Necessary and sufficient existence conditions for a boundary control of vibrations of a string and a spherical layer for small times

We find necessary and sufficient conditions for the existence of a boundary control of vibrations of a string or a spherical layer for critical and subcritical times. We completely analyze the existence of a boundary control of vibrations of a spherical layer by a force on two spheres. We find necessary and sufficient existence conditions for the control. Along with the control problem for vibrations of a spherical layer, we consider a similar control problem for string vibrations.     

State observability of elastic string vibrations under the boundary conditions of the first kind

We solve state observation problems for string vibrations, i.e., problems in which the initial conditions generating the observed string vibrations should be reconstructed from a given string state at two distinct time instants. The observed vibrations are described by the boundary value problem for the wave equation with homogeneous boundary conditions of the first kind. The observation problem is considered for classical and L ₂-generalized solutions of this boundary value problem.     

A Sturm-Liouville Inverse Spectral Problem with Boundary Conditions Depending on the Spectral Parameter

We consider a boundary value problem generated by the Sturm-Liouville equation on a finite interval. Both the equation and the boundary conditions depend quadratically on the spectral parameter. This boundary value problem occurs in the theory of small vibrations of a damped string. The inverse problem, i.e., the problem of recovering the equation and the boundary conditions from the given spectrum, is solved.     

Factorized equation of vibrations of a finite string with nonstationary second directional derivatives at the endpoints

We obtain closed-form recursion formulas for the classical solutions of a mixed problem for the general inhomogeneous factorized equation of vibrations of a bounded string with second directional derivatives in the boundary conditions, in which the coefficients multi-plying the first of the two directional derivatives are independent of time. We study the case of boundary conditions in which all first directional derivatives are not directed along the characteristics of the equation. We obtain necessary and sufficient conditions on the right-hand side and the initial and boundary data of the problem for its well-posed global solvability in the set of classical solutions.     

Boundary control of forced string vibrations by first directional derivatives on a short time interval

We solve a problem on the boundary control of forced vibrations of a homogeneous string by two first directional derivatives with time-dependent coefficients and noncharacteristic directions in the boundary conditions on a short time interval in the set of classical solutions. We obtain smoothness, coordination, and controllability conditions on the right-hand side of the equation and the initial and terminal data necessary and sufficient for the unique boundary control of the wave process.     

Necessary and sufficient conditions for the vibrations of a bounded string with directional derivatives in the boundary conditions

We obtain a closed-form expression for the classical solutions of a mixed problem describing the forced vibrations of a bounded string for two boundary modes with directional derivatives and time-dependent coefficients. We derive necessary and sufficient conditions on the right-hand side of the equation and the initial and boundary data.     

Analysis of vibrations in large flexible hybrid systems

The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions makes it impossible to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameters together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different characteristics take place, are defined. The dependence of the vibration frequencies on the physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allows one to control the frequency spectrum, in which excitation of vibrations is possible.     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

Solution of a mixed problem with oblique derivatives in the boundary conditions for the inhomogeneous string vibration equation without continuing the data

We find closed-form recursion formulas for the unique classical solution of a mixed problem describing forced vibrations of a bounded string under two boundary modes with timedependent oblique derivatives. The formulas do not use any continuation of the problem data. We obtain conditions on the right-hand side of the equation necessary and sufficient for the well-posed global solvability of the problem.     

Influence of boundary conditions and temperature of dissipative heating on active damping of forced axisymmetric resonant bending vibrations of circular viscoelastic plates by piezoelectric sensors and actuators

The problem of forced monoharmonic, axisymmetric, bending vibrations and dissipative heating of circular viscoelastic plates with piezoelectric sensors and actuators is considered. We describe the viscoelastic behavior of a passive (without the piezoeffect) and a piezoactive material according to the concept of complex modules depending on temperature. The nonlinear coupled problem of electrothermoviscoelasticity is solved by numerical methods. The influence of boundary conditions and temperature of dissipative heating on the active damping of forced resonant vibrations of circular viscoelastic plates using piezoelectric sensors and actuators is investigated.     

Application of the Laplace transform to the analysis of a mixed problem describing the vibrations of a compound rod

We suggest a method for finding a closed-form expression for the solution of the mixed problem on the vibrations excited in a compound rod by boundary conditions of the first, second, and third kind. The method uses the Laplace transform. We also suggest one possible numerical method for finding the solution.     

Observability of elastic vibrations of a beam

The observability problem for beam vibrations described by a fourth-order partial differential equation with various boundary conditions is considered. Dynamic observability problems are solved in terms of boundary conditions and observations of the beam state at certain fixed instants of time.     

Solution of problems of free torsional vibrations of thick-walled orthotropic inhomogeneous cylinders

With the use of the 3D theory of elasticity, we investigate the problem of free torsional vibrations of an anisotropic hollow cylinder with different boundary conditions at its end faces. We have proposed a numerical-analytic approach for the solution of this problem. The original partial differential equations of the theory of elasticity with the use of spline approximation and collocation are reduced to an eigenvalue problem for a system of ordinary differential equations of high order in the radial coordinate. This system is solved by the stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present numerical results for the case of orthotropic and inhomogeneous material of the cylinder for some kinds of boundary conditions.     

On the accumulation of eigenvalues of operator pencils connected with the problem of vibrations in a viscoelastic rod

In this paper, we study the problem of the boundary accumulation of a discrete spectrum, which is essential for a boundary-value problem of fourth order arising in the theory of small transverse vibrations in an inhomogeneous viscoelastic rod (a Kelvin—Voigt body). We establish conditions for such an accumulation and its asymptotics, which are expressed in terms of the coefficients defining the problem posed by the differential expression. The results obtained are illustrated by numerical computation data.     

High-frequency vibrations of isotropic plates

Using the method of A. L. Gol'denveizer we reduce the three-dimensional problem of high-frequency steady-state vibrations of a thin isotropic plate to a series of two-dimensional problems. We construct two recursive processes that describe the basic stress-strain state of the plate. It is established that the first coefficient of the asymptotic expansion of the natural frequency of vibrations is determined by the boundary conditions on the plane faces of the plate and is independent of the conditions on its lateral surface. Therefore to solve the problem it is necessary to take account of at least two approximations of the recursive processes. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 82–85.     

On the investigation of free vibrations of nonthin cylindrical shells of variable thickness by the spline-collocation method

For determining the dynamic characteristics of free vibrations of circular unclosed cylindrical shells of variable thickness in two coordinate directions, we have used the spline-collocation method together with the method of discrete orthogonalization. The problem has been solved within the framework of the refined Timoshenko–Mindlin theory. We have also investigated the influence of different laws of change in the shell thickness on the character of its natural vibrations. Our calculations have been carried out for different geometrical and elastic parameters of the shell under study and different boundary conditions.     

Optimization of a nonlocal boundary control of vibrations of a string with a fixed end for an arbitrary time interval that is a multiple of 2l

We consider the optimization problem for a nonlocal boundary control of vibrations of an elastic string with fixed right end for an arbitrary sufficiently large time interval that is a multiple of twice the string length. We prove the existence and uniqueness of a generalized solution of the first boundary value problem and indicate an explicit expression for the solution. The optimal control is found in closed form.     

Parametric Vibrations of a Three-Layer Conical Piezoshell

Based on the Kirchhoff-Love hypotheses and adequate supplementary hypotheses for the distribution of electric field quantities, a model for parametric vibrations of composite shells of revolution made of a passive (without a piezoeffect) middle layer and two active (with a piezoeffect) surface layers under the action of harmonic mechanical and electric loads is developed. The dissipative material properties are taken into account by linear viscoelastic models. Since the vibrations on the boundary of the main domain of dynamic instability (MDDI) are harmonic, the investigation of this domain, in a first approximation, is reduced to generalized eigenvalue problems, which are solved by the finite-element method. The problem on parametric vibrations of a three-layer conical shell under harmonic mechanical loading is considered. The influence of the shell thickness, dissipation, and electric boundary conditions on the MDDI is investigated. Two limiting cases of electric boundary conditions are considered, where the electrodes are short-circuited or not. The curves presented show a considerable influence of the electric boundary conditions on the characteristics of the MDDI, namely on its width and position on the frequency axis and on the critical parameter of excitation.     

On an optimal boundary control problem with a dynamic boundary condition

We are the first solve the optimization problem for a boundary displacement control of string vibrations at one endpoint in the case of a nonstationary boundary condition containing a directional derivative at the other endpoint.