Showing papers in "Quarterly of Applied Mathematics in 1976"

A Nonsimilar Moving Wall Boundary Layer Problem.

Abstract: : The nonsimilar moving wall stagnation point solution derived by Rott has been redefined so as to show that a second related solution exists. Rott's solution corresponds to a constant wall velocity with an external velocity proportional to the distance from the stagnation point. A counterpart solution corresponds to the case of a constant edge velocity proportional to the wall coordinate. Numerical and analytical solutions of the equations have defined the characteristics of the boundary layer including their integral properties.

On the asymptotic stability of oscillators with unbounded damping

Abstract: : Through a technique inspired by the invariance principle of LaSalle, a general growth condition on the damping coefficient h(t) of the equation 2nd derivative of x with respect to t + H(t)dx/dt + kx = 0, k > 0, h(t) > or = epsilon > 0, is given, which is sufficient for the global asymptotic stability of the origin, yet permits this coefficient to grow to infinity with time. The methods used do not depend on linearity, and are applied to obtain similar results to the nonlinear analog of this equation.

Variational-Lagrangian irreversible thermodynamics of nonlinear thermorheology

Abstract: A principle of virtual dissipation generalizing d'Alembert's principle to nonlinear irreversible thermodynamics provides a unifying foundation which leads to an extremely general variational-Lagrangian analysis of dissipative phenomena. Thus a synthesis is achieved between thermodynamics and classical mechanics. The present paper applies this principle to the nonlinear thermomechanics of continua with dissipation and heat conduction. Field equations, constitutive equations and Lagrangian equations with generalized coordinates are derived for nonlinear thermo- viscoelastcity, nonlinear thermoelasticity and heat conduction, plasticity, and com- pressible heat conducting fluids with Newtonian and non-Newtonian viscosity. The thermodynamics of instability is also analyzed from the same fundamental viewpoint. 1. Introduction. A Lagrangian-variational approach to irreversible thermo- dynamics was initiated by the author in 1954-55 (I, 21. It was developed mainly in the context of linearity and applied to thermoelasticity (3, 41 viscoelasticity (l, 2, 41, porous media (5), and initially stressed porous and continuous media (6, 71. The appli- cability of these methods to nonlinear problems was demonstrated in a variety of special cases, such as heat transfer (8), porous solids (9) and nonlinear thermoelasticity (lo). A treatment of nonlinear viscoelasticity based on a Lagrangian thermodynamic approach has also been presented by Schapery (ll). The theory embodied in the publications cited above provides a unified analysis based on Lagrangian formalism and generalized coordinates. Among many advantages, the equations have the same form in any coordinate system. Thus basic reciprocity properties of linear dissipative systems are immediately evident for a very large class of phenomena and boundary conditions. As a consequence, the proof of reciprocity properties does not have to be established for each particular case. Basic properties for systems with heredity are also obtained from the concept of internal coordinates and a general expression derived for the associated operator formalism. The corresponding

Distortionless wave propagation in inhomogeneous media and transmission lines

Abstract: Of concern are mechanical or electrical waves in a media which may be nonuniform and dissipative. The problem posed is to find conditions for the undistorted propagation of signals. The electrical transmission line is chosen as the general model. Along the length of the transmission line there are four functions which may be prescribed essentially arbitrarily. These are series resistance, series inductance, shunt conductance, and shunt capacitance. A differential equation is derived relating these functions which gives a necessary and sufficient requisite for distortionless transmission of a voltage wave. Various corollaries of this theorem are developed. For instance, it is shown that simultaneous voltage and current waves can be transmitted without distortion if and only if the characteristic impedance of the transmission line is positive at each point.

On convergence of the finite-element method for a class of elastic-plastic solids

Abstract: A proof of convergence of the finite-element method in rate-type, quasistatic boundary value problems is presented. The bodies considered may be discretely heterogeneous and elastically anisotropic, their plastic behavior governed by historydependent, piecewise-linear yield functions and fully coupled hardening rules. Elastic moduli are required to be positive-definite and plastic moduli nonnegative-definite. Precise and complete arguments are given in the case of bodies whose surfaces are piecewise plane.

A variational problem arising in the design of cooling fins

Abstract: The efficiency of a cooling fin of given weight is measured by the amount of heat dissipated per unit time by the fin. It is known that the efficiency of a given fin can be altered by changing the shape of the fin. In this paper we determine the shape of the most efficient fin of given weight and length, and thickness h.

Wave phenomena in inhomogeneous media

Abstract: The difference between the fundamental solutions of differential equations governing the same physical phenomenon in two different physical media is investigated. A stationary expression of this difference is established, leading to a RitzGalerkin procedure. The Ritz-Galerkin system is solved analytically, providing a representing series for the difference of the two fundamental solutions as a functional of either one or the other of these functions. The convergence of the series is a consequence of its construction itself. Physical examples are considered which show that the convergence rate can be partially controlled.

Application of Khas’minskii’s limit theorem to the buckling problem of a column with random initial deflections

Abstract: : An approximate asymptotic expression is obtained for the buckling load of an imperfect column resting on a nonlinear elastic foundation. The result holds for a large range of imperfection shapes, which are assumed to be stationary random functions of position. The asymptotic analysis is based on application of Khas'minskii's limit theorem to equations for the slowly varying part of the deflection of the column. Previous results obtained for Gaussian imperfection shapes are shown to be valid also for the larger class of random imperfections considered here. (Author)

A transformation property of the Boltzmann equation

Abstract: where t denotes the time, x< ,i = 1, 2, 3, denote the spatial coordinates, and c{ ,i = 1, 2, 3, denote the components of velocity. The usual summation convention is used, so that the repeated index i in (1.1) is summed from 1 to 3. The notation / = f(t, , cj is the phase density function whose value represents the number density of the gas molecules on the phase space at the time t. Specifically, if 5D is any domain in the phase space, then the expected number of gas molecules with their phase coordinates belonging to 3D at the time I is given by the integral

The stability of vector renewal equations pertaining to heterogeneous chemical reaction systems

Abstract: For a class of vector renewal equations arising in the theory of spatially nonuniform chemical reaction processes, stability conditions are given in terms of the physicochemical operators D and K. In particular, we provide conditions for the stability of an initially uniform, multicomponent film bounding a planar catalytic surface, and for the asymptotic stability of a reaction system composed of a population of catalyst particles. We obtain such results by identifying a sup norm which is naturally induced from the factors that symmetrize D and/or K. Further, we exhibit conditions under which a special class of vector renewal equations has positive solutions.

Stability under step loading of infinitely long columns with localized imperfections

Abstract: The dynamic buckling of a long column with small dimple imperfections resting on a nonlinear foundation and subjected to axial step-loading is studied using a formal multi-variable perturbation expansion. Simple asymptotic formulas are obtained for the dynamic buckling load and lateral deflection in terms of the Fourier transform of the imperfection. It is found that the static and dynamic buckling loads are equal. Introduction. The existence of small geometrical and physical imperfections in certain structures leads to large reductions in their buckling strengths. Such structures are known as \"imperfection-sensitive\". The first general static theory of the post-buckling behavior of these structures is the well-known theory of Ivoiter [1], Budiansky and Hutchinson [2, 3, 4] have extended this theory to dynamic buckling. These theories are based essentially on the assumption that the imperfections are in the shape of the classical buckling mode. In [5] the authors showed that this restrictive assumption need not be made to obtain asymptotic expressions for dynamic buckling loads. Here we consider an infinitely long column with an initial imperfection in the shape of a localized dimple. The column rests on a nonlinear elastic foundation and is subjected to an axial load. The static problem has been studied in [6] using equivalent linearization as well as a perturbation expansion involving double scaling in the spatial variable. If the initial imperfection is small, these two methods yield the same expression for the static buckling load in terms of the amplitude of the imperfection. We consider the extension of these results to time-dependent loadings. In this paper we present the case of suddenly applied loads that are subsequently maintained at a constant value. Differential equation. We consider an infinitely long column with a small localized initial imperfection resting on a nonlinear foundation which is subjected to an axial compressive load. The load is suddenly applied and thereafter maintained at a constant value. The nondimensional form of the equation for the lateral displacement w(x, t) of the column is wtl + wxxxx + 2\\wxx + w — aw3 = — 2\\tw0xx (1) * Received January 21, 1975. This work was supported in part by the National Science Foundation under Grant GP-33679X with Rensselaer Polytechnic Institute. 250 JOHN C. AMAZIGO AND DEBORAH F. LOCKHART where an alphabetic subscript denotes partial derivative, and the nondimensional axial coordinate x, lateral displacement w, axial load parameter X, stress-free initial displacement iv0 and time t are related to the corresponding physical quantities by x = (ki/EI)1/4X, w = (k,/kl)1/*W, X = P/2(EIk,)1/2 ew0 = {k3/k{)1/2Wa, t = (kJm)inT. Axial inertia and nonlinear geometric effects are neglected. The assumption is made that the initial displacements and velocities are zero. As shown in [5], this is equivalent to assuming that the nondimensional initial displacements and velocities are of the same order as the imperfections, e is a small imperfection parameter, EI is the bending stiffness of the column, P is the magnitude of the axial step loading applied at time T = 0, and m is the mass per unit length of the column. The column is restrained against additional lateral deflection W by a foundation that produces a restoring force per unit length kiW — ak3W3. a takes on the value 1 or —1 depending on whether the foundation behaves like a \"softening\" or \"hardening\" spring. We assume that the imperfection shape iv0 (x) is continuously differentiable and satisfies the exponential decay condition |w(.r)| < M exp ( —0 \\x\\)(M, 0 > 0). (2) The classical theory (linear, time-independent eigenvalue problem with w0 = 0) for any length column with simply supported ends consists of wxxxx + 2\\wxx + w = 0, w = wxx = 0 at x = 0, rir, where r is a measure of the length of the column. The eigenfunctions are wn(x) = sin (nx/r), n = 1, 2, with corresponding eigenvalues X. = i (r/n)\\ 1 + (n/r)*\\, n = 1, 2, • • • . For columns of length given by r an integer, the classical buckling load (lowest eigenvalue) is X = 1, corresponding to n = r. If r is not an integer, the buckling load Xc satisfies the inequality 0 < Xc — 1 < l/[2n(n + 1)], n < r < n + 1. Thus, for r J>> 1, \\c = 1 + 0(r~2), we consider the column to be of infinite length. The case for which r = 1 and the imperfection is arbitrary has been discussed in [5], Dynamic theory. The problem to be considered is wtt + wXIIX + 2Xw« + w — aw3 = —2\\ew0lI, — °° < x < <», t > 0, (3) iv, wx —» 0 as x —» ± <» , ^ w = w, = 0 at t = 0. We consider e <3C 1 and seek to determine the maximum value XD of X less than the classical buckling load Xc such that the deflection w vanishes as |.r| —> . This condition must be distinguished from the boundedness condition imposed for imperfections in the shape of classical buckling modes (see [4]) and for finite-length structures (see [5]). This condition was imposed in the solution of the time-independent problem [6]; however, STABILITY UNDER STEP LOADING 251 it was not specifically noted in that report. As in [6], a perturbation parameter 5 may be introduced and defined as 62 = 2(1 A). (5) As in this problem, we introduce a new variable f = Sx. As shown in the finite column problem [5], the dominant response depends only on the long-time scale r = 8t and not on the short-time scale t. We assume that w is a function of x, f, and t, write w(x, t) = u(x, f, t; §), and expand w and \\e in power series in 5: w(x, t) = u{x, f, r; 6) = 2 uM(x, f, t)S\", (6) n= 1 Xe^Ze'-'S\". (7) n = 1 Substituting these expansions into (3) and equating like powers of S leads to the sequence of equations: LuU> = ~2tn)W0rx , (8) Lu<2) = — 2t2)w0xx 4uJM,'((T), etc., 252 JOHN C. AMAZIGO AND DEBORAH F. LOCKHART where prime ' denotes differentiation with respect to the argument of the function. Thus with n = 1 /u>(co) = Hu'(co)/(co2 l)2 (18) where tf(n)(co) = 2eMaaH>0(u) + [/'\"\"\"] («2 2)[/<\"\"] + iw(co2 2)[/(\"'], n = 1, 2, 3, • • • . (19) Boundedness of (to) requires that //] _ [fl(.)]j [/(,)'] = -t[a(1)] + i[a(1)], (21) [/(,)\"] = [a] i[aw], where [a'1'] = [a(1>(0, r)] = aU)(0+, r) aa'(0\", r). (22) Since in general tf>0(l) 5^ 0, from (21) and (20) we obtain eU) = 0 and [a(1>(0, r)] = 0. Consequently fn)(x) = 0 and Uu,(:x, f, r) = a]~ [a(2)], [/(2)'] = -i[aw] + i[aw] [af(1)] [«f<,)], (2?) [/<2>\"] = [a<2)] + [a<2)] 2i[a{w] + 2i[ar\\, Um'\"} = i[a<2)] t[a<2)] + 3[ar(1>] + 3[ar(n], STABILITY UNDER STEP LOADING 253 Substituting (27) into (19) and (20) for n = 2 gives [«/\"] [«fU,(0, t)] = -*(2)®„(-l)/2, (28) [a<2>] a [a<2,(0, r)] = (—1) —®0'(—1))/2. We now examine Eq. (10) for w<3). In order for w<3> to be bounded in x, quantities on the right-hand side that give rise to secular terms in x must be eliminated. Thus aTT(1) 4arr(1) + aa) 3aaa)|a(1>|2 = 0. (29) (The complex conjugate of this equation is also asserted.) The corresponding initial conditions (derivable from (4)) and jump conditions previously stated are a(I)(f,0) = ara'(f,0) = 0, (30) [aa'(0, r)] = 0, (31) [<\"(0, r)] = -e(2,w>o(-l)/2. (32) Now, writing the complex constant — e<2)tf>0(—1)/2 in its polar form, namely — 6<2'i2)0(—l)/2 = A exp (—id) (33) where A and 8 are real, we observe that the solution to the problem (29)-(32) may be expressed as aU)(f, t) = a(f, r) exp (-id) (34) where a(f, t) is real. The problem for a(f, r) consists of arr — 4 an + a — 3 aa3 = 0, r>0, — «=

On the linear capillary gravity wave problem

Abstract: The two dimensional problem of the transient development of capillary-gravity waves on the free surface of an incompressible inviscid fluid is investigated. The fluid occupies the semi-infinite space bounded by a fixed vertical barrier and the cases of finite uniform depth and infinite depth are considered. A boundary condition is derived for the liquid-solid intersection and it is shown how this provides for a unique solution. The solution of the problem is obtained by using integral transforms and the asymptotic form of the free surface displacement is obtained.

Some exact solutions of Burgers-type equations

Abstract: A class of Burgers-type equations, reducible through a generalized Hopf-Cole transformation to a linear diffusion equation, are treated by similarity methods. New exact solutions of these equations are obtained and related to the wellknown solutions of the standard Burgers equation. Physical applications of these solutions are indicated.

Numerical study of periodic Hamiltonian systems by means of associated point mappings

Abstract: Among the problems which lead to the study of equations such as (1) can be found, for example, the study of the oscillations of a pendulum under parametric excitation [1], that of the oscillations of a satellite under the effect of periodic variations in its moments of inertia [2] and that of the motion of particles in alternating gradient accelerators [3]. In conservative dynamic systems with two degrees of freedom, studying movements around a periodic solution can amount to studying solutions of (1) around the origin [4], Since Poincar£ [5] it has been known that a point mapping linking the state (x, y) of the system between two instants separated by a time interval can be associated with (1). Seeking this transformation 3 can be justified by the fact that is it particularly suited to numerical treatment. The search for periodic solutions (sub-harmonic) amounts to determining the fixed points of the transformations 3\", n integer (positive) which in numerical terms is the solution of a two-variable, nonlinear system of algebraic equations. In actual practice, it is not possible, except for a few particular cases, to express the relationships which define 3 by means of classical functional analysis. Thus, it is the aim of this paper to present a method which makes it possible to determine a transformation 3* enabling one to carry out an approximate (in a sense which will be defined later and justified on the examples given) analysis of the solutions of the differential system (1). This method can be situated mid-way between that of direct simulation by numerical integration, which poses the problem of sensitivity to discretization, particularly crucial in the case of conservative equations, and the many analytical methods such as those based on successive approximations to determine asymptotic expansions of the solution when the nonlinear terms depend on a small parameter [1], Before outlining the plan of the paper, it is necessary to give the essential characteristics of the transformation 3 associated with (1). If the Hamiltonian H is analytical with respect to the variables x, y, then the associated transformation 3 is itself analytical and one-to-one and in addition it is area-preserving; the Jacobian, the determinant of the partial derivatives matrix, is 1.

Passivity and linear system stability

Abstract: Using the network concept of passivity (or positive realness), new criteria for stability and instability of linear systems (with time-varying coefficients) are derived.