Coordinate Transformations for Second-Order Systems: Part II Elementary Structure Preserving Transformations
SD Garvey (University of Nottingham), U Prells & MI Friswell (University of Wales Swansea)
Journal of Sound and Vibration, Vol. 258, No. 5, December 2002, pp. 911-930
It has been shown in a previous paper  that there is a real-valued transformation from the general N degree of freedom second order system to a second order system characterised by diagonal matrices. An immediate extension of this fact is that for any second order system, there is a set of real-valued transformations (the structure-preserving transformations) which transform this system to a different second order system having identical characteristic behaviour. There are several possible reasons why it may be very useful to achieve a particular structure in the transformed system. It is obvious that a diagonal structure is extremely useful and one method has been exposed  by which determining the diagonalising transformation from the solution of the usual (complex) eigenvalue-eigenvector problem.
This paper begins by outlining the usefulness of some other structures. Then it defines a class of elementary structure-preserving coordinate transformations that transform from one N degree of freedom second order system to another. The term elementary is applied because any one of these transformations is the minimum-rank modification of the identity transformation. The changes occurring in the system matrices as a result of the application of one such elementary transformation transpire to be very simple in form, they are low rank, and they can be computed very efficiently.
This paper provides the fundamental tools to enable the design of structure preserving coordinate transformations which transform a second order system originally characterised by three general matrices in stages into a mathematically-similar second order system characterised by three diagonal matrices. The procedure by which the individual elementary transformations are obtained is still under development and it is not discussed in this paper. However, an illustration is given of a 5 degree of freedom self-adjoint system being transformed into tridiagonal form.
This material has been published in the Journal of Sound and Vibration, Vol. 258, No. 5, December 2002, pp. 911-930, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Elsevier Science.
Link to paper using doi:10.1006/jsvi.2002.5166
Journal of Sound and Vibration on ScienceDirect