Random vibration of linear systems with singular matrices based on Kronecker canonical forms of matrix pencils

Abstract

A novel technique is developed for determining the stochastic response of linear dynamic systems with singular parameter matrices based on matrix pencil theoretical concepts and relying on Kronecker canonical forms (KCF). The herein developed solution technique can be construed as a generalization of the standard linear random vibration theory and tools to account for constraints in the system dynamics and for singular system parameter matrices. Further, in comparison with alternative generalized matrix inverse approaches providing a family of possible solutions, the KCF-based technique yields a unique solution. This is an additional significant advantage of the technique since the use of pseudo-inverses is circumvented, and the challenge of selecting an optimal solution among a family of possible ones is bypassed. Various diverse examples are considered for demonstrating the versatility and validity of the technique. These pertain to structural (multi-body) systems modeled by dependent degrees-of-freedom, energy harvesters with coupled electromechanical equations, and oscillators subject to non-white excitations described by additional auxiliary state equations acting as filters to white noise.