At the Advanced Photon Source (APS), a state-of-the-art synchrotron radiation facility at Argonne National Laboratory (ANL), high-precision optical positioning systems are needed to conduct a wide range of experiments utilizing the high-brilliance x-ray beam. The high-precision, multi-dimensional positioning capability required for these positioning systems may be compromised by vibratory motion. The vibratory dynamics of the complex kinematic joints and components that comprise these multibody structures are not easily described by simple theoretical models. A combined experimental and theoretical approach has been developed to predict the dynamic properties of the individual components and joints and of the assembled multibody system. A prototypical optical table has been analyzed as an example case.