Methods of component mode synthesis, such as Craig and Bampton reduction, are known to generally yield more accurate results in deformable multibody dynamics. The main shortcoming of those methods is that they are intuitively based. Recently Nikravesh developed a reduction method called mode condensation which is derived from the equations of motion and yields the same results as Craig and Bampton reduction. In this dissertation, it is proven that these two methods span the same column space; therefore, they should yield identical results. We propose that mode condensation provides an analytical justification for Craig and Bampton reduction. Test results suggest that Craig and Bampton reduction and mode condensation are appropriate for a broader range of applications because their column space matches up well with the conditions under which the deformable body is restrained. Although Guyan reduction preserves exact solutions for static problems, its applications shall be limited to low frequency excitation because of raised eigen-frequencies. Modal truncation is not recommended for use in multibody dynamic settings because it lacks the ability to receive forces and displacements at the moving boundary. Another issue addressed in this dissertation is the misconception that if mean axes are adopted as the moving reference frame, only free-free modes should be used for model reduction. It was not clear how a restrained deformable body with mean axes can be condensed properly. We have shown that the conventional (nodal-fixed) mode shapes can be used with mean axes as long as the transformation matrix has full rank and contains complete rigid-body mode shapes.