Determination of the physicomechanical characteristics of viscoelastic materials from the velocity of propagation and damping of longitudinal and transverse vibrations in a thin rod

The author examines the propagation of longitudinal and transverse waves in a thin viscoelastic rod. Relations are obtained for determining the components of the complex moduli and and Poisson's ratio from the propagation velocities and decrements of the longitudinal and transverse vibrations, together with the static and relaxational moduli of elasticity, the relaxation time, and the equivalent viscosity for materials whose law of deformation corresponds to a standard linear solid.Mekhanika Polimerov, Vol. 2, No. 3, pp. 453–460, 1966     

Boundary observability of elastic vibrations in a system of sequentially connected strings

The observability problem for the elastic vibrations of a system consisting of sequentially connected objects with distributed parameters is solved. An object with lumped parameters is connected to the system. The initial state of the system is recovered from observations on its boundary and at the connection point of the objects with distributed and lumped parameters.     

Mixed problems for the equation of longitudinal vibrations of a heterogeneous rod and for the equation of transverse vibrations of a heterogeneous string consisting of two segments with different densities and elasticities

Doklady Mathematics -     

Nonlinearly coupled in-plane and transverse vibrations of a spinning disk

Previous nonlinear spinning disk models neglected the in-plane inertia of the disk since this permits the use of a stress function. This paper aims to consider the effect of including the in-plane inertia of the disk on the resulting nonlinear dynamics and to construct approximate solutions that capture the new dynamics. The inclusion of the in-plane inertia results in a nonlinear coupling between the in-plane and transverse vibrations of the spinning disk. The full nonlinear partial differential equations are simplified to a simpler nonlinear two degrees of freedom model via the method of Galerkin. A canonical perturbation approach is used to derive an approximate solution to this simpler nonlinear problem. Numerical simulations are used to evaluate the effectiveness of the approximate solution. Through the use of these analytical and numerical tools, it becomes apparent that the inclusion of in-plane inertia gives rise to new phenomena such as internal resonance and the possibility of instability in the system that are not predicted if the in-plane inertia is ignored. It is also demonstrated that the canonical perturbation approach can be used to produce an effective approximate solution.     

Discrete transparent boundary conditions for the equation of rod transverse vibrations

Local perturbations of an infinitely long rod travel to infinity. On the contrary, in the case of a finite length of the rod, the perturbations reach its boundary and are reflected. The boundary conditions constructed here for the implicit difference scheme imitate the Cauchy problem and provide almost no reflection. These boundary conditions are non-local with respect to time, and their practical implementation requires additional calculations at every time step. To minimise them, a special rational approximation, similar to the Hermite - Padé approximation is used. Numerical experiments confirm the high “transparency” of these boundary conditions and determine the conditional stability regions for finite-difference scheme.     

The dual nature of the transverse vibrations of an elastic rod

The transverse vibrations of an elastic rod, to one of which displacements are applied while the other end is free, are investigated. It is assumed that the propagation velocity of the perturbations in the rod is finite. The unperturbed part performed rotational motion around the centre line. The angle of rotation is expressed by the angle of curvature of the centre line of the perturbed part of the rod. Two types of elastic vibrations are obtained: (1) the rod vibrates elastically due to displacements applied at the end, and (2) when performing rotational motion elastic vibrations and additional forces occur in the rod due to elasticity [1].     

The effect of stiffness on the transverse and longitudinal motions of a musical string: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The effect of stiffness on the propagation of longitudinal and transverse waves and vibrations in prestretched strings is considered. The contribution of the longitudinal and transverse components to the dynamic load is of the same order. The longitudinal vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena are possible. Low stiffness, which is characteristic for musical strings, leads to a small change in the frequencies of the whole spectrum of transverse and longitudinal vibrations, but to a considerable change in the shape of the string at strike and mounting points and on the transverse wave front.     

Steady longitudinal vibrations of the components of cylindrically anisotropic plates

Using the perturbation method we solve the problem of steady longitudinal vibrations of cylindrically anisotropic plates consisting of a finite number of circular rings welded together. A numerical study is carried out for a five-layer plate in the case when the outer boundary is subject to a load and the inner boundary is load-free. The graphs of the stress distributions are given. Two figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 30–34, 1991.     

Numerical solution of the optimal boundary control of transverse vibrations of a beam

Boundary control is an effective means for suppressing excessive structural vibrations. By introducing a quadratic index of performance in terms of displacement and velocity, as well as the control force, and an adjoint problem, it is possible to determine the optimal control. This optimal control is expressed in terms of the adjoint variable by utilizing a maximum principle. With the optimal control applied, the determination of the corresponding displacement and velocity is reduced to solving a set of partial differential equations involving the state variable, as well as the adjoint variable, subject to boundary, initial, and terminal conditions. The set of equations may not be separable and analytical solutions may only be found in special cases. Furthermore, the computational effort to determine an analytic solution may also be excessive. Herein a numerical algorithm is presented, which easily solves the optimal boundary control problem in the space‐time domain. An example of a continuous system is analyzed. This is the case of the vibrating cantilever beam. Using a finite element recurrence scheme, numerical solutions are obtained, which compare the behavior of the controlled and uncontrolled systems. Also, the analytic solution to the problem is compared with the results obtained using the numerical scheme presented. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 558–568, 1999     

Strong convergence of difference approximations in the problem of transverse vibrations of thin elastic plates

The problem of transverse vibrations of a thin elastic plate is considered. It is proved that the differential operators of the boundary value problem are regularly elliptic, and weak solutions are estimated. For a previously developed difference method, the solution to the difference problem is proved to converge strongly to a weak solution of the original differential problem and the rate of convergence is estimated.     

Waves and vibrations in strings with windings: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The equations of the propagation of transverse, twisting and longitudinal waves and vibrations are obtained, taking into account their interactions in musical strings with windings. Their solutions are obtained. The occurrence of transverse and twisting motions leads to the appearance of longitudinal motions, while the transverse and longitudinal components play the role of inducing forces for the twisting components. The contributions of the transverse, twisting and longitudinal components to the dynamic loading of the string are of the same order. The longitudinal-twisting vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena between the individual modes of these vibrations are possible.     

Inequalities between Dirichlet and Neumann eigenvalues of vibrating strings

The Dirichlet eigenvalues \({\{\lambda_{n}\}_{n=1}^{\infty}}\) and Neumann eigenvalues \({\{\mu_{n}\}_{n=1}^{\infty}}\) of the string equation \({\varphi'' (x) +\lambda \rho (x) \varphi(x) =0}\) are considered. It is known that \({ \mu_{n} < \lambda_{n} < \mu_{n+2}}\) for all n. The purpose of this paper is to provide conditions on the mass density \({\rho(x)}\) under which \({\lambda_{n} < \mu_{n+1}}\) or \({\mu_{n+1} < \lambda_{n}.}\)     

Optimal control of the transverse vibrations of a beam with a bound on the potential energy

The problem of the optimal control of the transverse vibrations of a uniform beam with a bound on the potential energy is considered. Approximation theory is used. It is shown that the optimal control exists, and a simple method for its computation is suggested.     

Magnetoelastic vibrations of an electrically conductive elastic layer in a longitudinal magnetic field

Based on the exact solution of a coupled problem of the magnetoelastic vibrations of an elastic electrically conductive layer (for a finite value of its electrical conductance) in a longitudinal magnetic field, we have established that, in the case of such vibrations of a thin plate, one can obtain correct results proceeding from Kirchhoff’s hypothesis and the hypothesis of magnetoelasticity of thin bodies.     

Interference transverse sh waves in a transversally,isotropic, inhomogeneous half space

In the paper Love waves in a transversally, Isotropic, inhomogeneous half space are investigated in the case where the plane of isotropy does not coincide with the boundary plane. The dependence on the angle between them is established to the solution obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 228–234, 1981.     

Statistical calculation of the strength of a circular plate subject to vibrations in a stochastic transverse transient field

A method for statistical calculation of the strength of a circular plate is introduced. A correlation function is constructed for the design stress at the danger point, whose location is determined from its extremum condition. Correlation functions for the design stresses and time-dependent amplitudes are constructed and used to determine the principal parameters of the statistical and dynamic strength of a plate in a statistical calculation. Zaporozhe State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 126–131, 1999.     

Substantiation of two-scale homogenization of the equations governing the longitudinal vibrations of a viscoelastoplastic Ishlinskii material

Initial-boundary value problems for the system of quasilinear operator-differential equations governing the longitudinal vibrations of a viscoelastoplastic Ishlinskii material with nonsmooth rapidly oscillating coefficients and initial data are investigated. The system involves the hysteresis Prandtl-Ishlinskii operator. Passage to the limit to initial-boundary value problems for the corresponding system of two-scale homogenized operator integro-differential equations is strictly substantiated globally in time without assuming that the data are small.     

Study of the effect of test piece geometry on the propagation velocity and logarithmic decrement of longitudinal vibrations in fluoroplastics

The effect of the geometry of teflon test pieces on the speed of sound and the logarithmic decrement of longitudinal vibrations has been investigated. Conditions for the determination of the speed of sound in an infinite medium and in a rod have been obtained. A method is given for determining the dynamic moduli of longitudinal elasticity, the shear modulus, and the dynamic Poisson's ratio on test pieces in the form of rods and prisms, starting with a length of 30 mm.Mekhanika Polimerov, Vol. 2, No. 4, pp. 557–564, 1966     

Optimal control of longitudinal vibrations of composite rods with the same wave propagation time in each part

We consider longitudinal elastic vibrations of a composite rod and find closedform expressions that describe optimal boundary controls bringing the rod from the quiescent state into a state with given displacement function φ(t) and velocity function ψ(t) in time T. We assume that the wave propagation time through each part of the rod is the same and T is a multiple of that time.