«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
To study the performance of TMD-based control, the total generalized mass of all TMDs are assumed to be 1% of that of the 1st symmetric bending mode of the bridge. Totally 12 identical TMDs are distributed evenly on the two sides in the middle area of the main 148 span. To save the installation space of TMDs, lever type of TMDs can be adopted and each of them has the mass of 11 ton for the Yichang Bridge. The basic distribution scheme and structure of the lever-type of TMDs are the same as that used by Gu et al. (2001). Figs. 6.9 and 6.10 show the response power spectral density of the 1st symmetric bending mode and the 1st symmetric torsional mode with and without control, respectively. For the original uncontrolled case, there are two peaks in the response curve of the bending mode. One corresponds to the modal frequency of the 1st symmetric bending mode, and the other corresponds to the modal frequency of the 1st symmetric torsional mode.
Fig. 6.11 Displacement Spectra with Torsional Mode Based Control (U= 40 m/s) 151 In Fig. 6.9, the parameters of the TMDs were designed for the 1st symmetric bending mode at U = 40 m/s in the conventional resonant control approach, and modal coupling effects were not considered in the TMD design. Similar to the 2DOF model, it was found that the TMDs designed based on the resonant reduction of the 1st symmetric bending mode are not very efficient in reducing the resonant part of the response for the 1st symmetric bending mode (first peak of Fig. 6.9(a)) due to its already high modal damping ratio. Moreover, they can hardly suppress the response component caused by coupling from the 1st symmetric torsional mode (the second peak of Fig. 6.9(a)). On the other hand, for the response component of the 1st symmetric torsional mode caused by coupling from the 1st symmetric bending mode (the first peak of Fig. 6.9(b)), this TMD design has some insignificant control effect. As expected, resonant component for 1st symmetric torsional mode (the second peak of Fig. 6.9(b)) is not efficiently suppressed.
In contrast, Fig. 6.10 shows that the TMDs designed for the 1st torsional mode are very efficient in not only reducing the resonant component of 1st symmetric torsional mode (the second peak of Fig. 6.10(b)), but also in reducing the second peak of Fig. 6.10(a), which is the response component of the 1st symmetric bending mode caused by coupling from the 1st torsional mode. However, it can also be found that they are not efficient for the response component of the 1st symmetric torsional mode caused by coupling from the 1st symmetric bending mode (the first peak of Fig. 6.10(b)).
6.4.2 Mechanism of Buffeting Control with Strong Coupling Effects
Based on the observations made above, to reduce the vibration in a case of strong modal coupling, TMDs can be designed to suppress peak values in the spectrum. It is obvious in Fig. 6.10 that, to control the response of the 1st symmetric bending mode, TMDs whose frequencies are close to that of the 1st symmetry torsional mode don’t suppress the resonant vibration of the bending mode, but suppress the modal coupling effect between these two modes (the second peak of Fig. 6.10(a)). Suppressing this coupling effect may be significant in reducing the overall vibration.
As has been discussed earlier, Eq (6.1) indicates that the resonant peak of the bending mode is difficult to control due to its high total damping ratio. This also implies that when strong aerodynamic coupling effect exists, the conventionally designed TMDs considering only resonant vibrations may not be efficient for control performance. Optimal variables of TMDs to control any given modes should be searched in a full range (not just near the modal frequencies) in order to optimally control the total response due to resonant vibration and coupled vibration.
For demonstration, two studies were conducted on the Yichang Suspension Bridge under varied wind speed for controlling the vibration of the 1st symmetric bending mode. The first one was to search the optimal TMD frequency considering only the resonant vibration.
The optimal TMD frequency was thus only searched around the modal frequency of the 1st symmetric bending mode. In the second one, the optimal TMD frequency was designed considering only modal coupling effect from the 1st symmetric torsional mode. The TMD frequency was thus searched and restricted around the modal frequency of the 1st symmetric torsional mode.
Fig. 6.12 Bending Mode Displacement Control Efficiency with Different Control Schemes For both cases, TMDs were designed under every wind speed (TMD has an optimal frequency under each wind velocity due to the aeroelastic effects) and corresponding control efficiency of the 1st symmetric bending mode is plotted into the two curves that are shown in Fig. 6.11. It is found that there is an intersection of the two control efficiency curves at the wind velocity near 60 m/s. These two curves suggest that, for optimal control efficiency on the 1st symmetric bending mode, the TMDs should be designed to control the resonant vibration (control efficiency is along point A to point B), then should be changed to control the vibration due to modal coupling when wind speed surpasses 60 m/s (control efficiency is along point B to point C). Ideally, if only a single-frequency TMD is designed, the TMD frequency should be adjustable to achieve optimal control efficiency under different wind speed, namely, the optimal control efficiency will be along points A, B, and C. Practically, for TMD unable to adjust the frequency, the optimal design of TMD relies on the wind speed under which the control is expected. For this particular example, if the control objective is to suppress the response under strong wind with velocity over 60 m/s, the TMD (no matter single or multiple) should be designed around the frequency of the 1st symmetric torsional mode since it will result in the best overall control performance. On the other hand, when the target wind speed is lower than 60 m/s for this problem, the conventional TMD design still have the better control performance. In this case, multiple TMDs can be used, some of them are designed for the 1st symmetric bending mode and some for the 1st symmetric torsional 153 mode. The mass distribution among multiple modes can be decided based on the different contribution to the overall bridge response from these modes. Since this is the issue about conventional resonant suppression design, it is not repeated here.
The above results indicate that for modern long-span bridges, there exists an alternative control approach for TMD design other than the conventional one that focuses only on resonant vibration. Optimal parameters for TMD design should be based on two control mechanisms, i.e., resonant and coupled vibrations. With the increase of modal coupling for ultra-long bridges, the proposed new control strategy may play much more important role in the coupled vibration control.
6.4.3 Dual-Objective Control of Coupled Vibration
TMDs have been proven effective in raising the critical flutter wind speed (Pourzeynali and Datta 2002; Gu et al. 1998). With the common adoption of streamlined cross sections for long-span bridges, coupled flutter is the dominant flutter instability. It has been found above that TMDs may also be effective in suppressing coupled buffeting response through reducing modal coupling effects. Similarly, it is also expected to be effective in improving the flutter stability through destroying the pre-existing modal coupling mechanism. For this reason, it is very logical to pursue dual-objective control of the TMDs, namely, suppressing buffeting response and improving flutter stability at the same time. To validate such a statement, the Yichang Suspension Bridge was also studied on the flutter stability with the TMDs designed for suppressing buffeting response considering the modal coupling effects. It was found that when TMDs are designed for buffeting response control at the wind speed of 65 m/s, the critical flutter wind speed can also be improved from 73 to 86 m/s. Such results are very promising for the extra long-span bridges where TMDs can be effectively adopted to enhance the flutter stability and to suppress the excessive wind-induced buffeting vibration.
6.5 Concluding Remarks Conventional TMD control approach usually focuses on suppressing the resonant vibration by supplying additional damping to the concerned modes. This approach could be inefficient for coupled vibration of long-span bridges in strong wind due to two reasons.
The first is the strong modal coupling effects in strong wind. For slender long-span bridges, the aeroelastic forces from the wind action often cause several vibration modes to couple together. Such coupling effect increases with the increase of wind speed. Coupled buffeting response of each mode usually consists of two major parts: one is resonant component associated with its modal frequency; the other part of response is due to the modal coupling with other modes. For bridges with weak modal coupling effects, the second part is trivial. However, for long-span bridges in high wind speed, modal coupling effects may become quite strong. The latter part of the response is no longer negligible and a control approach focusing on the first part may be inefficient.
The second reason is the increased total modal damping, caused by aeroelastic effects in strong wind, of the concerned modes. Even though damping helps reduce bridge vibration, satisfactory control performance may be extremely difficult to achieve by supplying
The present study proposes a new control approach that is to attenuate the modal coupling effects, in addition to suppressing the resonant vibration with TMDs. The vibration contributions to the total response from modal coupling (second part of Eq. (6.28)) can be significant at high wind velocity. Weakening the coupling effects with TMDs can significantly reduce the overall responses. The newly introduced control approach also enables a well-designed TMD system to be efficient in controlling buffeting vibration of coupled modes even with high modal damping under high wind velocity. For optimal control efficiency in applications, the TMD frequency needs to be adaptable in order to switch from resonant suppression to coupling suppression, or multiple frequency TMDs are needed in order to control both resonant and coupling effects.
The effects of TMDs on reducing both resonant and coupled vibrations have been demonstrated through the analytically derived closed-form solutions. Numerical analyses on a 2DOF model and an actual long-span bridge have validated that the new control approach may lead to more efficient control performance than the conventional resonant- suppression strategy when the coupling effects are significant and when the damping ratios of those modes of concern are high. Finally, the concept of dual-objective control, i.e., the TMD control system for both flutter stability improvement and buffeting response reduction, is briefly discussed.
7.1 Introduction Under wind excitations, long-span bridges exhibit complex aerodynamic behaviors.
Buffeting random response induced by the turbulence of airflow happens throughout the full range of wind speeds. As the wind speed increases, aerodynamic instability phenomena such as flutter may occur (Simiu and Scanlan 1996). Much research effort has been made towards mitigating excessive vibrations and improving aerodynamic stabilities for bridges during construction (Conti et al. 1996; Takeda et al. 1998) and at service stages (Gu et al. 1994; Wilde et al. 1999). Among all of the control procedures, dynamic energy absorbers such as tuned mass dampers (TMDs) were studied and adopted in suppressing excessive vibrations or maintaining the flutter stability of bridges (Gu et al. 1998).
In recent years, the importance of aeroelastic modal coupling to the bridge aerodynamic behaviors has been recognized (Tanaka et al. 1993; Bucher and Lin 1988; Lin and Yang 1983;
Miyata and Yamada 1999; Cai and Albrecht 2000). It has been concluded that the coupling tendency of two modes depends on their mode shapes and natural frequencies in still air as well as the flutter derivatives of the bridge section (Jain et al. 1996). The adoption of more slender deck and the increase of bridge span length tend to result in closer modal frequencies. As a result, modal coupling effects through aeroelastic forces in high wind speeds increase (Jain et al.
1998; Namini et al. 1992; Katsuchi et al. 1998; Thorbek and Hansen 1998).
The TMD is known to be effective in suppressing single-mode resonant vibrations when its frequency is tuned to the modal frequency of the structure. When the modal frequencies of the bridge are well separated and modal coupling effects are weak, each TMD is mainly designed to control a single-mode vibration while the effects from other modes on the control are omitted (Igusa and Xu 1991; Kareem and Kline 1995). Abe and Igusa (Abe and Igusa 1995) studied the performances of TMDs on a coupled system with closely-spaced natural frequencies. Through the assumption of very close frequencies, some analytical studies were given to the strongly coupled system. Studies on multi-mode wind-induced vibration controls are limited to the cases with very weak coupling effects (Chang et al. 2003) and few works have focused on the vibration controls of bridges with strong aeroelastic modal coupling.
Considering the complexity of bridge conditions under strong winds, an adjustable TMD system is desirable for the control system to be more robust and effective over various circumstances. However, the effects of system properties on the optimal variables of the TMDs have not been sufficiently addressed. Such study is extremely helpful in evaluating the control performance before the real control devices are designed in practice. It also helps in deciding, for the adaptive control system, what parameters of the bridge-flow system are to be monitored in a feed-back control. With such information, the number of variables to be monitored can accordingly be reduced to the least, through which the cost and complexity of the controller can also be minimized.