Excitation–response relationships for linear structural systems with singular parameter matrices: A periodized harmonic wavelet perspective


Most structural systems are subjected typically to stochastic excitations exhibiting strong variations both in the frequency and the time domains. In this regard, it is important to develop efficient and robust joint time-frequency analysis techniques for determining accurately the time-varying frequency content of the system response. In the past few decades, various standard concepts from random vibration theory have been generalized based on wavelets for practical applications. These wavelet-based techniques have been successfully used to address a wide range of problems, including evolutionary power spectrum (EPS) estimation, damage detection, system response analysis and system identification.

Among the known wavelet families, the generalized harmonic wavelets (GHWs) have emerged as an efficacious basis for expanding system excitation and response processes. This yields a system of algebraic equations to be solved for determining the associated wavelet coefficients and for evaluating the response process EPS. GHWs possess additional coefficients for decoupling and enhancing the resolution of the wavelet analysis in the frequency domain, making it an indispensable tool for structural dynamics applications.

Most recently, a new GHW-based input-output relationship has been developed to determine the response EPS of linear systems. This novel relationship not only avoided the common local stationarity assumption inherent in most early developments, but also improved the accuracy of the system response EPS estimate. This was achieved by obtaining in closed form the wavelet interaction coefficients at different scales and translation levels, and by employing periodized GHWs to address the non-orthogonality of the GHW basis on a finite time interval. Due to these benefits, this approach has been extended to different nonlinear systems.

Herein, PhD candidate George Pasparakis, Dr. Vasileios Fragkoulis and Professor Michael Beer from Leibniz Universität Hannover in collaboration with Professor Ioannis Kougioumtzoglou from Columbia University and Professor Fan Kong from Wuhan University of Technology further extended this novel technique to account for multi-degree-of-freedom (MDOF) linear structural systems with singular parameter matrices. An excitation-response relationship for such systems was derived based on periodized generalized harmonic wavelets for determining system response statistics in the joint time–frequency domain. To determine the response wavelet coefficients, a linear system of algebraic equations was derived and solved by employing the Moore–Penrose matrix inverse concept. Their work is published in the journal, Mechanical Systems and Signal Processing.

The authors demonstrated the effectiveness of the derived excitation-response relationships in determining the system response statistics in the joint time-frequency domain in a direct and straightforward manner. The proposed technique could be interpreted as a generalization of previous efforts found in the literature to account for the presence of singular parameter matrices in the governing equations of motion. Further, this technique overcomes the strong local stationarity assumption that limited the accuracy degree and application scope of the previously developed techniques.

In summary, the research team developed a new periodized GHWs-based technique for determining the joint time-frequency response of linear systems with singular parameter matrices. The derivation of the input-output relationships in the wavelet domain proved effective for accurately determining the EPS matrix of the system response. The reliability of this technique was validated by comparing the analytical results with pertinent Monte Carlo simulation data. This was accomplished by considering diverse numerical examples, such as energy harvesters whose dynamics are governed by coupled electromechanical equations. In a joint statement to Advances in Engineering, the authors explained their findings provide valuable insights for improving the performance of structural systems.

About the author

Mr. George Pasparakis holds a five-year Diploma in Mechanical Engineering from the Department of Mechanical Engineering of the University of Thessaly, Greece. He joined the Institute for Risk and Reliability of Leibniz University Hannover as an Early Stage Researcher under the Marie Skłodowska-Curie Actions Horizon 2020 Innovative Training Network: “Dynamic virtualisation: modelling performance of engineering structures” (DyVirt). Mr. Pasparakis’ research interests lie in the fields of computational stochastic dynamics, data driven uncertainty quantification of dynamical systems and probabilistic modelling of stochastic processes. More specifically, Mr. Pasparakis utilizes joint time-frequency analysis tools (e.g., wavelets), sparse representations (e.g., compressive sampling) and statistical inference methodologies for developing analytical and numerical schemes for nonstationary response characterization of complex engineering systems, for discovery of system governing equations purely from data as well as for stochastic field reconstruction and statistics estimation in the presence of missing data. His research finds applications and has a potential impact on transformative technologies, such as energy harvesting devices.

About the author

Prof. Ioannis A. Kougioumtzoglou received his five-year Diploma in Civil Engineering from the National Technical University of Athens (NTUA) in Greece (2007), and his M.Sc. (2009) and Ph.D. (2011) degrees in Civil Engineering from Rice University, TX, USA. He joined Columbia University in 2014, where he is currently an Associate Professor in the Department of Civil Engineering & Engineering Mechanics.

Prof. Kougioumtzoglou and his research group develop primarily analytic and numerical methodologies for stochastic response analysis, reliability assessment, and optimization of complex engineering systems and structures subject to uncertainties. These methodologies lead eventually to robust and efficient design of dynamic systems ranging from the nano-scale (e.g. nano-mechanical oscillators) to the macro-scale (e.g. energy harvesters and civil infrastructure systems). Specific theoretical research themes include nonlinear stochastic dynamics and path integrals, fractional calculus modeling, computational stochastic mechanics, data-driven uncertainty quantification methodologies, and signal processing techniques. Additional research endeavors with diverse applications in structural, earthquake, marine and biomedical engineering include uncertainty modeling and propagation via joint time-frequency analysis tools such as wavelets, as well as analysis of high-dimensional and/or incomplete data via sparse representations and compressive sampling.

He is the author of approximately 150 publications, including more than 75 technical papers in archival International Journals. Prof. Kougioumtzoglou was chosen in 2018 by the National Science Foundation (NSF) to receive the prestigious CAREER Award, which recognizes early stage scholars with high levels of promise and excellence. He is also the 2014 European Association of Structural Dynamics (EASD) Junior Research Prize recipient “for his innovative influence on the field of nonlinear stochastic dynamics”. Prof. Kougioumtzoglou is an Associate Editor for the ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems and an Editorial Board Member of the following Journals: Mechanical Systems and Signal Processing, Probabilistic Engineering Mechanics, and International Journal of Non-Linear Mechanics. He is also a co-Editor of the Encyclopedia of Earthquake Engineering (Springer), and has served as a Guest Editor for several Special Issues in International Journals. Prof. Kougioumtzoglou has co-chaired of the ASCE Engineering Mechanics Institute Conference 2021 and Probabilistic Mechanics & Reliability Conference 2021 (EMI 2021 / PMC 2021), and has served in the scientific and/or organizing committees of many international technical conferences. Prof. Kougioumtzoglou is a member both of the American Society of Civil Engineers (M.ASCE) and the American Society of Mechanical Engineers (M.ASME), while he currently serves as a member of the ASCE EMI committees on Dynamics and on Probabilistic Methods. He is a Licensed Professional Civil Engineer in Greece, and a Fellow of the Higher Education Academy (FHEA) in the UK.

About the author

Dr. Vasileios Fragkoulis received his five-year Diploma and his M.Sc. degree both in Applied Mathematical Sciences from the School of Applied Mathematical and Physical Sciences, National Technical University of Athens in Greece. He earned his Ph.D. degree in Engineering Mathematics and Uncertainty Quantification from the Department of Mathematics of the University of Liverpool, UK. He joined Leibniz University Hannover in 2018, where he is currently a Research Associate in the Institute for Risk and Reliability. Dr. Fragkoulis’ research interests focus on the general area of Engineering and Applied Mathematics, and more specifically on Stochastic Dynamics and Uncertainty Quantification methodologies with diverse applications in Civil/Mechanical Engineering and Engineering Mechanics. Analyzing and assessing the reliability of complex engineering systems and structures under the presence of uncertainties constitutes his main research theme. Specifically, utilizing stochastic and fractional calculi, wavelet theory and signal processing techniques, Dr. Fragkoulis develops analytical and/or numerical methodologies and techniques which introduce strong links between several research topics of Engineering and Applied Mathematics, and particularly, in the fields of Nonlinear Stochastic Dynamics/Mechanics. His research is funded by the German Research Foundation (DFG). He is an affiliate member of the American Society of Civil Engineers (ASCE) and an elected member of the Dynamics Committee and of the Probabilistic Methods Committee of the Engineering Mechanics Institute (EMI).

About the author

Prof. Fan Kong earned his bachelor’s (2004) and M. Sc. (2007) degrees in Civil Engineering from the Wuhan University of Technology in China, and his Ph.D. degree in Structural Engineering from Tongji University, Shanghai, China. He joined Wuhan University of Technology in 2013 as a Lecturer and then was promoted to an Associate Professor in 2015. He is currently a full Professor and Huangshan Outstanding Young Scholar in Hefei University of Technology. He visited Rice University working with Prof. Pol D. Spanos, as a visiting student and a visiting scholar, respectively, from 2009 to 2011 and from 2017 to 2018.

He works on challenging problems in stochastic dynamics, including systems with diverse nonlinearities and hysteresis, external excitation with time-frequency nonstationarity, and dynamic systems with fractional damping elements. He is the author of approximately 70 peer-reviewed journal papers, more than 30 of which are international journal papers. He is the principal investigator of more than ten scientific research projects, including one General Project funded by the National Natural Science Foundation of China (NSFC), one Young Scholar Project funded by NSFC, and one General Project funded by the Hubei Provincial Natural Science Foundation. Besides academic research, he actively engages in academic management. He serves as a Secretary of International Affairs in the College of Civil Engineering, Hefei University of Technology. He also serves as a Committee Member of the Dynamic Signal Processing Professional Committee of the Chinese Society for Vibration Engineering (CSVE), a Young Committee Member of two Professional Committees (Random Vibration Professional Committee and Structural Vibration Control and Health Monitoring Professional Committee) of CSVE. He is a Young Editorial Board Member of the following journals: Chinese Journal of Applied Mechanics, Journal of Basic Science and Engineering, Earthquake Engineering and Resilience. Prof. Kong has served in the scientific and/or organizing committees of many national or international academic conferences.

About the author

Michael Beer is Professor and Head of the Institute for Risk and Reliability, Leibniz Universität Hannover, Germany, since 2015. He is also part time Professor in the Institute for Risk and Uncertainty at the University of Liverpool and a Guest Professor in Tongji University and Tsinghua University, China. He is a co-director of the International Joint Research Center for Resilient Infrastructure and a member in the the International Joint Research Center for Engineering Reliability and Stochastic Mechanics at Tongji University, China. Dr. Beer obtained a doctoral degree from Technical University Dresden and pursued post-doctoral research at Rice University. From 2007 to 2011 Dr. Beer worked as an Assistant Professor at National University of Singapore. In 2011 he joined the University of Liverpool as Chair in Uncertainty in Engineering and Founding Director of the Institute for Risk and Uncertainty, where he established a large Doctoral Training Center on Quantification and Management of Risk & Uncertainty.

Dr. Beer’s research is focused on uncertainty quantification in engineering with emphasis on epistemic uncertainties and concepts of imprecise probabilities. Dr. Beer is Editor in Chief (joint) of the Encyclopedia of Earthquake Engineering, Editor in Chief of the ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, and Member of fifteen Editorial Boards including Probabilistic Engineering Mechanics, Computers & Structures, Structural Safety, Mechanical Systems and Signal Processing, Engineering Structures, and International Journal for Uncertainty Quantification.

He has won several awards including the Alfredo Ang Award on Risk Analysis and Management of Civil Infrastructure 2022 (ASCE), the CADLM PRIZE 2007 – Intelligent Optimal Design and a Certificate for Highly Cited Research in Structural Safety in 2016. His publications include a book, several monographs and a large number of journal and conference papers. He is a Fellow of the Alexander von Humboldt-Foundation, Member of the Board of Directors of the International Association for Probabilistic Safety Assessment and Management and a co-chair of the Risk and Resilience Measurements Committee (RRMC), Infrastructure Resilience Division, ASCE.


Pasparakis, G., Kougioumtzoglou, I. A., Fragkoulis, V. C., Kong, F., & Beer, M. (2022). Excitation–response relationships for linear structural systems with singular parameter matrices: A periodized harmonic wavelet perspectiveMechanical Systems and Signal Processing, 169, 108701.

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