Turbulence of the incoming wind field is of paramount
importance to the dynamic response of civil engineering structures. Hence reliable stochastic models of the turbulence should be available from which time series can be generated for dynamic response and
structural safety analysis. In the paper an empirical cross spectral
density function for the along-wind turbulence component over the wind field area is taken as the starting point. The spectrum is spatially
discretized in terms of a Hermitian cross-spectral density matrix for the turbulence state vector which turns out not to be positive
definite. Since the succeeding state space and ARMA modelling of
the turbulence rely on the positive definiteness of the cross-spectral
density matrix, the problem with the non-positive definiteness of such
matrices is at first addressed and suitable treatments regarding it are proposed. From the adjusted positive definite cross-spectral density
matrix a frequency response matrix is constructed which determines the turbulence vector as a linear filtration of Gaussian white noise.
Finally, an accurate state space modelling method is proposed which allows selection of an appropriate model order, and estimation of a state space model for the vector turbulence process incorporating its phase spectrum in one stage, and its results are compared with a conventional ARMA modelling method.
[1] M. Shinozuka and C.M. Jan, Digital Simulation of Random Processes
and its Applications. Journal of Sound and Vibration, 25(1), 111-128,1972.
[2] G. Solari and F.Tubino, A turbulence Model based on Principal Components.
Probabilistic Engineering Mechanics, 17, 327-335, 2002.
[3] A. Kareem, Numerical simulation of wind effects: A probabilistic perspective. Journal of Wind Engineering and Industrial Aerodynamics,
96, 1472-1497, 2008.
[4] X. Chen,A. Kareem, Aeroelastic analysis of bridges under multicorrelated
winds: integrated state-space approach. Journal of Engineering Mechanics ASCE, 127 (11), 1124-1134, 2001.
[5] J.C. Kaimal, J.C. Wyngaard and Y. Izumi, O.R. Cote, Spectral Characteristics
of Surface-Layer Turbulence. Quarterly Journal of the Royal
Meteorological Society, 98, 1972.
[6] M. Shiotani and Y. Iwayani, Correlation of Wind Velocities in Relation to
the Gust Loadings. Proceedings of the 3rd Conference on Wind Effects
on Buildings and Structures, Tokyo, 1971.
[7] E. Samaras, M. Shinozuka and A. Tsurui, ARMA representation of random processes. Journal of Engineering Mechanics ASCE, 111(3), 449461, 1985.
[8] A. Papoulis, Probability, Random Variables and Stochastic Processes,
2nd Ed. Mc Graw-Hill, 1984.
[9] W. Gersch and J. Yonemoto, Synthesis of multivariate random vibration
systems: A two-stage least squares AR-MA model approach. Journal of
Sound and Vibration, 52(4), 553-565, 1977.
[10] Y. Li and A. Kareem, ARMA systems in wind engineering. Probabilistic
Engineering Mechanics, 5(2), 50-59, 1990.
[11] P. Van Overschee and B. De Moor, Subspace Identification for Linear
Systems: Theory-Implementation-Applications, Dordrecht, Netherlands:
Kluwer Academic Publishers, 1996.
[12] H. Akaik, Stochastic theory of minimal realization, IEEE Transactions
on Automatic Control 19, 667-674, 1974.
[13] T. Katayama, Subspace Methods for System Identification, first ed.,
Springer, 2005.
[14] H. Akaik, Markovian representation of stochastic processes and its
application to the analysis of autoregressive moving-average processes,
Annals of the Institute of Statistical Mathematics 26(1), 363-387, 1974.
[1] M. Shinozuka and C.M. Jan, Digital Simulation of Random Processes
and its Applications. Journal of Sound and Vibration, 25(1), 111-128,1972.
[2] G. Solari and F.Tubino, A turbulence Model based on Principal Components.
Probabilistic Engineering Mechanics, 17, 327-335, 2002.
[3] A. Kareem, Numerical simulation of wind effects: A probabilistic perspective. Journal of Wind Engineering and Industrial Aerodynamics,
96, 1472-1497, 2008.
[4] X. Chen,A. Kareem, Aeroelastic analysis of bridges under multicorrelated
winds: integrated state-space approach. Journal of Engineering Mechanics ASCE, 127 (11), 1124-1134, 2001.
[5] J.C. Kaimal, J.C. Wyngaard and Y. Izumi, O.R. Cote, Spectral Characteristics
of Surface-Layer Turbulence. Quarterly Journal of the Royal
Meteorological Society, 98, 1972.
[6] M. Shiotani and Y. Iwayani, Correlation of Wind Velocities in Relation to
the Gust Loadings. Proceedings of the 3rd Conference on Wind Effects
on Buildings and Structures, Tokyo, 1971.
[7] E. Samaras, M. Shinozuka and A. Tsurui, ARMA representation of random processes. Journal of Engineering Mechanics ASCE, 111(3), 449461, 1985.
[8] A. Papoulis, Probability, Random Variables and Stochastic Processes,
2nd Ed. Mc Graw-Hill, 1984.
[9] W. Gersch and J. Yonemoto, Synthesis of multivariate random vibration
systems: A two-stage least squares AR-MA model approach. Journal of
Sound and Vibration, 52(4), 553-565, 1977.
[10] Y. Li and A. Kareem, ARMA systems in wind engineering. Probabilistic
Engineering Mechanics, 5(2), 50-59, 1990.
[11] P. Van Overschee and B. De Moor, Subspace Identification for Linear
Systems: Theory-Implementation-Applications, Dordrecht, Netherlands:
Kluwer Academic Publishers, 1996.
[12] H. Akaik, Stochastic theory of minimal realization, IEEE Transactions
on Automatic Control 19, 667-674, 1974.
[13] T. Katayama, Subspace Methods for System Identification, first ed.,
Springer, 2005.
[14] H. Akaik, Markovian representation of stochastic processes and its
application to the analysis of autoregressive moving-average processes,
Annals of the Institute of Statistical Mathematics 26(1), 363-387, 1974.
@article{"International Journal of Architectural, Civil and Construction Sciences:64616", author = "M. T. Sichani and B. J. Pedersen and S. R. K. Nielsen", title = "Stochastic Subspace Modelling of Turbulence", abstract = "Turbulence of the incoming wind field is of paramount
importance to the dynamic response of civil engineering structures. Hence reliable stochastic models of the turbulence should be available from which time series can be generated for dynamic response and
structural safety analysis. In the paper an empirical cross spectral
density function for the along-wind turbulence component over the wind field area is taken as the starting point. The spectrum is spatially
discretized in terms of a Hermitian cross-spectral density matrix for the turbulence state vector which turns out not to be positive
definite. Since the succeeding state space and ARMA modelling of
the turbulence rely on the positive definiteness of the cross-spectral
density matrix, the problem with the non-positive definiteness of such
matrices is at first addressed and suitable treatments regarding it are proposed. From the adjusted positive definite cross-spectral density
matrix a frequency response matrix is constructed which determines the turbulence vector as a linear filtration of Gaussian white noise.
Finally, an accurate state space modelling method is proposed which allows selection of an appropriate model order, and estimation of a state space model for the vector turbulence process incorporating its phase spectrum in one stage, and its results are compared with a conventional ARMA modelling method.", keywords = "Turbulence, wind turbine, complex coherence, state space modelling, ARMA modelling.", volume = "3", number = "10", pages = "448-9", }