Bond Graph Modeling Hydraulic systems Valve 1 R2R2R2R2 Pump P PRPR P0P0 L’Ecole Normale Supérieure de l‘ Enseignement Technique

Table of Contents I.INTRODUCTION II.BOND GRAPH ELEMENTS III. CAUSALITIES VI. EXAMPLES V. CONCLUSION  Passive elements  Active elements  Junctions

Bond graphs are a domain-independent graphical description of dynamic behavior of physical systems. This means that systems from different domains (cf. electrical, mechanical, hydraulic, acoustical, thermodynamic, material) are described in the same way. The basis is that bond graphs are based on energy and energy exchange.. Bond graphs for modelling and more… Because of its architectural representation, causal and structural properties, bond graph modelling is used not only for modelling but for :  Control analysis, diagnosis, supervision, alarm filtering  Automatic generation of dynamic modelling and supervision algorithms  Sizing  Used today by industrial companies (PSA, Renault, EDF, IFP, CEA, Airbus,…). I.INTRODUCTION

Notations (Pa) (m 3 /s) (Pa.s) (m 3 ) (W) (J) (Pa.s/m 3 ) (Pa.s 2 /m 3 ) (m 3 /Pa)

II. BOND GRAPH ELEMENTS ACTIVE ELEMENTS Generate and Provide a power to the system SfSe One port element R,C,I, Se,Sf 0,1 Tree ports element BOND GRAPH ELEMENTS PASSIVE ELEMENTS (transform received power into dissipated (R) or stored (C, I) energy RCI TF, GY Two ports element JUNCTIONS Connect different elements of the systems : are power conserving TF, GY0,1 They are not a material point (common effort (0) and common flow ((1) Energy transformation or transformation from one domaine to another

Passive elements Representation Definition The bond graph elements are called passive because they transform received power into dissipated power (R-element), stored under potential energy (C-element) or kinetic (I-element).

Relement A) R element ( hydraulic restriction) Representation p1p1 p2p2 C element (capacitance) Examples: B) C element (capacitance) Examples: tank, h p I element: c) I element: Inertance p1p1 p2p2 l

ACTIVATED ELEMENTS EFFORT AND FLOW SOURCES Se, Sf A source maintains one of power variables constant or a specified function of time no matter how large the other variable may be. Effort source Se 1. Effort source Se Generator of voltage, gravity force… f Se e Se = e(t) f MSe e Modulated effort source Modulated effort source u Se = e(t,u) Simple effort source Flow source Sf 2. Flow source Sf Current generator, applied velocity..

0 - JUNCTION “ Common effort junction” 0 - JUNCTION “ Common effort junction” Power conservation a i = +1 if 0 a i = -1 if 0 0 e1 e2 e3 e4 f1 f2f2 f4 Representation Representation f3 JUNCTIONS Defining relation Example of 0-junction

1 e1e1 e2e2 e3e3 e4e4 f1f1 f2f2 f4f4 1 - JUNCTION “ Common flow junction” 1 - JUNCTION “ Common flow junction” Representation Representation Defining relation Power conservation a i = +1 if 0 a i = -1 if 0 JUNCTIONS f3 Examples of 1-junction

III. CAUSALITIES Definition direction of the efforts and flows Causal analysis is the determination of the direction of the efforts and flows in a BG model. The result is a causal BG which can be considered as a compact block diagram. From causal BG we can directly derive an equivalent block diagram. It is algorithmic level of the modeling.

Convention A B e A B System A impose an effort e to system B e The causal stroke is placed near (respectively far from) the bond graph element for which the effort (respectively flow) in known. f Cause effect relation : effort pushes, response is a flow Indicated by causal stroke on a bond Effort pushes Flow points

Remarks about causalities  the orientation of the half arrow and the position of the causal stroke are independent e f A B e f A B A B e System A impose effort e to B A B f System A impose flow f to B e f

Integral and derivative causality Preferred (integral) Preferred (integral) causality causality f C e f eI e f C e f e fI f e Derivative

Causalities for 1-junction Only 1 bond without causal stroke near 1 - junction   Rule 1 e1e1 e2e2 e3e3 e4e4 f1f1 f2f2 f4f4 f3f3 Causal Bond Graph model Strong bond 1-Junction e1e1 e4e4 e3e3 f2f2 Block diagram

Causalities for 0-junction Only 1 causal stroke near 0 - junction   Rule 0 e1e1 e3e3 e4e4 f1f1 e2e2 f4f4 f3f3 f2f2 0-Junction f1f1 f4f4 f3f3 e2e2 Block diagram Strong bond

Hydraulic Bond Graphs In hydrology, the two adjugate variables are the pressure p and the volume flow q. Here, the pressure is considered the potential variable, whereas the volume flow plays the role of the flow variable. The capacitive storage describes the compressibility of the fluid as a function of the pressure, whereas the inductive storage models the inertia of the fluid in motion. P hydr = p · q [W] = [N/ m 2 ] · [m 3 / s] = kg · m -1 · s -2 ] · [m 3 · s -1 ] = [kg · m 2 · s -3 ]

Hydraulic Bond Graphs q in q out p dp dt = c · ( q in – q out ) p qq C : 1/c Compression: q  = k ·  p = k · ( p 1 – p 2 ) p1p1 Laminar Flow: q p2p2 pp q R : 1/k Turbulent Flow: pp q G : k p2p2 p1p1 q Hydro q  = k · sign(  p) ·   |  p|

Energy Conversion Beside from the elements that have been considered so far to describe the storage of energy ( C and I ) as well as its dissipation (conversion to heat) ( R ), two additional elements are needed, which describe the general energy conversion, namely the Transformer and the Gyrator. Whereas resistors describe the irreversible conversion of free energy into heat, transformers and gyrators are used to model reversible energy conversion phenomena between identical or different forms of energy.

Transformers f 1 e 1 f 2 e 2 TF m Transformation: e 1 = m · e 2 Energy Conservation: e 1 · f 1 = e 2 · f 2  (m ·e 2 ) · f 1 = e 2 · f 2  f 2 = m · f 1 (4) (3) (2) (1)  The transformer may either be described by means of equations (1) and (2) or using equations (1) and (4).

The Causality of the Transformer f 1 e 1 f 2 e 2 TF m e 1 = m · e 2 f 2 = m · f 1 f 1 e 1 f 2 e 2 TF m e 2 = e 1 / m f 1 = f 2 / m  As we have exactly one equation for the effort and another for the flow, it is mandatory that the transformer compute one effort variable and one flow variable. Hence there is one causality stroke at the TF element. Hydraulic Shock Absorber m = A

BUILDING HYDRAULIC MODELS 1.Fix for the fluid a power direction 2.For each distinct pressure establish a 0-junction (usually there are tank, compressibility, ….) 3.Place a 1-junction between two 0-junctions and attach to this junction components submitted to the pressure difference 4.Add pressure and flow sources 5.Assign power directions 6.Define all pressures relative to reference (usually atmospheric) pressure, and eliminate the reference 0-junction and its bonds 7.Simplify the bond graph

Inertia I Resistance R 1 Resistance R 2 Pump P1P1 P2P2 P3P3 P4P4 PatPat Se:P 1 0 P1P1 0 P2P2 0 P3P3 0 P4P4 PatPat R:R 1 I 1 R:R 2 C C Se:P 1 1 R:R 1 11 IR:R 2 0 C Se:P 1 R:R 1 1 I R:R 2 C VI. EXAMPLES Exemple(1)

Exemple(2 ) 0 C:C R R:R 1 Se:P PP I :  l/A 1 P P -P R PRPR PRPR Se:-P 0 P0P0 1 R:R 2 P R -P 0 Valve 1 R2R2R2R2 Pump P PRPR P0P0

CONCLUSION Why Bond graph is well suited Modelling Unified representation language Shows up explicitly the power flows Makes possible the energetic study Structures the modeling procedure Makes easier the dialog between specialists of differents physical domains Makes simpler the building of models for multi-disiplinary systems Shows up explicitly the cause - to efect relations (causality) Leads to a systematic writing of mathematical models (linear or non linear associate