Response determination of linear dynamical systems with singular matrices: A polynomial matrix theory approach

An approach is developed based on polynomial matrix theory for formulating the equations of motion and for determining the response of multi-degree-of-freedom (MDOF) linear dynamical systems with singular matrices and subject to linear constraints. This system modeling may appear for reasons such as utilizing redundant DOFs, and can be advantageous from a computational cost perspective, especially for complex (multi-body) systems. The herein developed approach can be construed as an alternative to the recently proposed methodology by Udwadia and coworkers, and has the significant advantage that it circumvents the use of pseudoinverses in determining the system response. In fact, based on the theoretical machinery of polynomial matrices, a closed form analytical solution is derived for the system response that involves non-singular matrices and relies on the use of a basis of the null space of the constraints matrix. Several structural/mechanical systems with singular matrices are included as examples for demonstrating the validity of the developed framework and for elucidating certain numerical aspects.