UC Berkeley

It has long been recognized that coordinate coupling in damped linear systems is a considerable barrier to analysis and design. In the absence of viscous damping, a linear system possesses classical normal modes, which constitute a linear coordinate transformation that decouples the undamped system. This process of decoupling the equation of motion of a dynamical system is a time-honored procedure termed modal analysis. A viscously damped linear system cannot be decoupled by modal analysis unless it also possesses a full set of classical normal modes, in which case the system is said to be classically damped. Rayleigh showed that a system is classically damped if its damping matrix is a linear combination of its inertia and stiffness matrices. Classical damping is routinely assumed in design and computations. Practically speaking, classical damping means that energy dissipation is almost uniformly distributed throughout the system. In general, this condition is not satisfied and thus damped linear systems cannot be decoupled by modal analysis. The purpose of this presentation is to report on a recently developed methodology to extend classical modal analysis to decouple any damped linear system (no restrictions). This method is based upon a new theory of phase synchronization, which compensates for time drifts caused by viscous damping and external excitation. A fast algorithm for decoupling any damped linear system is also described.