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«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»

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Similarly, the MCF values between Mode 5 (the 1st symmetric torsional mode) and the other modes are shown Fig. 2.6. It can be seen that Mode 3 (the 1st vertical symmetric mode) makes the largest contribution to the buffeting response of Mode 5. However, ϑ53 is significantly less than ϑ35 (see Fig. 2.4), suggesting that the aeroelastic modal coupling effects among modes may not have the property of reciprocity, i.e., ϑij may not be equal to ϑji. This is due to the well-known fact that the aeroelastic matrix is not symmetric.

–  –  –

The MCF values between Mode 6 (the 2nd symmetric torsional mode) and the other modes are shown in Fig. 2.7. While both Modes 3 and 5 make more significant contributions than the other modes, the absolute MCF values are very small, indicating that modal coupling between Mode 6 and the other modes are very weak and can thus be ignored.

As observed above, the MCF values for all the modes are relatively small in low wind velocity and increase with the wind velocity. This indicates that modal coupling effect is mostly due to the aeroelastic modal coupling effect since the structural coupling has nothing to do with wind velocity.

–  –  –

For the Yichang Suspension Bridge, the calculated MCF can clearly disclose the nature of modal coupling as well as the contribution of other modes to a given mode. Furthermore, the necessity to include a specific mode in the multimode analysis can be judged through the MCF values. For engineering practice, it can be concluded that Mode 3 (the 1st symmetric vertical bending mode) and Mode 5 (the 1st symmetric torsional mode) should be included in the coupled analysis, while the response of other modes can be solved in a mode-by-mode manner without considering modal coupling. In other words, for any other mode i except for modes 3 &5, ϑij = 0 can be used in Eq. (2.35). The strong coupling effect of buffeting response between Modes 3 and 5 indicates also a strong coupling tendency of flutter behavior between these two modes. This was verified in wind tunnel test showing strong coupling between these two modes for both buffeting and flutter behaviors (Lin et al. 1998).

2.4.4 Buffeting Prediction Using the Proposed Approximate Method

Considering only a limited number of modes, the approximate response spectrum for each mode of coupled buffeting can be obtained through Eq. (2.31) after omitting the third part of the formula. To verify the accuracy of mode selection using the MCF method discussed above, Figs. 2.8 to 10 show the normalized spectral response for Mode 3 (the 1st 27 symmetric vertical mode) and Mode 5 (the 1st symmetric torsional mode) under the wind velocity of 20, 40 and 70 m/s, respectively. The curve labelled as Coupled Multimode Analysis corresponds to a fully coupled analysis of all the seven modes. The curve labelled as Proposed Approximated Method corresponds to the MCF method, considering only the coupling effect between Modes 3 and 5. Finally, Uncoupled Single-mode Analysis corresponds to traditional mode-by-mode single-mode analysis.

–  –  –

1 0.1 0.01 0.001 0.0001 0.08 0.14 0.20 0.26 0.32 0.38

–  –  –

0.001 0.0001 0.08 0.14 0.20 0.26 0.32 0.38

–  –  –

1 0.1 0.01 0.001 0.08 0.14 0.20 0.26 0.32 0.38

–  –  –

0.001 0.0001 0.08 0.14 0.20 0.26 0.32 0.38

–  –  –

1 0.1 0.01 0.001 0.0001 0.08 0.14 0.20 0.26 0.32 0.38

–  –  –

0.01 0.001 0.0001 0.08 0.14 0.20 0.26 0.32 0.38

–  –  –

It is found in these figures that the Proposed Approximated Method predicts very close results to those of Coupled Multimode Analysis except for a shifting of peak frequency shown in Fig. 2.10, due to the approximation of K ≈ Kj discussed earlier. In this figure, the investigated wind velocity 70 m/s is much higher than the design wind velocity of 29 m/s.

Even though this observed shifting of the peak value frequency, the calculated RMS values of buffeting responses from these two methods are very close. This is because that the total areas under these two curves are about the same. The RMS errors for the vertical displacement of Mode 3 in terms of the generalized coordinate under wind velocities of 20 m/s, 40 m/s, and 70 m/s are 0 %, 1.1 %, and 3.1 %, respectively.

30 Results from Figs. 2.8 to 2.10 show two peak values for Mode 3 (top half of the figure).

The first one, corresponding to its own natural frequency, represents the resonant vibration of Mode 3. The second peak, corresponding to the natural frequency of Mode 5, represents the contribution of Mode 5 to Mode 3 vibration due to modal coupling. Comparison of these figures shows that the second peak value of Mode 3 increases with the wind velocity. At the wind velocity of 70 m/s shown in Fig. 2.10, the second peak value is even larger than the first one, indicating that the coupling effect is a significant, or even become a major, contributor to the buffeting response. In this case, ignoring the coupling effect of Mode 5 on Mode 3 would result in a significant error by comparing the curve labelled “Uncoupled Single-mode Analysis” and the curve labelled “Proposed Approximated Method.” This is due to the strong modal coupling effect as indicated by a large value of ϑ35 (see Fig. 2.4). In comparison, for the Mode 5 (bottom half of the figure), the difference between these two curves (and the other one) is trivial since the modal coupling effect is weak as indicated by a small value of ϑ53 (see Fig. 2.6).

The above calculation is based on the spectrum of individual mode in terms of generalized coordinates. To study the accuracy of the proposed MCF method in terms of the physical displacement, the total RMS values of buffeting response were calculated using Eq.

(2.35) under different wind velocities. The results of the MCF method (considering coupling effect between modes 3 and 5) are compared in Table 2.3 with that of fully coupled analysis (considering coupling effect among all of the 7 modes) and the uncoupled single-mode calculation (ignoring coupling effect among all of the 7 modes). Comparison of the results suggests that the proposed method results in an error of less than 5% in terms of the RMS of the total buffeting response, an accuracy good enough for engineering application.

2.5 Concluding Remarks

In the present study, a general modal coupling quantification method is introduced through analytical derivations of coupled multimode buffeting analysis. With the proposed Modal Coupling Factor, modal coupling effect between any two modes can be quantitatively assessed, which will help better understand modal coupling behavior of long-span bridges under wind action. Such assessment procedure will also help providing a quantitative guideline in selecting key modes that need to be included in coupled buffeting and flutter analyses. As seen in the numerical example, only as few as two key modes are necessary to be included into modal coupling analysis for the engineering practice. While the example does not necessarily represent the most common cases for long span bridges, it demonstrates that only the coupling effect among limited modes are really necessary to be considered in coupled analysis and selecting only the necessary modes will significantly reduce the calculation effort.

Since the MCF represents the inherent characteristics of modal coupling among modes, it can also provide useful information for flutter analysis. For modern bridges with streamlined section profiles, coupled modes instead of single-mode usually control flutter behaviors. Therefore, knowing the coupling characteristics among modes is extremely important in order to select appropriate modes for coupled flutter analysis and to better understand the aerodynamic behavior.

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20 0.12 0.13 0.13 0 0.003 0.003 0.003 0 40 0.30 0.40 0.39 2.6 0.01 0.011 0.011 0 70 0.48 0.99 0.95 4.2 0.025 0.030 0.029 3.5 (RMS displacement at the mid-point of main span for Yichang Bridge) By using the MCF, an approximate method for predicting the coupled multimode buffeting response was derived through a closed-form formula. Numerical results of a prototype bridge have proven that the proposed method is much more accurate than the traditional uncoupled single-mode method. That is especially true when the coupling effect is significant at high wind velocity. Difference of the predicted buffeting responses between the approximate and fully coupled analysis methods (considering the coupling of all modes) is less than 5%.

Another important potential application of the MCF values is in the design of adaptive control strategies. To achieve the optimal control efficiency, an adaptive control may not only aim at controlling a single-mode resonant vibration, but also at reducing coupled vibration by breaking the coupling mechanism. For this purpose, the coupling characteristics of modes need to be known in advance.



3.1 Introduction Long-span bridges are susceptible to wind actions and flutter and buffeting are their two common wind-induced phenomena. Buffeting is a random vibration caused by wind turbulence in a wide range of wind speeds. With the increase of wind speed to a critical one, the bridge vibration may become unstable or divergent - flutter (Scanlan 1978). This critical wind speed is called flutter wind speed. Flutter can occur in both laminar and turbulent winds.

In laminar flow, the bridge vibration prior to flutter is essentially a damped multifrequency free vibration, namely, a given initial vibration will decay to zero. When the wind speed increases to the flutter wind speed, the initial vibration (or self-excited vibration) would be amplified to become unstable. When the bridge starts to flutter due to the increased wind speed, it was observed that all modes respond to a single frequency that is called flutter frequency (Scanlan and Jones, 1990).

In turbulent flow, random buffeting response occurs before flutter. When wind speed is low, each individual mode vibrates mainly in a frequency around its natural frequency and the buffeting vibration is a multi-frequency vibration in nature. Keeping increase of wind speed will lead to a single-frequency dominated divergent buffeting response near the flutter velocity. The divergent buffeting response represents the instability of the bridge-flow system, which can also be interpreted as the occurrence of flutter (Cai et al. 1999). Therefore, physically, buffeting and flutter are two continuous dynamic phenomena induced by the same incoming wind flow. It is a continuous evolution process where a multi-frequency buffeting response develops into single-frequency flutter instability.

Previous flutter analysis usually focused only on finding the flutter wind speed and the corresponding flutter frequency. Except for some generic statements, how the multifrequency pre-flutter vibration turns into a single-frequency oscillatory vibration at the onset of flutter has not yet been well demonstrated numerically. The present study will simulate and discuss the two divergent vibration processes near flutter wind speed. The first case is from self-excited flutter and the second one is from random buffeting vibration. The simulated process will clearly demonstrate how the multi-frequency vibration process evolves into a single-frequency vibration, which will help clarify some confusing statements made in the literature and help engineers better understand the flutter mechanism.

3.2 Motivation of Present Research As discussed above, the bridge vibration is a multi-frequency vibration at low wind speeds. However, when the wind speed approaches the flutter wind speed, the multifrequency vibration merges into a single-frequency dominated flutter vibration.

Numerical simulations of the transition phenomenon from multi-frequency buffeting to single-frequency flutter have not been well introduced.

Flutter was classified by Scanlan (1987) as “stiffness-driven type” and “damping-driven type”. Classical aircraft-type flutter, called “stiffness-driven type” was believed to have typically two coupling modes coalesce to a single flutter frequency (Namini et al. 1992). On the other hand, single-degree-of-freedom, which was also called damping-driven flutter, has 33 different scenario (frequencies does not coalesce) (Scanlan 1987). These statements imply that the frequency of each mode changes to a single value at flutter for stiffness-driven flutter.

However, based on the results of eigenvalue analysis that will be shown later, the predicted modal oscillation frequencies of different modes are not necessarily the same when flutter occurs. Similar observation (oscillation frequencies of the modes are not the same at flutter) can also be made from the numerical results of other investigators (Namini et al. 1992). Chen et al. (2001) analyzed examples with different pair of frequencies for coupled modes.

“Veering” phenomena was observed when the frequencies are very close (extreme case). In the examples with not too close frequencies (like most realistic bridges), the frequencies of the coupled modes did not have the chance to coalesce. However, in the wind tunnel test as well as the observations of the Tacoma Narrows’ failure, the vibration is known to usually exhibit dominant torsion vibration with a single frequency right before the occurrence of flutter instability. The predicted different oscillation frequencies among modes seem to contradict with the observed “single-frequency” flutter vibration. As will be seen later from the numerical example, flutter is observed as a single-frequency vibration because that the modal coupling effects force all modes to respond to the oscillation frequency of the critical mode. The oscillation frequency of each mode does not necessarily merge or change to a single value. The “meeting” of the vertical and torsion frequencies may occur beyond the flutter velocity, not necessarily at the onset point of flutter as previously stated in some papers (Scanlan, 1978).

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