Subharmonic Oscillations of Order One Half

This paper deals with subharmonic oscillations of order one half, i.e., oscillations whose fundamental frequency is one half that of the applied force. The steady-state solutions are first sought for various combinations of the system parameters, and their stability is investigated. The transient state of the oscillations is then discussed by the phase-plane method. It is a distinctive feature of nonlinear systems that various types of steady-state response may take place even in the same system depending on different values of the initial conditions. Particular attention is, therefore, directed toward obtaining the relationship between the initial conditions and the resulting subharmonic responses. Several patterns of initial conditions leading to subharmonic responses are shown on the phase plane. The theoretical results are compared with the solutions obtained by analog-computer analysis and found to be in satisfactory agreement with them.