One important issue for the simulation of flexible multibody systems is the quality controlled reduction in the flexible bodies degrees of freedom. In this work, the procedure is based on knowledge about the error induced by model reduction. For modal reduction, no error bound is available. For Gramian matrix based reduction methods, analytical error bounds can be developed. However, due to numerical reasons, the dominant eigenvectors of the Gramian matrix have to be approximated. Within this paper, two different methods are presented for this purpose. For moment matching methods, the development of a priori error bounds is still an active field of research. In this paper, an error estimator based on a new second order adaptive global Arnoldi algorithm is introduced and further assists the user in the reduction process. We evaluate and compare those methods by reducing the flexible degrees of freedom of a rack used for active vibration damping of a scanning tunneling microscope.