Principles of Engineering System Design

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

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

3. 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.

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

5. p = e · dt 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) q = f · dt

6. e: Effort f: Flow e P = e · f f 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.

7. Modeling: Bond Graph Basics • effort/flow definitions in different engineering domains

8. I C R e1 e2 e2 = 1/m*e1 TF f1 = 1/m*f2 f1 f2 m e1 e2 f2 = 1/d*e1 GY f1 = 1/d*e2 f1 f2 d SF SE Modeling: Bond Graph Basic Elements • 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.

9. 5 4 0 1 1 11 3 13 2 12 Efforts are equal e1 = e2 = e3 = e4 = e5 Flows are equal f11 = f12 = f13 Flows sum to zero f1+ f2 = f3 + f4 + f5 Efforts sum to zero e11+ e12 = e13 Modeling: Bond Graph Basic Elements • Power Bonds Connect at Junctions. • There are two types of junctions, 0 and 1.

10. e f 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. • Mandatory Causality ( Sources, TF, GY, 0 and 1 Junctions) • Desired Causality (C and I elements) • Free Causality (R element)

11. The flow has to be computed on the right side. Se i 0 The source computes the effort. U u I 0 Sf The source computes the flow.  The causality of the sources is fixed. “Causalization” of the Sources U0 = f(t) I0 = f(t)

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

13. e e I f f 1 1 e sI sC e C f f Integral Causality (desired Causality) Integral causality is preferred when given a choice.

14. e2 e2 e2 = e1 e3 = e1 f1 = f2+ f3 f2 f2  e1 e1 1 0 e3 e3 f1 f1 f3 f3 f2= f1 f3 = f1 e1 = e2+ e3  “Causalization” of the Junctions 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.

15. Modelling Example Mechanical Systems Mass, Spring and Damper Syetms R F M C FR F Fm Equation Governing the system Fk

16. Bond Graph model Velocity Junction Reference Velocity=0 for this case Mass Spring Damper System Equations Final Bond Graph em fm e1 eR fR f1 ec fc

17. Simulation-Second order system

18. Modelling of underwater Robotic systems

19. Fx Fbx Tx Tbx Linear velocity of the base MSe Angular velocity to MSe 1 MSe point w.r.t MSe 1 first link of the Inertial frame manipulator MR MR Vx 1 I m+max Body fixed Ixx+Iax Body fixed I 1 wx linear angular velocity velocity wz*(m+maz) MTF 1 MTF 0 MTF wy*(m+may) MGY MGY MGY TFMV wz*(Izz+Iaz) MGY TFMV PV wy*(Iyy+Iay) Vy Vz wz 1 wy m+may I 1 MGY 1 I m+maz Iyy+Iay I 1 MGY 1 I Izz+Iaz wx*(m+max) wx*(Ixx+Iax) Euler angle MR MSe MR MTF Transformation MR MSe MR Fby MSe matrix MSe MSe Fbz MSe MSe MSe Tby Fz Ty Tbz Fy Tz 1 ò Euler angles

20. Tip velocity 1 of the manipulator Link 3 221 Pad I TF MTF MTF 0 TF1 Pad Joint velocity Se PV3 TFM3 Se 1 1 TF 0 1 1 MR TF2 TF MTF MTF 0 0 1 I AD R TF3 L3 PVM3 TFM3 m3+ma3 MTF Se 1 Tip velocity of Link2 Link 2 221 I TF MTF MTF 0 TF1 Pad Joint velocity Se PV2 TFM2 Se 1 1 TF 0 1 1 MR TF2 TF Angular velocity Pad MTF MTF 0 0 1 I R from previous link TF3 PVM2 TFM2 m2+ma2 AD MTF L2 Se Tip velocity 1 of link1 Iz1+Iaz1 Link1 1111asffa11 I Wx1 MR 1 angular velocity of the manipulator angular velocity of the manipulator Wz1*(Izz1+Iaz1) Wy1*(Iyy1+Iay1) MTF MTF 0 Pad MGY MGY Se PV1 TFM1 1 1 MR Wy1 MR MR 1 MGY 1 Pad MTF MTF 0 0 1 I Wx1*(Ixx1+Iax1) Wz1 PVM1 TFM1 m1+ma1 I I Iy1+Iay1 Ix1+Iax1 Se I TF TF1 Joint velocity Se 1 1 TF 0 TF2 Angular velocity TF from previous link R TF3 MTF Fx Fbx Tx Tbx MSe MSe 1 1 MSe MSe Linear velocity of the base Angular velocity to point w.r.t first link of the Vx Inertial frame manipulator I 1 MR m+max Body fixed Ixx+Iax Body fixed I 1 MR wx linear angular velocity velocity wz*(m+maz) 1 MTF MTF 0 MTF wy*(m+may) MGY MGY MGY TF wz*(Izz+Iaz) MGY TF PV wy*(Iyy+Iay) Vy Vz 1 wz wy m+may I 1 MGY 1 I m+maz Iyy+Iay I 1 MGY 1 I Izz+Iaz wx*(m+max) wx*(Ixx+Iax) Euler angle MR MSe MR MTF Transformation MR MSe MR Fby MSe matrix MSe MSe Fbz MSe MSe MSe Tby Fz Ty Tbz Fy Tz 1 ò Euler angles

21. 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.

22. 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.

23. 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.