Showing papers on "Asymptotic analysis published in 1974"
TL;DR: In this paper, the authors return to the asymptotic analysis, within the nonlinear equilibrium theory of compressible elastic solids, of the deformations and stresses near the tip of a traction-free crack in a slab of all around infinite extent under conditions of plane strain.
Abstract: In this paper we return to the asymptotic analysis, within the nonlinear equilibrium theory of compressible elastic solids, of the deformations and stresses near the tip of a traction-free crack in a slab of all-around infinite extent under conditions of plane strain. As before, the loading at infinity is taken to be one of uniform uni-axial tension at right angles to the faces of the crack. We show that once a restrictive assumption introduced at the start of our earlier asymptotic treatment of the problem is relinquished, certain perplexing anomalies encountered in the previous analysis no longer arise. The present reconsideration of the problem leads to modifications in the dominant-order results for the “secondary” deformations and stresses, while those pertaining to the physical quantities of primary interest remain unaffected. Furthermore, this investigation encompasses some higher-order considerations, which supply an essential clarification and improvement of the lowest-order asymptotic solution.
95 citations
78 citations
TL;DR: In this paper, the authors used Liapunov functions of Razumikhin type to give conditions sufficient for the stability of the zero state of a system of ordinary differential equations involving an interval of delay which becomes unbounded as t + + co.
44 citations
TL;DR: In this article, the authors investigate the properties of control rules in a linear regression model with one unlinomial-time parameter, where a dependent variable is set at certain levels in order to bring the dependent variable to a desired level.
Abstract: IK A~ULTIPERIODCONTROL PROBLEMS with unkilown parameters. curreilt decisioiis aKect iiot only current performance, but also the alnoulit of information that is obtained about the uiiknown paraineters. The purpose of this study is to investigate such aspects of m~~l t ipe r iod control in a siinple linear regression model with one unlino\\vn parameter, where t!le illdependent variable is set at certain levels in order to bring the dependent variable to soine desired level. The approach uses the methods and criteria of statistical estimation theory (such as stroiig consisteiicy and eliiciency) to jiivestigate the properties of various control rules. This approacl~ sceins particularly useful in coiitrol problems of this type wliere estimation of unknown parameters plays an important role. Previous i:zvestigatioiis of this type of illultiperiod control problem (Aoki [2]), Zellner [9],aild Prescott [ 5 ] ) have been from a Bayesian point of view. By specifying a loss fiu~~ction, prior distributioris on the paraineters, and a distribution for the random disturbance term, a Bayes co~ltrol rule call be calculated, in principle, with the methods of dy~ialllic programming. However, as these studies have shown, ca1culatio.11 or even characterization of Bayes co~itrol rules has proved quite difficult. The approach of this study is 11011-Bayesian. Thc methods and results should c o m p l e m e ~ ~ t the usual Bayeslail viewpoint in eventually leading to reasoiiabie decisioiis in practical probleins. In Section 2 the nlodel is introduced and two coiltrol rules are defined. In Sectio:~3 we prove that these control rilles converge with probability 1 to tlie value \\vI:icli would be used if the u~ilinown parameter were kliown with certainty. Iil Section 4 we derive tlie asyinptotic distribution of tlie coiitrol ruies aiitl parameter estimates, and in Section 5 we show that these coiitrol rules lead to parameter estilnates whicli have as small an asynlptotic variance as any other control rule in a fairly wide class. 111 particular this nieaiis that control rules which are designed for experimentation d o not give parameter estimates which are any better asy~ilptotically than tile inore simple control rulcs of this paper.
42 citations
TL;DR: In this article, existence and smoothness theorems for invariant manifolds characterized by asymptotic stability at specified rates are presented. But these results require roughly half of a hyperbolic structure, so various known results about stable and unstable manifolds are included as special cases.
Abstract: We announce existence and smoothness theorems for invariant manifolds characterized by asymptotic stability at specified rates. Our theorems require roughly half of a hyperbolic structure, so various known results about stable and unstable manifolds are included as special cases. Even under hyperbolicity assumptions, however, our results give new information about metric properties, invariant foliations, and asymptotic stability with asymptotic phase. Proofs will appear in [2].
38 citations
TL;DR: In this article, the buckling of finite circular cylindrical shells with random stress-free initial displacements was studied using a perturbation scheme developed in an earlier paper.
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.
20 citations
01 Jan 1974
16 citations
TL;DR: In this article, the authors considered the problem of finding the asymptotic stability of solutions of (1.1) when the kernel a(t) is known to have a specific (asymptotically stable) representation.
Abstract: and a(t) * a(O+), then the solution u(t) of (1.1) satisfies (1.2). In this paper, we are not concerned with sufficient conditions for the asymptotic stability of (1.1) but rather with the rate of decay of solutions of (1.1) when it is known that the equation is asymptotically stable. Thus, when the kernel a(t) is known to have a specific asymptotic representation, one seeks for asymptotic representation of the solution of (1.1). This approach has been investigated in a recent paper of
12 citations
TL;DR: A new computational method for calculating the asymptotic solution to the matrix Riccati equation is described, which is fast, efficient and gives all possible solutions to the Matrix quadratic form.
Abstract: This paper describes a new computational method for calculating the asymptotic solution to the matrix Riccati equation This method is fast, efficient and gives all possible solutions to the matrix quadratic form Matrix sign functions are used to find the asymptotic solutions
11 citations
11 citations
TL;DR: In this paper, the spectrum of weighted integral operators of the form (1) acting in the space is studied, and the asymptotic behavior of the spectrum is studied.
Abstract: In this paper the asymptotic behavior is studied of the spectrum of weighted integral operators of the form (1)acting in the space .Bibliography: 10 titles.
TL;DR: In this article, it was shown that even for some external lines on the mass shell, the procedure of dropping the mass-insertion term in the Callan-Symanzik equation is justified for the form factor at high squared momentum transfer in a certain class of models.
Abstract: We show that even for some external lines on the mass shell, the procedure of dropping the mass-insertion term in the Callan-Symanzik equation is justified for the form factor at high squared momentum transfer in a certain class of models. This provides a very quick method of summing leading contributions in perturbation theory, as well as summing the next-to-leading terms.
TL;DR: In this article, a multi-channel asymptotic method for the calculation of the free-free transitions of negative ions is presented, illustrated with an application to the negative lithium ion.
Abstract: Details are given of a multi-channel asymptotic method for the calculation of the free-free transitions of negative ions; the method is illustrated with an application to the negative lithium ion.
TL;DR: In this paper, the question of the construction of an inner state of stress of a thin plate under dynamic conditions is examined, and the possibility of considering states of stress with distinct variability in time and in the coordinates and with a distinct relationship between the displacement intensities, is taken into account.
TL;DR: In this paper, a formal, uniformly valid, asymptotic expansion of the Klein-Gordon equation with spatially varying coefficients is obtained with the help of two families of rays, and involving four functions : two successive Bessel functions of integer order and two new functions which are called the diffraction functions.
Abstract: The signaling problem for the one dimensional Klein-Gordon equation with spatially varying coefficients is analyzed. A formal, uniformly valid, asymptotic expansion of the solution is obtained with the help of two families of rays, and involving four functions : two successive Bessel functions of integer order and two new functions which we call the diffraction functions. The validity of the expansion is established when the coefficients in the Klein-Gordon equation are constants, and the results are applied to a signaling problem for a class of acoustic wave guides.
TL;DR: In this article, it was shown that when aperiodic loads of large intensity act on an elastic bar, the higher modes of stability loss have the highest rates of growth of deflections.
Abstract: It is established that when aperiodic loads of large intensity act on an elastic bar, the higher modes of stability loss have the highest rates of growth of deflections. A method is indicated for determining the numbers of these modes, when the effect of the longitudinal vibrations on the transverse vibrations is taken into account and when it is not taken into account. A comparison of the results obtained with results of other authors [1–7] is presented.
TL;DR: In this article, two forms of a unified type of Krylov-Bogoliubov method are considered for the purpose of deriving approximate solutions of non-linear ordinary differential equations that describe the underdamped and overdamped motion of systems subjected to step function excitation.
TL;DR: In this article, the stability problem for distributed parameter energy systems governed by a set of nonlinear, parabolic, partial differential equations is treated by the method of comparison theorems.
Abstract: The stability problem for distributed parameter energy systems governed by a set of nonlinear, parabolic, partial differential equations is treated by the method of comparison theorems. Sufficient conditions for asymptotic stability are presented in terms of the system parameters. The example of a system with temperature control is also presented.
01 Dec 1974
TL;DR: In this paper, an approximate nonasymptotic theory for self-sustaining supersonic laminar boundary layer interaction is presented based on the three-layer conceptual model first introduced by Lighthill (1953) to explain the upstream propagation of disturbances in linear shock wave boundary layer interactions where separation does not occur.
Abstract: : This investigation presents an approximate nonasymptotic theory for self-sustaining supersonic laminar boundary layer interaction which is based on the three-layer conceptual model first introduced by Lighthill (1953) to explain the upstream propagation of disturbances in linear shock wave boundary layer interactions where separation does not occur and recently extended by Stewartson and Williams (1969) (1973) for non-linear interactions using an asymptotic analysis valid for infinite Reynolds number. The practical shortcoming of the asymptotic analysis is that at the largest Reynolds number for which the boundary layer could be expected to remain laminar the viscous sublayer whose thickness is of 0(Reynolds number to the minus 1/8 times the boundary layer thickness) is of comparable thickness to the inviscid interaction layer and thus not a thin sublayer as required for the validity of an asymptotic theory.
TL;DR: In this article, a uniformly valid asymptotic expansion is obtained for the regular solution of a class of second-order linear differential equations with three transition points, a turning point and two regular singular points.
Abstract: A uniformly valid asymptotic expansion is obtained for the regular solution of a class of second‐order linear differential equations with three transition points‐a turning point and two regular singular points. The solution is found by matching three different solutions obtained using the Langer Transformation. The matching yields the eigenvalues and the eigenfunctions.
TL;DR: The method of asymptotic integration, described in this paper, allows an iteration procedure for the successive determination of a approximate solution of three-demensional equations of geometrically nonlinear theory for the shell continuum.
Abstract: The method of asymptotic integration, described in the paper, allows an iteration procedure for the successive determination of a approximate solution of three-demensional equations of geometrically nonlinear theory for the shell continuum. The classification of the shell problems is presented in terms of qualitative and quantitative characteristics of the shell data, and a possibility of estimating the influence of different types of external actions is indicated. On the basis of the results yielded by the asymptotic analysis, general information on the shell behavior may be deduced. A comparison of the predictions with the results of the numerical analysis is given for some plate and shell examples.
TL;DR: In this paper, the degree of singularity of a two point function on the light cone and the asymptotic behaviour of its Fourier transform were shown to be related.
01 Feb 1974
TL;DR: In this paper, Liapunov functions are constructed and used to prove stability theorems for critical autonomous systems in which the linear part of the right-hand side has a zero eigenvalue.
Abstract: Liapunov functions are constructed and used to prove stability theorems for critical autonomous systems in which the linear part of the right-hand side has a zero eigenvalue.
TL;DR: In this paper, an asymptotic method was proposed to study nonlinear partial differential equations for dispersive waves, modified by the addition of a small term, which may be a small damping, for example.