«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
In this chapter, a comprehensive investigation on the optimal variables of the adjustable TMD system is made. First, a general formulation of the multi-mode buffeting response control with multiple TMDs is developed. Second, a control strategy with “three-row” TMDs is 156 discussed especially to study the coupled vibration controls. Finally, the three most important factors of the bridge-flow system are studied numerically with the Humen Suspension Bridge built in China. This parametric study is conducted to investigate the factors of the bridge-flow system that will affect the optimal variables of TMDs as well as the control efficiency. These analytical results will be very useful in carrying out further studies of adaptive control strategy based on the “three-row” TMD model in order to “smartly” suppress the wind-induced vibrations.
7.2 Formulations of Multi-mode Coupled Vibration Control with TMDs
Consider a general case shown in Fig. 7.1. A bridge has multiple TMDs, displacement r(x, t), and wind forces consisting of buffeting force fb(x, t) and aeroelastic self-excited force f s (x, r, r). Assuming that a total number of n1 modes are included in the analysis and a total number of n2 TMDs are attached to the bridge deck at the location of xs (s = 1 to n2), the equation
of motion for the bridge-TMD system can be derived as:
ξ = generalized coordinate of the bridge; γ = coordinate of TMDs; a superscript prime “`” represents a derivative with respect to time t; U = mean velocity of the oncoming wind; B = bridge width; n1 = number of modes; n2 = number of TMDs; I = unit matrix; and Q b =
generalized buffeting force. The components of the matrices are:
where δ ij = Kronecker delta function that is equal to 1 if i = j and equal to 0 if i ≠ j ; ωi and ζ i = circular natural frequency and mechanical damping ratio of ith mode, respectively; ρ = air density; H*, Pi*, A* ( i = 1 − 6) = experimentally determined flutter derivatives; ζ st and ωst = i i damping ratio and circular natural frequency of the sth TMD, respectively; Ist = ms (the mass of the sth TMD) and ds = horizontal distance between the sth TMD and the torsion center of the cross-section (see Fig. 7.1).
where m(x) = mass per length of the deck for vertical and lateral bending modes; and I(x) = mass moment of inertia per length of the deck for torsion mode; l = bridge length.
The generalized inertia ratio between the sth TMD and the ith mode, µis, is defined as:
where η and G = Fourier transformation of η and G, respectively. The impedance matrix F has the general form as Fij = −ω2 M ij + i ⋅ ωCij ( ω) + Sij ( ω), where subscripts i and j = 1 to (n1+n2) and i = −1.
The mean square of displacements in vertical, lateral and torsion directions can be written
The power spectra of the wind velocity components in the horizontal and vertical
directions u and w in boundary layer can be expressed as (Simiu and Scanlan 1986):
where Suu and Sww = wind velocity spectrum in the horizontal and vertical direction, respectively; and Suw = cross spectrum, respectively.
Control efficiency of displacement at the location of x on the bridge span and in the direction of r is defined as
where σ r ( x ) and σ r ( x ) are the root-mean-square (RMS) of displacement after and before ˆ control at the location of x and in the direction of r, respectively. r=h, p or α representing vertical, lateral and torsion direction, respectively.
7.3 Parametrical Studies on “Three-row” TMD Control 7.3.1 “Three-row” TMD model According to the previous studies on modal coupling, there are only a limited number of modes prone to couple together (Katsuchi et al. 1998). Among all of the coupling cases for streamlined cross sections, the most common modal coupling is between vertical bending mode and torsion mode. Furthermore, in terms of the contribution of individual mode to the total buffeting response as well as the flutter occurrence, the vertical bending and torsion modes 161 usually play the major role. Hence, an appropriate control strategy of TMD system will be developed based on such observed characteristics. It is known that TMDs placed on the center line of the cross section normally have insignificant control effect on the torsion modes.
Therefore, we adopt three rows of TMDs: one along the center line of the cross section (named center row hereafter), and other two identical rows along the two sides of the cross section (named side rows hereafter) as shown in Fig. 7.2. This model is to literally separate the TMD control role into vertical bending and torsion modes since they are the main concern of windinduced vibrations. In other words, the center row TMD is mainly for vertical mode and the two side rows mainly for torsion mode. Such separation of TMD role is very rough and actually only accurate for the situation when modal coupling effect is weak under low wind speed. As will be found later, side rows of TMD will also contribute to the dynamic suppression vertical mode in the high wind speed when strong modal coupling exists.
The difference between the multiple-TMD and single-TMD placements is that the multiple placements are more robust in control since they cover a wider range of frequencies (Kareem and Kline 1995, Abe and Fujino 1994). Since the present study is only to disclose the nature of optimal variables for coupled vibration controls, only one TMD in each row is considered to reduce the complexity while without losing the generalities. Also, since the damping ratio of the TMD is not a very sensitive variable for TMD (Gu et al. 1994), damping ratios of all the TMDs are assumed to be the same as ζ t. The frequency and mass of the two identical TMDs on the side rows are assumed to be ω1 and m1, while the frequency and mass of the center TMD are assumed to be ω 2 and m2 (Figs. 7.2-7.3), respectively. The total generalized mass of TMDs, greatly related to the efficiency and the cost of the control system, is assumed to be 1% of that of the 1st bending mode of the bridge.
In the present study, two cases are considered in the analysis of the optimal variables of the TMDs. In Case 1, the TMD frequencies ω1 and ω 2 under a particular wind speed are set to be the optimal values based on single-torsion and single-bending mode vibration controls, respectively.
These optimal values were analytically derived by Fujino and Abe (1993). Under the condition of a given total mass of TMDs (1% of the 1st bending mode), the distribution of mass between m1 and m2 is varied and studied. This is to simulate the case when the TMD mass can be adjusted while the control objective of each TMD targets a particular mode (e.g. center row for the bending mode and side rows for the torsion mode). In Case 2, the total mass of the two side row TMDs (2m1) is set to be equal to the center one (m2), maintaining the total TMD mass the same as Case 1. Only the variables ω1, ω 2 and ζ t can be adjusted. This is to simulate a case that the mass of each TMD is fixed, while the frequency and damping ratio can be adjusted to obtain the optimal control performance.
7.3.2 Optimal Variables of “Three-row” TMDs
162 modal properties including modal damping and modal frequencies can be obtained using complex eigenvalue approach (Chen and Cai 2003). The results of modal properties are plotted in Fig. 7.5 and the flutter critical wind speed is identified as 87 m/s for Humen Bridge, which is very close to the result from wind tunnel test (Lin and Xiang 1995). Fig. 7.6 gives the buffeting response in vertical and torsional directions in the center of the mid-span section without control.
Results from coupled analysis based on the two modes (vertical bending and torsion) and from the single- mode analysis are compared. Strong coupling effects in high wind speed can be observed that the results of coupling analysis differ obviously from that of single-mode analysis when wind speed is high.
In following control studies, the chosen location of interests is at the edge of the mid-span section where the largest vertical displacements contributed by both the symmetric bending and torsion modes are expected. The vertical displacement in that location can combine the 163 contributions from the vertical bending mode as well as the torsion mode. The contribution to the vertical displacement at the edge of the mid-span section by the torsion mode is calculated with the torsion displacement multiple the half of bridge width. The TMDs are placed as shown in Fig. 7.3. The total generalized mass of TMDs, chosen as 1% of that corresponding to the 1st bending mode, is 80,000 kg. For Case 2, the center row and two side rows have the same mass, i.e., 2m1 = m2 = 40,000 kg. The horizontal distance from the side row TMD to the center of torsion ds equals to 14 m for Humen Bridge.
If only a single-mode-based vibration is considered, the optimal frequency of TMDs at
wind velocity 30 m/s can be obtained with formulas by Fujino and Abe (1993) as:
where, ωh and ωα = natural circular frequencies of vertical and torsion modes, respectively. As discussed earlier, these single-mode-based optimal frequencies are chosen for the TMDs in Case 1.
There exist many factors affecting the coupling effects among modes. From the existing knowledge of the modal coupling, the frequency ratio between the coupling-prone modes and the wind speed are the main possible factors that may affect the optimal variables of TMDs (Katsuchi et al. 1998). The wind speed greatly affects the aeroelastic modal coupling effects and affects the buffeting contribution from coupling-prone modes. These factors are discussed below.
7.3.3 Effect of Wind Speed
Aeroelastic coupling is of a great concern in wind-induced vibration of long-span bridges especially when wind speed is quite high. For most streamlined cross sections, the aeroelastic coupling is directly related to the wind speeds. When the total mass of TMDs is fixed as in Case 1, the predicted optimal mass distributions among the TMDs vary significantly for different wind speeds as shown in Fig. 7.7. At the wind speed of about 60 m/s, the masses of center and side TMDs are about equal (2m1/mtotal ≅ m2/mtotal ≅ 0.5). At low wind speed (say 40 m/s), much more mass needs to be assigned to m2 (m2/mtotal ≅ 0.85) with the bending-single-mode-based optimal frequency. However, at higher wind speed (say 80 m/s), much more mass should be assigned to m1 with the torsion-single-mode-based (2m1/mtotal ≅ 0.80). There are probably two reasons for such phenomena: one is in high wind speed, the contribution to the total vertical response at the edge of the mid-span section from the torsion mode increases; the other is that the resonant part of bending mode response with large modal damping (Fig. 7.6) is difficult to be suppressed.
Another part of bending response in high wind speed is due to the modal coupling effects between bending and torsion modes (Chen and Cai 2003), and it can be suppressed with TMD with torsion mode frequency. As stated earlier, the frequencies of TMD are set to be equal to the optimal frequency considering single-mode-based vibrations in Case 1. In terms of vertical response control at the edge of the mid-span section, it is noted that the dominant vibration mode changes from vertical to torsion mode at the wind speed of about 60 m/s.
164 For Case 2, since the mass of each TMD is fixed, namely 2m1 = m2, the optimal frequency changes with the wind speed especially when the wind speed is high as shown in Fig. 7.8, where ωh and ωα are the modal frequencies of bending and torsion modes considered in this study, namely, 0.17 and 0.36 Hz, respectively. Fig. 7.8 also indicates that with the increase of the wind speed, the optimal frequency of TMD in the center row may change dramatically from around the modal frequency of the bending mode (indicated by ω2/ωh =1.0) to around the torsion mode frequency (indicated by ω2/ωh = 2.0 since ωα is about twice of the bending frequency ωh). This drastic change occurs at the wind speed of 60 m/s where dominant vibration mode for the vertical response control at the edge of the mid-span section changes from bending mode to torsion mode.
Fig. 7.9 shows the respective optimal damping ratio of TMDs for Case 1 and Case 2.
Compared to other variables such as mass distribution and frequency, the damping ratio of the TMDs seems to be less sensitive to the change of wind speed. Therefore, it has relatively the least necessity to be adjusted in an adaptive control.
With the change of wind speed, the optimal control efficiency of the vertical displacement on the edge at the mid-span section. Rh also varies accordingly as shown in Fig. 7.10. It is observed that the control efficiency decreases with the increase of wind speed until around 60 m/s, where the dominant vibration control mode changes from bending mode to torsion mode.
Then, the control efficiency increases with the increase of wind speed. Such phenomenon cannot be observed in a single-mode-based control analysis. Hence, this observation is extremely important for the design of a special controller that, for example, controls bridge vibrations under hurricane-induced strong winds. An ideal TMD system in this case should target the bending vibration when wind speed is less than 60 m/s and then is adjusted to target the torsion mode vibration.
168 1.0 1.0
1.4 0.96 1.2 0.95 1.0
Fig. 7.10 Optimal control efficiency of “three-row” TMDs versus wind speed 170 1.0 1.0 0.8 0.8
7.3.4 Effect of Frequency Ratio With an optimal searching of the TMD mass under the condition of a fixed total mass, the optimal distribution of the total mass among the center and side TMDs has been obtained above.