Background; Resonant; Non Gaussian; MTSA: Multiple Timescale Spectral Analysis; Complete Cubic Combination; Cubic Root of the Sum of the Cubes
Abstract :
The paper develops an approximate semi-analytical solution for the computation of the third statistical cross-moments of modal responses in a stochastic dynamic analysis. These moments would require heavy twofold numerical integration in a general context but are drastically simplified in the proposed formulation by taking advantage of the assumed distinctness between the low characteristic frequency of the loading and the natural frequencies of the structure. This condition is typically respected and acknowledged in wind engineering where the buffeting analysis of large structures hinges on the Background/Resonant decomposition. As such, the proposed formulation extends to third statistical order the existing developments for the estimation of the modal variances and covariances. It allows the third order spectral analysis of large structures to be conducted within a reasonable amount of time. It also reveals the existence of three main components to the response: background, bi-resonant and tri-resonant. The latter one is specific to this very own problem and is shown to be important when the sum of two natural frequencies is equal to a third one, although the structural behavior is linear. Mathematics highlight this and other findings which are then illustrated on a minimum working example, easily reproducible by readers. Overall, it clearly demonstrates the benefits of the proposed decomposition in terms of both behavioral comprehension and computational consumption.
Disciplines :
Civil engineering
Author, co-author :
Denoël, Vincent ; Université de Liège - ULiège > Département ArGEnCo > Analyse sous actions aléatoires en génie civil
Esposito Marzino, Michele ; Université de Liège - ULiège > Département ArGEnCo > Analyse sous actions aléatoires en génie civil
Geuzaine, Margaux ; Université de Liège - ULiège > Département ArGEnCo > Analyse sous actions aléatoires en génie civil
Language :
English
Title :
A multiple timescale approach of bispectral correlation
Publication date :
January 2023
Journal title :
Journal of Wind Engineering and Industrial Aerodynamics
Bachmann, Hugo, Ammann, Walter, Vibrations in Structures: Induced by Man and Machines, Vol. 3. 1987, Iabse.
Blaise, Nicolas, Canor, Thomas, Denoël, Vincent, Reconstruction of the envelope of non-Gaussian structural responses with principal static wind loads. J. Wind Eng. Ind. Aerodyn. 149 (2016), 59–76.
Borino, G., Muscolino, G., Mode-superposition methods in dynamic analysis of classically and non-classically damped linear systems. Earthq. Eng. Struct. Dyn. 14:5 (1986), 705–717.
Canor, Thomas, Blaise, Nicolas, Denoël, Vincent, Efficient uncoupled stochastic analysis with non-proportional damping. J. Sound Vib. 331:24 (2012), 5283–5291.
Choi, Myoungkeun, Sweetman, Bert, The Hermite moment model for highly skewed response with application to tension leg platforms. J. Offshore Mech. Arct. Eng., 132(2), 2010.
Chopra, Anil K., Modal combination rules in response spectrum analysis: Early history. Earthq. Eng. Struct. Dyn. 50:2 (2021), 260–269.
Collis, W.B., White, P.R., Hammond, J.K., Higher-order spectra: the bispectrum and trispectrum. Mech. Syst. Signal Process. 12:3 (1998), 375–394.
Davenport, Alan Garnett, The application of statistical concepts to the wind loading of structures. Proc. Inst. Civ. Eng. 19:4 (1961), 449–472.
Davenport, Alan Garnett, Note on the distribution of the largest value of a random function with application to gust loading. Proc. Inst. Civ. Eng. 28:2 (1964), 187–196.
Denoël, Vincent, Estimation of modal correlation coefficients from background and resonant responses. Struct. Eng. Mech.: Int. J., 32(6), 2009.
Denoël, Vincent, On the background and biresonant components of the random response of single degree-of-freedom systems under non-Gaussian random loading. Eng. Struct. 33:8 (2011), 2271–2283.
Denoël, Vincent, Extension of the background/biresonant decomposition to the estimation of the kurtosis coefficient of the response. Uncertainty in Structural Dynamics, 2012.
Denoël, Vincent, 2013. Extension of Davenport's Background/Resonant decomposition for the estimation of higher response moments. In: 6th European-African Conference on Wind Engineering.
Denoël, Vincent, Carassale, Luigi, Response of an oscillator to a random quadratic velocity-feedback loading. J. Wind Eng. Ind. Aerodyn. 147 (2015), 330–344.
Denoël, Vincent, Degée, Hervé, Influence of the non-linearity of the aerodynamic coefficients on the skewness of the buffeting drag force. Wind Struct., 9(6), 2006.
Denoël, Vincent, Degée, Hervé, Asymptotic expansion of slightly coupled modal dynamic transfer functions. J. Sound Vib. 328:1–2 (2009), 1–8.
Der Kiureghian, Armen, Structural response to stationary excitation. J. Eng. Mech. Div. 106:6 (1980), 1195–1213.
Fan, Wenliang, Sheng, Xiangqian, Li, Zhengliang, Sun, Yi, The higher-order analysis method of statistics analysis for response of linear structure under stationary non-Gaussian excitation. Mech. Syst. Signal Process., 166, 2022, 108430.
Géradin, Michel, Rixen, Daniel J., Mechanical Vibrations: Theory and Application to Structural Dynamics. 2014, John Wiley & Sons.
Geuzaine, Margaux, Denoël, Vincent, 2020. Efficient estimation of the skewness of the response of a wave-excited oscillator. In: EURODYN 2020: XI International Conference on Structural Dynamics.
Geuzaine, Margaux, Esposito Marzino, Michele, Denoël, Vincent, Efficient Estimation of the Skewness of the Response of a Linear Oscillator Under Non-Gaussian Loading. 2020, Engineering Mechanics Institute conference.
Gusella, Vittorio, Materazzi, Annibale Luigi, Non-Gaussian response of MDOF wind-exposed structures: analysis by bicorrelation function and bispectrum. Meccanica 33:3 (1998), 299–307.
Holmes, J.D., Non-Gaussian characteristics of wind pressure fluctuations. J. Wind Eng. Ind. Aerodyn. 7 (1981), 103–108.
Holmes, John D., Wind Loading of Structures. 2007, CRC Press.
Howard, I.M., Higher-order spectral techniques for machine vibration condition monitoring. Proc. Inst. Mech. Eng. G 211:4 (1997), 211–219.
Ishikawa, T., A Study on Wind Load Estimation Method Considering Dynamic Effect for Overhead Transmission Lines. (Doctoral Thesis), 2004, Waseda University.
Kappos, Andreas, Dynamic Loading and Design of Structures. 2001, CRC Press.
Kareem, Ahsan, Wu, Teng, Wind-induced effects on bluff bodies in turbulent flows: Nonstationary, non-Gaussian and nonlinear features. J. Wind Eng. Ind. Aerodyn. 122 (2013), 21–37.
Kareem, A., Zhao, J., Analysis of non-Gaussian surge response of tension leg platforms under wind loads. J. Offshore Mech. Arct. Eng. 116:3 (1994), 137–144.
Kareem, Ahsan, Zhao, Jun, Tognarelli, Michael A., Surge response statistics of tension leg platforms under wind and wave loads: a statistical quadratization approach. Probab. Eng. Mech. 10:4 (1995), 225–240.
Kato, Shunji, Ando, S., Statistical Analysis of Low Frequency Responses of a Moored Floating Offshore Structure: 1st Report., 1986, Ship Research Institue of Japan, 17–58 (23).
Kevorkian, Jirayr, Cole, Julian D., Perturbation Methods in Applied Mathematics, Vol. 34. 2013, Springer Science & Business Media.
Li, Jie, Chen, Jianbing, Stochastic Dynamics of Structures. 2009, John Wiley & Sons.
Liepmann, Hans Wolfgang, On the application of statistical concepts to the buffeting problem. J. Aeronaut. Sci. 19:12 (1952), 793–800.
Lutes, Loren D., Sarkani, Shahram, Random Vibrations: Analysis of Structural and Mechanical Systems. 2004, Butterworth-Heinemann.
Nam, Sang-Won, Powers, Edward J., Application of higher order spectral analysis to cubically nonlinear system identification. IEEE Trans. Signal Process. 42:7 (1994), 1746–1765.
Paıdoussis, M.P., Li, G.X., Pipes conveying fluid: a model dynamical problem. J. Fluids Struct. 7:2 (1993), 137–204.
Peng, Xinlai, Yang, Luping, Gavanski, Eri, Gurley, Kurtis, Prevatt, David, A comparison of methods to estimate peak wind loads on buildings. J. Wind Eng. Ind. Aerodyn. 126 (2014), 11–23.
Piccardo, Giuseppe, Tubino, Federica, Equivalent spectral model and maximum dynamic response for the serviceability analysis of footbridges. Eng. Struct. 40 (2012), 445–456.
Preumont, André, Random vibration and spectral analysis. 1994.
Rice, Stephen O., Mathematical analysis of random noise. Bell Syst. Tech. J. 23:3 (1944), 282–332.
Stratonovich, Rouslan L., Topics in the Theory of Random Noise, Vol. 2. 1967, CRC Press.
Swami, Ananthram, Mendel, Jerry M., Nikias, Chrysostomos L., Higher-order spectral analysis toolbox. Mathw. Inc. 3 (1998), 22–26.
Wilson, Edward L., Der Kiureghian, Armen, Bayo, E.P., A replacement for the SRSS method in seismic analysis. Earthq. Eng. Struct. Dyn. 9:2 (1981), 187–192.
Xu, Zhiwei, Dai, Gonglian, Zhang, Limao, Chen, Y Frank, Flay, Richard GJ, Rao, Huiming, Effect of non-Gaussian turbulence on extreme buffeting response of a high-speed railway sea-crossing bridge. J. Wind Eng. Ind. Aerodyn., 224, 2022, 104981.
Yang, Luping, Gurley, Kurtis R., Prevatt, David O., Probabilistic modeling of wind pressure on low-rise buildings. J. Wind Eng. Ind. Aerodyn. 114 (2013), 18–26.
Yang, Qingshan, Tian, Yuji, A model of probability density function of non-Gaussian wind pressure with multiple samples. J. Wind Eng. Ind. Aerodyn. 140 (2015), 67–78.