A. K. Belyaev

Parametric resonances in the problem of longitudinal impact on a thin rod

The longitudinal impact on a thin elastic rod, which generates a periodic system of longitudinal waves in it, is considered. At definite values of the parameters of the problem in the linear approximation, these waves induce parametric resonances, which are accompanied by an unlimited increase in the amplitude of the transverse vibrations. To obtain finite values of the amplitudes, a quasilinear system is considered in which the effect of the transverse vibrations on the longitudinal vibrations is taken into account. This system was previously solved using the Bubnov–Galerkin method and beats accompanied by energy transfer between the transverse and longitudinal vibrations were obtained. In this work, an approximate analytical solution of the system has been derived that is based on double-scale expansions. A qualitative analysis of this solution has been carried out. An estimate of the maximum transverse bending has been obtained for various methods of loading. Both shortand long-term pulses have been considered. It has been shown that, in the case of a spontaneously applied long-term pulse that is lower than the Euler critical load, intensive transverse vibrations can occur.

Beating in the problem of longitudinal impact on a thin rod

The longitudinal impact on an elastic rod generating a periodic system of longitudinal waves in the rod, is considered. For certain values of the problem parameters in the linear approximation, these waves generate parametric resonances accompanied by an infinite increase in the transverse vibrations amplitude. To obtain the finite values of the amplitudes, a quasilinear system where the influence of transverse vibrations on the longitudinal ones is taken into account was considered. Earlier, this system was solved numerically by the Bubnov—Galerkin method and the beatings accompanied by energy exchange between the longitudinal and transverse vibrations were obtained. Here an approximate analytic solution of this system based on two-scale expansions is constructed. A qualitative analysis is performed. The maximum transverse deflection depending on the loading method is estimated.

The Ishlinskii—Lavrent’ev problem at the initial stage of motion

The dynamic loss of stability of a thin rod with hinge-supported edges under the action of an impact constant longitudinal load at the initial stage of motion, which is restricted to the time of longitudinalwave run along the rod length, is investigated. The transverse deflection is expanded in the Fourier series. The problem is solved in the linear approximation. The additional deflection is compared with the value of the initial disturbances.

Thin rod under longitudinal dynamic compression

The paper contains a short survey of the papers on the static and dynamic longitudinal compression of a thin rod initiated by Morozov and and carried out in 2009–2016 with his direct participation. We consider linear and nonlinear problems related to the propagation of longitudinal waves in a rod and the transverse vibrations generated by these waves; parametric resonances; beating due to energy exchange between longitudinal and transverse vibrations; the rod shape evolution as the load exceeds the Euler critical value; the possibility of buckling of the rod rectilinear shape under a load less than the Euler load; and the rod dynamics at the initial stage of motion. The prospects of further investigations related to the complication of the models are considered, in particular, the problem of longitudinal impact by a body on a rod and the transverse vibrations generated by it.

Rheological Model of Materials with Defects Containing Hydrogen

A two-continua model is constructed which allows one to describe the kinetics of hydrogen in metals. The developed rheological model is appropriate for estimation of the hydrogen transition from mobile to bonded state depending on the stress-state relation and description of the localization of the connected hydrogen that results in the material fracture.

Contact of Flexible Elastic Belt with Two Pulleys

The drive belt set on two pulleys is considered as a nonlinear elastic rod deforming in plane. The modern equations of the nonlinear theory of rods are used. The static frictionless contact problem for the rod is derived. The nonlinear boundary value problems for the ordinary differential equations are solved by the finite differences method and by the shooting method by means of computer mathematics. The belt shape and the stresses are determined in the nonlinear formulation which delivers the contact reaction and the contact area. The developed method allows performing calculations for any set of geometrical and stiffness parameters.

Initial stage of motion in the Lavrent’ev–Ishlinskii problem on longitudinal shock on a rod

The transverse motion of a thin rod under a sudden application of a prolonged longitudinal load at the initial stage of motion is considered. The introduction of self-similar variables makes it possible to propose a description of the transverse motion weakly dependent on the longitudinal deformation. Both single dents and periodic systems of dents are considered.

Rod Vibrations Caused by Axial Impact

The axial impact by an elastic body on an elastic-rod end with a fixed opposite end is considered. The propagation of elastic waves in the rod and the local deformations in the contact zone are taken into account. After recoil of the body, the rod performs free longitudinal vibrations, which can under certain conditions cause parametric transverse vibrations having the character of beats. Depending on the parameters of the problem, the collision time, the shock-pulse shape, and the greatest amplitude of the transverse vibrations under parametric resonance are determined.

Dynamics of a rod undergoing a longitudinal impact by a body

A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.

Buckling problem for a rod longitudinally compressed by a force smaller than the Euler critical force

It was earlier shown that a rod can buckle under the action of a sudden longitudinal load smaller than the Euler critical load. The buckling mechanism is related to excitation of periodic longitudinal waves generated in the rod by the sudden loading, which in turn lead to transverse parametric resonances. In the linear approximation, the transverse vibration amplitude increases unboundedly, and in the geometrically nonlinear approach, beats with energy exchange from longitudinal to transverse vibrations and back can arise. In this case, the transverse vibration amplitude can be significant. In the present paper, we study how this amplitude responds to the following two factors: the smoothness of application of the longitudinal force and the internal friction forces in the rod material.