Michael Y. Shatalov

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

Rotating structures and Bryan’s effect

In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates at a rate proportional to the rate of rotation. During investigations of the effect in various solid and fluid-filled objects of various shapes, an interesting commonality was found in connection with the gyroscopic effects of the rotating object. The effect has also been discussed in connection with a rotating fluid-filled wineglass. A linear theory is developed, assuming that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. The associated physics and mathematics are easy enough for undergraduate students to understand.

Dynamics of rotating and vibrating thin hemispherical shell with mass and damping imperfections and parametrically driven by discrete electrodes

A parametric electrode is normally used in hemispherical resonator gyroscopes, operating in the whole angle regime, for maintaining the amplitudes of vibratory patterns. It is well known that, due to variation of the gap between the resonator and parametric electrode, a drift of the vibrating pattern is obtained. This drift is similar to the gyro drift stipulated by the Q-factor imperfections and substantially deteriorates the quality of the hemispherical resonator gyroscope. In the present paper, we consider the methods of compensation of these drifts by means of special control of the vibrating pattern by a sectioned parametric electrode. Compensation of the drifts is achieved through amplitudes of voltage and phase manipulations at the sectioned parametric electrodes. The side effect of this control consisting of spurious splitting of frequencies of the vibrating pattern is discussed.

Application of the Heun functions to rotor vibratory gyroscope, their numerical computation and accuracy estimation

Abstract For an asymmetric rotor vibratory gyroscope that is oscillating in an elastic suspension means, the equation of motion is derived from the Eurler–Lagrange equation and the exact solution is obtained as Heun functions (Hfs). A fast and effective method for calculating Heun functions by direct calculation of solutions of the Heun differential equation (Hde) using standard numerical integration methods is developed. Three methods of accuracy check are employed in this case. The accuracy of the numerical solutions deteriorated in the vicinity of the singularity. To overcome this difficulty, an optimised method for calculating the Hfs is developed, which give a uniform accuracy of the calculated values on the interval. The optimised method for calculating the numerical Hfs by means of the solution of the governing initial value problem gave acceptable accuracy in modelling the behaviour of an asymmetric rotor gyroscope.

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.

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.

On Electronically Restoring an Imperfect Vibratory Gyroscope to an Ideal State

With regard to G.H. Bryan’s publication in 1890, we call the following Bryan’s law (or Bryan’s effect): “The vibration pattern of a revolving cylinder or bell revolves at a rate proportional to the inertial rotation rate of the cylinder or bell”. Bryan’s factor is the proportionality constant that can be theoretically calculated for an ideal vibratory gyroscope (VG). If a perfectly symmetric VG is not ideal, that is, if imperfections and damping are present, then the precession rate (pattern rotation rate) depends on a number of factors. Indeed it depends on the rotation rate of the vehicle it is attached to, mass-stiffness and symmetry imperfections as well as any anisotropic damping (linear or nonlinear) that may be present in the VG. Assuming perfect axissymmetry for the VG, we show how to negate the effects of manufacturing mass-stiffness imperfections as well as the effects of any type of tangentially anisotropic damping that might occur. We achieve this by showing exactly how to symmetrically arrange an electronic array about the symmetry axis. This array consists of curved capacitors under a mixture of a constant (fixed) charge and a small meander charge. We show exactly how the fixed voltage on the capacitor should be adjusted in order to eliminate the frequency split caused by the mass-stiffness imperfection. Furthermore, we show how the meander voltages of the capacitors should be adjusted in order to maintain principal vibration, eliminate quadrature vibration and restore spurious pattern drift in the VG so that it obeys Bryan’s law, restoring the precession rate to the ideal rate so that Bryan’s factor can be used for calibration purpose. Equations of motion are derived in the form of averaged ODEs that provide us insight into VG behaviour.

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.