Nonlinear characteristics of flutter and vibration bifurcation phenomenon of a centrally-slotted box deck and their mechanisms
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摘要: 为探究超大跨度缆索承重桥梁在大攻角范围内的颤振稳定性，通过节段模型风洞试验对中央开槽箱梁在风攻角±10°范围内的颤振非线性特性和振动分叉现象及其机理进行了研究。结果显示：当风攻角为−2°～10°时，节段模型系统未发生颤振；当风攻角为−3°和−4°时，观察到了含振动分叉的非线性颤振现象，且起振幅值随风速的增加而减小；当风攻角为−5°～−10°时，颤振无需人工激励就会自动发生。两种非线性颤振均为弯扭耦合颤振，并最终做极限环振动。非线性颤振的起振风速随着负攻角的增大而减小，耦合程度随着折减风速的增加而增加。系统等效阻尼比-振幅曲线可以很好地解释非线性颤振机理，曲线的零点为系统平衡点，其中斜率为正的零点为稳定平衡点，对应稳态振幅；斜率为负的零点为不稳定平衡点，对应起振振幅。对于含振动分叉的非线性颤振，系统存在一个稳定平衡点和一个不稳定平衡点；而对于无需人工初始激励的非线性颤振，系统只有一个稳定平衡点。Abstract: In order to investigate the flutter stability of super-long-span cables supported bridges at large angles of attack (AoAs), the nonlinear flutter characteristics, vibration bifurcation phenomenon and mechanism of a centrally-slotted box deck with AoAs ranging between ±10° were investigated through wind tunnel tests of the sectional model. The results show that no flutter occurs for the sectional model system with AoAs between −2°～10°, nonlinear flutter with vibration bifurcation is observed with AoAs of −3° and −4°, and flutter appears automatically without any artificial initial excitation with AoAs between −5° ～ −10°. Both kinds of flutter are coupled vertical bending and torsional motions and finally attain a stable state of the limit circle oscillation (LCO). The onset wind speed of such nonlinear flutter descends with the increasing absolute value of the negative AoA, and the coupling extent increases with the reduced wind speed. The mechanism of the nonlinear flutter can be well explained by the curve of the system damping ratio against the amplitude. With zero points of the curve representing equilibrium states of the system, a positive slope indicates the stable equilibrium, corresponding to the steady-state amplitude, and a negative slope for the unstable equilibrium and the onset amplitude. For the nonlinear flutter with vibration bifurcation, the system has one stable equilibrium point and one unstable equilibrium point, while for the flutter not requiring any artificial excitation, the system has only one stable equilibrium point.
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图 1 中央开槽箱梁示意图（单位：mm）
Figure 1. Diagram of centrally-slotted box bridge deck (unit: mm)
图 3 非线性颤振时的位移（
$\alpha_0 = -5^\circ, U^* = 7.33$ ）Figure 3. Displacement of the nonlinear flutter (
$\alpha_0 = -5^\circ, U^* = 7.33$ )
图 4 非线性颤振时位移相图（
$\alpha_0 = -5^\circ, U^* = 7.33$ ）Figure 4. Phase diagram of the nonlinear flutter (
$\alpha_0 = -5^\circ, U^* = 7.33$ )
图 5 非线性颤振时的扭转位移（
$\alpha_0 = -5^\circ, U^* = 7.33$ ）Figure 5. Torsional displacement of the nonlinear flutter(
$\alpha_0 = -5^\circ, U^* = 7.33$ )
图 6 非线性颤振时的扭转相图（
$\alpha_0 = -5^\circ, U^* = 7.33$ ）Figure 6. Torsional phase diagram of the nonlinear flutter (
$\alpha_0 = -5^\circ, U^* = 7.33$ )
图 7 稳态振幅随折减风速的变化
Figure 7. Steady-state amplitudes variation with the reduced wind speed
图 8 稳态频率随折减风速的变化
Figure 8. Steady-state frequency variation with the reduced wind speed
图 9 分叉振动前的扭转位移（
$\alpha_0= -4^\circ, U^* = 8.75$ ）Figure 9. Torsional displacement before the bifurcation vibration (
$\alpha_0 = -4^\circ, U^* = 8.75$ )
图 10 分叉振动前的扭转相图 (
$\alpha_0 = -4^\circ, U^* = 8.75$ )Figure 10. Torsional phase diagram before the bifurcation vibration (
$\alpha_0 = -4^\circ, U^* = 8.75$ )
图 11 分叉振动初期的扭转位移（
$\alpha_0 = -4^\circ, U^* = 9.04$ ）Figure 11. Torsional displacement at the initial stage of the bifurcation vibration (
$\alpha_0 = -4^\circ, U^* = 9.04$ )
图 12 分叉振动初期的扭转相图（
$\alpha_0 = -4^\circ, U^* = 9.04$ ）Figure 12. Torsional phase diagram at the initial stage of the bifurcation vibration (
$\alpha_0 = -4^\circ, U^* = 9.04$ )
图 13 分叉振动末期的扭转位移（
$\alpha_0 = -4^\circ, U^* = 9.53$ ）Figure 13. Torsional displacement at the final stage of the bifurcation vibration (
$\alpha_0 = -4^\circ, U^* = 9.53$ )
图 14 分叉振动末期的扭转相图（
$\alpha_0 = -4^\circ, U^* = 9.53$ ）Figure 14. Torsional phase diagram at the final stage of the bifurcation vibration (
$\alpha_0 = -4^\circ, U^* = 9.53$ )
图 15 分叉振动区间后的扭转位移（
$\alpha_0 = -4^\circ, U^* = 9.80$ ）Figure 15. Torsional displacement after the bifurcation vibration (
$\alpha_0 = -4^\circ, U^* = 9.80$ )
图 16 分叉振动区间后的扭转相图（
$\alpha_0 = -4^\circ, U^* = 9.80$ ）Figure 16. Torsional phase diagram after the bifurcation vibration (
$\alpha_0 = -4^\circ, U^* = 9.80$ )
图 17 非线性自限幅颤振起振振幅和稳态振幅随折减风速变化
Figure 17. Steady-state amplitude and onset amplitude variations with the reduced wind speed in the nonlinear self-limiting flutter
图 18 非线性颤振的偏心率均值随折算风速变化
Figure 18. Mean eccentricity ratio variation with the reduced wind speed in the nonlinear flutter
图 19 非线性颤振起始风速和对应偏心率均值随风攻角变化
Figure 19. Onset wind speed and corresponding mean eccentricity ratio variations with the wind attack angle in the nonlinear flutter
图 20 扭转振幅拟合结果(
$\alpha_0 = -5^{\circ}, U^* = 7.33$ )Figure 20. Fitted result of the torsional amplitude(
$ \alpha_0 = -5^{\circ}, U^* = 7.33 $ )
图 21 对数振幅拟合结果(
$\alpha_0 = -5^\circ, U^* = 7.33$ )Figure 21. Fitted result of the logarithmic amplitude(
$\alpha_0 = -5^\circ, U^* = 7.33$ )
图 22 系统等效阻尼比拟合结果（
$\alpha_0 = -5^\circ, U^* = 7.33$ ）Figure 22. Fitted result of the system equivalent damping ratio (
$ \alpha_0 = -5°, U^* = 7.33 $ )
图 23 系统等效阻尼比随扭转振幅变化曲线（
$\alpha_0 = -5^\circ$ ）Figure 23. System equivalent damping ratio variation with the torsional amplitude (
$\alpha_0 = -5^\circ$ )
图 24 系统等效阻尼比随扭转振幅变化曲线（
$\alpha_0 = -4^\circ$ ）Figure 24. System damping ratio variation with the torsional amplitude (
$\alpha_0 = -4^\circ$ )
图 25 气动阻尼比随扭转振幅变化（
$\alpha_0 = -5^\circ$ ）Figure 25. Aerodynamic damping ratio variation with the torsional amplitude (
$\alpha_0 = -5^\circ$ ) -
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