Fabien B. Analytical System Dynamics: Modeling and Simulation

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28 1 A Uniﬁed System Representation

A i

Ebers−Moll NPN Transistor Model

In this model the base current, i

, the emitter current, i

, and the collector

current, i

, are deﬁned in terms of the diode currents i

and i

as follows.

= i

− i

= i

− A

= −i

+ A

= I

αv

− 1)

= I

αv

− 1)

where A

, A

, I

and α are positive constants that depend on the

material properties of the transistor, v

is the base-emitter voltage, and

is the base-collector voltage.

It can be shown that A

= A

= I

(see M¨oschwitzer, 1991).

Hence, the behavior of the transistor can be described by the eﬀort con-

straint equations









−





−I

− I







αv

− 1

αv

− 1











. (1.19)

Using these equations we can assert that the transistor currents are a

function of the voltages v

and v

. Moreover, these equations imply

that the NPN transistor can be modeled using the equivalent circuit model

shown below.

1.2 System Components 29

Equivalent circuit model

In this model the currents i

, i

and i

are functions of the voltages v

and v

, which are considered to be eﬀort sources.

The operating modes of the NPN transistor are characterized by the volt-

ages v

and v

. In particular,

– if v

> 0 and v

< 0 the transistor is said to be in active normal

mode.

– if v

> 0 and v

> 0 the transistor is said to be in saturation mode.

– if v

< 0 and v

> 0 the transistor is said to be in active inverse

mode.

– if v

< 0 and v

< 0 the transistor is said to be in reverse mode.

The circuits considered in this text are constructed so the transistor op-

erates in active normal mode. In which case if v

 0 then, the term

αv

in the Ebers-Moll equation is negligible. We can therefore rewrite

the transistor eﬀort constraint equations as

= I

αv

− 1)

= A

αv

− 1)

= (1 − A

αv

− 1). (1.20)

If we deﬁne

β =

1 − A

it can be seen that

= A

= βi

Typically, A

ranges from 0.999 to 0.980, and saturation current I

1×10

−12

amp. As a result β ranges from 1000 to 50, approximately. Hence,

the last equation indicates that, in normal active mode, the collector and

emitter currents are almost equal. Also, the collector current is a signiﬁcant

ampliﬁcation of the base current.

• Operational Ampliﬁer

The ﬁgure (a) below shows the schematic of an operational ampliﬁer. This

30 1 A Uniﬁed System Representation

integrated circuit has input terminals 1 and 2, and output terminal 3.

The operational ampliﬁer can be considered to be a eﬀort regulated eﬀort

source. The input-output behavior can be described using the diagram (b).

(a)

vv v

(b)

1 2

+ + +

− − −

Speciﬁcally, the output voltage of the operational ampliﬁer, v

, is given by

= k

− v

where v

is the voltage input to terminal 1, v

is the voltage input to

terminal 2, and k

is the gain for the ampliﬁer. The gain for typical oper-

ational ampliﬁers is in the interval 10

≤ k

≤ 10

. In addition, the device

has very high input resistance (≈ 10

ohm), and the current input to the

device is negligible, i.e., i

≈ 0, and i

≈ 0.

• DC motor/generator

The schematic of a direct current (DC) motor/generator is shown below.

The essential electromechanical coupling in a DC motor/generator is given

τ = K

= K

ω,

where τ is the torque due to the current i, and K

is the torque constant.

The voltage v

(back electromotive force (emf)) is the result of the angular

velocity ω, and K

is the back emf constant. Thus, both the motor torque

τ and the generator voltage v

can be viewed are ﬂow regulated eﬀort

sources. Note that this model neglects the inductance and resistance of

the motor coils, as well as the inertia, damping and ﬂexibility of the rotor.

1.2 System Components 31

1.2.6 Paynter’s diagram and system models

The relationships between the fundamental system variables can be summa-

rized using Paynter’s diagram shown in Fig. 1.4. The diagram is a graph

V(q), V (e)

T(p), T (f)

D(f), D (e)

Fig. 1.4 Paynter’s Diagram

with the eﬀort e, ﬂow f, displacement q and momentum p at its vertexes.

The relationships between the fundamental variables are illustrated on the

line segments joining the variables. The line segment joining the kinematic

variables q and f shows the diﬀerential relationship between these variables.

Similarly, the line segment joining the kinetic variables p and e shows the

diﬀerential relationship between these variables. The energy stored by ideal

inductors is shown on the line segment joining the momentum (p) and ﬂow (f)

variables. The energy stored by ideal capacitors is shown on the line segment

joining the eﬀort (e) and displacement (q) variables. The energy dissipated

by ideal resistors is shown on the line segment joining the eﬀort (e) and ﬂow

(f) variables. This diagram is useful in making concrete the analogy that

exists between the diﬀerent engineering disciplines. We also note that, given

any pair of the fundamental system variables, we can use the relationships in

Paynter’s diagram to determine the unknown pair of fundamental variables.

For example, given the momentum p and displacement q, we can determine

the eﬀort and ﬂow as e =

and f =

, respectively.

In the development of a uniﬁed approach to modeling multidiscipline sys-

tems we have a choice as to which pair of the fundamental variables will be

treated as independent. Our approach, which is based on Lagrangian me-

chanics, uses the displacement and ﬂow variables (q, f) as the variables that

are used to described the equations of motion. In the bond graph approach to

system modeling the eﬀort and ﬂow (e, f) are used as the independent pair of

fundamental variables (see Karnopp, Margolis, and Rosenberg (1990)). The

linear graph approach to system modeling also uses the eﬀort and ﬂow (e, f)

as the independent pair of fundamental variables (see Rowell and Wormley

(1991)). A Hamiltonian approach to multidiscipline system modeling uses the

displacement and momentum (q, p) as the independent fundamental variables

32 1 A Uniﬁed System Representation

(see Layton and Fabien (1997)). The other variable pair combinations, i.e.,

(f, p), (e, p) and (e, q) are not often used to model multidiscipline systems.

References

1. S. H. Crandall, D. C. Karnopp, E. F. Kurtz, and D. C. Pridmore-Brown,

Dynamics of Mechanical and Electromechanical Systems, McGraw-Hill,

1968.

2. A. R. Hambley, Electrical Engineering: Principles & Applications, Prentice-

Hall, Inc., 1997.

3. E. Umez-Eronini, System Dynamics and Control, PWS Publishing, 1999.

4. R. W. Jensen and L. R. McNamee, Handbook of Circuit Analysis Lan-

guages and Techniques, Prentice-Hall, 1976.

5. D. C. Karnopp, D. L. Margolis, and R. C. Rosenberg, System Dynamics:

A Uniﬁed Approach, 2nd ed. Wiley & Sons, 1990.

6. R. A. Layton and B. C. Fabien, “Modeling and simulation of physical sys-

tems IV: An introduction to Hamiltonian DAEs,” IASTED International

Conference, Applied Modeling and Simulation, 264–268, 1997.

7. A. G. J. MacFarlane, Dynamical System Models, Harrap & Co., 1970.

8. L. Meirovitch, Methods of Analytical Dynamics, McGraw-Hill, 1970.

9. A. M¨oschwitzer, Semiconductor devices, circuits, and systems, Oxford

University Press, 1991.

10. M. Nelkon, Principles of Physics, 1973.

11. H. F. Olson, Dynamical Analogies, Van Norstrand, 1943.

12. H. M. Paynter, Analysis and Design of Engineering Systems, M.I.T.

Press, Cambridge, MA, 1961.

13. R. M. Rosenberg, Analytical Dynamics of Discrete Systems, Plenum

Press, 1977.

14. D. Rowell and D. N. Wormley, System Dynamics: An Introduction, De-

partment of Mechanical Engineering, MIT, 1991.

15. J. L. Shearer, A. T. Murphy, and H. H. Richardson, Introduction to Sys-

tem Dynamics, Addison-Wesley, 1967.

16. J. L. Shearer and B. T. Kulakowski, Dynamic Modeling and Control of

Engineering Systems, Macmillan, 1990.

17. R. J. Smith, Circuits devices and systems, John Wiley and Sons, 1976.

18. P. E. Wellstead, Introduction to Physical System Modelling, Academic

Press, 1979.

19. D. C. White, and H. H. Woodson, Electromechanical Energy Conversion,

John Wiley & Sons, 1959.

20. J. H. Williams, Jr., Fundamentals of Applied Dynamics, John Wiley &

Sons, 1996.

1.2 System Components 33

Problems

1. Determine a consistent set of English and SI units for the variables listed

in Table 1.1. Hence, show that the units of work, power and energy (Table

1.2) are equal in each discipline.

2. Verify the results given in Table 1.3, i.e., show that kinetic energy and the

kinetic coenergy are equal for a point mass, a rotor, a linear inductor and

a linear ﬂuid inertia. Also, assign consistent SI units to all the variable

in these expressions.

3. Verify the results given in Table 1.4, i.e., show that potential energy and

the potential coenergy are equal for a linear spring, a linear torsional

spring, a linear capacitor and a linear ﬂuid capacitor. Also, assign con-

sistent SI units to all the variable in these expressions.

4. Verify the results given in Table 1.5, i.e., show that content (Rayleigh

dissipation function) and the cocontent are equal for a linear damper, a

linear torsional damper, a linear resistor and a linear ﬂuid resistor. Also,

assign consistent SI units to all the variable in these expressions.

5. Consider a particle of mass m moving in a plane as shown below. Let

the velocity of the particle be ¯v = ˙x

i + ˙y

j, and the linear momentum be

¯p = m¯v.

a. Show that the kinetic coenergy is T

∗

(¯v) =

m¯v · ¯v =

m( ˙x

+ ˙y

b. Perform the coordinate transformation x = r cos θ, y = r sin θ, and

show that T

∗

(¯v) =

m( ˙r

+ r

). (Note that T

∗

is a function of the

displacement and ﬂow in this formulation.)

c. Suppose further that θ =

αt

+ ωt + θ

, where α, ω and θ

are con-

stants, and t is the time. Show that in general T

∗

can be a function of

displacement, ﬂow and time.

6. The ﬂux linkage for an electrical inductor is experimentally determined

to be λ = ki

, where k is a constant, and i is the current. Determine

expressions for the kinetic energy and kinetic coenergy for this device.

7. According to the special theory of relativity the mass of particle is given

m(t) =

1 − (v(t)/c)

where, the constant m

is rest mass, v(t) is the velocity of the particle, t

is the time, and the constant c is the speed of light.

34 1 A Uniﬁed System Representation

a. Show the the kinetic energy is T = (m(t) − m

b. Show that if v  c, T ≈ m

v(t)

/2.

8. A block of mass m is constrained to move on an incline. The incline is

rough and has a slope with angle α. A constant force, F , is applied to

the block at an angle θ.

The coeﬃcient of friction is µ, and the acceleration due to gravity acts

downward as shown. If the block starts from rest, ﬁnd an expression for

its velocity after it has moved a distance, s, up the incline. (Use the work

energy principle (1.10).)

9. The ﬂuid in the conical tank shown here has density ρ. The pressure at

the bottom of the tank is P , and the height of the ﬂuid in the tank is h.

a. Find the constitutive equation that relates the pressure P to the volume

V of ﬂuid in the tank.

b. Find an expression for the potential energy due to the volume of ﬂuid

stored in the tank.

10. The constitutive equation for a nonlinear spring is f = kx

where, f is

the applied force, k is a constant, and x is the net deﬂection of the spring.

Determine the potential energy and the potential coenergy.

11. A block of mass m fall a distance, h, onto an unstretched spring. The

spring has stiﬀness, k, and the acceleration due to gravity, g, acts down-

ward as shown.

1.2 System Components 35

If the block is released from rest: (a) what is the velocity the block just

before it hits the spring, and (b) what is the maximum deﬂection of the

spring?

12. Explain how you would modify the model of an ideal electrical trans-

former to include the eﬀect of the inductance and resistance of the coils.

13. Explain why ideal eﬀort and ﬂow sources do not exist. Illustrate the be-

havior you would expect from an actual eﬀort source as the systems ﬂow

demand increases. Use a typical electrochemical battery as an example.

(Plot the eﬀort versus ﬂow for an ideal eﬀorts source and an actual eﬀort

source.)

14. The circuit shown here satisﬁes the equations

R ˙q = v − v

˙q = I

αv

− 1),

where R = 10 ohm is the resistance, v is the voltage source, v

is the

diode voltage, I

is the diode saturation current, and α = 40 volts

−1

the diode junction constant. Obtain numerical solutions to these equa-

tions for 0 ≤ v ≤ 10, and plot ˙q versus v

for the following saturation

currents; I

= 10

−12

, 10

−14

, 10

−16

15. Sketch Paynter’s diagram for each of the energy disciplines. Label the

vertexes with the appropriate variable names.

16. For the systems shown below, write expressions for; (i) the kinetic en-

ergy and the kinetic coenergy, (ii) the potential energy and the potential

coenergy, and (iii) the content and the cocontent. For the systems with

applied eﬀorts write expressions for the work done on the system by these

eﬀorts. (See Section 3.1.1.)

36 1 A Uniﬁed System Representation

b k

m g

(gravity)

k b

(a) (b) (c)

(d) (e)

−

(f) (g)

−

(h)

(j)

(i)

C R

)l()k(

R C

Chapter 2

Kinematics

In analytical mechanics kinematics is called the study of the geometry of mo-

tion. For mechanical systems kinematic analysis involves the determination of

the position, velocity and acceleration of points in the system. Moreover, this

analysis takes place without regard for the forces that give rise to the motion.

In this text we will extend these concepts to multidiscipline systems using the

uniﬁed framework presented in Chapter 1. In particular, this chapter develops

techniques to help determine the displacement and ﬂow associated with the

components of the system. Section 2.1 considers the kinematics of mechani-

cal systems where the inertia elements are treated as point masses and rigid

bodies. Section 2.2 considers the mobility and kinematics of mechanisms. Fi-

nally, Section 2.3 presents the kinematic analysis of networks used to model

electrical, and ﬂuid systems. Using the results developed in this chapter we

will be able to write analytical expressions for the kinetic coenergy, the po-

tential energy and the dissipation function of multidiscipline systems, as well

as analytical expressions for the displacement and ﬂow constraints that may

exist in the system.

2.1 Mechanical Systems

2.1.1 A point moving in a ﬁxed frame

We begin the kinematic analysis of mechanical systems by considering the

motion of a point that moves relative to a ﬁxed reference frame. The point

in question could represent a point mass or a point on a rigid body. In either

case the objective of the kinematic analysis is to ﬁnd the position and velocity

(i.e., displacement and ﬂow) of the point with respect to some ﬁxed reference.

B. Fabien, Analytical System Dynamics: Modeling and Simulation, 37

DOI 10.1007/978-0-387-85605-6 2,

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