Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1985"

Multibond graph elements in physical systems theory

Abstract: The multibond graph notation turns out to be a natural and concise way to represent the behaviour of energy, power, entropy and other physical properties of macroscopic multiport systems. A global classification of the multiport elements in such a system is made on the basis of this (physical) behaviour in contrast with the usual classification on the basis of the (mathematical) form of the constitutive equations. Special attention is given to junction multiports.

Bond graph modeling and computer simulation of automotive torque converters

Abstract: A derivation of a set of four first-order nonlinear differential equations describing torque converter dynamics is given, along with the corresponding bond graph representation. The bond graph consists of an inertia-field and modulated gyrators which couple mechanical and hydraulic ports. A dualization of this structure replaces the original I-field by an IC-field, and all modulated gyrators by their partial duals—the modulated transformers. Further bond graph manipulations lead to torque converter—equivalent mechanical structures. The paper concludes with examples of static torque converter model validation and complete, dynamic model usage in the design of shift quality controllers for discrete ratio electronic transmissions.

Convex combinations of stable polynomials

Abstract: Sufficient conditions are given under which convex combinations of stable (complex and real) polynomials are stable.

Modelling the dynamics and kinematics of mechanical systems with multibond graphs

Abstract: A method to model mechanical systems with multibond graphs is described. The method is based on the description of the vector velocity relation of a moving point in a rotating system. This relation is incorporated in a bond graph. Since bond graphs are based on the power conserving concept, the force or momenta relations are then available too. By repeating the same bond graph structure for every point (which is of interest within a chosen coordinate frame), where each point has the same rotational velocity, a systematic way of modelling mechanical systems is achieved. It is explained how connected mechanical linkages have to be handled. Two simple examples are given.

Taylor series approach to system identification, analysis and optimal control

Abstract: The problems of system identification, analysis and optimal control have been recently studied using orthogonal functions. The specific orthogonal functions used up to now are the Walsh, the block-pulse, the Laguerre, the Legendre, the Chebyshev, the Hermite and the Fourier functions. In the present paper solutions to these problems are derived using the Taylor series expansion. The algorithms proposed here are similar to those already developed for the orthogonal functions; however, due to the simplicity of the operational matrix of integration, the Taylor series presents considerable computational advantages compared with the other polynomial series, provided that the input and the output signals may be assumed to be analytic functions of t .

Formula manipulation in the bond graph modelling and simulation of large mechanical systems

Abstract: A multibond graph element for a general single moving body is derived. A multibody system can easily be described as an interconnection of these elements. 3-D mechanical systems usually contain dependent inertias having both differential and integral causality. A method is described for the transformation of inertias with differential causality to an integral form, using formula manipulation. The program also helps to find experimentally the optimal choice for the generalized coordinates. The resulting explicit differential equation may be solved using a standard integration routine or simulation program.

Modelling complex vehicle systems using bond graphs

Abstract: In improving the performance of vehicles on the road, theoretical studies are needed which make use of dynamic models This paper directs the attention to the use of the bond graph technique as an aid in modelling the physical system Specific problems which may arise when the graph is prepared for the actual simulation phase are discussed Advanced modelling including 3-D dynamics of coupled bodies and the multiple transformations needed to describe the system from engine via driveline, axle, stub-axle, wheel to tyre-road contact on possibly non-level road surfaces are treated

Order Reduction of Multi-time Scale Systems Using Bond Graphs, the Reciprocal System and the Singular Perturbation Method

Abstract: The singular perturbation method applied to multi-time scale processes enables the reduction of dimensionality by considering only one part of the system—the slow or the fast part—depending on the frequency domain of interest. The fast and slow dynamics of bond graph models can be estimated by determination of causal loop-gains. We define here the notion of a reciprocal system which, with singular perturbation techniques, can obtain more accuracy on the fast time scale behaviour of the system.

Model reduction for a class of linear dynamic systems

Abstract: In this paper, a model reduction technique to remedy the singularity of reduced-order models is proposed. The approach adopted is based on the least-square fitting of time-moments of the system. The proposed method is also available to stabilize unstable reduced models. This method is superior to the existing techniques as shown in the numerical examples. It will be proved that this method always gives a stable reduced model assuming the original system is stable.

1985 Update of the bond graph bibliography

Abstract: This bibliography augments and includes the bibliographies of V. D. Gebben, published in the Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, Vol. 99, No. 2, pp. 143-145, 1977 and in the Journal of the Franklin Institute, Vol. 308, No. 3, pp. 361-369, 1979. The publications are listed alphabetically according to the year in which they have appeared. Although the authors have tried to produce an exhaustive list of bond graph publications, it is most likely that there still are oversights. Since it is our intention to make periodical updates of the bibliography readers are invited to send full bibliographical details of any missing or new bond graph or related publication, to the following address: Twente University of Technology, Department of Electrical Engineering, P.O. Box 217, 7500 AE Enschede, The Netherlands.

A survey of bond graph modelling for interacting lumped and distributed systems

Abstract: Through use of normal modes, bond graphs can be used to construct perhaps the most accurate low order models for linear distributed systems. By requiring relatively few equations, when compared to finite difference and finite element models, a physically understandable model results for design and automatic control applications. These bond graph distributed models can also be directly combined with bond graph models of lumped systems or other distributed systems to yield an overall system model with all the analytical and computational advantages that bond graph modelling affords. This paper surveys the development of bond graph modelling applied to interacting lumped and distributed systems.

Orientational analysis ― a supplement to dimensional analysis. I

Abstract: A method for extending the methods of dimensional analysis is based upon assigning “orientational” symbols to physical quantities such as area, force and angle which are spatially oriented. These symbols are shown to form a noncyclic group with four members, and they can be used to derive additional information that resolve problems incompletely solved by conventional dimensional analysis.

Multilinear theory of nonlinear networks

Abstract: A nonlinear network analysis procedure based on multilinear operators is developed in this paper. The procedure is equivalent to that using a multi-input, multi-output Volterra series and thus is most useful for slightly nonlinear networks. The new procedure, however, is simpler, computer implementable and lends a great deal of insight into network behavior. Some nonlinear circuit synthesis techniques are discussed using this insight. In addition, for the first time, a procedure for the incorporation of initial conditions is developed. It also is shown that the network response can be expressed as a linear combination of a set of waveforms. The network nonlinearities and input amplitude affect only the specific linear combination but not the waveforms themselves. These invariant waveforms are thus building blocks of the nonlinear network response. The manner in which the Volterra series converges and also the effect of the network nonlinearities and input amplitude upon the network response is discussed in terms of the invariant waveforms. Finally, the relation of the multilinear theory developed in this paper and the state representation is discussed. From this, a better understanding of both representations is obtained.

Computer generation of physical system differential equations using bond graphs

Abstract: A Computer Aided Modeling Program (CAMP) is developed to process a bond graph representation of a physical engineering system into suitable source input used by Digital Simulation Languages such as IBM DSL (Digital Simulation Language), CSMP (Continuous System Modeling Program), ACSL (Advanced Continuous Simulation Language) or CSSL (Continuous System Simulation Language). The basic philosophy and fundamental principles behind the design of CAMP are presented here. The preprocessed information includes the state variable representation of the system differential equations in first-order form. The appropriate input and interface software that simulation programs require is included in CAMP. The design and simulation of nonlinear systems and control systems has been investigated using this approach. The combination of a bond graph oriented program with a block diagram and FORTRAN oriented program is unique, and offers the best of both worlds, namely, the unified approach to modeling, automatic equation derivation from bond graphs and the extensive and flexible block modules of a continuous system simulator. A variety of cases are considered for mechanical, electromechanical, hydraulic and electrical systems.

Fair relation and modified synchronic distances in a petri net

Abstract: The notion of synchronic distances in a Petri net may serve as a fundamental basis to achieve synchronization without the use of a central clock in distributed processing systems. However, this notion is so far applicable to restricted subclasses of Petri nets. This paper introduces the concept of a fair relation and presents a modification and extension of synchronic distance measure. The measure can be applied to a wider class of Petri nets. As a result of our modification, it is now possible to say that the synchronic distance between two transitions is finite if and only if they are in a fair relation, i.e. there exists a positive integer k such that neither of them can fire more than k times without firing the other.

Orientational analysis, tensor analysis and the group properties of the SI supplementary units. II

Abstract: An extension of the methods of dimensional analysis to include the orientations of physical quantities shows that a useful approach is to assign orientations to the supplementary units, radian and steradian, and to require that physical equations be orientationally as well as dimensionally homogeneous. Orientational symbols representing the intuitive orientational character are shown to form a noncyclic Abelian group with four members. The methods of dimensional analysis derives from the fact that the laws of physics must be independent of the units; orientational analysis derives from the fact that they must also be independent of the coordinate system, i.e. they are expressible as tensor equations.

Osmosis and hydraulics by network thermodynamics and bond graphs

Abstract: In membranes with very small pores, equilibrium requires equal chemical potential, which leads to a pressure difference—the osmotic pressure, with different concentrations. Larger pores require equal pressures for equilibrium. The relations of these contradictory conditions are explored for varying diameters. A chemical boundary layer is introduced which follows chemical equilibrium even in large pores. Numerical values of the thickness of the boundary layer and of the chemical resistance are estimated from the diffusion coefficient in water.

The Use of Modal Analysis Concepts in the Simulation of Pipeline Transients

Abstract: In this article, the bond graph technique is used to build up a finite-order representation of a hydraulic network whose distributed character is inherent to the system. In a first step, the network model is set from the bond graphs of each line, which resulted from a modal approximation. The model obtained is then reduced so as only to contain the significant modes. In such a model, friction effects which are relatively weak compared to the dynamic effects are supposed to be linear. We deal with the problems of the causalities imposed by the subsystems linked to the network, then we shall suggest a technique using pressure andflow duality and conclude with a validated example.

Simulation of mixed bond graphs and block diagrams on personal computers using TUTSIM

Abstract: The TUTSIM simulation program for continuous dynamic systems accepts (nonlinear) block diagrams, bond graphs or a free mix of both. The simulation is “hands on” interactive, providing a direct contact with the model. The implementation of the program on existing personal computers (Apple II, IBM PC) requires small memory size and has a high computational speed, due to its assembler source code. A slower FORTRAN CP/M version is available. It is shown how bond graphs can be used as an input language. An example using bond graphs as a modelling tool is presented.

A mixed block diagram bond graph approach for biochemical models with mass action and rate law kinetics

Abstract: The use of a pseudo bond graph coupled with an algebraic block control structure is proposed to model biochemical systems with reactions defined by mass action and rate laws kinetics. The name “pseudo” refers to the assumption that the reactions are modelled by mass action kinetics with stoichiometric coefficients in power form. It is shown that the resulting model displays the topology of the stoichiometric reactions and of their enzymatic controls much more clearly than graphical methods used in chemistry. To illustrate the approach, several examples are presented: elementary enzymatic mechanisms, end product inhibition, enzymatic amplificatory cascade, simplified glycolysis. Finally, some “kinetic-like” models are presented: autocatalytic networks, Volterra systems, Goodwyn models of cellular controls.

Bond graph model of a reciprocating compressor

Abstract: In the present paper bond graph modelling techniques are used to develop a nonlinear model of a reciprocating compressor. Models are developed for different subsystems which are assembled into a final overall system bond graph. The orderly development of the governing state equations emanating from the bond graph is described. Some simulation results are included, which are shown to give excellent prediction of the real world phenomenon.

Identification of time-varying linear systems using orthogonal functions

Abstract: This paper considers the problem of identifying the parameters and initial conditions of systems described by linear differential equations with time-varying coefficients. A new approach is proposed which is based on the idea of using orthogonal functions to represent the input—output data, as well as the unknown time-varying parameters of the system. Using certain properties of the orthogonal functions, an algorithm is constructed which reduces the identification problem to that of solving a linear algebraic system of equations.

Bond graph models for electromagnetic actuators

Abstract: Rotary and linear electromechanical actuators are often used as the effectors for controlled systems and they often are important contributors to the dynamic response of the complete system. A generalized bond graph model for such actuators is developed which represents the three major types of forces or torques generated when coils, low reluctance elements and permanent magnets interact. A wide variety of actuators can bemodeled by versions of the basic bond graph which is simplifiable when some effects are neglible. The bond graph aids in understanding algebraic manipulations necessary to put the constitutive laws into forms compatible with other elements of the system for simulation or analysis.

Realizability of lossless reciprocal and nonreciprocal broadband matching networks

Abstract: Given the source and load impedances and a preassigned transducer power-gain characteristic, the necessary and sufficient conditions are given for the existence of a lossless reciprocal and a nonreciprocal equalizer which, when operating between the two given impedances, yields the desired transducer power-gain characteristic. It is shown that the realization of a broadband matching network can be accomplished by means of the driving-point synthesis using either a reciprocal or a nonreciprocal network.

Assigning causality in the case of algebraic loops

Abstract: When, during causality assignment in bond graph modelling, one is faced with a causality indetermination, one is told to assign “arbitrary” causalities. No rule is given to help except of course that of consistency. However, the different consistent causal assignments do not necessary lead to the same algorithm efficiency. Some very simple assignment rules can directly lead to an optimal solution. These rules are particularly useful when causalities are assigned by a computer program.

R-fields in the solution of implicit equations

Abstract: Bond graphs provide explicitly the topological and computational structures of a system. The topological structure describes how the components are assembled and the computational structure defines the set of mathematical equations defining the system behavior. When a topological structure corresponds to more than one computational structure, the mathematical equations are in implicit form. In this case, implicit methods of solution constitute the natural approach. This paper presents a method to deal with implicit equations based on R- field elements. It can be used to prepare a model with an implicit part to be simulated using explicit integration methods without introducing stiffness in the model. Simple examples from hemodynamics, simulated with THTSIM, are presented.

Recursive solution of the covariance equations for linear prediction

Abstract: An efficient recursive algorithm is presented to solve the “covariance” equations of the linear prediction modeling procedure. This algorithm is based on the conjugate direction optimization procedure and the expanding subspace theorem, and we show it is a natural extension as well as a geometric interpretation of the Levinson algorithm. The developed algorithm is simple to implement computationally, and can be extended to the multidimensional case.

Stable denominators for the simplification of z-Transfer Functions

Abstract: A well-known discrete stability test is used to derive from the denominator D ( z ) of a given stable high-order transfer function G ( z ), the denominator of a low-order approximant of G ( z ). The proposed method, based on the truncation and inversion of a continued fraction formed with the coefficients of D ( z ), yields a reduced denominator d ( z ) of degree, say m , which is always stable. Furthermore, depending on the neglected parts of the continued fraction, d ( z ) approximates m 1 and m 2 = m − m 1 zeros of D ( z ), located very near the points z =1 and z =-1, respectively. In the special case m 1 = m , d ( z ) is identical to the polynomial obtained by applying to D ( z ) the indirect technique, which combines the bilinear transformation with the Routh or the Schwarz approximation method.

On consistency of linear linearly constrained discrete time systems

Abstract: This paper addresses the problem of giving consistency conditions for constrained linear discrete time systems in state space form Conditions for various significant cases are given and analysed Finally, the structure of the set of initial states for which consistency prevails and the set of reachable states are studied