Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1965"

On the degrees of the vertices of a directed graph

Abstract: In a previous paper the realizability of a finite set of positive integers as the degrees of the vertices of a linear graph was discussed. Here we are concerned with the realizability of a finite set of pairs of non-negative integers { (di+, di−): i = 1, 2 …, n} as the degrees of the vertices of a directed graph. The directed graphs considered in this paper are allowed to have parallel elements but it is assumed to contain no self-loop elements. The integers di+ and di− specify the number of arrowheads directed toward and away from vertex di respectively. Other related problems such as; realizability of a given set of non-negative integer pairs as a connected directed graph, strongly connected directed graph, and cycleless directed graph are discussed. The problem of orienting a given graph and the Runyon problem are also considered.

A study of band saw vibrations

Abstract: Band saw flexural natural frequencies decrease continuously with increasing band velocity at a rate dependent upon the pulley mounting system—minimum if the band is allowed to extend under the dynamic loading with the initial static load held constant, and maximum if the band length is fixed. The Galerkin and the flexible band solutions provide bounding fundamental frequency approximations that are simple and accurate. Small periodic band tension variations may induce a large amplitude equilibrium as in the case of the classical string. However, the most probable occurrence of instability is not the classical string subharmonic (fundamental frequency equals 1 2 tension excitation frequency), but occurs at two other excitation frequencies that depend upon the band velocity and pulley mounting system.

A method of viscoelastic stress analysis using elastic solutions

Abstract: A method employing elastic solutions is proposed for solving the integral equations that arise in viscoelasticity. Two numerical examples are given to demonstrate its accuracy and ease of application. It is first shown that, in many cases, a viscoelastic response will be approximately equal to an elastic solution wherein elastic constants are replaced by time-dependent creep or relaxation functions. The familiar integral equations of linear thermo-viscoelasticity are reformulated in terms of dependent variables which satisfy homogeneous mechanical boundary conditions, and which are equal to the difference between these quasi-elastic solutions and the actual viscoelastic solutions. If it is found necessary, these equations can be solved by numerical techniques, such as finite-difference integration or successive approximations, whose methods reduce the elastic-viscoelastic difference equations to a sequence of elastic problems. Furthermore, they often can be solved more easily than the original viscoelastic equations. Following some considerations on superposition of the quasi-elastic and difference solutions in the presence of transient temperatures, two numerical examples are given: pressurization of a reinforced hollow cylinder with a moving boundary, and large deflection of a cantilever. In each case, the quasi-elastic approximation is found to be sufficiently accurate for most engineering purposes. Some concluding remarks relate to implications of the method in regard to photoviscoelasticity.

Optimal control of partially observable Markovian systems

Abstract: This paper discusses the discrete-time Bayesian optimal control of stochastic dynamic systems where some vectors, which augment the system state vectors and the observed state vectors by additional variables, constitute multi-dimensional Markov chains. Optimal control of such Markovian control systems is considered under the assumption that only a part of the components of such vectors is observed by the control system. Certain conditional probability densities needed in deriving optimal control policies are derived, and computational procedures which determine optimal control sequences are given.

Low energy electron diffraction study of the adsorption of oxygen on a (100) Tungsten surface

Abstract: Low energy electron diffraction studies of a (100) face of a tungsten crystal heated to about 1250°K in oxygen at a pressure of about 10 −7 torr, show that the adsorption proceeds through three distinct stages, each accompanied by some surface reconstruction. Diffraction patterns are interpreted to indicate that the first stage results in a surface structure containing one-half a monolayer of oxygen and the second stage in a structure containing three-quarters of a monolayer of oxygen. Diffraction features from the terminal structure strongly suggest a model wherein the surface has become faceted, the facets being planes of type 〈110〉. Beams from the individual 〈110〉 planes can be observed and, at certain electron voltages, all the (110) planes cooperate to produce a sharp bright beam. This latter beam is shown to be equivalent to a Bragg reflection from the bulk crystal. On this model we calculate the lattice parameter of the crystal to be 3.206 ± 0.01 A, close to but slightly larger than that of bulk tungsten. The adsorbed oxygen can be removed from the surface and the three structures traversed in the reverse order by heating the crystal to higher temperatures. No observable pressure increase accompanies the removal of the oxygen, for which fact possible explanations are discussed.

On polarity detection schemes with non-gaussian inputs

Abstract: The design of signal detectors which are optimum (in the Neyman-Pearson sense) requires a complete statistical description of the input processes. Certain non-optimum detectors, the “nonparametric detectors,” have the interesting property that their false alarm rates can be fixed for wide classes of inputs. These detectors are often easy to implement and are useful when the input statistics are not completely known. This paper treats two practical nonparametric detectors, the one-input sign detector and the two-input polarity coincidence correlator, both of which utilize only the polarity information about the input. The performance of these nonparametric detectors is evaluated for a wide class of non-gaussian input noises. Their performance is compared to that of their Neyman-Pearson counterparts designed on the assumption of gaussian stationary noises. The comparisons are based on asymptotic relative efficiency (a.r.e.). While the a.r.e. gives a close approximation to the actual relative efficiency for stationary gaussian inputs, significant differences result when considering non-gaussian inputs, particularly of the impulse type. For stationary gaussian inputs the sign detector is 64 per cent as efficient as its Neyman-Pearson counterpart, the mean detector. Under the same conditions the polarity coincidence correlator (PCC) is ,20 per cent as efficient as its Neyman-Pearson counterpart. It is demonstrated in this paper that these known results are relatively insensitive to small changes in the noise statistics. However, as the amplitude density of the noise processes become more sharply peaked and/or more asymmetrical, then the nonparametric detectors become more efficient than the Neyman-Pearson detectors designed for gaussian inputs. These types of noise processes resemble what is commonly called impulse noise.

Measurement of angular velocity and linear acceleration using linear accelerometers

Abstract: This paper describes the mathematical analysis for a method of measurement of angular velocity and linear acceleration by mounting three accelerometers on three mutually orthogonal discs rotating at a constant frequency. Various types of deterministic misalignment errors are investigated.

External cracks under longitudinal shear

Abstract: The theory of cracks under longitudinal shear, as discussed in previous publications (1, 2, 3), is extended to bodies containing external cracks. It is possible, by an adaptation of the Schwartz-Christofel transformation, to derive a general formula for finding the stress-intensity factor, a parameter governing the onset of rapid fracture in the theory of Griffith-Irwin (4). Specific examples of fundamental interest are solved in closed-form. The results are useful in predicting the remaining strength of structural members with crack-like imperfections.

Internal instability of anisotropic viscous and viscoelastic media under initial stress

Abstract: The internal instability of anisotropic viscoelastic media under initial stress is investigated in analogy with the phenomenon of internal buckling in elasticity by applying the author's principle of viscoelastic correspondence. The spreading of a disturbance in a medium of infinite extent is analyzed. A numerical solution is derived which describes the time history of an initial perturbation in a purely viscous anisotropic medium in compression and confined between parallel boundaries. It is shown that the system acts as a high-pass filter on perturbations of various wavelengths. In the range of significant instability the solution obtained by viscoelastic correspondence from the theory of elasticity is found to be applicable as an approximation to a viscous anisotropic fluid initially in a state of flow. An exact and independent treatment of the latter case is derived for comparison.

Large oscillations of an anisotropic triangular plate

Abstract: An analysis of large amplitude oscillations of right triangular anisotropic plates is given, based on von Karman governing equations generalized to the dynamical and rectilinearly orthotropic case. The coordinate functions represented in a separable form involve the deflection of the plate and the membrane stress function. Both the deflection and the stress satisfy the boundary conditions associated with a built-in contour, free to move in the normal inplane directions. By means of Galerkin procedure a nonlinear second order differential equation for the unknown time function is obtained and readily solved in terms of Jacobian elliptic functions. Period of linear and nonlinear oscillations as well as static nonlinear case are analyzed for various types of anisotropy, side ratios and values of the amplitude.

On the effective dominant pole of the distributed RC networks

Abstract: A technique is developed to determine the effective dominant pole of the distributed circuits The concept of the effective dominant pole facilitates enormously the time and frequency domain calculations Key features of the steady-state and transient response such as bandwidth, rise time and time delay, etc, of the network functions can be easily and accurately determined without much effort Although transfer functions of the exponentially tapered distributed RC network are considered in this paper, the method has a much broader application for all types of distributed RC circuits with arbitrary taper and diffused base transistors The method of approximation which makes use of the product expansion of transcendental functions permits the location of the effective dominant pole by means of a simple algebraic formula and to a degree of accuracy acceptable for most engineering purposes

Rational transform approximation via the Laguerre spectrum

Abstract: A method is presented for determination of an n th order rational transform approximation for a time function, given at least n + 1 of its Laguerre coefficients. The method is based on approximating the discrete set of Laguerre coefficients with a rational generating function. The method does not require predetermination of the poles; and allows the use of as many Laguerre coefficients as are available, without increasing the complexity of the model. Applications to time domain synthesis and transfer function identification are discussed.

Wave theories of traffic flow

Abstract: Two types of mathematical models are presented to explain the empirical observation that a form of “wave phenomenon” occurs when a long line of vehicles on a crowded highway is stopped or started by a signal. The interruption of the steady flow of vehicles initiates “waves of stopping” or “waves of starting.” The first model considered is a car-following model based on a system of linear differential-difference equations that contains two physically significant parameters. The second model adopts a macroscopic viewpoint in which a stream of traffic on a single-lane road may be represented by a compressible “traffic fluid” of density or concentration k and a rate of flow q. The analysis is based on the equation of continuity of the “traffic fluid” and a novel empirical “flow-concentration” relation. The behavior of small disturbances in the vicinity of steady flow is examined by the method of perturbations. The wave phenomena predicted by the first and second models are compared.

Elastic plastic analysis for hollow spherical media under uniform radial loading

Abstract: A practical iterative solution is applied to the problem of a spherical cavity in radially stressed infinite and finite spherical media under elastic-plastic conditions. Stress and strain concentration factors are given for various degrees of loading. Complete range of strain hardening is considered from the perfectly elastic to the perfectly plastic. Deformation and incremental theories are compared and found to be in complete agreement for the radial type of loading.

Effect of ionosphere on radar waveforms

Abstract: Consideration is given to the determination of the distortion introduced by the ionosphere on radar waveforms from which we can in turn obtain the effect of the resolution capabilities, the measurement capabilities and the dynamic range of a radar system. The analysis was carried out for a variety of waveforms: A rectangular pulse; a Gaussian pulse; an unweighted chirped pulse; a Taylor weighted chirped pulse, weighted to provide 40 db sidelobe level; and finally, an unweighted and Taylor weighted step frequency coded pulse, the weighting being used to provide 40 db sidelobe level. It was found that the available bandwidth of the ionosphere was dependent to some extent on the receiver processing used. In particular, it was found that if an unweighted chirp pulse is transmitted, and if on reception, this signal is Taylor weighted with the weighting being adjusted to give 40 db skirt levels, then the available bandwidth can be increased by 50 per cent over that for the case where no weighting is used in the receiver. The available bandwidth was also found to be dependent on the waveform transmitted. For example, if a chirp signal is transmitted and Taylor weighting used in the receiver, then the available bandwidth is at least 50 per cent greater than if a Gaussian pulse is transmitted. For a system carrier frequency of 2 kmc and an uncompensated receiver, it was found that the available bandwidth is 120 mc for a system using either a Gaussian pulse, an unweighted chirped pulse, or an unweighted step frequency coded signal. For the system using a chirped signal or a step frequency coded signal and Taylor weighting on reception, the available bandwidth was found to be 180 mc at the 2 kmc carrier frequency. A more complete summary of the results obtained on the available bandwidth for the various signals considered is given in Table I. The results given in the table apply for an uncompensated receiver. The resolution achievable by the radar will most of the time be one over the maximum available bandwidth indicated in the table.

On non negative difference analogues of elliptic differential equations

Abstract: Let L be the elliptic differential operator defined on region R by l[u(x,y)]≡ u xx +2b(x,y)u xy +c(x,y)u yy , b≢0 , where b(x, y) and c(x, y) are continuous and bounded on R. At any point (x, y) ϵ R, the ellipticity condition alone implies the existence of a nine point difference operator analogue L h of L which is of non-negative type. Moreover, L h is given constructively. For numerical considerations in the large, it is shown to be sufficient that the following inequality be valid: .

Minimum volumes to ensure homogeneity in certain conglomerates

Abstract: Conglomerates made up of aggregates enclosed by a binder are here idealized by a uniform matrix with spherical inclusions of different diameters and material properties. The usual notion of homogeneity involves common properties in all samples from the parent body. In fact this regularity does not occur and some statistical measure of homogeneity is necessary. Such a measure is here determined and methods of specifying sample dimensions for an acceptable standard of configurational homogeneity are indicated. For an equivalent standard of stiffness homogeneity a different sample volume is shown to be required and a measure of this property, associated with a sample volume, is developed. The inclusions of each sphere type are required to be distributed in the manner of independent Poisson series.

The effect of plasticity on the stress-distribution of thin notched plates in tension

Abstract: An explicit solution of the stress-distribution in notched plates under conditions of plane-stress is given in the elasto-plastic domain of straining. The material is considered as elastic-perfectly plastic and isotropic obeying the Mises yield condition and the Prandtl-Reuss incremental stress-strain relations. Elastic and plastic components of strains are determined by using information from birefringent coatings cemented on the surface of the specimens and observed in normal circularly or linearly polarized light. These measurements are complemented with data from an electrical analogy which yield the ϵx-strain distribution. Thin fat plates containing either symmetrical semi-circular grooves or deep notches of various flank angles are subjected to a uniform tension. Four cases of flank angles are considered lying between 0 and 120 deg. The minimum width of all plates is taken constant and equal to the half of the width of the plates. The analysis of the state of stress is restricted to the initial part of the loading path from an incipient plastic deformation up to an impending plastic flow. The variation of stress distribution with the evolution of plastic enclaves is studied and the influence of the severity of notches on the mode of spreading of plastic zones is determined.

On the stability of nonlinear sampled data feedback systems

Abstract: This work is a generalization of Tsypkin's stability criterion for a class of time-varying nonlinear sampled-data feedback systems. Some sufficient conditions for the response to any bounded input sequence to be bounded are presented. No assumptions are made concerning the internal dynamics of the linear subsystem, except that its input-output relation is of the form of convolution. The essence of the proof is to consider the nonlinear system as a perturbation of a stable linear system.

Limit analysis of simply supported conical shells subjected to uniform internal pressure

Abstract: A recapitulation of the basic concepts of the theory of limit analysis is followed by an upper and lower bound determination of the collapse load of a simply supported conical shell subjected to a uniform internal pressure. The bounds established coincide, thus representing an exact solution according to the assumed two-moment limited interaction yield criteria (1). The solution is then used to determine bounds on a shell following the Tresca yield criteria.

The dynamic stability of beam-columns with a fixed distance between supports

Abstract: This paper deals with the nonlinear vibrations of a two hinged beam-column subjected to a harmonic forcing function of any space distribution. Vibration in the fundamental mode is investigated and regions of dynamic instability are found. The perturbation solution used is suitable for both strong and weak nonlinear systems.

Analysis of high-voltage waveforms in X-ray units: I single-phase

Abstract: The equivalent circuits of an X-ray unit are first given utilizing linear lumped circuit elements, and then Hayashi's analysis of periodically interrupted electric circuits is applied. The analysis includes modified Heaviside operational calculus. Theoretical waveforms of the following cases coincide with experimental results: (1) normal case under resistive load, (2) tube-voltage waveforms are smoothed by the condenser-effect of high-voltage cables under resistive load, and (3) voltage-current curves of an X-ray tube are represented by two segments. In addition, two problems of applied voltage peaks are solved.

Mathematical formalism for probabilistic dynamical theories

Abstract: Classical statistical mechanics is formulated in terms of the characteristic function associated with a probability distribution in phase-space. Quantum statistical mechanics is formulated in terms of the hypercharacteristic function associated with an ensemble of Schrodinger wave functions. It is shown here that both probabilistic dynamical theories appear as special cases in the broader framework of a generalized statistical mechanics.