Coordinate coupling and the decoupling approximation in damped linear vibratory systems

An undamped linear vibratory system possesses classical normal modes, and that in each mode different parts of the system vibrate in a synchronous manner. The normal modes constitute a modal matrix, which defines a linear coordinate transformation that decouples the undamped system. This process of decoupling the equation of motion of a vibratory system is termed modal analysis. In the presence of damping, however, coordinate decoupling occurs only if the system is classically damped. Upon modal transformation, the system generally remains coupled by the off-diagonal elements of its modal damping matrix.; In the analysis of nonclassically damped systems, a common approximation is to ignore the off-diagonal elements of the modal damping matrix. This procedure is termed the decoupling approximation, which amounts to neglecting coupling of the principal coordinates. Intuitively, the errors of decoupling approximation should be small if either the off-diagonal elements of the modal damping matrix are small or if the natural frequencies of the system are well separated. Contrary to these widely accepted beliefs, examples are provided to demonstrate that neither of these two criteria would be sufficient for decoupling approximation. In fact, coupling effect can even increase as the off-diagonal elements of the modal damping matrix decrease in magnitude.; The purpose of this dissertation is to research into the characteristics of coordinate coupling in damped linear vibratory systems. The decoupling approximation is carefully re-examined in both the time and frequency domains. The reasons why the errors of the decoupling approximation can increase as the modal damping matrix becomes increasingly diagonal will be given. Sufficient conditions for the decoupling approximation will be determined. New coupling indices for determining modal coupling quantitatively will also be introduced. In addition, local and global sensitivity of the errors with respect to the off-diagonal elements of the modal damping matrix will be investigated.; Coordinate coupling plays a central role not only in vibration but also in numerical linear algebra and science. Research into the characteristics of coordinate coupling may lead to fundamental advances in vibratory systems and many other areas of physical science.