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

At the flutter critical velocity (87 m/s), the frequency of the torsion mode is 0.31Hz and vertical mode is 0.185 Hz. Flutter critical velocity predicted with the traditional iterative search method (Simiu and Scanlan 1986) is 91.0 m/s and measurement in wind tunnel test is 89 m/s (Lin and Xiang 1995). These comparisons have generally verified the present prediction.

While complex eigenvalue analysis is able to identify the critical point of flutter, oscillation modal frequencies for the two modes (0.31 vs. 0.185 Hz) do not approach and do not finally become the same value as stated in the literature. Though there is a tendency that the two frequency curves will approach or meet as the increase of the wind velocity, they do 40 not meet at the flutter wind velocity (they may meet beyond flutter wind velocity). When the natural frequencies of the coupled modes are close enough, the modal frequency curves may approach and intersect at the flutter wind velocity (This will be shown later with a simulated example). To further investigate the vibration characteristics and explain the discrepancy between the observations made above and the statement that flutter is a single-frequency vibration, time domain analysis is conducted below.

43 3.5.2 Time-Series and their Frequency Content Once the oscillating frequencies of vertical and torsion modes are predicted as shown in Bω Fig. 3.4, they are used to calculate the reduced frequency K = in order to quantify the U matrices A and B in Eq. (3.13). This state-space equation is then solved in the time domain.

The initial displacements of free vibration are assumed to be 0.01 m and 0.01 deg for vertical and torsion modes, respectively. Fig. 3.5 shows the time-series of the vertical and torsion vibrations at U = 30 m/s. It is obvious that they vibrate at two different frequencies, which is also clearly shown in Fig. 3.6, the spectra diagram. The first peak in Fig. 3.6(a) shows that the vertical mode vibrates mainly at its effective frequency that is predicted in the complex eigenvalue analysis. There exists also a small peak that corresponds to the torsion frequency due to the aerodynamic coupling between the two modes. Similarly, the second peak of Fig.

3.6(b) shows that the torsion mode vibrates mainly at its effective frequency with a very small peak at a frequency corresponding to that of vertical mode due to mode coupling.

Figs. 3.7 and 3.8 show the time-series and spectra at wind speed of 60 m/s. At this higher wind speed, the peaks due to modal coupling are more obvious, especially the second peak of vertical vibration shown in Fig. 3.8(a). The two modes still vibrate in different frequencies.

When the wind speed increases up to 87m/s, time-series in Fig. 3.9 show almost constant magnitude vibrations for both vertical and torsion modes, and both modes vibrate about the same frequency. These constant-amplitude vibrations suggest flutter critical condition and the flutter velocity can be identified as 87 m/s. The same vibration frequency of the two time-series at flutter (U = 87 m/s) is clearly shown through their spectral diagrams in Fig. 3.10. The two peaks correspond to the same frequency of 0.31 Hz. For vertical mode, the original dominant peak corresponding to its effective frequency of 0.175 Hz (Fig. 3.6(a)) becomes very trivial as shown in Fig. 3.10.

When the wind speed increases up to 87m/s, time-series in Fig. 3.9 show almost constant magnitude vibrations for both vertical and torsion modes, and both modes vibrate about the same frequency. These constant-amplitude vibrations suggest flutter critical condition and the flutter velocity can be identified as 87 m/s. The same vibration frequency of the two time-series at flutter (U = 87 m/s) is clearly shown through their spectral diagrams in Fig. 3.10. The two peaks correspond to the same frequency of 0.31 Hz. For vertical mode, the original dominant peak corresponding to its effective frequency of 0.175 Hz (Fig. 3.6(a)) becomes very trivial as shown in Fig. 3.10.

The above observations have numerically proven that the vertical and torsion modes exhibit the same vibration frequency at flutter. Such numerical results agree exactly the observation made in the wind-tunnel tests and the traditional belief that the coupled modes should have the same vibration frequency when flutter happens. However, though the vertical mode vibrates at the same frequency as the torsion mode at flutter, the “same frequency” is not due to the change of the effective oscillation frequency in vertical mode. The two effective oscillation frequencies never “meet” in this numerical example. This single frequency vibration is due to the magnification of the coupling effect at flutter so that the vibration corresponding to the torsion-induced coupling is dominant and that corresponding to the effective oscillation frequency of vertical mode is relatively too small to be seen by the observers. Therefore, a single-frequency vibration is observed at flutter.

44 0.40

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51 3.5.3 Vibration Characteristics of Buffeting With the same aerodynamic matrices as used for flutter analysis, buffeting response was obtained through a spectral analysis of the equations of motion.

Figs. 3.11 to 3.13 show the buffeting displacement spectra on the edge of the deck for the wind velocities of 30, 60 and 87 m/s, respectively. For comparison, the results of singlemode-based buffeting response are also shown in these figures. When the wind velocity is 30 m/s, vertical and torsion motions vibrate with different dominant frequencies (Fig. 3.11).

When the wind velocity increases to 60 m/s, Fig. 3.12 shows that vertical and torsion modes vibrate with stronger coupling effects; in other words, the magnitudes of the second peak of Fig. 3.12(a) and the first peak of the Fig. 3.12(b) increase obviously.

When wind velocity approaches the flutter critical velocity (U = 87 m/s), Fig. 3.13 shows that the vertical motion exhibits the same dominant frequency as the torsion mode.

The vibration corresponding to the effective oscillation frequency of vertical mode is greatly damped out (indicated by the flat peak at frequency near 0.175 Hz). The vibration of the vertical mode is mainly from modal coupling indicated by the sharp peak near 0.3 Hz.

Physically, the two modes vibrate in the same dominant frequency (about 0.3 Hz), i.e., the divergent buffeting vibration indicates flutter instability – a single frequency vibration.

Comparison between the results of buffeting analyses in Figs. 3.11 to 3.13 and those of flutter analyses in Figs. 3.6, 3.8 and 3.10 suggests that their vibration characteristics are similar, though not exactly the same due to the difference of vibration types (free vibration vs. forced vibration). Overall, the vibration characteristics of flutter and buffeting are well correlated.

In comparison, the results of single-mode-based buffeting analyses are significantly different from that of multi-mode coupled analyses at high wind velocity, especially at flutter critical wind velocity. Fig. 3.11 shows a small difference when wind speed is low (30 m/s) and Fig. 3.13 shows a very significant difference when wind speed is high (U = 87 m/s). It suggests that for future longer and slender bridges whose natural frequencies will even be lower, aerodynamic coupling effect will be more significant.

3.5.4 Discussions of Coupling Effect on Vibration Characteristics Due to the modal coupling among the coupling-prone modes, small peaks may be observed in the spectrum diagrams for low wind velocity, in addition to the resonant peaks corresponding to the effective modal frequencies. With the increase of wind speed, modal coupling among modes is strengthened by aeroelastic effects so that the original trivial peaks induced by modal coupling may gradually dominate the vibration and become sharp peaks in the spectral diagrams. Correspondingly, the once dominant peaks at the effective frequency may gradually be damped out and become trivial flat peaks (Figs. 3.11 to 3.13). In a twomode model with vertical and torsion modes, such phenomenon usually happens for vertical mode and finally the dominant frequencies for both modes become the same – close to the effective frequency of the torsion mode.

These observations indicate that the “same frequency” multimode flutter vibration is actually due to the effect of aerodynamic coupling. With the increase of wind velocity, the frequency of vertical mode increases and the frequency of torsion mode decreases correspondingly. However, the effective oscillation frequencies of vertical and torsion modes 52 1

0.1 55 are not necessary to “meet” at flutter critical wind speed as suggested in the literature.

Actually, the torsion vibration excites the vertical vibration through aerodynamic coupling at the effective oscillation frequency of torsion mode, and vice versa. When aerodynamic coupling is weak, the dominant vibration is around its effective frequency (Figs. 3.6 and 3.11). With the increase of wind velocity, the resonant vibration at the effective frequency is damped out gradually since the modal damping ratio of vertical mode increases significantly as shown in Fig. 3.4. The dominant vibration frequency thus changes from its own effective oscillation frequency to the effective oscillation frequency of torsion motion. Due to its low damping ratio, the torsion vibration usually keeps its effective oscillation frequency as the dominant one for the entire range of wind speeds.

These observations indicate that the “same frequency” multimode flutter vibration is actually due to the effect of aerodynamic coupling. With the increase of wind velocity, the frequency of vertical mode increases and the frequency of torsion mode decreases correspondingly. However, the effective oscillation frequencies of vertical and torsion modes are not necessary to “meet” at flutter critical wind speed as suggested in the literature.

Actually, the torsion vibration excites the vertical vibration through aerodynamic coupling at the effective oscillation frequency of torsion mode, and vice versa. When aerodynamic coupling is weak, the dominant vibration is around its effective frequency (Figs. 3.6 and 3.11). With the increase of wind velocity, the resonant vibration at the effective frequency is damped out gradually since the modal damping ratio of vertical mode increases significantly as shown in Fig. 3.4. The dominant vibration frequency thus changes from its own effective oscillation frequency to the effective oscillation frequency of torsion motion. Due to its low damping ratio, the torsion vibration usually keeps its effective oscillation frequency as the dominant one for the entire range of wind speeds.

As discussed earlier and demonstrated in Fig. 3.4, the vertical and torsion effective frequencies are not necessary to “meet” at flutter critical wind velocity. However, they do “meet” if the modal frequencies are in a specific ratio. To demonstrate this, the torsion natural frequency was reduced numerically to 66% of its original value and the vertical frequency was kept the same. As a result, a new complex eigenvalue analysis shows that the vertical and torsion frequencies meet at the flutter velocity of about 60 m/s as shown in Fig.

3.14. It can be seen from the figure that the two modal frequencies approach gradually together accompanying an increase of the damping ratio. When approaching the flutter wind velocity, the damping ratio of the vertical mode increases while the torsion mode decreases abruptly. When the two modal frequencies eventually “meet” together, the damping ratio of the torsion mode becomes negative, which leads to the flutter instability.

The motion of the example bridge consists of mainly three parts: (1) vertical motion at its effective frequency, (2) vertical motion excited at the frequency of torsion and (3) torsion motion excited at its effective frequency (torsion motion excited by vertical frequency is relatively small and neglected). Stronger coupling could be expected when two modal frequencies are close enough to be able to “meet” when flutter occurs. It also suggests that the natural frequency ratio of the torsion and vertical modes plays an important role in determining the coupling effect and the critical point of flutter.

3.6. Concluding Remarks With the increase of span length and slenderness, bridge flutter instability and buffeting response become important issues for future bridges. Coupling effect plays an extremely 56 important role in both flutter and buffeting of these future bridges. In this paper, a hybrid approach based on complex eigenvalue modal analysis and time domain analysis is used to better understand the natures of evolution process of bridge vibration. With this hybrid approach, the mechanism of the flutter occurrence and the actual transition process from multi-frequency dominated buffeting to the single-frequency dominated flutter are well illustrated through the numerical results of the Humen Bridge. The following conclusions can

**be drawn from the present study:**

(a) The proposed approach provides a convenient way to numerically replicate the transition of vibrations from multi-frequency buffeting or pre-flutter free vibration to a singlefrequency flutter. This transition has been observed in wind tunnel tests and is commonly accepted as a fact.

(b) The single-frequency vibration at flutter is not due to the “meeting” of the oscillation frequencies of all modes. It is due to the magnified modal coupling effect at flutter critical wind velocity. This coupling effect is very significant so that it dominates the vibration in uncritical mode. As a result, all modes are observed or felt by the instruments to vibrate in the same frequency as the effective oscillation frequency of the critical mode, though the actual effective frequency of each mode is different.

(c) The present study has shown that flutter and buffeting are well correlated in terms of their vibration characteristics. Flutter can actually be interpreted from divergent buffeting vibration. As shown numerically, by using the same aerodynamic matrices, the buffeting vibration automatically transits into a single-frequency flutter vibration at flutter critical wind velocity.

(d) Comparison of the results from the hybrid method with those from available experiments of the Humen Bridge has validated the proposed approach.