«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
So far, the nature of transition of frequencies during the flutter initiation process has not been satisfactorily explained. The writers believe that more specific numerical simulations and discussions are necessary to understand how the multi-frequency vibration always turns into a single-frequency one at flutter. The present work will help explain this phenomenon and better understand the evolution process from buffeting vibrations under strong wind to flutter occurrence. For this purpose, an examination of the frequency characteristics of bridge vibration in a full range of wind speeds is necessary.
Fig.3.1 2-D model of bridge section in incoming wind
3.3 Analytical Approach Flutter and buffeting analyses were carried out either in a time domain (Ding et al. 2000;
Boonyapinyo et al. 1999) or in a frequency domain (Scanlan and Jones 1990). In both methods, aerodynamic forces are defined (directly or indirectly) with the so-called flutter 34 derivatives that can be determined for each type of bridge deck through a specially designed wind tunnel experiment. To facilitate the presentation and discussion, a brief description of the related theoretical aspects of flutter and buffeting analyses is given below.
3.3.1 Equations of Motion For the convenience of theoretical presentation, a typical 2-dimensional section model (with a unit length of bridge deck) shown in Fig. 3.1 is used in the formulation of flutter and buffeting analyses. Two degree-of-freedom, i.e., vertical displacement h(x, t) and torsion displacement α(x, t), are considered here. The equations of motion for the section model can
be expressed as (Simiu and Scanlan 1986):
where the “dot” above the h and α represents a derivative with respect to time, m and I = generalized mass and generalized mass moment of inertia of a unit length of deck, respectively; ζα and ζh = structural damping ratios for torsion and vertical modes, respectively; ωα and ωh = natural circular frequencies for torsion and vertical modes, respectively; L ae and M ae = unit length self-excited lift force and pitch moment, respectively;
and L b and M b = unit length buffeting lift force and pitch moment, respectively. These aerodynamic forces are expressed as
35 and CM = static coefficients for lift force, drag force and pitch moment of bridge deck, respectively; and u(t) and w(t) = turbulent wind components in the lateral and vertical directions, respectively. A “prime” over the coefficients represents a derivative with respect to the wind attack angle.
The two vibration displacements can be expressed using modal superposition as:
where n = number of mode shapes; hi(x) and αi(x) = mode shapes in vertical and torsion directions, respectively; and ξi(t) and γi(t) = generalized coordinates in vertical and torsion directions, respectively.
Using Eqs. (3.2) to (3.7) and considering only the first vertical and first torsion modes for simplicity, Eq. (1) can be rewritten in a matrix form as
[ I] = unit matrix; and L = bridge length.
The equations of motion contain flutter derivatives that are functions of the reduced 36 frequency K. Therefore, even though bridge vibrations can be decoupled into modal vibrations through the modal superposition method, these flutter derivatives re-couple the equations of motion. Correspondingly, coupled modal vibrations have often been observed in wind tunnel tests, especially in high winds. In the coupled vibration, each mode vibrates about its oscillation frequency, while affected also by the other modes.
3.3.2 Complex Eigenvalue Analysis The flutter wind speed is commonly determined by eigenvalue method in the frequency domain, i.e., by iteratively searching for a pair of K and ω so that the determinant of the characteristic function of the equations of motion becomes zero (Simiu and Scanlan 1996).
For both buffeting and flutter analyses, one of the most important tasks is to quantify the aerodynamic damping and stiffness matrices, [C] and [K] in Eq. (3.8). These matrices include the contributions from aerodynamic forces, and are functions of the vibration circular frequencies corresponding to different modes. Iterative complex eigenvalue approach has been used to predict modal properties in a wide range of wind speeds, from which the occurrence of flutter can be determined and [C] and [K] can be quantified (Agar 1989;
Miyata and Yamada 1999). Once the damping [C] and stiffness [K] matrices in Eq. (3.8) are defined and wind buffeting force Q b is given, buffeting response can also be obtained in the frequency domain.
The flutter critical point can be identified by iteratively solving the complex eigenvalues in the state-space of Eq. (3.8) at each wind velocity. The Eq. (3.8) can be rewritten in the state-space as
where [ Φ] and [ λ ] diag = eigenvector and eigenvalue matrices of − A −1 B, respectively.
The Eq. (3.15) can be rewritten in a simplified form as
where λ i = ith eigenvalue, ω i and ζ i = modal oscillation frequency and damping ratio in ith mode, and j = unit imaginary number ( j = − 1 ).
Once the eigenvalues are iteratively predicted, the oscillation frequency ω and reduced frequency K are known at the given wind speed; and the damping and stiffness matrices [C] and [K] can then be calculated. The flutter wind speed and corresponding flutter frequency can be identified from the eigenvalue solutions at the condition that the total modal damping approaches zero.
3.3.3 Buffeting Analysis Buffeting is a multi-mode random vibration, where different ω values are used in the calculation of the reduced frequency K for each mode (Cai and Albrecht 2000). The ω value can be either the respective natural circular frequency of that particular mode or the aeroelastically modified natural frequency considering aeroelastic effects (Simiu and Scanlan 1986, 1996), i.e., the oscillation frequency predicted in the complex eigenvalue analysis discussed above. The choice of different ω values for each different mode is justified for buffeting analysis when the wind speed is lower than the flutter wind speed. When the wind speed approaches the flutter wind speed, the buffeting vibration will be dominated by the frequency of the critical mode. The linear buffeting theory has been well documented in the literature and is not repeated here.
3.3.4 Time-series Analysis It is noted that the aeroelastic forces in Eqs. (3.2) and (3.3) have frequency-dependent terms, K, Hi and Ai, as well as time-dependent terms, h(x, t) and α(x, t). Such a frequencydependence causes some difficulties to assemble the element matrices in finite element analysis (Cai and Albrecht 2000). In the previous time domain analysis, in order to eliminate the dependence on the frequency, indicial functions or rational functions were adopted to approximately curve fit the aerodynamic forces in Eqs. (3.2) to (3.5) into time-dependentonly functions (Bucher and Lin 1998).
In the present time-series study, it is proposed to first conduct the complex eigenvalue analysis before the time-series analysis is performed. In this way, the frequency-dependent components of flutter derivatives and K will be resolved from [C] and [K] matrices and Eq.
(3.8) will be time-dependent only. Consequently, to study the flutter process in time-series, a free vibration response can then be obtained through numerical integration of the homogeneous part of Eq. (3.8) (on the left hand side). Similarly, if buffeting forces (on the right hand side of Eq. (3.8)) are known, then buffeting response can be predicted using either time-series or spectral analysis.
3.4 Numerical Procedure For flutter in laminar flow, complex eigenvalue analysis will identify the flutter wind speed and frequency. However, we believe that simulation of the free vibration in time-series will disclose the evolution process of flutter that is in nature a self-excited vibration. The free vibration of the coupled system can be solved through a time integration of the equations of motion. The actual frequency characteristics of the bridge vibration can then be obtained 38 through a spectral analysis of the derived time-series response. The buffeting and flutter analyses at the same wind speed will then be compared to investigate their correlations.
This proposed numerical procedure consists of three steps - complex eigenvalue analysis, time-series analysis and spectral analysis. It is noted that neither complex eigenvalue analysis, time-series analysis, nor spectral analysis is new by itself, but their combined use in the present study can clearly demonstrate the flutter evolution process, which helps further understand the transition mechanism from the multi-frequency vibration to single-frequency flutter.
STEP 1 – COMPLEX EIGENVALUE SOLUTIONThe first step is to use the complex eigenvalue solution to predict the respective vibration damping ratio and oscillation frequency of each mode under any desired wind speed. These damping ratios and oscillation frequencies include the effect of aerodynamic forces. Though associated with vibration modes, these modal values are different from the traditional natural modal values in which no aerodynamic forces are involved. Therefore, these modal values that include aerodynamic effects are actually (and hereafter called) the effective modal values. These effective modal values are needed to quantify the aerodynamic matrices [C] and [K] in Eq. (3.8). Otherwise, matrices [C] and [K] include unknown variables, i.e., the effective oscillation frequency and damping ratio.
As discussed earlier, the effective oscillation frequency of each mode under each desired wind speed will be obtained with the complex eigenvalue solution of Eq. (3.14). This process is shown in Fig. 3.2 and briefly explained here. The whole process starts at zero wind speed, and the natural frequency and mechanical damping ratio of each mode are chosen as the initial values, ω0 and ζ 0. For the ith step, Ui = Ui-1 + ∆U and Ωi = ωi −1 are assumed.
With Ki = (B Ωi /Ui), the corresponding flutter derivatives can be decided from the experimentally measured flutter derivatives versus K curves. Using the complex eigenvalue analysis, modal oscillation frequency ωi and damping ratio ζi for each mode are predicted with Eqs. (3.16) and (3.17). The “bars” above ωi and ζ i denote that they are temporary values during the solution process. After a convergence is achieved by comparing ωi and Ωi, the effective modal properties at wind speed Ui can be obtained as ωi = ωi and ζ i = ζi. If the effective modal damping ratio ζ i is greater than zero, the solution process proceeds to the (i+1)th step. Otherwise, the flutter critical point is identified due to the non-positive ζ i and the flutter critical wind speed is predicted as Ucr = Ui-1. If no convergence has been achieved, i.e., ωi ≠ Ωi, then set Ωi = ωi and repeat the process until it converges.
STEP 2 - TIME DOMAIN ANALYSISThe second step is to carry out a time-domain analysis using the quantified equations of motion established in Step 1. This will provide a time-series for the spectral analysis of Step
3. Spectral analysis will reveal the frequency characteristics of vibration.
Effective oscillation frequencies from Step 1 are used to determine the respective Bω reduced frequency (K = ) for each mode under any wind speed. Accordingly, flutter U derivatives can be decided according to the respective K for each mode. The matrices, [C] in 39 Eq. (3.11) and [K] in Eq. (3.12), can thus be quantified and equations of motion, Eq. (3.8), is ready for a time domain analysis.
In the complex eigenvalue solution, flutter is identified when ζ i changes from positive to negative value. However, a free vibration response in time-series can better demonstrate the occurrence of flutter. As will be shown in the numerical example, time domain analysis can predict convergent, constant, and divergent amplitude vibrations that correspond to stable, critical, and unstable (flutter) conditions of the bridge, respectively. For this reason, a 5th order Runge-Kutta method is used to solve the equations of motion in the time domain.
STEP 3 - SPECTRAL ANALYSIS ON TIME-SERIESOnce the time-series of vibrations are available from Step 2, the third step is to conduct spectral analysis on the time-series of vibration response. Spectral analyses on these timeseries will reveal the vibration characteristics in the frequency domain. This information will help understand the true nature of the coalesced single-frequency vibration of flutter.
3.5 Numerical Example The Humen Bridge (Fig. 3.3) is a suspension bridge with a main span of 888 m located in Guang-Dong Province, China. The main data and main mode characteristics for the Humen Bridge are given in Tables 3.1 and 3.2, respectively (Lin and Xiang 1995). After examining its modal characteristics performed with finite element analysis, the first symmetric vertical mode (1st mode, hereafter simply called vertical mode) and the first symmetric torsion mode (4th mode, hereafter simply called torsion mode) are identified as two important modes and are thus used for both flutter and buffeting analyses. This numerical example is used to demonstrate the approach discussed above and to explain how the bridge vibration evolves from a multi-frequency vibration to a single-frequency flutter.
3.5.1 Complex Eigenvalue Analysis The predicted effective oscillation frequencies and damping ratios of vertical and torsion modes versus wind velocity are shown in Fig. 3.4 for the prototype Humen Bridge. It can be seen that with the increase of wind velocity, vertical mode frequency increases gradually while at the same time torsion mode frequency decreases gradually.
Correspondingly, the damping ratio of the vertical mode increases, while the damping ratio of the torsion mode remains about constant and decreases at higher wind velocity. Eventually, at the wind velocity of 87 m/s, the damping ratio of the torsion mode becomes zero (which represents the occurrence of flutter) and the damping ratio of vertical mode increases by about 20%.