RUNNING HEADING: ELEMENTARY SECOND ORDER TRANSFORMATIONS

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 coordinat