Reliability Assessment and Response Determination for Survival Probability of Nonlinear Systems Endowed with Fractional Derivative Operators by Galerkin Method
Spanos, Pol D.
Doctor of Philosophy
In this thesis a novel approximate analytical method is derived to assess the reliability of the first passage problem and determine the survival probability of randomly excited systems endowed with fractional derivative operators under Gaussian white noise excitation. Relying on the right hand definition of fractional derivative, the Caputo fractional derivative representation is implemented. The terms with fractional derivative operators are considered as perturbation element, which conveys the system into an equivalent nonlinear perfectly visco-elastic system. The response amplitude of the system is modeled as one dimensional Markovian process. Deterministic averaging and stochastic averaging technique together with statistical linearization methodology, which is based on the minimum mean square error estimation, are utilized to further quantify and simplify the response amplitude process. The approach yields the Backward Kolmogorov equation which governs the evolution of the survival probability of the linear and nonlinear fractional derivative systems. With separation of variable technique, the solution of linear systems can be determined. This constitutes a Sturm-Liouville problem with ensured properties for the sequence of eigenvalues and infinite set of eigenfunctions. In order to reduce the time complexity of the proposed algorithm in computation, the Galerkin formulation is applied for the system response derivation. Based on the orthogonality condition and the value of the barrier, the associated eigenvalues and eigenfunctions suggest the usage of the orthogonal basis for the Galerkin method. The Krommer confluent hypergeometric functions, related to the corresponding systems with integer order derivatives, are employed to specify the eigenfunctions. Reliability assessment and applications to the linear and nonlinear systems, including Van der Pol oscillator and Duffing oscillator, are investigated and determined. Designed mathematical experiments are carried out to validate the developed novel approximate analysis. Relying on the law of large numbers and ergodic theory, pertinent Monte Carlo simulations are devised for the validation procedure. Different parameters for the single degree of freedom linear and nonlinear systems, whose motions are governed by fractional derivative differential equation, are considered and discussed, such as truncated limit numbers in the series expansion of Galerkin method, values of the barriers, fractional derivative term coefficients, fractional derivative orders, small and large nonlinearities. In addition, the pilot study also includes the reliability assessment and response determination for the rational derivative and irrational derivative nonlinear systems. The reliability and accuracy of the devised novel approximate analytical approach are demonstrated by the detailed comparison and validation.