梁模态为主导的HGM内共振动力特性, 设定了1#研究工况.
1#研究工况(RC1#): 仅考虑拉索1阶模态, 针对DM1与DM2模型, 在主梁小里程桩号附近增加0.02 m初始位移, 改变σc1至分别与G7th, G9th模态频率满足“1:1”关系.
汇总RC1#工况条件下的C1#振动响应曲线如图10所示.
图10(a)和图10(b)中, DM1#模型的G7th是以主梁振动模态为主的混合整体模态, 主塔振动模态参与了整体模态运动, 同时也影响了拉索的内共振动力特性. 图10(c)和图10(d)中, G9th是完全以主梁振动模态为主的整体模态, 此时DM1#与DM2#系统拉索的内共振时程曲线动力特性相同. 为进一步明晰混合模态内共振时的能量传递路径, 采用无相位滤波法分离RC1#中C1#及其相连的塔段(P1#)和梁段(B2#)在主共振频率附近的振动响应信号, 汇总如表6所示.
表 6 DM1#中发生G7th和G9th内共振时, C1#振动子系统构件在各自主共振频率附近的分离响应信号Table 6. The separated signals around the main resonance frequency in the dynamic sub-system of C1# when the internal resonance of G7th -order or G9th -order occurring in DM1#When couples with G7th-order HGM in DM1# When couples with G9th-order HGM in DM1# 
表6中, 前25 s的分离信号存在滤波延迟, 以后75 s的滤波信号为研究对象. 当C1#局部模态与G7th混合模态“1:1”耦合时, P1#, C1#和B2#在主共振频率附近的响应信号此起彼伏、互相耦合, 表明共振时能量在三者间来回转换. 当与G9th混合模态“1:1”耦合时, C1#与B2#的振动响应曲线互相耦合, 而P1#与C1#的振动特性相同且其响应幅值与B2#存在数量级差异, 表明P1#的振动响应仅来源于塔−索端点耦合而未参与内共振, 此时系统能量在拉索与主梁间来回转换. 采用上述同样的滤波方法, 获取在各自主共振频率下运行时刻t1的结构振型如图11所示.
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11 DM1#和DM2#模型在RC1#工况下主共振频率对应的结构振动形态11. The structural dynamic configuration corresponding to the main resonance frequency in DM1# and DM2# under RC1#图11(a)显示, 由主塔第3阶面内模态和主梁第5阶面内竖向模态共同构成的混合整体模态激发了塔−索−梁耦合内共振, 此时主塔或主梁振动模态参与度系数Λp和Λb与原G7th的局部化程度系数Λp(G7)和Λb(G7)误差较小. 而在图11(b) ~ 图11(d)中, 由主梁的第5阶或第6阶面内竖向模态为主的整体模态激发了索−梁耦合内共振, 拉索与主梁间存在能量交换作用, 而主塔未参与整体运动. 为研究主塔振动模态对内共振的影响, 设定了2#研究工况.
2#研究工况(RC2#): 针对DM1与DM3模型, 在塔顶增加0.02 m初始位移, 改变σc1至分别与G8th, G11th模态频率满足“1:1”关系.
为简化表达, 汇总RC2#工况条件下的C1#振动响应曲线, 及其在各自主共振频率下运行时刻t1的结构振动形态分别如图12所示.
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图 12 RC2#工况下C1#时程曲线与对应的结构振动形态Figure 12. The time histories of C1# and the corresponding the structural dynamic configuration under RC2#图12结果再次验证了在HGM中, 非主导的主梁振动模态将参与由主塔振动模态主导的内共振能量转换, 并将改变拉索内共振响应曲线的“拍”频及振幅, 此时各构件的整体模态局部化程度系数是重要的量化因素.
3.3 HGM耦合内共振的影响性分析
为研究非主导的主梁振动模态在混合整体模态内共振下的影响效应, 设定了3#研究工况.
3#研究工况(RC3#): 针对DM1与DM2模型, 在主梁小里程桩号附近增加0.02 m初始位移, 分别改变σc1, σc2, σc3和σc4至与以主梁振动模态为主导的整体模态(G1th, G2th, G4th, G6th, G7th, G9th, G10th)频率满足1:1比值关系.
拉索的振动响应曲线(Aci)能一定程度上反映内共振时拉索动能变化规律, 定义$A^{\rm E}_{ci}$和$A^{\rm H}_{ci}$分别表示在EGM和HGM模型中拉索振动响应的最大值. 同时为尽量减少多自由度系统内复杂组合内共振引起的响应误差, 采用无相位滤波法分离各子工况下主共振频率附近的拉索响应, 汇总其响应振幅最大值曲线如图13所示.
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图 13 DM1#和DM2#模型在RC3#工况下拉索主共振频率附近的振幅最大值Figure 13. The maximum response of cables in DM1# and DM2# under RC3#图13中, 对于不同阶次的整体模态内共振, DM1#模型中的拉索共振响应最大值与DM2#的计算结果存在较大差异, 表明非主导的主塔振动模态参与并影响了HGM模态内共振, 其影响效应随阶次变化而不同. 其中, 对称布置的C1#与C4#(或C2#和C3#)基础构件参数、锚固位置对于整体模态振型有效质量相同, 因此对称索的各阶振幅最大值曲线趋势、误差数值基本一致. 对于同一侧的C1#与C2#(或C3#与C4#)各阶振幅最大值曲线变化趋势相似度较高但差值不同, 表明各阶非主导主塔模态对于任意拉索内共振的影响效应类型一致, 但影响效应大小随拉索变化而变化. 对于两种模型计算C1#(图13-A点)或C4#(图13-B点)结果差值较大的G6th内共振, 结合对应的响应与频谱图可得, DM1#中较长的C1#或C4#在整体模态激励作用下发生了组合内共振, 故而其主共振频率附近的模态响应弱于DM2#计算结果. 为研究非主导的主塔振动模态在混合整体模态内共振下的影响效应, 设定了4#研究工况.
4#研究工况(RC4#): 针对DM1与DM4模型, 在塔顶增加0.02 m初始位移, 分别改变σc1, σc2, σc3和σc4至与以主塔振动模态为主导的整体模态(G3rd, G5th, G8th, G11th)频率满足1:1比值关系.
同样采用无相位滤波法分离各子工况下主共振频率附近的拉索响应, 汇总其响应振幅最大值曲线如图14所示.
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图 14 DM1#和DM3#模型在RC4#工况下拉索主共振频率附近的振幅最大值Figure 14. The maximum response of cables in DM1# and DM3# under RC4#图14结果与图13结果规律基本一致, 即非主导模态的影响效应随阶次变化而变化, 且基本以DM3#模型的计算结果较大, 表明在相同初始条件下, 非主导构件模态的参与会减弱拉索的内共振响应, 且对于各阶的影响效应也随拉索锚固位置与基础参数的不同而不同. 为了能更进一步明晰HGM内共振模式的影响效应, 定义ηci(n)表示Ci#在第n阶EGM(AciE)和HGM(AciH)两者模型中振动响应最大值的误差率, 如下
$$ \eta _{ci}^{(n)} = \frac{{A_{ci}^{\rm{E}} - A_{ci}^{\rm{H}}}}{{A_{ci}^{\rm{E}}}} \times 100\% $$ (54) ηci(n) > 0表明非主导模态的影响效应为抑制作用, 反之为激励作用. 为了对比EGM和HGM两种模式内共振下的影响效应, 汇总C1#和C2#在G1st ~ G11th中两种模式的响应误差率如图15所示.
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图 15 同侧拉索在两类GM下振动响应最大值的误差率Figure 15. The error coefficient of same-side cables’ maximum response in these two GMs图15中, 除15(a)中的G6th外, EGM模态下的影响系数数值上接近零, 表明EGM模态下的内共振响应不受非主导构件的振动模态影响, 此时拉索模态与主导的主梁或主塔模态“1:1”耦合并激励产生剧烈内共振. 而在HGM模态下, 两索的影响系数变化趋势基本相同, 且除G1st外基本表现为抑制作用, 此时非主导模态参与了内共振并分走了一部分系统能量, 所以拉索的内共振响应幅值随之减小.
4. 结 论
本文基于离散的参数质量体系, 考虑了塔−索、索−梁间的动态关系与斜拉索初始垂度、大位移振动引起的几何非线性影响, 建立了新的塔−索−梁耦合面内整体动力学模型. 通过有限差分法代数转换了参数体系动力平衡方程的偏微分多项式, 采用模态拖拽法获得了结构运动方程与模态函数, 精细化分析了塔−梁模态耦合作用对整体模态的影响, 首次提出了完全整体模态和混合整体模态两种模式, 考虑拉索一阶模态并通过数值仿真开展了混合整体模态下的1:1内共振影响性分析, 得到结论如下.
(1)塔−梁模态耦合作用对本文结构的低阶对称及高阶整体模态无影响作用, 此时结构的整体模态完全由主塔或主梁振动模态主导, 为完全整体模态; 对结构的低阶反对称整体模态具有显著影响, 此时主塔以及主梁振动模态共同参与了该阶整体模态, 为混合整体模态.
(2)拉索局部模态与两种整体模态“1:1”耦合都将激励产生内共振, 而主塔主导或主梁主导的两种拉索内共振响应动力特性不同. 在同一阶混合整体模态下, 当主导模态的局部化程度系数靠近非主导模态局部化程度系数时, 该阶整体模态的主导模态将出现快速且连续的交换, 而拉索振动响应的动力特性也将随之改变.
(3)构件的模态局部化程度系数是两种模式内共振参与度的重要量化因素. 在完全整体模态内共振中, 非主导模态局部化程度系数接近0, 因此系统能量转换仅发生在拉索与主导模态间, 非主导模态不参与内共振; 而在混合整体模态内共振中, 系统能量转换发生在塔、索和梁局部模态间, 主导与非主导模态的结构振型参与度与各自局部化程度系数一致.
(4)针对本文固结体系四索结构的内共振分析发现, 由于各阶整体模态能量总数恒定, 混合整体模态内共振下非主导的结构模态参与了系统能量转换并将改变拉索振动响应的动力特性, 其影响效应随整体模态阶次与拉索构件参数变化而变化.
(5)本文建立的塔−索−梁耦合动力学模型为深入开展斜拉桥大系统非线性动力特性研究提供了一种可靠途径, 下一步将结合构件几何参数、各阶整体模态频率和振型以更准确研究和定义模态能量, 进而深入开展斜拉桥内共振的参数敏感性分析.
附录A
式(31)中, vpr, vci, vbj, φpr(n), φci(n), φbj(n), qpr, qpr, qpr, fpi(xci), fpi(xci), fpi(xci)等皆为形式相同的列向量, 为避免赘述, 在此仅展示vpr$\; $
$$ {{\boldsymbol{v}}_{pr}} = \left[ {\begin{array}{*{20}{c}} {{v_{p1}}\left( {{z_{p1}},t} \right)} \\ {{v_{p2}}\left( {{z_{p2}},t} \right)} \\ \vdots \\ {{v_{pR}}\left( {{z_{pR}},t} \right)} \end{array}} \right]\tag{A1} $$ 依据式(1)和拉索边界条件, 可得拉索的静平衡线形为二次抛物线线形, 其表达式为[19,24,26-27]
$$ {w_{ci}}\left( {{x_{ci}}} \right){\text{ = }}\frac{{{m_{ci}}g\cos {\theta _{ci}}}}{{2{H_{ci}}}}{x_{ci}} \cdot \left( {{l_{ci}} - {x_{ci}}} \right)\tag{A2} $$ 附录B
式(38) ~ 式(40)中, q相关参数表示为相同形式的R维、I维、J维的列向量, 为避免赘述, 仅取${\ddot{{{\boldsymbol{q}}}}}_{{b}{j}}$, ${\ddot{{{\boldsymbol{q}}}}}_{ci}$和${\ddot{{{\boldsymbol{q}}}}}_{bj}$如下所示
$$ {\ddot {\boldsymbol{q}}_{pr}} = {\left\{ {{{\ddot q}_{p1}},{{\ddot q}_{p2}}, \cdots ,{{\ddot q}_{pr}}, \cdots ,{{\ddot q}_{pR}}} \right\}^{\rm{T}}} \tag{B1}$$ $$ {\ddot {\boldsymbol{q}}_{ci}} = {\left\{ {{{\ddot q}_{c1}},{{\ddot q}_{c2}}, \cdots ,{{\ddot q}_{ci}}, \cdots ,{{\ddot q}_{cI}}} \right\}^{\rm{T}}} \tag{B2}$$ $$ {\ddot {\boldsymbol{q}}_{bj}} = {\left\{ {{{\ddot q}_{b1}},{{\ddot q}_{b2}}, \cdots ,{{\ddot q}_{bi}}, \cdots ,{{\ddot q}_{bJ}}} \right\}^{\rm{T}}} \tag{B3}$$ 除${\boldsymbol{\varPi }}_{pr,1}$, ${\boldsymbol{\varPi }}_{ci,1}$和${\boldsymbol{\varPi }}_{bj,1}$ 外, 其余$\boldsymbol{\varPi }$相关参数矩阵表示为相同形式的R阶、I阶、J阶的对角矩阵. 为避免赘述, 仅取${\boldsymbol{\varPi }}_{pr,1}$, ${\boldsymbol{\varPi }}_{ci,1}$和${\boldsymbol{\varPi }}_{bj,1}$如下所示
$$\qquad\qquad {\boldsymbol{\varPi } _{pr,1}} = \left[ {\begin{array}{*{20}{c}} {{\varPi _{p1,1}}}&{}&{} \\ {}& \ddots &{} \\ {}&{}&{{\varPi _{pR,1}}} \end{array}} \right] \tag{B4}$$ $$\qquad\qquad {\boldsymbol{\varPi } _{ci,1}} = \left[ {\begin{array}{*{20}{c}} {{\varPi _{c1,1}}}&{}&{} \\ {}& \ddots &{} \\ {}&{}&{{\varPi _{cI,1}}} \end{array}} \right]\tag{B5}$$ $$\qquad\qquad {\boldsymbol{\varPi } _{bj,1}} = \left[ {\begin{array}{*{20}{c}} {{\varPi _{b1,1}}}&{}&{} \\ {}& \ddots &{} \\ {}&{}&{{\varPi _{bJ,1}}} \end{array}} \right]\tag{B6} $$ ${{{\boldsymbol{\varOmega}} }}_{\mathit{c}\mathit{i}}$为拉索的局部模态对角矩阵, 其主元形式为
$$ {\varOmega _{ci}} = \frac{{512{E_{ci}}{A_{ci}}{f_{ci}}^2}}{{{\text{π} ^2}{L_{ci}}{l_{ci}}^3{m_{ci}}}} + \frac{{{\text{π} ^2}{H_{ci}}}}{{{l_{ci}}^2{m_{ci}}}} \tag{B7}$$ $\boldsymbol{\varPi }$相关参数矩阵的主元具体形式如下所示
$$ {\varPi _{ci,1}} = \frac{{2\sin {\theta _{ci}}}}{\text{π} } \tag{B8}$$ $$ {\varPi _{ci,2}} = - \frac{{2\cos {\theta _{ci}}}}{\text{π} } \tag{B9}$$ $$ {\varPi _{ci,3}} = \frac{{{c_{ci}}}}{{{m_{ci}}}} \tag{B10} $$ $$ {\varPi _{ci,4}} = \frac{{2{c_{ci}}\sin {\theta _{ci}}}}{{\text{π} {m_{ci}}}} \tag{B11} $$ $$ {\varPi _{ci,5}} = - \frac{{2{c_{ci}}\cos {\theta _{ci}}}}{{\text{π} {m_{ci}}}} \tag{B12}$$ $$ {\varPi }_{ci\text{, }6} = \frac{24\text{π} {E}_{ci}{A}_{ci}{f}_{ci}}{{L}_{ci}{l}_{ci}{}^{3}{m}_{ci}} \tag{B13}$$ $$ {\varPi _{ci,7}} = \frac{{{\text{π} ^4}{E_{ci}}{A_{ci}}}}{{4{L_{ci}}{l_{ci}}^3{m_{ci}}}} \tag{B14}$$ $$ {\varPi _{ci,8}} = \frac{{32{E_{ci}}{A_{ci}}{f_{ci}}\cos {\theta _{ci}}}}{{\text{π} {L_{ci}}{l_{ci}}^2{m_{ci}}}} \tag{B15} $$ $$ {\varPi _{ci,9}} = \frac{{16{E_{ci}}{A_{ci}}{f_{ci}}{{\sin }^2}{\theta _{ci}}}}{{\text{π} {L_{ci}}{l_{ci}}^3{m_{ci}}}}\tag{B16} $$ $$ {\varPi _{ci,10}} = - \frac{{32{E_{ci}}{A_{ci}}{f_{ci}}\sin {\theta _{ci}}}}{{\text{π} {L_{ci}}{l_{ci}}^2{m_{ci}}}} \tag{B17} $$ $$ {\varPi _{ci,11}} = \frac{{16{E_{ci}}{A_{ci}}{f_{ci}}{{\cos }^2}{\theta _{ci}}}}{{\text{π} {L_{ci}}{l_{ci}}^3{m_{ci}}}}\tag{B18} $$ $$ {\varPi _{ci,12}} = \frac{{{\text{π} ^2}{E_{ci}}{A_{ci}}\cos {\theta _{ci}}}}{{{L_{ci}}{l_{ci}}^2{m_{ci}}}} \tag{B19} $$ $$ {\varPi _{ci,13}} = - \frac{{{\text{π} ^2}{E_{ci}}{A_{ci}}\sin {\theta _{ci}}}}{{{L_{ci}}{l_{ci}}^2{m_{ci}}}} \tag{B20} $$ $$ {\varPi _{ci,14}} = \frac{{16{E_{ci}}{A_{ci}}{f_{ci}}\sin (2{\theta _{ci}})}}{{\text{π} {L_{ci}}{l_{ci}}^3{m_{ci}}}} \tag{B21} $$ $$ {\varPi _{ci,15}} = \frac{{{\text{π} ^2}{E_{ci}}{A_{ci}}{{\sin }^2}{\theta _{ci}}}}{{2{L_{ci}}{l_{ci}}^3{m_{ci}}}} \tag{B22} $$ $$ {\varPi _{ci,16}} = \frac{{{\text{π} ^2}{E_{ci}}{A_{ci}}{{\cos }^2}{\theta _{ci}}}}{{2{L_{ci}}{l_{ci}}^3{m_{ci}}}} \tag{B23} $$ $$ {\varPi _{ci,17}} = \frac{{{\text{π} ^2}{E_{ci}}{A_{ci}}\sin (2{\theta _{ci}})}}{{2{L_{ci}}{l_{ci}}^3{m_{ci}}}} \tag{B24}$$ $$ {\varPi _{pr,1}} = \frac{{{c_{pr}}}}{{{m_{pr}}}} \tag{B25} $$ $$ {\varPi _{pr,2}} = \frac{{{E_{ci}}{A_{ci}}{{\sin }^2}{\theta _{ci}}\cos {\theta _{ci}} \cdot \delta \left( {r - {r_i}} \right)}}{{2{l_{ci}}{L_{ci}}{m_{pr}}}} \tag{B26} $$ $$ {\varPi _{pr,3}} = \frac{{16{E_{ci}}{A_{ci}}{f_{ci}}\cos {\theta _{ci}} \cdot \delta \left( {r - {r_i}} \right)}}{{\text{π} {l_{ci}}{L_{ci}}{m_{pr}}}} \tag{B27} $$ $$ {\varPi _{pr,4}} = \frac{{{\text{π} ^2}{E_{ci}}{A_{ci}}\cos {\theta _{ci}} \cdot \delta \left( {r - {r_i}} \right)}}{{4{l_{ci}}{L_{ci}}{m_{pr}}}} \tag{B28} $$ $$ {\varPi _{pr,5}} = - \frac{{{E_{ci}}{A_{ci}}\sin (2{\theta _{ci}}) \cdot \delta \left( {r - {r_i}} \right)}}{{2{L_{ci}}{m_{pr}}}} \tag{B29} $$ $$ {\varPi _{pr,6}} = \frac{{{E_{ci}}{A_{ci}}{{\cos }^3}{\theta _{ci}} \cdot \delta \left( {r - {r_i}} \right)}}{{2{L_{ci}}{l_{ci}}{m_{pr}}}} \tag{B30} $$ $$ {\varPi _{pr,7}} = \frac{{{E_{ci}}{A_{ci}}\sin {\theta _{ci}}{{\cos }^2}{\theta _{ci}} \cdot \delta \left( {r - {r_i}} \right)}}{{{l_{ci}}{L_{ci}}{m_{pr}}}} \tag{B31} $$ $$ {\varPi _{bj,1}} = \frac{{{c_{bj}}}}{{{m_{bj}}}} \tag{B32} $$ $$ {\varPi _{bj,2}} = - \frac{{{E_{ci}}{A_{ci}}\sin {\theta _{ci}}{{\cos }^2}{\theta _{ci}} \cdot \delta \left( {j - {j_i}} \right)}}{{2{l_{ci}}{L_{ci}}{m_{bj}}}} \tag{B33} $$ $$ {\varPi _{bj,3}} = - \frac{{16{E_{ci}}{A_{ci}}{f_{ci}}\sin {\theta _{ci}} \cdot \delta \left( {j - {j_i}} \right)}}{{\text{π} {l_{ci}}{L_{ci}}{m_{bj}}}} \tag{B34} $$ $$ {\varPi _{bj,4}} = - \frac{{{\text{π} ^2}{E_{ci}}{A_{ci}}\sin {\theta _{ci}} \cdot \delta \left( {j - {j_i}} \right)}}{{4{l_{ci}}{L_{ci}}{m_{bj}}}} \tag{B35} $$ $$ {\varPi _{bj,5}} = - \frac{{{E_{ci}}{A_{ci}}\sin (2{\theta _{ci}}) \cdot \delta \left( {j - {j_i}} \right)}}{{2{L_{ci}}{m_{bj}}}} \tag{B36} $$ $$ {\varPi _{bj,6}} = - \frac{{{E_{ci}}{A_{ci}}{{\sin }^3}{\theta _{ci}} \cdot \delta \left( {j - {j_i}} \right)}}{{2{l_{ci}}{L_{ci}}{m_{bj}}}} \tag{B37} $$ $$ {\varPi _{bj,7}} = - \frac{{{E_{ci}}{A_{ci}}{{\sin }^2}{\theta _{ci}}\cos {\theta _{ci}} \cdot \delta \left( {j - {j_i}} \right)}}{{{l_{ci}}{L_{ci}}{m_{bj}}}} \tag{B38} $$ 为简化表达, 定义Ppm, Ppm,n, γpi, γbi, μpi和μbi表达式分别为
$$ {P_{pm}} = {E_{pm}}{I_{pm}} \tag{B39} $$ $$ {P_{pm,n}} = {E_{pm}}{I_{pm}} + {E_{pn}}{I_{pn}} \tag{B40} $$ $$ {\gamma _{pi}} = {H_{ci}}\sin {\theta _{ci}} \tag{B41} $$ $$ {\gamma _{bi}} = {H_{ci}}\cos {\theta _{ci}} \tag{B42} $$ $$ {\mu _{pi}} = {{{E_{ci}}{A_{ci}}{{\cos }^2}{\theta _{ci}}} \mathord{\left/ {\vphantom {{{E_{ci}}{A_{ci}}{{\cos }^2}{\theta _{ci}}} {{L_{ci}}}}} \right. } {{L_{ci}}}} \tag{B43} $$ $$ {\mu _{bi}} = {{{E_{ci}}{A_{ci}}{{\sin }^2}{\theta _{ci}}} \mathord{\left/ {\vphantom {{{E_{ci}}{A_{ci}}{{\sin }^2}{\theta _{ci}}} {{L_{ci}}}}} \right. } {{L_{ci}}}} \tag{B44} $$ Spr与Sbj, Dpr与Dbj, Cpr与Cbj形式相同, 仅角标(p, r或b, j)不同, 限于篇幅, 在此仅展示Spr, Dpr, Cpr为
$$\qquad\qquad\quad\qquad\qquad {{\boldsymbol{S}}_{pr}} = \left[ {\begin{array}{*{20}{c}} {{S_{pr}}(1,1)}&{ - 2{P_{p1,2}}}&{{P_{p2}}}&{}&{} \\ { - 2{P_{p1,2}}}&{{S_{pr}}(2,2)}&{ - 2{P_{pr,r + 1}}}&{{P_{pr + 1}}}&{} \\ {{P_{pr - 2}}}&{ - 2{P_{pr - 1,r}}}&{{S_{pr}}(r,r)}& \ddots & \ddots \\ {}& \ddots & \ddots & \ddots &{ - 2{P_{pR - 1,R}}} \\ {}&{}&{{P_{pR - 1}}}&{ - 2{P_{pR - 1,R}}}&{{S_{pr}}(R,R)} \end{array}} \right] \tag{B45} $$ $$\qquad\qquad\quad\qquad\qquad {{\boldsymbol{D}}_{pr}} = \left[ {\begin{array}{*{20}{c}} 0&0&{}&{}&{}&{} \\ \ddots & \ddots & \ddots &{}&{}&{} \\ {}&0&{\dfrac{{ - {\gamma _{p{r_1}}}}}{{{m_{p{r_1}}}{d_p}}}}&{\dfrac{{{\gamma _{p{r_1}}}}}{{{m_{p{r_1}}}{d_p}}}}&{}&{} \\ {}&{}&{\dfrac{{\displaystyle\sum\limits_{i = 2}^{i - 1} {{\gamma _{p{r_i}}}} }}{{{m_{p{r_i}}}{d_p}}}}&{\dfrac{{ - 2\displaystyle\sum\limits_{i = 2}^{i - 1} {{\gamma _{p{r_i}}}} - {\gamma _{p{r_i}}}}}{{{m_{p{r_i}}}{d_p}}}}&{\dfrac{{\displaystyle\sum\limits_{i = 2}^i {{\gamma _{p{r_i}}}} }}{{{m_{p{r_i}}}{d_p}}}}&{} \\ {}&{}&{}& \ddots & \ddots & \ddots \\ {}&{}&{}&{}&{\dfrac{{\displaystyle\sum\limits_{i = 1}^I {{\gamma _{p{r_i}}}} }}{{{m_{pR}}{d_p}}}}&{ - \dfrac{{\displaystyle\sum\limits_{i = 1}^I {{\gamma _{p{r_i}}}} }}{{{m_{pR}}{d_p}}}} \end{array}} \right] \tag{B46} $$ $$\qquad\qquad\quad\qquad\qquad {{\boldsymbol{C}}_{pr}} = \left[ {\begin{array}{*{20}{c}} 0&{}&{}&{}&{}&{}&{} \\ {}& \ddots &{}&{}&{}&{}&{} \\ {}&{}&{\dfrac{{{\mu _{p{r_1}}}}}{{{m_{p{r_1}}}}}}&{}&{}&{}&{} \\ {}&{}&{}& \ddots &{}&{}&{} \\ {}&{}&{}&{}&{\dfrac{{{\mu _{p{r_i}}}}}{{{m_{p{r_i}}}}}}&{}&{} \\ {}&{}&{}&{}&{}& \ddots &{} \\ {}&{}&{}&{}&{}&{}&0 \end{array}} \right] \tag{B47} $$ 式中
$$\qquad\qquad\qquad\qquad\qquad {S_{pr}}(r,r) = \left\{\begin{aligned} & {{\left( {{E_{pr - 1}}{I_{pr - 1}} + 4{E_{pr}}{I_{pr}} + {E_{pr + 1}}{I_{pr + 1}}} \right)} / ({{m_{pr}}{d_p}^3})},2 < r < R \\ & {{{E_{pr}}{I_{pr}}} / ({{m_{pr}}{d_p}^3})},r = 1 \\ & {{\left( {4{E_{pr}}{I_{pr}} + {E_{pr + 1}}{I_{pr + 1}}} \right)} / ({{m_{pr}}{d_p}^3})},r = 2 \\ & {{\left( {{E_{pr - 1}}{I_{pr - 1}} + 3{E_{pr}}{I_{pr}}} \right)} / ({{m_{pr}}{d_p}^3})},r = R\end{aligned} \right. \tag{B48}$$ $$\qquad\qquad\qquad\qquad\qquad {S_{bj}}(j,j) = \left\{\begin{aligned} & {\left( {{E_{bj - 1}}{I_{bj - 1}} + 4{E_{bj}}{I_{bj}} + {E_{bj + 1}}{I_{bj + 1}}} \right)/({m_{bj}}{d_b}^3),1 < j < J} \\ & \left( {4{E_{bj}}{I_{bj}} + {E_{bj + 1}}{I_{bj + 1}}} \right)/({m_{bj}}{d_b}^3),j = 1 \\ & \left( {{E_{bj - 1}}{I_{bj - 1}} + 4{E_{bj}}{I_{bj}}} \right)/({m_{bj}}{d_b}^3),j = J \end{aligned} \right. \tag{B49} $$ 塔的重力影响矩阵Gpr为
$$ {{\boldsymbol{G}}_{pr}} = \left[ {\begin{array}{*{20}{c}} 0&{\dfrac{g}{{{d_p}}}}&{}&{} \\ {\dfrac{{{m_{p1}}g}}{{{m_{p2}}{d_p}}}}&{ - \dfrac{{{m_{p1}}g + {m_{p2}}g}}{{{m_{p2}}{d_p}}}}&{\dfrac{{\displaystyle\sum\limits_{r = 1}^2 {{m_{pr}}g} }}{{{m_{p2}}{d_p}}}}&{} \\ {}&{\dfrac{{\displaystyle\sum\limits_{r = 1}^{r - 1} {{m_{pr}}g} }}{{{m_{pr}}{d_p}}}}&{ - \dfrac{{2\displaystyle\sum\limits_{r = 1}^{r - 1} {{m_{pr}}g + {m_{pr}}{g}} }}{{{m_{pr}}{d_p}}}}& \ddots \\ {}&{}&{\dfrac{{\displaystyle\sum\limits_{r = 1}^{R - 1} {{m_{pr}}g} }}{{{m_{pR}}d_p}}}&{ - \dfrac{{\displaystyle\sum\limits_{r = 1}^{R - 1} {{m_{pr}}g} }}{{{m_{pR}}d_p}}} \end{array}} \right] \tag{B50} $$ $$ {{\boldsymbol{\eta}} _{pk}} = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{D}}_{bj}}\left( {{j_k},{j_k} - 1} \right)} \\ {{{\boldsymbol{D}}_{bj}}\left( {{j_k},{j_k}} \right)} \\ {{{\boldsymbol{D}}_{bj}}\left( {{j_k},{j_k} + 1} \right)} \end{array}} \right] \cdot {\left[ {\begin{array}{*{20}{c}} {{q_{{j_k} - 1}}} \\ {{q_{{j_k}}}} \\ {{q_{{j_k} + 1}}} \end{array}} \right]^{\text{T}}} \tag{B51} $$ $$ {{\boldsymbol{\eta}} _{bk}} = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{G}}_{pr}}\left( {{r_k},{r_k} - 1} \right) + {{\boldsymbol{D}}_{pr}}\left( {{r_k},{r_k} - 1} \right)} \\ {{{\boldsymbol{G}}_{pr}}\left( {{r_k},{r_k}} \right) + {{\boldsymbol{D}}_{pr}}\left( {{r_k},{r_k}} \right)} \\ {{{\boldsymbol{G}}_{pr}}\left( {{r_k},{r_k} + 1} \right) + {{\boldsymbol{D}}_{pr}}\left( {{r_k},{r_k} + 1} \right)} \end{array}} \right] \cdot {\left[ {\begin{array}{*{20}{c}} {{{{q}}_{{r_k} - 1}}} \\ {{{{q}}_{{r_k}}}} \\ {{{{q}}_{{r_k} + 1}}} \end{array}} \right]^{\text{T}}} \tag{B52} $$ -
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图 1 离散化的斜拉桥面内整体动力学模型
Figure 1. The in-plane full bridge dynamic model of a cable-stayed bridge
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图 5 G1st ~ G9th结构的整体模态频率变化曲线
Figure 5. The trends of the structural global modal frequencies of G1st ~ G9th
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图 6 斜拉桥面内固有振动模态的细分流程
Figure 6. The reduced process of the in-plane natural vibration modes of cable-stayed bridges
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图 7 结构N5th ~ N9th整体模态频率随C1#的垂度与索力变化曲线
Figure 7. The trends of the global modal frequencies from N5th to N9th with the sag and force of C1#
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图 8 局部−整体模态耦合激励产生“1:1”内共振现象
Figure 8. The phenomenon of the 1:1 internal resonance excited by the local-global modal interactions
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图 9 结构G7th模态参数随主塔单位长度质量变化规律
Figure 9. The trend of the G7th -order modal property varied with the mass per unit length of the pylon
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11 DM1#和DM2#模型在RC1#工况下主共振频率对应的结构振动形态
11. The structural dynamic configuration corresponding to the main resonance frequency in DM1# and DM2# under RC1#
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图 12 RC2#工况下C1#时程曲线与对应的结构振动形态
Figure 12. The time histories of C1# and the corresponding the structural dynamic configuration under RC2#
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图 13 DM1#和DM2#模型在RC3#工况下拉索主共振频率附近的振幅最大值
Figure 13. The maximum response of cables in DM1# and DM2# under RC3#
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图 14 DM1#和DM3#模型在RC4#工况下拉索主共振频率附近的振幅最大值
Figure 14. The maximum response of cables in DM1# and DM3# under RC4#
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图 15 同侧拉索在两类GM下振动响应最大值的误差率
Figure 15. The error coefficient of same-side cables’ maximum response in these two GMs
表 1 拉索参数
Table 1 Parameters of cables
No. mci/(kg·m−1) Eci/Pa Aci/cm2 Hci/N lci/m θci/(°) $ {z}_{p{r}_{i}} $/m $ {x}_{b{j}_{i}} $/m Ωci/Hz C1# 68.1 2.0 × 1011 66.8 8.0 × 105 42.52 0.56 28 8 1.42 C2# 60.1 2.0 × 1011 27.5 6.2 × 105 28.84 0.98 24 24 1.87 C3# 60.1 2.0 × 1011 27.5 6.2 × 105 28.84 2.16 24 56 1.87 C4# 68.1 2.0 × 1011 66.8 8.0 × 105 42.52 2.42 28 72 1.42 
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表 2 主塔及主梁参数
Table 2 Parameters of the pylon and beam
Component μm/(kg·m−1) E/Pa I/m4 L/m Lk/m pylon 8142.97 3.45 × 1010 0.21 40 32 beam 12819.29 3.45 × 1010 0.41 80 40
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表 4 两种方法得到的结构体系整体模态频率
Table 4 Global modal frequencies obtained via these two methods
Mode no. FEM/Hz EMP/Hz Error/% d = 1 m d = 4 m d = 1 m d = 4 m G1st 1.17 1.13 1.12 −2.8 −4.2 G2nd 2.07 2.06 2.03 −0.6 −1.9 G3rd 2.66 2.66 2.52 −0.1 −5.2 G4th 3.75 3.76 3.61 −0.3 −3.6 G5th 4.55 4.51 4.38 −0.8 −3.7
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表 5 3种索−支撑简化动力学模型
Table 5 Three dynamic models reduced from the cable-support structures
Sketch Instruction The upper end of the cable is anchored on the pylon while lower end is anchored on the beam. The upper end of the cable is consolidation while the lower end is anchored on the beam. The upper end of the cable is anchored on the pylon while the lower end is consolidation. pylon-cable-beam modal coupling cable-beam modal coupling pylon-cable modal coupling Abbreviation DM1# DM2# DM3#
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表 6 DM1#中发生G7th和G9th内共振时, C1#振动子系统构件在各自主共振频率附近的分离响应信号
Table 6 The separated signals around the main resonance frequency in the dynamic sub-system of C1# when the internal resonance of G7th -order or G9th -order occurring in DM1#
When couples with G7th-order HGM in DM1# When couples with G9th-order HGM in DM1#
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1. 陈柯帆，李源，卢涛，贺拴海，宋一凡，杨鹏，任祥伟. 考虑模态振型影响的斜拉桥非线性内共振能量转换研究. 中国公路学报. 2024(09): 157-170 . 
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