Linear Radom Vibration of Structural Systems with Singular Mass Matrices

A framework is developed for determining the stochastic response of linear multi-degree-of-freedom (MDOF) structural systems with singular matrices. This system modeling can arise when using more than the minimum number of coordinates, and can be advantageous, for instance, in cases of complex multibody systems whose dynamics satisfy a number of constraints. In such cases the explicit formulation of the equations of motion can be a nontrivial task, whereas the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, standard concepts, relationships, and equations of the linear random vibration theory are extended and generalized herein to account for systems with singular matrices. Adopting a state-variable formulation, equations governing the system response mean vector and covariance matrix are formed and solved. Further, it is shown that a complex modal analysis treatment, unlike the standard system modeling case, does not lead to decoupling of the equations of motion. However, relying on a singular value decomposition of the system transition matrix significantly facilitates the efficient computation of the system response statistics. A linear structural system with singular matrices is considered as a numerical example for demonstrating the applicability of the methodology and for elucidating certain related numerical aspects

Response Determination of Nonlinear Systems with Singular Matrices Subject to Combined Stochastic and Deterministic Excitations

A new technique is proposed for determining the response of multi-degree-of-freedom nonlinear systems with singular parameter matrices subject to combined stochastic and deterministic excitations. Singular matrices in the governing equations of motion potentially account for the presence of constraint equations in the system. They also appear when a redundant coordinates modeling is adopted to derive the equations of motion of complex multibody systems. Since the system is subject to both stochastic and deterministic excitations, its response also has two components, namely a deterministic and a stochastic component. Therefore, using the harmonic balance method to treat the deterministic component leads to an overdetermined system of equations to be solved for computing the associated coefficients. Then the generalized statistical linearization method for deriving the stochastic response of nonlinear systems with singular matrices, in conjunction with an averaging treatment, are utilized to determine the stochastic component of the response. The validity of the proposed technique is demonstrated by pertinent numerical examples

Survival probability determination of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique based on a combination of statistical linearization and stochastic averaging is developed for determining the survival probability of stochastically excited nonlinear/hysteretic oscillators with fractional derivative elements. Specifically, approximate closed form expressions are derived for the oscillator non-stationary marginal, transition, and joint response amplitude probability density functions (PDF) and, ultimately, for the time-dependent oscillator survival probability. Notably, the technique can treat a wide range of nonlinear/hysteretic response behaviors and can account even for evolutionary excitation power spectra with time-dependent frequency content. Further, the corresponding computational cost is kept at a minimum level since it relates, in essence, only to the numerical integration of a deterministic nonlinear differential equation governing approximately the evolution in time of the oscillator response variance. Overall, the developed technique can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the equation of motion. The numerical examples include a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative terms. The accuracy degree of the technique is assessed by comparisons with pertinent Monte Carlo simulation data

Positive Extension Of Linear Operators

49 σ.Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) "Εφαρμοσμένες Μαθηματικές Επιστήμες"Σκοπός αυτής της εργασίας είναι η μελέτη και παρουσίαση του προβλήματος της θετικής επέκτασης ενός (θετικού) γραμμικού τελεστή. Μιλώντας για θετική επέκταση, εννοούμε γραμμική επέκταση ενός θετικού γραμμικού τελεστή από υπόχωρο ενός μερικά διατεταγμένου χώρου σε ολόκληρο τον χώρο.Our aim is to study and perform some results about the problem of positive extension of linear operators. Talking about positive extension, we mean linear extension of a positive operator defined on a subspace of a partially ordered vector space, to the whole space, such that the extension operator remains positive.Βασίλειος Χρ. Φραγκούλη

Non-stationary response determination of nonlinear systems subjected to combined deterministic and evolutionary stochastic excitations

A semi-analytical method is proposed for determining the response of a lightly damped single-degree-of-freedom nonlinear system subjected to combined deterministic and non-stationary stochastic excitations. This is attained by combining the stochastic averaging and statistical linearization methodologies. Specifically, first, the system response is decomposed into two components, namely the deterministic and the stochastic parts. This leads to a set of coupled differential sub-equations governing, respectively, the deterministic and the stochastic component of the response. Next, aiming at solving the set of differential sub-equations, an additional expression is derived by applying the statistical linearization methodology, followed by the application of a stochastic averaging step to the stochastic sub-equations. Therefore, an equivalent time-varying linear system is defined for the original nonlinear system. The stochastic averaging method is then applied to the linearized system for reducing its order, and thus, its complexity from a solution perspective. In this regard, an additional equation is derived, which connects the deterministic and stochastic components of the response. The latter and the deterministic differential sub-equations are solved simultaneously for determining the system response. A single-degree-of-freedom nonlinear system exhibiting quadratic and cubic nonlinear stiffness is considered for assessing the reliability of the proposed technique. The obtained results are compared with pertinent Monte-Carlo simulation estimates

Effect of the Preparation Method on the Physicochemical Properties and the CO Oxidation Performance of Nanostructured CeO2/TiO2 Oxides

Ceria-based mixed oxides have been widely studied in catalysis due to their unique surface and redox properties, with implications in numerous energy- and environmental-related applications. In this regard, the rational design of ceria-based composites by means of advanced synthetic routes has gained particular attention. In the present work, ceria–titania composites were synthesized by four different methods (precipitation, hydrothermal in one and two steps, Stöber) and their effect on the physicochemical characteristics and the CO oxidation performance was investigated. A thorough characterization study, including N2 adsorption-desorption, X-ray diffraction (XRD), scanning electron microscopy with energy dispersive X-ray spectroscopy (SEM/EDS), transmission electron microscopy (TEM) and H2 temperature-programmed reduction (H2-TPR) was performed. Ceria–titania samples prepared by the Stöber method, exhibited the optimum CO oxidation performance, followed by samples prepared by the hydrothermal method in one step, whereas the precipitation method led to almost inactive oxides. CeO2/TiO2 samples synthesized by the Stöber method display a rod-like morphology of ceria nanoparticles with a uniform distribution of TiO2, leading to enhanced reducibility and oxygen storage capacity (OSC). A linear relationship was disclosed among the catalytic performance of the samples prepared by different methods and the abundance of reducible oxygen species