Resonant phenomena of elastic bodies that perform bending and torsion vibrations
2018
The method of study of the influence of torsional oscillations of one-dimensional
models of nonlinear elastic bodies, along which moves with a constant velocity continuous flow of
inelastic homogeneous medium, into bending, is developed. It is believed that information on
torsional oscillations is known from empirical studies. Based on the latter, the refined model of the
dynamics of the process of the investigated object is constructed. The latter is a boundary value
problem for nonlinear nonautonomous differential equations with partial derivatives. The imposed
restrictions on power factors and the main parameters of torsional oscillations allow for the
analytical study of the dynamics of the process to use the basic ideas of the asymptotic integration of
equations with partial derivatives. With their help, we obtain a two-parameter set of solutions that
describe the determinant parameters of bending vibrations of an elastic body. It is established that
for the considered elastic body there can be resonance oscillations, which are caused not only by
external factors, but also by internal – torsional oscillations. Regarding the law of the change in the
basic parameters of the dynamics of the bending motion of an elastic body, its rotation around the
vertical axis reduces the frequency of its own flexural oscillations of the body, and even small
torsional oscillations cause an additional periodic action on the transverse. In connection with the
above bending vibrations of the elastic body, which performs complex oscillations (torsion and
bending), resonances are possible both at the frequency of the external periodic perturbation and at
the frequencies of the torsional oscillations (internal resonances).
The amplitude of the transition through the resonance: a) at the basic frequency of external
perturbation takes less value for elastic bodies of greater flexural rigidity and for higher values of the
relative motion of the medium; b) at the frequency of torsional oscillations for larger values of the
angular velocity takes more importance; c) with “fast” transition through resonance at the frequency
of external or internal perturbation is less than with “slow”. The obtained results can serve as the
basis for the choice of operating parameters of elastic elements of machines that carry out complex
oscillations.
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