ing Volterra system (up to the 3 rd order) is carried out adopting the topological assemblage scheme proposed in [3] for scalar systems and generalize

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1 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September -6, Non-linear buffeting and flutter analysis of bridges: a frequency domain approach uigi Carassale a, Teng Wu b, Ahsan Kareem b a University of enova, via Montallegro enova, Italy b University of Notre Dame, 56 Fitzpatrick Hall, Notre Dame, USA ABSTRACT: A frequency domain approach, hich is based on the Volterra series expansion, for non-linear bridge aerodynamics is proposed in this paper. The Volterra frequency-response functions (VFRFs) are constructed utilizing the topological assemblage scheme and identified through a full-time-domain non-linear bridge aerodynamic analysis frameork. A todimensional numerical example of a long-span bridge is presented. The results sho good comparison beteen the time domain simulations and the proposed frequency-domain model for non-linear bridge aerodynamics. KEYWORDS: Non-linearity; Bridge; Buffeting; Flutter; Volterra series INTRODUCTION Wind-induced forces on bridges are traditionally represented as the sum of buffeting forces, related to the incoming ind velocity fluctuation, and the self-excited forces, generated by the bridge motion. Both these forces are modelled by linear operators (ith memory) hose Frequency Response Functions (FRF) are estimated through specific ind-tunnel tests. On the other hand, hen a fluid-structure interaction problem is characterized by high reduced velocity (i.e. the structure time scale is much sloer than the fluid time scale) the quasi-steady assumption is invoked and the ind action is represented as a non-linear memory-less transformation of the incoming ind velocity and structural motion, hich are often combined in the so-called effective angle of attack. Several experimental experiences suggest that, in some cases, both the mentioned approaches may be inadequate due to the simultaneous presence of both significant nonlinearities and memory effects. With a specific reference to the case of long-span bridges, some attempts to fill this gap have been made adopting a band-superposition approach in hich the quasi-steady model is used to represent the lo-frequency part of the forces, hile the high-frequency part (both buffeting and self-excited) is modelled through the usual linear models ith memory, hose parameters are updated according to the instantaneous lo-frequency effective angle of attack []. Folloing this formulation, the high-frequency response is provided by a differential equation hose parameters depend on both time and frequency. To solve this problem ith more efficient computational procedures, Chen & Kareem [] transformed the problem into a full-time-domain formulation by means of a rational-function approximation and solved the equations of motion in the time domain through an integrate statespace approach. This solution, though mathematically rigorous, consists essentially of a calculation procedure and not a model, thus does not assist in the qualitative assessment of the problem. In order to circumvent the limitations of the to above approaches, the concept of harmonicband superposition is revised proposing a full-frequency-domain approach based on the Volterra series expansion of the dynamical systems representing both buffeting and self-excited forces. To this purpose, the governing equations (e.g. []) are re-casted into a block-diagram format (Fig. ) invoking, henever necessary, polynomial approximations. The synthesis of the result- 453

2 ing Volterra system (up to the 3 rd order) is carried out adopting the topological assemblage scheme proposed in [3] for scalar systems and generalized in [4] to the case of multi-variate systems. Section provides some background on Volterra series, ith particular reference to its multivariate form [4]; Section 3 describes the non-linear aeroelastic bridge model assumed as reference; Section 4 shos the synthesis of its Volterra series approximation; Section 5 shos its numerical application for the dynamic analysis of a long-span bridge. VOTERRA SERIES: THEORETICA BACKROUND et us consider the nonlinear system represented by the folloing equation: x() t H u() t () u(t) and x(t) being vectors ith size n and m, respectively, representing the input and the output; t is the time. If the operator H [] is time-invariant and has finite-memory, its output x(t) can be expressed, far enough from the initial conditions, through the Volterra series expansion [e.g. 3, 4]: t t x h τ u dτ r τ r here = [ ] T is a vector containing the integration variables; the functions h have values in mn and are called Volterra kernels. The product operator is interpreted as a sequence of Kronecker products, i.e.: r tr t t u u u (3) The th -order term of the Volterra series, h, is a constant independent of the input; the st - order term is the convolution integral typical of the linear dynamical systems, ith h being the impulse response function. The higher-order terms are multiple convolutions involving products of the input values for different time delays. A Volterra system is entirely determined by its constant output and its Volterra kernels. An alternative representation is provided, in the frequency domain, by the Volterra frequencyresponse functions (VFRF), the multi-dimensional Fourier transforms of the Volterra kernels: T iω τ e H Ω h τ dτ τ here =[ ] T is a vector containing the circular frequency values corresponding to,, in the Fourier transform pair. The VFRFs are functions ith values in mn. () (4) 3 NON-INEAR AEROEASTIC BRIDE MODE In this section the non-linear model for the calculation of the aeroelastic response of bridges described in [] is briefly recalled. For simplicity, the original model is reduced disregarding the effect of the longitudinal turbulence and of the say degree of freedom; besides, the aerodynamic load is applied to the bridge deck according to the strip theory. These quite strict hypotheses 454

3 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September -6, could be easily relaxed and are here adopted only because the added formal complexity introduced by a more sophisticated bridge model may hide the conceptual structure of the proposed technique. According to this formulation, the dynamic response of the bridge is provided by the differential equation: Mx Cx Kx f f f (5) H se here x = [h, ] T is the displacement vector containing the heave displacement h (positive donards) and the torsional rotation (positive nose up), as shon in Fig. ; the response x is divided into a lo-frequency component x = [h, ] T and a high-frequency component x H = [h H, H ] T, separated by the frequency n c. M, C and K are the mass, viscous damping and stiffness matrixes, respectively; f, f H and f se are, respectively, the lo-frequency force modeled through a quasi-steady non-linear approach, the high-frequency buffeting force and the selfexcited force. U M h D Figure. Bridge deck, incoming ind velocity, generalized forces and displacements. The lo-frequency force is defined as: f BlVr e M R C (6) here is the air density, B the deck idth, l the length of the strip on hich the load is applied, V r is the ind-structure relative velocity expressed as: V U h mb (7) r here U is the mean ind velocity, is the lo-frequency vertical turbulence (i.e. the vertical turbulence,, lo-pass filtered at the frequency n = n c ); m B is the leg for the reduction of the apparent velocity field generated by the torsional velocity; R is a rotation matrix defined as sin cos R (8) here is the apparent (lo-frequency) ind angle h mb U arctan The vector C = [C D, -C, BC M ] T contains the aerodynamic static coefficients, estimated for the effective angle of attack e = through static ind tunnel tests. The high-frequency buffeting force is modeled as: (9) 455

4 fh UBlB H; e () Where H is the high-frequency vertical turbulence (i.e. high-pass filtered at n = n c ) and B is a linear operator hose Frequency-Response Function (FRF) depends of the lo-frequency response through the effective angle of attack e and is expressed as C B CD e e ; e ; e C MeM ; () e here C and C M are the prime derivatives of C and C M, respectively; and M are the admittance functions eighting the effect of the vertical turbulence on the lift force and torsional moment, respectively; = n is the circular frequency. Since e is variable in time due to and x, the operator B may be interpreted as a time-variant linear operator. The self-excited forces are defined through the model: fse BlA xh; e () here A is a linear operator hose FRF depends on e and is defined as * * * * 4 ; ei ; e 3 ; ei ; e * * * * 4 ; ei ; e 3 ; ei ; e H k H k B H k H k A ; e (3) B A k A k B A k A k here H * (k, ) and A * (k, ) are the flutter derivatives estimated at the reduced frequency k = B/U and for a mean angle of attack corresponding to e. ikeise the operator B, also A is linear, but is time-variant because of the dependency on e. 4 SYNTHESIS OF A 3 RD -ORDER VOTERRA SYSTEM In this section, the model described above is synthesized into a 3 rd -order Volterra system hose FRFs are calculated adopting the topological assemblage scheme described in [3, 4]. Figure shos a block diagram of the hole system. The input is separated into and H through the lo-pass filter P and the high-pass filters P H, hose FRFs are P () and P H (), respectively. The output x is generated by the sum x and x H obtained, respectively, as the outputs of the lofrequency and high-frequency stages of the system. A channel delivers the effective angle of attack e from the lo-frequency stage of the system to its high-frequency stage. The rectangular boxes represent operators defined through a constitutive equation; the boxes ith rounded corners are operators defined through experimental data; the triangles are gain blocks; b = [ ]; b = [ m B]. The operator D represents the left-hand side of Eq. (5) and its FRF is D() = - M ic K. The operators A and B (=,,) are linear operators hose FRFs, A and B, are obtained by a polynomial approximation of A and B given by Eq. () and (3) e e e e ; ; ; A A B B (4) A and B ( =,..,) are the FRFs of the time-invariant linear operators A and B and represented in Figure. et us assume that the lo-frequency and the high-frequency stages of the system can be represented by the operators 456

5 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September -6, x ; x H (5) H and assume that they can be expanded into convergent Volterra series, at least for the range of input amplitude relevant for the application. U P P H H d dt x b x b atan U e R C r c V r Bl High freq. H B buffeting y o freq. quasi-steady f D - x x e e H e H x e H B B A BlU Self-excited f H D - x H x e H A B l f se x H A Figure. Block diagram of the nonlinear aeroelastic bridge model. (a) x d dt P x b atan U (b) x b E e (c) E e R C r c Y y (d) x d dt x b U V V r (d) V Y V r y Bl F f Figure 3. Block diagram of the sub-systems composing the lo-frequency stage. 457

6 4. o-frequency stage Due to its topology, the identification of the system must start from the synthesis of the components of. et be the operator providing given, described by the block diagram reported in Figure 3a. Its VFRFs can be expressed as the sum of a direct term (d) that does not depends on and a feedback term (f) that is linear in [3, 4]. d d P U 3 3 i,,3 U d Ω P i b r r r 3U r r f Ω Ω bω here =. Figure 3b shos the block diagram of the operator E providing the effective angle of attack e. Its VFRFs E = E (d) E (f) are given as: E b E E d d Ω Ω f f Ω Ω b Ω,, 3 here is the angle of attack obtained for =. Figure 3c shos the block diagram of the operator Y providing the vector y = R()C( e ). The matrix R and the vector C are approximated by 3 rd -order polynomials of and e, respectively. 3 3 ; R R C C (8) e e The VFRFs of Y are then in the form: Y R C d d d Y RC R C d d d Y Ω RC Ω R C R ΩC d d d Y 3 Ω3 RC 3 Ω3 R, C 3 R C, 3 R3 Ω3C f f f Y Ω RC Ω R Ω C,, 3 (6) (7) (9) here R R R are given as ( d) ( f ) 458

7 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September -6, R R R d d R d R Ω R d d, R Ω R Ω R R f f,, 3 R Ω R Ω () ( d) ( f ) The VFRFs C C C can be obtained through Eq. () by replacing R ith C and ith E. Figure 3d shos the block diagram of the operator V providing V r. Its VFRFs are given as V U V V P P P V Ω b b b Ω i P b, b b, Figure 3e shos the operator F providing the lo-frequency force f. Its VFRFs F = F (d) F (f) are given as: F BlVY d d F BlVY d d F Ω Bl VY ΩVΩY () d d F 3 Ω3 Bl VY3 Ω3V, Y3V 3Ω3Y f f F Ω BlVY Ω,, 3 The VFRFs of the operator providing the lo-frequency response can be obtained by equating the VFRFs of the operators D[ []] and F, leading to the equations: D F D Ω Ω F Ω F Ω f d,,3 The th -order equation is non-linear due to the dependency of the polynomial approximation of the aerodynamic coefficients from the (Eq. (8)). The other equations are linear in (F (f) is linear in ) and contain loer-order solutions ( k ith k < ) in the term F (d). 4. High-frequency stage The synthesis of the high-frequency stage of the system requires the identification of the operators A and B providing, respectively the high-frequency buffeting force f H and the selfexcited force f se (Figure 4). Their VFRFs, respectively, A A A ( d) ( f ) and B are given as () (3) 459

8 A d A d E A Ω A Ω H d E,, E E E A Ω A Ω H H A Ω H f,, 3 A Ω A Ω H Ω (4) B B B Ω B Ω B Ω B Ω B Ω B PH PH E P E, P E E H 3 3 H 3 The FRFs H of H can be obtained by equating, order by order, the VFRFs of the systems D[H []] and A B, leading to the equations: f d,, 3 D Ω H Ω A Ω A Ω B Ω (6) ( ) hich can be solved ith respect to H since A f is linear in H. It can be observed that the th - order high-frequency response is influenced by the lo-frequency response up to the order - through the VFRFs E k (k =,..,-). (5) H E e x H e x H x x H e H A A A BlU A f se P H E e H H e H e H B B B B l B f H Figure 4. High-frequency stage. 5 NUMERICA APPICATION The accuracy of the Volterra series representation of the non-linear aeroelastic bridge model described in Section 3 is demonstrated through the calculation of the response of a long-span bridge. The static coefficients of the cross section are measured for angle of attack beteen - and ith step. These data are approximated through 3 rd -order polynomials through a mean-square error minimization procedure [3] (Figure 5a). The admittance function is modeled 46

9 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September -6, through the Sears function and is independent of the angle of attack. The flutter derivatives are estimated for = -3,, 3 and are interpolated, for each reduced velocity value, through a nd - order polynomial (Figure 5b). The natural frequency of the to considered modes is n =.9 Hz and n =.53 Hz. The frequency value dividing lo and high-frequency components is n c =. Hz. Figure 6 shos the lo-frequency response calculated through a Volterra model ith order from to 3, compared ith the exact non-linear response calculated by a standard ODE solver. The result of the 3 rd -order Volterra series appears quite accurate, in particular if compared to the linear model ( st -order Volterra series) hich fails in reproducing the torsional response mainly due to its implicit symmetry. Figure 7 shos the high-frequency response calculated through a Volterra model ith order from to 3, compared ith the solution obtained by the time-domain procedure proposed in []. Also for this response component the 3 rd -order Volterra series approximation provides a response very close to the time-domain solution...8. C D.4 C D, C M C M C C (a) (b) Figure 5. Static aerodynamic coefficients (a); flutter derivative H * function of the mean angle of attack..8 h (m) st order nd order 3 rd order Exact t (s) Figure 6. o-frequency response. 46

10 h H (m) st order nd order 3 rd order Exact H t (s) Figure 7. High-frequency response. 6 CONCUSIONS The proposed model is essentially the translation of an existing analysis procedure to the frequency domain, and therefore it inherits all its conceptual advantages and limitations (in particular, it is based on the hypothesis of small high-frequency oscillation of the bridge around a slo, possibly large, variation of the effective angle of attack). The added value of this ne formulation is mostly related to its potential for deriving qualitative interpretations through this model. Besides, the functional structure of the proposed model may serve as a scaffolding to construct a fully-nonlinear model based on the Volterra series representation, to be identified from ad-hoc experimental procedures. From a computational point of vie, the implementation of the model in the frameork of the Volterra series enables the formulation of a very efficient frequency-domain solution based on the concept of Associated inear Equations (AE). Accordingly, the model defined herein can be implemented through six linear frequency-domain equations that can be conveniently solvedin a cascade manner. ACKNOWEDEMENT The support of this proect as made possible by NSF rant # CMMI 9 88 REFERENCES.Diana, M. Falco, S. Bruni, A. Cigada,.. arose, A. Damsgaard, A. Collina, Comparisons beteen ind tunnel tests on a full aeroelastic model of the proposed bridge over Stretto di Messina and numerical results. J. Ind. Aerodyn., (995), -3. X. Chen, A. Kareem, Aeroelastic analysis of bridges: effects of turbulence and aerodynamic nonlinearities, J. Engrg. Mech. ASCE, 9 (3), Carassale, A. Kareem, Modeling nonlinear systems by Volterra Series, J. Engrg. Mech. ASCE, 36 (), Carassale, A. Kareem, Synthesis of multi-input Volterra systems by a topological assemblage scheme, In Proc. Conf. on Non-linear Dynamics and Diagnosis, Marrakech, Morrocco,. 46