Resonant phenomena of elastic bodies that perform bending and torsion vibrations

Abstract

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.

Asymptotic method in investigation of complex nonlinear oscillations of elastic bodies

“…However, to study the complex oscillations of even the simplest physical models of structural elements, there are mathematical difficulties. To overcome them in recent years, has gained a new impetus the development of the method, which is based on the use of partial information about the dynamics of the process [15,16]. Further use of it in a mathematical model makes it simpler, and on the other hand, it allows to trace such complex processes as resonant phenomena caused by external factors and the process of interaction of some oscillations with others (internal resonances [13,17]).…”

Methodology for Increasing the Efficiency of Dynamic Process Calculations in Elastic Elements of Complex Engineering Constructions

“…Some experimental and theoretical studies show that components of vibrations of structural elements, in some cases interacting with each other, can lead to significant dynamic loads in them. These loads increase rapidly in magnitude in the so-called resonant cases, when the frequencies of the component oscillations (bending, longitudinal, torsional) or external and natural frequencies are related by rational relations [2,9] (i.e., oscillations in bodies with material properties that are or are nearly linearly elastic). To prevent them means to extend the service life of elastic bodies, i.e., structures as a whole, only on the basis of mathematical models adequate to the dynamic process and their solutions.…”

“…To prevent them means to extend the service life of elastic bodies, i.e., structures as a whole, only on the basis of mathematical models adequate to the dynamic process and their solutions. If such phenomena under the action of external perturbation for both quasilinear and nonlinear systems with concentrated masses have found a relatively proper development (see, for example [1,9]), the issues of internal resonances for systems with distributed parameters were considered only in some works [9,10]. To prevent such phenomena, it is necessary to know in advance about the reaction of the object under study to a particular type of disturbance, as well as the result of the interaction of certain types of oscillations.…”

Advancing asymptotic approaches to studying the longitudinal and torsional oscillations of a moving beam

“…With the development of new technology, with increasing speeds, that is, during the transition to high-speed engineering, the role of fluctuations in the elements of mechanisms has become especially relevant. In particular, this concerns the processes of oscillation of the drilling part of the rig (column) when drilling oil and gas fields [1], fluctuations in the ring combustion chamber in cars [2], vibrations in the elements of power plants [3]. The stricter operational requirements for the safe and efficient operation of modern machines lead to the fact that increased attention in engineering calculations is paid to solving problems associated with longitudinal and torsional oscillations.…”