 # The numerical solution of certain problems in elastodynamics

Blair, James McBeth (1967) The numerical solution of certain problems in elastodynamics. PhD thesis, University of Glasgow.

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## Abstract

The method of integral transforms can provide the solution of a differential equation satisfying prescribed boundary conditions if certain requirements involving the boundary conditions and the transforms are met. The three requirements are stated explicitly in the thesis for the differential equation L gamma = f on a finite domain R, where L is a matrix and gamma and f are columns. In general not all the requirements can be satisfied for the equations of elasticity. If one particular requirement is relaxed, then the transform procedure may be applied in such a way as to reproduce in R a formal series solution of the above differential equation without reference to the boundary conditions, and the solution is a general solution in that sense. When the solution is applied to a particular set of boundary conditions there results an infinite system of simultaneous linear equations in an infinite number of unknowns, whose solution yields a solution of the differential equation. The infinite system is given formally in chapter I for the equations of elasticity. Theoretical results pertaining to the solution of infinite systems of equations and approximate methods of solution are known, and chapter II is devoted to a statement and a discussion of those results which are relevant to the problems of the thesis. A deficiency in the existing theory is noted. The application of the above approach to the solution of some specific vibration problems in elastodynamics is given in chapters and IV, A study of the vibrating elastic parallelepiped with clamped edges provides an indication of the rate of convergence of the approximate numerical solution for different dimensions, and allows some conclusions to be drawn about the value of the method as a practical numerical procedure. The numerical results are compared with those obtained by another method due to V.V. Bolotin. The approximate solution of the infinite system of equations is justified in terms of the theory in chapter II. Two problems of the axially symmetric vibrations of elastic rods are investigated in chapter IV. The first rod has all its bounding surfaces stress-free, while the second rod has one of its plane ends clamped and the remaining- surfaces stress-free. Numerical results are presented for both problems. Those for the first case are compared with existing theoretical and experimental values, and they are shown to be the most accurate yet available. No other results for the second problem have been found in the literature. The infinite system of equations is studied in both cases, the conclusions being less satisfactory than for the parallelepiped, as certain questions remain unanswered. In chapter V consider an initial-value problem in which a stress pulse is suddenly applied to one end of an elastic rod. The solution is expressed as an infinite sum over the solutions for the free- free rod, using the method of eigenfunction expansions. The motion of the free end of the rod is computed using the finite set of eigenfunctions in Chapter IV, and the resulting solution is sufficiently accurate to show the successive reflections of the initial pulse as it traverses the rod. Some aspects of the solution are discussed by comparing it with the solution of the analogous problem for an infinite slab.

Item Type: Thesis (PhD) Doctoral Adviser: D C Gilles Applied mathematics 1967 Enlighten Team glathesis:1967-73637 Copyright of this thesis is held by the author. 14 Jun 2019 08:56 14 Jun 2019 08:56 http://theses.gla.ac.uk/id/eprint/73637 View Item