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Principles of Engineering System Design Dr T Asokan asok@iitm.ac.in

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Principles of Engineering System Design Dr T Asokan asok@iitm.ac.in Bond Graph Modelling of Dynamic systems

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Physical System Modelling Bond Graph Method The exchange of power between two parts of a system has an invariant characteristic. The flow of power is represented by a Bond Effort and Flow are the two components of power.

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Physical System Engineering Model Differential Equations Output Simulation Language Block Diagrams Physical System Engineering Model Bond Graph Output Software(Computer Generated Differential Equations) Classical approach for modeling of physical system Bond Graph Modeling

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Generalised Variables p = e · dt q = f · dt Power variables: Effort, denoted as e(t); Flow, denoted as f(t) Energy variables: Momentum, denoted as p(t); Displacement, denoted as q(t) The following relations can be derived: Power = e(t) * f(t)

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October 2, 2008 Energy Flow The modeling of physical systems by means of bond graphs operates on a graphical description of energy flows. The energy flows are represented as directed harpoons. The two adjugate variables, which are responsible for the energy flow, are annotated above (intensive: potential variable, “e”) and below (extensive: flow variable, “f”) the harpoon. The hook of the harpoon always points to the left, and the term “above” refers to the side with the hook. e f P = e · f e: Effort f: Flow

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Modeling: Bond Graph Basics effort/flow definitions in different engineering domains Effort eFlow f ElectricalVoltage [V]Current [A] TranslationalForce [N]Velocity [m/s] RotationalTorque [N*m]Angular Velocity [rad/sec] HydraulicPressure [N/m 2 ]Volumetric Flow [m 3 /sec] ChemicalChemical Potential [J/mole] Molar Flow [mole/sec] ThermodynamicTemperature [K] Entropy Flow dS/dt [W/K]

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I for elect. inductance, or mech. Mass C for elect. capacitance, or mech. compliance R for elect. resistance, or mech. viscous friction TF represents a transformer GY represents a gyrator SE represents an effort source. SF represents a flow source. Modeling: Bond Graph Basic Elements I C R TF m e1 f1 e2 f2 e2 = 1/m*e1 f1 = 1/m*f2 GY e1 f1 e2 f2 d f2 = 1/d*e1 f1 = 1/d*e2 SESF

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Modeling: Bond Graph Basic Elements Power Bonds Connect at Junctions. There are two types of junctions, 0 and 1. 01 1 2 3 4 5 11 12 13 Efforts are equal e1 = e2 = e3 = e4 = e5 Flows sum to zero f1+ f2 = f3 + f4 + f5 Flows are equal f11 = f12 = f13 Efforts sum to zero e11+ e12 = e13

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Causal Bond Graphs Every bond defines two separate variables, the effort e and the flow f. Consequently, we need two equations to compute values for these two variables. It turns out that it is always possible to compute one of the two variables at each side of the bond. A vertical bar symbolizes the side where the flow is being computed. e f Mandatory Causality ( Sources, TF, GY, 0 and 1 Junctions) Desired Causality (C and I elements) Free Causality (R element)

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“Causalization” of the Sources U 0 = f(t) I 0 = f(t) U 0 i Se Sf u I 0 The source computes the effort. The flow has to be computed on the right side. The source computes the flow. The causality of the sources is fixed.

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“Causalization” of the Passive Elements u i R u = R · i u i R i = u / R u i C du/dt = i / C u i I di/dt = u / I The causality of resistors is free. The causality of the storage elements is determined by the desire to use integrators instead of differentiators.

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Integral Causality (desired Causality) e f I e f sIsI 1 f e C f e sCsC 1 Integral causality is preferred when given a choice.

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“Causalization” of the Junctions 0 e1e1 e2e2 e3e3 f1f1 f2f2 f3f3 e 2 = e 1 e 3 = e 1 f 1 = f 2 + f 3 1 e1e1 e2e2 e3e3 f1f1 f2f2 f3f3 f 2 = f 1 f 3 = f 1 e 1 = e 2 + e 3 Junctions of type 0 have only one flow equation, and therefore, they must have exactly one causality bar. Junctions of type 1 have only one effort equation, and therefore, they must have exactly (n-1) causality bars.

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Modelling Example Mechanical Systems R CM F Equation Governing the system F FRFR FmFm FkFk Mass, Spring and Damper Syetms

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Final Bond Graph e1 emem ecec f1 fmfm eReR fc fRfR System Equations Bond Graph model Reference Velocity=0 for this case Velocity Junction Damper Spring Mass

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Simulation-Second order system

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Modelling of underwater Robotic systems

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Euler angles Linear velocity of the base point w.r.t Inertial frame Tz Ty Tx Ixx+Iax wx*(Ixx+Iax) wz*(Izz+Iaz) wy*(Iyy+Iay) wz wx wy Izz+Iaz Iyy+Iay PV TFMV Body fixed angular velocity Body fixed linear velocity m+max m+may m+maz Euler angle Transformation matrix wx*(m+max) wy*(m+may) wz*(m+maz) Vx VyVz Angular velocity to first link of the manipulator Tbx Tby Tbz Fz Fy FxFbx Fbz Fby MTF0 MGY II I 11 1 MSe MTF MSe MGY II I 11 1 MSe 1 MTF 1 1 ò MSe 1 MR 1

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Angular velocity from previous link Joint velocity angular velocity of the manipulator PV1TFM1 PVM1 Ix1+Iax1Iy1+Iay1 Iz1+Iaz1 m1+ma1 Tip velocity of link1 TF1 TF2 TF3 Pad Euler angles Linear velocity of the base point w.r.t Inertial frame Tz Ty Tx Ixx+Iax wx*(Ixx+Iax) wz*(Izz+Iaz) wy*(Iyy+Iay) wz wx wy Izz+Iaz Iyy+Iay PV TF Body fixed angular velocity Body fixed linear velocity m+max m+may m+maz Euler angle Transformation matrix wx*(m+max) wy*(m+may) wz*(m+maz) Vx VyVz Angular velocity to first link of the manipulator Tbx Tby Tbz Fz Fy FxFbx Fbz Fby Wx1*(Ixx1+Iax1) Wz1*(Izz1+Iaz1)Wy1*(Iyy1+Iay1) Wz1 Wx1 Wy1 Angular velocity from previous link Joint velocity angular velocity of the manipulator PV2TFM2 PVM2m2+ma2 Tip velocity of Link2 TF1 TF2 TF3 Pad Joint velocity PV3TFM3 PVM3m3+ma3 Tip velocity of the manipulator TF1 TF2 TF3 Pad Se1 R I 1 TF 0 1 MGY II I 11 1 1 MTF 0 0I0 1 1 Se MR MTF0 MGY II I 11 1 MSe MTF MSe MGY II I 11 1 MSe 1 MTF 1 1 ò MSe MR 11 Se1 R I 1 TF 01 1 MTF 0 0I0 1 1 Se MR AD L2 Se1 R I 1 TF 011 MTF 0 0I0 1 1 Se MR Pad AD L3 Link1 1111as ffa11 Link 2 221 Link 3 221

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Advantages and disadvantages of modelling and simulation Advantages Virtual experiments (i.e. simulations) require less resources Some system states cannot be brought about in the real system, or at least not in a non-destructive manner ( crash test, deformations etc.) All aspects of virtual experiments are repeatable, something that either cannot be guaranteed for the real system or would involve considerable cost. Simulated models are generally fully monitorable. All output variables and internal states are available. In some cases an experiment is ruled out for moral reasons, for example experiments on humans in the field of medical technology.

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Disadvantages: Each virtual experiment requires a complete, validated and verified modelling of the system. The accuracy with which details are reproduced and the simulation speed of the models is limited by the power of the computer used for the simulation.

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SUMMARY Modelling and simulation plays a vital role in various stages of the system design Data Modelling, Process Modelling and Behavior modelling helps in the early stages to understand the system behavior and simulate scenarios Dynamic system models help in understanding the dynamic behavior of hardware systems and their performance in the time domain and frequency domain. Physical system based methods like bond graph method helps in modelling and simulation of muti-domain engineering systems.

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