The Decoupling of Linear Dynamical Systems
Decoupling a second-order linear dynamical system requires that one develop a transformation that simultaneously diagonalizes the coefficient matrices that define the system in terms of its distribution of inertia and viscoelasticity. A traditional approach to decoupling a viscously damped system uses the eigenvectors of the corresponding undamped system to diagonalize the mass, damping, and stiffness matrices through a real congruence transformation in the configuration space, a process known as classical modal analysis. However, it is well known that classical modal analysis fails to decouple a linear dynamical system if its damping matrix does not satisfy a commutativity relationship involving the system matrices. Such a system is said to be non-classically damped. We demonstrate that it is possible to decouple any non-classically damped system in the configuration and state spaces through generally time-dependent transformations constructed using spectral data obtained from the solution of a quadratic eigenvalue problem.
When a non-classically damped system has complex but non-defective eigenvalues, the effect of non-classical damping is that it introduces constant phase shifts among the components of the system's free response. Decoupling of free vibration in the configuration space is achieved through a real, linear, time-shifting transformation that eliminates these phase differences, yielding classical modes of vibration. This decoupling transformation, referred to as phase synchronization, preserves both the eigenvalues and their multiplicities. When cast in a state space form, the transformation between the coupled and decoupled systems is real, linear, but time-invariant. Through the concept of real quadratic conjugation, we illustrate that there is no fundamental difference in the representation of the free response of a system with complex eigenvalues and one with real eigenvalues, and thus systems with non-defective real eigenvalues can also be decoupled by phase synchronization. When phase synchronization is extended to forced systems, the decoupling transformation in both the configuration and state spaces is nonlinear and depends continuously on the applied excitation.
If a non-classically damped system is defective, it may only be partially decoupled if one insists on preserving the geometric multiplicities of the defective eigenvalues. We present the first systematic effort to decouple defective systems in free or forced vibration by not demanding invariance of the geometric multiplicities. In the course of this development, the notion of critical damping in multi-degree-of-freedom systems is clarified and expanded. It is shown that the decoupling of defective systems is a rather delicate procedure that depends on the multiplicities of the system eigenvalues. A generalized state space-based decoupling transformation is developed that relates the response of any non-classically damped system to that of its decoupled form. In principle, one could extract from the state space a decoupling transformation in the configuration space, but it generally does not have an explicit form. The decoupling transformation in both the configuration and state spaces is real and time-dependent. Several numerical examples are provided to illustrate the theoretical developments.