Second order to second order balancing for index-1 vibrational systems

Linearizing constraint equations of motion around equilibrium points in mechanics or coupling electrical and mechanical parts in mechatronics one obtains large sparse second-order index-1 differential algebraic (DAE) models. To get reduced order models of such systems, first they can be rewritten into first-order models. Then, model reduction techniques are applied to these first order representations to get a reduced first-order index-1 or standard state space model. Unfortunately, it is not possible to go back to second order formulation from these reduced systems, though it is often desirable to work with second order surrogate models. In this paper, we present algorithms to retrieve reduced index-1 DAE or standard second order ODE systems and apply these to a micro-mechanical piezo-actuated structural FEM model of a certain building block of a machine tool. Numerical results illustrate the efficiency of the techniques.