Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1963"
248 citations
TL;DR: In this article, a mathematical model for the oscillations of an infinite and finite threadline between a fixed eyelet and the traverse device is presented. But the analysis of the oscillation of a string as it is traversed and wound on a bobbin is not considered.
Abstract: The characterization and analysis of the oscillation of a string as it is traversed and wound on a bobbin are considered. A mathematical model is formulated and solved for the oscillations of an infinite and finite threadline between a fixed eyelet and the traverse device. The methods of D'Alembert and of characteristics are applied and the importance of the ratio of the winding speed to the string's characteristic wave velocity is demonstrated.
136 citations
TL;DR: In this article, a unified theory for the time-optimal control of general linear plants with generalized constraints on the control variable is developed, and the geometrical approach taken provides new insight and interpretation for some results obtained by methods of functional analysis.
Abstract: A unified theory for the time-optimal control of general linear plants with generalized constraints on the control variable is developed. The geometrical approach taken provides new insight and interpretation for some results obtained by methods of functional analysis. Theorems stating necessary and sufficient conditions for the existence and uniqueness of the optimal control function, u 0 , and explicit formulae for u 0 , are derived by exploiting the topological properties of the reachable region—the set of the output points ( n -vectors) that can be reached using constrained controls. The detailed study of the conditions for the occurrence of corners and flat portions in the reachable region clarifies some “degenerate” cases for amplitude and area constraints, which have hitherto remained obscure. The synthesis problem is briefly discussed.
43 citations
TL;DR: In this paper, a yield condition which takes into account the influence of hydrostatic stresses is discussed, and an application to the determination of the burst pressure of a perfectly plastic spherical shell is given.
Abstract: The paper discusses a yield condition which takes into consideration the influence of hydrostatic stresses. It is shown that many experimental observations made in plasticity can be described satisfactorily by such a yield condition. An application to the determination of the burst pressure of a perfectly plastic spherical shell is given.
40 citations
TL;DR: In this paper, a method of formulating a system of first derivative-explicit differential equations for RLC graphs containing voltage and current sources is given, based on properties of the f-circuit, f-seg, and element equations of the graph G and tree T.
Abstract: A method of formulating a system of first derivative-explicit differential equations for RLC graphs containing voltage and current sources is given. It is shown that for any connected RLC graph G containing no circuits of voltage sources and no segs (or cut sets) of current sources, there exists a tree T such that all voltage sources and as many C-elements as possible are branches of T and all current sources and as many L-elements as possible are chords of T. The method of formulation is based on properties of the f-circuit, f-seg (or cut set) and element equations of the graph G and tree T. By combining the algebraic-differential system of equations of G (f-circuit, f-seg and element equations), a detailed expression of the first derivative-explicit system of differential equations is obtained. The number of equations is equal to the number of C-elements in the tree T plus the number of L-elements in the complement of T. Detailed expressions for the initial conditions (t = 0 +) for the first derivative explicit system of differential equations for G in terms of the arbitrarily specified t = 0 − values of the variables are also given.
27 citations
TL;DR: In this article, the author's general equations for the mechanics of continua under initial stress are applied to the formulation of a rigorous theory of stability of multilayered elastic media in a state of finite initial strain.
Abstract: The writer's general equations for the mechanics of continua under initial stress are applied to the formulation of a rigorous theory of stability of multilayered elastic media in a state of finite initial strain. The medium is assumed incompressible. It is either isotropic or anisotropic. The problem is analyzed in the context of the writer's earlier discussions showing the existence of internal and interfacial instability. The results provide a rigorous solution of the problem of buckling of sandwich plates. Recurrence equations are derived for the interfacial displacements. It is shown that they are equivalent to a variational principle expressed in terms of these displacements. A matrix multiplication procedure is also developed for automatic computing of critical values when a large number of layers is involved.
23 citations
TL;DR: In this paper, the decay of a heterogeneous population of particles is used to estimate the distribution function of the population, which is the Laplace transform of that distribution function which relates the number of particles to the diffusion coefficients of the particles.
Abstract: The decay of a heterogeneous population of particles such as occurs when a heterogeneous aerosol is decaying by diffusion of particles to the containing walls may be used to estimate the distribution function of the heterogeneous population. By making observations as the decay proceeds, it is possible to obtain a “decay curve” which is the Laplace transform of that distribution function which relates the number of particles to the diffusion coefficients of the particles. The decay curve has been evaluated for a number of hypothetical distributions; it is shown that with the decay curve as a starting point, it is possible to obtain a numerical inversion of the transform which yields the distribution function with fair accuracy.
22 citations
TL;DR: In this article, a model is proposed for time-varying sequential machines, which is essentially a generalized version of the conventional fixed (i.e., time-invariant) sequential machine.
Abstract: In this paper a model is proposed for time-varying sequential machines, which is essentially a generalized version of the conventional fixed (i.e., time-invariant) sequential machine. It is shown how a time-varying machine can be characterized by means of matrices, and how simple matrix operations can be employed to compute the behavior of the machine at arbitrary times. Equivalence notions are developed which facilitate the comparison of states and machines in the time-varying case. The special class of periodic machines is investigated in detail. A procedure is proposed for establishing the equivalence or nonequivalence of two given periodic machines. It is shown that every periodic machine has a fixed representation whose minimal form is unique up to isomorphism. Algorithms are formulated for constructing fixed representations for periodic machines and periodic representations for fixed machines. Switching from one representation to another offers greater design flexibility, with the possibility of trading off memory capacity for logical complexity.
21 citations
TL;DR: In this paper, a systematic procedure is developed to generate Liapunov functions of third and higher-order nonlinear systems, which are represented in quadratic form, which involves a symmetric matrix S whose elements contains functions of the state variables.
Abstract: A systematic procedure is developed to generate Liapunov functions of third and higher order nonlinear systems. The Liapunov functions V are represented in quadratic form which involves a symmetric matrix S whose elements contains functions of the state variables. The given nonlinear system is characterized by a matrix A whose elements are specified by the nonlinearities. The problem is to get a Liapunov stability matrix T which makes V negative semidefinite. It is shown that T = B′A, where B′ is the transpose of matrix B, which defines ▿V and is obtainable from S. The method is illustrated by a fourth order example and two-third order examples. Conditions of stability are derived from the T and S matrices.
21 citations
TL;DR: In this article, the stability of a non-reciprocal n-port is determined by finding a reciprocal n -port having identical stability characterizations, and the necessary and sufficient condition for the stability is formulated in terms of its driving point impedance at each of its ports when all the other (n − 1) ports are terminated in arbitrary passive impedances.
Abstract: A linear active nonreciprocal n -port is defined to be stable if the port currents are zero under all passive terminations. In this paper the stability of the prescribed nonreciprocal n -port is ascertained by finding a reciprocal n -port having identical stability characterizations. Compact stability criteria have been obtained for a general class of nonreciprocal n -port with its prescribed n × n impedance matrix in the tridiagonal form of a matrix of Jacobi. The necessary and sufficient condition for the stability of an n -port can also be formulated in terms of its driving-point impedance at each of its ports when all the other ( n − 1) ports are terminated in arbitrary passive impedances. By using the impedance transformation property of the linear n -port and an iterative procedure of the bilinear transformation, stability criterion for a general nonreciprocal n -port can be obtained.
20 citations
TL;DR: The equivalence between Liapunov's Second Method and the Routh-Hurwitz Criterion for linear systems is established in this article, where a transformation matrix Q is developed which transforms the system matrix A into a special matrix R called a Routh Canonical matrix.
Abstract: In this paper the equivalence between Liapunov's Second Method and the Routh-Hurwitz Criterion for Linear Systems is established. A transformation matrix Q is developed which transforms the system matrix A into a special matrix R called a Routh Canonical matrix. The elements of R matrix are closely related to the elements in the first column of Routh's arrays. The conditions of stability on R matrix from Liapunov's Second Method are the same as the Routh-Hurwitz Conditions. The treatment is extended to nonlinear systems. As a result of this transformation, the linear terms are automatically removed from further consideration and only the nonlinear terms remain. Method of analysis is explained with the help of examples.
TL;DR: In this article, the theory of stability of multilayered continua is extended to include the effect of gravity and the case of viscoelastic materials, and a general theorem is derived for the conditions under which only real values are possible for the characteristic exponents of the stability problem.
Abstract: The theory of stability of multilayered continua is extended to include the effect of gravity and the case of viscoelastic materials. It is also applied to obtain numerical solutions for the buckling of the anisotropic plate in finite elasticity with free boundaries or embedded in an infinite medium. In the multilayered system it is shown that the effect of gravity forces may be included by a very simple process leading to a matrix multiplication scheme for the solution of the characteristic stability problem and to a new variational principle. By the correspondence principle the theory is immediately extended to the stability problem of multilayered viscoelastic media, and a general theorem is derived for the conditions under which only real values are possible for the characteristic exponents of the stability problem. As an example of gravity instability, two cases are solved numerically for purely viscous or elastic layers. The theory yields the solution for a large class of problems of technological and geophysical interest.
TL;DR: In this paper, the equations of linear and angular momentum are derived for any system whose mass varies with time, and the derivations are somewhat unique but the angular momentum equation is generalized so that an arbitrary reference origin can be used.
Abstract: The equations of linear and angular momentum are derived for any system whose mass varies with time. The equations themselves are rather well known but the derivations are somewhat unique, and the angular momentum equation is generalized so that an arbitrary reference origin can be used. Throughout the derivations, a deliberate effort has been made to emphasize the physics of the problem.
TL;DR: In this paper, small perturbations along the optimal state and control action trajectories are analyzed and their effects on the deviations from the optimal trajectory or decrease of the performance index in optimal processes are discussed.
Abstract: This work analyzes small perturbations along the optimal state and control action trajectories, and discusses their effects on the deviations from the optimal state trajectory or decrease of the performance index in optimal processes. This is done by developing the linearized adjoint model to the optimal dynamics of the process and applying the adjoint computational technique. It is shown that in this way the computational procedure is greatly reduced, indicating the effectiveness of adjoint analysis.
TL;DR: In this paper, a comparison of the analytic signal defined by Gabor with the more conventional "exponential" representation for wide-band radio frequency signals of finite duration is made.
Abstract: A comparison is made of the “analytic signal” defined by Gabor with the more conventional “exponential” representation for wide-band radio frequency signals of finite duration. It is shown that the rms difference between them, with respect to both modulus and phase, is bounded by the fractional spectral energy of the “exponential” representation in the left half frequency plane. The difference in each case is shown to be small in most practical applications for fractional bandwidths up to one half.
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TL;DR: The Matric Computor is an analytic machine for the solution of matric problems, including differential and algebraic systems, through the synthesis of an electronic admittance network whose adjustable parameters are in 1-to-1 reciprocal correspondence with matric mathematical equations.
Abstract: The Matric Computor is an analytic machine for the solution of matric problems, including differential and algebraic systems. The theoretical basis of the machine is the synthesis of an electronic admittance network whose adjustable parameters are in 1-to-1 reciprocal correspondence with matric mathematical equations. Matric entries are represented through digitally-adjusted admittances permanently interconnected through a system of entry and trace amplifiers. The prescribed mathematical functions are represented by current sources, and the dynamical voltage responses automatically generated by the network yields the solution-vector of the mathematical matric problem.
TL;DR: The free and forced vibration of a string with a concentrated mass attached at the middle is formulated and solved with the aid of the Dirac δ-function.
Abstract: The free and forced vibration of a string with a concentrated mass attached at the middle is formulated and solved with the aid of the Dirac δ-function. The method developed is extended to the case where many masses are attached to the string. Compactness in presentation and manipulation is achieved by this new method.
TL;DR: Applications of symbolic functions to represent concentrated forces and moments as distributed loads are illustrated by two examples, a statically indeterminate problem and a vibration problem.
Abstract: This paper shows how symbolic functions are used to represent concentrated forces and moments as distributed loads. Applications of such representations are illustrated by two examples, a statically indeterminate problem and a vibration problem.
TL;DR: In this paper, a procedure for determining the excitation parameters of a linear array of isotropic sources such that the signal-to-noise ratio is maximized is presented.
Abstract: A procedure has been developed which determines the excitation parameters of a linear array of isotropic sources such that the signal-to-noise ratio is maximized. The signal is considered to be a point source and the noise has a known distribution in space.
TL;DR: In this paper, a well-ordered branch of the Lorentz transformation for velocities in excess of the velocity of light is introduced, and the branching of the transformation may be determined so that the elapsed time along the path of such a motion remains positive.
Abstract: Conformal transformations in two dimensions provide a simple extension of the Lorentz transformation. The velocity of light appears in such transformations as a singular velocity rather than as an upper limit for the velocity. A well-ordered branch of the theory exists for velocities in excess of the velocity of light. If the velocity of a point exceeds the singular velocity in an inertial system, then the conformal representation of the motion is no longer uniform, but contains a folded region. However, the branching of the transformation may be determined so that the elapsed time along the path of such a motion remains positive. Kinematic relations on the other side of the singular velocity seem to complement the usual results of relativity theory in an interesting way. Thus it is known that motion at the speed of light occurs along a null geodesic, and hence corresponds in a certain sense to motion at infinite velocity (that is, in the sense of proper time elapsed). The complementary relation is that a motion of infinite velocity corresponds in the same sense to motion at the speed of light.
TL;DR: In this article, the authors present the analysis of a viscous model for the study of the impact of plates by projectiles under conditions which would lead to failure of the plate by the formation of a plug.
Abstract: This paper presents the analysis of a viscous model for the study of the impact of plates by projectiles under conditions which would lead to failure of the plate by the formation of a plug. The impact is represented by a velocity uniformly distributed over a circular area on the plate surface. Only the vertical shearing stress is considered and it is assumed to depend only on the radial coordinate. The stress, velocity and displacement profiles are calculated for the viscous model. The calculated displacement profiles are compared with experimental profiles determined from photographs of an actual plugging experiment.
TL;DR: Analytic methods based on graph theory are developed to deal with communication nets with switching, given a communication system of stations, fixed finite capacity channels, and specified types of message and channel switching.
Abstract: Analytic methods based on graph theory are developed to deal with communication nets with switching. In particular, given a communication system of stations, fixed finite capacity channels, and specified types of message and channel switching, physically significant criteria of optimality for switching patterns are first defined. Next, algorithms are given for optimal channel switching patterns for several special cases—the most important being when all channel capacities are equal. Based on the results for these special cases, two rapid methods are given for achieving approximations of optimal channel switching patterns. Finally, with these approximations as a first step, a general algorithm is developed.
TL;DR: In this article, a new method of determining the deflection and the natural frequencies of vibration for beams with many elastic supports is presented, which avoids the necessity of having to analyze the beam in many spans, consequently achieving economy of thought.
Abstract: A new method of determining the deflection and the natural frequencies of vibration for beams with many elastic supports is presented. The basic idea of the method is the use of the Dirac delta-function to introduce the concentrated loads applied to the beam by the elastic supports in the governing differential equation. This approach avoids the necessity of having to analyze the beam in many spans, consequently achieving economy of thought.
TL;DR: In this article, the equivalence classes of (n, k) switching networks are derived for the important groups encountered in switching theory, and the number of equivalence classifiers is derived for each of these groups.
Abstract: An ( n , k ) switching network is defined as an n -input, k -output network such that associated with each output is a Boolean transmission function of the n -inputs. If we allow a group U on the inputs and a group b on the outputs, then the family of networks is decomposed into equivalence classes. In this paper the number of equivalence classes is derived for the important groups encountered in switching theory.
TL;DR: In this paper, a general numerical integration procedure for the calculation of dynamic response to impulsive loads is presented, which is completely developed for a single-degree-of-freedom system and extended to systems with multiple degrees of freedom.
Abstract: A general numerical integration procedure for the calculation of dynamic response to impulsive loads is presented. The procedure is completely developed for a single-degree-of-freedom system and extended to systems with multiple degrees of freedom. Numerical examples are given in which a comparison is made between approximate and exact solutions. The accuracy obtained is well within acceptable limits for most applications.
TL;DR: Differential equations for Hertz potentials in inhomogeneous media in three coordinate systems (namely, rectangular, circular cylindrical, and spherical) have been derived in this article.
Abstract: Differential equations for Hertz potentials in inhomogeneous media in three coordinate systems (namely, rectangular, circular cylindrical, and spherical) have been derived Suitable assumptions have been made so that a typical differential equation for a Hertz potential, φ, reduces to a simple form, (▿ 2 + K 2 ϵ + f ) φ = 0, where K 2 = ω 2 μ 0 ϵ 0 , and f is a known function of a single coordinate with respect to which the relative dielectric constant of the medium, ϵ, varies The results are summarized in a table which also enables one to formulate various problems of wave propagation in inhomogeneous media, including electromagnetic sources
TL;DR: In this paper, the authors present analyses of creep in annular plates subjected to uniform lateral pressures, based upon creep-flow laws associated with the maximum shearing stress criterion, assuming that the creep rate is a power function of moment multiplied by a function of time.
Abstract: This paper presents analyses of creep in annular plates subjected to uniform lateral pressures. These analyses are based upon creep-flow laws associated with the maximum shearing stress criterion. It is assumed that the creep rate is a power function of moment multiplied by a function of time. An annular plate with free inner edge and simply supported outer edge is analyzed in detail for moments and deformations. The results for plates with three other specific edge conditions are also given.
TL;DR: In this article, the current flux function defined by Stokes for incompressible (solenoidal) velocity vector fields is generalized to axi-symmetric field problems and a table useful for many physical problems is presented along with some examples of magnetic field-problems solved by use of the magnetic flux function.
Abstract: The current flux function defined by Stokes for incompressible (solenoidal) velocity vector fields is generalized to othe axi-symmetric field problems. A table useful for many physical problems is presented along with some examples of magnetic field-problems solved by use of the magnetic flux function. This function gives the magnetic lines of force in the field directly.