Modeling and simulation of large-scale, linear, multiport systems

Automatica, Vol. 9, pp. 87-95. Pergamon Press, 1973. Printed in Great Britain.

Modeling and Simulation of Large-Scale, Linear, Multiport Systems* Modelage et Simulation de Systbmes lin6aires, Multi-Orifice, de grande Echelle Modellierung und Simulation von umfangreichen Linearen multi6ffnungssystemen

Mo,zIe~HpoaanriemIapOroMacmTa6nofiJIHHeHHOH

CklCTeMBI C MHOFOqHC.rleHHBIMI, I

Bxo~aMI4 R O N A L D C. R O S E N B E R G t

Bond graph models of linear multiport systems may be simulated by a program, ENPORT, which features a choice of physically-significant state variables and an ability to simulate systems containing static storage subfields. Summary--A large number of physical and engineering systems may be represented directly in terms of component energy characteristics and their power interactions. When the system elements are modeled as energetic multiports, and their interconnections by power bonds, then the bond graph language is a natural one for describing the entire system. Bond graphs may be written for dynamic systems involving various energy types, such as electrical, mechanical, fluid and thermal; all energy types may be coexistent. Useful modeling elements include multiport storages, dissipators, and junction elements and transducers, as well as sources. Bond graph models of linear multiport systems may be transformed to state-space form by a powerful algorithm based upon operational causality. From the state-space equations, dynamic responses may be obtained by the matrix exponential technique, thereby allowing the direct digital simulation of linear multiport models. The ENPORT program is a realization of the bond graph reduction algorithm. It is a principal purpose of this paper to describe the procedure upon which ENPORT is based, and to present some results. Important features of ENPORT are its choice of physically significant state variables, its use of operational causality to obtain an orderly formulation of system equations, and its ability to handle systems containing static storage subfields.

vibrations in a rocket or similar u n t o w a r d effects. To maintain control over these creations that is as effective as possible, it is necessary to develop tools for modeling, analysis, and simulation equal to the challenge. This paper reports on a novel t o o l - - t h e E N P O R T p r o g r a m - - d e s i g n e d to achieve for the engineer an effective partnership with the computer in the study o f large-scale, linear, multiport systems. The three principal steps to be taken by the engineer-computer combination are: (1) to develop a mathematical model in some suitable form; (2) to select a formulation, typically in terms o f state-space differential equations; and (3) to effect a calculation based on the formulation that yields the dynamic behavior o f desired response variables. In the procedure outlined in this paper the first step is taken by the engineer, who develops a multiport in b o n d graph terms [1]. A brief description o f such models is given in the next section. The next two steps are performed by the computer, which selects state variables o f physical significance, formulates a set o f state-space equations, and calculates the response indicated by the formulation. The third section o f the paper describes the selection, formulation, and calculation procedure; and the fourth section shows two examples o f the use o f the program. The first example illustrates the power o f the method in studying potentially complex transducer systems with multiport fields. The second example shows how systems containing statically-constrained energy variables are treated.

INTRODUCTION EACH new year brings an increase in the size and complexity o f the physical and engineering systems collectively conceived, designed, analyzed and implemented. On occasion there are possible behaviors o f a system that have escaped notice in the preliminary stages only to emerge in operation, producing electric power blackout or unwanted * Received 20 December 1971 ; revised 7 June •972. The original version of this paper was presented at the IFAC Symposium on Digital Simulation of Continuous Processes (DISCOP) which was held in Gyt)r, Hungary during September 1971. It was recommended for publication in revised form by Associate Editor L. Meier . . . . . t Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan 48823. 87

88

RONALD C. ROSENBERG M U L T I P O R T M O D E L S A N D BOND G R A P H S

A large number of physical and engineering systems may be represented directly in terms of component energy characteristics and their power interactions. When the system elements are modeled as energetic multiports and their interconnections by power bonds, then the bond graph language is a natural one for describing the entire system. Bond graphs may be written for dynamic systems involving various energy types, such as electrical, mechanical, fluid and thermal; all energy types may be coexistent. Useful modeling elements include multiport storages, dissipators, and junction elements and transducers, as well as sources. There are nine basic multiports useful in a wide variety of physical and engineering applications. In order to describe them succinctly it is necessary to define four generalized power-energy variables, as follows: (1) effort, examples of which are voltage, force, torque, pressure and temperature; (2) flow, examples of which are current, velocity, angular velocity, volume flow and heat flow; (3) momentum, the time integral of effort, examples of which are flux linkage, linear momentum, angular momentum and pressure momentum; and (4) displacement, the time integral of flow, examples of which are charge, displacement, rotation, volume and heat energy. These variables are denoted by e(effort), f(flow), p (momentum) and q (displacement). The bond is used to denote appropriate pairs of effort and flow variables, whose product gives a scalar power. Several examples of particular bonds are /3

electrical circuit:

-----r

i F mechanical translation: V

The relations implied by the bonds above are Pelectrical(t) = v(t)*

i(t),

Pmeeh. t . . . . l.(t) =

F(t)* v(t),

Pmech. rot.(t) =

Z(t)* ~ ( t ) ,

Pflold(t) =

P(t)* Q(t),

and Phe,t, ..... = T(t)* Q(t). The half-arrow is used to denote the reference direction for positive power. The heat transfer bond is a pseudo-energetic one, but it is very useful in conduction heat transfer, and so is included. The nine basic multiports are classified according to their energy characteristics as follows: (1) there are two conservative storage elements, the inertance, written I,--, defined by p=I*f; and the capacitance, written C A, defined by q = C*e. Examples of I element behavior include mechanical inertia, electrical inductance, and fluid inertia. Examples of C element behavior include mechanical compliance, electrical capacitance, fluid capacitance, both storage and compressibility, and thermal capacitance. (2) there is one dissipation element, the resistance written R,---, defined by e =R*f Examples of R element behavior include mechanical dissipation due to friction, electrical resistance, viscous fluid losses, and thermal resistance to heat flow. (3) there are two source elements, the effort source, written S E T , defined by e=e(t); and the .[tow source, written SF-,, defined by f=l(t). Examples of effort source (SE) behavior include force sources, e.g. gravity, torque sources, voltage sources, pressure sources, e.g. reservoir, and temperature "sources" or conditions imposed on the system. Flow source (SF)behavior is represented by velocity sources, e.g. a shake table, current sources, fluid flow sources, e.g. a positive displacement pump, and heat sources, e.g. electrically-resistive heating. Sources are always approximate models of the environment of a system, and must be used with care. (4) there are two ideal two-ports, the gyrator, written

mechanical rotation:

1

2

-,GY-,, defined by e l = r ' f 2 and e2=r*J~; and the trans-

P

former, written

fluid circuit:

l

Q T heat transfer:

.

Q

2

-,TF-,, defined by el=m'e2 and f2=m*J~. Examples of transformer (TF) behavior are ideal mechanical levers and gear pairs, ideal electrical transformer

Modeling and simulation of large-scale, linear, multiport systems action, and ideal fluid area-change effects, without losses. Gyrator (GY) behavior shows up principally as ideal power transduction, e.g. electro-mechanical motors and fluid-mechanical pumps, between different energy domains, and in rigid body mechanics, representing gyrational coupling, see, for example, Ref. [1], p. 72 ft. (5) there are two ideal junctions, the flow

junction, written 1 ~ 0

3 ~

,

2I defined by e l = e 2 = e 3 and fl+f2=f3; and the effort junction, written 1

3

----~ 1 -----r

2t defined by J'l =f2 =fa and el + e2 = e3. Effort and flow junctions represent the most abstract phase of multiport system behavior; namely, the structure or coupling effects. Examples of flow junctions (O's) are electrical parallel connections where the voltage is common, the currents sum to zero, and hydraulic pipe tees where pressure is common, the flows sum to zero. Examples of effort junctions (l's) include electrical series connections where current is common to all ports, voltages add to zero and mechanical elements joined such that their motions are common, and their forces add to zero. It should be noted that only the linear, constantcoefficient form of each element has been presented here. Each of the elements I, C, R, G Y and TF has a general nonlinear form. Also, only the oneport form of the 1, C and R elements has been shown, although each can have n ports. In such a case the element is characterized by an n-by-n matrix. Each of the ideal junctions, 0 and 1, may have n ports, in which case the appropriate identity relation is repeated ( n - 1 ) times, and the summation relation has n terms. Any element with more than one port can join more than one energy domain, although the most common case has the gyrator or transformer as the coupling element. By considering that a set of bonds can represent several distinct energy domains, it becomes apparent that the bond graph can be useful in making unified graphical representations of complex engineering systems. The rest of this paper discusses the simulation of bond graph models, but more on the modeling aspect is to be found in the literature. [1-4]. In addition, a concise definition of the bond graph language is presented as part of a special collection on bond graph modeling [9], to which the interested reader is referred.

89

SIMULATION OF LINEAR, CONSTANTCOEFFICIENT MULTIPORT SYSTEMS

Bond graph models of linear multiport systems may be transformed to state-space form by a powerful algorithm based upon operational causality [5]. From the state-space equations, dynamic responses may be obtained by the matrix exponential calculation technique, thereby allowing the direct digital simulation of linear multiport models. Operational causality is a concept of assigning a sense of"cause-effect" or "input-output" to each of the signals (i.e. efforts and flows) in the system. This is done in bond graphs in a unique fashion, in that each bond is marked with a single stroke, showing the causal sense of both signals on that bond simultaneously. The two signals are directed in opposite senses, thus establishing a bilateral actionHreaction pattern as shown below. All relations among signals may be organized compatibly before any relations are written by assigning causality to the graph as a whole. e

e

A--IB implies IAI~IBI f

f

e

A-B f

acausal bond e

e

AI--B implies IAI~-IB I . f

eau sal bonds

f

signal pairs

The ENPORT program is a realization of the bond graph reduction algorithm applied to linear, constant-coefficient systems. The principal feature of the design is the use of system vectors and matrices to make a complete initial formulation, which is subsequently reduced to the desired statespace form. All aspects of the formulation and reduction procedure may be anticipated by the proper interpretation of operational causality applied to the bond graph before equations are generated. For example, static constraints among energy variables will show up as causality is assigned. In following a description of the system design it may be helpful to keep in mind the picture shown in Fig. 1. Part (a) is a symbolic bond graph representation of the major fields of a multiport system; namely, the independent and dependent storage fields, the source field, the dissipation field, and the junction structure. Notice that each basic multiport element belongs to only one field except for C and I elements. In part (b) of Fig. I, two complementary vectors are associated with each field, one vector of inputs and one vector of outputs.

90

RONALD C. ROSENBERG ,4},'

Sit

I I i

%, ,:1 v c e

Yield

~/' , z

_-A,

I~Ht
i

L

I ~1

l,

]I,

Lib

- T .

. ~

g 1<

.

.

I)issi],a!;,

J,

the junction structure matrix;

L,

the dissipation matrix; and

S,

the storage field matrix which is subsequently partitioned into subarrays for the independent and dependent fields.

)

2_

[l?del)enr/~ Ht hlorauc . . i ic]d

?

__~-I l

I ~

The principal matrices are:

q-

l :/ ! ' i e l d

]

--7

!

I ) e p ( n c h t:t Storage } ie]c]

± R

With these vectors and matrices in mind, consider the basic ENPORT procedure described in Fig. 2. The first three blocks are the heart of the preliminary procedure, which allow the subsequent operations to be done in straightforward fashion. In particular, it is the results of block (3) that permit the selection and definition of the various system vectors in block (4), from which all else follows. In block (5) the equations for all of the elements of the junction structure are generated in mutuallycompatible fashion, and they are reduced to a system matrix expressing junction outputs in terms of junction inputs. Block (6) uses the relation for each field element (/, C, R, P) in a form compatible with the rest of the system equations to generate the field matrices. (:)

FIG. 1. Principal fields in linear multiport systems. (a) symbolic bond graph representation, (b) vectors associated with the fields.

r

Cz)

P, eacl t h u b,,nd g r a p h d a t a . O [ ~ a n i z ~ t]l~ ~ r , q ~lrucltlt',r. t t

i Classib i

defhl,~

time derivative of the independent energy variables (X 3;

i

5L

i

~

'i

i

-

i

Formulate f i e l d (t]u;ttil,,%:S hi;" s t . r a e ( " and dissipation IRI fields

C6)

!

I

.

.

.

Ii

.

foartition eliminate

L . . . . .......

C8)

I I

C9)

I

foartition eliminate • ...... f°rint

i

"

Zd"

(t:,

I)

i i

- I Z i,

the un
i t

J

!

. . . .

the jtm~ti,m matrix, and l the dependent ~ariables. ] energy ....... J [ _

the resull~: s t a t e v e e r , r , X: s , u r :~ ~ ( , t,, -, t ;

?,L, r ' . E'

/

i

(10)

''

I i J

--J---

.

U)

'

J

}, o l ' n ~ - u l a t e t h e i H n L t i H n M p H , t'~t'c e ( ~tatt,,l~5, and ~ t a i a a r c, "e i l ,,n r \. • :

Evaluate

tin, matriza:~t,

x

~,xi,,~ ~ I, L/

[ .....'~2'"""'"!2: " |

r ,:

J

.

] k'

I

\

~

,,i,h

r I

I

FIG. 2. The basic ENPORT procedure.

Z , the independent coenergy variables; )(~, time derivative of the dependent energy variables (X~); Z~, the dependent coenergy variables;

2

e a c h b,uwl, a n d th,~ s , , , s t c ~ ~ , ' * l , , r s ,

];i',al~at,

2,

ahd

I

(4)

The input and output vectors are ordered corresponding to the bonds from which they come. The scalar product of the vectors gives the power associated with the field. For the junction structure the input vector is the composite of the field output vectors, and the junction output vector becomes the input vectors to the various fields. Each field and the junction structure is characterized by a matrix, whose size is governed by the number of field ports. The most general form of constantcoefficient linear system can be treated by the current procedure, and time-varying systems can be handled by an extension to the program. The principal vectors are:

b , , n d ~rap~ ~::Hd, [ I>nv;H!l,l,""

:"~

U, the source vector; V, the source complement vector, of no significance analytically;

Blocks (7) and (8) use matrix partitioning and inversion to eliminate unwanted vectors until only the independent energy vector, which is now the state vector, the source vector, and possibly the source vector time derivative, remain, related by the A, B and E matrices. The system equations are

Di, input vector to the dissipation field; and

nOW :

Do, output vector from the dissipation field.

.,~ = A'X, + B* U + E* 0.

(1)

Modeling and simulation of large-scale, linear, multiport systems One way to calculate the time response from equation (1) is to use the well-known matrix exponential method as discussed, for example, in TotJ [6]. Block (10) indicates that a time interval,x, must be specified; then the M and N matrices may be evaluated. When the initial condition vector, X(0), and the source vector, U(t), are given, the results may be marched out at intervals of z, as block (11) indicates. In reviewing the procedure described in Fig. 2, it is important to note that the only information the person supplies to the computer is that required to make a physically-meaningful and complete problem statement, with the exception of the time interval which could be chosen internally. The necessary data includes the bond graph model, the element parameters, the source vector, and the initial conditions. This type of direct simulation capability has been available to designers of electric circuits for some time; for example, see BASHKOW [7]. It is now available to designers of complex engineering systems working in a variety of energy domains. TWO EXAMPLES OF ENPORT SIMULATION An illustration of the type of multiport system that can be studied using bond graphs and ENPORT is given in Fig. 3. Part (a) shows a beam-block transducer system that involves electrical and mechanical energies of several types. A bond graph model is shown in Fig. 3(b). Reading the graph from right to left, it says that power proceeds from the voltage source (SE7) through the electro-mechanical transduction (GY8 9) to the block. Electrical coil inductance (I 5)and resistance

~ block ~

_

coii

sour~e

pe r ~qanent magnet

(~)

c

'eft

1

91

(R6) are included. The block has two inertias, one in rotation about its center of mass (I3)and one in linear translation (14). Coupling between the beam and block occurs at two ports, bonds 1 and 2, and the beam is modeled as a two-port elastic field (C12). The motion of the block center-of-mass is related to the motion of the beam-block connection point by a lever-arm, associated with TF11 12. More information on the generation of this particular bond graph model is contained in Ref. [1], pp. 65 and 66. A more general discussion is to be found in Ref. [3]. (A new collection of bond graph models in a wide variety of applications is to be published in the Trans. ASME, Journal of Dynamic Systems, Measurement and Control in September 1972.) The information required for specifying the problem to ENPORT is shown in Fig. 4. All input data is marked with a vertical solidus on the left. The bond graph and the constant parameters are described in 12 lines, one per multiport element. The integers following each element are its incident bonds, and the parameter(s) implied by the element are listed next. Observe that the C1 2 element requires four parameters, which is actually a 2 × 2 array, being a two-port field. The G Y and TF elements each require only one parameter, because of their special definitions. The ideal junction elements 0 and 1 require no parameters. Given the data just described, ENPORT proceeds to analyze the graph structure, assign power directions to the bonds, and define the system vectors. For example, the independent energy variables are Q1 (vertical beam displacement), Q2 (beam rotation), P3 (angular momentum of the block), P4 (vertical momentum of the block) and P5 (flux linkage in the coil). This vector is also the state vector since all the variables are statically independent. The source variable vector contains one component, E7, the input voltage. The next results are the A and B matrices, at which point the current information is exhausted, as indicated in Fig. 4(b). Further input, again shown by a left vertical solidus in Fig. 4(b), includes a description of required output variables, as distinct from the particular vectors chosen by ENPORT, the time interval and initial conditions. In addition, the source characteristics are specified, by a data card if constant, or by a subroutine if time-varying. The particular data in the example implies the following conditions: (1) time from 0.0 to 10.0 in steps of 0.1;

7-

Z

Z

1~.

(b)

FIG. 3. A beam-block transducer system. (a) the beamblock transducer schematic, (b) bond graph model for (a).

(2) there is one output variable, the block displacement, constructed from the state variables and sources, as Yl = 1.0x 1+ 1.5x2 + 0.0u 1 .

92

RONALD C ROSENBERG

//XEQ EN3AI BEAM-BLOCK GRAPH

EX ELECTROMEGHANICAL 2

]

7. 0 1.0 2.0 3.0 6.0

3 4

I

7856 89 4910 lO 12 1 ll 12 Z311 7

i

GY 0 rF i SE GREND SETUP DONE TIIE GRAPH

}[AS

BOND

TRANSDUCER 4. 0

SYSTEM

SET-UP CONTINUES FOR BEAM-BLOCK EI,E{;TROMECHANH:AI,

-- ]6

4, "

9.0

THE o. o. 6. -7.

00 0. 0(]0E 00 0. 0 0 0 E 00 -3. 0 0 0 E

0{} 00 00

1. q(}0E 1. 0 0 0 E 0. 0 0 0 E

000E

00

00

o.

4. 0 0 0 E

000E 000E 000E O00E 1. 000E

0. 0. 0. 0.

12 N O D E S

9

l0 11 12

1

TF 0

0 1 TF

AND

12

~/XEQ DISK 5

BONDS.

00 00 00 00 00

SLDFQ 1

1

FX 1

0.1

0.(I

1,0 0.0 1o.o o. o

1.5

0.(}

0.0

0.0

~. o

o. o

o. o

o. o

l

STEADY-STATE 22 26(}6E 00

:' OF SYSTEM EQUATIONS ELECTROMEGHANICAL

THERE

ARE

BONDS 12 Ii

INTERNAL I0 9 8

BEGINS FOR TRANSDUCER

SYSTEM

- - 16.

Y \'ECTOR

7 FIELD

INDEPENDENT

TO

BONDS

AND

THE

JUNCTION

ENERGY

P3

IJi}

:L 3 2 2 3 E - 0 6

5.5355E-01

5 JUNCTION

YBLS

=

: 1: ;:

m

END

: ',::<: :::::::::::::::::::::: :::: b><<

TIME [}m oo()oo E 1. 0000010 3' 0 0 0 0 0 E • 00000E 4. 0 0 0 0 0 E q. 0 0 0 0 0 E 6. 0 0 0 0 0 E 7. 0 0 0 0 0 E 8. 0 0 0 0 0 E % 00000E

1

E6 S O U R C E %rBLS =

P , I , O C E I~RIN'F V A R I A B L E S

BONDS.

PS

P4

DISSIPATION VBI~

-

.'\ . . . P,E A M D I S P I , A C E M E N T B ... B E A M A N G L E (2 " " B L O C K D I S P L A C E M E N T

;:

STRUCTURE

5

=

IS

BEAM

;::

QZ

]0.

GY

::{

Q1

0(]0E

STEADY-STATE X V E C T O R IS q.319 E-0] I. 1 6 2 4 E 00 ;.]316E

SETUP FORMULATION BEAM-BLOCK

O0 2. d)(}t)/.: ,}t} 0, o~,oH {}c] 00 0• 0 0 U E r3rJ D. [[ll[]~] {ill 0{) I). (]CWE o0 {i. <]lJftI,] (H) 00 o, 0 0 0 E t}c) I. [ } I ) E :11

T H E ]3 IvIA'PR[X

R. I

8

000E 000E 500E

1.5

1 SE 1 GY

7

-- 16

5.0

POWER FROM TO 0 C 1 G 1 I 1 I

l z 3 4

T R A N S D 1 [;}H! ; Y S T E M

A MATRIX

1

c](] 00 t) 0 oo 00 oo 00 00 00 00

::: : :: <: : :< : ::<<:x

VECTOR (I! Cmm 0()00DE 0l) i. { t96ZE-O l ' m6 1 0 7 0 E - - 0 1 {. (>)G/6E-t}I 6. 06893E-(~I 4.71643E-(]] ~. q S 6 0 9 E - 0 ] S. g 0 0 1 7 E - 0 1 ~. ~ ] 0 0 E t;l 5. ~47 {GB-i)]

: b: : x<< : :

VECTOR (2) N. I)01)00E (]0 (. 77 i 1 7 E - 0 ] '. d T 2 ' ) S E - 0 ] 1. l q 3 2 8 E oo I. ()7483E 00 I. 18130}] oo 1. l Z 6 3 5 E 00 1. ] 6 2 8 3 E 00 [. 1 4 7 9 0 E Or) 1. 1 ; 2 4 7 E {}o

< :

:,: ' :: : :

~,E C T O R (I 0()000E i: ] 5 5 6 2 E l, ~qql) l E ,!, D } q 5 4 E ,'. 21')14E ,i, Z 4 5 ()0 E Z. 2 4 8 1 3 E 2. 2 6 4 2 7 E ?.. 2 q 4 q S E 2. lb ~4Sb]

({I 0(J 00 00 0o 00 (h] 00

00 (m r~{l

E7

Fio. 4. Sample ENPORT study of the beam-block problem. (3) the constant input voltage has a value of 10.0; and (4) all initial conditions are zero. With these data, E N P O R T calculates the steadystate results for the state-vector and the output vector. The data indicate that the beam deflects vertically and has a non-zero slope, the inertia stops moving, and the coil reaches a constant state of flux. A plot of some important geometric variables is given in Fig. 5.

mechanism. In Fig. 6(a) a simple lever mechanism is shown. A bond graph for the lever model is given in Fig. 6(b), in which a source of force (SE) pushes on an ideal lever (TF) to which an inertia (L) a spring (C,) and a damper (R) are connected. A simulation of this problem involves only a damped second-order system.

I ~

PLOT-* PLOT-* PLOI-I

O O P

TIME STEP 1,20E÷OO

I,O00E-OI

=

(a}

I

*~+++*++++ ++¢+ *÷

11 1

I

-3,00E-01

+

]

!b)

+¢+~*+++ ¢÷+++

+*+ *+

*

II

2,OOE-O1

*+ ÷÷+

*

7,OOE-Ol +

1'1' - -

"

I VERSUS T I M E , 2 VERSUS T I M E , 4 VERSUS l I M E ,

÷ I 1 I

SE F(t.

÷

***

111111111

~

~.+.+ I I ¢

*

***

** *~

***

***'~

*1

*

+

o*

o

....

:***o*

.....

II

IIIIi IIIIIIII--+--IIIII--*---IIIIIII IIIiii I

G

FIG. 5. Plot of selected results for the beam-block problem. {~}

A second example, intended to illustrate the automatic treatment of systems with staticallyconstrained energy variables, involves a lever

(,11

Fro. 6. A lever mechanism with inertia load. (a) lever mechanism model, (b) bond graph for (a), (c) mechanism model with load added, (d) bond graph for (c).

R

Modeling and simulation of large-scale, linear, multiport systems When an inertial load is put on the lever mechanism, as shown in Fig. 6(c), the corresponding bond graph has an inertia (•4) added, as in part (d). In this case it will turn out that the two inertias are statically-constrained. While this observation is made readily for such a simple problem, in a complicated engineering system, especially one assembled from several subsystems, such static constraints are not so easy to discover. ENPORT will discover all such implied constraints, inform the user, make a particular formulation, and obtain a solution. For an interesting discussion of the problem of sub-assemblies, see KOENIG et aL [8]. The input data needed to specify the system structure and constant parameters are shown in Fig. 7, to the right of the first vertical solidus. The graph structure is analyzed and reference power directions are assigned. In the identification of system vectors the independent energy variables are taken to be P2, the linear momentum of inertia 2, and Q3, the displacement of spring 3. This vector is the state vector. The other energy variable, P4, the momentum of inertia 4, is classified as dependent, meaning it is statically-related to the state

// X E Q E N 3 A I FX LEVER MECHANISM WITH REDUNDANT GRAPH I Z 2.0 C 3 8.0 I 4 3.0 R S Z. 0 1 1 2 6 TF 6 7 2, S SE I I 7435 GREND SETUP DONE THE GRAPH

HAS

8 NODES

AND

I N E R T I A S -- 17

vector. The source vector has one component, E l , the input force. Various intermediate results are available as the formulation proceeds. The influence matrix shown in Fig. 7 gives the values of the internal variables in the junction structure in terms of the junction structure input vector. The reduced junction matrix gives the values of the junction structure output vector in terms of the input vector. Finally the A, B and E matrices are calculated. The dimension of the A matrix, 2 x 2, indicates a state vector with two elements, which corresponds to the independent energy vector defined previously. When additional data is given, indicated by the lines to the right of the second vertical solidus in Fig. 7, further calculations are possible. The conditions specified are: (1) time from 0.0 to 7.5 in steps of 0.1; (2) there are no output variables other than the state variables; (3) the constant force source has a value of 10.0; and (4) the initial conditions are zero.

THE INELUENCE 0. 00 I. 00 Z. 50 0. 00 0. 00 Z. 50 l . O0 O. O0

MATRIX I. 00 0. 00 Z. 50 O. O0

IS .., I. 00 0. 00 Z. 50 O. O0

SET-UP CONTINUES LEVER MECHANISN[

FOR WITH REDUNDANT

7 BONDS.

POWER FROM SE 1 1 1 1 I ~F

6 7

-4.838E 00 5. 000E 00

TO I I C I i% TF

I. 935E-01 0. 000E 00

i

//

FORMULATION OF SYSTEM EQUATIONS LEVER MECHANISM WITH REDUNDANT

BONDS 7

INDEPENDENT P2

5 FIELD BONDS

INTERNAL 6

2 JUNCTION

BONDS.

XEQ

DI%E 0

SLDFQ

1

FX

i

1~:~o

o o

0.0

0.0

VBLS

VBLS

= Z

= 1

4 =

5 VBLS

75

STRUCTURE

Q3

SOURCE

E

BEGINS FOR I N E R T I A S -- 17,

STEADY-STATE X-VECTO1% 5. Z6~4E-Q7 4. 9999E-01

ENERGY

DISSIPATION VBLS E

AND

TO THE JUNCTION

-DEPENDENT'ENERGY P

-3.870E 00 0. 000E 00

THE B NiKTRIX

SETUP

THERE.ARE

0. 00 0. 00 0. 00 O. O0

T H E R E D U C E D J U N C T I O N M A T R I X IS ... 0. 0000 -Z. 5000 -2. 5000 -2. 5000 Z. 5000 0. 0000 0. 0000 0. 0000 Z. 5000 0. 0000 0. 0000 0. 0000 Z. 5000 0. 0000 0. 0000 0. 0000 l . 0000 O. 0000 O. 0000 O. 0000

THE A MATRIX" BOND I Z 3 4 S

93

= I

I

FIG. 7. Sample ENPORT study of the lever mechanism,

IS

I. 0000 0. 0000 0. 0000 0. 0000 O. 0000

I N E R T I A S -- 17

94

RONALD C. ROSENBERG

The steady-state results show that the momentum of inertia two approaches zero, and the spring deflection approaches 0.5. Finally, a plot of selected results is given in Fig. 8, clearly indicating the over-damped second-order nature of the mechanism response. PLOT-~ PLOT-*

P Q

TIME STEP = 9.60E-01

*

5,60E-01

I I I I * I I I

2 VERSUS TIME. 3 VERSUS TIME. 1.OOOE-O1

÷÷

÷ ÷ ÷

1.60E-oI 1.÷

-2,Z*0E-01

ao

[3] D. C. KARNOPP: Power-conserving transformatnons: Physical interpretations and applications using bond graphs. J. Franklin Inst. 288, 175-201 (1969). [4] D. C. KARNOPP and R. C. ROSENBERG: Application of bond graph techniques to the study of vehicle drive line dynamics. Trans. ASME, J. Basic t:'ngng 355-362 (1970). [5] R. C. ROSENBERG: State-space formulation for bond graph models of multiport-systems. Trans. A S M E Dynamic Syst. Measure. Control 35-40 (1971). [6] J. T. Tou: Modern Control Theory, Chapter 3. McGrawHill, New York (1964). [7] T. R. BASHKOW (Editor): Engineering Applications' of Digital Computers, Chapter 6. Academic Press, New York (1968). [8] H. E. KOENIG et al. : Analysis of Discrete Physical Systems, Chapter 8. McGraw-Hill, New York (1967). [9] R. C. ROSENBERGand D. C. KARNOPP: A definition of the bond graph language, a special collection on bond graph modeling. Trans. ASME, J. Dynamic Syst. Measure. Control (1972).

I ÷

F~6.8. Plot of selected results for the lever mechanism.

CONCLUSIONS

The principal purpose of this paper has been to describe the ENPORT procedurefor the simulation of multiport dynamic systems, and to present some examples of its use. Important features of ENPORT are its selection of physically-significant state variables from the energy-variable set, its use of operational causality to obtain an orderly formulation of system equations, its ability to handle systems containing statically-constrained energy elements, and the processing of multiport field elements. ENPORT carries out a direct simulation of the physical system modeled as a bond graph, and can be extended to include linear multiport elements other than the basic nine described in this paper. Major extensions to the system are planned; these include storage and retrieval capability for subgraphs, which is useful for treating engineering multiports. Modifications to permit the simulation of certain classes of nonlinear systems are in the design stage. Much of the current graph-handling procedure is directly applicable to nonlinear system simulation. Acknowledgement--The author wishes to acknowledge the major influence that Professor H. M. Paynter, inventor of the bond graph language, has had on this work. It was he who first proposed the creation of an n-port, or ENPORT, program (in 1963), and his contributions have been helpful in guiding the work through several stages of evolution. REFERENCES [1] D. C. KARNOPP and R. C. ROSENBERG: Analysis and Simulation of Multiport Systems. M.I.T. Press, Massachusetts (1968). [21 H. M. PAYNTER and D. C. KARNOPP: Design and Control of Multiport Engineering Systems, pp. 443-454, Proceedings of IFAC Tokyo Symposium on Systems Engineering for Control System Design (1965).

R6sum6--Un grand nombre de syst6mes physiques et d'engineering peuvent ~tre repr6sent6s directement en fonction de caract6ristiques d'6nergie de composants et de l'interaction de leurs puissance. Lorsque les 61ements du syst~me sont model6s comme multi-orifices 6nerg6tiques, et leurs interconnections par des liens de puissance, le langage du graphique de puissance est alors naturel pour d6crire le syst6me entier. Des graphiques de liens peuvent Etre dessin6s pour des syst6mes dynamiques impliquant divers types d'6nergie soit 61ectrique, soit m6canique, fluide ou thermique; tousles types d'6nergie peuvent coexister. Les 616ments utiles de modelage comprennent les emmagasinages multi-orifice, les dissipateurs, les 616ments de jonction et les transducteurs et aussi les sources. Des modules de graphiques de liens de systemes lin6aires multi-orifice peuvent 6tre transform6s en forme d'espace d'6tat par un algorithme puissant has6 sur une causalit6 op6rationelle. Des 6quations d'espace d'6tat, des r6ponses dynamiques peuvent 6tre obtenues par la technique exponentielle de matrice, rendant ainsi possible la simulation digitale directe de mod61es lin6aires multi-orifice. Le programme ENPORT est une r6alisation de l'algorithme de r6duction du graphique de lien. L'objet principal de ce texte est de d6crire la proc6dure selon laquelle ENPORT est fond6e et de pr6senter quelques r6sultats. Parmi les caract6ristiques importantes de ENPORT, citons le choix de variables d'6tat physiquement signifiants, l'emploi de la causalit6 op6rationelle pour obtenir une formulation ordonn6e d'6quations de syst~me, et sa possibilit6 d'utiliser des syst6mes contenant des sous-domaines d'emmagasinage statique. Zus~immenfiissung--Eine grosse Zahl physikalischer und technischer Systeme k6nnen direkt in Form yon Charakteristiken der Bestandteilenergy und ihrer Kr~ftewechselwirkungen dargestellt werden. Wenn die Systemelemente als Energiemulti0ffnungen modelliert werden und ihre Verbindungen miteinander durch bindende Kr/ifte, dannist die Sprache der Bindungsdarstellung eine natiJrliche, um das ganze System zu beschreiben. Bindungskurven k/Snnen for dynamische, mit verschiedenen Energiearten verbundene Systeme dargestellt werden, wie elektrische, mechanische, Fliissigkeits- und W/irmeenergiearten. Alle Energiearten k~Snnen zu gleicher Zeit bestehen. Zu ni.itzlichen modellelementen gehiSren Multi0ffffnungsspeicher, Ableiter, Verbindungselemente wie Messumformer und auch Quellen. Bindungskurvenmodelle von linearen Multi6ffnungssystemen k~Snnen auf Zustand-Raumform durch einen kr/iftigen Algorithmus umgeformt werden, auf Grund yon betriebsm~ssigem Kausalzusammenhang. Von den Zustands-Raumgleichungen k6nnen dynamische Reaktionen durch die Exponentialtechnik der Matrix erhalten werden, wodurch die direkte Digitalsimulation linearer Multi/3ffnungsmodelle m6giich wird.

Modeling and simulation of large-scale, linear, multiport systems D a s E N P O R T P r o g r a m m ist eine Realisation des Reduzierungsalgorithmus der Bindungskurve. Der H a u p t zweck dieses Berichts ist, das Verfahren zu beschreiben, a u f das sich E N P O R T grfindet und einige Resultate wiederzugeben. Wichtige Merkmale von E N P O R T sind seine Wahl von physikalisch wichtigen Zustandsver/inderlichen, seine Verwendung betrieblichen Kausalzusammenhangs, eine geordnete Formulierung yon Systemgleichungen zu erhalten und seine Fahigeit, Systeme mit statischen Subfeldspeichern zu behandeln. Pe3]oMe--Eonbmoe qHC.rlO d~H3HqeCKHX H TeXHO-rlOFHtleCICJeIX CHCTeM MOFyT ~bITb 17pe~CTaB.rleHbl HeI/ocpe~CTBeHHO 3HeFeTHqeCKHMH xapaKTepHCTHKaMH KOMHOHeHTOBH a3aH/ 3HepreTHqeCKHM~I nX Cn.q. B cnyqae MO~/e~rlpoBannfl 3.rleMeHTOB CriCTeMbI B qbopMe 3HepFeTHtleCKHX MHOFOqItC/ .rleHHblX BXO~OB H HX B3aHMOCB~I3H B ~opMe CHnOBnX CBfl3e~, ~I3HK ~HarpaMM CBfl3e~ RBJI~IeTC~I IIpHpo~HblM fl3HKOM ~.rlfl OnHCaHHa tteno~ CHCTeMbI. ~narpaMMbi CB~3e~ MOFyT 61aITb Har~HCaHbI ~/YI~t~I~,IHaMt~qeCKHXCHCTeM BKJ~roqafl B ce6~l pa3J~nqHble THHbI 3HeprHn, r a t Ha nprtMep 3neKTpHHeCKyrO, MeXaHI,lqeCKy~O, rn//ponHeBMaTn~ecxyK) H TepMn~ecKy~o; Bce THrtb~ 9Heprrin MOFyT cocymeCTBOBaTb.

95

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