Stephan V. Joubert

Closed solutions of boundary-value problems of coupled thermoelasticity

Coupled equations of thermoelasticity take into account the effect of nonuniform heating on the medium deformation and that of the dilatation rate on the temperature distribution. As a rule, the coupling coefficients are small and it is assumed, sometimes without proper justification, that the effect of the dilatation rate on the heat conduction process can be neglected. The aim of the present paper is to construct analytical solutions of some model boundary-value problems for a thermoelastic bounded body and to determine the body characteristic dimensions and the medium thermomechanical moduli forwhich it is necessary to take into account that the temperature and displacement fields are coupled. We consider some models constructed on the basis of the Fourier heat conduction law and the generalized Cattaneo-Jeffreys law in which the heat flux inertia is taken into account. The solution is constructed as an expansion in a biorthogonal system of eigenfunctions of the nonself-adjoint operator pencil generated by the coupled equations of motion and heat conduction. For the model problem, we choose a special class of boundary conditions that allows us to exactly determine the pencil eigenvalues.

Energy and the nonsymmetric nonlinear spring

We give aderivation of the nonlinear spring equation mx - f(x) = 0 where f(x) represents the restoring force and m is the mass of a weight attached to the spring. This derivation shows that even powers may appear in the Maclaurin expansion of the restoring force function. We demonstrate how the ‘energy approach’ facilitates the investigation of the resultant motions and makes phase plane analysis easy by reducing the characterization of critical points to determining the sign of asecond-order partial derivative of the energy function. This circumvents the sometimes difficult eigenvalue determination of the linearization procedure commonly taught. This approach is no more difficult than the approach usually takes in textbooks for much simpler equations and it leads to interesting harmonic oscillator examples and problems suitable for undergraduate research.

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.

A storm in a wineglass

The qualitative effect of a major disturbance such as an earthquake or a hurricane or, on a lesser scale, a powerboat moving along the edge of a bay, harbor, or lake, can be observed in a partially filled wineglass. We simulate this small-scale disturbance and do a quantitative analysis to explain how a resonance can occur on the liquid surface in the wineglass. An explanation is also given why such circumstances can occur in scaled-up situations such as bays, rivers, and harbors.

Square Waves from a Black Box

where a > 0. Here the mass on the spring has been normalized to 1 and the restoring force is taken to be the odd function ax bx 3. If b > 0, then the spring is called hard and all solutions are oscillatory and periodic; if b < 0, then the spring is called soft and for sufficiently small initial values, the solutions are oscillatory and periodic, but for other initial values the solutions grow without bound. For more details on this example and similar ones see [1] and [2]. Consider the soft spring equation

Analysing manufacturing imperfections in a spherical vibratory gyroscope

In a recent article in the Journal of Sound and Vibration (JSV) we discussed the influence of mass imperfections and isotropic (viscous) damping on the vibrating pattern of a slowly rotating spherical body. Using the mathematical tools developed in the JSV article, in addition to mass imperfections, we demonstrate how to introduce prestress using a non-linear theory of elasticity and how to introduce anisotropic (viscous) damping into the equations of motion using Rayleigh dissipation. The equations of motion thus obtained demonstrate that a “precession wave” is generated within the sphere and indicate what control measures (if any) might be taken in order to approximate an “ideal” situation where the vibration pattern within the spherical body rotates at a rate (called the precession rotation rate) that is proportional to the slow rotation rate of the sphere. The constant of proportionality referred to above is called Bryans constant and is used to calibrate the hemispherical resonator gyroscopes that are used in the space shuttle. The equations of motion demonstrate that four “slow” variables are present namely the principal and quadrature amplitudes of vibration, the precession angle and a phase angle. It appears that each slow variable is affected by mass-stiffness imperfections and\or constant prestress and anisotropic damping. The phenomenon of “beats” is predicted and a numerical experiment indicates that two capture effects are possible with precession angle and that the same is true for the phase angle.

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.

Nonlinear resonance and Duffing's spring equation II

The paper discusses the boundary in the frequency–amplitude plane for boundedness of solutions to the forced spring Duffing type equation For fixed initial conditions and for representative fixed values of the parameter ϵ, the results are reported of a systematic numerical investigation into the global stability of solutions to the initial value problem as the parameters F and ω are allowed to vary. This can be interpreted as varying the forcing amplitude and forcing frequency to a nonlinear spring problem and asking for the threshold between bounded oscillatory responses and unbounded unstable responses. These preliminary results indicate that the low resonance frequency (to two decimal places) is independent of the value of ϵ and that near a higher jump frequency phenomena the behaviour of solutions is very unstable. Computer laboratory problems suitable for student research and small group projects are included.

The singing wineglass: an exercise in mathematical modelling

Lecturers in mathematical modelling courses are always on the lookout for new examples to illustrate the modelling process. A physical phenomenon, documented as early as the nineteenth century, was recalled: when a wineglass ‘sings’, waves are visible on the surface of the wine. These surface waves are used as an exercise in mathematical modelling. Based on assumptions about the wine in the glass and observations illustrated with photographs, a mathematical problem is set up. This problem includes a non-homogeneous Neumann boundary condition on the lateral side of the glass. The solution to the mathematical problem is animated using Mathematica™. The predictions of the model are tested by comparing them with the known facts. The predictions of the model agree with the actual observations.

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.