{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T01:17:07Z","timestamp":1769822227690,"version":"3.49.0"},"reference-count":18,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T00:00:00Z","timestamp":1709596800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Research Council","award":["101007595"],"award-info":[{"award-number":["101007595"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Random vibration analysis is a mathematical tool that offers great advantages in predicting the mechanical response of structural systems subjected to external dynamic loads whose nature is intrinsically stochastic, as in cases of sea waves, wind pressure, and vibrations due to road asperity. Using random vibration analysis is possible, when the input is properly modeled as a stochastic process, to derive pieces of information about the structural response with a high quality (if compared with other tools), especially in terms of reliability prevision. Moreover, the random vibration approach is quite complex in cases of non-linearity cases, as well as for non-stationary inputs, as in cases of seismic events. For non-stationary inputs, the assessment of second-order spectral moments requires resolving the Lyapunov matrix differential equation. In this research, a numerical procedure is proposed, providing an expression of response in the state-space that, to our best knowledge, has not yet been presented in the literature, by using a formal justification in accordance with earthquake input modeled as a modulated white noise with evolutive parameters. The computational efforts are reduced by considering the symmetry feature of the covariance matrix. The adopted approach is applied to analyze a multi-story building, aiming to determine the reliability related to the maximum inter-story displacement surpassing a specified acceptable threshold. The building is presumed to experience seismic input characterized by a non-stationary process in both amplitude and frequency, utilizing a general Kanai\u2013Tajimi earthquake input stationary model. The adopted case study is modeled in the form of a multi-degree-of-freedom plane shear frame system.<\/jats:p>","DOI":"10.3390\/computation12030050","type":"journal-article","created":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T05:31:19Z","timestamp":1709616679000},"page":"50","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Numerical Covariance Evaluation for Linear Structures Subject to Non-Stationary Random Inputs"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6077-8338","authenticated-orcid":false,"given":"M.","family":"Domaneschi","sequence":"first","affiliation":[{"name":"Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Turin, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9344-6006","authenticated-orcid":false,"given":"R.","family":"Cucuzza","sequence":"additional","affiliation":[{"name":"Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Turin, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0928-0606","authenticated-orcid":false,"given":"L.","family":"Sardone","sequence":"additional","affiliation":[{"name":"Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Turin, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-5907-7297","authenticated-orcid":false,"given":"S. Londo\u00f1o","family":"Lopez","sequence":"additional","affiliation":[{"name":"Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Turin, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8393-724X","authenticated-orcid":false,"given":"M.","family":"Movahedi","sequence":"additional","affiliation":[{"name":"Department of Structural and Geotechnical Engineering, Sz\u00e9chenyi Istv\u00e1n University, Gy\u0151r, Hungary"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8472-2956","authenticated-orcid":false,"given":"G. C.","family":"Marano","sequence":"additional","affiliation":[{"name":"Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Turin, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"796","DOI":"10.1016\/j.apm.2023.05.022","article-title":"Hybrid Reliability-Based Sequential Optimization for PID Vibratory Controller Design Considering Interval and Fuzzy Mixed Uncertainties","volume":"122","author":"Liu","year":"2023","journal-title":"Appl. Math. Model."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"102313","DOI":"10.1016\/j.strusafe.2022.102313","article-title":"Dynamic reliability analysis of stochastic structures under non-stationary random excitations based on an explicit time-domain method","volume":"101","author":"Huang","year":"2023","journal-title":"Struct. Saf."},{"key":"ref_3","unstructured":"Soong, T.T., and Grigoriu, M. (1993). Random Vibration of Mechanical and Structural Systems, Prentice Hall."},{"key":"ref_4","unstructured":"Wax, N. (1967). Probabilistic Theory of Structural Dynamics, Krieger Publishing, Co., Inc.. Stochastic Processes."},{"key":"ref_5","unstructured":"Lutes, L.D., and Sarkani, S. (1997). Stochastic Analysis of Structural and Mechanical Vibration, Prentice-Hall Inc."},{"key":"ref_6","unstructured":"Nigam, N.C., and Narayanan, S. (1994). Application of Random Vibration, Narosa Publishing House."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"111111","DOI":"10.1016\/j.engstruct.2020.111111","article-title":"Topology optimization of buildings subjected to stochastic base excitation","volume":"23","author":"Gomez","year":"2020","journal-title":"Eng. Struct."},{"key":"ref_8","unstructured":"Spencer, B.F., Gomez, F., and Xu, J. (2016). Topology Optimization for Stochastically Excited Structures, ISSRI."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Kressner, D. (2008, January 3\u20135). Memory-efficient Krylov subspace techniques for solving large-scale Lyapunov equations. Proceedings of the 2008 IEEE International Conference on Computer-Aided Control Systems, San Antonio, TX, USA.","DOI":"10.1109\/CACSD.2008.4627370"},{"key":"ref_10","unstructured":"Saad, Y. (1989). Numerical Solution of Large Lyapunov Equations, Birkhauser. Signal Process Scattering Oper Theory Numer Methods, Proc MTNS-89."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1016\/j.laa.2004.11.004","article-title":"Projection methods for large Lyapunov matrix equations","volume":"415","author":"Jbilou","year":"2006","journal-title":"Linear Algebra Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106467","DOI":"10.1016\/j.soildyn.2020.106467","article-title":"Generation of fully non-stationary random processes consistent with target seismic accelerograms","volume":"141","author":"Muscolino","year":"2021","journal-title":"Soil Dyn. Earthq. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1450044","DOI":"10.1142\/S0219455414500448","article-title":"Integration algorithm for covariance Nonstationary dynamic analysis of SDOF systems using equivalent stochastic linearization","volume":"15","author":"Marano","year":"2015","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3685091","DOI":"10.1155\/2018\/3685091","article-title":"Nonstationary First Threshold Crossing Reliability for Linear System Excited by Modulated Gaussian Process","volume":"2018","author":"Greco","year":"2018","journal-title":"Shock. Vib."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1016\/j.soildyn.2011.05.014","article-title":"An evolutionary stochastic ground-motion model defined by a seismological scenario and local site conditions","volume":"31","author":"Sgobba","year":"2011","journal-title":"Soil Dyn. Earthq. Eng."},{"key":"ref_16","unstructured":"Crandall, S.H., and Mark, W.D. (1963). Random Vibration in Mechanical Systems, Academic Press."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1061\/JMCEA3.0000463","article-title":"Periodic response of a general yielding structure","volume":"90","author":"Jennings","year":"1964","journal-title":"J. Eng. Mech. Div."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1115\/1.3423521","article-title":"On the Distribution of the First-Passage Time for Normal Stationary Random Processes","volume":"42","author":"Vanmarcke","year":"1975","journal-title":"ASME J. Appl. Mech. March"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/3\/50\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:09:27Z","timestamp":1760105367000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/3\/50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,5]]},"references-count":18,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["computation12030050"],"URL":"https:\/\/doi.org\/10.3390\/computation12030050","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,5]]}}}