An asymptotic approach to the problem of the free oscillations of a beam

Equations describing the free small longitudinal and transverse oscillations of a straight elastic beam of rectangular cross section are obtained using the plane linear theory of elasticity and the method of integrodifferential relations. The initial system of partial differential equations is reduced to a system of ordinary linear differential equations with constant coefficients. The effect of the geometrical and elastic characteristics of the beam on the frequency and form of the natural oscillations is investigated. For longitudinal motions it is shown that different types of natural displacements and internal stresses of the beam exist. For transverse oscillations, it is found that there are frequency zones corresponding to different forms of the solutions of the characteristic equation obtained using the proposed model.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

Oscillations of a rigid body containing an elastic element with distributed parameters

The motions of a hybrid (discrete-continual) system, consisting of a carrier rigid body and an elastic element with distributed parameters fastened to it are investigated. Two types of fastening are considered: (1) both ends are clamped, and (2) one of the ends is clamped while the other is free. A closed system of integro-differential equations is obtained which describes the state of the system under arbitrary initial conditions and forces applied to the rigid body. The perturbed motion of the rigid body in the case of a quasi-linear restoring force is investigated using asymptotic methods. The motions are studied both when there is internal resonance between the oscillations of the rigid body and the natural oscillations of the element, and when there are no such resonances. Qualitative effects are found.     

Eine genäherte Behandlung des schweren symmetrischen Kreisels in nicht-Eulerschen Koordinaten

Summary The gyroscope has many technical applications as an essential member in a system with several degrees of freedom, the (dynamic) stability of which with respect to a certain equilibrium position may be of particular interest. To investigate the stability of a system by the well-known method of small oscillations, the movements of the different members must be described by convenient coordinates. As to the gyroscope, it is necessary for this purpose to replaceEuler's angles by other coordinates, with which the rigorous equations of its movement are deduced. Assuming only small displacements of the axis of symmetry from its undisturbed equilibrium position, and by making the problem dimensionless, a simple solution can be obtained for the nutation and precession of a heavy gyroscope both as a symmetrical top and as a gyroscopic pendulum. Further, the damping of the gyroscope by its movement around the equilibrium position has also been considered in order to be able to estimate its influence on the stability.     

The equations of the perturbed motion of a rocket as a thin-walled structure with a liquid

The perturbed motion of a rocket as an elastic thin-walled structure with compartments partially filled with liquid propellant is considered. It is assumed that the normal modes of the hydroelastic oscillations of the rocket are determined under the condition that the velocity potential on the free surface of the liquid is equal to zero and with standard remaining conditions. Certain features of these modes with zero fundamental frequencies are pointed out and the “loss” of mass effect associated with this is explained. Equations are derived for the perturbed motion of a rocket taking account of the hydroelastic oscillations of its structure and the oscillations of the liquid with deviations of the free surface from the equilibrium position under the action of mass forces. The coefficients of these equations, characterizing the relation between the different type of oscillations, are expressed in terms of known hydrodynamic parameters and the values of the oscillation modes at certain points.     

Free oscillation of a curvilinear rod

A method and finite-element algorithm are presented for determining the frequencies and form of free oscillations of elastic curvilinear rods with finite rigidity under deflection. Numerical examples are introduced in order to investigate the dependence of the precision of the calculated frequencies and the form of the oscillations on the method of division into finite elements.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 72–76, 1989.     

Elastic Oscillations of a Half-Space Excited by Free Oscillations of a Heavy Fluid on Its Surface

An approximate mathematical model is constructed to account for the presence of long-period elastic deformations of the Earth's surface as a result of free oscillations of closed, bounded bodies of water (inland seas). The model provides a basis (ignoring the Earth's rotation) for calculating the frequencies and corresponding waveforms of free oscillations of the Black Sea and the Sea of Azov, which are treated as isolated basins of shallow, variable depth. The theoretical amplitudes of the components of the strain tensor and other elastic characteristics of the medium are calculated at a given point of the Earth's surface (recording instrument station).     

Single-axis vibratory gyroscopes

The paper deals with the principles of single-axis vibratory gyroscope operation. A mathematical model for a single-mass vibratory gyroscope and some modes of its operations are considered and relations between measured values (angle of rotation or angular rate) and sensitive mass generalized coordinates are described. Some details for the forced oscillations mode used in measuring of the angular rate and for two modes of natural oscillations used in measuring of the rotational angle and angular rate are described. A new method for identification of anisoelasticity in a single-axis vibratory gyroscope is suggested. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 8, pp. 149–163, 2005.     

Homogenization of a spectral problem for a pile foundation structure

The subject of this paper is the study of free oscillations of a composite material consisting of an elastic medium and a large number of elastic piles periodically immersed in a part of the medium. The ends of piles lying on the boundary of the medium are clamped to a force-free rigid plate. Such a partially composite structure arises in pile foundation mechanics. Assuming that the period of pile structure tends to zero, we obtain the macroscopic equations for free oscillations by homogenization method and prove the corresponding convergence result. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Coupled Frequencies of a Fluid in Circular Cylindrical Tank with an Elastic Inclusion in its Roof

A closed upright rigid circular cylindrical tank is filled with a compressible and inviscid fluid as a part of the tank roof is an elastic plate. It is supposed that the tank roof and the elastic plate are eccentrically and the plate is clamped. The problem about the determination of the free coupling vibrations of the received hydroelastic system is considered. Using the Bubnov‐Galerkin method, the frequency equation is obtained. Some numerical examples are made and the results show the influence of the sizes of the radius of the elastic plate as well as of the other parameters of the hydroelastic system on its coupled frequencies.     

Numerical simulation of vortex induced vibrations and its control by suction and blowing

The vortex formation and shedding behind bluff structures is influenced by fluid flow parameters such as, Reynolds number, surface roughness, turbulence level, etc. and structural parameters such as, mass ratio, frequency ratio, damping ratio, etc. When a structure is flexibly mounted, the Kármán vortex street formed behind the structure gives rise to vortex induced oscillations. The control of these flow induced vibrations is of paramount practical importance for a wide range of designs. An analysis of flow patterns behind these structures would enable better understanding of wake properties and their control. In the present study, flow past a smooth circular cylinder is numerically simulated by coupling the mass, momentum conservation equations along with a dynamical evolution equation for the structure. An active flow control strategy based on zero net mass injection is designed and implemented to assess its efficacy. A three actuator system in the form of suction and blowing slots are positioned on the cylinder surface. A single blowing slot is located on the leeward side of the cylinder, while two suction slots are positioned at an angle α = 100°. This system is found to effectively annihilate the vortex induced oscillations, when the quantum of actuations is about three times the free stream velocity. The dynamic adaptability of the proposed control strategy and its ability to suppress vortex induced oscillations is verified. The exact quantum of actuation involved in wake control is achieved by integrating a control equation to decide the actuator response in the form of a closed loop feed back system. Simulations are extended to high Reynolds number flows by employing eddy viscosity based turbulence models. The three actuator system is found to effectively suppress vortex induced oscillations.     

Free oscillations of plastic structures taking into account the elastic and damping properties of the material

The problem of the free oscillations of a plastic bar with a weight at one end is examined. A method is proposed for calculating the free oscillations of plastic structures, taking into account the special elastic properties and hysteresis losses of plastics. The original equation for the motion of the system, simplified by certain transformations and assumptions, is solved to a first approximation by the Krylov-Bogolyubov asymptotic method (following G. S. Pisarenko). The calculated data are in good agreement with the experimental findings.Mekhanika polimerov, Vol. 1, No. 1, pp. 136–145, 1965     

Frequencies of Free Acoustic Oscillations in a Fluid Separating Rigid and Elastic Tube Walls of Finite Length

A solution of the problem of determining the frequencies and mode shapes of free nonsymmetric oscillations in an annular volume filled with an ideal compressible fluid is constructed. The inner tube and the end plane walls are ideally rigid. A thin elastic shell with edges clamped to the end walls is located on the outer tube boundary. A phenomenon of a decrease in the fundamental frequency as the thickness of a fluid layer adjacent to the elastic wall decreases is confirmed. Bibliography: 8 titles.     

The transmission of a flexural-gravitational wave through several straight obstacles in a floating plate

Periodic wave processes in a thin elastic plate floating on the surface of an incompressible fluid of finite depth are studied. The plate completely covers the fluid surface and executes flexural oscillations under the action of gravitational waves in the fluid. The system of free oscillations in the plate is disrupted along a set of parallel lines. Rigid clamping of the plate, a sliding fastening and an infinitesimally narrow slit are considered as such disruptions. The apparatus used to construct the solution is quite general, and other disruptions in the elastic properties of a plate or its reinforcement, that are realized with linear boundary-contact conditions, can be treated in a similar way. The transmission and reflection of a harmonic flexural-gravitational wave, that is orthogonally incident on the inhomogeneities in the plate, are studied. Exact analytical representations of the wave fields in the plate and in fluid are obtained and the transmission and reflection coefficients for the incident flexural-gravitational wave are determined. The forces developed in the fastenings are found.     

Variational setting of the problem on the oscillations of a fluid in a vessel with an elastic inclusion

The problem on the oscillations of an ideal incompressible fluid in a moving rectangular vessel is studied. One wall of the vessel contains an elastic inclusion. The problem involves two free boundaries—the free surface of the fluid and the surface of the elastic inclusion. It is suggested to solve this problem by using a functional whose variation leads to differential equations with nonlinear kinematic and dynamical conditions on the free boundaries. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Transverse Oscillations of a Cylindrical Vessel with a Two-Layer Liquid Separated by an Elastic Membrane

A solution is given for the transverse oscillations, under elastic forces, of a cylindrical vessel containing an ideal two-layer liquid with elastic membranes at the free and inner surfaces of the stratified liquid. The following limiting cases for the elastic membrane are examined: it lies only at the free surface of a uniform and two-layer liquid; it separates a two-layer liquid; it exists at both the free and the inner surfaces of a two-layer liquid. The numerically calculated dependences of the first eigenfrequency on the tension of the membrane, liquid densities, and filling depth are analyzed.     

Free localized vibrations of a long double-walled carbon nanotube introduced into an inhomogeneous elastic medium

Based on modified Flügge equations and nonlocal elasticity theory, free axisymmetric oscillations of a long double-walled carbon nanotube embedded into an inhomogeneous elastic medium is studied. The ambient medium is simulated by the Winkler foundation. Van der Waals forces are introduced in order to take into account the interaction between the nanotube walls. Using Tovstik’s asymptotic method, eigenmodes are constructed in the form of functions that decay far from the line on the surface of the outer wall, on which the modulus of subgrade reaction has a local minimum. Eigenmodes and eigenfrequencies corresponding to the coand counterdirected wall motions are found. It has been found that introducing a nonlocality parameter into the model results in eigenmodes that are not inherent in macroscale shells. In particular, an increase in the stretching force leads first to greater localization of vibrations and increase in the amplitudes of tangential atomic oscillations and, second, to reduction in the frequencies in the case when the tube lies in a sufficiently stiff medium.     

Application of the Averaging Method to the Investigation of Nonlinear Wave Processes in Elastic Systems with Circular Symmetry

We apply asymptotic methods of nonlinear mechanics (the Bogolyubov–Mitropol'skii averaging method) to the construction of approximate solutions of a system of nonlinear equations describing wave processes in elastic systems with circular symmetry. As an example, we study the dynamics of interaction of two flexural waves that propagate in a cylindrical shell under the conditions of free oscillations and periodic excitation.     

Vibrations of a cylinder in a concentric vessel filled with a two-layer fluid

A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found.     

Nonlinear Oscillations of a Spring Pendulum at the 1 : 1 : 2 Resonance: Theory,Experiment, and Physical Analogies

Nonlinear spatial oscillations of a material point on a weightless elastic suspension are considered. The frequency of vertical oscillations is assumed to be equal to the doubled swinging frequency (the 1 : 1 : 2 resonance). In this case, vertical oscillations are unstable, which leads to the transfer of the energy of vertical oscillations to the swinging energy of the pendulum. Vertical oscillations of the material point cease, and, after a certain period of time, the pendulum starts swinging in a vertical plane. This swinging is also unstable, which leads to the back transfer of energy to the vertical oscillation mode, and again vertical oscillations occur. However, after the second transfer of the energy of vertical oscillations to the pendulum swinging energy, the apparent plane of swinging is rotated through a certain angle. These phenomena are described analytically: the period of energy transfer, the time variations of the amplitudes of both modes, and the change of the angle of the apparent plane of oscillations are determined. The analytic dependence of the semiaxes of the ellipse and the angle of precession on time agrees with high degree of accuracy with numerical calculations and is confirmed experimentally. In addition, the problem of forced oscillations of a spring pendulum in the presence of friction is considered, for which an asymptotic solution is constructed by the averaging method. An analogy is established between the nonlinear problems for free and forced oscillations of a pendulum and for deformation oscillations of a gas bubble. The transfer of the energy of radial oscillations to a resonance deformation mode leads to an anomalous increase in its amplitude and, as a consequence, to the break-up of a bubble.