«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
Fig. 5.12 shows the lateral and yawing displacements and the corresponding steering angle for vehicles with 50 m/s (112 mph) driving speed when wind speed is 5 m/s (11 mph). Lateral and yawing responses are all lower than the accident criteria even the driving speed is quite high. But the steering angle should be changed with a frequency about 0.8 Hz (it takes 500m/(50m/s) = 10s to finish the roughly 8 steering cycles), higher than the case when wind speed and driving speed all equals to 15 m/s as shown in Fig. 5.11 and lower than the case with high wind speed (35m/s) as shown in Fig. 5.10. The phenomenon suggests that in low wind speed, the vehicle can theoretically be driven in a quite high speed. However, more attention on appropriately adjusting the steering angle is necessary to keep the high driving speed safe. Since long time of adjusting the steering angle in a high frequency will quickly accumulate the driver fatigue and then jeopardize the driving safety, high driving speed is practically not safe for drivers. The reaction force ratios as shown in Fig. 5.13 suggest that the two front wheels lose some reaction forces and the vehicle body is not in the same equilibrium condition as the static situation even though the wind speed is quite low.
5.4.4 Accident Driving Speed In the transportation practice, lowering the driving speed limit or close the bridge or highway is one common option to ensure the vehicle safety under strong winds. A suitable speed limit is of utmost importance to the drivers and the traffic administrators. Accident-related responses of the truck in several typical situations are studied in the previous section for demonstration. For vehicles running on the bridge with wind, it is desirable to know the highest allowable driving speed under any particular wind speed to avoid risks of accidents. Such critical driving speed is called “accident driving speed” in the present study. The three types of typical accidents (overturning accident, rotational accident, and side slipping accident) may happen
0.20 0.16 0.12 0.08 0.04 0.00 0 100 200 300 400 500
-0.06 0 100 200 300 400 500
1.0 0.5 0.0
-0.08 0 100 200 300 400 500
Figure 5.13 Wheel reaction forces of the truck when U=5 and V=50 m/s (λ1=0.
2, λ2=0.3) In order to predict the accident driving speed for different wind speeds for the truck on the prototype bridge, the accident driving speeds are searched under different wind speeds. Under each wind speed, the driving speed is increased in a step of 1.0 m/s. In each step, the accidentrelated response and reaction forces are predicted to check if any of the three accidents may occur during the driving process on the bridge. Keep increasing the driving speed to next step if no accident happens during that period. The lowest driving speed under which at least one accident criterion is not satisfied even after different variables (λ1 and λ2 here) of the steering angle model are tried can be identified as the “accident driving speed”. Repeating such process under different wind speeds, a curve for accident driving speeds versus wind speeds can be drawn from the results as shown in Fig. 5.14.
As shown in Fig. 5.18, the accident driving speed generally decreases with the increase of wind speed. When the wind speed increases from 5 m/s up to about 20 m/s, the accident driving 128 speed decreases gradually from 60 m/s to about 30 m/s. When the wind speed keeps increasing, the accident driving speed drops to zero. This phenomenon suggests that when wind is not so strong, lowering the driving speed can maintain the safety of the vehicles on the bridge.
However, when wind speed reaches the upper limit, solely lowering driving speed cannot avoid the accident occurrence. In other words, a still truck will also be blown off under some high wind speed, which agrees with the common sense. It is also found in this example that overturning accidents are most likely to happen when wind speed is over 20 m/s, while side slipping accidents are most likely when wind speed is lower than 20 m/s.
For comparison purpose, accident analysis is also conducted for the same truck on the road, and the results are also plotted in Fig. 5.18. It shows that vehicles on the road have higher accident driving speed, and the upper limit wind speed under which the truck cannot keep safe is the same. At this maximum wind speed (about 35 m/s), the driving speeds are approaching zero no matter on the road or on the bridge, which means the truck can not safely move on the bridge or on the road.
45 30 15 0
Figure 5.14 Accident driving speed versus wind speed
5.5 Concluding Remarks In the present study, an assessment model for vehicle accidents on bridges and on roads under wind action is introduced. The proposed model starts with a full interaction analysis between the bridge and the vehicle, which predicts, in addition to the bridge vibration, the vehicle response in the directions of vertical, rolling and rotation under the wind action and road roughness. Such vehicle and bridge vibration information is carried over to the following 129 accident analysis of the vehicle only. With given accident criteria, the accident driving speed can then be predicted under any wind speed.
The following conclusions can be made after the numerical example with a truck model moving on the prototype bridge. It is noted that some of these conclusions are “common sense” qualitatively, but are quantitatively verified through numerical simulation, which provides a base to make scientific decisions for traffic management in windy environments.
1. The proposed accident analysis model can be used to predict the accident-related response. With suggested accident criteria and driving behavior model, the accident risks can be assessed.
2. Lowering driving speed is effective to lower the accident risk only if the wind speed is not extremely high. Setting suitable driving speed limit is important to decrease the likeliness of accident occurrence.
3. When wind speed reaches high to some extent, the vehicle should not be on the bridge no matter what driving speed it has. Rational critical wind speed limit should be set to decide when to close the bridge. In the present study, 32 m/s (71 mph) is the critical wind speed limit based on numerical simulation. Actual limits can be set by considering also other factors.
4. Vehicles on the bridge are more vulnerable to accidents than on the road. Usually lower driving speed limits for vehicles on the bridge than on the road should be rationally defined to avoid the accident when strong wind speed exists.
5. Overturning is most likely to happen on the bridge for high-sided vehicles, like trucks and tractor-trailer. Windward rear wheel is mostly likely to initiate the accident.
6. The present study is to build up the framework for the accident analysis and more insightful studies on the driver behavior model and accident criterion are necessary.
6.1 Introduction Long-span bridges undergoing wind excitation exhibit complex dynamic behaviors.
Buffeting vibrations induced by wind turbulence happen throughout the full range of wind speed. As the wind speed increases, aerodynamic instabilities such as flutter may occur at high wind speed (Simiu and Scanlan 1996). Much research effort has been made in mitigating excessive buffeting vibrations and improving aerodynamic stabilities for long-span bridges during construction (Conti et al. 1996; Takeda et al. 1998) and at service (Pourzeynali and Datta 2002; Omenzetter et al. 2002; Miyata and Yamada 1999). Among all of the control procedures, dynamic energy absorbers such as tuned mass dampers (TMDs) have been studied in suppressing the excessive dynamic buffeting (Gu et al. 2001) or enhancing the flutter stability of bridges (Gu et al. 1998; Pourzeynali and Datta 2002). As traditional control devices, the dynamic energy absorbers dissipate external energy through providing supplemental damping to the modes of concern (Abe and Igusa 1995; Kareem and Kline 1995).
Jain et al. (1998) analyzed the effects of modal damping on bridge performance of aeroelasticity. It was found that supplemental damping provided through appropriate external dampers could certainly increase the flutter stability and reduce the buffeting response of long-span bridges. In a conventional TMD control design, the TMD frequency is designed or tuned to the modal frequency of the fundamental mode (Fujino and Abe 1993) in order to reduce the so-called resonant vibration and this method is thus called resonant-suppression approach here.
When the modal coupling among the modes is weak, the bridge can be regarded as a simple combination of many single Degree-Of-Freedom (DOF) systems and single mode analysis is usually applicable (Lin and Yang 1983). In such a case, an equivalent damping ratio ζe can be adopted to assess the control efficiency and performance for each individual mode. When white noise excitation was assumed, the root- mean-square (RMS) ratio between
the controlled and uncontrolled vibrations was derived as (Fujino and Abe 1993):
where S0 = spectral density of white noise excitation; Ms = generalized modal mass;
and ωs, ζs and ζe = modal circular frequency, modal damping ratio, and equivalent damping ratio of TMD, respectively.
It is well-known that wind-induced aeroelastic effects result in additional aerodynamic damping and stiffness for long-span bridges (Tanaka et al. 1993). The additional aerodynamic damping and stiffness may vary with wind speed and be different for various bridges.
Typically, the aerodynamic damping increases with the increase of wind speed for bending modes but decreases for torsion modes. Consequently, the modal damping ratio ζs for the 131 mode of concern, which consists of mechanical and aerodynamic damping, could increase and be very high under strong wind. As indicated in Eq. (6.1), the control efficiency decreases with the increase of modal damping ratio ζs. It also implies that, for coupled mode vibrations of long-span bridges, the control efficiency of buffeting response of bending mode decreases with the increase of wind velocity since the aerodynamic damping of bending modes usually increases with the wind speed as mentioned above. Since bending modes usually contribute significantly to the overall buffeting response among all of the modes, the decreased control efficiency in bending modes may deteriorate the overall control efficiency of the bridge vibration.
The adoption of slender deck and the increase of bridge span lengths tend to make the frequencies of modes closer, which increases modal coupling effects through aeroelastic effects in high wind velocity (Bucher and Lin 1988; Jain et al. 1996; Katsuchi et al. 1998;
Thorbek and Hansen 1998; Cai and Albrecht 2000). Modal coupling effects due to strong wind may result in a significant additional component to the buffeting response of each individual mode, compared with the cases of weak modal coupling. Accordingly, a more efficient control approach than the traditional resonant-suppression method may exist for the coupled buffeting control of bridges in strong wind.
The present study aims at introducing an alternative TMD design approach, which is based on suppression of modal coupling effect among modes under strong wind.
Conventional resonant suppression TMD design idea has difficulty to achieve satisfactory control effects because strong modal coupling under strong wind causes high damping ratio of some concerned modes. Different from conventional resonant suppression through supplying additional damping, the proposed design approach is to attain control efficiency under strong wind through suppression modal coupling effects among several coupling-prone modes. With the proposed control approach, a well-designed TMD system can efficiently suppress wind-induced vibrations for the strongly-coupled modes even in high wind speed.
Poorer control performance may otherwise be anticipated for TMDs designed based on the conventional resonant-suppression approach.
To better develop and explain the new control approach, approximated closed-form solutions of coupled buffeting response were first derived for a multi-mode coupled bridge system attached with arbitrary number of TMDs. This derivation clearly shows the contributions of all components of the response and indicates how TMDs can be designed to control each part. Examples of a two Degree-Of-Freedom (2DOF) model and a long-span prototype bridge were then used to further demonstrate and validate the efficiency of TMDs on the coupled response control. Finally, the applicability of dual-objective control with passive TMD system designed based on the new control approach was briefly discussed.
6.2 Closed-Form Solution of Bridge-TMD System
To better understand the coupled vibrations and the interaction of the bridge-TMD system, closed-form solutions are derived below. This derivation will give insights and facilitate the discussion in developing a new TMD control approach for coupled vibrations in strong wind.
132 Linda and Donald (1998) once analysed the coupling problem of sound-induced vibrations and proposed an approximated closed-form formation for acoustics. By using a similar procedure, a closed-form formulation is derived below for a coupled wind-induced vibration of a bridge-TMD system. To keep the integrity of the derivation, some well-known formulas are necessarily revisited below.
For a bridge under wind action with a displacement of r(x, t), the buffeting and aeroelastic forces are expressed as functions of the displacement r(x, t) and location ordinate x as fb(x, t) and f s ( x, r, r), respectively. Assume that a total number of n2 TMDs are attached to the bridge at the location of xp (p = 1 to n2), then the equation of motion is derived as
where L [⋅] and D [⋅] = elastic and viscous damping operators; ρ(x) = mass density; δ(⋅) = Dirac delta function; and fTMDp = reaction force from the pth TMD on the bridge.