Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1990"

Finite-difference models of ordinary differential equations: influence of denominator functions

Abstract: This paper discusses the influence on the solutions of finite-difference schemes of using a variety of denominator functions in the discrete modeling of the derivative for any ordinary differential equation. The results obtained are a consequence of using a generalized definition of the first derivative. A particular example of the linear decay equation is used to illustrate in detail the various solution possibilities that can occur.

Bond graph models for electrochemical energy storage : electrical, chemical and thermal effects

Abstract: Bond graph models for chemical kinetics are extended to electrochemical systems. Although many electrochemical systems can be considered to function in a constant temperature environment, high-power energy storage systems, such as electric vehicle batteries, undergo drastic changes in temperature in operation. Therefore the bond graph models are also extended to couple with thermal models driven by energy dissipation of a chemical or electrical nature. Electrochemical-thermal bond graph models are readily coupled to standard bond graph models for electrical and mechanical components of an overall system such as an electric vehicle.

Model reduction by minimization of integral square error performance indices

Abstract: A model reduction method based on the minimization of output response deviations is presented. In the proposed method, both the poles and zeros of the reduced model are considered to be free parameters and are obtained by minimizing the integral square error in impulse or step responses. When tested with a random input, the reduced models derived from impulse response deviations outperform those designed to minimize step response deviations. Comparison to existing methods shows that the proposed procedure yields better reduced models.

The operational matrix of integration for Bessel functions

Abstract: A general expression for the operational matrix of integration P for the case of Bessel functions is derived. Using this P, several problems such as identification, analysis and optimal control may be studied. Examples are included to illustrate the theoretical results.

The mathematical foundations of bond graphs—III. Matroid theory

Abstract: The cycle and co-cycle matroids of a bond graph are defined using chain group matroids derived from the cycle and co-cycle vector spaces of a bond graph. The relationship between these structures is investigated and various results are proved. A precise equivalence is defined for bond graphs. Duality theory is seen to be very clear in the context of bond graph matroids.

Optimum shapes of convective pin fins with variable heat transfer coefficient

Abstract: Principles of variational calculus are used to determine the shapes of convective pin fins that maximize heat dissipation, given the amount of fin material. The analysis considers the convective heat transfer coefficient h to depend on the fin diameter D according to the relationship h ∝ 1/Dn, where n takes on values from 0.2 to 0.5 depending on the type of pin fin configuration and flow condition (1: AIChEJ, Vol. 29, p. 1043, 1983). The Euler equations, which are nonlinear and coupled, are formulated and solved for the cases of both length and weight constraints as well as only weight constraint. The resulting quadrature formulae are represented in the form of a convinient design plot, from which the optimum design parameters may be obtained and used to determine the fin and temperature profiles as well as the cooling performance. The solutions under both constraints yield considerably simple results for the case of only weight constraint, which corresponds to the diameter and excess temperature of the fin tip being zero. An important result is that the Schmidt criterion (2: Z. Verein. Deutsch Ing., Vol. 70, pp. 885, 947, 1926) of a linear temperature profile also holds for pin fins of specified weight with a variable heat transfer coefficient. Finally, by using Pontryagin's minimum principle (3: The Mathematical theory of Optimal Processes, Wiley, New York, 1962), it is demonstrated that the problems of maximizing cooling (for a given fin weight) and minimizing weight (for a given fin cooling) are identical as both are governed by the same optimum design equations.

The mathematical foundations of bond graphs—IV. Matrix representations and causality

Abstract: The development of a mathematical theory for bond graphs continues with an analysis of two areas crucial to the derivation of system equations from a bond graph model. Matrix representations of bond graph matroids are examined and used to provide a rigorous proof of the mathematical equivalence of the linear graph and bond graph modelling methods. The procedure of selecting causality by the method of causal strokes is discussed. This is shown to be a device for choosing a base of the cycle matroid of a bond graph, and a sufficient condition is proved for when the method gives a base. An example is given of a bond graph with a causal loop which corresponds to a valid causal assignment. Various combinatorial formulae are proved concerning ranks and dimensions. Formulae for the ranks of effort and flow matrices and the number of independent equations obtained from a bond graph are consequences of the results.

Coupled integral equation approach for solving phase-change problems in a finite slab

Abstract: The recently advanced Coupled Integral Equation approach for solving phase-change problems of semi-infinite medium is extended to develop an approximate analytic solution for melting or solidification in a slab of finite thickness subjected to a time-varying prescribed temperature at one surface and a constant prescribed temperature at the other. The accuracy of the analytic expressions obtained in this manner is examined by comparing the present results to those reported in the literature. The influence of physical parameters such as the Stefan number and the applied surface temperature on the interface position and boundary heat fluxes is examined.

Stability of a general predator—Prey model

Abstract: In this paper, we propose a general predator—prey model which includes the Lotka—Volterra model, Gause model, generalized Gause model, Hsu model (1978), Kuang- Freedman model (1988), etc. as special cases. Several results on the local and global stability are obtained concerning this model. These results are generalizations of those obtained by Hsu (1978), Cheng (1981), Liou and Cheng (1988).

Full rubs, bouncing and quasi chaotic orbits in rotating equipment

Abstract: In order to improve performance, closer tolerances are usually required in high performance turbomachinery. This often results in an increase in rotor casing/real rub sensitivities. The objective of this paper is to investigate the effects of mass, support stiffness and blade stiffness during a rotor-casing rub event. Special emphasis will be given to defining the tuning/detuning effects of the system during such variations in mass and stiffness. The overall model will incorporate the influence of : (i) casing and rotor inertia, (ii) casing and rotor support stiffness, (iii) contact friction induced during rub interaction, (iv) single and multiple blade contact, as well as (v) lateral and radial blade stiffness effects. The main thrust will be investigate the different regimes of rubbing, i.e. the development of full rubs, rigid bouncing and essentially chaotic behavior due to the changes in rotor-to-casing mass and stiffness ratios, as well as blade stiffness and friction effects.

Solution of linear two-point boundary value problems and optimal control of time-varying systems by shifted Chebyshev approximations

Abstract: A method for finding the solution of a linear two-point boundary value problem with time-varying coefficients is discussed. Properties of shifted Chebyshev series are first briefly presented and the transformation matrix relating the back vector to the current time vector together with the operational matrix are utilized to solve the two-point boundary value problems. This approach can be applied to obtain the optimal control of linear time-varying systems subject to quadratic cost criteria. An illustrative example is given.

Application of graph theory to the mathematical modelling of a class of rigid body systems

Abstract: The process of determining equations of motion of a system consisting of rigid bodies and springs is often extremely laborious. In this paper, a method is presented whereby all terms of a system of equations for planar rigid body systems are calculated systematically. It is assumed that the system of rigid bodies has a topological tree structure. The method is based on a graph-theoretical approach and is consistent with that for the calculation of kinetic and potential energies presented by Arczewski ( J. Franklin Inst. , Vol. 324, pp. 351–367 and pp. 369–386, 1987). The proof of the method and an example are provided.

On the vibrations of overdamped systems

Abstract: Vibration systems are characterized by the two parameters: ωn, the undamped natural frequency, and ζ, the damping ratio. In the case of lightly damped systems, these parameters are determined by means of the logarithmic decrement. A simple method is presented here which applies to overdamped systems, ζ>1. The method involves simple measurements, and a criterion for the accuracy of the method is given.

Chaos and limit cycle in Duffing's equation

Abstract: Duffing's equation was first given in 1918. Recently CHAOS was found and studied by using analog and digital computers. The present paper reports the chaotic behavior for a wide range of parameters and limit cycle in Duffing's equation. Some discussion on the chaotic behavior and its benefit is included.

Chaos in Van der Pol's equation

Abstract: Van der Pol's Equation was first given in 1926. It gives limit cycles. The present paper reports the chaotic behavior of modified Van der Pol's Equation with forcing function. In three of six cases, CHAOS is found, while three other cases give limit cycles.

Energetically consistent bond graph models in electromechanical energy conversion

Abstract: Linear and rotary motors using permanent magnet field excitation can often be modeled using power-conservative gyrator elements based on the Lorentz force law and Faraday's voltage law. A common extension of such models using modulated gyrators is shown to be inconsistent with storage field models based on a stored energy state function. Although the modulated gyrator models have intuitive appeal and are useful in certain restricted circumstances, the storage field models are preferable energetically.

Robust Schur stability of a complex-coefficient polynomials set with coefficients in a diamond

Abstract: Strict Schur property of a complex-coefficient family of polynomials with the transformed coefficients varying in a diamond is considered. It is proved that the checking of eight edge polynomials provides necessary and sufficient conditions for the strict Schur property of the transformed family of perturbed-coefficient polynomials, and a sufficient condition for the original family. The case of polynomials with real coefficients falls out as a special case, and the approach given also applies to a far wider class of regions in coefficient space than those represented either by boxed domain or diamond. An illustrative example is given.

A new orthogonal series approach to sensitivity analysis

Abstract: The problem of trajectory and output sensitivity analysis of linear time-invariant systems is studied using a new orthogonal series method. Simple expressions are derived for the determination of the coefficient sensitivity matrices involving multiplications of matrices of small dimensions. No solution of algebraic equations is required, hence no matrix inversion is needed here compared to known techniques. The present results, therefore, reduce the computational effort involved and increase accuracy due to round-off errors.

Information storage in high-order neural networks with unequal neural activity

Abstract: Neural networks with high-order interactions only have been shown to be sufficient to provide satisfactory attractivity to the stored patterns and error corrections. Such interactions increase the storage capacity of the networks and allow one to solve a class of problems which are intractable with standard networks. In this paper we analyse the capacity of these higher-order networks by the statistical method and show why the probability of the states of neurons being active and passive can always be chosen equal, i.e. with a probability of 0.5 each.

On model reduction by L2-optimal pole retention

Abstract: The L 2 -optimal reduced model retaining a given number of poles of the original system is determined in an efficient way. In particular, an expression for evaluating the minimum of the impulse response error norm for any choice of retained poles is derived in terms of the original system parameters, which allows us to find the truly dominant poles in a simple manner. Then, the corresponding optimal model numerator is obtained using an order- recursive procedure. An interesting interpolation property of the reduced model is also pointed out. An example shows how the method operates.

Finding roots by deflated polynomial approximation

Abstract: A numerical technique is presented which evaluates the roots of polynomials with real coefficients. Features of the method include no complex arithmetic requirements, no need to guess at initial quadratic factor estimates, multiple or nearly equal roots being easily dealt with and a high degree of flexibility in coping with non-convergent iterations. The method is simple to use and is based upon a Routh Array-type algorithm familiar to control engineers. Numerical examples demonstrate its application to various polynomials.

Analysis of second-order multivariable linear systems

Abstract: Efficient methods are discussed for the solution of second-order state-variable equations and their application to the study of dynamical multivariable linear systems. Two different approaches are presented. A major one is based on the matrix Laplace transform and matrix polynomial factorization. Several cases are considered, and suitable expressions for the second-order matrix polynomial factorization are developed. Particular relations between system element parameters are discussed, and some simple criteria for stability are derived.

Model matching control for multivariable systems—I. Stable plants

Abstract: A conceptually lucid approach to practical controller design is presented for Multiple-Input-Multiple-Output (MIMO) systems. Algebraic manipulations are used to parameterize reference models which can be matched by a controlled plant with total stability. Novel techniques allow unprecedented flexibility in reference model selection. Controller synthesis is performed in the frequency domain. The plant models may be stable or unstable, possess nonminimum-phase zeros and need not have the same number of outputs as inputs. Examples are provided at each step of the theoretical development in order to illustrate the key concepts. Part I deals with stable plants and Part II extends the design approach to plants which may be open-loop unstable

A network approach to interpolation with symmetric positive-real matrices

Abstract: Employing Hazony's theory of the cascade synthesis of passive n -ports, a simple new procedure of interpolation with symmetric positive-real matrices is presented. The problem related to interpolation with rational lossless positive-real matrices is also discussed. The results of the paper can be considered as a matrix extension of the Youla-Saito scalar interpolation theory.

The near time-optimal motion control of robotic manipulators

Abstract: The development of strategies for near time-optimal motion control of robotic manipulators is discussed. A PUMA 560 three degree-of-freedom industrial robot is used for computer simulation of algorithm. A quintic polynomial describes the trajectory of the robot end-effector traversing a specified path in Cartesian space. It is shown that a significant increase in speed and accuracy of the manipulator can be achieved with the use of the proposed algorithm.

Partial conservativity and its applications to multiphase oscillators and other oscillators

Abstract: The well-known Barkhausen Criterion deals with designing oscillators which can be regarded either as “partially conservative” or as (“fully”) conservative. The present paper develops the concept of partial conservativity and shows how to distinguish between the two types of systems. It demonstrates that the concept is on the one hand related to practical aspects of well-established oscillators (phase-shift oscillator, and Colpitts and Hartley oscillators). On the other hand, the concept is helpful in developing new types of multiphase oscillators with potential practical advantages for power electronics and for feeding phased array antennae.

Identification of non-stationary continuous systems using modulating functions

Abstract: The identification of non-stationary multi-input multi-output continuous systems by the use of trigonometric modulating functions is considered. The time variable matrices to be identified are developed in polynomial form, and the fact that the product of a modulating function with these series also gives a modulating function is discussed. The proposed method avoids the necessity of knowledge and estimation of the initial conditions and signal derivatives.

An efficient algorithm for the design of circular symmetric linear phase recursive digital filters with separable denominator transfer function

Abstract: In this paper a method is presented for the generation of 1-variable HP using the properties of the positive definite and positive semi-definite matrices. We have also presented a method for the design of 2-D recursive digital filter with separable denominator transfer function and octagonal symmetry satisfying a prescribed magnitude and group-delay specifications. The method consists of two steps. In the first step the two 1-D all pole filter is used to approximate the group-delay and magnitude responses along ω1−ω2 axis while in the second step the numerator coefficients are used to approximate the overall magnitude response.

Determination of vibration parameters in moderately damped systems

Abstract: The undamped natural frequency ωn and the damping ratio ζ can be determined from a plot of the free motion of the system. If the damping is light, ζ < 0.2, the logarithmic decrement can be used to determine the coefficients. Systems with moderate damping are those in the range 0.2 < ζ < 1.2. A method for determining vibration parameters for systems with moderate damping is presented here. The method is easy to apply and while it is approximate, it is extremely accurate throughout the entire range of moderate damping.