The phasing effect of the slot/magnet combination on rigid-elastic vibration is addressed by incorporating the cyclic symmetry of permanent magnet (PM) motors. Expanding research is also carried out to achieve more general findings in rotary power-transmission systems widely available in practical engineering. To these aims, model-free analysis is used to deal with the effect via superposition treatment. The results imply that the vibration induced by temporal-spatial excitation can be classified into rotational, translational, and balanced modes, all of which have rigid and elastic vibrations having specific base and/or contaminated deflections, and the elastic vibration can be of the standing, forward traveling, and backward traveling waves. These modes can be suppressed or excited depending on whether particular algebraic relationships are satisfied by slot/magnet combination, excitation order, and base and contaminated wave numbers. Since the analysis is independent of any models, specified magnetic force, and rigid-elastic vibration, analytical results regarding the expected relationships can be naturally created due to the structural and force symmetries of the PM motors. Because of this, similar results can be found for other rotary systems basically consisting of a rotary rotor and a stationary stator both having equally-spaced features, apart from the PM motors, typically including the turbine machines having fluid field and planetary gears with a mechanical contact. As an engineering application, the proposed method can serve as a fundamental tool when predicting or even suppressing the possible excitations associated with particular vibration modes in the mechanical and electrical designs of the symmetric systems. The superposition effect and analytical predictions are verified by the finite element method and strict comparisons against those from disk-shaped structures in an existing study.