«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
By numerically varying the torsion/bending modal frequencies ratio of the bridge, the effects of the bridge frequency ratio on the TMD optimal variables are studied below considering the wind speed of 30 m/s. It should be noted that for Case 1, the ratios between the TMD and the bridge modal frequencies are fixed as shown in Eqs. (7.34) and (7.35), i.e., fixed to the optimal frequency for the single-mode-based vibration control. Therefore, the numerical values of the TMD frequencies ω1 and ω2 vary with the change of the natural frequencies ωh and ωα accordingly.
Fig. 7.11 shows that the optimal mass of each TMD depends on the frequency ratio between the torsion and bending modes ωα/ωh. When the value of ωα/ωh is around 1.15, 2m1/mtotal is approximately equal to m2/mtotal. When ωα/ωh is around 1.76, about 95% of the total mass should be allocated to m2 (that targets the bending vibration) for the most efficient control.
171 0.99 0.97
Fig. 7.12 Optimal frequencies of TMDs versus frequency ratio of coupled modes (Case 2) Fig. 7.12 shows the optimal frequencies of the TMDs versus the ωα/ωh ratio when wind speed is 30m/s, considering Case 2 where 2m1 = m2. It can be found from the figure that the ωα/ωh ratio affects the optimal frequency of the TMDs for coupled buffeting control. When the frequencies of bending and torsion modes are well separated (with high ωα/ωh values, say 1.6), the optimal frequencies of TMDs shown in Fig. 7.12 are quite close to those of single-modebased cases shown in Eqs. (7.34) and (7.35). This finding justifies the common assumptions that the control strategy for weakly coupled vibration can be simplified as that of single-mode-based control.
Fig. 7.13 shows that the optimal damping ratio of the TMD is also affected by the ratio of ωα/ωh for both Cases 1 and 2. With the increase of ωα/ωh, the damping ratio of the TMDs approaches to that of single-mode-based case (Fujino and Abe 1993). The control efficiency of the vertical buffeting vibration (the total vertical displacement from both the vertical vibration of the bending mode and the rotation of the torsion mode) at the edge of the mid-span section is shown in Fig. 7.14. With the increase of ωα/ωh, the control efficiency decreases and approach to that of a single-mode-based control.
The results discussed above show that the “three-row” TMD control system has the collaborative control effect for the vibration when the frequencies of coupling-prone modes are close, i.e., when the ratio of ωα/ωh is low. It has been found that when the ratio of ωα/ωh is low, 172 the difference of the optimal variables for the multi-mode-based and those for single-mode-based controls is significant. When the modal frequencies are well separated (with high ωα/ωh values), both the optimal variables and control efficiency approach to those of single-mode-based controls. In this case, the TMDs can be designed based on the single-mode control without significantly scarifying the accuracy compared to that of multi-mode based control.
7.3.5 Effect of Modal Contributions Different buffeting response contributions among modes actually indicate the energy distributions of modes and can be varied through changing the static force coefficients in the buffeting force terms, Eqs. (7.19) to (7.21). To simulate the relative contributions among modes, the static force coefficient Cm that is related to the contribution of torsion mode is increased numerically with an amplification factor β. Through changing the quantity of buffeting force for torsion mode, the relative contribution to buffeting response among modes can be adjusted.
Again, the wind speed is fixed at 30 m/s in the following discussions.
Fig. 7.15 shows the optimal mass distributions of the TMDs versus the amplification factor β. With the increase of the torsion mode contribution due to the increase of β, the optimal mass m1 that is responsible for the control of torsion mode response increases. The mass of TMD for torsion mode (2m1) is the same as that for bending mode (m2) when β is about 7. It can also be 173 found in Fig. 7.15 that when β approaches 20, essentially all the mass should be assigned to side rows in order to control the vibrations from the torsion mode. When β is less than 3, majority of the mass should be assigned to center row in order to control the vibration from the bending mode at the given wind velocity of 30 m/s.
40 36 32 28
Fig. 7.16 shows the dependency of the optimal frequencies ratio ω1 / ωα and ω2 / ωh on the factor β. With the increase of the β, ω2/ωh increases from about 0.982 up to about 0.986 and keeps stable, while ω1/ωα decreases from about 0.969 to 0.966. These variations are essentially insignificant.
Fig. 7.17 shows the optimal damping ratio of the TMDs versus β. Since the damping ratio of the TMDs has relatively insignificant effect on the control efficiency of the whole control system, such a variation range of the optimal damping ratio is not that critical. Fig. 7.18 shows that with the optimal variables of TMDs, the control efficiency increases quickly with the increase of buffeting contribution of the torsion mode, i.e., the increase of β. This implies, for this particular case, that the buffeting response induced by the torsion mode is easier to control compared with that induced by bending mode.
174 1.0 1.0
0.2 0.2 0.0 0.0
Fig. 7.15 Optimal mass distribution of TMDs versus amplification factor β (Case 1) 0.969 0.986
0.967 0.983 0.982 0.966
Fig. 7.16 Optimal frequencies ratio of TMDs versus amplification factor β (Case 2) 175 0.07
Fig. 7.18 Optimal control efficiency of “three-row” TMDs versus amplification factor β 176 In summary, different modal contributions of the buffeting vibrations has less significant effect on the optimal frequency and damping ratio if the mass of each TMD is fixed. On the other hand, the optimal mass for each row varies with the change of the contribution factor β, meaning that mass allocation among rows depends on the contribution factor β to some extent.
Different bridges have different contributions of buffeting response from each individual mode.
Since other variables depend less significantly on the contribution factor β, perhaps only the mass allocation should be made specifically for each bridge. Such feature is helpful to build a general control scheme with robust control performance for different bridges.
7.4 Concluding Remarks
The mathematical formulation of the bridge-TMD system is developed and a “three-row” TMD strategy is discussed. In this strategy, conceptually, the center TMDs are mainly to control the vibration from bending modes and the side TMDs are for the vibration from torsion modes.
However, in high wind speed when strong modal coupling effect exists, the side TMDs will also suppress the coupling part of the bending mode response. The optimal variables of the TMDs are predicted based on multi-mode coupled vibrations, instead of single-mode-based mode-by-mode
analysis. The following conclusions can be drawn through the present study:
1. Wind speed has significant effect on the optimal variables of TMDs, especially when wind speed is high. To efficiently control buffeting vibration over a wide range of wind speeds, an adaptive semi-active TMD control system that can adjust the optimal variables is necessary.
2. The modal frequency ratio between the torsion and bending modes has large effect on the optimal frequencies of the TMDs as well as the mass distribution when the total mass of the TMDs is fixed. When the frequencies of the coupling-prone modes are close, the optimal variables of the TMDs based on multi-mode coupled vibration control are significantly different from those of single-mode-based control. In these cases, a specific design for coupled vibration control should be considered. When the frequencies of the coupling-prone modes are well separated (weakly coupled vibrations), the optimal variables of the TMDs are close to those of single-mode-based control. In this case, a control strategy based on the single-mode vibration can be used in practice.
3. The change of buffeting response contribution from the torsion and bending modes has relatively less significant effect on the optimal frequency and damping ratio of the TMDs, while it has significant effect on the mass distribution among the “three-row” TMDs.
4. The present finding verifies the common assumption that single-mode-based control strategy can be used for bridges with well-separated modal frequencies. However, for couplingprone bridges with low frequency ratio, the control strategy should be based on the analysis of coupled vibrations. Many modern long-span bridges may fall in this category.
CHAPTER 8. WIND VIBRATION MITIGATION OF LONG-SPAN BRIDGES INHURRICANES
8.1 Introduction Despite the massive population growth in the south and southeast along the hurricane coast of the United States, the transportation infrastructure has not increased its capacity accordingly. Longspan bridges are usually the backbones of transportation lines along the coastal areas. When hurricane is approaching, these long-span bridges sometimes have to be closed in order to ensure the safety of the bridge as well as the transportation on them due to excessive wind-induced vibrations, which however greatly reduces the capability of hurricane evacuation through the bridges.
To date, bridge vibration controls in high wind speeds have not been adequately addressed.
Most previous control work dealt with the bridge buffeting under moderate wind speeds (Gu et al.
2001), along with some cases of flutter controls in high wind speeds (Phongkumsing et al. 2001).
While active control devices may provide satisfactory multi-objective control performance in a full range of wind speeds (Gu et al. 2002), their dependence on external energy supply has hindered their applications to the disaster evacuations. Recently, some aerodynamic controls using flaps were proposed to control flutter instability (Wilde et al. 1999). However, their applicability to buffeting control has not been reported and established.
Spring-Damper–Subsystems (SDSs) is a mechanical model, which includes the spring, damper and mass blocks. Tuned Mass Damper is a typical example of SDSs and many other passive controllers, such as Tuned Liquid Dampers, Tuned Liquid Column Dampers, can also be simply modeled as equivalent SDS. Even vehicles on bridges can also be very roughly treated as a sort of Spring-Damper–Subsystems (SDSs) to the bridge (Guo and Xu 2001; Park et al. 2001; Zaman et al.
1996). The SDS is used here as a general terminology to differentiate with Tuned Mass Dampers (TMDs). The objective of the present study is to investigate the effects of different SDSs (with different vibration frequencies) on the bridge performance during hurricane evacuations and develop a truck-type of movable passive SDS. The passive nature makes the control approach more reliable than the active one, considering the reality that power may not be available during the hurricane disasters.
The temporary/movable SDS can be conveniently driven on the existing bridges when necessary, and be removed when it is not needed.
It has been reported that the gust wind speed during hurricanes could be up to 60-80 m/s or more in the United States and other areas (Schroeder et al. 1998; Sharma and Richards 1999).
Though the duration may be short, the consequence of such strong winds may be catastrophic for both the safety of the bridge and the safety of traffic on the bridge.
Flutter instability problem is the most critical wind-related issue for long-span bridges. It has been known that hurricane-induced strong winds have much higher turbulence intensity than that of moderate winds. The effects of turbulence on the flutter stability are still controversial (Cai et al.
1999 a, b) and nonlinear aerodynamic analysis by Chen et al. (2000) confirmed that turbulence might destabilize the bridge. Before fully understanding the turbulence effect, raising the flutter instability limit of the bridge to a conservative level, as proposed in the present study, seems to be an appropriate way to maintain the safety of the bridge.
178 For service performance, the main threat to the bridge and vehicles is the excessive acceleration response in the vertical and lateral directions when the bridge is subjected to strong winds. Lateral large acceleration may cause the overturning or loss of control of the vehicles (Baker 1994; Sigbjornsson and Snabjornsson 1998). The excessive acceleration in vertical direction may cause the discomfort problems of drivers and passengers (Irwin 1991). Therefore, effectively reducing the acceleration response of the bridge may maximize the transportation capacity and possibly save lives and properties in hurricane-prone areas.
Irwin (1991) suggested the following control guidelines:
When wind speed U ≤ 13 m/s, peak vertical acceleration should be no more than
0.05g (g = gravity acceleration) When wind speed U 13 m/s, peak vertical acceleration should be no more than 0.1g
8.2 Equations of Motion of Bridge-SDS System For the model shown in Fig. 8.1-8.2, assuming that a bridge has a displacement of r(x, t) consisting of h(x) in vertical direction, p(x) in lateral direction, and α(x) in torsion direction, and wind forces consisting of buffeting force fb(x, t) and the aeroelastic self-excited force f s ( x, r, r).
Assuming also that a total number of n1 modes are included in the analysis and a total number of n2 SDSs are attached to the bridge at location xs (s =1 to n2). For each mode, vibrations in three directions h, p and α are considered. The equations of motion for the bridge-SDS system can be derived as Mη′′ + Cη′ + Sη = G (8.1)