Time-domain and frequency-domain approaches to identification of bridge flutter derivatives

Abstract

Flutter derivatives are essential for flutter analysis of long-span bridges, and they are generally identified from the vibration testing data of a sectional model suspended in a wind tunnel. Making use of the forced vibration testing data of three sectional models, namely, a thin-plate model, a nearly streamlined model, and a bluff-body model, a comparative study was made to identify the flutter derivatives of each model by using a time-domain method and a frequency-domain method. It was shown that all the flutter derivatives of the thin-plate model identified with the frequency-domain method and time-domain method, respectively, agree very well. Moreover, some of the flutter derivatives of each of the other two models identified with the two methods deviate to some extent. More precisely, the frequency-domain method usually results in smooth curves of the flutter derivatives. The formulation of time-domain method makes the identification results of flutter derivatives relatively sensitive to the signal phase lag between vibration state vector and aerodynamic forces and also prone to be disturbed by noise and nonlinearity.

Keywords

long-span bridges / wind-induced vibration / flutter derivatives / forced vibration test / time-domain method / frequency-domain method

Introduction

The wind-induced flutter and flutter derivatives concepts for long-span bridges originate from the aeroelasticity of aircrafts. The theories and experimental techniques for the flutter of the aircrafts with airfoil (or streamlined) cross section have been very mature. However, the bridge girder used for long-span bridges often has a blunt shape such as trussed girder and box girder, which is far away from the streamlined shape. Moreover, the identification of the flutter derivatives associated with bluff body is still confronted with many practical problems. For example, the identification results of flutter derivatives sometimes show significant fluctuation (or nonsmoothness) and may have poor test repetition. One basic assumption often adopted in flutter analysis of long-span bridges is that self-excited aeroelastic forces are linear or almost linear functions in terms of flutter derivatives [1]. However, through forced vibration testing of the sectional model in the wind tunnel, it was reported that the self-excitation aerodynamic forces of bluff girder have notable nonlinearity, even when the sectional model was driven by sinusoidal excitation with small amplitude [2,3]. Therefore, it is only a gross approximation to express the self-excited aeroelastic forces with the linear functions in terms of flutter derivatives, and this approximation may cause the identified flutter derivatives quite different for different identification methods.

Up to the present, the major approach to obtain the flutter derivatives of bridge sections is still through wind tunnel test of sectional models, although computational fluid dynamics (CFD) has advanced greatly during the past two decades [4]. It includes two stages: the first stage is to measure the time-history vibration signals of sectional models by using a free vibration test [1] and/or a forced vibration test [3,5,6], and the second stage is to extract the self-excited aeroelastic forces from vibration data and then to identify the flutter derivatives according to the relation between flutter derivatives and aeroelastic forces. The identification of flutter derivatives may be accomplished with two approaches, namely, the frequency-domain approach and the time-domain approaches. Therefore, there are four possible method combinations for identifying the flutter derivatives identification as follows:

1) Time-domain method combined with free vibration test data.

2) Time-domain method combined with forced vibration test data.

3) Frequency-domain method combined with free vibration test data.

4) Frequency-domain method combined with forced vibration test data.

It is well known that the identification accuracy of the flutter derivatives depends on careful preparation of the experimental setup and reliable signal processing methods. A set of forced vibration equipment developed by Chen et al. [3], which can drive the sectional model in a sinusoidal motion with constant amplitude, was used in this study to obtain the testing data of vibration displacements and aeroelastic forces of three sectional models, namely, a thin-plate model, a nearly streamlined model, and a bluff-body model. Making use of the same forced vibration data, the flutter derivatives identified with a frequency-domain approach is compared with those identified with a time-domain approach. Discussions on the notable discrepancies in identified flutter derivatives between the two methods are made, and remedies are suggested to improve the identification accuracy.

Time and frequency-domain approaches to identification of flutter derivatives

Self-excited aerodynamic forces and flutter derivatives

Consider that a rigid sectional model is immersed in a two-dimensional flow field. It is assumed that the oncoming flow is uniform with a velocity of U and an attack angle of α₀, which is sufficiently far from the model. The model vibrates around the equilibrium position with small amplitude and a frequency of ω, in two directions, namely, the heaving h and pitching α. According to Scanlan et al. [1], the self-excited aeroelastic forces acting on the model is approximately expressed as linear functions in terms of the nondimensional flutter derivatives of H_i^*,A_i^*, i=1,2,3,4, as

(1a)L={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}B)\left(k{H}_{\mathrm{1}}^{\ast }{\displaystyle \frac{\dot{h}}{U}}+k{H}_{\mathrm{2}}^{\ast }{\displaystyle \frac{\dot{\alpha }}{U}}+k^{\mathrm{2}}{H}_{\mathrm{3}}^{\ast }\alpha +k^{\mathrm{2}}{H}_{\mathrm{4}}^{\ast }{\displaystyle \frac{h}{B}}\right),

(1b)M={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}{B}^{\mathrm{2}})\left(k{A}_{\mathrm{1}}^{\ast }{\displaystyle \frac{\dot{h}}{U}}+k{A}_{\mathrm{2}}^{\ast }{\displaystyle \frac{\dot{\alpha }}{U}}+k^{\mathrm{2}}{A}_{\mathrm{1}}^{\ast }\alpha +k^{\mathrm{2}}{A}_{\mathrm{4}}^{\ast }{\displaystyle \frac{h}{B}}\right),

where ρ is the air density; U is the velocity of the oncoming flow; k=ωB/U is the reduced frequency; and ω is the circular frequency. When the testing data of displacements and velocity and aeroelastic forces are known, the flutter derivatives may be identified from Eq. (1) by using a time-domain or a frequency-domain approach.

Forced vibration testing of sectional models in wind tunnel

The wind tunnel used in this experimental study has a test section of 1.0 m wide, 0.8 m high, and 2.0 m long. The wind speed available for the test ranges from 0 m/s to 60 m/s. A vibration excitation system was designed and built to suspend and drive the sectional model in sinusoidal motion. To simplify its mechanism, this system currently can only generate a heaving motion or a pitching motion but not both motions simultaneously. Figure 1 provides a sketch of the vibration excitation system in the heaving mode. As shown in Fig. 1, rods 1-4 are devised to make synchronization in the heaving (vertical) motion under this configuration; therefore, the driving system can generate sinusoidal heaving motion of the sectional model. When rods 5 and 6 were removed, and rods 1 and 2 were fixed, as shown in Fig. 2, the vibration excitation system can drive the sectional model in pitching motion. More detailed information on forced vibration testing apparatus is given in Chen et al. [3].

Forces acting on the sectional model were measured using four axial-force transducers. Each of these transducers was connected between the suspension rod and sectional model. The movements of the sectional model were measured by four accelerometers installed on each rod close to the force transducer. The force and displacement measurements were first passed through low-pass filters with a cut-off frequency of 30 Hz and then recorded by the INV306 data acquisition system.

To identify flutter derivatives associated with heaving motion, it is assumed that the axial forces in rods 1-4 at wind speed U are measured as f₁, f₂, f₃, and f₄, respectively. Thus, the total lift force L_U and pitching moment M_U are

(2)L_U=f_{\mathrm{1}}+f_{\mathrm{2}}+f_{\mathrm{3}}+f_{\mathrm{4}},

(3)M_U=(f_{\mathrm{2}}+f_{\mathrm{4}})R.

It can be noticed that the life force L_U and pitching moment M_U is comprised of the wind-induced aeroelastic forces and the initial forces (including lift force and pitching moment) at zero wind speeds, which are denoted as L₀ and M₀, and are irrelevant with the flutter derivatives. After removing forces at zero wind speed, the aeroelastic lift force L(t) and pitching moment M(t) acting on a unit length of the sectional model are given by

(4)L(t)=(L_U-L_{\mathrm{0}})/D=(f_{\mathrm{1}}+f_{\mathrm{2}}+f_{\mathrm{3}}+f_{\mathrm{4}}-f_{\mathrm{10}}-f_{\mathrm{20}}-f_{\mathrm{30}}-f_{\mathrm{40}})/D,

(5)M(t)=(M_U-M_{\mathrm{0}})/D=(f_{\mathrm{2}}+f_{\mathrm{4}}-f_{\mathrm{20}}-f_{\mathrm{40}})R/D,

where D is the length of the sectional model; and f₁₀-f₄₀ are measured forces in rods 1-4 at zero wind speed, respectively.

Time-domain identification method

Assume that the sectional model is driven in a purely vertical forced vibration at a frequency of ω under wind velocity of U in wind tunnel, that is, \alpha =\dot{\alpha }=\mathrm{0}. We can record displacement signal h(t), velocity signal \dot{h}(t) of model motion, and derive the aerodynamic forces signal L(t) and M(t) of model. These signals are discretely acquired in computers by an AD transformer and are denoted as h_i,{\dot{h}}_i, L_i, M_i, i=1,2,…,N, where N is the total point of the whole time history. It can be seen from Eq. (1) that there are only four flutter derivatives H₁^*, H₄^*, A₁^*, A₄^* related to vertical degree of freedom h. The residual between the measured forces and the forces expressed by flutter derivatives at every discrete time point i is given as

(6a)\delta {(L)}_i=L_i-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}B)\left(k{H}_{\mathrm{1}}^{\ast }{\displaystyle \frac{{\dot{h}}_i}{U}}+k^{\mathrm{2}}{H}_{\mathrm{4}}^{\ast }{\displaystyle \frac{h_i}{B}}\right),

(6b)\delta {(M)}_i=L_i-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}{B}^{\mathrm{2}})\left(k{A}_{\mathrm{1}}^{\ast }{\displaystyle \frac{{\dot{h}}_i}{U}}+k^{\mathrm{2}}{A}_{\mathrm{4}}^{\ast }{\displaystyle \frac{h_i}{B}}\right).

It can be seen from Eq. (6a) that the squared sum of residual is

(7){\mathrm{\Delta }}^{\mathrm{2}}={\displaystyle \sum _{i=\mathrm{1}}^N{L}_i-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}B)\left(k{H}_{\mathrm{1}}^{\ast }{\displaystyle \frac{{\dot{h}}_i}{BU}}+k^{\mathrm{2}}{H}_{\mathrm{4}}^{\ast }{\displaystyle \frac{h_i}{B}}\right).}

By setting the partial differentiation of Δ² with respect to H₁^*, H₄^* being zero, the following system of equations is obtained:

(8)\rho B^{\mathrm{2}}\omega \left(\begin{array}{ll}{\displaystyle \sum _{i=\mathrm{1}}^N{\dot{h}}_i^{\mathrm{2}}}& \omega {\displaystyle \sum _{i=\mathrm{1}}^N{h}_i{\dot{h}}_i}\\ {\displaystyle \sum _{i=\mathrm{1}}^N{h}_i{\dot{h}}_i}& \omega {\displaystyle \sum _{i=\mathrm{1}}^N{h}_i^{\mathrm{2}}}\end{array}\right)\left(\begin{array}{l}{H}_{\mathrm{1}}^{\ast }\\ H_{\mathrm{4}}^{\ast }\end{array}\right)=\left(\begin{array}{l}{\displaystyle \sum _{i=\mathrm{1}}^N{L}_i{\dot{h}}_i}\\ {\displaystyle \sum _{i=\mathrm{1}}^N{L}_i{h}_i}\end{array}\right)\mathrm{.}

The flutter derivatives H₁^*, H₄^* may be solved from above system of equations. Two other flutter derivatives of A₁^*, A₄^* can also be similarly obtained from Eq. (6b). By repeating this procedure for the testing data at different wind velocities, a series of curves describing these four flutter derivatives versus reduced wind speed is obtained.

Similarly, assume that the sectional model is driven by a pitching motion, namely, h=\dot{h}=\mathrm{0}, the remaining four flutter derivatives H₂^*, H₃^*, A₂^*, A₃^* related to torsional motion can be obtained.

Frequency domain identification method based on forced vibration

Assume that the sinusoidal heaving and pitching motions have the forms of

(9)h=h_{\mathrm{0}}{\mathrm{e}}^{i{\omega }_{\mathrm{0}}t},\alpha ={\alpha }_{\mathrm{0}}{\mathrm{e}}^{i{\omega }_{\mathrm{0}}t},

where h₀ and α₀ are amplitudes of the sinusoidal heaving and pitching motions, respectively; and ω₀ is circular frequency. Substituting Eq. (9) into Eq. (1) and applying Fourier transform, one has

(10)L(\omega )={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}B)\left[k{H}_{\mathrm{1}}^{\ast }{\displaystyle \frac{i{\omega }_{\mathrm{0}}{h}_{\mathrm{0}}}{U}}+k{H}_{\mathrm{2}}^{\ast }{\displaystyle \frac{i{\omega }_{\mathrm{0}}B{\alpha }_{\mathrm{0}}}{U}}+k^{\mathrm{2}}{H}_{\mathrm{3}}^{\ast }{\partial }_{\mathrm{0}}+k^{\mathrm{2}}{H}_{\mathrm{4}}^{\ast }{\displaystyle \frac{h_{\mathrm{0}}}{B}}\right]\delta (w-w_{\mathrm{0}}),

(11)M(\omega )={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}{B}^{\mathrm{2}})\left[k{A}_{\mathrm{1}}^{\ast }{\displaystyle \frac{i{\omega }_{\mathrm{0}}{h}_{\mathrm{0}}}{U}}+k{A}_{\mathrm{2}}^{\ast }{\displaystyle \frac{i{\omega }_{\mathrm{0}}B{\alpha }_{\mathrm{0}}}{U}}+k^{\mathrm{2}}{A}_{\mathrm{3}}^{\ast }{\partial }_{\mathrm{0}}+k^{\mathrm{2}}{A}_{\mathrm{4}}^{\ast }{\displaystyle \frac{h_{\mathrm{0}}}{B}}\right]\delta (w-w_{\mathrm{0}}),

in which L(ω) and M(ω) are Fourier transforms of L(t) and M(t), respectively; and δ(·) is Dirac function. Noting that the reduced frequency k at the frequency of ω₀ is given by k₀=Bω₀/U, Eqs. (9) and (10) can be further simplified after separation of the real and imaginary parts of L(ω) and M(ω) as follows:

(12)\mathrm{Re}[L(k_{\mathrm{0}})]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}B)k_{\mathrm{0}}^{\mathrm{2}}\left[H_{\mathrm{3}}^{\ast }(k_{\mathrm{0}}){\alpha }_{\mathrm{0}}+H_{\mathrm{4}}^{\ast }(k_{\mathrm{0}}){\displaystyle \frac{h_{\mathrm{0}}}{B}}\right],

(13)\mathrm{Im}[L(k_{\mathrm{0}})]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}B)k_{\mathrm{0}}^{\mathrm{2}}\left[H_{\mathrm{2}}^{\ast }(k_{\mathrm{0}}){\alpha }_{\mathrm{0}}+H_{\mathrm{1}}^{\ast }(k_{\mathrm{0}}){\displaystyle \frac{h_{\mathrm{0}}}{B}}\right],

(14)\mathrm{Re}[M(k_{\mathrm{0}})]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}{B}^{\mathrm{2}})k_{\mathrm{0}}^{\mathrm{2}}\left[A_{\mathrm{3}}^{\ast }(k_{\mathrm{0}}){\alpha }_{\mathrm{0}}+A_{\mathrm{4}}^{\ast }(k_{\mathrm{0}}){\displaystyle \frac{h_{\mathrm{0}}}{B}}\right],

(15)\mathrm{Im}[M(k_{\mathrm{0}})]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}(\mathrm{2}{B}^{\mathrm{2}})k_{\mathrm{0}}^{\mathrm{2}}\left[A_{\mathrm{2}}^{\ast }(k_{\mathrm{0}}){\alpha }_{\mathrm{0}}+A_{\mathrm{1}}^{\ast }(k_{\mathrm{0}}){\displaystyle \frac{h_{\mathrm{0}}}{B}}\right].

As mentioned in the previous section, the vibration excitation system utilized in the experiment can only generate a heaving or a pitching motion in one test. Therefore, two sets of tests are needed to identify the eight flutter derivatives given in Eq. (1). In the first set of tests, the sectional model was driven by a sinusoidal heaving motion; in the second set of tests, the sectional model was excited by a sinusoidal pitching motion.

For the first set of tests, the pitching and heaving displacements of the sectional modal are given by

(16)\begin{array}{ll}h(t)=h_{\mathrm{0}}\sin {\omega }_{\mathrm{0}}t,& \alpha (t)=\mathrm{0.}\end{array}

The aerodynamic lift forces L(k₀) and pitching moment M(k₀) can be obtained from the FFT of the experimental data which are given by

(17){L(k_{\mathrm{0}})=F[(f_{\mathrm{1}}+f_{\mathrm{2}}+f_{\mathrm{3}}+f_{\mathrm{4}}-f_{\mathrm{10}}-f_{\mathrm{20}}-f_{\mathrm{30}}-f_{\mathrm{40}})/D]|}_{\omega ={\omega }_{\mathrm{0}}},

(18){M(k_{\mathrm{0}})=F[((f_{\mathrm{2}}+f_{\mathrm{4}}-f_{\mathrm{20}}-f_{\mathrm{40}})R)/D]|}_{\omega ={\omega }_{\mathrm{0}}},

where F(·) is operator of the Fourier transform. After combining Eqs. (12)-(15), four of the flutter derivatives can then be identified as follows:

(19)H_{\mathrm{1}}^{\ast }(k_{\mathrm{0}})={\displaystyle \frac{\mathrm{1}}{\rho D{h}_{\mathrm{0}}{B}^{\mathrm{2}}{\omega }_{\mathrm{0}}^{\mathrm{2}}}}\mathrm{Im}\{{F(f_{\mathrm{1}}+f_{\mathrm{2}}+f_{\mathrm{3}}+f_{\mathrm{4}}-f_{\mathrm{10}}-f_{\mathrm{20}}-f_{\mathrm{30}}-f_{\mathrm{40}})|}_{\omega ={\omega }_{\mathrm{0}}}\},

(20)H_{\mathrm{4}}^{\ast }(k_{\mathrm{0}})={\displaystyle \frac{\mathrm{1}}{\rho D{h}_{\mathrm{0}}{B}^{\mathrm{2}}{\omega }_{\mathrm{0}}^{\mathrm{2}}}}\mathrm{Re}\{{F(f_{\mathrm{1}}+f_{\mathrm{2}}+f_{\mathrm{3}}+f_{\mathrm{4}}-f_{\mathrm{10}}-f_{\mathrm{20}}-f_{\mathrm{30}}-f_{\mathrm{40}})|}_{\omega ={\omega }_{\mathrm{0}}}\},

(21)A_{\mathrm{1}}^{\ast }(k_{\mathrm{0}})={\displaystyle \frac{R}{\rho D{h}_{\mathrm{0}}{B}^{\mathrm{3}}{\omega }_{\mathrm{0}}^{\mathrm{2}}}}\mathrm{Im}\{{F(f_{\mathrm{2}}+f_{\mathrm{4}}-f_{\mathrm{20}}-f_{\mathrm{40}})|}_{\omega ={\omega }_{\mathrm{0}}}\},

(22)A_{\mathrm{4}}^{\ast }(k_{\mathrm{0}})={\displaystyle \frac{R}{\rho D{h}_{\mathrm{0}}{B}^{\mathrm{3}}{\omega }_{\mathrm{0}}^{\mathrm{2}}}}\mathrm{Re}\{{F(f_{\mathrm{2}}+f_{\mathrm{4}}-f_{\mathrm{20}}-f_{\mathrm{40}})|}_{\omega ={\omega }_{\mathrm{0}}}\},

For the second set of tests, the heaving and pitching displacements of the sectional model are given by

(23)\begin{array}{ll}h(t)=\mathrm{0},& \alpha (t)={\alpha }_{\mathrm{0}}\sin {\omega }_{\mathrm{0}}t\end{array}.

Similarly, the rest four flutter derivatives can be identified:

(24)H_{\mathrm{2}}^{\ast }={\displaystyle \frac{\mathrm{1}}{\rho D{\alpha }_{\mathrm{0}}{B}^{\mathrm{3}}{\omega }_{\mathrm{0}}^{\mathrm{2}}}}\mathrm{Im}\{{F(f_{\mathrm{1}}+f_{\mathrm{2}}+f_{\mathrm{3}}+f_{\mathrm{4}}-f_{\mathrm{10}}-f_{\mathrm{20}}-f_{\mathrm{30}}-f_{\mathrm{40}})|}_{\omega ={\omega }_{\mathrm{0}}}\},

(25)H_{\mathrm{3}}^{\ast }={\displaystyle \frac{\mathrm{1}}{\rho D{\alpha }_{\mathrm{0}}{B}^{\mathrm{3}}{\omega }_{\mathrm{0}}^{\mathrm{2}}}}\mathrm{Re}\{{F(f_{\mathrm{1}}+f_{\mathrm{2}}+f_{\mathrm{3}}+f_{\mathrm{4}}-f_{\mathrm{10}}-f_{\mathrm{20}}-f_{\mathrm{30}}-f_{\mathrm{40}})|}_{\omega ={\omega }_{\mathrm{0}}}\},

(26)A_{\mathrm{2}}^{\ast }={\displaystyle \frac{R}{\rho D{\alpha }_{\mathrm{0}}{B}^{\mathrm{4}}{\omega }_{\mathrm{0}}^{\mathrm{2}}}}\mathrm{Im}\{{F(f_{\mathrm{2}}+f_{\mathrm{4}}-f_{\mathrm{20}}-f_{\mathrm{40}})|}_{\omega ={\omega }_{\mathrm{0}}}\},

(27)A_{\mathrm{3}}^{\ast }={\displaystyle \frac{R}{\rho D{\alpha }_{\mathrm{0}}{B}^{\mathrm{4}}{\omega }_{\mathrm{0}}^{\mathrm{2}}}}\mathrm{Re}\{{F(f_{\mathrm{2}}+f_{\mathrm{4}}-f_{\mathrm{20}}-f_{\mathrm{40}})|}_{\omega ={\omega }_{\mathrm{0}}}\}.

In each set of tests, the wind speed U is increased from zero to the desired maximum wind speed in discrete intervals with equal spacing. Note that at a given wind speed, the same test was repeated three times and the experimentally obtained flutter derivatives were averaged to reduce measurement error.

The signal processing program based on frequency-domain method and time-domain method was implemented in MATLAB. Then, flutter derivatives curves identified by the two methods can be obtained from the same experimental data to compare the coherence of the results.

Results and analyses of identification

Three types of sectional models were tested in the wind tunnel using the forced vibration testing apparatus. These models were chosen such that their shapes varied from a streamlined section to a bluff section. The schematic of these models are shown in Fig. 3. The first sectional model, TP22, is a very thin hexagonal plate; the second model, HM13, is a streamlined model of a steel-box main girder of a suspension bridge; and the third model, DT09, is a bluff model.

The effect of the driving frequency and amplitude on the results of flutter derivatives was investigated in Ref. [3]. This study is to verify the consistency in identification results of flutter derivatives by using the time- and frequency-domain methods. Figures 4-6 provide a comparison of the identified flutter derivatives of these three sectional models at a zero degree attack angle. The testing amplitudes and frequencies are noted on each figure.

In the frequency domain method, flutter derivatives were identified by performing FFT transform for signals and then extract the first-order harmonic component with a frequency equal to the excitation frequency. This procedure is equal to pose a first-order harmonic filter and get rid of the disturbance due to measurement noises and possible higher order harmonic components of the signals. Therefore, the flutter derivatives curves seem relatively smooth. FFT transformation verified that measured aerodynamic forces do contain large higher order harmonic component for those bluff bodies [3]. Therefore, the frequency-domain approach may result in a truncation error of self-excited aerodynamic forces. A mathematical model has been presented in Ref. [7] to explain this truncation error.

Solving Eq. (8) in time-domain method requires the displacement and velocity history signals of sectional models to be known, but it is generally more difficult to measure the displacement and velocity than to measure the acceleration. In this study, both the displacement and velocity of vibration motions are estimated from the measured acceleration responses. The velocity signals are obtained by dividing acceleration signals with circular frequency ω and made a phase lag of 900 to the divided signals; the displacement signals were computed by multiplying acceleration signals with-1/ω². The identified curves of flutter derivatives with such estimated displacement and velocity signals were smooth in most case and were consistent with those obtained with the frequency domain method.

Generally speaking, the flutter derivatives curves obtained with time-domain approach were less smooth than those obtained with frequency-domain approach, which showed small amplitude wave in Figs. 5 and 6. In some cases, there also may be a global shift, as shown in Fig. 7, and a sudden shift of a special experimental point, as in Fig. 8.

The underlying reason causing the fluctuation of identified flutter derivatives was discussed in some detail here. The left-hand side of Eq. (8) of the time-domain method is a 2×2 matrix about displacement and velocity. It is known theoretically that the undiagonal coefficients of the matrix will be nearly zero as the computed displacement and velocity signals always have a phase difference of 900, which is also validated numerically in this study. The diagonal coefficients were computed as very stable for different repetitive testing data. Therefore, the major factor leading to the identification instability is the right-hand-side component of Eq. (8), which is the summation of product of aerodynamic forces and displacement or velocity. The value of the summation significantly relies on the phase difference between the two signals. Additionally, as shown in Eqs. (4) and (5), the aerodynamic forces was obtained through subtracting the measured force at zero wind velocity from those measured at wind velocity U. Thus, there is an issue to normalize the two sets of forces with a same phase. If the calculation of phase lag is less accurate due to measurement noise, a large error even as fitting failure may appear. Though many measures had been taken to assure the accuracy of right-hand-side component of Eq. (8), this issue cannot be satisfactorily solved. There were five cases with phenomena of fluctuation and shift in total 24 experiment cases. Therefore, it requires improvement in experimental setup and numerical procedure.

Conclusions

In this study, the time-domain method and frequency-domain methods are developed to identify the flutter derivatives based on forced vibration data. It was shown that all the flutter derivatives of the thin-plate model identified with the frequency-domain method and time-domain method, respectively, agree very well, and some of the flutter derivatives of each of the other two models identified with the two methods deviate to some extent.

The frequency-domain method usually results in smooth curves of the flutter derivatives because it includes a filtering process but at the cost of introducing a truncation error. The formulation of time-domain method makes the identification results of flutter derivatives relatively sensitive to the signal phase lag between vibration responses and aerodynamic forces and also prone to be disturbed by noise and nonlinearity.

References

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[2] Falco M, Curami A, Zasso A. Nonlinear effects in sectional model aeroelastic parameters identification. Journal of Wind Engineering and Industrial Aerodynamics, 1992, 42: 1321-1332

[3] Chen Z Q, Yu X D, Yang G, Spencer B F, Jr. Wind-induced self-excited loads on bridges. Journal of Structural Engineering, ASCE, 2005, 131: 1783-1793

[4] Larsen A, Walther J H. Aeroelastic analysis of bridge girder sections based on discrete vortex simulations. Journal of Wind Engineering and Industrial Aerodynamics, 1997, 67-68: 253-265

[5] Matsumoto M, Shiraishi N, Shirato H, Shigetaka K, Niihara Y. Aerodynamic derivatives of coupled/hybrid flutter of fundamental structural sections. Journal of Wind Engineering and Industrial Aerodynamics, 1993, 49: 575-584

[6] Li Q C. Measuring flutter derivatives for bridge sectional models in water channel. Journal of Engineering Mechanics, ASCE, 1995, 121: 90-101

[7] Chen Zhengqing. Time domain method and error analysis to identify flutter derivatives of forced vibration. In: Proceedings of the 11th National Conference of Structural Wind Engineering, Sangya, China, 2003 (in Chinese)

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