Abstract: To investigate the repeated frequency condition (RFC) for torsional vibration of shafts’ system, the transfer matrix method was adopted. Firstly, the transfer relationship from the boundary to engaging disks of double shafts’ system(DSS) was constructed. Secondly, the RFC of DSS was deduced out and the methods to select mode shape were presented. Finally, the relationship was extended to multilevel transmission system (MTS), and the RFC of this system was explored. The conclusions is this: 1) the necessary RFC requires the existence of joint engaging couple (JEC);2) for DSS, the sufficient is the number of boundary transfer factors (fB) larger than 2;3) the whole system can be split into independent groups, the total multiplicity is the sum of independent solution number of every group, the latter is the number of independent fB=0 inside the group minus 1.

Key words: vibration, natural frequency, torsional vibration, shaft system