Showing papers in "Quarterly of Applied Mathematics in 1974"

Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations

Abstract: : In the report the authors study harmonic properties of sequences (e - (lambda sub k)t) of real exponential functions. Linear independence results, including estimates on the norms of biorthogonal functions, are obtained in the space (L sup 2) over the interval = or > 0, but 0. The results are uniform in that they depend only upon certain separation requirements on the (lambda sub k) rather than upon the individual sequence (lambda sub k). The results are used to study the boundary value controllability of the heat equation in the unit ball of (R sup n). (Author)

Axisymmetric stagnation flow on a cylinder

Abstract: Due to the inherent nonlinearity of the Navier-Stokes equations, there exist onlythree exact solutions of stagnation flows: Hiemenz [1] found a solution to the twodimensional stagnation flow against a plate, Homann [2] investigated the axisymmetric stagnation flow, also against a plate, and Howarth [3] and Davey [4] extended the results to unsymmetric cases. The present note presents a new exact solution, namely, axisymmetric stagnation flow on an infinite circular cylinder. Fig. 1 shows a cylinder described by r = a in the cylindrical polar coordinates. The flow is axisymmetric about the z axis and also symmetric to the z = 0 plane. The stagnation \"line\" is at z = 0, r = a. This flow may be useful in certain cooling processes. Let u and w be the velocities in the directions r and z respectively. If the flow is inviscid, the potential velocity and pressure distribution in the neighborhood of the stagnation line are u = — k(r — a /r), (1) w = 2 kz,

Stability of Pure Homogeneous Deformations of an Elastic Cube under Dead Loading

Abstract: In a previous paper [1], the problem was considered of the pure homogeneous deformation of a unit cube of incompressible neo-Hookean elastic material by three pairs of equal and opposite forces acting normally on the faces of the cube and distributed uniformly over them. It was found that, for certain specified values of the forces, more than one equilibrium state of pure homogeneous deformation can exist. The stability of each of these states was investigated, with respect to superposed infinitesimal pure homogeneous deformations, with the same principal directions as the equilibrium state. It was found that for certain ranges of values of the applied forces, more than one equilibrium state of pure homogeneous deformation which is stable in this sense can exist. Which of these stable states is actually attained in practice will depend on the order in which the forces are applied.

Asymptotic analysis of the buckling of externally pressurized cylinders with random imperfections

Abstract: The buckling of finite circular cylindrical shells with random stress-free initial displacements which are subjected to lateral or hydrostatic pressure is studied using a perturbation scheme developed in an earlier paper [1], A simple approximate asymptotic expression is obtained for the buckling load for small magnitudes of the imperfection. This result is compared with earlier results obtained for localized imperfections and imperfections in the shape of the linear buckling mode. Introduction. It is generally recognized that the buckling loads of some elastic structures are substantially reduced by the presence of nonuniformities in these structures. These nonuniformities or imperfections may be in the elastic or geometric properties of the structure. In [7, 8], Koiter developed a general theory of post-buckling behavior and derived simple asymptotic formulae for the buckling load of a class of elastic structures with imperfections in the shape of their classical (linear) buckling modes. In [5] Budiansky and Amazigo applied a reworked version [6] of Koiter's theory in deriving an asymptotic formula for the buckling load of externally pressurized cylinders. Furthermore they derived the range of values of a length parameter Z, introduced by Batdorf [4], for which the cylinder is sensitive to imperfection in the shape of the classical buckling mode. In a more recent study [3], Amazigo and Fraser derive similar results for cylinders with localized or dimple imperfections and obtained the same range of values of Z for imperfection-sensitivity. It is clear that in general the imperfections in structures are stochastic rather than deterministic. Here we assume that the imperfections are Gaussian and obtain an asymptotic formula for the buckling load. The perturbation scheme used here was developed in [1J. It is found that the range of values of Z for imperfection-sensitivity remains the same and the loss in the buckling load for the three types of imperfections parallels that obtained for columns on nonlinear foundations [1, 2], Kdrmdn-Donnell equations. A cylindrical shell is characterized by its outward radial displacement W(X, Y) and an Airy stress function F(X, F) where X and Y are the cartesian coordinates in the axial and circumferential directions. The membrane stress resultants Nx , NY , Nxy are given by Nx = F,YY , NY = F,Xx , and NXy = — F,xy where ( ),Y = d( )/dY , etc. Introducing the effect of a stress-free initial outward * Received November 20, 1972. This work was supported in part by the National Science Foundation under Grant GP-33679X.

Applications of invariant transformations in one-dimensional non-steady gasdynamics

Abstract: Certain reciprocal and adjoint transformations available for onedimensional non-steady gasdynamic flow are applied to an existing solution to construct new exact solutions of the governing equations. The particle trajectories and the pressure-density relations on these trajectories are calculated. An application of the adjoint transformation in studying the flow between a piston and a non-uniform shock wave is indicated.

Plane Waves in Linear Viscoelastic Materials

Abstract: In this paper we discuss the propagation of plane sinusoidal waves in linear viscoelastic materials, both anisotropic and isotropic. Unlike the usual discussions (see, for example, [1]), we do not here assume that the planes of constant amplitude and constant phase are parallel. We do, however, assume that the imaginary parts of the complex moduli are small compared with their real parts and that correspondingly the magnitude of the imaginary part of the slowness vector is small compared with that of the real part. This implies that the attenuation of the wave is small in distances of travel of the order of a wavelength.

Multiple Fourier analysis in rectifier problems. II

Abstract: The nonlinear problem of the multiple Fourier analysis of the output from a cut-off power-law rectifier responding to a two-frequency input, reviewed in general in Part I of this study [1], is further scrutinized here for the special case of a zero-power-law device; i.e., a bang-bang device or a total limiter. Solutions for the modulation product amplitudes or multiple Fourier coefficients as in Part I appear as Bennett functions, and line graphs of the first fifteen basic functions for the problem are given. The new functions Amnw(h, 7c) studied, being based on a discontinuous device, then, together with the functions Amnn)(h, k) studied in Part I, provide approximate solutions to the two-frequency modulation product problem for an arbitrary piecewise continuous nonlinear modulator, and the solution for this general problem is outlined. Finally, numerical tables of the zeroth-kind functions Amnm (h, k) graphed have been prepared and arc available separately in the United States and Great Britain. As before, the entire theory is based on the original multiple Fourier methods introduced by Bennett in 1933 and 1947.

On Extremum Properties of the Generalized Rayleigh Quotient Associated with Flutter Instability

Abstract: : The extremum properties of the generalized Rayleigh quotient related to flutter instability are investigated. It is shown that, in addition to the well-known stationary property, under centain circumstances the quotient exhibits maximum-minimum properties which are in contrast to those of the classical Rayleigh quotient. One consequence is that a Rayleigh-Ritz type of stability analysis using these results leads to a lower bound as opposed to an upper bound in the classical case. The results are applied to multiple-parameter systems and a physical interpretation is given for the generalized Rayleigh quotient, leading to the proof of a convexity theorem. (Author)

Motion of an explosive-induced plane shock wave

Abstract: A plane, unsupported, Chapman-Jouguet detonation in a condensed explosive drives a decelerating shock into a semi-infinite inert of lower shock impedance. A previously reported exact solution for a portion of this one-dimensional time-dependent problem is extended to the entire flow field, and some numerical results are given. The solution has the form of a small set of first-order ordinary differential equations for the shock, and a similar set for each particle path.

On the non-uniqueness of elastic rotations for deformations of materials with elastic range

Abstract: In this note precise definitions of the concepts of elastic and permanent deformation are used to establish non-uniqueness of elastic rotations for deformations of materials with elastic range.

Transitions and stability in the nonlinear buckling of elastic plates.

Abstract: Part