Free vibration predicted using forced oscillation in the lock-in region

The rationale of using the results from the transverse forced oscillation of a body in the lock-in region to predict the corresponding free vibration is provided based on the mathematical analysis and verified through the numerical results at Re = 106 obtained from the immersed boundary-lattice Boltzmann method. It is also shown through mathematical analysis that when the structural damping is fixed, if the body mass and stiffness vary together following a particular relationship, the free motion will remain the same. With this conclusion, the damping ratio is redefined using the motion frequency of the body instead of the commonly adopted natural frequency of the body. As a result, it is shown that when a cylinder is in periodic free motion, its motion will remain the same if the following two conditions are satisfied: (1) the combined mass-damping parameter remains unchanged and (2) the variations of body mass and stiffness follow a particular pattern. In this sense, the sinusoidal free motion will rely only on the combined mass-damping parameter, even at the low mass region. This is different from the previous result based on the conventionally defined mass-damping parameter. Motion amplitude and frequency contours are plotted against the damping and stiffness components based on the results of forced motion. From these, the sinusoidal free motion results can be predicted and their physics is discussed.