chapter 3 modeling physical systems: problemsbryant/courses/me344/downloadfiles/chapter3... ·...

Chapter 3

Modeling Physical Systems:Problems

x

V

k

+ -

external circuit

moving part stationary

i

Figure 3.1: A relay.

1. Relays are electromechanical switches that open and close another circuit. To start

your car, a relay engages the starter and battery when you turn the ignition key. In

figure 3.1, voltage V or current i activates the relay’s coil, and the resultant elec-

tromagnetic force pulls the relay contactors together. Construct a bond graph/word

bondgraphthatshowshowpowerflowsinthisdevice. Includethesource, energizing

coil, losses in the coil’s resistance, electromechanical conversion, and mechanical

elements of the relay, such as mass and the retraction spring. Indicate the direction

1

2 CHAPTER 3. MODELING PHYSICAL SYSTEMS: PROBLEMS

of all power flows.

motorrotor

motorstator

stator

switchmotor-fanshaft

fan blades

air flow

110 V AC

Figure 3.2: Table fan with motor, shaft, and fan blades.

2. A table fan consists of power cord (with resistance), switch, electric motor, shaft,

and fan blades. Rotation of the rotor causes the fan blades to move air for cooling.

Construct a complete bond graph of the table fan. Include all elements, correct

structure, power flows, causality, and labels on bonds sufficient to extract the state

equations.

3. Construct a word bond graph thatdescribes conversion ofwind power into electrical

power, via the windmill shown in figure 3.3.

4. Derive an expression for the electrical energy stored in the coil of figure 3.1, if the

coil is energizedby current i. Selectan appropriate element (R, I,orC) to modelthe

coil, and determine if the element is linear or nonlinear. Flux linkage λ and current

i through the coil are linked via

λ =n2A

x/µair + `/µFei (3.1)

where n is the number of turns in the coil, A and ` are the cross sectional area and

circumferential length of the relay’s flux return path, µair and µFe are magnetic

permeabilities of air and iron, and x is the size of the contactor gap.

3

generator

windpropeller

wind

electrical

Figure 3.3: Windmill with propeller, generator, and power cable.

5. In the amusement park ride “Brain Centerfuge,” cars A and A’ rotate about point P

at 10 RPM, see figure 3.4. With passengers aboard, each car weighs about 500 kg.

Estimate the kinetic energy stored in the rotor under normal operating conditions.

6. In an elevator system shown in figure 3.5, a circuit (power source V and line resis-

tance R) energizes an electric motor with gearbox which drives a pulley and cable

system. A cable spools over a winch, runs around pulleys atop the elevator car, be-

neaththeplatform, andatopthecounterweights, andterminates beneaththetop floor

platform. (The counterweights balance the elevator car such that the total potential

energy between the two is always constant). Rotation of the winch pulls the cable,

which simultaneously lifts the elevator and lowers the counterweight. Construct a

word bond graph for the elevator system. Indicate the direction of all power flows.

7. Suppose the elevator in figure 3.5 is initially at rest, but the motor draws 25 amperes

current at 200 volts for 5 seconds. If the elevator car and counterweight each weigh

1000 kg, what is the maximum speed the elevator can attain? The counterweights

balance the elevator car so that the total potential energy between counterweights

and elevator car is constant. Hint: Assume no power or energy losses and apply

P = dE/dt.


20 m

A A'

P

20 cm

20 cm

Figure 3.4: Amusement park ride.

8. The engine in an airplane twirls its propeller. The propeller induces airflow over

the wing, which generates lift on the wings and thrust on the plane. Starting with

the engine, construct a word bond graph for the airplane system, i.e., show where

the power goes. Include inertia of the propeller, mechanical losses, and airflow over

the wings, fluid losses, motion of the plane, and altitude of the plane. Indicate the

direction of all power flows.

9. Suppose the airplane is initially at rest, and the engine produces 400 hp at 3,600

rpm for 5 minutes. If the airplane weighs 8000 lb, what is the maximum speed the

airplane can attain? Also, what is the maximum altitude? Note: 1hp ≈ 0.75kW,

1kg ≈ 2.2lb. Hint: Assume no losses, all power into the relevant quantity, and

apply P = dE/dt .

10. An important effect in the dynamics of an anti-friction (ball) bearing involves com-

pression of the balls. When a force F = Cδ3/2 presses two balls together, the

centers approach each other by the amount δ = xo − x ≥ 0, see figure 3.6. De-

rive an expression for the energy storedor power dissipated during the compression,

select an appropriate element (R, I, or C) to model the compression process, and

determine if the element is linear or nonlinear.

11. A Belleville spring (or washer) with has an approximately parabolic force F =F (δ)versusdeflection δ relationshowninfigure3.7. Thespring,initiallyunloaded,

is compressed 0.1 cm. Roughly estimate the energy stored or dissipated during

the compression process, select an appropriate modeling element (R, I, or C), and

determine if the element is linear or nonlinear.

12. The flux linkage λ of a nonlinear inductor depends upon its current i according

5

motor

gearboxwinch/spool

elevator car

counterweights

pulley

cable

elevator carguide rail

R

V+

-

ground

counterweightguide rail

cable fixedto platform

pulleypulley

Figure 3.5: An elevator.

to λ(i) = (2/π)Lio arctan[i/io] , where io is a constant current, and L is a

constant. Assumingthecurrentinitiallyzero,afterenergizingtheinductorto i = io,

determine the total kinetic energy stored.

13. A crane lifts objects by pulling a cable over pulleys. The cable winds around a

spool on the crane’s winch. As the spool rotates, cable is brought in (lifting) or let

out (lowering). The diameter D on the spool about which the cable winds and the

rotational inertia J of the winch vary with the amount of cable present on the spool.

Without cable, the spool has diameter Ds and inertia Js. Derive an expression for

the nonlinear rotational inertia J = J(θ) of the winch’s spool as a function of the

angular displacement θ of the spool. Assume the cable has diameter d and linear

density η [kg m−1], and packs perfectly over the width w of the spool. HINT:

Estimate J from Js and the layers of cable on the spool.

14. Friction force between bodies in sliding contact depends in a nonlinear manner on

the relative speed of sliding v between the bodies. As figure 3.8 shows, the friction

force Fµ =Fd + (Fs − Fd)e−|v|/vd

sgn(v) varies between static friction

force Fs and dynamic friction force Fd. Here vd is a characteristic velocity. The


x

x

o

F F

FF

Figure 3.6: Balls under compression.

F (N)

δ (cm)0 0.1

1000

2000

F F

Figure 3.7: Belleville spring or washer.

“sign” function sgn (v)—equal to +1 when its argument is positive and -1 when

its argument is negative—appears because friction forces always oppose relative

motion. Select an appropriate bond graph element to model this friction law, label

the effort and flow on the bonds, assign a causality appropriate to this law, and

estimate the instantaneous power. Why is this element dissipative?.

15. The diode pictured in figure 3.9 responds to voltage V with the current i shown in

the graph. Determine the bond graph element that best models this diode.

16. Construct a complete bond graph for the circuit of figure 3.10. Include all pertinent

bond graph elements, and indicate direction of all power flows. Then use the bond

graph to answer the following questions: a.) How many energy storage elements

are present in this system? b.) How many independent energy storage elements are

present in this system? c.) What is the dynamic order of the system?

7

Fs

v

Fd

-Fs-Fd

Friction force Fµ

vd

-vd

Figure 3.8: Simple constitutive relation for friction force.

+ -v

i

i

v

Figure 3.9: Diode.

17. Repeat the instructions of problem 16 for the circuit of figure 3.11.

18. Repeat the instructions of problem 16 for the circuit of figure 3.12.

19. Shown in figure 3.13 is a passive network bandpass filterwhich rejects high and low

frequencies, but passes intermediate frequencies. This is an example of a π-filter

design: the legs of the “π” have capacitances C2 and inductances L2 in parallel,

while the bar of the “π” has capacitance C1 and inductance L1 in series . Develop

a bond graph for this circuit.

20. A block (m = 10 kg) initially at rest is pulled by force F (t) = 100N, see figure

3.14. As the block slides over the flat, motion is resisted by tangential force Tacting between the block and flat. For the sliding block, construct a bond graph

and determine the block speed v, for: a) a lubricated surface, where T = bv,

and b = 50Nm−1s; b) a dry surface, where T = µmg, with dynamic friction

coefficient µ = 0.5 and g = 9.8ms−2.

21. Develop a bond graph for the two degree of freedom (DOF) system consisting of

two sliding masses m1 and m2 and two springs k1 and k2 forced by f(t), see


+

-

V

R1

L

C1 R2 I

Figure 3.10: Electric circuit.

+

-

V

R2R1

C1 C2

L 1

L 2n I


figure 3.15. Assume the friction (force Fµ = µdmg, where µd is the coefficient

of dynamic friction) between the mass and the sliding surface.

22. A gearbox with input shaft, layshaft, and output shaft is driven by an input torque,

see figure 3.16. The gear on the input shaft has 20 teeth, and its mating gear on the

layshaft has 25 teeth. The other gear on thelayshaft has 20 teeth,and its matinggear

on the output shaft has 25 teeth. Construct a complete bond graph for the gearbox

system. Assume the shafts to be rigid, i.e., no storage of energy in shaft torsion,

account for all inertia, and include losses in the bearings.

23. The drive train on an automobile with manual transmission consists of motor, fly-

wheel, clutch, gears, and shafts, see figure 3.17. The clutch is essentially two plates

9

+

-

V

R1L

C1

R2

n


L1C1

+

-

L2 C2 L2 C2e in

R load

e out

+

-

Figure 3.13: A passive Pi-filter bandpass circuit.

with dry friction between; here friction forces transmit torque between plates. At

the end of the drive train is the vehicle’s inertia. Construct a complete bond graph

for the drive train system. Assume the shafts to be rigid, i.e., no storage of energy

in shaft torsion, account for all inertia, and include losses in the bearings and the

clutch. At the end of the drive train is the vehicle’s inertia. Construct a complete

bond graph for the drive train system.

24. A motor propels a motorcycle. Attached to the motors crankshaft is a flywheel

(not shown) and a friction clutch/ gearbox. The gearbox changes gear ratios. On

the output shaft of the clutch/gearbox is a sprocket, which by a chain drive rotates

the rear wheel via another sprocket on the rear wheel. Rolling of the rear wheel

movesthe motorcycleforward. Construct a complete bondgraph for themotorcycle

system, to describe the vehicles forward motion propelled by the motor. Include


m F(t)

v

Figure 3.14: Sliding block.

m1

m2

k2

k1

y1

y2

f(t)

friction

Figure 3.15: Sliding masses.

effects ofinertia ofthe motorscrankshaft andflywheel, effects ofthe friction clutch,

the gearbox, transmission of power by the chain drive and sprockets, inertia of the

driven rear wheel, and losses in wheel bearings. Also convert the rotational power

of the rear wheel into translational power of forward motion. Include effects of

motorcycle and rider mass, and losses due to wind.

25. The conical tank shown in figure 3.19 has radius r = r(z) = z, where z marks

distance from the bottom of the tank. If v is the fluid volume in the tank, determine

the potential energy stored for a fluid level z = h.

26. Construct a complete bond graph for the system of fluid tanks fed by sources u1,

u2, and u3 and depicted in figure 3.20. Assume long constricted pipes. Determine

the dynamic order of the system. Then extract the state equations.

27. An electric car is powered by a battery of constant voltage, which energizes a motor

with resistance Rm and inductance Lm on its input circuit. The lossless motor,

with torque to current T = Ki and back emf to motor speed Vm = Kω, drives

11

input torquegears

output torque

layshaft

output shaftinput shaft

Figure 3.16: Gearbox with input shaft, layshaft, and output shaft.

flywheel

motor

vehicleinertia

clutch

friction

gears

Figure 3.17: Automobile transmission system schematic.

the transmission with friction clutch, gears, and shafts. The automobile with mass

M moves when its wheels with diameter D rotate. Assume all shafts rigid, and

account for bearing friction and inertia of wheels a.) Bond graph the electric car

system. Include all important elements, bonds, power flow arrows, causality, and

labels on bonds sufficient to extract state equations. b.) What is the order of this

dynamic system?

28. During braking, the regenerative brakes on a hybrid automobile converts the car’s

kinetic energy into electrical power, which is then routed and stored in the bat-tery. Attached to anddrivenby each ofthe fourwheels (figure 3.22showsonly two

wheels) is a motor-generator with back emf to motor speed Vm = KΩ and torque

to current T = Ki. Bond graph the regenerative brake system and state the order

of this dynamic system. The automobile has mass M and the wheels have diameter

D. Assume all shafts and axles rigid, but account for bearing friction and inertia

of wheels and motors. Here Rm and Lm represent motor/generator resistance and

inductance, respectively.


wheel

chain

sprocketsprocket

clutch

motor

road

Figure 3.18: Motorcycle Drive Train.

h

r(h) = h

z

Figure 3.19: A conically shaped fluid tanks, with sources.

29. Construct a complete bond graph for the elevator system shown in figure 3.5, given

the conditions of problem 6.

30. The tubular loudspeaker system shown in figure 3.23 is activated by voltage E(t)applied to an N -turn voice coil of diameter D. Magnetic force F = (BNπD)I ,

induced by permanent magnet flux B interacting with voice coil current I , causes

displacement x(t) of the plunger/cone assembly. The electrical to mechanical

power conversion also induces a back emf Eb = (BNπD)dx/dt in the coil.

Cone motions x(t) then impels sound (acoustic) flow v(t) = Adx/dt down the

acoustic tube. Here A is the speaker cone area.

(a) Construct a bond graph for the tubular loudspeaker system. Include voice

coil resistance and inductance; the mechanical mass, stiffness and damping

inherent in the plunger/cone assembly; and fluidic long pipe effects down the

length of the acoustic tube. Assume lossless electrical to mechanical, and

mechanical to hydraulic power conversions.

13

R12R23

h1 = v1/A1 h2 = v2/A2 h3 = v3/A3 u3

u1

h4 = v4/A4R14

I14

u2

I23I12

Figure 3.20: A system of interconnected fluid tanks, with sources.

automobile

motoribattery +

-

Rm

Lm

frictionclutch

gears

vehicle wheelsdiameter D

Figure 3.21: An automobile with electric drive.

(b) Labelthebondgraph,includingsourcesandenergystorageelements: express

effortsandflowsonbondstoallenergystorageelementsintermsofrespective

energy variables.

31. Arack-pinionmechanismdepictedinfigure3.24controlsmotionsofthecarriageona

machinetool. ADCmotorenergizedbyacurrentsource is(t) drivesthemechanism.

Figure 3.24 also showsthe nominal values ofthe system’s components. Construct a

bondgraphforthe system,including power flows,labels onenergy storageelements,

and causal strokes. Include motor inertia Jm, motor damping Bm, motor constant

km, shafttorsionalstiffness ks, gear(pinion)inertia Jp, andlumpedbarandcarriage

mass mb + mc. The speed at the end of the shaft is Ω1, the shaft torque Ts, the


battery

+

-

motor/generator

vehicle wheelsdiameter D

iRm Lm

automobile

Figure 3.22: An automobile with regenerative braking.

load velocity v2, the supply current is.

32. Amechanicalspeedometermeasuresvehiclespeed. Rotationsofashaftconnectedto

thevehiclewheelstwistacable; theoppositeendofthecableinducesaninertiainside

the speedometer to rotate. Torque from the inertia deflects a spring. Deflections

of a readout needle connected to the spring are approximately proportional to the

vehicle’s speed.

(a) Sketch the speedometer design. Show all important elements. Indicate all

input and output variables.

(b) Constructabondgraphofthespeedometersystem,includingcomponents/effects

as bond graph elements, power flows, causal strokes, and label bonds to all

sources and energy storage elements. Finally, state the system order.

33. Bond graph the pumping/storage system that consists of a circuit connected to an

electric motor that drives a pump via shaft 1 mounted on bearings, pulley 1 (with

inertia), v-belt (assume no slip and/or stretch), pulley 2 (with inertia), and shaft 2

mounted on bearings. The pump forces water from the lake through a long pipe into

a storage tank. Include all important components/effects as bond graph elements,

indicate the direction of all power flows, label bonds off all sources and Energy

StorageElements (intermsofenergy variables), show causality, andstatethesystem

order.

34. Assuminglinearelements,extractstateequationsforthewaterturbinedrivensystem

shown in figure 34, and then put them into matrix form. Note: Flow source Q(t)

15

speaker cone(area A)

voice coil(N-turns &diameter D)

+-

E(t)

plunger/conemotions x(t)

I

permanent magnet

ACOUSTIC IMPEDANCE MATCHING TUBE

SOUND WAVESv(t) = A dx/dt

Figure 3.23: Loudspeaker mounted onto a tube for acoustic impedancematching.

is injected into the tank. The turbine, which relates shaft speed N to pipe flow Qp

via N = µQp, is lossless. You may neglect the inertance of the short pipe.

35. Construct a complete bond graph for the system shown in figure 35. Then extract

the state equations. Note: Voltagesource V (t) energizes the lossless electric motor

(with i = µT , where T is the motor torque) which induces shaft rotation. The

shaft with inertia J transmits power to the pump, which causes volumetric flow rate

Q = βw (w = shaft speed). Qs(t) is a flow source.


DC motor

Jm = 0.0075 [N-m-sec2/rad]

Bm = 0.03 [N-m-sec/rad]

km = 1.0 [N-m/amp]

compliant shaftks = 8500 [N-m/rad]

pinion

Jp = 0.0025 [N-m-sec2/rad]

rp = 10.0 [cm]

rack (rigid)

mb = 2.0 [N-sec2/m]

currentsupplyis(t)

load

mc = 60.0 [N-sec2/m]

latheguide

v2Ω1

Figure 3.24: Schematic representation of motor-driven rack-pinionmechanism.

long pipe withlinear Ip and Rp

tank 1 Qout(t)

pump

water supply

C+

-V

R L

motor

pulley with inertia J1 and radius R1i

+

-

e

belt drive: smallmass & no slip

pulley withinertia J2 andradius R2

bearings with b

Q(t)

tank

short pipewithconstrictionresistance R J

b

losslessturbine:N = µ Qp

17

+

-

V

R

L

motor

shaft shaft inertia

bearings

pipe

storagetankpump

watersupply

J

b+

-

e

i

Qs(t)

chapter 3 modeling physical systems: problemsbryant/courses/me344/downloadfiles/chapter3... ·...

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