A nonlinear convolution scheme to simulate bridge aerodynamics

Computers and Structures 128 (2013) 259–271

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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

A nonlinear convolution scheme to simulate bridge aerodynamics Teng Wu ⇑, Ahsan Kareem Nathaz Modeling Laboratory, University of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, IN, USA

a r t i c l e

i n f o

Article history: Received 17 January 2013 Accepted 11 June 2013 Available online 13 September 2013 Keywords: Bridge aerodynamics Nonlinearity Convolution Volterra

a b s t r a c t A linear convolution scheme involving first-order (linear) kernels for linear bridge aerodynamics is first reviewed and the significance of the selection of proper input parameters is emphasized. Following the concept of nonlinear indicial response function, the linear convolution scheme is extended to the nonlinear convolution scheme involving higher-order (nonlinear) kernels for the treatment of nonlinear bridge aerodynamics using a ‘‘peeling-an-onion’’ type procedure. Utilizing an impulse function as input, a comprehensive kernel identification scheme is developed. A numerical example of a long-span suspension bridge is investigated to verify the fidelity of the proposed nonlinear convolution scheme. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction A main source of nonlinearity in bridge aerodynamics results from flow separation around the deck. For streamlined sections like an airfoil, flow separation occurs only in the case of large angles of attack (dynamic stall) or the shock motions in the transonic region (the shock motion itself also induces nonlinearity). For bluff sections like a bridge deck, flow separation is prevalent as the fluid motion around the deck cannot negotiate sudden changes in the deck profile. The resulting nonlinearity can be viewed from four viewpoints: (i) non-proportional relationship between amplitudes of input and output; (ii) single-frequency input exciting multiple frequencies; (iii) amplitude dependence of aerodynamic and aeroelastic forces and (iv) hysteretic behavior of aerodynamic forces versus angles of attack [1]. Nonlinear effects are usually exploited to offer a possible explanation for any differences observed between the linear analysis results and experiments [2] although it is difficult to delineate their relative contributions. In order to take into account the increasing nonlinear behavior of bridge aerodynamics observed in wind-tunnel tests, several numerical schemes such as the ‘‘band superposition’’ [3], ‘‘hybrid’’ [4], ‘‘rheological’’ [5] and ‘‘artificial neural network’’ [6] have been proposed over the last decade to advance conventional linear analysis framework [7,8]. Generally, these numerical schemes have been unable to represent completely nonlinear bridge aerodynamics [6,9], which limits their utility and calls for a comprehensive nonlinear analysis framework. The consideration of nonlinearity is usually carried out in the time domain benefitting from its ability to take into account the ⇑ Corresponding author. Tel.: +1 574 904 4290. E-mail addresses: [email protected] (T. Wu), [email protected] (A. Kareem). 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.06.004

nonlinear effects readily. In the time domain, the convolution of a linear kernel, e.g., the unit-step response function, is well known as the Duhamel’s integral. In this study, the linear convolution scheme concerning first-order kernels for linear analysis of bridge aerodynamics is reviewed together with a selection of proper input variables. Then, it is extended to the nonlinear convolution scheme involving higher-order kernels for nonlinear analysis of bridges under winds based on the concept of nonlinear indicial response function. A nonlinear convolution scheme is represented utilizing a Volterra-type formalism, which ensures convergence of its truncated form. To facilitate this formalism, a comprehensive kernel identification scheme is developed utilizing the impulse function as input. Finally, a numerical example of a long-span suspension bridge with vertical and torsional degrees of freedom is investigated to verify the fidelity of the simulation based on the proposed nonlinear convolution scheme, where the amplitude dependence of kernels is also discussed.

2. Linear convolution scheme This study focuses on the simulation based on a two-dimensional (2-D) representation of the deck and the strip theory.

2.1. Input information of bridge aerodynamics The selection of proper input variables for bridge aerodynamics based on convolution integrals is a critical issue. In the case of gust-induced effects, the input information is straightforward, i.e., the gust fluctuations in each degree of freedom. However, in the case of motion-induced effects, the input information is often misunderstood in bridge aerodynamics.

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It is well known that the motion-induced forces on a bridge deck are dependent on the orientation of local coordinate system and the relative motion between the wind and the deck [10]. In order to simplify the analysis, the case of a bridge deck moving in the absence of flow is first investigated (Fig. 1). As shown in Fig. 1, the orientation of local coordinate system (as denoted by the subscript ‘‘b’’) can be represented by the angle h defined as the pitching angle, which is the angle between the local coordinate abscissa axis xb and the global coordinate abscissa axis x; while the relative motion between the wind and the deck can be decomposed into the relative translatory motion represented by ~ V, which is the time derivS, and the relative rotational motion ative of the displacement ~ represented by h_ (velocity of angle of pitch). ~ V is represented by the velocity magnitude V and velocity direction c, which is the angle between the velocity direction and the global coordinate abscissa axis x. It is noted that a = c + h, where a defined as the angle of attack, is the angle between the velocity direction and the local coordinate abscissa axis xb. As a result, the motion-induced forces on the bridge deck are functions of V, h, h_ and a . For a bridge deck harmonically oscillating in vertical and torsional degrees of freedom under a stationary uniform wind flow (constant wind velocity), translation direction c is a constant. For _ such a case, a and V can be represented by the variable h and h, respectively, where h is the vertical translatory displacement of the deck. Hence, the bridge deck motion can be decomposed into _ h and h, _ as shown in Fig. 2, which contribute to the motion-inh, duced forces on the bridge deck. As indicated in Fig. 2, the translatory motion coupled with h has a negative equivalent that is _ hence, combination of motions h and h_ presents a coupled with h, pure harmonic rotary oscillation. It should be noted that, for the streamlined cross section, the contributions to the motion-induced forces from the translatory motion h_ and the orientation of the

y

V

S

2.2. Convolution scheme with indicial response function With the inclusion of chord-wise correlation, the gust-induced effects (mainly vertical) are directly related to the motion-induced case [12] and are not presented here for the sake of brevity. Generally, the motion-induced forces are expressed as

_ h; _ tÞ FðtÞ ¼ f ðh; h;

ð1Þ

where F denotes motion-induced forces, i.e., the motion-induced lift force L or torsional moment M; f represents a general nonlinear function. The linear part of the Taylor expansion of the nonlinear motion-induced lift force increment DL(t) and torsional moment increment DM(t), due to an infinitesimal change in the input variables at time s, could be represented as [13]

"( !)   ) (  @Lðt; sÞ dh @Lðt; sÞ dh_ þ Ds Ds ds @h _ h_ @ h_ h¼ h¼h0 ds 0 ( !)#  @Lðt; sÞ dh_ ð2aÞ Ds þ _h¼h_ 0 ds @ h_

DLðtÞ ¼ 

"( !)   ) (  @Mðt; sÞ dh @Mðt; sÞ dh_ DMðtÞ ¼ Ds Ds þ ds @h _ h_ @ h_ h¼ h¼h0 ds 0 ( !)#  @Mðt; sÞ dh_ ð2bÞ Ds þ ds _ h_ @ h_ h¼ 0

yb

ob

cross section h are identical since the flow passing by the structure is always attached on the solid surface. Whereas, for the bluff cross section, the contribution from h becomes complicated due to the flow separation. There is no clear explanation to demonstrate the assumption of an equivalent physical origin for these two contributions to the aeroelastic forces. Besides, as the harmonic oscillation is applied to the deck, the identified contribution of the orientation of the cross section h in a wind tunnel naturally involves the apparent moment of inertia effects of the rotational motion. Different values of the identified flutter derivatives in the wind tunnel corresponding to the translatory motion h_ and orientation of the cross section h also indicate that it is necessary to separate their contributions in bridge aerodynamics and to retain both of them as input variables [11].

γ θ

α

xb

o

x Fig. 1. Arbitrary motion of bridge deck.

_ or h) _ denotes the rate of where oy(t, s)/ox (y = L, or M and x = h, h, change of F(t) with input at time s. It is obvious that there are two time scales involved in describing the time dependent characteristics of linear wind–bridge interactions, i.e., the time s at which the boundary conditions (input variables) change and the time t at which the lift force or torsional moment is measured. As a time invariant system, the time dependent characteristics of the linear wind–bridge interactions could be described only using one time scale, i.e., the time difference (t  s) representing duration since the change in boundary conditions. Hence, as Dh,

h h

h

θ

θ t

h t

Fig. 2. Decomposition of bridge deck motion.

t

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Dh_ and Dh_ converge towards infinitesimal, the accumulated lift force or torsional moment on the bridge deck up to t is given with the linear convolution integral [14]    Z s 1 dC L LðsÞ ¼  qU 2 2b ULh ðsÞhð0Þ þ ULh ðs  rÞh0 ðrÞdr 2 dh 0 !  Z s 0 00 dC L h ð0Þ h ðrÞ 0 ðsÞ 0 ðs  rÞ þ d U þ U r h h 0 þ CD Lb Lb b b d hb 0    Z s dC L 0 00 0 0 þ C U ðsÞh ð0Þ þ U ðs  r Þh ð r Þd r ð3aÞ þ D Lh Lh dh0 0

   Z s 1 dC qU 2 2b2 M UMh ðsÞhð0Þ þ UMh ðs  rÞh0 ðrÞdr 2 dh 0   Z s 0 00 dC M h ð0Þ h ðrÞ dr þ UMh0 ðs  rÞ þ h0 UMh0 ðsÞ b b b b db 0   Z s dC M ð3bÞ UMh0 ðs  rÞh00 ðrÞdr þ 0 UMh0 ðsÞh0 ð0Þ þ dh 0

MðsÞ ¼

where CM, CL, CD are steady-state moment, lift and drag coefficients, respectively; b is the half width of the bridge deck; s and r represent non-dimensional time corresponding to t and s, respectively; ULh, ULh0 =b , ULh0 , UMh, UMh0 =b and UMh0 denote nondimensional rate of change of F(s) with input at nondimensional time r, which are also known as nondimensional motion-induced indicial (unit-step) response functions. As indicated in Eq. (3), there are totally six unit-step response functions for simulating the motion-induced lift force or torsional moment on the bridge deck. On the other hand, it is noted that the motion-induced lift force on the thin airfoil only depends on the downwash (the vertical component of the flow) at the 3/4 chord point of the airfoil (rear neutral point). Furthermore, the resultant lift force on the airfoil always acts at the 1/4 chord point of the airfoil (forward neutral point). Hence, in the case of a thin airfoil with small disturbance, only one unit-step response function, namely the lift force induced by the unit-step function of angle of attack is sufficient to characterize the motion-induced effects. This unique unit-step response function is usually called Wagner liftgrowth function Uw(s), which was theoretically derived by Wagner [15] for a thin airfoil. In the case of a bridge deck, there is no analytical expression for these unit-step response functions, hence, they need to be directly identified in a wind tunnel or evaluated using computational fluid dynamics (CFD) (e.g., [10,16]). Alternatively, these could be identified indirectly using the indicial or rational function approximations (e.g., [17,4,14]).

 n 1 2 dC M ~ Mh fhð0Þ; sghð0Þ MðsÞ ¼ qU 2 2b U 2 dh  Z s ~ Mh fhðrÞ; s  rgh0 ðrÞdr U þ

ð4bÞ

0

~ Lh and U ~ Mh are functionals of the input variable h(r). In orwhere U der to improve the simulation accuracy of nonlinear bridge aerodynamics, Wu and Kareem [9] developed a ‘‘modified hybrid’’ scheme, where the motion-induced unit-step response functions are dependent on h(r) and its derivative with time. Furthermore, Tobak and Pearson [18] considered a more generalized situation where the motion-induced unit-step response functions depended not only on the motion h(r) at time r at which the unit-step input was made, but also on all the past values of h. Accordingly, the motion-induced nonlinear lift force and torsional moment should be expressed [18]

 1 dC L n ~ LðsÞ ¼  qU 2 2b ULh fhð0Þ; s; 0ghð0Þ 2 dh  Z s ~ Lh fhðjÞ; s; rgh0 ðrÞdr U þ

ð5aÞ

0

 1 dC n ~ qU 2 2b2 M U Mh fhð0Þ; s; 0ghð0Þ 2 dh  Z s ~ Mh fhðjÞ; s; rgh0 ðrÞdr U þ

MðsÞ ¼

ð5bÞ

0

~ Lh fhðjÞ; s; rg and U ~ Mh fhðjÞ; s; rg (0 6 j 6 s) denote the nonwhere U linear indicial (unit-step) response functions. The nonlinear unitstep response function is a functional and could be defined by the functional derivative (Fréchet derivative) with respect to the input h [19]

DLðsÞ Dh   LfhðjÞ þ Hðj  rÞDhg  LfhðjÞg ¼ limDh!0 Dh

~ Lh fhðjÞ; s; rg ¼ limDh!0 U

ð6aÞ

DMðsÞ Dh   MfhðjÞ þ Hðj  rÞDhg  MfhðjÞg ¼ limDh!0 Dh

~ Mh fhðjÞ; s; rg ¼ limDh!0 U

ð6bÞ where the input incremental Dh is applied at time s = r and H denotes the Heaviside step function. 3.2. Nonlinear convolution scheme involving higher-order kernels

3. Nonlinear convolution scheme In this study, the discussion on the simulation of nonlinear bridge aerodynamics involves only one input variable to illustrate the concept clearly without the loss of generality. 3.1. Nonlinear indicial response function In order to capture the nonlinear bridge aerodynamics features, a ‘‘hybrid’’ scheme was proposed by Chen and Kareem [4], in which the motion-induced unit-step response functions were amplitude dependent. Hence, if only h is involved as the input variable, the motion-induced nonlinear lift force and torsional moment are given by LðsÞ ¼ 

   Z s 1 2 dC ~ ~ Lh fhðrÞ; s  rgh0 ðrÞdr qU 2b L U U Lh fhð0Þ;sghð0Þ þ 2 dh 0

The concept of nonlinear unit-step response function offers a general framework to simulate nonlinear aerodynamics, however, its translation to applications is intractable. One possible approach to obtain a practical nonlinear analysis framework for practical problems is to utilize a ‘‘peeling-an-onion’’ type approach, in which the nonlinear effects of wind–bridge interactions are extracted from a nonlinear unit-step response function using a ‘‘step by step’’ procedure. Obviously, if the motion-induced effects are linear and time~ Lh fhðjÞ; s; rg and U ~ Mh fhðjÞ; s; rg reduce to ULh(s  r) invariant, U and UMh(s  r), respectively, which are characterized by one time scale (s  r). Hence, the nonlinear unit-step response functions could be expressed as



~ Lh fhðjÞ; s; rg ¼ ULh ðs  rÞ þ U

ð4aÞ

Z 0

s

~ non fhðjÞ; s; rgh0 ðj1 Þdj1 U Lh

 ð7aÞ

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Time-Invariant Nonlinear Aerodynamic System

Step 1

Extract effects of one time scale

Linear Effects

Truncated Effects

Step 2

Extract coupling effects of two time scales

2nd Nonlinear Effects

Linear Effects

Truncated Effects

Step 3

Extract coupling effects of three time scales

2nd Nonlinear Effects

Linear Effects

3rd Nonlinear Effects

Truncated Effects

Step n

Extract coupling effects of n time scales

2nd Nonlinear Effects

Linear Effects

3rd Nonlinear Effects

Truncated Effects

Fig. 3. A schematic of the proposed ‘‘peeling-an-onion’’ type approach.



~ Mh fhðjÞ; s; rg ¼ UMh ðs  rÞ þ U

Z

s 0

~ non fhðjÞ; s; rgh0 ðj1 Þdj1 U Mh



~ non fhðj1 Þ; s; rg and U ~ non fhðj1 Þ; s; rg with respect to the input U Lh1 Mh1 h could be

ð7bÞ where it is assumed that h(0) = 0 to provide a parsimonious model. It should be noted that there are infinite time scales in nonlinear unit-step response functions given in Eq. (7) since j may attain an arbitrary value in the time interval [0, s]. Suppose one extracts a part of the time-invariant nonlinear coupling effects due to two time scales from the remaining nonlinear unit-step response func~ non fhðjÞ; s; rg and U ~ non fhðjÞ; s; rg by adding a unit-step functions U Lh Mh tion input at time j1, a possible form of the nonlinear functional ~ non fhðjÞ; s; rg with respect to the input h, sim~ non fhðjÞ; s; rg and U U Lh Mh ilar to the expressions of Eq. (7), could be

~ non fhðjÞ; s; rg U Lh 

¼ ULh2 ðs  r; s  j1 Þ þ

Z

~ non fhðjÞ; s; rg U Mh 

¼ UMh2 ðs  r; s  j1 Þ þ

s

0

Z 0

~ non fhðj1 Þ; s; rgh0 ðj2 Þdj2 U Lh1



~ non fhðj1 Þ; s; rg ¼ ULh ðs  r; s  j1 ; s  j2 Þ U Lh1 3 þ

Z

s

0

~ non fhðj2 Þ; s; rgh0 ðj3 Þdj3 U Lh2

Z

þ

s

0

~ non fhðj2 Þ; s; rgh0 ðj3 Þdj3 U Mh2

 ð9bÞ

By analogy, one can obtain



~ non fhðji Þ; s; rg ¼ ULh ðs  r; s  j1 ; . . . ; s  jiþ1 Þ U Lhi iþ2



~ non fhðj1 Þ; s; rgh0 ðj2 Þdj2 U Mh1

ð9aÞ



~ non fhðj1 Þ; s; rg ¼ UMh ðs  r; s  j1 ; s  j2 Þ U Mh1 3

þ

Z

s

0

ð8aÞ



~ non fhðjðiþ1Þ Þ; s; rgh0 ðjiþ2 Þdjiþ2 ð10aÞ U Lhðiþ1Þ



~ non fhðji Þ; s; rg ¼ UMh ðs  r;s  j1 .. .; s  jiþ1 Þ U Mhi iþ2 s



þ



s 0

ð8bÞ

where the subscript ‘‘1’’ denotes step 1 and indicates that there are some limitations added to the nonlinear unit-step response functions and the variable j due to the extraction of information from the previous nonlinear unit-step response functions. Furthermore, one could extracts a part of the time-invariant nonlinear coupling effects due to three time scales from the remaining nonlinear ~ non fhðj1 Þ; s; rg unit-step response functions U and Lh1 ~ non fhðj1 Þ; s; rg by adding a unit-step function input at time j2, U Mh1 hence, a possible form of the nonlinear functional

Z



0 ~ non U Mhðiþ1Þ fhðjðiþ1Þ Þ; s; rgh ðjiþ2 Þdjiþ2 ð10bÞ

where i = 1, . . ., +1. Substituting Eqs. (7), (8) and (10) into Eq. (5), one can obtain a possible form of the nonlinear lift force and torsional moment as  Z s 1 dC L LðsÞ ¼  qU 2 2b ULh ðs  rÞh0 ðrÞdr 2 dh 0 Z sZ s ULh2 ðs  r; s  j1 Þh0 ðrÞh0 ðj1 Þdrdj1 þ 0 0  Z sZ sZ s þ UMh3 ðs  r;s  j1 ;s  j2 Þh0 ðrÞh0 ðj1 Þh0 ðj2 Þdrdj1 dj2 þ   ð11aÞ 0

0

0

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Z

s 1 2 2 dC M qU 2b UMh ðs  rÞh0 ðrÞdr 2 dh 0 Z sZ s UMh2 ðs  r;s  j1 Þh0 ðrÞh0 ðj1 Þdrdj1 þ 0 0  Z sZ sZ s þ UMh3 ðs  r; s  j1 ;s  j2 Þh0 ðrÞh0 ðj1 Þh0 ðj2 Þdrdj1 dj2 þ  ð11bÞ

MðsÞ ¼

0

0

0

A typical ‘‘peeling-an-onion’’ procedure to analyze a time-invariant nonlinear aerodynamic system is summarized in Fig. 3. It is interesting to note that based on the relationship between the unit-step response function U(s) and unit-impulse response _ ðsÞ (where d(s) is a Dirac delta function) function IðsÞ ¼ Uð0ÞdðsÞ þ U the nonlinear convolution integrals of Eq. (11) could be represented by a Volterra-type formalism Z s 1 LðsÞ ¼  qU 2 2b ILh ðs  rÞhðrÞdr 2 0 Z sZ s ILh2 ðs  r; s  j1 ÞhðrÞhðj1 Þdrdj1 þ 0 0  Z sZ sZ s þ IMh3 ðs  r;s  j1 ;s  j2 ÞhðrÞhðj1 Þhðj2 Þdrdj1 dj2 þ   0

0

4. Kernel identification scheme

ð12aÞ

0

Wind–bridge interactions have a fading memory, which suggests that the wind-induced forces on a bridge deck are not dependent on the infinite past inputs. Boyd [23] demonstrated that the finite-order truncated Volterra series will converge in the simulation of a fading memory nonlinear system. Hence, the nonlinear bridge aerodynamics could be simulated using finite terms of the Volterra series, which indicates that finite steps of the ‘‘peelingan-onion’’ procedure may be carried out. In this study, the seond-order truncated Volterra series is applied to simulate

Z s 1 2 2 qU 2b UMh ðs  rÞh0 ðrÞdr 2 0 Z sZ s UMh2 ðs  r;s  j1 Þh0 ðrÞh0 ðj1 Þdrdj1 þ 0 0  Z sZ sZ s þ UMh3 ðs  r; s  j1 ;s  j2 Þh0 ðrÞh0 ðj1 Þh0 ðj2 Þdrdj1 dj2 þ  ð12bÞ

MðsÞ ¼

0

procedure is enlightened by the ‘‘successive substitution’’ procedure, which is utilized to derive the Volterra equation of the second kind [20]. The mathematical properties of the Volterra series are discussed comprehensively by Volterra [21]. The basic premise of the Volterra theory of nonlinear systems is that a large class of nonlinear systems can be approximated as a sum of multidimensional convolution integrals of increasing order [21]. On the other hand, it should be noted that the ‘‘peeling-an-onion’’ procedure, utilized for deriving Eq. (12), suggests that there may be some information missed at each step in this procedure. Actually, it has been demonstrated that the Volterra series is a subset of the nonlinear functional expansion of the nonlinear unit-step response functions [22]. Besides, in the case of the multiple input variables, the principle of superposition is not valid in the nonlinear situation, hence, cross terms which involve coupling effects of different input variables should be added to accurately simulate nonlinear aerodynamics.

0

0

where the steady-state information is included in the unit-impulse response functions. The abovementioned ‘‘peeling-an-onion’’

14

x 10

-3

2.5

x 10

-4

1.5

6 4 2 0

1 0.5 0

-2 -4 0

200

400 600 Time step [n]

800

x 10

-4

2

2

2

8

Second order kernel h11[n1,n2]

2

10

diagonal values of h11[n,n]

1

First-order kernel h11[n]

12

-0.5 0

1000

200

400

600

800

1

0

-1 1000 Tim 500 es tep [ n]

1000

Time step [n]

1000

0 0

500 n] ep [ e st Tim

(a) Motion-induced direct-kernels -3

2

2

diagonal values of h21[n,n]

1

First-order kernel h21[n]

-5

0

0.5

0

-0.5

-1

-1.5 0

x 10

200

400 600 Time step [n]

800

1000

2

x 10

Second order kernel h21[n1,n2]

1

-2 -4 -6

-5

2 0 -2 -4

-6 1000

-8 -10 0

x 10

1000 T im

200

400

600

800

1000

es

500 tep [n]

Time step [n]

(b) Motion-induced cross-kernels Fig. 4. Identified motion-induced kernels of the nonlinear wind–bridge interaction system.

0 0

500 n] ep [ e st Tim

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nonlinear bridge aerodynamics as the first step beyond conventional linear simulation. To this end, the motion-induced nonlinear lift force and torsional moment could be expressed as

Z s 1 LðsÞ ¼  qU 2 2b ILh ðs  rÞhðrÞdr 2 0  Z sZ s þ ILh2 ðs  r; s  j1 ÞhðrÞhðj1 Þdrdj1 0

ð13aÞ

0

Z s 1 qU 2 2b2 UMh ðs  rÞh0 ðrÞdr 2 0  Z sZ s þ UMh2 ðs  r; s  j1 Þh0 ðrÞh0 ðj1 Þdrdj1

MðsÞ ¼

0

ð13bÞ

0

In order to simplify the presentation of the proposed kernel identification scheme, a typical single-input single-output (SISO) secondorder Volterra system is expressed as [24,25]

yðtÞ ¼

Z

t

h1 ðt  sÞxðsÞds þ

Z

0

0

t

Z

t

h2 ðt  s1 ; t

0

 s2 Þxðs1 Þxðs2 Þds1 ds2

ð14Þ

where x(t) and y(t) are input and output of the system, respectively; h1 represents the first-order (linear) kernel which describes the linear behavior of the system; h2 the second-order (nonlinear) kernel which represents the nonlinear behavior existing in the system. There are various approaches to identify the Volterra kernels. In this study, the identification of Volterra kernels with impulse function as inputs is introduced, which is based on the earlier work by Rugh [24]. Suppose two input signals are applied to the second-order terms of the Volterra sereis, the nonlinear response is represented as below

0.1

θ (radians)

0.05

0

-0.05

-0.1 0

2

4

6

8

10 Time (s)

12

14

16

18

20

14

16

18

20

14

16

18

20

(a) Harmonic torsional motion input x1

Torsional displacement (radians)

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 0

Exact solution Linear Approximation Nonlinear Approximation 2

4

6

8

10 Time (s)

12

(b) Torsional output y1

Vertical displacement (m)

0.02

Exact solution Linear Approximation Nonlinear Approximation

0.01

0

-0.01

-0.02 0

2

4

6

8

10 Time (s)

12

(c) Vertical output y2 Fig. 5. Motion input and corresponding outputs of the nonlinear wind–bridge interaction system.

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T. Wu, A. Kareem / Computers and Structures 128 (2013) 259–271 -3

1

-5

0.5

12 2

diagonal values of h [n,n]

1

First-order kernel h12[n]

0.5

x 10

0

-0.5

-1

-1.5 0

200

400 600 Time step [n]

800

0

-0.5

-1

-1.5 0

1000

x 10

200

400 600 Time step [n]

800

-5

1 0.5

2

x 10

Second order kernel h12[n1,n2]

1

0 -0.5 -1 -1.5 1000 Tim 500 es tep [n]

1000

1000

0 0

500 n] ep [ e st Tim

(a) Gust-induced cross-kernels -3

14

x 10

2.5

x 10

-4

Second order kernel h22[n1,n2]

12

2

8 6 4 2 0

2

diagonal values of h22[n,n]

1

First-order kernel h22[n]

2 10

1.5 1 0.5

200

400 600 Time step [n]

800

1000

-0.5 0

-5

15 10 5 0

-5 1000

0

-2 -4 0

x 10

200

400 600 Time step [n]

800

Tim e s 500 tep [n]

1000

1000

0 0

T

500 [n ] step ime

(b) Gust-induced direct-kernels Fig. 6. Identified gust-induced kernels of the nonlinear wind–bridge interaction system. (a) Gust-induced cross-kernels; (b) gust-induced direct-kernels.

y2 ½x1 ðtÞ þ x2 ðtÞ ¼

Z

1

Z

1

1

Eq. (15) is employed to calculate the second-order kernel, which could be expressed as

h2 ðs1 ; s2 Þ½x1 ðt  s1 Þ

1

þx2 ðt  s1 Þ½x1 ðt  s2 Þ þ x2 ðt  s2 Þds1 ds2 Z 1Z 1 ¼ h2 ðs1 ; s2 Þx1 ðt  s1 Þx1 ðt  s2 Þds1 ds2 1 1 Z 1 Z 1 þ h2 ðs1 ; s2 Þx1 ðt  s1 Þx2 ðt  s2 Þds1 ds2 1 Z1 Z 1 1 þ h2 ðs1 ; s2 Þx2 ðt  s1 Þx1 ðt  s2 Þds1 ds2 1 Z1 1 Z 1 þ h2 ðs1 ; s2 Þx2 ðt  s1 Þx2 ðt  s2 Þds1 ds2 1

1 ½y ½b dðt  s1 Þ þ b2 dðt  s2 Þ 2b1 b2 2 1 y2 ½b1 dðt  s1 Þ  y2 ½b2 dðt  s2 Þ

h2 ðt  s1 ; t  s2 Þ ¼

For an inhomogeneous system, the responses resulting from the first-order and higher-order terms have to be considered simultaneously. There is a typical linear relationship involving the first-order kernel with the impulse function inputs, given by

y1 ½x1 ðtÞ þ x2 ðtÞ  y1 ½x1 ðtÞ  y1 ½x2 ðtÞ ¼ 0

1

¼ y2 ½x1 ðtÞ þ y2 ½x2 ðtÞ Z 1Z 1 þ2 h2 ðs1 ; s2 Þx1 ðt  s1 Þx2 ðt  s2 Þds1 ds2 ð15Þ 1

1

where the subscript ‘‘2’’ with the output y(t) indicates the response resulting from the second-order of nonlinear origin. The symmetric property of the kernels is used here to obtain the final result. Suppose the input signals are impulse functions, i.e.

xe ðtÞ ¼ be dðt  se Þ

ð16Þ

h2 ðt  s1 ; t  s2 Þ ¼

1 ½y½b1 dðt  s1 Þ þ b2 dðt  s2 Þ 2b1 b2  y½b1 dðt  s1 Þ  y½b2 dðt  s2 Þ

Z

Hence, a system of equations is obtained as

ð17Þ

1

Z

1

1

Z

1

1

h2 ðs1 ; s2 Þdðt  s1 Þdðt  s2 Þds1 ds2 ¼ h2 ðt  s1 ; t  s2 Þ

ð18Þ

ð21Þ

Using the property of power series shared by the Volterra series, the first-order kernel could be obtained with an interpolation scheme. Suppose a new impulse function ad(t  s), where a is not equal to unit, is applied to the system, the corresponding response is

y½adðt  sÞ ¼ ah1 ðt  sÞ þ a2 h2 ðt  s; t  sÞ

1

ð20Þ

As a result, Eq. (19) could be presented as a more practical form

where e equals to 1 or 2; b1 and b2 are selected constants. Utilizing the property of the impulse function

h1 ðsÞdðt  s1 Þds1 ¼ h1 ðt  s1 Þ

ð19Þ



y½dðt  sÞ 1 1 h1 ðt  sÞ

y½adðt  sÞ ¼ a a2 h ðt  s; t  sÞ 2 Solving Eq. (23) for kernels gives

ð22Þ

ð23Þ

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T. Wu, A. Kareem / Computers and Structures 128 (2013) 259–271

h1 ðt  sÞ ¼

1  2 a y½dðt  sÞ  y½adðt  sÞ a a

ð24Þ

2

h2 ðt  s; t  sÞ ¼

1

a  a2

ðay½dðt  sÞ  y½adðt  sÞÞ

ð25Þ

With the Eqs. (21) and (24), for the case of b1 = 1, b2 = 1 and a = 2, a popular formulation used for the identification of first- and secondorder kernels is obtained as following [24]

1 h1 ðt  s1 Þ ¼ 2y½dðt  s1 Þ  y½2dðt  s1 Þ 2

ð26Þ

1 ½y½dðt  s1 Þ þ dðt  s2 Þ  y½dðt  s1 Þ 2 y½dðt  s2 Þ

h2 ðt  s1 ; t  s2 Þ ¼

ð27Þ

It should be noted that a nonlinear system usually features inputamplitude dependence if it is simulated utilizing a truncated nonlinear model. Hence, in the kernel identification of a truncated Volterra series based model, the selection of the parameters a, b1 and b2 is significantly important and is dependent on the modeled nonlinear system. This will be investigated in the following section by a numerical example. Besides, Eq. (24) indicates that the first-order kernel may be also input-amplitude dependent and could be utilized to measure the degree of discrepancy to the linear superposition principle which is only valid for a linear system. The inputamplitude dependent property of the first-order kernel also suggests that the first-order kernel for a nonlinear system may be different from the linear unit-impulse response function describing purely a linear system, while the nontrivial values of the higher-order kernels directly present the nonlinearity of a system.

Vertical wind velocity (m/s)

0.02

0.015

0.01

0.005

0 0

2

4

6

8

10

12

14

16

18

20

Time (s)

(a) White-noise vertical gust input x2 Torsional displacement (radians)

0

x 10

-3

-1

Exact solution Linear Approximation Nonlinear Approximation

-2 -3 -4 -5 -6 -7 -8 -9 0

2

4

6

8

10

12

14

16

18

20

Time (s)

(b) Torsional output y1 Vertical displacement (m)

0.07 0.06 0.05 0.04 0.03 0.02

Exact solution Linear Approximation Nonlinear Approximation

0.01 0 0

2

4

6

8

10

12

14

16

Time (s)

(c) Vertical output y2 Fig. 7. Gust input and corresponding outputs of the nonlinear wind–bridge interaction system.

18

20

267

T. Wu, A. Kareem / Computers and Structures 128 (2013) 259–271

5. Numerical example 5.1. Mathematical model Since the catastrophic failure of the Tacoma Narrows Bridge, the aerodynamic behavior of long-span bridges has been drawing remarkable attention in the engineering and science fields. Pugsley [26] made a prophetic suggestion that the investigation of wind– bridge interactions might rely on the experimental approach. Following this concept, Scanlan and his coworkers (e.g., [8]) set up a so-called semi-empirical analysis framework based on the aerodynamic coefficients which were measured from wind-tunnel tests. An alternate approach, as considered in several studies, models the wind-induced issues by investigating the interactions between the convected vortex and bridge deck (e.g., [27]). However, the semi-analytical schemes employed in such analysis fail to take into account nonlinear effects. CFD based approaches show promise to model nonlinear bridge aerodyamics, but they have their own lim-

itations at this juncture stemming from the lack of robust turbulence models for engineering applications and extensive computational demands. On the other hand, some attempts have been made to look for possible mathematical expressions, involving nonlinear considerations, for wind–bridge interactions (e.g., [28]). In this study, a nonlinear analytical model is introduced to describe the simplified nonlinear wind–bridge interactions, based on which the kernel identification scheme with impulse function inputs, the fidelity of the simulation based on nonlinear convolution integrals and the amplitude dependence of kernels are discussed. The proposed nonlinear model is not very sensitive to the initial conditions as compared to the model developed by McKenna and Tuama [28], which is given here 2

I1

d y1

dy þ c1 1 þ kðy1 þ ey2 þ e1 y21 Þ þ dy1 y2 2 dt dt þ f1 cosðy1 Þðc=gÞ½expðgðy2  j sinðy1 ÞÞÞ  expðgðy2 þ j sinðy1 ÞÞÞ ¼ x1 ðtÞ

-3

x 10

1 α=1.2 α=1.5 α=1.7 α=2 α=3 α=4 α=5 α=6 α=7 α=8 α=9 α=10

8

1

First-order kernel h11[n]

10

6 4

x 10

-3

0.5

1

12

First-order kernel h21[n]

14

ð28aÞ

2

0 α=1.2 α=1.5 α=1.7 α=2 α=3 α=4 α=5 α=6 α=7 α=8 α=9 α=10

-0.5

0

-1

-2 -4 0

200

400

600

800

1000

-1.5 0

200

Time step [n]

400

600

800

1000

Time step [n]

(a) Motion-induced first-order kernels 1

x 10

-3

14

x 10

-3

α=1.2 α=1.5 α=1.7 α=2 α=3 α=4 α=5 α=6 α=7 α=8 α=9 α=10

12 10 8

1

0 α=1.2 α=1.5 α=1.7 α=2 α=3 α=4 α=5 α=6 α=7 α=8 α=9 α=10

-0.5

-1

-1.5 0

First-order kernel h21[n]

First-order kernel h12 [n] 1

0.5

200

400 600 Time step [n]

800

1000

6 4 2 0 -2 -4 0

200

(b) Gust-induced first-order kernels Fig. 8. Amplitude dependence of first-order kernels.

400 600 Time step [n]

800

1000

268

T. Wu, A. Kareem / Computers and Structures 128 (2013) 259–271 2

d y2

þ c2

2

dy2 þ kðey1 þ y2 þ e1 y22 Þ þ dy1 y2 þ f2 ðc=gÞ dt

dt  ½expðgðy2  j sinðy1 ÞÞÞ þ expðgðy2 þ j sinðy1 ÞÞÞ

¼ kx2 ðtÞ

ð28bÞ

where y1 and y2 represent the torsional and vertical displacement, respectively; I1 and m2 are moment of inertia and mass of the bridge deck, respectively; c1 and c2 are the viscosity damping in the torsional and vertical degrees of freedom, respectively; x1(t) is motion input (namely torsional motion of the bridge deck) while x2(t) is gust input (namely vertical wind fluctuations); k, e, e1, d, f1, f2, c, g, j and k are physical constants whose values are based on the bridge structural and aerodynamic properties. Specifically, these values are tuned to mimic amplitudes and frequencies of large torsional and vertical oscillations as noted in the original report of the Tacoma Narrows disaster [29]. In order to focus on a single-input multi-output (SIMO) case, x1(t) and x2(t) are not set equal to nonzero value concurrently. It should be noted that the linear and

nonlinear unit-impulse response functions are bridge deck responses (displacements) instead of the motion-induced or gust-induced aerodynamic forces, hence, the identified kernels herein actually integrated the contributions from both the aerodynamics and structural dynamics perspectives. 5.2. Simulation results Suppose the motion input (unit-impulse function of the angle of attack) is applied to this nonlinear wind–bridge interaction system. The first-order, the diagonal term of second-order and the secondorder motion-induced kernels are shown in Fig. 4(a) and (b) for a specific group of parameters (Dt = 0.02 s). The superscripts i1i2 represent the kernel of output i1 under input i2. The identified kernel is called as direct-kernel if i1 is equal to i2 or cross-kernel if i1 is not equal to i2. As shown in the figure, the motion-induced second-order kernel is several orders of magnitude smaller than that of the motion-induced first-order kernel. Besides, both motion-induced first- and second-order impulse response functions decay rapidly

-5

10

x 10

x 10

2

-5

β1=0.2, β2=1 β1=0.5, β2=1

1

β1=0.7, β2=1

0

β1=1, β2=1

6

diagonal values of h21[n,n]

β1=1, β2=2 β1=10, β2=15

4

-1

2

β1=3, β2=7

2

diagonal values of h11[n,n]

8

2

0

-2 β1=0.2, β2=1

-3

β1=0.5, β2=1

-4

β1=0.7, β2=1

-5

β1=1, β2=1

-6

β1=3, β2=7

β1=1, β2=2

-2

β1=10, β2=15 -4 0

200

400

600

Time step [n]

8

x 10

800

-7 0

1000

200

(a) Motion-induced second-order kernels

-6

8

x 10

400

600

β1=0.2, β2=1 β1=0.5, β2=1 6

β1=0.7, β2=1

β1=0.7, β2=1 β1=1, β2=1

β1=1, β2=2 β1=10, β2=15

2

0

-2

-4 0

β1=1, β2=2

4

β1=3, β2=7

2

β1=3, β2=7

2

diagonal values of h12[n,n]

β1=1, β2=1 4

1000

-6

β1=0.5, β2=1 6

800

Time step [n]

β1=0.2, β2=1

diagonal values of h22[n,n]

m2

β1=10, β2=15 2

0

-2

200

400

600

800

1000

0

200

Time step [n]

400

600

Time step [n]

(b) Gust-induced second-order kernels Fig. 9. Amplitude dependence of second-order kernels.

800

1000

269

T. Wu, A. Kareem / Computers and Structures 128 (2013) 259–271

with time, which suggests computational efficiency of the Volterra series based reduced-order modeling for the motion-induced bridge aerodynamics. The torsional harmonic input signal (simulation of a torsional motion input) is applied to verify the identified motion-induced impulse response functions and to show the necessity of using higher-order motion-induced impulse response functions for the simulation of motion-induced responses in this nonlinear system. Fig. 5(a) presents the time history of the torsional harmonic input; Fig. 5(b) and (c) show the corresponding torsional and vertical responses, respectively. The linear approximation response in the figures indicates that the response is calculated only involving the first-order kernel; the nonlinear approximation denotes the response is obtained by utilizing both the first- and second-order kernels; the exact solution represents the response is calculated using the fourth-order Runge–Kutta scheme. As presented in Fig. 5(b), there is notable difference in the torsional output y1 (amplitude) between the linear approximation response and exact solution, while the nonlinear approximated response collapses to the exact solution. This observation suggests that the second-order kernel is necessary and sufficient to simulate the nonlinear torsional-motion-induced torsional response. On the other hand, in the case of vertical output y2, as shown in Fig. 5(c), the linear response shows notable discrepancy (both amplitude and phase) as compared to the exact solution. Although the inclusion of the second-order kernel significantly enhances the simulation accuracy of the vertical response (both amplitude and phase) for this nonlinear wind–bridge interaction system, there remains some discrepancy between the nonlinear approximation and exact solution. This observation indicates that the second-order kernel is necessary but may not be sufficient to simulate the nonlinear torsional-motion-induced vertical response of a bridge sensitive to flutter conditions. Suppose the gust input (unit-impulse function of vertical fluctuations) is applied to this nonlinear wind–bridge interaction system. The first-order, the diagonal term of second-order and the secondorder gust-induced kernels are shown in Fig. 6(a) and (b) for the same group of parameters (time interval Dt = 0.02 s). As shown in the figure, the gust-induced second-order kernel is several orders of magnitude smaller than that of the gust-induced first-order kernel. Besides, both gust-induced first- and second-order impulse response functions decay significantly rapidly with time, which suggests computational efficiency of the Volterra series based reduced-order modeling for gust-induced bridge aerodynamics. It should be noted that, although the gust-induced linear cross-ker12 21 nel h1 is close to the motion-induced linear cross-kernel h1 , they are not exactly the same.

The vertical random input signal (simulation of a vertical gust input) is applied to verify the identified gust-induced impulse response functions and to identify the need for using higher-order gust-induced impulse response functions for the simulation of gust-induced responses in this nonlinear system. Fig. 7(a) presents the time history of vertical random input; Fig. 7(b) and (c) show the corresponding torsional and vertical responses, respectively. As presented in Fig. 7(b), there is no significant difference among the linear approximation, nonlinear approximation and the exact solution, which indicates that the torsional output y1 may be appropriately simulated using a linear model. On the other hand, in the case of vertical output y2, as shown in Fig. 7(c), the linear response shows notable discrepancy as compared to the exact solution while there is insignificant difference between the nonlinear approximation response and exact solution. This suggests that the second-order kernel is necessary and sufficient to simulate the nonlinear vertical-gust-induced-vertical response. 5.3. Discussion on amplitude dependence As mentioned in the preceding discussion, due to possible contributions of the truncated terms of Volterra series to nonlinear bridge aerodynamics, the identified kernels may be amplitude dependent. The amplitude dependence of the first-order kernels is investigated using Eq. (24), where various values of a are selected. Fig. 8(a) and (b) present the motion-induced and gust-induced first-order kernels in torsional and vertical degrees of freedom, respectively. As indicated in the figures, the amplitude dependence of the first-order kernels are negligible in this study. The amplitude dependence of the second-order kernels is investigated using Eq. (21), where various values of b1 and b2 are selected. Fig. 9 (a) and (b) present the motion-induced and gustinduced second-order kernels (diagonal values) in torsional and vertical degrees of freedom, respectively. As noted from the figures, there are substantial changes among the identified second-order kernels with different values of b1 and b2. This indicates that the amplitude dependence of the second-order kernels are significant. It should be noted that, even though both the linear and nonlinear approximations present small discrepancy as compared to the exact solution, as in the case of the gust-induced torsional response, the amplitude dependence of the corresponding second-order kernels are very notable. The weak nonlinearity in bridge aerodynamics of this specific numerical example, where the second-order kernels are several orders of magnitude smaller than that of the first-order kernels, may have been the reason for this observation.

0.25

0.45

0.2

0.35

Errors in vertical DOF (%)

Errors in torsional DOF (%)

0.4

0.15

0.1

0.05

0.3 0.25 0.2 0.15 0.1 0.05

0 (0.2, 1)

(0.5, 1)

(0.7, 1)

(1, 1) β1, β2

(1, 2)

(3, 7)

(10, 15)

0 (0.2, 1)

(0.5, 1)

Fig. 10. Errors of motion-induced responses.

(0.7, 1)

(1, 1) β1, β2

(1, 2)

(3, 7)

(10, 15)

T. Wu, A. Kareem / Computers and Structures 128 (2013) 259–271

1.4

1.4

1.2

1.2 Errors in vertical DOF (%)

Errors in torsional DOF (%)

270

1 0.8 0.6 0.4 0.2 0 (0.2, 1)

1 0.8 0.6 0.4 0.2

(0.5, 1)

(0.7, 1)

(1, 1) β1, β2

(1, 2)

(3, 7)

(10, 15)

0 (0.2, 1)

(0.5, 1)

(0.7, 1)

(1, 1) β1, β2

(1, 2)

(3, 7)

(10, 15)

Fig. 11. Errors of gust-induced responses.

Fig. 10 presents the errors of the simulated nonlinear motioninduced responses with the kernels identified using various values of b1 and b2, where the error is defined as

error ¼

 X yexact ½n  yappr ½n 2 ðyexact ½nÞ2

ð29Þ

As indicated in Fig. 10, the errors of the nonlinear motion-induced torsional and vertical responses are monotonically increasing functions with the values of b1 and b2. Fig. 11 presents the errors of the simulated nonlinear gust-induced responses with the kernels identified using various values of b1 and b2. As indicated in Fig. 11, the errors of the nonlinear gust-induced torsional and vertical responses are smallest for b1 equal to b2. It should be noted that the smallest values of b1 and b2 result in the largest errors in this case, which may indicate that the magnitudes of the impulse function inputs are insufficient to excite the nonlinear bridge aerodynamics. Since the amplitude dependence of the kernels and corresponding simulation results are case sensitive, additional examples are needed for further delineation of these observations. 6. Concluding remarks The nonlinear simulation of bridge aerodynamics is comprehensively investigated within the framework of a nonlinear convolution scheme. A group of parameters, namely the translatory _ angle of pitch (or angle of attack) h and angular velocity motion h, h_ for the motion-induced effects and the fluctuation component of wind in each degree of freedom for the gust-induced effects are used as input variables for the convolution integrals. The flow separation around the bridge deck is a main physical basis for treating h_ and h as two independent variables in bridge aerodynamics. The linear convolution scheme concerning the first-order kernels for linear analysis of bridge aerodynamics is reviewed. Based on the concept of nonlinear indicial response function, the nonlinear convolution scheme involving higher-order kernels for nonlinear analysis of bridge aerodynamics is formulated with a ‘‘peelingan-onion’’ type procedure. The nonlinear convolution scheme is represented utilizing a Volterra-type formalism, hence, the convergence of its truncated expression is guaranteed. A comprehensive kernel identification scheme is developed based on the impulse function input. This also facilitates the evaluation of amplitude dependency of kernels. A numerical example of a long-span suspension bridge with vertical and torsional degrees of freedom is investigated to verify the fidelity of simulation based on the proposed nonlinear convolution scheme for nonlinear bridge aerodynamics. The mathemat-

ical model of this numerical example is designed to mimic amplitudes and frequencies of the large torsional and vertical oscillations of the Tacoma Narrows Bridge near collapse. It is shown that accurate simulations of the motion-induced torsional (direct), motion-induced vertical (cross) and gust-induced vertical (direct) responses need to include the nonlinear convolution scheme, while the gust-induced torsional (cross) response could be simulated adequately using the linear convolution scheme. The amplitude dependency of kernels is investigated based on the proposed comprehensive kernel identification scheme. Linear kernels show negligible amplitude dependence, whereas nonlinear kernels exhibit significant dependence on the amplitude. Acknowledgements The support for this project provided by the NSF Grant # CMMI 09-28282 is gratefully acknowledged. References [1] Wu T, Kareem A. Aerodynamics and Aeroelasticity of Cable-Supported Bridges: Identification of Nonlinear Features. J Eng Mech 2013. http://dx.doi.org/ 10.1061/(ASCE)EM.1943-7889.0000615. [2] Dowell EH, Ilgamov M. Studies in nonlinear aeroelasticity. New York: SpringerVerlag; 1988. [3] Diana G, Falco M, Bruni S, Cigada A, Larose GL, Damsgaard A, et al. Comparisons between wind tunnel tests on a full aeroelastic model of the proposed bridge over Stretto di Messina and numerical results. J Wind Eng Ind Aerodyn 1995;54–55:101–13. [4] Chen X, Kareem A. Aeroelastic analysis of bridges: effects of turbulence and aerodynamic nonlinearities. J Eng Mech 2003;129(8):885–95. [5] Diana G, Rocchi D, Argentini T, Muggiasca S. Aerodynamic instability of a bridge deck section model: linear and nonlinear approach to force modeling. J Wind Eng Ind Aerodyn 2010;98(6–7):363–74. [6] Wu T, Kareem A. Modeling hysteretic nonlinear behavior of bridge aerodynamics via cellular automata nested neural network. J Wind Eng Ind Aerodyn 2011;99(4):378–88. [7] Davenport AG. Buffeting of a suspension bridge by storm winds. J Struct Div ASCE 1962;88(3):233–68. [8] Scanlan RH, Tomko JJ. Airfoil and bridge deck flutter derivatives. J Soil Mech Found Div 1971;97(EM6):1717–37. [9] Wu T, Kareem A. Bridge aerodynamics and aeroelasticity: a comparison of modeling schemes. J Fluids Struct 2012 [under review]. [10] Yoshimura T, Nakamura Y. On the indicial aerodynamic moment responses of bridge deck sections. In: Proceedings of the fifth international conference on wind engineering, vol. 2. Fort Collins, CO, USA; 1979. pp. 877–885. [11] Scanlan RH, Jones NP, Singh L. Inter-relations among flutter derivatives. J Wind Eng Ind Aerodyn 1997;69–71:829–37. [12] von Kármán Th, Sears WR. Airfoil theory for non-uniform motion. J Aero Sci 1938;5(10):379–90. [13] Nixon D. Alternative methods for modeling unsteady transonic flows. In: Nixon D, editor. Unsteady transonic aerodynamics, AIAA Inc., Washington, DC; 1989. pp. 349–376. [Chapter 8]. [14] Wu T, Kareem A. Bridge aerodynamics in time domain: indicial and impulse responses. In: Proceedings of 2012 joint conference of the engineering mechanics institute and the 11th ASCE joint specialty conference on

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