Igor Fedotov

Longitudinal Vibration of Isotropic Solid Rods: From Classical to Modern Theories

Longitudinal waves are broadly used for the purposes of non-destructive evaluation of materials and for generation and sensing of acoustic vibration of surrounding medium by means of transducers. Many mathematical models describing longitudinal wave propagation in solids have been derived in order to analyse the effects of different materials and geometries on vibration characteristics without the need for costly experimental studies. The propagation of elastic waves in solids has seen particular interest since the end of the 19th century. The solution for three dimensional wave propagation in solids was derived independently by L. Pochhammer in 1876 and by C. Chree in 1889. The solution describes torsional, longitudinal and flexural wave propagation in cylindrical rods of infinite length and is known as the Pochhammer-Chree solution (Achenbach, 1999:242-246; Graff, 1991:470473). The Pochhammer-Chree solution is valid for an infinite bar with simple cylindrical geometry only. For even slightly more complex geometry, such as conical, exponential or catenoidal, no exact analytical solution exists. The need for useful analytical results for bars with more complex geometries fuelled the development of one dimensional approximate theories during the 20th century. The exact Pochhammer-Chree solution has typically been used as a reference result in order to make deductions regarding the accuracy of the approximate theories and the limits of their application (Fedotov et al., 2009). The classical approximate theory of longitudinal vibration of rods was developed during the 18th century by J. D’Alembert, D. Bernoulli, L. Euler and J. Lagrange. This theory is based on the analysis of the one dimensional wave equation and is applicable for long and relatively thin rods vibrating at low frequencies. Lateral effects and corresponding lateral and axial shear modes are fully neglected in the frames of this theory. The classical theory gained universal acceptance due to its simplicity, especially for engineering applications. It is broadly used for design of low frequency mechanical waveguides such as ultrasonic transducers, mechanical filters, multi-stepped vibrating structures, etc. J. Rayleigh was the first who recognised the importance of the lateral effects and analysed the influence of the lateral inertia on longitudinal vibration of rods. This result was briefly exposed in Rayleigh’s famous book “The Theory of Sound”, first published in 1877 (Rayleigh, 1945:251-252). A. Love in his “Treatise on the Mathematical Theory of Elasticity”, first published in 1892 (Love, 2009:408-409), further developed this theory, which is now

Longitudinal vibrations of a Rayleigh-Bishop rod

Doklady Physics, 2010, Vol. 55(12) pp 609–614. Copyright: Pleiades Publishing Ltd. 2010. Original Russian Text Copyright: I.A. Fedotov, A.D. Polyanin, M.Yu. Shatalov, E.M. Tenkam, 2010, published in Doklady Akademii Nauk, 2010, Vol. 435(5) pp. 613–618. [ABSTRACT ONLY]

THE QUASILINEARITY OF SOME FUNCTIONALS ASSOCIATED WITH THE RIEMANN–STIELTJES INTEGRAL

The superadditivity and subadditivity of some functionals associated with the Riemann–Stieltjes integral are established. Applications in connection to Ostrowski’s and the generalized trapezoidal inequalities and for special means are provided.

Method of Multiple Orthogonalities for Vibration Problems

The modeling of vibration problems is of great importance in engineering and mathematical physics. A widely spread method of analyzing such problems is the variational method. The simplest and advanced vibration models are represented using the examples of a long and thick rod. Two kinds of eigenfunction orthogonality are proved and the corresponding norms are used to derive Green’s function that gives rise to the analytical solution of these problems. The method can be easily generalized to a broad class of hyperbolic problems.

Diffraction and Dirchlet problem for parameter-elliptic convolution operators with constant symbols

In this paper we evaluate the difference between the inverse operators of a Dirichlet problem and of a diffraction problem for parameter-elliptic convolution operators with constant symbols. We prove that the inverse operator of a Dirichlet problem can be obtained as a limit case of such a diffraction problem.

The Dynamics of an Accreting Vibrating Rod

A linear model of longitudinal vibration is formulated for a viscoelastic rod subjected to external harmonic excitation within the framework of the classical theory of vibrating rods. It is assumed that the rod has a time-dependent variable length and cross-section. A mixed problem of dynamics is formulated, which contains non-conventional fixed-free boundary conditions with the coordinate on the right-hand side of the rod being dependent on time. A special transformation of variables eliminates the dependence of the right-hand side coordinate of the boundary conditions on time. The transformation substantially simplifies the boundary conditions, converting them to the classical fixed-free boundary conditions. The simplification of the boundary conditions is, in turn, exacerbated by the equation of rod motion because it becomes a linear partial differential equation with variable coefficients containing some additional terms. The proposed solution of this equation is built in terms of a trigonometric series with time-dependent coefficients, where the spatial components satisfy the boundary conditions. In this case the original partial differential equation is converted into an infinite system of coupled ordinary differential equations with corresponding initial conditions. Truncation of the system produces an initial problem which is solved numerically. The corresponding truncated trigonometric series rapidly converges to the solution. The solutions are built for different combinations of the parameters of the varying rod. It is shown that for lightly damped rods, the amplitudes of different modes are mainly defined by free solutions of the initial problem. The notion of generated equations of the system is introduced. Free solutions can be obtained from the generating equations of the coupled system of ordinary differential equations. Moreover, exact solutions of the generating equations are built in terms of the elementary Kummer and confluent Heun functions. These exact solutions give one proper insight into the dynamic processes governing vibrations of the varying lightly damped rods. In the case of heavily damped coefficients, free vibration of the rod is rapidly suppressed and the amplitude behaviour of the modes on a finite time interval is defined by the excitation force. For example, in the case of a linearly growing rod of constant volume, the amplitude of the equivalent excitation force also grows proportionally to time. Owing to this effect, the amplitudes of the particular modes, in turn, are linearly increased with time.

Estimating the parameters of chemical kinetics equations from the partial information about their solution

The inverse problem of identifying the parameters of sets of ordinary differential equations using experimental measurements of three functions that correspond to some components in the vector solution of a set is considered. A private case that is important for applications of chemical and biochemical kinetics when reduced equations linearly depend on the combinations of initial unknown parameters has been studied. An analysis and the numerical results are presented for two types of sets of chemical kinetics equations, such as the Lotka–Volterra model that describes the coexistence of a predator and a prey and the chemical kinetics equations that model enzyme catalysts reactions, including the Michaelis–Menten equations. The search for unknown parameters is confined to the problem of minimizing a quadratic function. In this case, the reduced differential equations of systems are used instead of their vector solutions, which are unknown in most cases. The cases of both stable and unstable search for unknown parameters are analyzed.

Approximating the Stieltjes integral via a weighted trapezoidal rule with applications

Abstract In this paper we provide sharp error bounds in approximating the weighted Riemann–Stieltjes integral ∫ a b f ( t ) g ( t ) d α ( t ) by the weighted trapezoidal rule f ( a ) + f ( b ) 2 ∫ a b g ( t ) d α ( t ) . Applications for continuous functions of selfadjoint operators in complex Hilbert spaces are given as well.

One-dimensional diffusion model in an inhomogeneous region

A one-dimensional model is developed to describe atomic diffusion in a graphite tube atomizer for electrothermal atomic adsorption spectrometry. The underlying idea of the model is the solution of an inhomogeneous one-dimensional diffusion equation, with the diffusion coefficient being a function of temperature over the entire inhomogeneous region. An analytical solution of the problem is obtained in the form of a Green’s function.