The document discusses the double modal transformation technique for analyzing the dynamic response of linear structures subjected to stochastic loading processes. This technique transforms both the structural equations of motion and the loading process to decouple them. The method is demonstrated on two classic wind engineering problems: alongwind and vortex-induced crosswind response of slender structures.

1. DOUBLE MODAL TRANSFORMATION AND WIND<br />ENGINEERING APPLICATIONS<br />By Luigi Carassale,1 Giuseppe Piccardo,2 and<br />Giovanni Solari,3<br />Modal transformation techniques are usually adopted in structural dynamics with the aim of decoupling<br />the equations of motion. They are based on the search for an abstract space in which the solution of<br />the problem results simplified. Analogous transformation techniques have recently been developed with the aim<br />of defining a space where a multivariate stochastic process is expressed by a linear combination of one-variate<br />uncorrelated processes. This paper proposes a method, called double modal transformation, by which the dynamic<br />analysis of a linear structure is carried out through the simultaneous transformation of the equations of motion<br />and the loading process. By adopting this technique, the structural response is obtained through a double series<br />expansion in which structural and loading modal contributions are superimposed. Its effectiveness and application<br />are discussed with reference to two classic wind engineering problems—the alongwind response and the vortexinduced<br />crosswind response of slender structures—which provide a wide panorama of the most relevant properties<br />of this procedure.<br />Wind Modes for Structural Dynamics: A Continuous Approach<br />L. Carassale, G. Solari<br />Load on structural systems is often represented by a multi-dimensional and/or multi-variate random process. The cross-correlation often existing between loading components acting in different points of the structure introduces conceptual and computational difficulties in many practical problems. It is the case, for example, of the projection of the external load on the vibration modes in the modal analysis of linear systems or of the simulation of multi-correlated time series for a Monte Carlo-based analysis of nonlinear structures. The use of the Proper Orthogonal Decomposition (POD) introduces some formal simplifications in the solution of the aforementioned problems, but requires the evaluation of the eigenquantities of some statistical representations of the loading process. The knowledge of such quantities in analytic form yields computational advantages and enables important physical interpretations. In the present paper, an analytic expression of POD is developed for a class of processes, which includes models usually adopted to represent the atmospheric turbulence. Examples of linear analysis of a wind-excited slender structure and of simulation of turbulence fields are presented.<br /> MACROBUTTON MTEditEquationSection Equation Section 1 SEQ MTEqn MERGEFORMAT SEQ MTSec 1 MERGEFORMAT POD-Based Filters for the Representation of Random Loads on Structures<br />Luigi Carassale<br />Random loads on structures are often represented as stationary Gaussian multi-variate random processes, whose components are correlated with each other according to characteristic patterns. Numerous techniques usually applied in structural dynamics require having the input constituted by a vector of independent random processes or even white noises. The application of such techniques in many practical contexts requires the introduction of suitable pre-filters to reproduce the exact correlation and harmonic content of the real external excitation. The present paper describes the development and the implementation in the time and in the frequency domain of digital filters based on the Proper Orthogonal Decomposition (POD). Two different approximated approaches are discussed and verified through numerical examples.<br /> MACROBUTTON MTEditEquationSection Equation Section 1 SEQ MTEqn MERGEFORMAT SEQ MTSec 1 MERGEFORMAT Aeroelastic Forces on Yawed Circular Cylinders:Quasi-steady Modeling and Aerodynamic Instability<br />Luigi Carassale*, Andrea Freda, Giuseppe Piccardo<br />Quasi-steady approaches have been often adopted to model wind forces on moving cylinders in cross-flow and to study instability conditions of rigid cylinders supported by visco-elastic devices. Recently, much attention has been devoted to the experimental study of inclined and/or yawed circular cylinders detecting dynamical phenomena such as galloping-like instability, but, at the present state-of-the-art, no mathematical model is able to recognize or predict satisfactorily this behaviour. The present paper presents a generalization of the quasi-steady approach for the definition of the flow-induced forces on yawed and inclined circular cylinders. The proposed model is able to replicate experimental behaviour and to predict the galloping instability observed during a series of recent wind-tunnel tests.<br />(a) (b)<br />Monte Carlo Simulation of Wind Velocity Fields on Complex Structures<br />Luigi Carassale* and Giovanni Solari<br />Monte Carlo simulation is becoming a fundamental tool for the design of complex and important wind-excited structures. A common application regards the time-domain dynamic analysis of multi-dof nonlinear structures whose excitation is calculated on the base of simulated wind velocity time-histories. The present paper describes a methodology for the simulation of wind velocity fields over large domains, possibly in zones characterised by complex topography. The modelling of turbulence in non-homogeneous flow condition and some computational aspects related to its simulation are discussed, proposing some strategies for reducing the calculation time. The simulation procedure is applied to the case of the Messina Strait bridge for which the three components of turbulence are simulated over a domain composed by 351 nodes.<br />a.1b.1a.2b.2a.3b.3a.4b.4<br />Proper Orthogonal Decomposition in Wind Engineering.<br />Part 1: A State-of-the-Art and Some Prospects<br />Giovanni Solari, Luigi Carassale and Federica Tubino<br />The Proper Orthogonal Decomposition (POD) is a statistical method particularly suitable and versatile for dealing with many problems concerning wind engineering and several other scientific and humanist fields. POD represents a random process as a linear combination of deterministic functions, the POD modes, modulated by uncorrelated random coefficients, the principal components. It owes its popularity to the property that only few terms of the series are usually needed to capture the most energetic coherent structures of the process, and a link often exists between each dominant mode and the main mechanisms of the phenomenon. For this reason, POD modes are normally used to identify low-dimensional subspaces appropriate for the construction of reduced models. This paper provides a state-of-the-art and some prospects on POD, with special regard to its framework and applications in wind engineering. A wide bibliography is also reported.<br />Proper Orthogonal Decomposition in Wind Engineering.<br />Part 2: Theoretical Aspects and Some Applications<br />Luigi Carassale, Giovanni Solari and Federica Tubino<br />Few mathematical methods attracted theoretical and applied researches, both in the scientific and humanist fields, as the Proper Orthogonal Decomposition (POD) made throughout the last century. However, most of these fields often developed POD in autonomous ways and with different names, discovering more and more times what other scholars already knew in different sectors. This situation originated a broad band of methods and applications, whose collation requires working out a comprehensive viewpoint on the representation problem for random quantities.<br />Based on these premises, this paper provides and discusses the theoretical foundations of POD in a homogeneous framework, emphasising the link between its general position and formulation and its prevalent use in wind engineering. Referring to this framework, some applications recently developed at the University of Genoa are shown and revised. General remarks and some prospects are finally drawn.<br />