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. Coordinate transformations can be defined which result in more general definitions of the state-vector. This paper discusses the general case of coordinate 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 characterised 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.