Influences of Attack Angles on Aerodynamic Derivatives and Flutter Characteristics of Flat Box Girders
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摘要:
以南京第四长江大桥扁平箱梁为研究对象，通过节段模型自由振动风洞试验详细测试了模型在不同风攻角下的颤振响应，探讨了系统非稳态及稳态临界振幅随风速的演化规律。首先，基于颤振响应振幅包络，结合Hilbert变换，识别了系统振幅依存的模态阻尼，并初步阐释了颤振形态随风攻角转变的机理。其次，提取了系统在不同风攻角下的模态参数，基于双模态耦合闭合解法，识别了断面在不同风攻角下的非线性颤振导数，研究了关键颤振导数振幅依存性随风攻角变化的规律及对断面颤振形态和特性的潜在影响。最后，通过逐项拆解模态阻尼，深入剖析了风攻角对非耦合及耦合气动阻尼的影响，并阐明了分项阻尼导致系统颤振性能差异性的动力学机理。
Abstract:The flutter responses of the Nanjing No.4 bridge flat box girder under different wind attack angles were tested in detail through sectional model tests. The evolution of unsteady and steady critical amplitudes at different wind speeds was discussed. Based on the amplitude envelope of the flutter response and the Hilbert transform, the amplitude-dependent modal damping of the system was identified, and the mechanism of the flutter mode change with the wind angle of attack was initially explained. Secondly, the modal parameters of the system under different wind attack angles were extracted. With the bimodal coupled flutter analysis method, the nonlinear flutter derivatives of the section under different wind attack angles were identified, and the change law for the amplitude dependence of the key flutter derivatives on the wind attack angle and the potential influence on the section flutter morphology and characteristics, were studied. Finally, the effects of the wind attack angle on the uncoupled and coupled aerodynamic damping were analyzed through analyses of the modal damping subterms one by one, and the dynamic mechanism of the differential flutter performance caused by fractional damping was illustrated.
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Key words:
- flat box girder /
- soft flutter /
- flutter derivative /
- aerodynamic damping
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图 1 南京第四长江大桥主桥布置图(单位：m)
Figure 1. The layout of the main bridge of the Nanjing No.4 bridge (unit: m)
图 2 节段模型桥梁断面图，B/H=10.95 （单位: mm）
Figure 2. The cross-section of the model, B/H=10.95 (unit: mm)
图 3 不同攻角下稳态振幅随风速的变化曲线：(a) 非正攻角下临界振幅随风速的变化曲线；(b) 正攻角下稳态振幅随风速的变化曲线
Figure 3. Curves of amplitude varying with the wind speed at different angles of attack: (a) curves of critical amplitude varying with the wind speed at non-positive angles of attack; (b) curves of steady-state amplitude varying with the wind speed at positive angles of attack
图 4 不同激励下的时程发展曲线(−5°攻角，U=15 m/s)：(a)小激励下衰减的时程曲线；(b)大激励下发散的时程曲线
Figure 4. Time history development curves under different excitations (attack angle of −5°, U=15 m/s): (a) the damped time history curve under the small excitation; (b) the divergent time history curve under the large excitation
图 5 不同激励下的时程发展曲线(5°攻角，U=11.5 m/s)：(a) 无激励下增长至稳定的时程曲线；(b) 大激励下衰减至稳定的时程曲线
Figure 5. Time history development curves under different excitations (attack angle of 5°, U=11.5 m/s): (a) the growth-to-stability time history curve without excitation; (b) the damping-to-stability time curve under the large excitation
图 6 不同激励下的时程发展曲线(0°攻角，U=17 m/s)：(a)小激励下衰减至零的时程曲线；(b)大激励下增长至稳定的时程曲线
Figure 6. Time history development curves under different excitations (attack angle of 0°, U=17 m/s): (a) the damped time history curve under the small excitation; (b) the growth-to-stability time history curve under the large excitation
图 7 非正攻角下不同风速时，阻尼随振幅的变化曲线：(a) U=14 m/s；(b) U=16 m/s
Figure 7. Damping curves varying with the amplitude at non-positive attack angles and different wind speeds: (a) U=14 m/s; (b) U=16 m/s
图 8 正攻角下不同风速时，阻尼随振幅的变化曲线：(a) U=10 m/s；(b) U=13 m/s
Figure 8. Damping curves varying with the amplitude at positive attack angles and different wind speeds: (a) U=10 m/s; (b) U=13 m/s
图 9 不同风攻角下气动参数随风速的变化关系：(a) 频率；(b) 振幅比；(c) 相位差
Figure 9. Aerodynamic parameters varying with the wind speed at different attack angles: (a) the frequency; (b) the amplitude ratio; (c) the phase difference
图 10 非耦合颤振导数识别结果：(a) −5°攻角下的颤振导数
$ A_2^* $ ；(b) −3°攻角下的颤振导数$ A_2^* $ ；(c) 0°攻角下的颤振导数$ A_2^* $ ；(d) 3°攻角下的颤振导数$ A_2^* $ ；(e) 5°攻角下的颤振导数$ A_2^* $ ；(f) 不同攻角下的颤振导数$ A_3^* $ Figure 10. Evolution of uncoupled flutter derivatives: (a)
$ A_2^* $ under a wind attack angle of −5°; (b)$ A_2^* $ under a wind attack angle of −3°; (c)$ A_2^* $ under a wind attack angle of 0°; (d)$ A_2^* $ under a wind attack angle of 3°; (e)$ A_2^* $ under a wind attack angle of 5°; (f)$ A_3^* $ at different wind attack angles
图 11 不同攻角下耦合颤振导数的识别结果：(a) 不同攻角下的颤振导数
$ H_2^* $ ；(b) 不同攻角下的颤振导数$ H_3^* $ Figure 11. Evolution of coupled flutter derivatives: (a)
$ H_2^* $ values at different wind attack angles; (b)$ H_3^* $ values at different wind attack angles
图 12 非正攻角下各阻尼项随振幅变化曲线：（a） 14 m/s下不同攻角的耦合气动阻尼和结构阻尼；（b） 16 m/s下不同攻角的耦合气动阻尼和结构阻尼；（c） 14 m/s下不同攻角的非耦合气动阻尼；（d） 16 m/s下不同攻角的非耦合气动阻尼
Figure 12. The damping term curves varying with the amplitude at non-positive angle of attack: (a) the coupled aerodynamic damping and the structural damping at different attack angles (U=14 m/s); (b) the coupled aerodynamic damping and the structural damping at different attack angles (U=16 m/s); (c) the uncoupled aerodynamic damping at different attack angles (U=14 m/s); (d) the uncoupled aerodynamic damping at different attack angles (U=16 m/s)
图 13 正攻角下各阻尼项随风速变化曲线：(a) 10 m/s不同攻角的耦合气动阻尼和结构阻尼；(b)13 m/s不同攻角的耦合气动阻尼和结构阻尼；(c) 10 m/s不同攻角的非耦合气动阻尼；(d) 13 m/s不同攻角的非耦合气动阻尼
Figure 13. The damping term curves varying with the amplitude under positive angles of attack: (a) the coupled aerodynamic damping and the structural damping at different attack angles (U=10 m/s); (b) the coupled aerodynamic damping and the structural damping at different attack angles (U=13 m/s); (c) the uncoupled aerodynamic damping at different attack angles (U=10 m/s); (d) the uncoupled aerodynamic damping at different attack angles (U=13 m/s)
表 1 基础试验参数
Table 1. Basic test parameters
$ m $/(kg/m) $ I $/(kg·m2/m) $ {\omega _{h0}} $/(rad/s) $ {\omega _{\alpha 0}} $/(rad/s) $ {\xi _{h0}} $ $ {\xi _{\alpha 0}} $ 9.29 0.345 14.20 37.20 0.0035 0.0030 
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表 2 不同攻角下
$ A_{\text{3}}^* $ 取值（$ U/(fB) = 12 $ ）Table 2.
$ A_{\text{3}}^* $ values at different angles of attack（$ U/(fB) = 12 $ ）attack angle −5° −3° 0° 3° 5° $ A_{\text{3}}^* $ 29 22 20.5 21.5 27.5
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表 3 非正攻角下耦合气动阻尼各子项
Table 3. Sub-terms of the coupled aerodynamic damping at non-positive angles of attack
sub-term 14 m/s 16 m/s −5° −3° 0° −5° −3° 0° $ - 0.5\upsilon \mu $ −1.5E + 5 1.5E + 5 1.49E + 5 1.49E + 5 1.49E + 5 1.49E + 5 $ {\left( {\dfrac{{{\omega _2}}}{{{\omega _1}}}} \right)^2}{\left[ {1 - {{\left( {\dfrac{{{\omega _2}}}{{{\omega _1}}}} \right)}^2}} \right]^{ - 1}} $ 1.78 1.74 1.74 1.86 1.81 1.78 $ {[ {{{( {H_2^*} )}^2} + {{( {H_3^*} )}^2}} ]^{1/2}} $ 44.26 46.9 47.8 56.5 63.6 63.6 $ {[ {{{( {A_1^*} )}^2} + {{( {A_4^*} )}^2}} ]^{1/2}} $ 6.26 4.64 4.38 7.98 6.32 5.19 $\sin ( { {\psi ^\prime } } )$ 0.99 0.98 0.99 0.99 0.97 0.98
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表 4 正攻角下耦合项气动阻尼各子项
Table 4. Sub-terms of the coupled aerodynamic damping at positive angles of attack
sub-term 10 m/s 13 m/s 3° 5° 3° 5° $ - 0.5\upsilon \mu $ −1.49E + 5 −1.49E + 5 −1.49E + 5 −1.49E + 5 $ {\left( {\dfrac{{{\omega _2}}}{{{\omega _1}}}} \right)^2}{\left[ {1 - {{\left( {\dfrac{{{\omega _2}}}{{{\omega _1}}}} \right)}^2}} \right]^{ - 1}} $ 1.68 1.69 1.72 1.76 $ {[ {{{( {H_2^*} )}^2} + {{( {H_3^*} )}^2}} ]^{1/2}} $ 21.5 18.77 42.6 39.0 $ {[ {{{( {A_1^*} )}^2} + {{( {A_4^*} )}^2}} ]^{1/2}} $ 2.56 3.30 3.75 5.60 $\sin ( { {\psi ^\prime } } )$ 0.99 0.99 0.99 0.99
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