N.G. Stephen

On static analysis of finite repetitive structures by discrete Fourier transform

Functional solutions for the static response of beam- and plate-like repetitive lattice structures are obtained by discrete Fourier transform. The governing equation is set up as a single operator form with the physical stiffness operator acting as a convolution sum and containing a matrix kernel, which relates to the mechanical properties of the lattice. Boundary conditions do not affect the equation form, and are taken into account at a subsequent stage of the analysis. The technique of virtual load and substructure is proposed to formally close the repetitive lattice into a cyclic structure, and to assure the equivalence of responses of the modified cyclic and original repetitive lattices. A discrete periodic Greens function is introduced for the modified structure, and the final displacement solutions are written as convolution sums over the Greens function and the actual external and virtual loads. Several example problems illustrate the approach.

Design of a 100 kVA high temperature superconducting demonstration synchronous generator

The paper presents the main features of a 100 kVA high temperature superconducting (HTS) demonstrator generator, which is designed and being built at the University of Southampton. The generator is a 2-pole synchronous machine with a conventional 3-phase stator and a HTS rotor operating in the temperature range 57–77 K using either liquid nitrogen down to 65 K or liquid air down to 57 K. Liquid air has not been used before in the refrigeration of HTS devices but has recently been commercialised by BOC as a safe alternative to nitrogen for use in freezing of food. The generator will use an existing stator with a bore of 330 mm. The rotor is designed with a magnetic core (invar) to reduce the magnetising current and the field in the coils. For ease of manufacture, a hybrid salient pole construction is used, and the superconducting winding consists of twelve 50-turn identical flat coils. Magnetic invar rings will be used between adjacent HTS coils of the winding to divert the normal component of the magnetic field away from the Bi2223 superconducting tapes. To avoid excessive eddy-current losses in the rotor pole faces, a cold copper screen will be placed around the rotor core to exclude ac magnetic fields.

Dynamic response of spinning tapered Timoshenko beams using modal reduction

A method for dynamic response analysis of spinning tapered Timoshenko beams utilizing the finite element method is developed. The equations of motion are derived to include the effects of Coriolis forces, shear deformation, rotary inertia, hub radius, taper ratios and angular setting of the beam. Modal transformations from the space of nodal coordinates to the space of modal coordinates are invoked to alleviate the problem of large dimensionality resulting from the finite element discretization. Both planar and complex modal transformations are presented and applied. The reduced order modal form of equations of motion is computer generated, integrated forward in time, and the system dynamic response is evaluated. Numerical results and comparisons with the full order model (FOM) are presented to demonstrate the accuracy of the reduced order model (ROM).

On the maximum power transfer theorem within electromechanical systems

Abstract Conditions for maximum power transfer are considered within electrical, mechanical, and electromechanical systems. The familiar concept of resistance load matching within an electrical system is extended to a mechanical system and is 50 per cent efficient. For electromechanical systems, the transfer of maximum power from one domain (the electrical, say) to the other (mechanical, say) is also 50 per cent efficient. However, when losses occur in both the electrical and mechanical domains, the concept of load matching must be applied within the domain to which power is being delivered; efficiency is now <50 per cent, which is most easily seen from the source domain.

Initial tension in randomly disordered periodic lattices

This paper is concerned with probabilistic analysis of initial member stress in geometrically imperfect regular lattice structures with periodic boundary conditions. Spatial invariance of the corresponding statistical parameters is shown to arise on the Born-von Karman domains. This allows analytical treatment of the problem, where the parameters of stress distribution are obtained in a closed form. Several benchmark problems with beam- and plate-like lattices are considered, and the results are verified by the direct Monte–Carlo simulations. Behaviour of the standard deviation as a function of lattice repetitive cell number is investigated, and dependence on the lattice structural redundancy is pointed out.

Characteristic solutions for the statics of repetitive beam-like trusses

Abstract This paper concerns two major points: (1) decomposition of functional solutions for the static response of repetitive pin-jointed beam trusses under end loadings into spectrum of elementary function modes; and (2) a mathematical classification of the last. The governing finite difference equation of statics is written as a single matrix form by considering the stiffness matrix of a representative substructure. It is shown that its general solution can be spanned by only 2R individual modes, where R is the number of degrees of freedom for a typical nodal pattern inside the truss. These modes are divided into two primary classes: transfer and localised. A unique set of “canonical” transfer solutions is found by a method based on Jordan decomposition of the transfer matrix. Also, a technique of constructing transfer matrices for a wide class of trusses is presented. The canonical modes can be further subclassified as exponential, polynomial and quasi-polynomial. The complete set of 2R canonical transfer and localised modes uniquely represents the basic structural response behaviour, and gives a basis for the characteristic (non-harmonic) expansion of static solutions. Several illustrative examples are considered.

Macaulay's Method for a Timoshenko Beam

The Macaulay bracket notation is familiar to many engineers for the deflection analysis of a Euler–Bernoulli beam subject to multiple or discontinuous loads. An expression for the internal bending moment, and hence curvature, is valid at all locations along the beam, and the deflection curve can be calculated by integrating twice with respect to the axial coordinate. The notation obviates the need for matching of multiple constants of integration for the various sections of the beam. Here, the method is extended to a Timoshenko beam, which includes the additional deflection due to shear. This requires an additional expression for the shearing force, also valid at all locations along the beam.

Transfer matrix analysis of the elastostatics of one-dimensional repetitive structures

Transfer matrices are used widely for the dynamic analysis of engineering structures, increasingly so for static analysis, and are particularly useful in the treatment of repetitive structures for which, in general, the behaviour of a complete structure can be determined through the analysis of a single repeating cell, together with boundary conditions if the structure is not of infinite extent. For elastostatic analyses, non-unity eigenvalues of the transfer matrix of a repeating cell are the rates of decay of self-equilibrated loading, as anticipated by Saint-Venants principle. Multiple unity eigenvalues pertain to the transmission of load, e.g. tension, or bending moment, and equivalent (homogenized) continuum properties, such as cross-sectional area, second moment of area and Poissons ratio, can be determined from the associated eigen- and principal vectors. Various disparate results, the majority new, others drawn from diverse sources, are presented. These include calculation of principal vectors using the Moore–Penrose inverse, bi- and symplectic orthogonality and relationship with the reciprocal theorem, restrictions on complex unity eigenvalues, effect of cell left-to-right symmetry on both the stiffness and transfer matrices, eigenvalue veering in the absence of translational symmetry and limitations on possible Jordan canonical forms. It is shown that only a repeating unity eigenvalue can lead to a non-trivial Jordan block form, so degenerate decay modes cannot exist. The present elastostatic analysis complements Langleys (Langley 1996 Proc. R. Soc. A 452, 1631–1648) transfer matrix analysis of wave motion energetics.

On transfer matrix eigenanalysis of pin-jointed frameworks

Eigenanalysis of the state vector transfer matrix has previously been employed to obtain Saint-Venant decay rates and continuum beam properties of a repetitive pin-jointed framework. Decay eigenvalues occur as reciprocal pairs, the transfer matrix being symplectic, and three of the unity, transmission, eigenvalues pertain to the trivial rigid body displacements. By setting displacement or force components equal to zero at the remote right-hand end of the structure and, through use of a recurrence relationship, new displacement transfer matrices, S or C, are derived for the generic cell; these are one-half of the original size, well conditioned, and the redundant information is eliminated. The former, S, requires a large value of the recurrence index, i, to achieve accurate eigenvalues while the latter, C, retains trivial information pertaining to the rigid body displacements. An alternative force transfer matrix, M, derived from S, retains the maximum amount of relevant information and converges quickly. The method suppresses the redundant right-to-left decay eigenvectors, and calculation of the transmission vectors of tension, bending moment and shearing force is simplified by the need to calculate just one principal vector rather than four for the original eigenproblem. Finally, these transmission vectors are employed to determine the continuum beam properties of the framework.

Saint-Venant decay rates: A procedure for the prism of general cross-section

A procedure previously developed for the determination of decay rates for self-equilibrated loadings at one end of a pin-jointed framework consisting of repeated identical cells, wherein the decay factors are the eigenvalues of the single cell transfer matrix, is here further developed and applied to a prismatic continuum beam of general cross-section. A sectional length of beam is treated within ANSYS finite element code as a super element; nodes at both ends of the section are treated as master nodes and the stiffness matrix relating forces and displacements at these master nodes is constructed within ANSYS. Manipulation of this stiffness matrix within MATLAB gives the transfer matrix from which the eigenvalues and eigenvectors may be readily determined. Accuracy of the method is assessed by treating the plane strain strip, the plane strain sandwich strip, and the rod of circular cross-section, representing a selection of the examples for which exact analytical solutions are available, and is found to be very good in all cases.