CO-ORDINATE TRANSFORMATIONS FOR SECOND ORDER SYSTEMS. PART I: GENERAL TRANSFORMATIONS

When the dynamics of any general second order system are cast in a state-space format, the initial choice of the state vector usually comprises one partition representing system displacements and another representing system velocities. Co-ordinate transformations can be defined which result in more general definitions of the state vector. This paper discusses the general case of co-ordinate transformations of state-space representations for second order systems. It identifies one extremely important subset of such coordinate transformations—namely the set of structure-preserving transformations for second order systems—and it highlights the importance of these. It shows that one particular structure-preserving transformation results in a new system characterized by real diagonal matrices and presents a forceful case that this structure-preserving transformation should be considered to be the fundamental definition for the characteristic behaviour of general second order systems—in preference to the eigenvalue–eigenvector solutions conventionally accepted.