Modeling the resonance of a swinging spring based on the synthesis of a motion trajectory of its load

The paper reports a technique for building the resonance trajectories of the motion of a swinging spring load. A swinging spring is the kind of a mathematical pendulum consisting of a point load attached to a weightless spring. The other end of the spring is fixed immovably. We have considered the pendulum-like spring oscillations in a vertical plane provided its axis straightness is maintained. Calculations have been performed based on the solutions to a system of differential equations with components that include values for the frequency values of vertical and horizontal displacements of a point on a spring. The relevance of the subject is predetermined by the necessity to study the technological processes of dynamic systems when the nonlinearly connected oscillatory components of the system exchange energy. Using a swinging spring phenomenon illustrates the exchange of energies between the transverse (pendulum) and longitudinal (spring) oscillations. In this case, we also take into consideration the influence of the initial conditions for initiating oscillations. Of particular importance is to study the resonance state of a swinging spring when the frequency of longitudinal oscillations differs by a multiple number of times from the frequency of transverse oscillations. In addition to a common «classic» case (resonance 2:1), there is a need to consider cases with different values for the frequency ratio. The result is the derived geometric shapes of the motion trajectory of a swinging spring load that correspond to the patterns in the state of its resonance. The results obtained in the current paper make it possible, by using a computer, to synthesize the motion trajectory of a swinging spring load that would match the assigned frequency ratio of longitudinal and transverse oscillations. For this purpose, in addition to basic parameters (a load’s mass, rigidity of the spring, its length in a no-load state), we added the initial values for the parameters during oscillation initiation. Specifically, the «starting» coordinates for a load position, and the initial load motion velocities in the direction of the coordinate axes. We have considered examples of building a load motion’s trajectories for cases of resonances the type of 2:1, 7:3; 9:4; and 11:2. The results obtained are illustrated by the computerized animations of oscillations of appropriate swinging springs for different cases of resonance. The results could be used as a paradigm in order to study the nonlinear connected systems, as well as in the calculation of variants for mechanical devices where springs affect the oscillation of their elements. Additionally, for cases when the technology of using mechanical devices necessitates abandoning the chaotic movements of loads in order to ensure the periodic trajectories of their displacements.