The purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non-oscillatory free vibration or in forced vibration. Based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations, the concept of damped modes of imaginary vibration is introduced. By phase synchronization of these real and physically excitable modes, a time-varying transformation is constructed to decouple non-oscillatory free vibration. When time drifts caused by viscous damping and by external excitation are both accounted for, a time-varying decoupling transformation for forced vibration is derived. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the second and final part of a solution to the "classical decoupling problem." Together with an earlier paper, a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system. © 2010 Elsevier Ltd. All rights reserved.

The purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in oscillatory free vibration. Based upon an exposition of how viscous damping causes phase drifts in the components of a linear system, the concept of non-classically damped modes of vibration is introduced. These damped modes are real and physically excitable. By synchronizing the phase angles in each damped mode, a time-varying transformation is constructed to decouple damped oscillatory free vibration. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the first part of a solution to the "classical decoupling problem" of linear systems. © 2009 Elsevier Ltd. All rights reserved.