Torsional vibrations of anisotropic rods

By the method of eigenfunction expansions we solve the dynamic problem of the torsion of an anisotropic rod. As an example we study a prismatic rod with a square cross section. The natural vibration frequencies are computed. Two figures. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 104–107.     

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.     

Brownian fluctuations in space-time with applications to vibrations of rods

In this paper Brownian fluctuations in space-time are considered. Time is assumed to run alternately forward and backward, the alternance being marked by a Poisson process with rate λ. It is shown that the law of this motion is a solution of a fourth-order partial differential equation. Furthermore the law of this movement in the presence of an absorbing barrier is derived. The equation ruling the movement analysed, when λ = 0 and is submitted to the change t' = −it, reduces to the equation of vibrations of rods. This fact is exploited to obtain the solution of boundary value problems concerning the equation of vibrating beams by means of Brownian motion techniques.     

Theorems on asymptotics for singular Sturm-Liouville operators with various boundary conditions

We consider the Sturm-Liouville operator L(y) = ?d ² y/dx ² + q(x)y in the space L ₂[0, π], where the potential q(x) is a complex-valued distribution of the first order of singularity; i.e., q(x) = u′(x), u ∈ L ²[0, π]. (Here the derivative is understood in the sense of distributions.) We obtain asymptotic formulas for the eigenvalues and eigenfunctions of the operator in the case of the Neumann-Dirichlet conditions [y ^[1](0) = 0, y(π) = 0] and Neumann conditions [y ^[1](0) = 0, y ^[1](π) = 0] and refine similar formulas for all types of boundary conditions. The leading and second terms of asymptotics are found in closed form.     

Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

This study presents a direct comparison of measured and predicted nonlinear vibrations of a clamped–clamped steel beam with non-ideal boundary conditions. A multi-harmonic comparison of simulations with measurements is performed in the vicinity of the primary resonance. First of all, a nonlinear analytical model of the beam is developed taking into account non-ideal boundary conditions. Three simulation methods are implemented to investigate the nonlinear behavior of the clamped–clamped beam. The method of multiple scales is used to compute an analytical expression of the frequency response which enables an easy updating of the model. Then, two numerical methods, the Harmonic Balance Method and a time-integration method with shooting algorithm, are employed and compared one with each other. The Harmonic Balance Method enables to simulate the vibrational stationary response of a nonlinear system projected on several harmonics. This study then proposes a method to compare numerical simulations with measurements of all these harmonics. A signal analysis tool is developed to extract the system harmonics’ frequency responses from the temporal signal of a swept sine experiment. An evolutionary updating algorithm (Covariance Matrix Adaptation Evolution Strategy), coupled with highly selective filters is used to identify both fundamental frequency and harmonic amplitudes in the temporal signal, at every moment. This tool enables to extract the harmonic amplitudes of the output signal as well as the input signal. The input of the Harmonic Balance Method can then be either an ideal mono-harmonic signal or a multi-harmonic experimental signal. Finally, the present work focuses on the comparison of experimental and simulated results. From experimental output harmonics and numerical simulations, it is shown that it is possible to distinguish the nonlinearities of the clamped–clamped beam and the effect of the non-ideal input signal.     

General stability of composite panels reinforced with flexible rods taking account of the side boundary conditions

Reinforced panels are the basic load-bearing elements of various structures. Optimization of massive structures requires consideration of deformation of the panel cross-sections. This is particularly important in determining the bearing strength at buckling. The load scheme, conditions for fixation of the panel cross-section, and bend-torsional stiffness taking account of the deformation of the rod cross-section affect the buckling load in real structures. The stress distribution prior to buckling must be known to solve the buckling problem properly. The stress in the panel is proportional to the active load. The stress distribution is assumed to be known according to our previous method [1]. The load scheme and panel dimensions are shown in Fig. 1. The stress distribution in the panel prior to buckling can be found using Eqs. (1)-(3). A view of the cross-section is given in Fig. 1. The displacements in the panel at buckling for the boundary area are found using Eqs. (4)-(6), while the stresses in the skin and stiffness are found using Eq. (7). Roots k₁ and k₂ are those of the characteristic equation and is a dimensionless coordinate. The problem was solved using variational theory. The potential energy is given by Eqs. (8) and (9) by orihogonalization of Eqs. (5). The basic equations are converted to Eqs. (10) by evaluation of the components in Eqs. (8) and (9). Its calculation (11) gives the compression load. Optimization of parameter gives the critical strength P₁ = 6.93 kN (without taking account of the boundary area) and P₂ = 5.31 kN (taking account of the boundary area).Translated from Mekhanika Kompozitnikh Materialov, Vol. 30, No. 4, pp. 540–546, July–August, 1994.     

Existence and uniqueness of solutions of hyperbolic equations in divergence form with various boundary conditions on various parts of the boundary

We prove the existence and uniqueness of an energy class solution of an initial–boundary value problem for a semilinear equation in divergence form. We consider the case in which an inhomogeneous third boundary condition is posed on one part of the lateral surface of the cylinder in which the equation is studied and the homogeneous Dirichlet boundary condition is posed on the other part of the lateral surface.     

On analytical solutions of vibrations of rods with variable cross sections

Analytical solutions of several rods whose cross-sections vary in the axial direction are considered. The analysis in this paper uses two transformations that exist in the literature to help transform the equation of motion of the rod into a form similar to that of the one-dimensional Schroedinger equation where the shape of the cross-section of the rod is governed by a potential function that satisfies a second order differential equation. By solving this second order differential equation, it is possible to obtain a class of shapes that have a common form of solution that are determined from the well known solution of the Schroedinger equation. There are very few analytical solutions available in the literature especially when the area of cross-section varies. In this paper, it is shown that the set of available solutions for variable cross-sectional rods are particular cases of the general analysis. In addition, analytical solutions to several new variable cross-sections are considered.This paper also considers an existing transformation that transforms the Sturm-Liouville equation to a particular form of the equation of motion under consideration. So, tracing the analysis backwards, for any second order differential equation for which existence of solutions are guaranteed, and that can be reduced in the above manner, it is possible to determine the form of the cross-section of the rod. In this paper, several shapes of the cross-section are investigated when the equation of motion is described by special functions, the Legendre, Hermite and Laugurre functions. The analysis given here is a general one and is applicable when the equation of motion is described by other special functions.     

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.     

Fractional boundary value problems with integral boundary conditions

In this article, we study a type of nonlinear fractional boundary value problem with integral boundary conditions. By constructing an associated Green's function, applying spectral theory and using fixed point theory on cones, we obtain criteria for the existence, multiplicity and nonexistence of positive solutions.     

Second-order boundary value problems with integral boundary conditions

In this paper we investigate the existence of positive solutions of nonlocal second-order boundary value problems with integral boundary conditions.     

Impulsive boundary value problems with nonlinear boundary conditions

In this paper, the upper and lower solution method and Schauder’s fixed point theorem are employed in the study of boundary value problems for a class of second-order impulsive ordinary differential equations with nonlinear boundary conditions. We prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem.     

A degenerate diffusion problem with dynamical boundary conditions Diffusion problem with dynamical boundary conditions

The purpose of this paper is to study the existence, the uniqueness and the limit in , as of solutions of general initial-boundary-value problems of the form and in a bounded domain with dynamical boundary conditions of the form Received: 5 December 2000 / Revised version: 20 November 2001 / Published online: 4 April 2002