Co-Ordinate Transformations for Second Order Systems. Part Ii: Elementary Structure-Preserving Transformations

The decoupling of defective linear dynamical systems in free motion

“…It has been illustrated in the literature that it is possible to decouple a non-defective system (1) while preserving both the algebraic and geometric multiplicities of the associated eigenvalues (i.e., the decoupling transformation is strictly isospectral). Phase synchronization [39,40] in the n-dimensional configuration space is an example of how decoupling is achieved in such a way, as is the method of structure-preserving transformations in the 2n-dimensional state space proposed by Garvey and others [31][32][33][34]. If one insists that geometric multiplicities be preserved, then a defective system (1) may only be partially decoupled [46].…”

“…Moreover, unlike classical modal analysis, complex modal analysis provides little in the way of physical insight since the complex congruence transformation involved generally makes it impossible to identify the 2n state variables with displacements and velocities. Alternative methods for decoupling include the recently proposed structure-preserving transformations by Garvey and others [31][32][33][34], but the case of defective systems (that is, those systems with eigenvalues which do not have corresponding eigenvectors) is not considered. In fact, it is quite common in the literature to avoid the issue of systems with defective eigenvalues, a trend likely motivated by the practical reason that it is rare to obtain exactly repeated eigenvalues.…”

The decoupling of second-order linear systems with a singular mass matrix

“…Similarly, Newland demonstrated that the original n-degree-of-freedom, differential-algebraic system (1) may be rewritten as a first-order differential equation through a reduction method (see [Chapter 6,14]), but his approach is limited to homogenous systems with a mass matrix of rank nÀ1. Finally, Garvey et al [15,16] have recently proposed decoupling through what they term "structure-preserving transformations." While their work focuses on the case when the mass matrix M is invertible, the authors do briefly mention how their decoupling methodology may be extended to include the case when M is singular.…”

The decoupling of damped linear systems in free or forced vibration

“…A traditional approach, as emphasized by Lancaster [29][30][31][32][33], is to address this problem as a reduction of quadratic pencils of matrices. Garvey and others [34][35][36][37][38] recently diagonalized a class of matrix pencils by transformations in state space. To be sure, the problem of simultaneous diagonalization of matrices can also be interpreted from other perspectives [39][40][41][42].…”

“…Phase synchronization of a non-oscillatory system amounts to introducing complex phase shifts to each damped mode in Eq. (35) …”