Showing papers in "Quarterly of Applied Mathematics in 1958"

On a routing problem

Abstract: : Given a set of N cities, with every two linked by a road, and the times required to traverse these roads, we wish to determine the path from one given city to another given city which minimizes the travel time. The times are not directly proportional to the distances due to varying quality of roads, and v varying quantities of traffic. The functional equation technique of dynamic programming, combined with approximation in policy space, yield an iterative algorithm which converges after at most (N-1) iterations.

On the application of infinite systems of ordinary differential equations to perturbations of plane Poiseuille flow

Abstract: Introduction. While considerable progress has been made in recent years in various stability problems of hydrodynamics (see C. C. Lin [1] and the bibliography contained in the reference) and in the discussion of fully developed turbulence (see G. K. Batchelor [2]), the problems associated with transition from laminar to turbulent motion are in a less satisfactory state. To be sure the phenomenological theory of H. Emmons [3, 4] shows considerable promise, but it replaces explicit dependence on the hydrodynamical equations by general stochastic and probabilistic considerations. Unfortunately, the usual laminar stability theory of the Orr-Sommerfeld equation seems incapable of reasonable extension to the nonlinear regime. In addition, even the linear stability analysis is very difficult and requires delicate mathematical consideration. In contrast to this, the techniques used by E. Hopf [5] to establish the existence of a weak solution for all time of the Navier-Stokes equations in a bounded region are in principle simple and constructive, and it therefore seems reasonable to investigate their practicality for, first, the investigation of the stability problems of hydrodynamics and, second, any insight they might be capable of providing for the transistion problem. In essence, Hopf's method consists of proving the existence of a complete orthonormal set of functions of the space variables which have divergence zero and which vanish outside sets of compact support. A solution to the Navier-Stokes equations is then sought as an orthogonal series expansion with unknown coefficients of time. This, when substituted into the Navier-Stokes equations and sampled by all members of the orthogonal set, leads to an infinite set of ordinary nonlinear differential equations in the infinite number of unknown time dependent coefficients. The existence of a solution of this system is then established by a process of successive approximations involving a square truncation of the system. The idea of the weak solution and Hopf's treatment of the boundary conditions closely parallel those of the direct methods of the calculus of variation. Certainly, if for a given problem, a sufficiently judicious choice of a complete orthonormal set of functions in the space variables could be made so that the main remaining features of the Navier-Stokes equations were to be contained in a relatively few non-

A free boundary value problem for the heat equation

Abstract: Differentiation is denoted by a subscript whether it is a partial derivative of a function of two variables or an ordinary derivative of a function of one variable. This problem with A = 0 has been discussed by several authors, see for example [1, 2, 6], but thus far no existence proof has been established**. The problem describes the physical phenomena of evaporation, fusion, sublimation, etc. For example with B = 0, (1) could refer to the following situation. A long metal rod insulated at the sides has begun to melt at one end (x = 0). The layer of liquid metal is A units deep and has some initial temperature distribution, F(x). The critical temperature Tc is the melting point of the metal. At x = 0 heat is applied to the rod at a known rate proportional

Stability limits for a clamped spherical shell segment under uniform pressure

Abstract: An integration procedure for the differential equations for the finite deflections of clamped shallow spherical shells under uniform pressure is developed. Stability limits for the clamped shell are obtained for a range of the central height to thickness ratio from about 1 to 35. This serves to correct and extend previously known stability limits for this problem.

On the heat transfer to constant-property laminar boundary layer with power-function free-stream velocity and wall-temperature distributions

Abstract: where /3 = 2m/(m + 1), a is the non-dimensional velocity gradient at the wall (usually expressed as a = /"(0)),

Some new techniques in the dynamic programming solution of variational problems

Abstract: : It was seen that the numerical solution of a problem involving N state variables depended upon the computation of sequences of functions of N variables. This fact made the method routine only for the case where N = 1 or 2, with grave difficulties arising in the general case. In the paper, it is indicated how to overcome this difficulty for a large class of problems in which the underlying equations and the criterion function are linear, although the restraints on the forcing functions may be nonlinear, corresponding say to energy considerations. Finally, it is briefly indicated how the method of successive approximations may be combined with the foregoing techniques to reduce general variational problems, in which the equations and criterion function are nonlinear, to sequences of problems which can be solved numerically by means of sequences of functions of one variable.

On finite expansion of a hole in a thin infinite plate

Abstract: A complete solution for all values of applied pressure is obtained for the expansion of a hole of zero radius in an initially uniform infinite sheet. The analysis is compared with those of previous investigators and found to be simpler and/or more complete. It is shown that the results are applicable to the expansion of a finite hole in a uniform sheet and to theexpansion of a hole in a tapered sheet.

Transverse velocities in fully developed flows

Abstract: 5. Conclusion. In view of the results obtained above the following conclusions, at least for the examples discussed, seem to be justified: (a) /* is not significantly better than g; (b) the improvement h due to the further smoothing of /* may be worthwhile; (c) F* is about as good as while

On transfer functions and transients

Abstract: In the first part of this paper the concept of the positive real function is generalized so that it is applicable to transfer functions and the functions, satisfying this generalized concept, are arranged into classes. Some tests are then developed which may be used to determine whether a transfer function belongs to a particular class. It is also shown that if transfer functions have certain general forms then they will automatically be members of one of the classes. Finally, several properties of the phase functions for such system functions are developed. The second part of the paper considers the impulse and step responses corresponding to these transfer functions. It is found that these transient responses are bounded and moreover the rise time and settling time of the step response are found to be greater than lower bounds which depend on the amount of the maximum overshoot (or undershoot) . These results are also generalizations of the restrictions developed on the transient responses of positive real system functions and are considerably stronger. As the difference in degree between the numerator and denominator of the transfer function increases, the magnitudes of these lower bounds also increase. Introduction. The question of what restrictions exist on the transient responses of various classes of networks has been treated in a series of papers [1-3]. In particular, bounds on the impulse and step responses have been given when the corresponding frequency responses are restricted to being of constant sign or monotonic in a semiinfinite interval. An example of this is that, if a rational system function, Z(s), is positive real and has one more pole than zero, then the absolute value of the impulse response is never greater than l/C which is the constant multiplier of the system function. These results lead in turn to lower bounds on the rise time or settling time of the step response of such networks when the overshoot and undershoot is specified. Recently, one of these results has been improved by Ovseyevich [4]. The improvement is on the bounds developed on the impulse response of the positive real system functions in the interval (0, T) when the bounds on that response in the interval (T, ») are known. A related problem has been considered by Cutteridge [5] who determines the optimum rise time of passive two-terminal networks under various restrictions. For instance, considering only the step responses which are monotonic or have only a small overshoot, the optimum rise time is determined when the system function has only two zeros and three poles. This paper deals with a generalization of these results which is applicable to transfer functions. In particular, certain classes of functions are defined from the system functions having any number of poles in excess of zeros such that the corresponding transient responses are restricted in a manner similar to the restrictions existing on the transient responses of positive real functions. The strength of the restriction on the transient responses increases as the excess of poles over zeros increases. In the first part of this paper, these classes of system functions are defined. The *Received September 3, 1957. 274 ARMEN H. ZEMANIAN [Vol. XVI, No. 3 definition may be considered a generalization of the concept of positive real system functions which is suitable for application to transfer functions. Various tests are then developed which determine whether a system function is a member of any class. One immediate result is that, if the system function has poles only in the left hand complex frequency plane and no zeros anywhere except at infinityN(that is, if the system function is the reciprocal of a Hurwitz polynomial), then it will be a member of one of the classes. Furthermore, certain properties of the phase functions for such system functions are also developed. In the second section the restrictions on the transient responses for the defined classes of functions are derived. These constitute bounds on the impulse responses and the step responses. From these, lower bounds are developed on the rise time and settling time of the step response when the overshoot and undershoot are given and, conversely, the specification of the rise time or settling time fixes the lower bounds on the maximum overshoot or undershoot. Part I. A generalization of the concept of positive real functions. The systems considered in this paper are lumped, linear, fixed, finite and stable systems so that the system functions1 Z(s) have the following rational form where s is the complex frequency variable a + jut, the coefficients and the constant multiplier K are real numbers, and n and m are positive integers (m > n). 7/„\ _ vs" + Q~-is" 1 + ■ • • + flo _ ^ N(s) . . Z{s) K gm + + ... + bo K D(s)' (1) The term "stable system" is taken to mean a system whose response will eventually become arbitrarily small once the input is removed. Thus the polynomial D(s) is a Hurwitz polynomial all of whose roots have a negative (non-zero) real part. The special classes of functions that are of interest here will be defined after a few preliminary remarks. Let ZQ(s) = (-j)a [ ds„_i f dsQ-2 ■ f Z{s0) ds0 , (2) J CO J CO J CO where the real parts of the complex variables s0 , s, , • • • , , s are all non-negative and q < m — n — 1. Letting st = ) and of Z,(jw) may be obtained from the real and imaginary parts, R(o>) and /(&>), of Z(joj) by (3) and (4). Za(ju) = Rq (w) + jIQ(u) /CO ft 0) q — l /»«t dua-i / dw,-2 • • • / R(w0) do)o (3) CO J — CO J — 00 /CO /» Cd q — 1 /* 0) X do3a-1 J dua-2 J IM dwo . (4) It is evident from the following argument that these successive integrations may be performed m — n — 1 times. Since Z(s) is analytic for 0, the integral of Z(s) between two given points will yield the same result for all paths between these points which do not enter the left half s plane. Furthermore, the inverse power series expansion of Z(s), 'These system functions may be impedances, admittances or transfer functions that are ratios of currents or voltages. 1958] ON TRANSFER FUNCTIONS AND TRANSIENTS 275 which holds for | s | > M, is i ^, (5) H = m-n $ where = K and M is a positive number greater than the distance from the origin to the pole of Z{s) farthest away from the origin. Since this series converges uniformly for | s | > M, it may be integrated term by term yielding a series which also converges uniformly in the same region. This process may be continued q times where q < m — n — 1. Thus the successive integrations of Z(s) given by (2) yield unique functions ZQ(s) which are all analytic for 0. Now the aforementioned classes of functions Z(s) given by (1) are defined as follows. Definition. Z(s) will be called a class k junction, where k = m — n, if one of the following inequalities holds for — «> < « < +00 • For k = 2c + 1 (v = 0, 1, 2, • • •), (-1)'/?*_»(«) > 0 (6) and, for k = 2v (v = 1, 2, 3, • • •)> (-ir.7._,(«) > o. (7) The class k functions have the interesting property that they are related to the positive real functions in the region where they are defined (that is, in the right half s plane and on the imaginary axis). Theorem 1. If the system function Z(s) is a class k function, then ( — Zm_„_1(s) is a positive real function for 0 and the constant multiplier K is positive. Proof. The inequalities (6) and (7) state that the real part of ( — is non-negative for all co. Moreover Zm-»_i(s) is analytic for c > 0 since Z(s) is analytic in this region. Thus by the minimax theorem, the real part of ( —Zm_„_x(s) is non-negative for a > 0. Therefore to prove the first part of the theorem it need only be shown that ( — Zm-n-i(s) is real for s = 0. But 'Z.-.-.W = (-1)-—1 f" d(Tm_n_2 r " 3 ••• r ZMdao J co J oo J oo and the right hand side of this expression is a real function of a real variable. Finally the series expansion of Z(s) given by (5) may be integrated term by term q times according to (2) for q < m — n — 1. The resulting expression for Z„(s), which holds for [ s | > M, is " f t, h-Dfa.Vfc-j?;<8) Thus the first term of the inverse power series expansion of ( — ' Zm^„.1(s) is Km.n (m — n — l)!s ' where Km-n = K. Moreover for s real and sufficiently large, this first term becomes the dominant term. Therefore K must be positive. This completes the proof. A property of the functions Z,(s) which will be used subsequently is the fact that their real and imaginary parts are either odd or even. This is stated by Lemma 2. To prove this however, Lemma 1 will be needed. 276 ARMEN H. ZEMANIAN [Vol. XVI, No. 3 Lemma 1. Let h(u) be even (or odd) and integrable for — m < c < +°°; let q(u) = h(u) du; and let q(°°) = 0. Then q(cc) is odd (or even). Proof. q(co) = / h(u) du+ h(u) du = 0. J — 00 J u

On an inequality of Liapounoff

Abstract: and is a non-vanishing constant on some ^-interval. If such a solution x(t) of (1) is disregarded [and (1) cannot have two, linearly independent, such solutions in any case], then, besides the continuity of pit), only the assumption pit) =; 0 will be needed. Only real-valued solutions xit) will be considered, and the trivial solution, xit) = 0, will be excluded. It is clear from (1) that x\"it) ^ 0 or x\"(i) g 0 at a given t according as xit) < 0 or xit) > 0 at that t. Ifence the graph of x = xit) must turn its concavity toward the 2-axis at every t. Since the clustering of zeros of the derivative x'it) has been excluded, it follows that the zeros of xit) separate, and are separated by, the zeros of x'it) [provided that either xit) or x'it) has at least two zeros]. Let a closed ^-interval [c, d] be called a primitive interval of xit) if neither xit) nor x'it) has any zero in the interior of [c, d], Such an interval [c. d] will be called a complete primitive interval of xit) if for no e > 0 is [c — e, d + «] a primitive interval of xit), that is, if xit) ^ 0 and x'it) ^ 0 for c < t < d but either x(c) = 0 and x'id) = 0 or x'(c) = 0 and xid) — 0. Note that x'ic) ^ 0 and xid) 0 in the first case, and that a:(c) 0 and x'id) ^ 0 in the second case, since the simultaneous vanishing of xit) and x'it) leads to the trivial solution. 2. The purpose of this note is to show that, owing to the concept of a primitive interval, a theorem of Liapounoff (see below) can be extended from his \"disconjugate\" case to the general case, as follows: If pit) is continuous and non-negative on a closed t-interval [a, b\\, and if [a, 6] consists of exactly n primitive intervals of some solution xit) of (1), then

On the equidistribution of pseudo-random numbers

Abstract: A method is found for evaluating the coefficients of a sum of generalized spherical harmonics so that the sum will be simultaneously invariant under two rotation groups. The coefficients for a sum of ordinary spherical harmonics invariant under each individual group must first be known. *Reeeived May 15, 1957.

The rectification of non-Gaussian noise

Abstract: Section

Solutions of the Helmholtz equation for a class of non-separable cylindrical and rotational coordinate systems

Abstract: Abstract. In cylindrical and rotational coordinate systems, one of the variables can be separated out of the Helmholtz equation, leaving a second order partial differential equation in two variables. For a class of the coordinate systems, this equation is reducible to a recurrence set of ordinary differential equations in one variable, which are solvable by ordinary methods.

On isoperimetric inequalities in plasticity

Abstract: Hence, if P„) = 0 on each boundary, J vanishes for all admissible choices of Ta/>. Hence, by the converse theorem mentioned above, t'af is derivable from a single valued displacement. The same is then necessarily true of = ea0 + t'afi, and hence

On the periodic solutions of the forced oscillator equation

Abstract: Introduction. The phenomenon of \"subharmonic vibrations\" has been considered by many authors. The research on the subject goes back nearly one hundred years, beginning (probably) with Melde [14]f, Helmholtz [5], and Lord Rayleigh [17], all before the turn of the century, and continuing uninterruptedly to the present. For a fairly complete bibliography on this subject, the reader is referred to the classical paper on non-linear engineering problems by von Karman [8], and the books by Stoker [20], Minorski [15] and McLachlan [13]. In this paper, we shall discuss a single degree of freedom system whose mechanical model might be a mass under the action of an \"elastic force\" (linear or non-linear, restoring and/or exciting) and of a simple harmonic forcing function of frequency a. Moreover, we assume that small quantities of positive damping are present in the physical system, but absent from the equation of motion. Such damping can be shown, in general, to reduce the free vibrations to negligibly small amplitudes in a finite time. Consequently, the periodic solutions of this system are those usually referred to as \"steady state vibrations.\" We are assuming here that the effect of small damping on the motion is slight, but later on we shall prove this to be the case. It follows that the equation of motion of the system is the oscillator equation

Air drag effect on a satellite orbit described by difference equations in the revolution number

Abstract: For a satellite on an orbit which is affected by air drag, difference equations are derived whose solutions express changes in orbital size and shape and give the satellite's time behavior as functions of revolution number. The results are obtained from a first order perturbation theory using a small air drag parameter. Introduction. The behavior of a satellite orbit under the effects of air drag have been discussed by Petersen [l]1, the present author [2], and others. Using the method of variation of parameters, I derived some approximate expressions for the decay of eccentricity with radius, the decay of radius with revolution number, and the growth of time with revolution number. Those results appear to be especially useful for eccentricities which are quite small, say e < 0.01, although a complete investigation of their range of validity has not been investigated. An alternative approach is available which may be preferable for larger eccentricities. It consists of a perturbation method in a small parameter, carried only to the first approximation, followed by the derivation of an exact relationship between values at the beginning and the end of any orbital revolution. Thus one obtains a set of difference equations in the values of the problem's dependent variables which occur at each full orbital revolution. Because the approach leads to difference equations, it appears well suited to numerical treatment. It offers a significant advantage even when a high speed digital computer is available, for it obviates the numerical integration around each orbital revolution which can impose a significant burden on machine storage capacity and result in a prohibitively long solution time. It is clear that in a conventional numerical solution in which the satellite is followed step-wise around each revolution, even the finest practicable integration interval and the greatest care in handling errors may not avoid a significant accumulation of error over tens of thousands of revolutions. On the other hand, the difference equation method developed here allows the errors for any complete revolution to be made as small as desired without excessive difficulty. Outline of the method. Like [2], this treatment begins with the drag law and equations of motion discussed by Petersen [l]2. The following notation is used: r radial distance from center of earth to satellite 13 angular advance of satellite from arbitrary interially-fixed reference line in plane of orbit K product of earth mass and constant of gravitation •Received March 28, 1957; revised manuscript received August 27, 1957. 'Numbers in square brackets refer to the bibliography at the end of the paper. These apply only to the case where the aerodynamic force acts only as a drag along the line of the velocity vector. Lift forces are assumed absent. Other assumptions are given with the notation below. 132 ROBERT E. ROBERSON [Vol. XVI, No. 2 CD drag coefficient, assumed constant A satellite projected area on plane normal to velocity vector, assumed constant3 m satellite mass a equatorial radius of earth h altitude, r — a (neglecting oblateness of earth) p(h) standard stable atmospheric density function In these terms, the equations of motion are P\" _ ^ = -£ ~ P(h)rY2 + r^T', (1) r/3\" + 2r'f}' = P{hW(r2 + r°/3\"2),/2. (2) It is convenient to introduce auxiliary dimensionless variables £ = a/r, (3) V = Ka/m\\ (4) and to regard them as functions of 0. Primes henceforth denote differentiation with respect to 0 as the new independent variable. Also define the parameter.