On The Decoupling Approximation in Damped Linear Systems

Abstract

The principal coordinates of a non-classically damped linear system are coupled by non-zero off-diagonal elements of the modal damping matrix. In the analysis of non-classically damped systems, a common approximation is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is widely believed that if the modal damping matrix is diagonally dominant, then errors due to the decoupling approximation must be small. In addition, it is intuitively accepted that the more diagonal the modal damping matrix, the smaller will be the errors due to the decoupling approximation. Two numerical indices are proposed in this paper to measure quantitatively the degree of being diagonal in modal damping. It is demonstrated that, over a finite range, errors due to the decoupling approximation can continuously increase while the modal damping matrix becomes more and more diagonal with its off-diagonal elements decreasing in magnitude continuously. An explanation for this unexpected behavior is offered. Within a practical range of engineering applications, diagonal dominance of the modal damping matrix may not be sufficient for neglecting modal coupling in a damped system.