Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1972"
TL;DR: Two small-gain theorems and a passivity theorem yielding stability results for general feedback systems were shown to be equivalent in this article, and two small gain theorem and a small gain theorem were also shown to yield stability.
Abstract: Two small-gain theorems and a passivity theorem yielding stability results for general feedback systems are shown to be equivalent.
109 citations
TL;DR: In this article, a linear first-order partial differential equation is derived for the age density k(x,t) of a biological population, which contains factors which are time-dependent functions defining attrition and immigration.
Abstract: A linear first-order partial differential equation is derived for the age density k(x,t) of a biological population. It contains factors which are time-dependent functions defining attrition and immigration. Emigration may be included in the attrition function. The general solution k(x,t) of the differential equation contains an arbitrary function that is identified as the birth rate B(t) of the community. A Volterra integral equation for B(t) arises if the total population N of the community is planned to be a prescribed function of time. Also, projection of the population on the basis of a given net maternity function and a given immigration function requires the solution B(t) of a linear integral equation. This equation can be transformed to a Volterra type, but another form in which the limits of integration are constant bounds of the fertility period has some advantages. Some properties of solutions of the integral equation are discussed, and effects of stepwise discontinuities in the kernel are considered.
55 citations
TL;DR: The long history of contamination flashover is reviewed in this article, followed by a discussion of some of the physical processes involved in flashover, and several flashover theories are critically examined, and recent experimental results are presented.
Abstract: The long, unpleasant history of contamination flashover is reviewed briefly, followed by a discussion of some of the physical processes involved. The numerous flashover theories are critically examined, and recent experimental results are presented. Although there is at present no comprehensive flashover theory, it has nevertheless been possible to improve insulator design over the years by a judicious combination of intuition, experiment and physical insight. Some existing designs as well as some proposed modifications are examined in the light of present theoretical understanding, and suggestions are made for future research in areas where more complete knowledge might lead to better insulation.
47 citations
TL;DR: In this article, the root clustering problem can be expressed in terms of positive definite (p.d.) or positive innerwise matrices, where the entries in these matrices are rational functions of the coefficients of the polynomial.
Abstract: The general problem of root-clustering and root-distribution of a polynomial in a certain region Γ in the complex plane has been investigated in this paper. The region Γ is general and includes all the previously investigated regions. For the root-clustering problem, it is shown that by using a certain transformation, the necessary and sufficient condition can be represented either in terms of positive definite (p.d.) or positive innerwise (p.i.) matrices. The entries in these matrices are rational functions of the coefficients of the polynomial. The connection between p.d. and p.i. matrices is established in terms of matrix multiplication. The investigations of this paper are quite general in that all the previously known criteria for root-clustering can be obtained from the general formulation. Furthermore, this study shows that the “Inner” matrix representation offers unification, simplification and a systematic way for studying the mentioned problems.
30 citations
27 citations
TL;DR: In this article, a new incremental theory for solving the problem of partially yielded cylinders was developed for predicting stress and strain distribution of a thick-walled cylinder subjected to nonproportionate loading.
Abstract: A new incremental theory has been developed for solving the problem of partially yielded thick-walled cylinders. Incremental stresses and strains are directly used as variables, hence numerical differentiation in the evaluation of stresses and strains is not required. The stresses and strains in all principal directions can be computed at the same time from governing equations for each increment of loading. Since the consideration of loading history is involved, the present theory is particularly suitable for predicting stress and strain distribution of a thick-walled cylinder subjected to nonproportionate loading.
22 citations
TL;DR: In this article, the authors developed a new approach to Newtonian dynamics in which the equations of motion are difference equations and the solutions of these equations are discrete functions, which is in complete harmony with modern digital computer capability and methodology.
Abstract: In this self-contained paper, we develop a new approach to Newtonian dynamics in which the equations of motion are difference equations and in which the solutions of these equations are discrete functions. Such an approach is in complete harmony with modern digital computer capability and methodology (3).
21 citations
TL;DR: In this paper, it was shown that the ideal low-pass filter which provides the fastest monotonic step-response for a prescribed bandwidth is the prolate filter, which is obtained from the autocorrelation of the zero-order, zero degree angular prolate spheroidal wave function.
Abstract: An ideal lowpass filter permits no signal transmission outside of a prescribed frequency band centered around the origin. In this paper, it is shown that the ideal lowpass filter which provides the fastest monotonic step-response for a prescribed bandwidth is the prolate filter. The system-function of this circuit is obtained from the autocorrelation of the zero-order, zero degree angular prolate spheroidal wave function. The bandwidth and risetimes of the filter are related through the largest eigenvalue of the same wave function. The computation and realization of the optimum frequency- and time-responses is discussed in detail and is illustrated by a numerical example. It is also shown how the asymptotic forms of the optimum system function degenerate, for very small or very large risetimes, into the well-known Hadamard and Gaussian filter functions, respectively.
18 citations
TL;DR: In this article, the equations of motion consisting of continuity equations, momentum equations and mass-tension relations are derived for a string which has prescribed axial motion at its two ends.
Abstract: The equations of motion consisting of continuity equations, momentum equations and mass-tension relations are derived for a string which has prescribed axial motion at its two ends. These equations are shown to be fully hyperbolic in nature and are then solved by the method of characteristics. The nature and physical interpretation of the characteristics lines are discussed in detail and the analysis is illustrated by an example.
18 citations
16 citations
TL;DR: In this article, a synthesis procedure is presented whereby the network function T (s) can be realized as an active RC multiport network with grounded ports, based on V(s) = T (S) U (s), where T is a q × p matrix of real rational functions of the complex variable s, the realization requires a minimum number of grounded capacitors and no more than 2 (p+n) inverting, grounded voltage amplifiers or p+n differential output.
Abstract: A synthesis procedure–easily implemented as a digital computer program– is presented whereby the network function T (s) can be realized as an active RC multiport network with grounded ports. Based on V (s) = T (s) U (s), where T (s) is a q × p matrix of real rational functions of the complex variable s, the realization requires a minimum number of grounded capacitors–n = degree { T (s)}–and no more than 2 (p+n) inverting, grounded voltage amplifiers or p+n differential output, grounded voltage amplifiers. Note: These properties of the realization are desirable if the network is to be fabricated as an integrated circuit.
TL;DR: In this article, two lower bound methods are applied to obtain bounds to frequencies of bending vibration of a rotating uniform beam simply supported at one end, free at the other, and numerical results are given.
Abstract: Two lower bound methods are applied to obtain bounds to frequencies of bending vibration of a rotating uniform beam simply supported at one end, free at the other. These methods, in conjunction with the well-known Rayleigh–Ritz procedure, can also be used to provide rigorous upper and lower bounds to a wide variety of vibration problems. The virtues of the methods are discussed and numerical results are given.
TL;DR: In this article, the explicit form of Bashkow's A-matrix for RLCM networks with ideal transformers is obtained by associating with each transformer a magnetic graph; the transformer constraint equations then become associated with cutsets and loops of this graph.
Abstract: The explicit form of Bashkow's A-matrix is obtained for RLCM networks containing ideal transformers The constraints introduced are handled by associating with each transformer a magnetic graph; the transformer constraint equations then become associated with cutsets and loops of this graph Necessary conditions on the topology of RLCT networks for the existence of a unique solution are derived, and the concept of a maximal proper tree is introduced Starting from a maximal proper tree, the explicit form of the A-matrix is obtained by showing that the constraint equations for an RLCMT network can be written in a similar form to those for an RLCM network This enables the elimination process, derived earlier for the RLCM case, to be extended to the RLCMT case
TL;DR: In this paper, the behavior of electromagnetic waves in irrotational moving media is studied based on Maxwell's equations and Minkowski's constitutive relations taken to the first order in the velocity.
Abstract: The behavior of electromagnetic waves in irrotational moving media is studied. The model is based on Maxwell's equations and Minkowski's constitutive relations taken to the first order in the velocity. Accordingly, the effect of acceleration on the electromagnetic fields is neglected. The new velocity effects are investigated by considering simple propagation and scattering problems: scattering in the presence of fluid sources and boundaries, rotating media and a uniform flow perturbed by the presence of a circular cylinder and a sphere.
TL;DR: In this article, structural properties of sensitivity systems, i.e., systems generating the state sensitivity coefficients of a linear time-invariant system (basic system), are presented.
Abstract: Structural properties of “sensitivity systems”, i.e. systems generating the state sensitivity coefficients of a linear time-invariant system (“basic system”) are presented. A formula for the eigenvalues sensitivity coefficients is first derived and the multiplicity of eigenvalues, in the minimal polynomial of the Sensitivity System is discussed. Then, a necessary and sufficient condition for the cyclicity of the Sensitivity System is given and uncontrollability of the Sensitivity System is proved whenever the number of parameters is sufficiently large. All the results enable conclusions about structural properties of the Sensitivity System in terms of the structure of the basic system.
TL;DR: In this paper, a statistical analysis is presented for steady creep bending in a beam, where the material parameters are random processes along the neutral axis arising from random fluctuations in temperature and imperfection density.
Abstract: A statistical analysis is presented for steady creep bending in a beam, where the material parameters are random processes along the neutral axis arising from random fluctuations in temperature and imperfection density. Analytical and numerical results are presented for the statistical properties of the normal stress and lateral velocity. It is found that whereas the normal stress shows very little random fluctuation the lateral velocity is highly random. The statistics of the lateral velocity generally show a significant dependence on the nominal value of the creep power and on the position along the neutral axis.
TL;DR: In this paper, the effect of imperfectly bonded laminar composite is examined in terms of the intensification of the torsional stresses operative near the imperfection which is assumed to be a circular shaped area.
Abstract: The effect of an imperfectly bonded laminar composite is examined in terms of the intensification of the torsional stresses operative near the imperfection which is assumed to be a circular shaped area. The laminar composite is modelled by four layers of different materials with the two outer layers being infinite in height and debonding occurs at the interface of the two inner layers. The analysis based on the application of Hankel transforms and the solution of a pair of dual integral equations can be easily extended to a multilayered system. Depending on the size of the layer thickness relative to the radius of the debonded area, delamination may take place either in a stable or unstable fashion. The analytical results also indicate that the influence of lamination tends to lower the stress intensity around an interface imperfection as compared to the stress state in a homogeneous solid containing the same imperfection. Numerical results are obtained for two special laminate geometries and discussed with reference to the pertinent parameters used in the current theory of fracture mechanics.
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TL;DR: In this article, the authors reviewed the scope and history of the problem of atmospheric and industrial pollution in open air transmission at the ultra high voltage (UHV) level and reviewed the present knowledge of the physical nature of the phenomenon involved.
Abstract: The phenomena of atmospheric and industrial pollution stands as the major problem for, and perhaps deterent to, the steady advancement of open air transmission at the ultra high voltage (UHV) level. This paper discusses this problem in three parts. In Part I, the scope and history of the problem is reviewed, and its importance is stressed. Part II reviews the present knowledge of the physical nature of the phenomenon involved. Part III covers the testing of polluted insulator strings, and discusses methods in use today. The importance of the parameters of the test voltage source is emphasized. The influence of switching impulse tests is reviewed. The paper closes with some of the remedies currently in practice.
TL;DR: In this paper, the sequential gradient-restoration algorithm for minimizing a functional subject to certain differential equations and boundary conditions was developed, which includes the alternate succession of gradient phases and restoration phases.
Abstract: In Ref. 1, the sequential gradient-restoration algorithm for minimizing a functional subject to certain differential equations and boundary conditions was developed. This algorithm, here called Algorithm (α), includes the alternate succession of gradient phases and restoration phases. In the gradient phase, the first-order change of the functional is minimized subject to the linearized differential equations, the linearized boundary conditions and a quadratic constraint on the variations of the control and the parameter. In the restoration phase, the differential equations and the boundary conditions are restored to a predetermined degree of accuracy, subject to the least-square change of the control and the parameter. In this paper, several modifications and extensions of Algorithm (α) are studied. In parameter is modified by the inclusion of two positive-definite weighting matrices W1(t) and W2. In Algorithm (γ), an intermediate phase is interposed between the gradient phase and the restoration phase of Algorithm (α); in this intermediate phase, the control is reset at that value which minimizes the Hamiltonian. In Algorithm (δ), an intermediate phase is interposed between the gradient phase and the restoration phase of Algorithm (β); in this intermediate phase, the control is reset at that value which minimizes the Hamiltonian. Several numerical examples are developed, and it is shown that, if the matrix W1(t) is the matrix of the second derivatives of the Hamiltonian with respect to the control, Algorithm (β) exhibits better convergence characteristics than Algorithm (α). On the other hand, Algorithms (γ) and (δ) are erratic in their behaviour: their convergence characteristics are found to be better or worse than those of Algorithm (α), depending on the particular problem.
TL;DR: In this paper, an up-to-date summary of the contamination research at Project UHV is presented with results of flashover tests on various types of insulators with discussions of test methods.
Abstract: This paper describes an up-to-date summary of the contamination research at Project UHV. Results of flashover tests on various types of insulators are presented with discussions of test methods. Two important factors—flashover probability on multiple strings, and nonlinearity of flashover strength with insulator length—areconcentratively discussed. The latter is a particularly important problem for UHV line insulation. A practical design method for line insulation is developed on a statistical basis.
TL;DR: In this article, a two-dimensional reinforced quartz-phenolic composite material is subjected to a high-amplitude stress-wave loading condition in uniaxial strain to determine its dynamic fracture behavior.
Abstract: A two-dimensionally reinforced quartz-phenolic composite material is subjected to a high-amplitude stress-wave loading condition in uniaxial strain to determine its dynamic fracture behavior. This loading condition is achieved by using exploding-foil plate-impact techniques. The direction of wave propagation through the thickness of the material is normal to the principal direction of reinforcement. Two failure levels are defined: the visual threshold, and a microscopic threshold represented as the lower limit of connected delamination. These data indicate that the macroscopic and microscopic spallation thresholds of the laminated, 0-degree shingle-angle, quartz-phenolic composite are sensitive to the tensile-pulse duration, amplitude, impulse and shape at the spall-plane location. The time-dependent aspects of the spallation threshold of two-dimensionally reinforced quartz phenolic are found to obey failure theories which are rate-process oriented, and which reflect the effects of tensile-pulse duration, peak tensile stress, tensile impulse and tensile pulse shape.
TL;DR: Every linear sequential circuit defined over a residue class ring Zm, where m = pr11 pr22...prkk (the pi being distinct primes), is isomorphic to a parallel connection of k LSC's over Zpiri.
Abstract: In this paper we show that every linear sequential circuit (LSC) defined over a residue class ring Zm, where m = pr11 pr22...prkk (the pi being distinct primes), is isomorphic to a parallel connection of k LSC's over Zpiri (i = 1,2,...,k); each LSC over Zpiri is, in turn, isomorphic to a cascade connection of ri LSC's over Zpi (that is, over the Galois field GF(pi)), where the cascaded stages are separated by a (generally nonlinear) delay-free logic. Thus, every LSC over a residue class ring is isomorphic to an interconnection of LSC's over Galois fields.
TL;DR: In this paper, a max-min problem in the realm of optimum beam design is formulated and thoroughly investigated from a dynamic programming point of view, and the conditions of optimality can be directly derived from the Hamilton-Jacobi-Bellman equation of the process.
Abstract: A max—min problem in the realm of optimum beam design is formulated and thoroughly investigated from a dynamic programming point of view. It is shown that the conditions of optimality can be directly derived from the Hamilton—Jacobi—Bellman equation of the process. The classical Euler—Lagrange equations for the beam are derived from the fundamental partial differential equation. It is shown that the conditions of optimality associated with the minimum operation are local expressions of the theorem of Castigliano. An analytical solution for the unconstrained optimum cantilever laying on elastic foundation is presented, and a method of successive approximations consisting in a stable, two-sweep iterative procedure, is developed. Numerical examples are given.
TL;DR: In this paper, the Taylor-Cauchy transform was used to solve the Boole and Riccati equations with the help of continued fractions, and the results were compared with the series solutions (which may be recognized in closed form for simple cases) obtained by a class of nonlinear varying-parameter systems.
Abstract: This note follows the work of Lagrange and Euler and gives the essential steps for reaching a solution of the Boole and the Riccati equations with the help of continued fractions. The results are compared with the series solutions (which may be recognized in closed form for simple cases) obtained by the Taylor-Cauchy transform method for a class of nonlinear varying-parameter systems.
TL;DR: In this paper, the procedures used by the Florida Power & Light Company which have resulted in 6 years of virtually "flashover proof" substations are discussed. And the successful prevention of flashovers due to contamination in its 69, 138 and 240 kV substations and switching stations is described.
Abstract: This paper discusses in detail the procedures used by the Florida Power & Light Company which have resulted in 6 years of virtually “flashover proof” substations. The successful prevention of flashovers due to contamination in its 69, 138 and 240 kV substations and switching stations is described.