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Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1977"


Journal ArticleDOI
TL;DR: In this paper, the Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems and a new set of orthogonal functions is derived from Walsh functions.
Abstract: The Walsh operational matrix for performing integration and solving state equations is generalized to fractional calculus for investigating distributed systems. A new set of orthogonal functions is derived from Walsh functions. By using the new functions, the generalized Walsh operational matrices corresponding to √s, √(s2 + 1), e-s and e-√s etc. are established. Several distributed parameter problems are solved by the new approach.

207 citations



Journal ArticleDOI
TL;DR: It is shown by a numerical example that the robust filter can be very useful in maintaining a reasonable error performance over the whole of the classes of PSD's.
Abstract: The performance of minimum mean-square-error estimation filters for signals in additive noise can deteriorate considerably for deviations of the actual signal and noise power spectral densities (PSD's) from assumed, nominal densities. We consider two classes of PSD's which are useful models for the signal and noise when their PSD's are not precisely known. For these classes, robust filters which are saddlepoints for mean-square- error performance are derived. The robust filters achieve their worst performance for pairs of least-favorable signal and noise PSD's for which they are the optimum filters. It is shown by a numerical example that the robust filter can be very useful in maintaining a reasonable error performance over the whole of the classes of PSD's.

104 citations



Journal ArticleDOI
TL;DR: In this paper, a vector space and its dual are associated with a nonoriented graph and the vectors of each subspace share the topological properties of one of the four entities singled out.
Abstract: Four topological entities are necessary for a complete description of a network graph, as required by orthogonal network theory: the seg, the circ and two others, introduced in this paper, as complementary to the seg and the circ respectively. A vector space and its dual are associated with a nonoriented graph. Both spaces are decomposed in two complementary subspaces. The vectors of each subspace share the topological properties of one of the four entities singled out. The results obtained are extended to a directed graph by associating two dual Z-modules with it. The given description of the network graph is perfectly functional to the orthogonal network theory.

35 citations


Journal ArticleDOI
TL;DR: In this article, a procedure is developed whereby the original equations are reduced to a form suitable for modal decomposition and the resulting modes are reinterpreted in bond graph form with the resulting model being an extremely accurate system representation while requiring only a fraction of the original number of equations.
Abstract: Bond graphs are an extremely useful modeling procedure for representing the actual energy exchange mechanisms of interacting dynamic systems. Governing state equations are straightforwardly obtained from the bond graph; however, for large structures, a restrictively large number of equations can result. A procedure is developed whereby the original equations are reduced to a form suitable for modal decomposition. The resulting modes are reinterpreted in bond graph form with the resulting model being an extremely accurate system representation while requiring only a fraction of the original number of equations. The procedure is demonstrated through example.

30 citations


Journal ArticleDOI
TL;DR: Applying the theory of local tests, general criteria are derived for the optimal selection of quantizer parameters for the large-sample-size case and are shown to lead to that quantizer-decision system which is most efficient asymptotically.
Abstract: The problem of designing quantizers for use in decision-making systems is considered. Applying the theory of local tests, general criteria are derived for the optimal selection of quantizer parameters for the large-sample-size case. These criteria agree with previously established results based on optimization in terms of distance measures and are shown also to lead to that quantizer-decision system which is most efficient asymptotically. To illustrate the design procedure, several applications to signal detection are discussed.

28 citations


Journal ArticleDOI
TL;DR: In this article, the relations between the power and energy in a non-linear physical system and the analogous quantities associated with variables representing small deviations from steady-state values are studied.
Abstract: Although no truly linear physical systems exist, linearized system models are very often used in practice to study the dynamics of real systems and components. Here, the relations between the power and energy in a non-linear physical system and the analogous quantities associated with variables representing small deviations from steady-state values are studied. Sometimes a linearized system is energetically similar to the non-linear system from which it was derived, and in other cases, new types of energy elements appear which were not present in the original system. In order to show the structure of the systems, the results are presented in both equation form and in the form of bond graphs which are unique in exhibiting the power and energy structure of system models.

26 citations


Journal ArticleDOI
TL;DR: In this paper, the mass bond has special properties which are explained by entropy and volume stripping and indicate the reason why the derivative of the free enthalpy, not the internal energy, is the driving effort of chemical reactions.
Abstract: Thermodynamic systems with variable mass, like liquid/vapor equilibria and chemical reactions, are represented as networks having discrete elements and connections using bond graph symbols. The mass bond has special properties which are explained by entropy and volume stripping and indicate the reason why the derivative of the free enthalpy, not the internal energy, is the driving effort of chemical reactions. Chemical friction is represented by RS-fields that dissipate power which is different from the observed heat rate of chemical reactions. Different reticulations apply to reactions near and far from chemical equilibrium; their relation to experimental reaction kinetics and order of reaction is discussed.

25 citations


Journal ArticleDOI
TL;DR: A new approach for time-domain analysis and design of lumped networks is considered, where the lumped elements are modeled by transmission-line sections or stubs and the modeled network is analysed by theTLM method, which provides an exact solution to the model.
Abstract: A new approach for time-domain analysis and design of lumped networks is considered. The lumped elements are modeled by transmission-line sections or stubs and the modeled network is analysed by the transmission-line matrix (TLM) method, which provides an exact solution to the model. Compensation of errors arising in modeling the network elements is discussed. Sensitivities w.r.t. design variables can easily be obtained and thus used in optimization. Sensitivities w.r.t. time and the time step are also obtained and used to improve the model's response.

25 citations


Journal ArticleDOI
TL;DR: The necessary and sufficient conditions for vibrational stabilizability of linear dynamic systems are found and basic relations of the vibrational control method and the optimal shape of vibrations are determined.
Abstract: The theory of vibrational control is developed. The necessary and sufficient conditions for vibrational stabilizability of linear dynamic systems are found. Basic relations of the vibrational control method and the optimal shape of vibrations are determined. Unlike conventional methods, based on feedback or feedforward principles, the method of the paper does not require measurement of the deviations or disturbances.

Journal ArticleDOI
TL;DR: In this paper, a two-dimensional wave digital filter of the recursive type was obtained from a doubly terminated LC-ladder network in two variables by replacing each series or shunt arm element of the ladder by its equivalent digital two-port.
Abstract: This paper proposes a method of obtaining a two-dimensional wave digital filter of the recursive type from a doubly terminated LC-ladder network in two variables by replacing each series or shunt arm element of the ladder by its equivalent digital two-port. A number of realizations of the wave digital two-ports, which are canonic in multipliers, have been obtained. An example of a circularly symmetric low-pass two-dimensional digital filter is considered using these realizations. The sensitivity of this filter with respect to the multiplier coefficient changes due to finite word length is compared with that of the direct realization. It is found that the wave digital filter appears to be a more desirable form of implementation than the conventional cascade form.

Journal ArticleDOI
TL;DR: In this paper, the problem of minimizing the maximum temperature of a purely conducting circular fin with heat generation via a variational technique was studied and an exact solution for an arbitrary heat source distribution and a general non-linear heat conductivity was obtained.
Abstract: The problem of minimizing the maximum temperature of a purely conducting circular fin with heat generation is studied via a variational technique. An exact solution is obtained for an arbitrary heat source distribution and a general non-linear heat conductivity. The problem of minimum weight design of convecting circular fin is then considered. An exact solution for temperature distribution and shape of the optimum fin with constant heat generation is obtained and discussed.

Journal ArticleDOI
TL;DR: In this paper, the Taylor series expansion of the Chebyshev polynomials was used for large amplitude motions of circular plates under transient loads, with and without damping.
Abstract: The Chebyshev polynomials have been applied to the large amplitude motions of circular plates under transient loads, with and without damping. The non- linear differential equations are linearized by using Taylor series expansions for one of the terms. It is shown that there is good agreement between the results obtained by the present technique and the available results. The advantage of this technique is essentially due to the fact that the Chebyshev polynomials are rapidly converging polynomials. It is shown that very accurate results can be obtained with only four terms of the Chebyshev series which may not be possible with conventional methods.

Journal ArticleDOI
TL;DR: In this article, a root exclusion criterion for root exclusion from a region composed as a union of elemental regions (discs and halfplanes) is established. But the root exclusion test is based on the equivalence of the strict Hurwitz property of a polynomial and the positivity of a set of inners associated with it.
Abstract: A criterion for root exclusion from a region composed as a union of elemental regions—discs and halfplanes—is established. Associated with each elemental region is a derived polynomial which must be by the criterion a strictly Hurwitz polynomial. The criterion is the basis for a root exclusion test, obtained by invoking the equivalence of the strict Hurwitz property of a polynomial and the positivity of a set of inners [equivalently, Hurwitz] determinants associated with the polynomial.

Journal ArticleDOI
TL;DR: In this paper, it is shown that a class of pursuit-evasion games can be treated more simply by the geometrical approach with the topological properties of the reachable region and by the method of functional analysis.
Abstract: In this paper it is shown that a class of pursuit-evasion games can be treated more simply by the geometrical approach with the topological properties of the reachable region and by the method of functional analysis. Conditions for capture and optimal strategies are derived. A numerical example is given to show both the optimal open-loop strategies of players and the optimal trajectories of the game.

Journal ArticleDOI
M. Surdin1
TL;DR: In this article, the effects of the magnetic field of a rotating body were detected and their properties investigated, based on a model proposed by the author, an interpretation of the experimental results is given.
Abstract: A laboratory experiment devoted to the measurement of the magnetic field created by a rotating body is described. Using an original method of detection, the effects of the magnetic field of a rotating body were detected and their properties investigated. Based on a model proposed by the author, an interpretation of the experimental results is given.

Journal ArticleDOI
TL;DR: In this paper, a set of constitutive equations for polycrystalline plasticity is derived using arguments based directly on the dislocation processes involved, which can model such phenomena as the development of anisotropy with plastic deformation, and material hardening or softening.
Abstract: A set of constitutive equations for polycrystalline plasticity is derived using arguments based directly on the dislocation processes involved. Distributed glide-plane orientations and Burgers-vector directions facilitate handling of the polycrystalline structure, and they yield equations involving probability distributions for variables that are directly related to measurable dislocation quantities. When the motion of the dislocations is isochoric, the tensorial character of the plastic strain rate is shown to be entirely determined by a second-rank symmetric tensor directly related to ordinary elements of crystallographic glide. This same tensor is also shown to determine the resolved shear stress acting on a dislocation in the direction of its Burgers vector, a quantity critical to the determination of the dislocation speed. Evolutionary equations for the dislocation density and the mobile fraction of dislocations are developed to complete the material description. The resulting theory, which allows for the production and interaction of non-uniform dislocation distributions, can model such phenomena as the development of anisotropy with plastic deformation, and material hardening or softening.

Journal ArticleDOI
TL;DR: In this paper, the time-history of the performance of a system is treated as a stochastic corrective process, in which deterioration due to aging is counteracted at brief maintenance checks.
Abstract: The time–history of the performance of a system is treated as a stochastic corrective process, in which deterioration due to aging is counteracted at brief maintenance checks. Using a diffusion approximation for the deterioration, simple models are proposed for describing maintenance either by component replacement or by performance restoration. Equilibrium solutions of the models show that the performance has a probability distribution with exponential tails: the uncritical use of Gaussians can grossly underestimate the probability of poor performance. The proposed models are supported by recent observational evidence on aircraft track-keeping errors, which are shown to follow the modified exponential distribution derived here. The analysis also brings out the relation between the deterioration characteristics of the system and the intensity of the maintenance effort required to achieve a given performance reliability.


Journal ArticleDOI
TL;DR: In this paper, the Carson-Cambi equation (1+e cos t)y + py = 0, referred to as the second-order differential equation with a periodic coefficient associated with the second derivative was examined.
Abstract: The periodic differential equation (1+e cos t)y + py = 0, hereby termed the Carson–Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided |e|<1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p–e plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to e3. Two-timing theory is employed to obtain solutions for values of k near the critical points k = ±12, ±32, ±52. Three-timing is employed to extend the solution near k = ±12. The solutions of the Carson–Cambi equation are compared with the solutions of the corresponding Mathieu equation.

Journal ArticleDOI
TL;DR: In this article, a reduction method was proposed for deriving an approximate formula for estimating the natural frequency of an orthotropic plate by using the results of an isotropic one.
Abstract: A reduction method is proposed for deriving an approximate formula for estimating the natural frequency of an orthotropic plate by using the results of an isotropic one. To justify the method, approximate formulae for estimating the natural frequency of clamped orthotropic rectangular and elliptical plates are derived from results previously reported for clamped isotropic rectangular and elliptical ones, respectively. The accuracy of the approximate formulae is discussed.

Journal ArticleDOI
Chih-Bing Ling1
TL;DR: In this article, the convergence of the successive approximations in Howland's solution of an infinite perforated strip containing a symmetrically-located circular hole under longitudinal tension can be improved considerably by using the well known solution of a finite-sized plate as the approximation of zero order.
Abstract: The convergence of the successive approximations in Howland's solution of an infinite perforated strip containing a symmetrically-located circular hole under longitudinal tension can be improved considerably by using the well known solution of an infinite perforated plate as the approximation of zero order. Further, by basing on a somewhat different physical concept, a corresponding direct solution of the problem can be formulated. Such a solution is presented in this paper. Numerical results are obtained from the solution over a wider range of values of the radius of hole, a feature hardly attainable in the previously known solutions of the problem. Further extension of the method of solution to the transverse bending problem as well as to the case of an unsymmetrically-located hole is also possible.


Journal ArticleDOI
TL;DR: An iterative technique for the derivation of a nearly optimal feedback control law for continuous dynamic systems with separable cost functions is developed based on the fundamental concepts of multilevel control.
Abstract: An iterative technique for the derivation of a nearly optimal feedback control law for continuous dynamic systems with separable cost functions is developed based on the fundamental concepts of multilevel control. The developed technique comprises two major and subsequent stages. In the first stage, a two-level optimization structure is devised using the coordination by control method. An identification processthen follows in order to determine the constant gains of the dynamic controller utilizing the discrete quasilinearization algorithm. A three-level control structure is thus provided for the overall technique which can adaptively compute the unknown gains of the dynamic controller.


Journal ArticleDOI
R.A. Rink1
TL;DR: Using the Krylov-Bogoliubov method for obtaining analytical solutions to systems with small nonlinearities, a procedure is employed to determine the initial amplitude and phase in terms of the initial displacement and velocity.
Abstract: Using the Krylov–Bogoliubov method for obtaining analytical solutions to systems with small non-linearities, a procedure is employed to determine the initial amplitude and phase in terms of the initial displacement and velocity. Equations representing the time rate of change of amplitude and phase are used directly. Whether the corresponding linear equations of the non-linear system has purely imaginary, complex conjugate or real roots, the same procedure can be applied. An example is given which demonstrates the initial amplitude and phase change for various higher order approximations.

Journal ArticleDOI
TL;DR: In this article, the propagation of elastic waves in a heterogeneous bar of variable cross-sectional area is investigated via use of the method of characteristics andthe Laplace transform technique, where the Young's modulus and density are assumed to be representable as either power law or exponential distributions in the axial coordinate.
Abstract: The propagation of elastic waves in a heterogeneous bar of variable cross-sectional area is investigated via use of the method of characteristics andthe Laplace transform technique. The Young's modulus and density are assumed to be representable as either power law or exponential distributions in the axial coordinate. The transform method is used to establish an infinite number of multi-parameter solutions in closed form for either a stress, velocity or displacement type boundary condition. The numerical characteristic computations show excellent agreement when compared to the transform solutions, and are then used to obtain additional solutions not attainable by the transform method. Detailed results and conclusions for a bar of ogival cross-section are given for a wide range of inhomogeneity.


Journal ArticleDOI
Gerald Rosen1
TL;DR: In this article, the linear manifold of solutions to a generic system of reaction-diffusion equations in the neighborhood of a constant uniform (equilibrium) solution has been studied for the case of two reacting and diffusing molecular or biological species, where the normal gradient of any species concentration function is zero at all boundary points.
Abstract: A systematic study is presented for the linear manifold of solutions to a generic system of reaction–diffusion equations in the neighborhood of a constant uniform (equilibrium) solution. The theory pertains directly to an arbitrary number of reacting and diffusing molecular or biological species in an arbitrary bounded spatial (1-, 2- or 3-dimensional) region with an impermeable boundary, so that the normal gradient of any species concentration function is zero at all boundary points. The stability analysis developed by previous authors is streamlined here for the case of two reacting and diffusing species, worked out completely for the case of three species, and made more amenable to specialized treatment for cases with four or more species. With the use of modern algebraic computational methods, explicit analytical general solutions to the linearized reaction–diffusion equations are derived for certain classes of model theories. These results either apply directly or admit extension to a wide range of practical reaction–diffusion problems in physical chemistry and biology.