P. E. Tovstik

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.

Generalized Timoshenko–Reissner model for a multilayer plate

A multilayer plate with isotropic (or transversally isotropic) layers strongly differing in rigidity is considered. This plate is reduced to an equivalent homogeneous transversally isotropic Timoshenko–Reissner plate whose deflections and free transverse vibration frequencies are close to those of the multilayer plate. By comparison with the exact solution of test three-dimensional problems of elasticity, the error of the proposed method is estimated both for the static problem and for free vibrations. This comparison can readily be carried out for the hinged edges of the plate, and explicit approximate formulas are obtained for the vibration frequencies. The scope of the proposed model turned out to be rather wide (the Young moduli of soft and rigid layers can differ by a factor of 1000). In the case of boundary conditions other than hinged support, a closed-form solution cannot be constructed in general. For rigidly fixed edges, the asymptotic method proposed by V. V. Bolotin is generalized to the case of a Timoshenko–Reissner plate.

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.

The Timoshenko–Reissner generalized model of a plate highly nonuniform in thickness

A thin plate fabricated of material that is transversally isotropic and nonuniform in thickness is considered. The model of the monolayer transversally homogeneous isotropic plate, which is approximately equivalent to a thickness-nonuniform plate in the deflection and in the lowest frequencies of free vibrations, is constructed. The range of applicability of the model constructed is very wide. The main result of this study is a formula for calculating the transverse-shear rigidity of an equivalent transversally isotropic plate.

The impact of the shape of the spectral density of random wave disturbance on the vibrations of a fixed sea-based offshore platform

It was generally assumed earlier in the study of the impact of random wave disturbance on fixed sea-based offshore platforms (FSOP) that the spectral density of wind-driven gravitational waves has the shape described at the Second International Ship Structures Congress held in Delft, Netherlands, in 1964. Since that time, a number of analytical expressions for the description of the spectral density have been suggested. In this paper we study the amplitude of the FSOP vibrations as a function of the shape of the spectral density in the framework of a model with one degree of freedom. Four shapes of the spectral density are considered. We compare the levels of the FSOP vibrations caused by wave disturbance with these shapes of the spectral density with each other and with the level of vibrations caused by harmonic wave disturbance. It has been established that the off-resonance amplitude of the FSOP vibrations depends mostly on the average wavelength and wave height (or on related parameters such as the average frequency and the standard deviation). In the vicinity of resonance, the amplitude of vibrations was proved to noticeably depend on the shape of the spectral density.

Initial supercritical behavior of buckled transversely isotropic elastic medium

The paper is concerned with a transversely isotropic homogeneous elastic medium subjected to uniform compression in the isotropy plane. The medium becomes unstable in the sense of Hadamard [1] at a definite level of initial strain. The critical strain is established to be uniquely determinate from the system of equations of bifurcation of equilibrium; however, there are many modes of buckling corresponding to this strain. A solution of the system of equations of bifurcation is built in the form of doubly periodic functions sinr1x1sinr2x2. The uncertainty of the mode of buckling consists in the fact that the wave numbers r1 and r2 remain arbitrary. In order to determine the relationship between the wave numbers we examine the initial supercritical behavior of the material. It turns out that the only possible modes are the chess-board mode (with r1 = r2) and the corrugation-type mode (when one of the wave numbers r1 or r2 vanishes). The initial supercritical equilibrium is shown as being stable.

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.

Two-dimensional model of plate made of anisotropic inhomogeneous material

Two-dimensional equations of thin elastic plate made of anisotropic inhomogeneous material are delivered by using asymptotic expansion in powers of the relative plate thickness. In the general case anisotropy is described by 21 elastic moduli. Partial cases of elastic moduli are also discussed. The proposed model is compared with the classic models of Kirchhof–Love and of Timoshenko–Reissner.

On the frequency spectrum of free vibrations of membranes and plates in contact with a fluid

A parallelepiped-shaped container, which is completely filled with a perfect incompressible fluid, is considered. The container is covered with an elastic lid, which is modeled by a membrane or a constant-thickness plate. The other faces of the container are nondeformable. The frequency spectrum of small free vibrations of the lid has been obtained taking into account the apparent mass of the fluid the movement of which is assumed to be potential. The main specific feature of the problem formulation is that the volume of the fluid under the cover remains unchanged in the course of vibrations. As a result, the shape of the deflection of the lid should satisfy the equation of constraint, which follows from the condition of preservation of the volume of the fluid under the lid.