  In this paper, we consider the problem of finding surrogate models for
large-scale second-order linear time-invariant systems with inhomogeneous
initial conditions. For this class of systems, the superposition principle
allows us to decompose the system behavior into three independent components.
The first behavior corresponds to the transfer between the input and output
having zero initial conditions. In contrast, the other two correspond to the
transfer between the initial position and the initial velocity conditions
having zero input, respectively. Based on this superposition of systems, our
goal is to propose model reduction schemes allowing to preserve the
second-order structure in the surrogate models. To this aim, we introduce
tailored second-order Gramians for each system component and compute them
numerically, solving Lyapunov equations. As a consequence, two methodologies
are proposed. The first one consists in reducing each of the components
independently using a suitable balanced truncation procedure. The sum of these
reduced systems provides an approximation of the original system. This
methodology allows flexibility on the order of the reduced-order model. The
second proposed methodology consists in extracting the dominant subspaces from
the sum of Gramians to build the projection matrices leading to a surrogate
model. Additionally, we discuss error bounds for the overall output
approximation. Finally, the proposed methods are illustrated by means of
benchmark problems.
