Transcript Slide 1
Principles of Engineering System Design Dr T Asokan [email protected] Principles of Engineering System Design Bond Graph Modelling of Dynamic systems Dr T Asokan [email protected] 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. Classical approach for modeling of physical system Bond Graph Modeling Physical System Physical System Engineering Model Engineering Model Differential Equations Bond Graph Block Diagrams Software(Computer Generated Differential Equations) Simulation Language Output Output Generalised Variables 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) p = e · dt q = f · dt Energy Flow The modeling of physical systems by means of bond graphs operates on a graphical description of energy flows. e f P=e·f e: Effort f: Flow 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. October 2, 2008 Modeling: Bond Graph Basics • effort/flow definitions in different engineering domains Effort e Flow f Voltage [V] Current [A] Force [N] Velocity [m/s] Rotational Torque [N*m] Angular Velocity [rad/sec] Hydraulic Pressure [N/m2] Volumetric Flow [m3/sec] Chemical Chemical Potential [J/mole] Molar Flow [mole/sec] Temperature [K] Entropy Flow dS/dt [W/K] Electrical Translational Thermodynamic Modeling: Bond Graph Basic Elements • I for elect. inductance, or mech. Mass I • C for elect. capacitance, or mech. compliance C • R for elect. resistance, or mech. viscous friction R • TF represents a transformer e1 • GY represents a gyrator e1 f1 f1 • SE represents an effort source. SE • SF represents a flow source. SF TF e2 m f2 GY d e2 f2 e2 = 1/m*e1 f1 = 1/m*f2 f2 = 1/d*e1 f1 = 1/d*e2 Modeling: Bond Graph Basic Elements • Power Bonds Connect at Junctions. • There are two types of junctions, 0 and 1. 5 1 4 0 3 11 1 2 13 12 Efforts are equal Flows are equal e1 = e2 = e3 = e4 = e5 f11 = f12 = f13 Flows sum to zero Efforts sum to zero f1+ f2 = f3 + f4 + f5 e11+ e12 = e13 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) “Causalization” of the Sources The flow has to be computed on the right side. Se Sf U 0 U0 = f(t) i u The source computes the effort. I0 = f(t) I0 The source computes the flow. The causality of the sources is fixed. “Causalization” of the Passive Elements u i u R i u=R·i R i=u/R The causality of resistors is free. u i u C i du/dt = i / C I di/dt = u / I The causality of the storage elements is determined by the desire to use integrators instead of differentiators. Integral Causality (desired Causality) e I f 1 f edt I f F ma e f 1 sI e e C f 1 sC 1 e fdt C Integral causality is preferred when given a choice. “Causalization” of the Junctions e2 e1 f1 f2 0 f3 e3 e2 = e1 e3 = e1 f1 = f2 + f3 Junctions of type 0 have only one flow equation, and therefore, they must have exactly one causality bar. e2 e1 f1 f2 1 f3 e3 f2 = f1 f3 = f1 e1 = e2+ e3 Junctions of type 1 have only one effort equation, and therefore, they must have exactly (n-1) causality bars. Modelling Example Mechanical Systems Mass, Spring and Damper Syetms x m x 0 R F M C FR Equation Governing the system Mxm Rx m Kxm F F Fm Fk Bond Graph model Mass I Se 1 Velocity Junction 0 Spring 1 Reference Velocity=0 for this case Sf 1 Damper C R System Equations Final Bond Graph Fm FR Fc F I e1 Se f1 em fm eR 1 ec fc C fR R 1 ec f c dt K x m dt x m C Fc Kx m Simulation-Second order system Modelling of underwater Robotic systems Tx MSe Tbx MSe 1 MR Ixx+Iax I 1 wz*(Izz+Iaz)MGY Iyy+Iay I wy 1 MSe Ty MGY wy*(Iyy+Iay) MSe Tbz wz 1 I Izz+Iaz MR MSe Tz Fx MSe Linear velocity of the base point w.r.t Inertial frame MTF PV 1 Euler angle MTF Transformation matrix 1 Euler angles MTF TFMV 0 Vx m+maxI Body fixed linear velocity 1 MSe Tby 1 Body fixed angular velocity wx MGY wx*(Ixx+Iax) MR Angular velocity to first link of the manipulator wz*(m+maz) MGY MTF TFMV m+mayI MSe Fby Vy 1 MSe Fy Fbx MSe MR 1 MGY wy*(m+may) Vz MGY 1 wx*(m+max) MR MSe Fbz I MR MSe Fz m+maz Tip velocity 1 of the manipulator Link 3 221 Pad I TF TF1 Se 1 1 MTF Joint velocity TF 0 1 1 MTF 0 PV3 TFM3 MTF MTF PVM3 TFM3 Pad Se MR TF2 TF R AD L3 TF3 0 0 1 I m3+ma3 MTF Se I Se Link 2 221 TF1 1 TF 1 Tip velocity of Link2 1 MTF 0 TF MTF Joint velocity 0 1 1 TF2 Angular velocity from previous link TF R TF3 Pad PV2 TFM2 MTF MTF PVM2 TFM2 Pad 0 Se 0 1 I m2+ma2 AD L2 MTF MR Se Tip velocity of link1 1 MTF 0 Iz1+Iaz1 I Link1 1111as MR ffa11 Wz1*(Izz1+Iaz1) 1 Wx1 angular velocity of the manipulator angular velocity of the manipulator Wy1*(Iyy1+Iay1) MGY MGY 1 Wy1 MR 1 MGY 1 Wx1*(Ixx1+Iax1) I Wz1 MR Pad I Iy1+Iay1 MTF PV1 TFM1 MTF MTF PVM1 TFM1 Pad 0 Se 0 MR 1 I m1+ma1 Ix1+Iax1 I Se TF TF1 Se 1 1 1 Joint velocity TF 0 TF2 Angular velocity from previous link TF R TF3 MTF Tx Tbx MSe Ixx+Iax I 1 MSe 1 wx MR 1 wz*(Izz+Iaz) MGY Linear velocity of the base point w.r.t Inertial frame Angular velocity to first link of the manipulator Body fixed angular velocity MGY 1 MTF MTF PV TF 0 I wy 1 MGY wx*(Ixx+Iax) wz 1 I Izz+Iaz MR MR MSe Ty MSe Tbz MSe Tby 1 Euler angle MTF Transformation matrix MSe MTF 1 Euler angles 1 MR MGY TF m+may I Vy 1 MGY MR MSe MGY wx*(m+max) MSe Fby Fy Tz MSe wz*(m+maz) wy*(Iyy+Iay) Iyy+Iay Fbx Vx m+max I Body fixed linear velocity Fx MSe wy*(m+may) Vz 1 MR MSe Fbz I m+maz MSe Fz 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. 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. 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.