Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1967"
349 citations
325 citations
TL;DR: The problem of interpolation with positive-real functions is solved within a network theoretic framework and applied to several situations of engineering interest and questions pertaining to realizations employing a minimum number of reactances are studied in great detail.
Abstract: The problem of interpolation with positive-real functions is solved within a network theoretic framework and applied to several situations of engineering interest. Questions pertaining to realizations employing a minimum number of reactances are studied in great detail.
159 citations
137 citations
66 citations
TL;DR: This paper proposes a efficient search strategy, suitable for programming on a digital computer, which will yield the solution in a reasonable time when locating the ground track of an aircraft, given the elevation of the terrain below the aircraft during flight.
Abstract: Ground track location on contour map for given terrain elevation profile, using digital computer
65 citations
TL;DR: The c.
Abstract: The c. 1959 mathematical model for human operator control dynamics has been validated and extended to produce a practically complete mathematical description of manual control dynamics for single-loop systems. This model is essential to the analytical design of closed-loop man-machine systems, and it facilitates understanding of the human as a control device. An extensive number of selected experiments using 9 subjects, 4 forms of plant dynamics of general applicability, and 3 principal forcing functions, yielded definitive describing function data over a frequency range of two decades including system crossover. Models were constructed at three levels of detail: 1) a crossover model which is easily and usefully applied; 2) an extended crossover model which accounts more adequately for low frequency lags and plant dynamics; and 3) a precision model which provides a description so detailed that inferences can be drawn about neuromuscular functions. The resulting adaptive, optimalizing c. 1965 human operator mathematical model is presented, with a detailed summary of its adjustments for proper application.
51 citations
TL;DR: A dynamic program for obtaining optimum flows is devised and demonstrated by an example, and the max-flow min-cut Theorem of Ford and Fulkerson is extended for general communication networks.
Abstract: The concept of optimum flows is introduced and its properties in a general communication network where flow suffers loss or gains during transmission through channels are investigated. A set of necessary and sufficient conditions for optimality is obtained, and based on this a dynamic program for obtaining optimum flows is devised and demonstrated by an example. Finally the max-flow min-cut Theorem of Ford and Fulkerson is extended for general communication networks.
41 citations
TL;DR: In this article, an analog computer program is developed which is capable of solving a wide range of one-dimensional melting problems with constant, or time-dependent boundary conditions, applied to a study of the melting of a semi-infinite solid subjected to a constant heat input.
Abstract: An analog computer program is developed which is capable of solving a wide range of one-dimensional melting problems with constant, or time-dependent boundary conditions. This program is applied to a study of the melting of a semi-infinite solid subjected to a constant heat input. Calculations describe the temperature distribution and depth of melting for metallic materials. It is shown that when temperature gradients are allowed to develop in the melt (i.e., when the molten material does not mix or flow) vaporization at the surface limits the melting process and the maximum depth of melting. In such cases, the ratio of the maximum depth of melting to the extent of the heat affected region of the parent material is considerably less than unity for most materials. This suggests that mixing in the molten material may be essential in applications requiring minimization of the heat affected zone relative to fusion penetration.
35 citations
TL;DR: In this paper, the problem of the torsional vibrations of a thin strip moving with constant speed in a longitudinal direction is formulated and solved, and the effect of a point load acting on one of the thin edges is included.
Abstract: The problem of the torsional vibrations of a thin strip moving with constant speed in a longitudinal direction is formulated and solved. The effect of a point load acting on one of the thin edges is included. Some results of a parametric study show the effects of various parameters on the torsional frequency, and the occurrence of torsional buckling is predicted. In addition, an interesting behavior is shown in the case where the torsional wave propagation speed in the region “up-stream” from the point load is greater than in the “downstream” region.
23 citations
TL;DR: In this article, an algorithm is developed to generate a class of matrices N which, when non-singular, reduce a constant square matrix A by a similarity transformation C = NAN−1 to the companion form.
Abstract: An algorithm is developed to generate a class of matrices N which, when non-singular, reduce a constant square matrix A by a similarity transformation C = NAN−1 to the Companion form. The coefficient of the characteristic equation |sI − A| = 0 are displayed in the last row of C: stability of the multivariable system x = Ax may thus be determined by the Routh-Hurwitz procedure. If the system is stable, Liapunov functions may be generated by further transformation of C to Routh or Schwarz canonical forms. The former is particularly useful for calculation of quadratic functionals of the response stemming from non-zero initial conditions x(0). The algorithm is used to produce a generalized Vandermonde matrix which relates a companion matrix with repeated eigenvalues to the Jordan canonical form.
TL;DR: In this article, an analysis based on elasticity principles is presented for determining the stresses, strains and displacements in the vicinity of a fiber embedded in an elastic medium, where Love's solution for a single force in an infinite elastic domain is used to calculate the interaction forces between fiber and matrix at discrete points.
Abstract: An analysis based on elasticity principles is presented for determining the stresses, strains and displacements in the vicinity of a fiber embedded in an elastic medium. Use of Love's solution for a single force in an infinite elastic domain is used to calculate the interaction forces between fiber and matrix at discrete points. The investigation deals with the overall problem of multi-fiber reinforcement and interaction, and may be used for any given dimension of fiber reinforcement.
TL;DR: In this paper, the Volterra functional analysis of nonlinear time-varying systems with deterministic and stochastic inputs is presented, and illustrative examples are shown.
Abstract: This paper gives the Volterra functional analysis of nonlinear time-varying systems with deterministic and stochastic inputs. Representative classes of systems are studied, and illustrative examples are shown. A comparison of the Volterra Functional Method with the Transform Ensemble Method is given in the Appendix.
TL;DR: Dynamic responses of cylindrical shells under axisymmetric surface tractions, solving free vibration problem as discussed by the authors, have been shown to solve the problem of free vibration in a 3D model.
Abstract: Dynamic responses of cylindrical shells under axisymmetric surface tractions, solving free vibration problem
TL;DR: In this article, complementary variational principles can be applied in the analysis of dynamic systems, and it is shown how to obtain a complementary formulation of the problem by means of a repeated application of the Legendre transformation either to the system Lagrangian or, in the case of a system with ignorable coordinates, to the Routhian.
Abstract: Two, complementary, variational principles can be applied in the analysis of dynamic systems. Often it appears to be difficult to formulate a. given system in terms of the so-called “complementary variational principle.” In the case of a linear, finite dimensional system, however, we can always obtain a complementary formulation of the problem by means of a repeated application of the Legendre transformation either to the system Lagrangian or, in the case of a system with ignorable coordinates, to the system Routhian.
TL;DR: In this article, the antenna pattern synthesis problem for a finite line source is formulated and treated in the space of absolutely square-integrable functions, and an examination is made of the condition that must be imposed on the desired radiation pattern so that the problem is solvable.
Abstract: The antenna pattern synthesis problem for a finite line source is formulated and treated in the space of absolutely square-integrable functions. An examination is made of the condition that must be imposed on the desired radiation pattern so that the problem is solvable. The optimum aperture distribution is determined by the variational technique, subject to the requirement that the approximating radiation patterns should have a superdirective ratio less than or equal to some fixed positive value. The optimality of the solution is demonstrated by considering the sufficient condition. The error criterion is measured in the mean-square sense.
TL;DR: In this paper, a variational formulation is presented which has as its Euler-Lagrange equation the fundamental partial differential equation of diffusion, which is applicable for both time-dependent and time-independent boundary conditions and for a diffusion coefficient which is a function of the concentration.
Abstract: A variational formulation is presented which has as its Euler-Lagrange equation the fundamental partial differential equation of diffusion. This variational expression is applicable for both time-dependent and time-independent boundary conditions and for a diffusion coefficient which is a function of the concentration. This variational technique is applied to the problem of unidimensional diffusion in a semi-infinite isotropic medium where the diffusion coefficient is strongly dependent upon the concentration of the diffuser. To solve the extremum problem, a system of “natural” finite difference equations is established as a consequence of the variational expression. Curves of concentration ratio vs distance are calculated and compared with a formal exact solution for a specific diffusion coefficient.
TL;DR: In this paper, the analysis and solution of variationally-derived finite difference equations in mixed boundary value problems of plane thermoelastic stress and thermo-elastic strain are discussed.
Abstract: This paper is concerned with the analysis and solution of variationally-derived finite difference equations in mixed boundary value problems of plane thermoelastic stress and thermoelastic strain. The relationship between these equations at the boundary and the natural boundary conditions is derived, and convergence to the explicit boundary equations with decreasing grid size is shown. Efficient decomposition and solution techniques for the positive definite, quasi-tridiagonal coefficients matrix are presented. Numerical studies are included for a finite, thin plate subjected to a parabolic temperature field.
TL;DR: In this article, the authors developed complementary forms of the variational principle for heat convection, which are applicable to non-homogeneous fluids with temperature dependent properties and include the case of turbulent flow.
Abstract: Complementary forms of the variational principle for heat convection are developed. They are applicable to non-homogeneous fluids with temperature dependent properties and include the case of turbulent flow. For linear problems following a general procedure introduced by the author for non-selfadjoint operators, the variational principle is expressed in operational symbolism which includes implicitly a convolution form.
TL;DR: In this paper, a method for transformation of a particular operator equation into a set of 2n linear differential equations is described, which is applicable to both continuous and discrete linear systems.
Abstract: Minimum energy problems in Hilbert function space are formulated. The methods presented are applicable to both continuous and discrete linear systems. Transformation of a particular operator equation into a set of 2n linear differential equations is described. Applications to physical systems are illustrated.
TL;DR: In this paper, a generalized ergodic theorem is proven for a class of stationary random processes, and four lemmas are proven for linear and non-linear nonlinear operations on these processes.
Abstract: A generalized ergodic theorem is proven for a class of stationary random processes. According to this theorem a strictly stationary random process with finite mean, mx and variance σx2 is strictly ergodic with probability-one if limτ→∞Rx(τ) = mx2 is satisfied where Rx(τ) is the probability correlation function of the process.
In addition to the theorem four lemmas are proven. According to these lemmas, linear operations on ergodic processes are themselves ergodic, zero-memory nonlinear operations on ergodic processes are ergodic, and linear combinations of ergodic processes are ergodic. Implied from these lemmas is the result that both separable and nonseparable nonlinear operations on ergodic processes are ergodic.
TL;DR: An approach based on the assumption that the branch flows are observable but their probability distributions are unknown is offered, which is optimum in the sense that the net is improved at minimum cost while the statistical errors in the procedure are controlled.
Abstract: In a communication network, the existing traffic within the branches is not deterministic. As part of the synthesis problem, it is necessary to determine the probability that a particular flow rate can be attained between a given pair of stations. If this probability is inadequate, we must improve the net with minimum cost. This paper offers an approach based on the assumption that the branch flows are observable but their probability distributions are unknown. An analysis technique that uses statistical hypothesis testing is developed. Following the statistical analysis procedure, a synthesis technique is presented which is optimum in the sense that the net is improved at minimum cost while the statistical errors in the procedure are controlled. The synthesis procedure utilizes linear integer programming and seems to be feasible for reasonably “large” communication networks.