«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
While coupled multimode analysis is more accurate in predicting the wind-induced vibration than the single-mode analysis, it is still advantageous and desirable to develop a more convenient method that can balance the simplicity of single-mode analysis and the accuracy of coupled multimode analysis. This method can be used for practical applications or in cases when a complicated coupled multimode analysis is not desirable, such as in a preliminary analysis.
In the present study, firstly, a closed-form formulation of coupled multimode response is derived where only the primary modal coupling effects are considered while the trivial secondary ones are ignored. As will be seen later, this approximate method gives much more accurate results than the traditional single-mode analysis and agrees well with the coupled multimode analysis results. Secondly, the tendency of modal coupling effect is quantitatively assessed by using a modal coupling factor (MCF). Though the derivation of MCF is based on buffeting analysis, it can also disclose the nature of modal coupling mechanism that will govern the flutter behavior of long-span bridges. Lastly, another important application of the MCF is in the design of vibration control strategies (Gu et al. 2002a). An optimal control design may call for suppressing the response induced by modal coupling effect other than traditional resonant component in high wind speed. To develop such control strategy, it is essential to know the coupling characteristics of modes in advance.
2.2 Mathematical Formulations 2.2.1 Coupled Multimode Buffeting Analysis For the convenience of discussion, the multimode analysis procedure (Jain et al. 1996) is briefly reviewed below. Deflection components of a bridge can be represented in terms of the generalized coordinate ξ i ( t ) as
where ξ = vector of generalized coordinates; the superscript prime represents a derivative with respect to dimensionless time parameter t = ( Ut / b ) ; I = identity matrix; Q b = vector of normalized buffeting force (Jain et al. 1996); and the general terms of matrices A and B are
Kronecker delta function (=1 if i = j or = 0 if i ≠ j ); ωi, ζ i and Ii = circular natural frequency, damping ratio and generalized modal mass of i-th mode, respectively; ρ = air density; U = mean velocity of the oncoming wind; K( = bω / U = 2πf b / U ) = reduced
where E * = complex conjugate transpose of E ; and S Q bQ b is the spectrum of buffeting force (Jain et al. 1996).
The mean-square values of physical displacements can be obtained as
2.2.2 Simplification of Coupled Multimode Buffeting Analysis Generally, the equations of motion, Eq. (2.3), are coupled and can only be solved simultaneously. By ignoring modal coupling effects among modes, i.e., ignoring the offdiagonal elements in matrices A and B of Eq. (2.3), Eq. (2.12) reduces to
where (and hereafter) the superscript ‘un’ stands for the single-mode result from uncoupled single-mode solution.
As the combination of the resonant response and background response, the single-mode uncoupled response can be written in the approximated closed-form as (Simiu and Scanlan 1996).
Eq. (2.13) is usually called SRSS single-mode method based on traditional mode-bymode single-mode buffeting analysis procedure. Eqs. (2.15) and (2.18) are practical formulas after fair approximations are made on Eq. (2.13). As will be seen in the numerical example, the accuracy of such results at high wind velocity is significantly scarified due to ignoring modal coupling effects. To consider modal coupling effect in the buffeting analysis, coupled multimode simultaneous equations need to be solved (Chen et al. 2000). In the following part, a new approach to decoupling those equations will be developed. The new approximate solution will not only lay a foundation to develop the MCF for modal coupling quantification but also provide a method to predict the coupled multimode response based on SRSS results without solving complicated simultaneous equations.
To continue the derivation, the i-th equation is extracted from Eq. (2.10) and is rewritten as
When off-diagonal terms in matrices A and B are dropped, Eq. (2.3) is decoupled.
Correspondingly, Dij = 0 (if i ≠ j) and the uncoupled single-mode response of mode i can be easily derived from Eq. (2.19) as
where the superscript ‘un’ in ξiun (K ) stands for the result from uncoupled single-mode solution.
Defining a non-dimensional parameter
According to the definition in Eq. (2.22), ξi (K) and ξ j (K) equal to 1 if the modal coupling effect is entirely omitted. Otherwise, there exists the condition that ε 1. When the i-th mode is under study, the resonant reduced frequency equals to Ki that is usually different from Kj. For ε satisfying the condition ε 1, following equation can be approximately
derived (Linda and Donald, 1998):
In the above process, coupling effects with the order of O(ε2) are ignored. When the response of the i-th mode is to be determined, the coupling between the i-th mode and any other mode is defined here as the primary modal coupling; the coupling effects between arbitrary j-th mode and any other mode (except for i-th mode) is defined as secondary modal coupling. Physically, the above simplification process ignores the secondary coupling effect while including the primary coupling effect. Since only the high order small terms are ignored, the accuracy of the solution should not be significantly scarified. The corresponding solution is much more accurate, as will be seen in the numerical example, than the traditional uncoupled single-mode analysis with Eqs. (2.13) and (2.18).
where the first part of Eq. (2.30) represents the mean square of primary response which is the summation of each single-mode response in r direction and the second part represents the secondary response which is the summation of cross-modal response in r direction.
The power spectral density (PSD) for the generalized i-th mode displacement ξi can be
derived from Eq. (2.27) as:
It can be seen from Eq. (2.31) that the approximated response spectrum of mode i consists of three parts. The first part is the response spectrum of mode i from the traditional uncoupled single-mode analysis, in which the modal coupling effect with other modes is completely neglected. The second part is the response contributed by other modes due to modal coupling and is written as a linear combination of the single-mode response spectra of each mode. The third part is related to the cross-modal buffeting force spectrum between mode i and other modes. Since cross-modal buffeting force spectrum has small effect on aeroelastic coupling, it is negligible (Jain et al. 1996). Similarly, cross-modal response shown in Eq. (2.33), namely the second part of Eq. (2.30), is usually also omitted since the contribution to the total response is normally insignificant (Katsuchi et al. 1999). The numerical verification on such approximations will be made in the example described later.
2.3.1 RMS Response of Coupled Analysis After omitting trivial terms as discussed above, the mean-square values of physical displacements can be obtained using Eqs. (2.30) and (2.31).
An examination of βij (K) shows that there exists a peak value in the curve of βij (K) when K = Ki. However, βij (K) decays fast when K is away from Ki. Since the frequencies of bridge modes are usually distinctly separated, βij (K) in Eqs. (2.31) and (2.34) can be approximated as βij (Kj), i.e., using the modal natural frequency to replace the oscillation frequency. This is similar to the approach used by Simiu and Scanlan (1996) and Cai et al.
In the premise of neglecting the cross-modal buffeting spectrum, namely, the third part of Eq. (2.31), the physical root-mean-square (RMS) response in r motion direction can be finally expressed from Eq. (2.34) as
Above formula can be used for engineers to conveniently, yet reliably, calculate the coupled buffeting response. It will be seen in the following example that Eq. (2.35) is much more accurate than the previous traditional mode-by-mode single-mode analysis shown in
2.3.2 Modal Coupling Factor (MCF) When the background response is omitted for simplicity to assess MCF, the MCF in Eq.
(2.36) can be simplified as a simple closed-form like
due to modal coupling effect; it is here named Modal Coupling Factor (MCF). Since MCF represents the relative significance of modal coupling, this information gives a convenient way to quantitatively assess the degree and prone of modal coupling between any pair of modes under study.
2.4 Numerical Example 2.4.1 Prototype Bridge The developed procedure was applied to the analysis of Yichang Suspension Bridge that is located in the south of China with a main span length of 960m and two side spans of 245m each. The deck elevation is 50m above the sea level, the design wind speed is 29 m/s, and the predicted critical flutter wind velocity Ucr is 73 m/s (Lin et al. 1998). The major parameters along with some other information are summarized in Table 2.1.
The characteristics of natural modes were predicted with finite element methods. To facilitate the discussion, only the seven important modes are presented here for the windinduced vibration analysis. These seven modes are numbered from 1 to 7 in sequence corresponding to one lateral bending, two vertical symmetric bending, one vertical asymmetric bending, two torsional symmetric, and one torsional asymmetric modes as given in Table 2.2. These modes are plotted in Fig. 2.1 for the main span. Flutter derivatives are the other important information for aeroelastic analysis. Eight flutter derivatives of Yichang * Bridge are shown in Figs. 2.2 and 2.3 (Lin et al. 1998). The relatively large values of A1 (representing the effect of vertical vibration on torsional vibration) and H* (representing the 2 effect of torsional vibration on vertical vibration) is an indication of possible strong coupling between vertical and torsional modes. However, this kind of observation is very preliminary and many other factors, such as modal characteristics, wind velocities, and their combinations, will affect the modal coupling. A more rational quantification method for modal coupling effect, as has been developed in the present study, is necessary.
2.4.2 Summary of Numerical Procedure The whole procedure of assessing modal coupling and predicting the coupled buffeting
response with the proposed approximate method can be described as:
Firstly, the mean square response for mode i is calculated using traditional single-mode analysis method, i.e., Eqs. (2.13) or (2.18). From these results, it can be decided preliminarily which modes are the main contributors to the physical response. These mode numbers are recorded as the selected i mode for following response analysis in each motion direction.
get ϑij using Eq. (2.36) or Eq. (2.37); ϑij is used to assess the tendency of coupling between mode j and mode i. Since ϑij is directly related to the contribution to the response of mode i due to modal coupling effect of mode j, a preset threshold value of ϑij can be used to select
major sources of flutter instability, it can also be used to decide which modes are necessary to be included into flutter analysis.
Lastly, for any motion in r direction, with selected mode i in step one and corresponding ϑij obtained in step two, approximate RMS response can be easily derived with Eq. (2.35).
2.4.3 Assessment of Modal Coupling Effect Using MCF ϑij Only the results for the mid-point of the central span are presented here since this location is usually the most important one to study the vibration of the bridge. The MCF values, ϑij, were calculated with Eq. (2.36) for different wind velocities. For the sake of brevity, only the coupling effects between the four typical modes (Modes 3 to 6) and the other modes (Modes 1 to 7) are shown in Figs. 2.4 to 2.7, respectively. In these figures, the xaxis represents the ratio between the wind speed and flutter critical wind speed (Ucr = 73 m/s). The log scale is used for the y-axis to fit all curves in the figures.
Fig. 2.4 shows the coupling effects between Mode 3 (the 1st symmetric vertical mode) and the other modes, denoted as ϑ3 j (j = 1 to 7). It was found that the values of ϑ3 j are all very small when the wind velocity is lower than 10 m/s, but goes higher when the wind velocity increases. The MCF between Modes 3 and 5, ϑ35, is significantly larger than other ϑ3 j. When wind velocity is close to 70 m/s, ϑ35 is about three orders larger than the second largest value ϑ34. The results conclude that Mode 5 (the 1st symmetric torsion mode) contributes most to the buffeting response of the coupling part of Mode 3 (Again, each mode’s vibration consists of one part from single-mode vibration and another part from mode coupling, as discussed earlier). Other modes contribute insignificantly (10-5-10-4 level) to the response of Mode 3 and can thus be ignored in calculating the buffeting response of the coupling part of Mode 3.
24 Fig. 2.5 shows the MCF values between Mode 4 (the 2nd vertical symmetric mode) and the other modes. It can be seen that Mode 3 (the 1st vertical symmetric mode) contributes relatively large to the buffeting response of the coupling part of Mode 4. However, the value of ϑ4 j is very small for all the modes, indicating a weak coupling between Mode 4 and the other modes.