Resonant and Bifurcation Oscillations of the Rod with Regard to the Resistance Forces and Relaxation Properties of the Medium

Abstract

A mathematical model of elastic oscillations of a rod under the influence of an external harmonic load, taking into account the relaxation properties and forces of the medium resistance, has been developed. The derivation of the differential equation of the model is based on taking into account the time dependence of the stresses and strains in the formula of Hooke’s law, which, when presented in this way, coincides with the formula of the complicated Maxwell and Kelvin-Voigt models. The study of the model using numerical method showed that when the frequency of the natural oscillations of the rod coincides with the frequency of the external load oscillations (if the resistance of the medium and its relaxation properties are not taken into account), the amplitude of the oscillations (resonance) increases unlimited in time. When taking into account the resistance and relaxation properties of the medium at resonant frequencies, the amplitude of oscillations stabilizes on a value depending on the values of the resistance and relaxation coefficients. At frequencies close to resonant, bifurcation oscillations (beats) are observed, at which there is a periodic increase and decrease of the amplitude of oscillations. At frequencies substantially different from resonant ones, in the case of taking into account resistance forces and relaxation properties of materials, bifurcation oscillations are not observed. In this case, the amplitude of oscillations is stabilized in time at a value depending on the amplitude of oscillations of the external load, the resistance coefficient and the relaxation coefficients.