Moment-matching model order reduction of second-order dynamical systems with multiple inputs and multiple outputs (MIMO) is considered for damped or proportionally damped systems. It is well-known that moment-matching can be efficiently and numerically sound implemented by Krylov subspace methods. For second-order systems second-order Krylov subspaces and an appropriate second-order Krylov method have to be employed.

As model reduction of linear first-order systems is much further developed and understood, it is tempting to transform the original second-order system to a mathematically equivalent first-order system and to employ known model order reduction methods. But this approach doubles the size of the matrices to be considered. Moreover, it may be difficult to retrieve the second-order reduced system from the first-order reduced one.

For the problem investigated, it turns out that the second-order Krylov subspaces are identical to certain first-order ones, so that no linearization is necessary, but a first-order model reduction method can be employed. The problem size does not increase.

The discussion will be restricted to rational block Arnoldi methods, in particular the global Arnoldi method. A new model reduction algorithm for second order MIMO systems is proposed which automatically generates a reduced system of given order approximating the transfer function in the lower range of frequencies. It determines the expansion points iteratively and the number of moments matched per expansion point adaptively. An extension of the proposed method to a fully automatic one also iteratively determining the order of the reduced system is work in progress.

Numerical examples comparing our results to modal reduction for a problem arising in the numerical simulation of mechanical structures are presented.