Impulse Moments Influence on Single-Frequency Torsional Oscillations of Nonlinear Flexible Bodies

Abstract

A method of analytical study of the influence of impulse moments on nonlinear torsional oscillations of a homogeneous constant cross-section of a body under classical boundary conditions of the first, second, and third types have been developed. When the flexible material properties meet the body close to the power law of flexibility, mathematical models of the process have been obtained. They are the boundary value problems for an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of pulse momentum on the oscillatory process. Under the effect of periodic pulse momentum on a flexible body, resonant and non-resonant processes are possible, and resonant processes occur when the amplitude of natural oscillations approaches a fixed value. The peculiarities of resonant oscillations are established. Relative torsional oscillations of a nonlinear flexible body that rotates around the axis with a constant portable angular velocity are considered, taking into account the periodic action of pulse momentum acting in a fixed cross-section. It is shown that: the amplitude of passing through the main resonance is more significant for larger values of the nonlinearity parameter; the case of the action of pulse momentum closer to the middle of the body. If the initial perturbation amplitude is less than the amplitude at which the resonance occurs in the presence of only internal forces of viscous friction, the external periodic impulse moments of resonance processes have occurred. The reliability of the calculation formulas is confirmed by the fact that in the extreme case known from the literature, resources are obtained related to the dynamics of the respective objects under the continuous action of autonomous perturbation.