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Title: Random vibration of systems with singular matrices
Author: Fragkoulis, Vasileios C.
ISNI:       0000 0004 6496 4191
Awarding Body: University of Liverpool
Current Institution: University of Liverpool
Date of Award: 2017
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In the area of stochastic engineering dynamics, a flourishing field of research has been connected to assessing the reliability of systems subjected to stochastic excitations. In particular, the development of analytical and numerical methodologies for the response statistics determination of multi-degree-of-freedom (MDOF) systems with potentially singular matrices are of high interest. These singular matrices can appear naturally in the systems governing equations of motion, for instance due to coupling of electro-mechanical equations in energy harvesting applications, or they are related to FEM modeling; they also appear due to a redundant degrees-of-freedom (DOF) modeling of the systems equation of motion. In the later case, for reasons pertaining to a less labor intensive formulation of the systems governing equations of motion, especially in case of large-scale MDOF systems, and/or from a computational efficiency perspective, the system governing equations of motion are derived by utilizing a redundant DOFs modeling. This results in equations of motion with singular mass, damping and stiffness matrices. Taking also into account that the classical state and frequency domain analysis methodologies for deriving the system stochastic response, have been developed ad hoc for the case of systems with non-singular matrices, the necessity for developing a framework for treating systems with singular matrices arises. A novel Moore-Penrose (M-P) generalized matrix inverse based framework is developed for circumventing the difficulties arising from the redundant DOFs modeling of the systems governing equations of motion. The standard time and frequency domain analysis treatments have been extended to account for linear systems with singular matrices. A M-P based solution framework for the systems mean vector and covariance matrix is determined, first by solving the equations derived after the application of the standard state-variable formulation. By following a frequency domain analysis the corresponding mean vector and covariance matrices are derived. In the latter case, a M-P based expression is obtained for the system frequency response function (FRF) matrix, and subsequently utilizing the relationship that connects the impulse response function of the system excitation to the corresponding of its response, a M-P solution for the system response power spectrum is derived. Next, the classical statistical linearization approximate methodology is generalized to account for nonlinear systems with singular matrices. Adopting a redundant DOFs modeling for the derivation of the systems governing equations of motion, and relying on the concept of the M-P generalized matrix inverse, the extended time and frequency domain analysis treatment are applied for deriving the response statistics of systems subjected to stochastic excitations. Working on the time domain, a family of optimal and response dependent equivalent linear matrices is derived. Extending a classical excitation-response relationship of the random vibration theory, and taking into account the aforementioned family of matrices, results in an iterative determination of the system response mean vector and covariance matrix. It is proved that setting the arbitrary element in the M-P solution for the equivalent linear matrices equal to zero yields a mean square error at least as low as the error corresponding to any non-zero value of the arbitrary element. The M-P based frequency domain analysis treatment also yields an iterative determination of the system response mean vector and covariance matrix. The generalization of a widely utilized formula that facilitates the application of statistical linearization is also given.
Supervisor: Pantelous, A. ; Kougioumtzoglou, I. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral