Peter Bövik

Autobalancing of Rotors

It is shown that plane rotors with one or two particles free to move, subject to viscous damping,, in a groove on the rotor exhibit autobalancing, a property attributed to non-autonomous systems possessing an hyperbolic stable fixed point in the (non-extended) phase space for an open domain in parameter space. The analysis is also extended to non-plane rotors. Dynamical systems of this kind may be represented by perturbed Hamiltonian systems and averaging methods may be applied. Here the driving is considered in the form of a constraint and the particular structure emerging is found to be most easily analyzed by using the method of multiple scales. With this method it is also shown that non-plane rotors are autobalancing. The first order solutions showing how this physically six-dimensional problem approaches the fixed point are also given.

A model of ultrasonic nondestructive testing for internal and subsurface cracks

The scattering of elastic waves in a half-space containing a striplike crack is investigated. As a special case it seems that the crack may be surface breaking. A surface integral equation with the half-space Green tensor is employed. The key point of the method is the expansion of the Green tensor in Fourier representations with the free part of the Green tensor expanded in the crack coordinate system and the half-space part in the half-space coordinate system. The integral equation is discretized by expanding the crack opening displacement in terms of Chebyshew functions having the correct square root behavior along the crack edges. The incident field is emitted from an ultrasonic probe and a recent model for this is employed. The signal response in another (or the same) probe is modeled by a reciprocity argument and the stationary phase approximation is employed to simplify the final answer, which is thus only valid in the far field of the probes (yielding essentially a spherical wave). Numerical resul...

A Comparison of Exact First Order and Spring Boundary Conditions for Scattering by Thin Layers

In scattering problems for time-harmonic elastic waves, thin elastic layers are often of interest, e.g., in laminates. Various ways of substituting such layers by some effective boundary conditions have been proposed, and these are briefly reviewed. A rational way of obtaining boundary conditions that are exact to first order in the layer thickness is then described. For a thin spherical layer numerical comparisons are performed between these “exact” first order boundary conditions, the commonly used spring boundary conditions and the exact solution, and it is shown that the “exact” boundary conditions are far superior to the spring boundary conditions in most situations. A drawback with the “exact” boundary conditions is that they are quite complicated.

Effective boundary conditions for the scattering of two-dimensional SH waves from a curved thin elastic layer

The problem of scattering of time-harmonic elastodynamic sh waves from a curved thin elastic layer is addressed in the present paper. In the literature there are a number of suggestions on how to model a thin elastic layer by means of approximate boundary conditions. The ones that seem to be most commonly used are the spring contact boundary conditions (BCS). These type of BCS are also used to model a number of other flaw types, including, e. g. a thin layer of viscous fluid or a distribution of microcracks over a surface. However, in some recent papers on the scattering from inclusions surrounded by thin elastic interphase layers, it has been shown that the spring contact BCS do not lead to correct results when used in the context of thin elastic layers. The reason for this is that the spring contact BCS do not incorporate certain effects, both physical and geometrical, which are of the same order of magnitude as the ones that are in fact taken into account by these BCS. In the present paper we analyse in detail what the appropriate BCS for two-dimensional (2D) SH wave scattering from a thin curved elastic layer are. We show that when all effects to lowest order in the layer thickness are included, we get BCS that have a number of nice features, including the one of implying that no scattering occurs when the layer has material parameters identical to the ones in the matrix medium. The jump in displacement is written in terms of the normal derivatives, and the jump in normal derivative is written in terms of displacements and their second-order tangential derivatives. The BCS are derived even for the case where the layer lies between two different media, and for the case of a planar layer with anisotropy. The energy conservation properties of the BCS are analysed. To test the approximate BCS we check them against the exact solution for the case of scattering from a layer of circular cross-section by a 2D point source outside the layer. We also compare the results against the commonly used spring contact model, as well as against an improved version of it.

Ultrasonic scattering by a side-drilled hole

The scattering of elastic waves by a side-drilled hole (sdh) i.e. a circular cylindrical cavity, is considered. The scattering of plane or cylindrical waves by an sdh is an old subject; here the T matrix solution is adopted. The elastic waves are excited by an ultrasonic probe and a model of such a probe is thus used. The waves from the probes are expressed as a Fourier transform, i.e. as a superposition of plane waves. These plane waves are then transformed to the cylindrical system centred at the sdh. To obtain the signal in a receiving ultrasonic probe an electromechanical reciprocity relation is used. The signal response is obtained as a double wavenumber integral and an azimuthal summation. In the far field the integrals can be calculated approximately by the stationary phase approximation. Some numerical examples are given, in particular concentrating on when this approximation is valid.

Integral equation method for evaluation of eddy-current impedance of a tilted, surface-breaking crack

An integral equation method for solving the eddy-current nondestructive evaluation problem for a flat, tilted, and surface-breaking crack in a conducting half-space is presented. The method involves use of a half-space Greens tensor and the Bowler potential. This potential describes the jump in the electric field over the crack and is expanded in basis functions related to the Chebyshev polynomials, being a more analytical approach than the commonly used boundary element method. In the method, the scatterer defines a transformation operator to be applied on the incoming field. This is practical in simulations of the eddy-current inspection where this operator is independent of the position of the probe. The numerical calculations of the change in impedance due to the crack are compared to a Finite Element model of the problem and good agreement is found.

A Complete Model of Ultrasonic NDE for Internal and Surface-Breaking Cracks

Ultrasonic nondestructive testing and evaluation (NDT and NDE) of materials and structures plays an increasingly important role for the safety and integrity in some advanced engineering applications, e.g., in the nuclear power and aeroplane industries. This has led to advances in the mathematical modelling of such problems. To make a realistic model of the whole testing situation has many benefits. It can greatly reduce costly experimental work with various test blocks and components with known defects. It shows that all important parameters in the testing have been identified and can give a good physical “feeling”. A model can also be very useful in the construction and evaluation of various test routines as it can ascertain whether a postulated defect can be detected or not.