Eigenderivative analysis of asymmetric non‐conservative systems

In general, the damping matrix of a dynamic system or structure is such that it can not be simultaneously diagonalized with the mass and stiffness matrices by any linear transformation. For this reason the eigenvalues and eigenvectors and consequently their derivatives become complex. Expressions for the first‐ and second‐order derivatives of the eigenvalues and eigenvectors of these linear, non‐conservative systems are given. Traditional restrictions of symmetry and positive definiteness have not been imposed on the mass, damping and stiffness matrices. The results are derived in terms of the eigenvalues and left and right eigenvectors of the second‐order system so that the undesirable use of the first‐order representation of the equations of motion can be avoided. The usefulness of the derived expressions is demonstrated by considering a non‐proportionally damped two degree‐of‐freedom symmetric system, and a damped rigid rotor on flexible supports.