DOUBLE MODAL TRANSFORMATION AND WIND ENGINEERING APPLICATIONS

Modal transformation techniques are usually adopted in structural dynamics with the aim of de- coupling the equations of motion. They are based on the search for an abstract space in which the solution of the problem results simplified. Analogous transformation techniques have recently been developed with the aim of defining a space where a multivariate stochastic process is expressed by a linear combination of one-variate uncorrelated processes. This paper proposes a method, called double modal transformation, by which the dynamic analysis of a linear structure is carried out through the simultaneous transformation of the equations of motion and the loading process. By adopting this technique, the structural response is obtained through a double series expansion in which structural and loading modal contributions are superimposed. Its effectiveness and application are discussed with reference to two classic wind engineering problems—the alongwind response and the vortex- induced crosswind response of slender structures—which provide a wide panorama of the most relevant prop- erties of this procedure. The dynamic analysis of structures is generally carried out by transforming the equations of motion from the initial La- grangian space into a new space characterized by particular properties. The choice of the transformation is usually driven by the necessity of reducing the computational size of the problem and by the opportunity of representing the system selecting a set of parameters with a suitable mechanical mean- ing. The classic modal analysis (Hurty and Rubinstein 1964) performs both these tasks by defining a principal space in which the motion of the structure can be represented through a limited number of principal coordinates. Under suitable con- ditions concerning the damping (Caughey 1960), moreover, such coordinates are orthogonal in a mechanical sense, i.e., are governed by decoupled equations. Load terms do not take part in the definition of the transformation law and assume, in the new space, a pure mathematical meaning. If the load is a mul- tivariate stochastic process, as happens for the wind, its pro- jection on the transformed space is numerically very burden- some. The loading process may be transformed following analo- gous principles of the modal analysis by the proper orthogonal

[1]  T. Caughey,et al.  Classical Normal Modes in Damped Linear Dynamic Systems , 1960 .

[2]  Giovanni Solari,et al.  Gust-Excited Vibrations , 1994 .

[3]  Ahsan Kareem,et al.  Simulation of Multivariate Random Processes: Hybrid DFT and Digital Filtering Approach , 1993 .

[4]  Ahsan Kareem,et al.  On Stochastic Decomposition and Its Applications in Probabilistic Dynamics , 1990 .

[5]  A. Pisano,et al.  Multivariate stochastic wave generation , 1996 .

[6]  Ahsan Kareem,et al.  Simulation of Multivariate Nonstationary Random Processes: Hybrid DFT and Digital Filtering Approach , 1997 .

[7]  M. D. Paola,et al.  Digital simulation of wind field velocity , 1998 .

[8]  I. Elishakoff,et al.  Probabilistic Methods in the Theory of Structures , 1984 .

[9]  Ira B. Schwartz,et al.  Dynamics of Large Scale Coupled Structural/ Mechanical Systems: A Singular Perturbation/ Proper Orthogonal Decomposition Approach , 1999, SIAM J. Appl. Math..

[10]  K. McKee.  Book Reviews : Wind Effects On Structures: E. Simiu and R.H. Scanlan John Wiley & Sons, New York, New York 1980, $33.50 , 1981 .

[11]  B. Lee.  The effect of turbulence on the surface pressure field of a square prism , 1975, Journal of Fluid Mechanics.

[12]  Giuseppe Rega,et al.  Characterizing bifurcations and classes of motion in the transition to chaos through 3D-tori of a continuous experimental system in solid mechanics , 2000 .

[13]  Giovanni Solari,et al.  Generalized equivalent spectrum technique , 1998 .

[14]  B. J. Vickery,et al.  Lift or Across-Wind Response to Tapered Stacks , 1972 .

[15]  Ahsan Kareem,et al.  PRESSURE FLUCTUATIONS ON A SQUARE BUILDING MODEL IN BOUNDARY-LAYER FLOWS , 1984 .

[16]  Michel Loève,et al.  Probability Theory I , 1977 .

[17]  Jagmohan Humar,et al.  Dynamic of structures , 1989 .

[18]  J. D. Holmes,et al.  Use of eigenvalues in the covariance integration method for determination of wind load effects , 1983 .

[19]  G. Solari,et al.  Gust buffeting. I: Peak wind velocity and equivalent pressure; II: Dynamic alongwind response. , 1993 .

[20]  Giovanni Solari,et al.  3D Wind-Excited Response of Slender Structures: Closed-Form Solution , 2000 .

[21]  Ahsan Kareem,et al.  Simulation of Multivariate Nonstationary Random Processes by FFT , 1991 .

[22]  J. Lumley.  Stochastic tools in turbulence , 1970 .

[23]  E. Vanmarcke.  Properties of Spectral Moments with Applications to Random Vibration , 1972 .

[24]  C. Scruton,et al.  Wind Effects on Structures , 1970 .

[25]  Giovanni Solari,et al.  Mathematical model to predict 3-D wind loading on buildings , 1985 .

[26]  L. Carassale,et al.  Modal transformation tools in structural dynamics and wind engineering , 2000 .