Deterministic and Random Vibration of Linear Systems with Singular Parameter Matrices and Fractional Derivative Terms

Both time- and frequency-domain solution techniques are developed for determining the response of linear multi-degree-of-freedom systems exhibiting singular parameter matrices and endowed with derivative terms of noninteger orders modeled as rational numbers. This is done based on the Moore-Penrose matrix inverse theory, in conjunction with a state variable formulation and with a complex modal analysis treatment. It is worth noting that, for the class of systems considered herein, this treatment also yields decoupled governing equations, thus facilitating further their numerical solution. Next, a generalization of the standard frequency-domain input-output (excitation-response) relationship is derived based on an appropriately defined frequency response function. This spectral relationship is further extended to account also for stochastic excitation vector processes described by power spectral density matrices. Two illustrative examples are considered for demonstrating the validity of the herein developed technique and of the derived input-output relationships.