MODELING OF TRANSDUCERS WITH BOND GRAPHS

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MODELING OF TRANSDUCERS WITH BOND GRAPHS

S. Hanish

To cite this version:

S. Hanish. MODELING OF TRANSDUCERS WITH BOND GRAPHS. Journal de Physique Collo-

ques, 1990, 51 (C2), pp.C2-559-C2-562. �10.1051/jphyscol:19902131�. �jpa-00230426�

COLLOQUE DE PHYSIQUE

Colloque C2, supplkment au n02, Tome 51, Fgvrier 1990 ler Congres Franqais dlAcoustique 1990

MODELING OF TRANSDUCERS WITH BOND GRAPHS

S. H A N I S H

U.S. Naval Research Laboratory, Washington D C 20375, U.S.A.

This paper is devoted to an expositio~~ of the basic ele~nents of bond graph theory. Examples of bond graph modeling of selected acoustic transducers are preseated. By them it is show11 that bond graph mod- eling has notable advantages over the col~ventio~~al procedure of using electrical aaalogies.

1 Introduction

An acoustic transducer is a device that accepts input power in a variety of forms (electrical,mechanical,hydraulic,pneumatic,thermal) and delivers out- put power in acoustic form (or vice versa). These transducers, complicated in inner structure though they lnay be, can be lnodeled in first approxil~ia- tion as siniple input/outoul systems,viz.

power source i n -+ junction structure -+ load out

Power sources are of several kinds: an effort source S,,which delivers a con- stant magnitude of effort regardless of the junction structure; a flow source Sf which delivers a constant magnitude of flow, regardless of the junction structure; an illipedance sourceSimp which delivers power at a constant ra- tio of effort to flow; and an admittance source S,d,which delivers power at a constant ratio of flow to effort. Junction structures are of two kinds: a

"zero-junction" which acts as a point of algebraic flow summation at con- stant effort; and a "one-junction" which acts as a point of algebraic effort summation at constant flow. Loads are of three kinds: (1) capacitances C~,namely those loads that store effort; inertances 1,nanlely those loads that store flow; (3) resistances R, narnely those loads that dissipate power. In bond graph theory,all of these elementary pieces of the input/output system are joined together (i.e."bonded) by straight lines, each tipped with a half arrow head to indicate direction of power flow. The rules governing the join- ing of loads to junction structures fornls a major part of the theory. This is discussed next.

2 Graphs of junction structures and their loads

2.1 Luiilped para~llete~ loads

A load bonded to a junction signifies a mathelnatical relation of effort to flow on the joining bond. When the load receives an effort it responds by sending back a. flow. This excJiange is indicated by placing a perpendicular stroke on the bond adjacent to the load. When the load receives a flow, it responds by sending back an effort. This exchange is indicated by placing the perpen- dicular stroke on the bond adjacent to the junction, As applied to elements C:,I,R the corresponding graphs take the following forms: (1) capacitance loading,relating two flows t o one effort,(a)(b),or relating two efforts to one flow,(c)(d). ( 2 ) inertance loading,relating two efforts to one flow,(e)(f),or re- lating two flows to one effort,(g)(h). (3) resistive loading,relating two flows to one effort,(i)(j),or relating two efforts to one flow,(k)(l).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19902131

C2-560 COLLOQUE DE PHYSIQUE

2.1.1 C,I,R fields

A load striictiire in which one oiit,put power variable is a function of several input power variables is called a field. Ele~ilents C!,I,R can forni fields: ( I ) a C-field is a load structure in which a single output effort is a function of several input integrated flows (a),or a single output flow is a function of sev- eral input differentiated efforts (1)).(2) an I-field is a load structure in which a single output flow is a fiu~ction of several input integrated efforts (c),or a single output effort is a function of several input differentiated flows(d). ( 3 ) a R-field is a load structure in which a single output effort is a function of several input flows (e),or a single output flow is a function of several input efforts(f).

2.1.2 Tra~lsfor~nation of power variables OII a bond

The relation of power variables on a bond can be changed by transform- ers(TF) and gyrators (GY). These are coded each in two for~ns:(a)(b) for

transtormers and (c)(d) for gyrators.

Here

a p ,

2.1.3 Multiport transformers

A dynamical relation ainong power variables in a multiport representation of a transducer that leads to a set of sili~ultlrneous equations is coded in a n~ultiport transformer,

2.2 Distributed parameter loads

These are dynamical loads that can he expanded in fans of lumped pa- rameter resonant or antiresonant modes.They are of four kinds: (1) "one- one"fans,signifying expansion at constant flow in a fan of antiresonant modes.

(2)"one-zeroVfans,signifying expansion at constant flow in a fan of resonant

~ ~ ~ o d e s . (3) "zero-one"fans,signifying expansion at constant effort in a fan of antiresonant modes. (4) "zero-zero" fans, signifying expansion at constant effort in a fan of resonant modes.

Expansions (l)(2)are called,for convenience,a 2-1oad;expansions (3)(4), a Y-1oad.Their graphs are,

2.2.1 Coupled modes

Coupled modes are best explained by two examples (a)(b): In (a)the two modes shown are coupled by capacitance; in (b),by inertance.

cq ^0-0

\ /

I I -1 I

3 Bond graphs of elementary mecanical struc- tures

3.1 A single mass/single spring systern

C2-562 COLLOQUE DE PHYSIQUE

Bonds 1,5 are unoccupied. We occupy bond 5 by driving the Illass with a force source (SF). Leaving bond 1 unoccupied,we elinlinate it and coalesce bonds 2,3: Alternatively,we occupy bond 1 by driving the spring with a.

velocity source (S,). Leaving bolld 5 unoccupied,we eliminate it and coalesce l,onds 4,5:

The equations of lnotion are obtained by swnnling forces a t the one junctions. Tile equations of velocity constraint are obtained by sulnlning velocity a t the 0-junction.

4 Bond graph of a piezoelectric underwater sound projector

The projector is desgned to operate in a single longitudinal lnode at reso- nance: A possible bond graph is obtained by inspection.

Q

8

c, p

E ⁰

0

1. n~ass,head end cap; 2. head neck,compliance and damper; 3. ra- diation load,mass and resistance; 4. housing seal,con~pliance and damper;

5. external support,compliance; 6. friction,housing/ball~t mass; 7. tail mass; 8. oil fill,compliance and damper; 9. mass,tail end cap; 10.21 = jZotankl/2; 11.Z2 = Zo(l/jainkl);12. Z.,electrical impedance; 13. TF, electronlechanical transformner; 14. S,, electrical source voltage. 15. bolt, con~pliance ; 16. housing, compliance.

Reference

[ I ] " I n t r o d u c t i o n t o P h y s i c a l System Dynamics". R. ^C. Rosenberg, D. Karnopp, McGraw B i l l , New York, 1983.