Congruent coordinate transformations are used to convert second-order models to a form in which the mass, damping and stiffness matrices can be interpreted as a passive mechanical system. For those systems which can be constructed from interconnected mass, stiffness and damping elements, it is shown that the input-output preserving transformations can be parameterized by an orthogonal matrix whose dimension corresponds to the number of internal masses – those masses at which an input is not applied nor an output measured. Only a subset of these transformations result in mechanically realizable models. For models with a small number of internal masses, complete discrete mapping of the transformation space is possible permitting enumeration of all mechanically realizable models sharing the original model’s input-output behavior. When the number of internal masses is large, a nonlinear search of transformation space can be employed to identify mechanically realizable models. Applications include scale model vibration testing of complicated structures and the design of electro-mechanical filters.