Diagonal dominance of damping and the decoupling approximation in linear vibratory systems

A common approximation in the analysis of non-classically damped systems is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation. It is generally believed that errors due to the decoupling approximation should be negligible if the modal damping matrix is diagonally dominant. In addition, the errors are expected to decrease as the modal damping matrix becomes more diagonally dominant. It is shown numerically in this paper that, over a finite range, errors due to the decoupling approximation can increase monotonically at any specified rate while the modal damping matrix becomes more diagonally dominant with its off-diagonal elements decreasing continuously in magnitude. An explanation for these unexpected drifts of decoupling errors is provided with the use of complex coupling coefficients. Small off-diagonal elements in the modal damping matrix are not sufficient to ensure small errors due to the decoupling approximation. Any error-criterion based solely upon the diagonal dominance of the modal damping matrix would not be accurate.