Data-driven stochastic subspace identification of flutter derivatives of bridge decks

Abstract

Most of the previous studies on flutter derivatives have used deterministic system identification techniques, in which the buffeting forces and the associated responses are considered as noises. In this paper, one of the most advanced stochastic system identification, the data-driven stochastic subspace identification technique (SSI-DATA) was proposed to extract the flutter derivatives of bridge decks from the buffeting test results. An advantage of the stochastic method is that it considers the buffeting forces and the responses as inputs rather than as noises. Numerical simulations and wind tunnel tests of a streamlined thin plate model conducted under a smooth flow by the free decay and the buffeting tests were used to validate the applicability of the SSI-DATA method. The results were compared with those from the widely used covariance-driven SSI method. Wind tunnel tests of a two-edge girder blunt type of Industrial-Ring-Road Bridge deck (IRR) were then conducted under both smooth and turbulent flows. The identified flutter derivatives of the thin plate model based on the SSI-DATA technique agree well with those obtained theoretically. The results from the thin plate and the IRR Bridge deck helped validate the reliability and applicability of the SSI-DATA technique to various experimental methods and wind flow conditions. The results for the two-edge girder blunt type section show that applying the SSI-DATA yields better results than those of the SSI-COV. The results also indicate that turbulence tends to delay the onset of flutter compared with the smooth flow case.

Introduction

Long-span cable-supported bridges are highly susceptible to wind excitations because of their inherent flexibility and low structural damping. Wind loads play an important role in the design of these structures. The actions of wind load are broadly divided into aerostatic and aerodynamic loads. Effects of aerostatic wind load are given by Boonyapinyo et al., 1994, Boonyapinyo et al., 2006, among others. The wind-induced aerodynamic force can be divided into two parts: a buffeting force that depends on turbulence of the incoming flow, and an aeroelastic force that originates from the interactions between the airflow and the bridge motion. The motion-dependent forces feed back into the dynamics of the bridge as aerodynamic damping and stiffness; the effect is termed “aeroelasticity” and is commonly described via “flutter derivatives”. The problems of aeroelastic phenomena, including vortex-induced vibrations, galloping, flutter, and buffeting, may have serious effects on safety and serviceability of the bridges. Among these, flutter is the most serious wind-induced vibration of bridges and may destroy the bridges due to diverging motions in either single or torsion–bending coupled mode. Notorious examples of the flutter phenomenon are the failures of the Brighton Chain Pier Bridge in 1836 and the original Tacoma Narrow Bridge in 1940. The flutter derivatives depend primarily on cross-sectional shape of bridges and reduced velocity. Nevertheless, no theoretical values exist for these derivatives for various bridge shapes except only for a simple thin plate section. A major research tool in these studies is, therefore, a wind tunnel test, in which a geometrically and aerodynamically representative scale model of a length of a bridge deck is built, mounted, and then tested in a wind tunnel (Hjorth-Hansen, 1992). The flutter derivatives are non-dimensional functions of reduced wind speed and geometry of the bridge; therefore they can be applied directly to the full-scale bridge in a piecewise manner.

The experimental methods used for determination of flutter derivatives can be grouped under two types, i.e. forced (Falco et al., 1992, Larose, 1997, Haan, 2000, Hatanaka and Tanaka, 2002, Chen and Yu, 2002, Sarkar et al., 2009) and free vibration methods (Scanlan and Tomko, 1971, Poulsen et al., 1992, Sarkar et al., 1994, Gu et al., 2000). For the forced vibration method, it is possible to identify flutter derivatives from the difference of inertial and excitation forces of a model forced to vibrate (Brownjohn and Jakobsen, 2001). The free vibration method, on the other hand, does not force any prescribed motion on the model, but rather allows the fluid structure interaction to drive the motion. Having less emphasis on elaborate equipments required, time, and the amount of work involved, the free vibration method seems to be more tractable than the forced method. In the determination of flutter derivatives by the free vibration method, the system identification method is the most important part required to extract these parameters from the response output of the section model. The free vibration method depends on the system identification techniques and can be classified into two types, i.e. the free decay and the buffeting tests. In the free decay test method, the bridge deck is given initial vertical and torsional displacements. The flutter derivatives are based on the transient (i.e. free decay) behavior that occurs when the bridge deck is released. The buffeting test, on the other hand, uses only steady random responses (i.e. buffeting responses) of bridge deck under wind flow without any initial displacement given to the model. Compared with the free decay method, the buffeting test is simpler in the test methodology, more cost effective, and more closely related to the real bridge behaviors under wind flow, but with the disadvantage that the outputs appear random-like. This makes the parameters extraction more difficult and a more advanced system identification technique is required.

In most of the previous studies, flutter derivatives were estimated by the deterministic system identification techniques, which can be applied to the free decay method only. Examples of the previous deterministic system identification techniques that were applied to the free decay method included Scanlan’s method (Scanlan and Tomko, 1971), Poulsen’s method (Poulsen et al., 1992), the Extended Kalman Filter Algorithm (Yamada et al., 1992), the Modified Ibrahim Time Domain method (MITD; Sarkar et al., 1994), the Unified Least Square method (ULS; Gu et al., 2000), and the Iterative Least Square method (ILS; Chowdhurry and Sarkar, 2003). In these system identification techniques, the buffeting forces and their responses are regarded as external noises; the identification process then requires many iterations. It is also confronted with difficulties at high wind speeds, where the initial free decay is drowned by buffeting responses. At high reduced wind speed, vertical bending motion of the structure will decay rapidly due to the effect of vertical aerodynamic damping, and thus length of decaying time history available for system identifications will decrease. This causes more difficulties to the deterministic system identification techniques (Gu and Qin, 2004). In the case of turbulent flow, the presence of turbulence in the flow is equivalent to a more noisy-input signal to the deterministic system identification. This made the extraction process more complicated and most likely reduced accuracy of the flutter derivatives identified (Sarkar et al., 1994). In addition, due to the test technique, the free decay method is impractical to determine flutter derivatives of real bridges in the field.

On the other hand, the buffeting test uses random responses data of bridge motion from wind turbulence only. This mechanism is more closely related to a real bridge under wind flow and is applicable to real prototype bridges. The buffeting method is simpler than the free decay method since no operator interrupts in exciting the model. However, as wind is the only excited source, it results in low signal-to-noise ratio, especially at low velocity, and therefore a very effective system identification technique is required. None of the aforementioned system identification techniques is applicable to the buffeting responses tests. System identification techniques can be divided into two groups, i.e. deterministic and stochastic.

If the stochastic system identification technique (Juang and Pappa, 1985, Overschee, 1991, Peeters, 1999) is employed to estimate the flutter derivatives of a bridge deck from their steady random responses under the action of turbulent wind, the above-mentioned shortcomings of the deterministic system identification technique can be overcome. The reason is that the random aerodynamic loads are regarded as inputs rather than noises, so that the identification model is more consistent with the phenomenon being analyzed. Hence, the signal-to-noise ratio is not affected by wind speed, and flutter derivatives at high reduced wind speeds are more readily available. These aspects give the stochastic system identification methods an advantage over the deterministic system identification.

Many stochastic system identification methods have been developed during the past decades, among which the stochastic subspace identification (SSI in short; Overschee, 1991, Peeters, 1999) has proven to be a method that is very appropriate for civil engineering. The merit points of SSI are: (1) assumptions of inputs are congruent with practical wind-induced aerodynamic forces, i.e. stationary and independent of the outputs; (2) identified modes are given in frequency stabilization diagram, from which the operator can easily distinguish structural modes from the computational ones; (3) since the maximum order of the model is changeable for the operator, a relatively large model order will give an exit for noise, which in some cases can dramatically improve the quality of the identified modal parameters; and (4) mode shapes are simultaneously available with the poles, without requiring a second step to identify them. There are two kinds of SSI methods, one is data-driven, and the other is covariance-driven.

The similarity of the covariance- and the data-driven SSI methods is that they both are aimed to cancel out the (uncorrelated) noise using stochastic realization. In the SSI-COV algorithm, raw time histories are converted to covariances of the Toeplitz matrix. The implementation of SSI-COV consists of estimating the covariances, computing the singular value decomposition (SVD) of the Toeplitz matrix, truncating the SVD to the model order n, estimating the observability and the controllability matrices by splitting the SVD into two parts, and finally estimating the system matrices (A,C). A and C are discrete form of state and output matrices in the state space equation, respectively. The modal parameters are then found from A and C. Gu and Qin (2004) applied the SSI-COV to extract six derivatives (H₁⁎–H₃*, A₁*–A₃*). Mishra et al. (2006) used the SSI-COV to extract 18 flutter derivatives from wind tunnel tests, but the identified flutter derivatives seem to be scattered.

As opposed to SSI-COV, the data-driven stochastic subspace identification (SSI-DATA) avoids computation of covariances between the outputs, since error and noises may be squared up from the covariance estimation (Golub and Van Loan, 1996). It is replaced by projecting the row space of the future outputs into the row space of the past outputs. This projection is computed favorably from the numerically robust square root algorithm, i.e. QR factorization. Theoretically, the numerical behavior of SSI-DATA should then be better than that of SSI-COV (Peeters and De Roeck, 2001). However, very few researchers, if any, have applied the SSI-DATA for identification of flutter derivatives of bridge decks.

In this paper, the data-driven stochastic subspace identification method is proposed to estimate flutter derivatives from random responses (buffeting) under the action of smooth and turbulent wind. Tests are also carried out with the free decay method (single and two degrees of freedom) in order to examine the robustness of the present technique that the results are not affected by test methods used. To validate the applicability of the present technique, numerical simulations were firstly performed. Then, sectional-model tests of a thin plate model, which is the only section that theoretical flutter derivatives exist for, were performed under smooth flow. Encouraged by the success in the evaluation process, flutter derivatives of a real bridge were determined. The two-edge-girder type blunt section model of Industrial-Ring-Road Bridge (IRR in short), a cable-supported bridge with a main span of 398 m in the Samutprakan province of Thailand, was tested both in smooth and the turbulent flows. Tests were conducted in TU-AIT Boundary Layer Wind Tunnel in Thammasat University, the longest and the largest wind tunnel in Thailand.