高层建筑风振预测

Predicting wind-induced vibrations of high-rise buildings usingunsteady CFD and modal analysisYue Zhang a,n,Wagdi G.Habashi a,Rooh A.Khurram ba Computational Fluid Dynamics Laboratory,Departmentof Mechanical Engineering,McGill University,Montreal,Canada H3A2S6b Supercomputing Laboratory,King Abdullah University of Science and Technology,Thuwal23955-6900,Saudi Arabiaa r t i c l e i n f oArticle history:Received31March2014Received in revised form15September2014Accepted8November2014Available online25November2014Keywords:CFDWind-induced vibrationsZonal Detached-Eddy SimulationInflow turbulencegenerationModal analysisCAARC standard tall building modela b s t r a c tThis paper investigates the wind-induced vibration of the CAARC standard tall building model,viaunsteady Computational Fluid Dynamics(CFD)and a structural modal analysis.In this numericalprocedure,the natural unsteady wind in the atmospheric boundary layer is modeled with an artificialinflow turbulence generation method.Then,the turbulentflow is simulated by the second mode of afluid–structure interaction mechanism is determined by empirical functions extracted from wind tunnelexperiments.Eventually,theflow solutions and the structural responses in terms of mean and root meansquare quantities are compared with experimental measurements,over a wide range of reducedvelocities.The significance of turbulent inflow conditions and aeroelastic effects is highlighted.Thecurrent methodology provides predictions of good accuracy and can be considered as a preliminarydesign tool to evaluate the unsteady wind effects on tall buildings.&2014Elsevier Ltd.All rights reserved.1.IntroductionWith rapid developments in materials and construction tech-niques,modern high-rise buildings are built with lighter weightand lower structural damping.The high-rise buildings arethus prone to oscillations excited by unsteady wind loads.Themechanism of wind-induced vibration can be classified intodominated in a single directiontorsional direction),or develop into complex hybrid patternsunder certain wind conditions.Thus,wind-induced vibrationsare an important issue for designing tall buildings and have tobe correctly assessed in order to guarantee structural safety andoccupant comfort under strong winds.The conventional wind tunnel remains the most important toolfor the study of wind effects on buildings.The high frequency forcebalance(HFFB)method is most used in the early design period(Xie and Irwin,1998;Lam et al.,2011;Bernardini et al.,2012).However,it only works well for the fundamental modes ofvibration and it is difficult to achieve sufficient stiffness forrequired for HFPI.However,the number of taps may be limitedsuch aeroelastic models.In addition to wind tunnel techniques,computational simula-tion has become a popular and progressively reliable tool in thefluidflow putational Fluid Dynamics(CFD)simula-tions provide abundantflow information that can be visualized atany instant in time,and most importantly,solve wind-relatedwindflows over blunt bodies poses special challenges in terms ofContents lists available at ScienceDirectjournal homepage:/locate/jweiaJournal of Wind Engineeringand Industrial Aerodynamics/10.1016/j.jweia.2014.11.0080167-6105/&2014Elsevier Ltd.All rights reserved.n Corresponding author.E-mail address:yue.zhang@cfdlab.mcgill.ca(Y.Zhang).J.Wind Eng.Ind.Aerodyn.136(2015)165–179turbulence modeling.Reynolds-Averaged Navier–Stokes(RANS) turbulence models predict the time-averagedflow solutions. Several RANS turbulence models were used in the CWE applica-tions(Murakami,1998;Yang et al.,2009;Aubéet al.,2010;Lateb et al.,2013).Nevertheless,the Reynolds-averaging processfilters out thefluctuating information offlow that plays an important role in the dynamic motions of rge-Eddy Simulation(LES)is a promising unsteady turbulence modeling approach,in which large eddies of turbulence are resolved explicitly and small eddies are modeled by subgrid-scale models.LES requires afine grid inside the attached boundary layer to capture the small turbulent structures near the walls,which dramatically increases the com-putational cost for high Reynolds numberflows.While LES is a powerful research tool,it is not yet suitable for producing numerical results in a timeframe and at a cost comparable to the wind tunnel.Detached-Eddy Simulation(DES)is an increasingly popular hybrid RANS/LES technique capable of predicting,at an affordable cost,massively separated high Reynolds numberflows (Spalart et al.,1997).The principle is to treat the boundary layer by a RANS model,and other regions by a LES model.Some recent improvements of DES include Delayed DES(DDES)(Spalart et al., 2006),Zonal DES(ZDES)(Deck,2012)and so on(Shur et al.,2008; Riou et al.,2009).Previous applications of hybrid RANS/LES models in CWE are found for:surface mounted cube in atmo-sphericflow(Haupt et al.,2011),modeling of natural wind and aerodynamics of tall buildings(Zhang et al.,2013),and Fluid–Structure Interaction(FSI)simulation of wind-induced responses of high-rise building(Zhang et al.,2012).Accurate wind environment modeling is a prerequisite for determining the unsteady wind loads on the high-rise buildings. However,generation of randomflowfields,such as the inflow boundary condition,is a challenge for unsteady numerical simula-tions.It is considered as one of the three key numerical issues for LES in CWE and other CFD applications(Tamura,2008).The inflow turbulence generator needs to satisfy several important character-istics of wind,such as mean velocity,turbulence intensity,power spectral density,and spatial correlation,which may vary according to various terrain types like seashores,openfields,small towns or urban cities.Recycling methods form thefirst type of turbulence generation methods.They require an auxiliary computational domain to drive and recycle turbulence.The generated turbulence is then introduced to the main simulation domain.Liu and Pletcher(2006)reviewed the recycling methods and some success was achieved in the CWE context(Nozawa and Tamura,2002; Kataoka,2008).Nevertheless,the difficulty in controlling the turbulence characteristics and significant increase of computa-tional cost in the precursor become major drawbacks of these methods.Synthesized methods,another type of turbulence gen-eration methods,generate independently the artificial velocity field that follows prescribed turbulence features,and imposed them at the inlet of the computational domain.As no external domain is required,the computational cost is less than recycling methods.Moreover,turbulent features are easier to control by direct manipulation of formulation parameters.The challenge of synthesized methods is to satisfy the divergence-free condition for the artificial velocityfield.Several studies have attempted to solve this issue(Kondo et al.,1997;Smirnov et al.,2001;Huang et al., 2010;Yu and Bai,2014).A recent discretizing and synthesizing randomflow generation(DSRFG)method(Huang et al.,2010)is adopted in the current work to model the natural unsteady wind in the atmospheric boundary layer.It has been successfully combined with a DES approach in a previous study to generate the desired wind environment(Zhang et al.,2013).Aeroelastic effects play an important role in the wind-induced vibrations of high-rise buildings.The interaction between aero-dynamic forces and induced motions leads to amplified vibrations when the aerodynamic damping becomes negative.In FSI compu-tations,motion-induced forces are considered in an implicit manner.In the current study,however,the aerodynamic damping is explicitly addressed to represent the aeroelastic effects.Gabbai and Simiu(2010)proposed an iterative procedure to estimate the along-wind aerodynamic damping for tall buildings.A prerequisite of this method is the availability of the unsteady aerodynamic surface pressure measurement.Cao et al.(2012)summarized an empirical along-wind aerodynamic damping formula for isolated rectangular high-rise buildings,based on37wind tunnel mea-surements of aeroelastic models.Several factors were considered, including the mass density ratio,reduced velocity,structural damping ratio and aspect ratio.In the across-wind direction, building vibration is mainly induced by vortex shedding in the wakeflow.Aerodynamic damping is more likely to become negative in this case and result in unstable oscillations.Cheng et al.(2002)studied the across-wind response and aerodynamic damping of an isolated square-shaped high-rise building.They presented different empirical models in the aerodynamically stable,unstable and divergence regimes.Chen(2013)considered the across-wind aerodynamic damping as a polynomial function of the time-dependent velocity,and/or displacement of vibration,to address the nonlinear behavior.Many earlier studies were also found(e.g.Vickery and Steckley,1993;Marukawa et al.,1996; Watanabe et al.,1997).The empirical functions proposed by Cao et al.(2012)and Cheng et al.(2002)are adopted in the current simulation to determine the aerodynamic damping ratio of tall buildings in the along-wind and across-wind directions, respectively.The high-rise building investigated in the present work is the Commonwealth Advisory Aeronautical Research Council(CAARC) standard tall building model.Wind-induced vibrations of the CAARC building have been widely studied via wind tunnel experi-ments.Melbourne(1980)compared the responses of aeroelastic CAARC building model from several wind tunnel experiments and proposedfitting functions for mean and root mean square(RMS) displacements at top of building.Tanaka and Lawen(1986)carried out an aeroelastic study of the CAARC model with an extremely small geometric scale of1:1000.It was found that no particular error of response was observed.Goliger and Milford(1988) investigated the sensitivity of geometric scale and turbulence intensity on the response of the CAARC building.They found that the geometric scale had negligible influence,but turbulence intensity had a more noticeable effect.Thepmongkorn et al. (1999)conducted response measurements using a based hinged aeroelastic model able to consider coupled translational–torsional motion.The significant peak in the across-wind RMS response of the CAARC building was attributed to the vortex shedding reso-nance.Tang and Kwok(2004)conducted a comprehensive study to investigate the interference excitation mechanisms on the translational and torsional responses of the CAARC building. It was noticed that the along-wind,across-wind and torsional responses could be largely induced by the wake of an upstream interfering building.In addition,several numerical studies have also attempted to predict the wind-induced response of the CAARC building.Braun and Awruch(2009)conducted a numerical simulation on the aerodynamic and aeroelastic behavior of the CAARC building using a partitioned FSI technique.The influence of structural damping and lock-in phenomenon were demonstrated.However,the RMS quantities of response were not well predicted,especially in the across-wind direction.The possible reasons of under-prediction were attributed to the inappropriate modeling of structure, unequal natural frequencies in two principal directions and different inflow boundary conditions.Zhang et al.(2012)studied the wind-induced response of the CAARC building using aY.Zhang et al./J.Wind Eng.Ind.Aerodyn.136(2015)165–179 166two-way loosely coupled FSI.However,it was realized that the total computational cost of FSI simulation is rather high.If complete wind-excited behaviors were studied over a wide range of reduced velocities,the estimated total simulation time can take up months even on supercomputers.Therefore,a numerical strategy with a high degree of accuracy as well as affordable timeframe is desired for the assessment of wind-induced vibra-tions of tall buildings.In the present work,a cost-effective numerical strategy is proposed by solving the unsteadyflow only once,while repeatedly using the corresponding aerodynamic forces for structural dynamics analysis,over a wide range of reduced velocities ðV r¼U H=f BÞ.The wind velocity U H at the building height isfixed, and the reduced velocity is modified with the change of natural frequency f instead of U H.This strategy differs from the FSI simulations whereflow solutions are solved as many times as the structural dynamics computations are performed.Since the flow solution consumes most of the computational time,the current strategy is expected to significantly reduce cost as com-pared to FSI simulations.Theflow solutions and the aerodynamic forces are computed via a combination of an advanced inflow turbulence generator,ZDES turbulence modeling approach,and a conservativefluid-solid load transfer scheme.The dynamic motion of the structure is solved with the modal analysis,assuming that the material is linear elastic and the mode shapes are independent to each other.The aeroelastic effects are represented by empirical aerodynamic damping functions.The numerical results are com-pared with available wind tunnel measurements.Several impor-tant issues regarding the significance of turbulent inflow condition and aerodynamic damping are discussed.2.Numerical methodologies2.1.Turbulence modelingIn the wind engineering applications for blunt bodies,the accurate prediction offlow separation and reattachment is vital. The turbulence model employed in the present study is the second mode of ZDES approach(Deck,2012),which utilizes RANS and LES in the near-wall and separated regions,respectively.The ZDES formulation can be derived from the Spalart–Allmaras one-equation turbulence model by modifying the wall distance.The non-dimensional form of the Spalart–Allmaras turbulence model (Spalart and Allmaras,1992)is as follows:∂~ν∂t þu j∂~ν∂x j¼1σRe1∂∂x kνþ~νðÞ∂~ν∂x k!þC b2∂~ν∂x k∂~ν∂x k&'þC b11Àft2ÀÁ~S~νÀ11C w1f wÀC b1κf t2~ν 2þRe1ft1ΔU2ð1ÞThe terms appearing in the transport Eq.(1)are the timederivative,convective,diffusion,production,destruction,and source terms.~v is a working variable and turbulent eddy viscosity v t is calculated by multiplying~v by a near-wall function f v1,i.e. v t¼f v1~v.The details of function definitions and closure coeffi-cients are referred to the original paper(Spalart and Allmaras, 1992)and are not repeated here.In the original DES method(Spalart et al.,1997),the wall distance d is replaced with a new variable~d,which is a key parameter that controls switching from RANS to LES.The new length scale~d is determined by a minimum function of wall distance and subgrid length scale,i.e.~d¼minðd;C DESΔmaxÞwhere C DES has a constant value of0.65andΔmax¼maxðΔx;Δy;ΔzÞisthe maximum dimension of the element.Since anisotropic grids are commonly used near the walls,the length scale~d will be equal to d near the wall and switch to C DESΔmax in the far-field.It can be shown that when balancing the destruction and production terms of the Spalart–Allmaras turbulence model,the eddy viscosity~v becomes proportional to SΔ2max,which is similar to the LES Smagorinsky subgrid scale(SGS)model:v SGS p SΔ2,where S is the local deformation rate andΔis the grid spacing.The original DES can exhibit an incorrect behavior in thick boundary layers and shallow separation regions when the LES branch of DES is activated within the boundary layer.In this scenario,the grids in the boundary layer are notfine enough to resolve the Reynolds stresses.As the insufficiently resolved Reynolds stresses cannot fully replace the modeled Reynolds stresses,the eddy viscosity and therefore the modeled Reynolds stress will be reduced.This phenomenon is referred to as modeled-stress depletion(MSD)(Spalart et al.,2006).Eventually, the depleted stresses will reduce the skin friction and lead to grid-induced separation.In order tofix the problem of MSD,Spalart et al.(2006) proposed a DDES approach,in which the length scale in the turbulence model depends not only on the grid size but also on theflow solution.This modification allows a more precise delinea-tion of boundary layer and efficiently prevents premature separa-tion.In another study,Deck(2005)proposed a ZDES approach,in which RANS and DES domains are selected manually by the user before the simulation.Recently,Deck(2012)improved the ZDES method and presented an efficient generalized formulation,which aims at performing hybrid RANS/LES for both internal and external aerodynamics problems.The second mode of this ZDES method is employed in the current work.The length scale is defined as follows:~d¼dÀfdmax0;dÀC DES~ΔII DESð2Þwhere~ΔIIDES¼Δmax if f d o f d0Δvol orΔw if f d4f d0(ð3Þin which f d0is user-defined variable and chosen as f d0¼0:8 according to aflat plat boundary layer calculation(Riou et al., 2009).This formulation works in a non-zonal manner and the major difference with DDES is the definition of subgrid length scale~ΔII DES.It is argued that the subgrid length scaleΔmax¼maxðΔx;Δy;ΔzÞentering DDES is physically justified to shield the boundary layer but is definitely not a good subgrid length scale in the LES sense(Deck,2012).The use of the maximum dimension of the gridΔmax may lead to a slow delay in the formation of instabilities in the free shear layers.Chauvet et al.(2007)proposed a more efficient definition of the subgrid length scale,based on the time-dependent vorticityfieldΔw¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN2xΔyΔzþN2yΔzΔxþN2zΔxΔyqð4Þwhere N¼ω=‖ω‖is the unit vector in which components ðN x;N y;N zÞgive the orientation of the vorticityω.This definition is aimed at preventing a delayed development of instabilities in the shear layer,and a subsequent late transition to a fully turbulentflow.This new definitionΔωis adopted in our ZDES computations.The transport equation of ZDES is solved via a weak-Galerkin Finite Element Method.Stabilization terms are calculated via a variational multiscale(VMS)approach(Khurram et al.,2012).The VMS is based on a decomposition of the scalarfield into coarse (resolved)andfine(unresolved)scales.Modeling of the unre-solved scales corrects the lack of stability of the standard Galerkin formulation.The stabilization terms appear naturally and are freeY.Zhang et al./J.Wind Eng.Ind.Aerodyn.136(2015)165–179167of user-de fined coef ficients.The resulting formulation provides an effective stabilization for turbulent computations,where production-dominated effects strongly in fluence boundary layer prediction.The details of derivation are referred to the paper of Khurram et al.(2012).2.2.In flow turbulence generationThe DSRFG method (Huang et al.,2010)is adopted in this study to generate arti ficial turbulence as the inlet boundary condition.It is designed for LES as a general in flow turbulence generator able to create spatially correlated homogeneous-isotropic or inhomogeneous-anisotropic turbulence satisfying prescribed characteristics.To synthe-size an inhomogeneous and anisotropic turbulent flow field,such as the natural wind in the atmospheric boundary layer,the unsteady velocity vector field is expressed asu x ;t ðÞ¼∑k maxm ¼k 0∑m ¼k 0n ¼1p m ;n cos ~k m ;n U ~x þωm ;n t þq m ;n sin ~k m ;n U ~xþωm ;n t hi ð5Þwherep m ;n i ¼sign r m ;n i ÀÁffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4N E i k m ðÞr m ;n i ÀÁ21þr m ;n iÀÁ2v uu t ;q m ;ni ¼sign r m ;n i ÀÁffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4E i k m ðÞ11þr m ;niÀÁs :ð6ÞThe parameters in Eqs.(5)and (6)areu ¼u ;v ;w f g ;x ¼x ;y ;z ÈÉ;~x ¼x L s;~k m ;n ¼k n ;n k 0;k m ;n ¼k m ωm ;n A N 0;2πf m ÀÁ;f m ¼k m U avg ;r m ;niA N 0;1ðÞ:ð7Þr m ;niis a set of pseudo-random numbers following a normal distribution with a mean value of 0and a standard deviation of 1.E ðk m Þis the energy spectrum value and N is the sampling number for each wave number k m .L s is a length scale of the turbulence that controls the spatial correlation.U avg is the mean velocity magni-tude.k m ;nis the wave number,the distribution of which can be determined by solving the following equations:k m ;n U p m ;n ¼0k m ;nU q m ;n ¼0k m ;n ¼k m8><>:ð8ÞA major challenge of the synthesized turbulence generation methods is to satisfy the governing equations of fluid flow,which takes an important role in diminishing the nonphysical transition region between the inlet flow-field and the solution provided by the Navier –Stokes solver inside the computational domain (Smirnov et al.,2001).If the random data is not generated from the governing flow equations themselves,the velocity fluctuations may not have the organized structures of a turbulent atmospheric boundary layer,resulting in questionable wind-induced responses (Wang et al.,2013).The basic premise of synthesized methods is to satisfy the continuity equation,which can be interpreted as the divergence-free condition for incompressible flows.The atmo-spheric flow in the current problem is considered incompressible because the Mach number is very low (less than 0.05),and the variance of density is insigni ficant (varying between 1.222and 1.226kg/m 3).An attractive point of this DSRFG method is its rigorous enforcement of the divergence-free condition without any additional steps (Huang et al.,2010).The divergence of this velocity field is∇U u x ;t ðÞ¼∑k max m ¼k 0∑m ¼k 0n ¼1Àk m ;n U p m ;n k 0L s sin ~k m ;n U ~x þωm ;n t þkm ;nU q m ;n k 0L scos ~k m ;n U ~xþωm ;n t !:ð9ÞAccording to Eq.(8),k m ;nU p m ;n ¼0and k m ;nU q m ;n ¼0.Hence,the divergence of the velocity field is reduced to zero.Satisfying the moment and energy equations,however,is a more daunting task for synthesized methods,and is the subject of intensive research.The synthetized unsteady velocity field is considered as the inlet boundary condition for the numerical wind tunnel.It can be stored in a database and reused for a similar wind environment.2.3.Fluid-solid load transfer schemeThe computation of real-time vibration of a tall building needs the time history of external forces transferred to the structural nodes.In the current problem,the external forces come from the wind loads,i.e.flow pressure and shear stresses exerted on the surfaces of buildings.The accurate transfer of these loads from fluid to structure is important.In most cases,the spatial discreti-zation of fluid and solid domains is different due to disparate resolution requirements,which results in non-matching nodal positions of the fluid and solid mesh points at the interface.It is highly desirable that the load transfer schemes should be numeri-cally accurate and physically conservative.In other words,the total loads need to be completely transmitted from fluid to solid with low numerical error.The common load transfer schemes include node-projection (Farhat et al.,1998),quadrature-projection (Cebral and Löhner,1997),and common-re finement based (Jiao and Heath,2004)schemes.Jaiman et al.(2005)reviewed and assessed these load transfer schemes in terms of accuracy and computational cost,via several two-dimensional fluid-solid interaction problems.It was shown that the common-re finement based scheme exhib-ited generally lower numerical error with negligibly higher cost.However,an additional dif ficulty of this scheme was highlighted for three-dimensional problems when creating a common re fine-ment surface.In the present study,a conservative quadrature-project scheme (Cebral and Löhner,1997)is adopted,due to its simple implementation for three-dimensional simulations and relatively low numerical error.Suppose t s ðx Þand t f ðx Þare the traction fields (including pressure and shear stresses)over the solid and fluid surface interface boundary Γ,respectively.The traction fields can be approximated by the summation of nodal values multiplied by the shape functions t s x ðÞ%∑n sj ¼1N j s x ðÞ~t j s ;t f x ðÞ%∑n fj ¼1N j fx ðÞ~t j f :ð10ÞTo minimize the residual of t s Àt f ,the problem is solved withGalerkin weighted residual method by multiplying a set of weighting functions (same as structural shape functions,i.e.W i ¼N i s )and integrating over the surface interface Γ,as follows:∬ΓN i s x ðÞt s x ðÞd Γ¼∬ΓN i s x ðÞt f x ðÞd Γð11ÞThis equality can be discretized into a linear system and expressed in the matrix form M ij s h i ~t s ÈɼR s f g ð12ÞwhereM ij s h i¼∬ΓN i s x ðÞN js x ðÞd Γð13ÞR s f g ¼∬ΓN i s x ðÞt f x ðÞd Γð14ÞThe right hand side vector can be calculated via looping over the fluid surface elements and integrating over the GaussianY.Zhang et al./J.Wind Eng.Ind.Aerodyn.136(2015)165–179168quadrature points in those elementsR i s ¼∑f luid ∑n g ¼1N is x g ÀÁAW g ðÞt f x gÀÁð15Þwhere N i s ðx g Þis the shape function of solid surface elementevaluated at the quadrature points,A is the area of the fluid surface element,W ðg Þis the weight of the quadrature points,and t f ðx g Þis the traction vector evaluated at the quadrature points.One way to enhance the accuracy is to utilize higher-order Gaussian integration by placing more points on the fluid surface.In this study,seven Gaussian points are scattered over each fluid surface element.At the interface,the fluid surface is discretized with three-node triangular elements and the solid surface is discretized with four-node quadrilateral elements.The load transfer procedure is illu-strated in Fig.1and the procedure is summarized as follows:(1)Loop over all fluid surface elements at the interface and findthe matching solid surface element.In some cases,the fluid surface element cannot be entirely immersed in the solid surface element.An accurate way to handle this issue is to recursively subdivide the fluid surface element into smaller ones and introduce additional nodes.However,the in fluence of overlaying can be neglected since the size of fluid surface elements are considerably small compared to the solid surface elements.(2)Calculate the matrix ½M ij s and the right hand side vector f R s g ofthe local fluid surface element,and solve the linear system.The contribution of the local fluid surface element to thecorresponding solid nodal traction vector f ~ts g is obtained.(3)On each solid node,summarize all traction vectors f ~ts g from various fluid elements to obtain the total nodal forces.2.4.Structural modeling and modal analysisThe structure of the CAARC standard tall building model is discretized and modeled with three-dimensional two-node frame elements,with six degrees-of-freedom at each node.The basic assumption for this element is that structural material is linearly elastic.Nonlinear behavior due to large displacements or rotations is not considered.The governing equation of the discretized structural dynamic problem isM ½ €y t ðÞÈÉþC ½ _y t ðÞÈÉþK ½ y t ðÞÈɼF t ðÞÈÉð16Þwhere ½M is the mass matrix,½C is the damping matrix,½K is the stiffness matrix,f F ðt Þg is the unsteady wind load vector from the flow solution,and f y ðt Þg is the general displacement vector including translational displacement as well as the rotational angles at each node.The dimension of matrices ½M ,½C ,and ½K is n Ân ,where n is the number of total degree-of-freedom.The gravity forces are omitted in this computation.The first step of the modal analysis is to express the displace-ment vector f y ðt Þg by a linear combination of mode shape matrix ½Φ and generalized non-dimensional displacement vector f q ðt Þg ,as followsy t ðÞÈÉ¼Φ½ q t ðÞÈÉ;ð17Þin which ½Φ ¼½Φ1;Φ2;:::;Φm with m being the total numberof mode shapes involved in this computation.Multiplying the transpose matrix ½Φ T on both sides of the governing equation,the new form becomes M ÂÀq t ðÞÈÉþC h i _q t ðÞÈÉþK ÂÃq t ðÞÈɼQ t ðÞÈÉ;ð18Þwith generalized matrices ½M ¼½Φ T ½M ½Φ ,½C ¼½Φ T ½C ½Φ ,½K ¼½Φ T ½K ½Φ ,and generalized load vector ½Q ðt Þ ¼½Φ T f F ðt Þg .The generalized matrices are diagonal matrices by virtue of the linearly independent property of the mode shapes.Therefore,this system can be decoupled into a series of ordinary differential equations €q j þ2ξj ωj _q j þω2j q j ¼f j t ðÞ;j ¼1;2;…;m ;ð19Þwhere ξj is the critical damping ratio that includes structural andaerodynamic damping,and ωj ¼2πn j is the natural angular frequencies of structure.Mathematically,square of the natural angular frequencies ω2j and mode shape vector f ϕj g represent eigenvalues and eigenvectors of the system,respectively.TheycanFig.1.Illustration of the quadrature-projectscheme.putational domain.Y.Zhang et al./J.Wind Eng.Ind.Aerodyn.136(2015)165–179169。