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摘要: 利用非线性理论和混沌时间序列分析方法, 建立了桥梁风致振动的数学模型, 开发了计算桥梁振动加速度时间序列Lyapunov指数的MATLAB程序, 进行了桥梁涡振和颤振的风洞试验, 分析了不同风攻角下的桥梁风致振动的阻尼比、Lyapunov指数与风速的关系以及涡振振幅与风速的关系, 研究了桥梁颤振和涡振的混沌特性。试验结果表明: 在颤振试验中, 当风速小于颤振临界风速15.5m·s-1时, Lyapunov指数小于0, Lyapunov指数与阻尼比存在很大的相关性, 当风速从3m·s-1增大为18m·s-1时, 相空间逐渐发散; 在涡振试验中, 当风速从4.5m·s-1增大至8.5m·s-1时, Lyapunov指数大于0, 桥梁发生明显涡振, 并由多频振动逐渐转变为单频振动, 相空间变为一个较为理想的圆。桥梁的涡振与颤振均属于混沌现象, 低风速下的Lyapunov指数可用来预测高风速下的风致振动, 并且利用相空间也能识别涡振与颤振。Abstract: According to nonlinear theory and chaotic time series analysis method, the mathematical model of bridge wind-induced vibration was built.The MATLAB program for calculating the Lyapunov exponent of bridge vibration acceleration time series was developed, and the flutter and vortex vibration were tested in wind tunnel.Under various wind attack angles, the damping ratios of bridge wind-induced vibrations, the relationships between Lyapunov exponents and wind speeds, and the relationships between vortex vibration amplitudes and wind speeds were analyzed, and the chaos characteristics of flutter and vortex vibration were studied.Test result indicates when wind speed is less than critical wind speed (15.5 m·s-1), the Lyapunov exponent is negative in flutter test, and the close correlation between Lyapunov exponent and damping ratio is found.When wind speed increases from 3 m·s-1 to 18 m·s-1, the phase space becomes divergent gradually.In vortex vibration test, when wind speed increases from 4.5 m·s-1 to 8.5 m·s-1, the Lyapunov exponent is more than 0, obvious vortex vibration happens, and multi-frequency vibration turns to single frequency vibration gradually.The phase space also becomes an ideal circle.Both flutter and vortex vibration are chaos phenomena.Lyapunov exponent at low wind speed can be used to predict the wind-induced vibrations at high windspeed, and the phase space can also be used to explain flutter and vortex vibration.
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图 1 颤振试验中阻尼比与风速的关系
Figure 1. Relationships between damp ratios and wind speeds in flutter test
图 2 涡振试验中振幅与风速的关系
Figure 2. Relationships between vortex vibration amplitudes and wind speeds in vortex vibration test
图 4 颤振试验中5°风攻角的Lyapunov指数
Figure 4. Lyapunov exponents at wind attack angle of 5° in flutter test
图 5 颤振试验中5°风攻角的阻尼比
Figure 5. Damping ratios at wind attack angle of 5° in flutter test
图 6 颤振试验中3°风攻角的Lyapunov指数
Figure 6. Lyapunov exponents at wind attack angle of 3° in flutter test
图 7 颤振试验中3°风攻角的阻尼比
Figure 7. Damping ratios at wind attack angle of 3° in flutter test
图 8 颤振试验中0°风攻角的Lyapunov指数
Figure 8. Lyapunov exponents at wind attack angle of 0° in flutter test
图 9 颤振试验中0°风攻角的阻尼比
Figure 9. Damping ratios at wind attack angle of 0° in flutter test
图 10 颤振试验中-3°风攻角的Lyapunov指数
Figure 10. Lyapunov exponents at wind attack angle of -3° in flutter test
图 11 颤振试验中-3°风攻角的阻尼比
Figure 11. Damping ratios at wind attack angle of -3° in flutter test
图 12 颤振试验中-5°风攻角的Lyapunov指数
Figure 12. Lyapunov exponents at wind attack angle of -5° in flutter test
图 13 颤振试验中-5°风攻角的阻尼比
Figure 13. Damping ratios at wind attack angle of -5° in flutter test
图 14 颤振试验中风速为3m·s-1时的加速度时程曲线
Figure 14. Acceleration time-history curve at wind speed of 3 m·s-1 in flutter test
图 15 颤振试验中风速为3m·s-1时的相空间
Figure 15. Phase space at wind speed of 3 m·s-1 in flutter test
图 16 颤振试验中风速为18m·s-1时的加速度时程曲线
Figure 16. Acceleration time-history curve at wind speed of 18 m·s-1 in flutter test
图 17 颤振试验中风速为18m·s-1的相空间
Figure 17. Phase space at wind speed of 18 m·s-1 in flutter test
图 18 涡振试验中5°风攻角的Lyapunov指数
Figure 18. Lyapunov exponents at wind attack angle of 5° in vortex vibration test
图 19 涡振试验中5°风攻角的振幅
Figure 19. Amplitudes at wind attack angle of 5° in vortex vibration test
图 20 涡振试验中3°风攻角的Lyapunov指数
Figure 20. Lyapunov exponents at wind attack angle of 3° in vortex vibration test
图 21 涡振试验中3°风攻角的振幅
Figure 21. Amplitudes at wind attack angle of 3° in vortex vibration test
图 22 涡振试验中0°风攻角的Lyapunov指数
Figure 22. Lyapunov exponents at wind attack angle of 0° in vortex vibration test
图 23 涡振试验中0°风攻角的振幅
Figure 23. Amplitudes at wind attack angle of 0° in vortex vibration test
图 24 涡振试验中-3°风攻角的Lyapunov指数
Figure 24. Lyapunov exponents at wind attack angle of -3° in vortex vibration test
图 25 涡振试验中-3°风攻角的振幅
Figure 25. Amplitudes at wind attack angle of -3° in vortex vibration test
图 26 涡振试验中-5°风攻角的Lyapunov指数
Figure 26. Lyapunov exponents at wind attack angle of -5° in vortex vibration test
图 27 涡振试验中-5°风攻角的振幅
Figure 27. Amplitudes at wind attack angle of -5° in vortex vibration test
图 28 涡振试验中风速为4.5 m·s-1时的加速度时程曲线
Figure 28. Acceleration time-history curve at wind speed of 4.5 m·s-1 in vortex vibration test
图 29 涡振试验中风速为4.5m·s-1的相空间
Figure 29. Phase space at wind speed of 4.5 m·s-1 in vortex vibration test
图 30 涡振试验中风速为8.5m·s-1时的加速度时程曲线
Figure 30. Acceleration time-history curve at wind speed of 8.5 m·s-1 in vortex vibration test
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