Longitudinal oscillations and flights of the string pendulum driven by a periodic force

The longitudinal oscillations of a string pendulum are discussed. The regime in which the oscillating bob executes free flight under gravity is studied. This regime arises if, during the motion in the upward direction, bob reaches a height at which no elastic force acts on bob, and its further motion occurs only under gravity. Thus, the string pendulum executes a motion of two types: the harmonic oscillations under the action of elastic force and free flights under gravity. In doing so, the height of flights can considerably exceed the amplitude of the forced oscillations, i.e., a noticeable amplitude amplification in the longitudinal oscillations of a string pendulum takes place. A mechanism of a resonance phenomenon related to the excitation of free oscillations is described. These free oscillations arising after each bob flight are the channel for energy pumping into the system in question. It turns out that the frequency of the above resonance is equal to the fundamental frequency of the string, and the resonance amplitude of the flights is proportional to the square of amplitude of regular forced oscillations.