Sparse representations and quadratic approximations in path integral techniques for stochastic response analysis of diverse systems/structures

Uncertainty propagation in engineering mechanics and dynamics is a highly challenging problem that requires development of analytical/numerical techniques for determining the stochastic response of complex engineering systems. In this regard, although Monte Carlo simulation (MCS) has been the most versatile technique for addressing the above problem, it can become computationally daunting when faced with high-dimensional systems or with computing very low probability events. Thus, there is a demand for pursuing more computationally efficient methodologies. Recently, a Wiener path integral (WPI) technique, whose origins can be found in theoretical physics, has been developed in the field of engineering dynamics for determining the response transition probability density function (PDF) of nonlinear oscillators subject to non-white, non-Gaussian and non-stationary excitation processes. In the present work, the Wiener path integral technique is enhanced, extended and generalized with respect to three main aspects; namely, versatility, computational efficiency and accuracy. Specifically, the need for increasingly sophisticated modeling of excitations has led recently to the utilization of fractional calculus, which can be construed as a generalization of classical calculus. Motivated by the above developments, the WPI technique is extended herein to account for stochastic excitations modeled via fractional-order filters. To this aim, relying on a variational formulation and on the most probable path approximation yields a deterministic fractional boundary value problem to be solved numerically for obtaining the oscillator joint response PDF. Further, appropriate multi-dimensional bases are constructed for approximating, in a computationally efficient manner, the non-stationary joint response PDF. In this regard, two distinct approaches are pursued. The first employs expansions based on Kronecker products of bases (e.g., wavelets), while the second utilizes representations based on positive definite functions. Next, the localization capabilities of the WPI technique are exploited for determining PDF points in the joint space-time domain to be used for evaluating the expansion coefficients at a relatively low computational cost. Subsequently, compressive sampling procedures are employed in conjunction with group sparsity concepts and appropriate optimization algorithms for decreasing even further the associated computational cost. It is shown that the herein developed enhancement renders the technique capable of treating readily relatively high-dimensional stochastic systems. More importantly, it is shown that this enhancement in computational efficiency becomes more prevalent as the number of stochastic dimensions increases; thus, rendering the herein proposed sparse representation approach indispensable, especially for high-dimensional systems. Next, a quadratic approximation of the WPI is developed for enhancing the accuracy degree of the technique. Concisely, following a functional series expansion, higher-order terms are accounted for, which is equivalent to considering not only the most probable path but also fluctuations around it. These fluctuations are incorporated into a state-dependent factor by which the exponential part of each PDF value is multiplied. This localization of the state-dependent factor yields superior accuracy as compared to the standard most probable path WPI approximation where the factor is constant and state-invariant. An additional advantage relates to efficient structural reliability assessment, and in particular, to direct estimation of low probability events (e.g., failure probabilities), without possessing the complete transition PDF. Overall, the developments in this thesis render the WPI technique a potent tool for determining, in a reliable manner and with a minimal computational cost, the stochastic response of nonlinear oscillators subject to an extended range of excitation processes. Several numerical examples, pertaining to both nonlinear dynamical systems subject to external excitations and to a special class of engineering mechanics problems with stochastic media properties, are considered for demonstrating the reliability of the developed techniques. In all cases, the degree of accuracy and the computational efficiency exhibited are assessed by comparisons with pertinent MCS data

Nonlinear Dynamics

This volume covers a diverse collection of topics dealing with some of the fundamental concepts and applications embodied in the study of nonlinear dynamics. Each of the 15 chapters contained in this compendium generally fit into one of five topical areas: physics applications, nonlinear oscillators, electrical and mechanical systems, biological and behavioral applications or random processes. The authors of these chapters have contributed a stimulating cross section of new results, which provide a fertile spectrum of ideas that will inspire both seasoned researches and students

Identification et optimisation des propriétés dynamiques des matériaux viscoélastiques

In the automotive, railway and aerospace industries, interior noise is an important consideration in design and operation. Among the available technologies to reduce the structure-borne vibration and noise, the use of Metal-Polymer Sandwich (MPS) panels is attracting more interest from Original Equipment Manufacturers (OEMs). As for constrained-layer damping (CLD) treatments, besides developing more accurate models (theoretical and finite element) to simulate the vibroacoustic performance, it is very important to 'accurately identify the properties of the constituent materials of an MPS. Since the core materials in MPS exhibit viscoelastic properties which vary significantly with temperature and frequency, it is necessary to develop experimental and/or optimization methods to characterize the dynamic properties so that they may be well matched to specific noise and vibration control applications. This is the objective of this thesis.In this thesis, a simple free-free beam based setup, together with an identification algorithm has been developed to identify the dynamic properties of core materials from the measured frequency response functions. The setup involved circumventing some drawbacks of the traditional clamped-free setup.In particular, a new optimization method is brought' forward wherein a four-parameter fractional derivative model plus a three-parameter Williams-Landel-Ferry (WLF) equation are used to describe the temperature and frequency dependent behaviour of core materials. Therefore, few parameters are optimized for the temperature and frequency dependent properties. The objective function in the optimization is based on the so-called amplitude correlation coefficient which can be calculated directly by the frequency response functions. The normal mode superposition method taking the added mass into account, as well as Ross-Kerwin-Ungar (RKU) equations, as a solver, is used to calculate the predicted frequency response functions. The Pattern Search algorithm is used to find the best values of design parameters. This algorithm is a global optimization algorithm and is less sensitive to the initial values of design parameters. Numerical examples and tests on several MPS panels were used to validate the free-free setup and optimization method by systematic comparison with the ASTM E756-04 Standard and with DMA when the latter is possible. However, with some MPS panels, the proposed method failed to provide satisfactory results. It was further postulated that the manufacturing process of these MPS panels may somehow have modified the properties or the constitutive law of the polymer itself

Theoretical Approaches in Non-Linear Dynamical Systems

From Preface: The 15th International Conference „Dynamical Systems - Theory and Applications” (DSTA 2019, 2-5 December, 2019, Lodz, Poland) gathered a numerous group of outstanding scientists and engineers who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without great effort of the staff of the Department of Automation, Biomechanics and Mechatronics of the Lodz University of Technology. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our event was attended by over 180 researchers from 35 countries all over the world, who decided to share the results of their research and experience in different fields related to dynamical systems. This year, the DSTA Conference Proceedings were split into two volumes entitled „Theoretical Approaches in Non-Linear Dynamical Systems” and „Applicable Solutions in Non-Linear Dynamical Systems”. In addition, DSTA 2019 resulted in three volumes of Springer Proceedings in Mathematics and Statistics entitled „Control and Stability of Dynamical Systems”, „Mathematical and Numerical Approaches in Dynamical Systems” and „Dynamical Systems in Mechatronics and Life Sciences”. Also, many outstanding papers will be recommended to special issues of renowned scientific journals.Cover design: Kaźmierczak, MarekTechnical editor: Kaźmierczak, Mare

15th Conference on Dynamical Systems Theory and Applications DSTA 2019 ABSTRACTS

From Preface: This is the fifteen time when the conference „Dynamical Systems – Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and the Ministry of Science and Higher Education. It is a great pleasure that our invitation has been accepted by so many people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcome nearly 255 persons from 47 countries all over the world. They decided to share the results of their research and many years experiences in the discipline of dynamical systems by submitting many very interesting papers. This booklet contains a collection of 338 abstracts, which have gained the acceptance of referees and have been qualified for publication in the conference edited books.Technical editor and cover design: Kaźmierczak, MarekCover design: Ogińska, Ewelina; Kaźmierczak, Mare