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Some fundamental theorems concerning the relationship between the junction structure of bond graphs and the effort and flow equations, are presented. Necessary and sufficient conditions are stipulated to guarantee the correct number of constraint equations. The structure and rank of the coefficient matrices of the effort and flow equations are examined. An orthogonality relationship between effort and flow equations is established. The development yields the result that the number of effort and flow equations corresponding to a causal assignment are sufficient and their coefficient matrices are of maximum rank.

A method is presented for the formulation of mathematical models of nonlinear systems. The method employs as the underlying principle the relaxation of the requirement of explicit causality at 0- and 1-junctions. Instead, causal assignments are made which are consistent with the computing causality of the nonlinear elements, thus leading to causal violations in the conventional sense. A classification of causal violations is identified which provides information useful in the systematic derivation of algebraic equations. A most attractive aspect of the method is the fact that causal relaxation is required only when needed, and only to the extent needed.

The notion of a chemical reaction network is made precise in terms of electrical network theory by demonstrating that the mathematical formulation of chemical reactions can be viewed as a subset of abstract circuit theory. An algorithm is presented for constructing linear graph and bond graph representations of chemical reaction systems. Input-output considerations are treated, using a network model of open reaction systems. Network and graphical models are also employed to elucidate the mathematical structure of cellular transport and cellular transport-reaction systems.

The dynamic responses of geared torsional systems are analyzed with the delay-bond graph technique. By transforming the power variables into torsional wave variables, the torsional elements are modeled as transmission line elements. The nonlinear elements, e.g., varying tooth stiffness, gear-tooth backlash, and nonlinear damping, are incorporated into the ideal transmission line element. A computational algorithm is established where the state variables of the system are expressed in terms of wave scattering variables and the dynamic responses are then obtained in both time and space domains. The simulation results of several simple examples of linear and nonlinear geared torsional systems are presented to demonstrate the feasibility of this algorithm.

Amongst the available graph theories the one making use of “bond graphs” has been considered the most suitable in network thermodynamics for representing a wide range of physico-chemical processes in the form of networks. In this article a complete representation of chemical reactions, both far from- and near-equilibrium, by bond graphs, is proposed. In addition, a new proof of the Tellegen theorem is given, derived directly from the properties of bond graphs. A new insight into the general meaning of Tellegen's theorem in variable networks and its relevance to biological networks is thus provided. The structure of the network being represented by new elements — i.e. the junctions — in bond graphs, time variations of these elements have the meaning of changes in the structure itself, to that the Tellegen theorem appears as an invariance relation in networks where the structure is allowed to change.

A large number of physical and engineering systems may be represented directly in terms of component energy characteristics and their power interactions. When the system elements are modeled as energetic multiports, and their interconnections by power bonds, then the bond graph language is a natural one for describing the entire system. Bond graphs may be written for dynamic systems involving various energy types, such as electrical, mechanical, fluid and thermal; all energy types may be coexistent. Useful modeling elements include multiport storages, dissipators, and junction elements and transducers, as well as sources. Bond graph models of linear multiport systems may be transformed to state-space form by a powerful algorithm based upon operational causality. From the state-space equations, dynamic responses may be obtained by the matrix exponential technique, thereby allowing the direct digital simulation of linear multiport models. The ENPORT program is a realization of the bond graph reduction algorithm. It is a principal purpose of this paper to describe the procedure upon which ENPORT is based, and to present some results. Important features of ENPORT are its choice of physically significant state variables, its use of operational causality to obtain an orderly formulation of system equations, and its ability to handle systems containing static storage subfields.

The problem of deriving a suitable mathematical model for complex devices is discussed. A small vibratory air pump is used as the medium of presentation. The modeling process begins with the basic coupling structure of the device. In a logical step-by-step procedure the initial model is built up to satisfy a number of functional considerations inherent to the device, such as the resonance behavior, input impedance, output impedance, and internal dissipation. At each step in the modeling process the completeness and suitability of the model is examined. Bond graphs drawn for the successively larger and more complex model clearly predict the shortcomings of the partial model and point the way to the next step. It is evident that the principle of causal relations forms a most important guiding element in the modeling process. The final model is in the form of a set of linear state equations, and scaling of the A-matrix indicates the relative importance of parameters when experimental values are substituted for literals.

A novel procedure for systematically generating state-space equations for multiport systems is presented. The method is based upon a bond graph representation of the system and causal manipulation of the field equations. Principal advantages of the method are the ability to anticipate formulation properties before writing equations, the availability of a simple check for correctness of the initial system relations, and the specification of a systematic reduction procedure for obtaining state-space equations in terms of energy variables.

A nonlinear analytical model for the dominant low-frequency dynamic behavior of flexible-skirt air cushion suspensions is developed using bond graph notation. The model includes nonlinear fluid source and orifice characteristics, and modulation of cushion fluid capacitance and sealing gap fluid resistance by changes in geometry and cushion pressure. Bond graphs are presented for the fully nonlinear cushion and for the linearized small perturbation case. Both the nonlinear and linearized equations for the heave dynamics of an air cushion vehicle are developed and solved by machine computation for a realistic range of operating conditions. By comparing the linear and nonlinear solutions, it is concluded that the linear model is suitable for the preliminary design and optimization of air cushion vehicle suspensions. The full nonlinear simulation is prudent, however, to check a tentative design for the effects of unusually large guideway irregularities or external load disturbances.

This paper considers the application of signal flow graph techniques to the problem of characterizing sequential circuits by regular expressions. It is shown that the methods of signal flow graph theory, with the proper interpretation, apply to state diagrams of sequential circuits. The use of these methods leads to a simple algorithm for obtaining a regular expression describing the behavior of a sequential circuit directly from its state diagram.

Topological representations, using the bond graph rotation developed in Part 1, are presented for several common thermodynamic couplings. Examples of more complex systems are presented, including diffusion with chemical reaction, relaxation oscillations in electrochemical systems and nonlinear phenomena in charged membrane structures.

The modeling process whereby distributed parameter fluid systems are described approximately by lumped parameter models is discussed using bond graph techniques. It is shown that despite the analogies which exist between some fluid systems and other physical systems, the lumping process for fluid systems introduces forms of nonlinear behavior not often encountered in other types of systems. Many such effects may be traced to the desire to use control volumes and Eulerian rather than Lagrangian descriptions of the fluid systems.

This paper describes a problem-oriented programming system in which students can construct models of physical systems. The "bond-graph" notation is used for representing physical and engineering components and their interconnections, whether as a general and descriptive model or a specific and quantitative one. Using a simple linear notation the student enters the model into the computer via a standard keyboard and observes the behavior simulated by the ENPORT program. The results are given in function values and simple curves that reveal certain well known but instructional aspects of physics. A description is given of an instructional monitor that is capable of evaluating the student's model by comparing it to a "reference" model constructed by a teacher. The problem of interpreting differences between bond graphs in a way that aids student learning was not solved, but the student is given full access to ENPORT to bypass the problems of effective monitoring.

The “memristor”, first defined by L. Chua for electrical circuits, is proposed as a new bond graph element, on an equal footing with R, L, & C, and having some unique modelling capabilities for nonlinear systems.

General bond graph methods for the description, analysis, and simulation of dynamic systems are illustrated through the study of vehicle drive line dynamics. Emphasis is placed upon the problem of assembling a compatible and efficient system description from multiport models of the system components. Examples show how state space descriptions for analysis and block diagrams for analog simulation may be obtained systematically from bond graphs. Digital simulation is conveniently accomplished using the ENPORT programs, which accept bond graphs directly.

Two types of bond graphs which imbed Lagrange’s equations and variables are found for systems conceived in terms of energy and power. The basic form also imbeds simpler transformed equations and more primitive variables. The graphs enhance one’s grasp of both energetic structure and functional behavior, including nonlinearities, and give state space formulations.

The Shannon-Mason loop rule permits direct reduction of a linear constant-parameter signal flow graph to a transfer function. Signal flow graphs can be constructed from bond graphs or sets of equations. Application of the loop rule to the parent bond graphs, however, with the aid of certain rules, is shown to be quicker and less prone to error. Also, four invariant classes of bond graph meshes are distinguished, with implications in physical analogies and in computation.

This paper aims to simulate and control a three-phase Brushless DC Motor. Bond Graph method has been used to obtain fast and simple dynamic model. The system has been controlled by classical PID controller. All the paper results were fulfilled using LabVIEW program.

The domain of engineering, well founded on quantitative physical laws, is generally considered an eldorado for model builders. Nevertheless, even in the field of engineering, performing the abstraction from a real world system to a mathematical model may be difficult.