Reduction of models of large scale lumped structures using normal modes and bond graphs

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.     

Pseudo bond graphs for generalized compartmental models in engineering and physiology

True bond graphs, which use effort and flow variables whose product is power, can in principle be used to describe all types of physical systems. However, many system models do not use power variables and yet can be represented usefully as pseudo bond graphs. Pseudo bond graphs have been used particularly for open systems in which it is convenient to consider control volumes or compartments with boundaries across which mass can flow. In this paper, we show how the bond graph methods used for conductive and convective heat transfer can be generalized to account for diffusion, convection, and accumulation of a variety of physical quantities and how pseudo bond graphs can aid in constructing and representing such models. These models are known in mathematical biology as “compartmental models” and it is a main contribution of this paper to show that the same pseudo bond graphs apply to thermofluid and physiological dynamic models. The bond graphs build in some conservation principles automatically and yet have the flexibility to incorporate general multiport laws when necessary. Thus the pseudo bond graphs can exhibit system structure as do other network graphs and are very general in nature.     

Real-time dynamic modeling of hydrogen PEMFCs

A control-oriented dynamic model of proton exchange membrane fuel cells (PEMFCs) has been developed in this paper. It aims to generate system dynamics with real-time performance for the micro-chip controller design. A unified bond graph approach was employed that integrates physical and chemical domains in the fuel cell operation, including fluid dynamics, heat transfer, and electrochemistry effects. In this paper, two major bond graphs, one for thermofluids, the other for the electrochemical system, were constructed. They are inter-connected to interpret the highly nonlinear transport and reaction system dynamics. The nonlinear simulation on a personal computer (PC) is about four times faster than the realistic operation. A step response test shows that the start-up time of an example PEMFC is about 5 s from ambient conditions. Further frequency-response test in the operation region shows that the bandwidth is near 2 rad/s.     

Finite mode bond graph representation of vehicle-guideway interaction problems

Classical modal analysis techniques for the prediction of vehicle-guideway dynamics are developed and interpreted with bond graph representations. Bond graphs are shown to allow easy generalization to multiple span guideways incorporating virtually any dynamic boundary conditions at the support locations. In addition, non-linear vehicle models can be used with any vehicle displacement-time history desired. Straightforward formulation procedures are also provided through the bond graph representation.The analysis procedure is demonstrated for a two-span Bernoulli-Euler Guideway with first-order dynamic boundary conditions. The results for a vehicle traveling at different speeds are shown to compare favorably with current literature.     

Bond graph fluid line models for inclusion with dynamic systems simulations

A procedure is presented whereby the control volume equations for one-dimensional, compressible gas dynamics are cast into first-order, state variable form. These equations are interpreted using causal bond graphs. The resulting bond graph is shown to reduce to the classic I-C chain under acoustic constraints and to a more recently developed model of low speed thermal energy transport subject to associated constraints.Through example it is demonstrated that the control volume bond graph is easily coupled to an overall system model and thus can be digitally simulated as part of the overall nonlinear state space representation. The result is that a very accurate gas dynamic model can be coupled with an overall dynamic system model without requiring a prohibitively large number of equations.     

Bond graph based sensitivity and uncertainty analysis modelling for micro-scale multiphysics robust engineering design

Components within micro-scale engineering systems are often at the limits of commercial miniaturization and this can cause unexpected behavior and variation in performance. As such, modelling and analysis of system robustness plays an important role in product development. Here, schematic bond graphs are used as a front end in a sensitivity analysis based strategy for modelling robustness in multi-physics micro-scale engineering systems. As an example, the analysis is applied to a behind-the-ear (BTE) hearing aid.By using bond graphs to model power flow through components within different physical domains of the hearing aid, a set of differential equations to describe the system dynamics is collated. Based on these equations, sensitivity analysis calculations are used to approximately model the nature and the sources of output uncertainty during system operation. These calculations represent a robustness evaluation of the current hearing aid design and offer a means of identifying potential for improved designs of multiphysics systems by way of key parameter identification.     

An Algorithm for Incorporating Subsystem Models into Overall Dynamic System Models

Bond graphs are used to construct finite mode representations of inherently distributed systems. These systems are, perhaps, only part of an overall dynamic system. The “causal” information provided by the bond graph permits the derivation of an automatable algorithm which produces the state equations as well as all output variables associated with the finite modes. The procedure requires only the a priori knowledge of modal masses, frequencies, and associated mode shapes for general boundary conditions of the distributed parts of the system. Thus, the algorithm is applicable to any multidimensional distributed system which is representable by normal modes.     

Bond graphs and linear graphs

Bond graphs have proven to be useful in modeling a wide variety of physical systems where multiports are needed to represent coupling phenomena. Here we show the relationship between bond graphs and more conventional notations utilizing connection N-ports and system graphs. This provides a rigorous foundation for bond graph manipulations and highlights certain difficulties with sign conventions and casual assignments that arise when junction structure loops are present.     

Physical parameter sensitivity of system eigenvalues and physical model reduction

The identification of subsystems and/or components that is related to a given eigenvalue of the overall system is a challenging and important topic. The use of special structure of the system matrices obtained busing bond graphs can result in identifying subsystems and/or components that affect a given eigenvalue of an overall system. This paper, by making use of a set of theorems and definitions proposes an efficient procedure for this purpose. The basic procedure is based upon the calculation of sensitivity of eigenvalues. The so-called “effect” matrices are produced that indicates the relative importance of physical parameters on a selected eigenvalue. In addition to the relative importance, the effect matrix is used for an efficient physical model reduction procedure. Furthermore, reasons of different dynamic behavior of a system can be explained. Use of effect matrices also improves the physical model reduction method based on decomposition procedures. Three examples are given to illustrate the approach and its consequences.     

Energy conversion in biological systems—part II. heat production in brown adipose tissue

This study involves the application of bond graphs to the generation of heat by cells in brown adipose tissue. In the first portion of this paper, data are presented to show the types of signals that can be measured at several sites in the system controlling thermogenesis—signals travelling from the brain to the tissue as well as signals over a feedback pathway from spinal cord temperature receptors back to the brain. Bond graphs are developed for several components of the system with an emphasis being placed on graphs describing the alternative mechanisms proposed for cellular energy conversion. The power flow is calculated using ENPORT for one mechanism for different values of plasma membrane resistance.     

Output consensus of heterogeneous linear systems with quantized information

In this paper, we design two distributed output consensus controllers for heterogeneous linear systems based on internal model principle and then study the quantization effect on the controllers when uniform quantizers are used in the communication channels. The first controller considers the general situation when the internal model state matrix of the system may be unstable and the communication graphs are strongly connected directed graphs. We prove that the bound of the consensus error is proportional to the quantizer parameter with a coefficient related to the size of the network and the property of the communication graphs. The second controller considers the situation when the internal model state matrix is neutrally stable and the communication graphs are undirected connected graphs. In this case, we derive a better bound of the consensus error which is proportional to the quantizer parameter and the coefficient is unrelated to the size of the network when the linear systems are homogeneous. Simulation examples are provided to illustrate the theoretical results.     

The energetic structure of multi-body dynamic systems

Dynamic systems containing particles and rigid bodies capable of movement in two- or three-dimensions may be represented by equations of motion in several basic forms. In all cases geometric nonlinearities are involved and terms arise in the equations which are difficult to understand and interpret. The equations often conceal the basic structure of the dynamic model of the system since only combinations of parameters and combined effects of various constraints finally appear.In this paper the basic relations for the dynamic elements as well as the transformations among internal forces and velocities are depicted using bond graphs. The energetic structure of the system is thus exhibited in multiport form. This aids in the exploration of alternative equation formulation and in understanding of the assumptions involved in any particular equation set. Further, a bond graph model is easily coupled with nonmechanical dynamic models through force or motion generators.     

基于多色图理论的供应链建模新方法

企业为了降低供应链的运作成本、提高顾客服务水平,需要经常性的对供应链网络进行重新设计。但是,由于一个供应链网络中涉及的影响因素众多,因此运用传统的方法很难对一个供应链网络进行全面的分析。本文通过对现有供应链模型的分析,提出一种基于多色图的供应链建模方法,用以反映供应链中完整的成员信息,并给出该模型的具体操作步骤。     

Pseudo bond graphs of circulating Fluids with application to solar Heating design

Two different bond-graph schemes for the modeling of circulating fluids are discussed and compared. The first uses a heuristic approach wherein the flow path is broken into segments. Each segment has separate hydraulic and thermal submodels which are linked with active bonds. The model is general enough to handle natural convective flows, heat storage, conduction, forced convective heat transfer and heat energy sources. The second method, applicable to pumped flows, uses a modal analysis approach to derive state equations for the modal temperatures, the time varying coefficients of the spatially varying mode shapes in a separation of variables solution for the temperature partial differential equation. Two examples are worked to illustrate the models. Numerical simulation results are presented using the first method and a simple analytic solution given to illustrate the second.     

Modeling of musculoskeletal structure and function using a modular bond graph approach

The dynamic effects of muscle strength, timing of muscle activations, and body geometry have been modeled for a wide variety of human activities. These types of models require the development of complex system equations that account for the effects of rigid-body dynamics, musculotendon actuators, passive and active resistance to motion, and other physiological structures. One way in which model refinement can be expedited is through the use of bond graph modeling techniques. While bond graph techniques have been used extensively in a broad variety of applications, they have been used only sparingly in the field of biomechanics, despite the potential suitability of a modular, multidomain approach to the modeling of musculoskeletal function. In the current paper, bond graph modules representing muscle function and rigid-body motions of underlying bone structures are introduced. The system equations generated with the use of these models are equivalent to those developed with more traditional techniques, but the modules can be more easily used in conjunction with control models of neuromuscular function for the simulation of overall dynamic motor performance.     

Bending-torsional flutter of a cantilevered wing containing a tip mass and subjected to a transverse follower force

This paper is a study of the bending-torsional flutter of a cantilevered wing subjected to a follower force, and containing a lumped mass, at the free end. In addition, a distributed aerodynamic loading is introduced along the wing. This results in a set of nonself-adjoint differential equations with variable, complex coefficients whose solutions are obtainable only in series form. Using the Frobenius method, a direct procedure is employed which retains the exact expression of the Theodorsen function and the unknown coefficients are evaluated on a computer which numerically converge to any prescribed accuracy. It is found that, as a result of the interaction of the two sources of non-conservative loadings, the follower force reduces considerably the critical speed of flows in all cases studied. An increase in the tip mass, however, has a stabilizing influence. The effect of structural damping is also examined and it is shown that internal damping forces may have pronounced influences on the flutter speed of the system.     

Analysis of switching devices with bond graphs

Modelling and simulation of switching devices with bond graphs is a subject which has no totally satisfying solutions. In this paper, switching devices are represented by flow or effort sources, with a variable circuit topology at switching time. For the simulation, the usual method uses causality resistors to insure integral causality to energy storage elements. The choice of those resistors is quite arbitrary and can lead to stiff systems. We propose, for linear systems, an approach without such resistors. When components lose the integral causality, the order of the state vector changes, provoking the use of pulse variables. A solution is proposed to compute the amplitude of the pulse and the value of the new variables.     

Optimal boundary control of heat conduction problems on an infinite time domain by control parameterization

The present paper deals with an optimal boundary control problem in which the process of systems under consideration is governed by a linear parabolic partial differential equation over an infinite time interval. The objective of the paper is to determine the optimal boundary control that minimize a given energy-based performance measure. The performance measure is specified as a quadratic functional of displacement and a suitable penalty term involving the boundary controls. In order to determine the optimal boundary controls, the problem with boundary controls are converted into a problem with distributed controls. The modal space technique is then used to reduce the system into the optimal control of time invariant lumped parameter system. The associated system of uncoupled first order initial value problems is solved in terms of controllers. Next step deals with the computation of the control and trajectory of the linear time-invariant lumped parameter. For this we approximate the controllers by a finite number of orthogonal exponential zero-interpolants over the interval [0,∞). The resultant performance index after using the optimality condition leads to a system of linear algebraic equations. The suggested technique is easy to implement on digital computer. We provide a numerical example to demonstrate the applicability and efficiency of the proposed approach.     

Modeling of Star-shaped and Parallel-wire Carrier Distribution Systems

It is often desirable to represent carrier distribution systems by networks so that analysis can be made to predict the behavior of these systems. Modeling of two canonical carrier distribution systems, the star-shaped system and the parallel-wire system by networks was carried out in this paper. The representation is given in terms of lumped parameters when the lengths of the carrier lines are short and it is given in terms of lumped and distributed parameters when the lengths of the carrier lines are long.     

Distributed average tracking for uncertain directed multiagent networks

This paper studies a distributed average tracking problem of uncertain multiagent systems over directed graphs. The basic idea is to establish an appropriate reference model by introducing an adaptive control scheme. The output of the reference model is the average of multiple time-varying reference signals. This implies that the reference model partially solves the problem of distributed average tracking of directed graphs. Moreover, an appropriate control law is designed to make the output of the multiagent system with uncertain parameters approaches the average of the reference signals from the reference model. The feasibility of the proposed scheme is theoretically proved and the effectiveness of the scheme is verified through a simulation example.