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211Figure17.19.2:

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17.20 Pistonwith2ndOrderTranslationHydraulic Piston

P,QA

M

x

K

BRollers

Figure17.20.1: MassSpringDamperSystemwithHydraulicPiston

ThefigureaboveshowsalargestagewithmassM(kg)drivenbyahydraulicpiston. ThehydraulicpistonisdrivenbyapressuresourceP(t)(N/m2)withavolumeflowQ(t) (m3/s)(theflowresistanceof

the

piping

is

ignored).

The

piston

has

aforce/pressure

relationship

of

F=P*A,

and

thus

Q

=

Ax.

(a) FindthedifferentialequationforthissystemintermsofaninputP(t)andanoutputx. Note:

Ignoretherollerdynamics.(b) GivenM=5000kg,K=250,000N/m,B=75,000N*s/m,andA=0.1m2 showthatthetransfer

functionfromthepressure inputPtoflowQ(thefluidicadmittance) isgivenbyQ 0.01s

(s) =P 5000s2+ 75,000s+250,000

(c) ThesystemhasbeenheldinequilibriumwithP=500kPa=500,000N/m2 whenthepressureis

removed

suddenly

at

t=0.

That

is,

P(t)=P0(1-us(t)).

Find

and

make

adimensioned

plot

ofQ(t). Usethetransferfunctionfrompartb)forthiscalculationeven ifyouwerenotabletoderiveit.

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17.21 PoleZeroBodeMatching

Six pole/zero plots labeled A-F are shown below. The following page has eight Bode plots labeled 1-8.

Match each of the six pole/zero plots to its corresponding Bode plot. Your solution should consist of

a list of letters A through F with the corresponding Bode plot number directly next to it.

No partial credit will be given.

50

A 100 B50

0Im 0Im

-50

-100

-100 -80 -60-50

-40Re

-20 0 -100 -80 -60 -40 -20 0Re

100

C 100 D50 50

0Im 0Im

-50 -50

-100

-300 -250 -200-100

-150Re

-100 -50 0 -50 -40 -30 -20 -10 0 10Re

50

E40

60 F20

0Im 0Im

-20

-40

-150 -100-50

Re-50 0 -50 -40 -30 -20 -10 0 10

-60

Re

Figure17.21.1:

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1-1

0 2 0-2

LogMagnitude

-1

LogMagnitude

Phaseo

-3

-180

-90

0

101

101

102

102

Frequency r/s

103

103

Phaseo

10

0

-2

100

-180

-90

0

10

1

101

Frequency r/s

10

2

102

3-1

41

LogMagnitude

Phaseo

100

-2

LogMagnitude

0

101

102

103 100

0

90

Phaseo

101 102 103

100

-90

101

102

Frequency r/s10

3 100

0

101

102

Frequency r/s10

3

Phaseo

510

0

-1

0

1

Log

Magnitude

100

-180

-90

0

101

102

101

102

Frequency r/s

103

103

Phaseo

6-4

-3-2-10

Log

Magnitude

-360

-270

-180

-90

0

101

101

Frequency r/s

102

102

7-1

0

LogM

agnitude

80

LogM

agnitude

-2 -1

Phaseo

-3

-180

-90

0

101

101

102

102

103

103

Phaseo

-90

0

101

102

101

102

103

103

Frequency r/s Frequency r/s

Figure17.21.2:214

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17.22 PowerSemiconductorThermalProblemThe thermal system shown below represents large power semiconductor device which is capturedbetweentwoplates. ThedevicehasathermalcapacitanceC1[J/K],anditstemperatureisT1[K].Thermal resistances R1[K/W] and R2[K/W] connect the device to the upper and lower platesrespectively. TheseplatesaremaintainedatambienttemperatureTA[K]. Powerdissipationinthedeviceismodeledasheatflowqin[W]intothedevice. Assumenoheatflowoccursthroughthesidewallsofthedevice.

Figure17.22.1: Schematicofthermalconfiguration.

(a) Writethegoverningdifferentialequation intermsofT1 andqin.(b) Assume that in steady state, qin = qo, a constant. What is the steady state temperature

T1ss?

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17.23 SubmersibleCapsuleHoistSystemA submersible capsule requires in case of malfunction a rescue hoist system, as shown in Figure17.23.1. Thecapsulemass isM,andthehoistcablestiffness isassumedconstantandofvalueK.Thedrumhas inertiaJ,andradiusr. Thedrum isdrivenbyavelocitysourcew(t). Thecapsuleis subject to a drag force equal toFb = bv(t) and to a flotation force Fw. Note: The system isdesigned inawaythatthehoistcablewillbealways intension,undernormaloperation.

Figure17.23.1: Capsulehoistsystem

(a)

Whenthe

capsule

is

at

rest

the

cable

is

extended

by

an

amount

e. Find

eas

afunction

of

M

andFw.

(b) FindthetransferfunctionV(s)/W(s).(c) Given M = 10000 [kg] and the bode plot shown on Figure 17.23.2, try to find the values

of J, r, K, and B. Use the correct units for all values, and indicate what value cannot bedetermined.

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Figure17.23.2: BodePlotofV(s)/W(s)

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17.25 VaccineCoolerYoure a 2.009 student trying to verify the feasibility of a portable vaccine cooler. The vaccinechamberofthecoolerisintheshapeofacubeandisshownbelowinFigure17.25.1. Thevaccineswithinthechamberhavemassm and specific heat c. The combined effectiveresistance of allsixwalls is given by R. Tvac is the temperature of the vaccines within the cooler, and Tamb is thetemperatureoutsidethecooler. Theheatflowintothevaccinechamberduetothecoolingsystemisgivenbyqin. Thewidthofeachsurroundingwall isL.

QIN

4AMB

M C

24VAC Q

OUT

Figure17.25.1: Cross-sectionofacubicalportablevaccinecooler

(a) Derivetheordinarydifferentialequationthatdescribesthissystem.(b) You are told that the mass of the vaccines is 1 kg, their specific heat is 4.2 kJ/(kgK), and

thetotal

effective

thermal

resistance

resistance

around

the

vaccine

chamber

is

given

by

the

followingequation:

KR= 40 L

WmIfthewidthofthesurroundingwallsis2.5cm,theambienttemperatureis25C,thevaccinechamber is initially2C,andthecoolingsystem isoff,how longwill ittake forthevaccinechambertoreachatemperatureof10C?

(c) Youvebeenswampedwithworkfor2.003,andyour2.009teammakesanincredibleamountofprogresswithoutyou. Youcomeback2weekslaterandfindanalmostworkingprototypecooler. The cooler uses a non-linear on/off controller with a dead band. On a workbenchyouseethefollowingplots. Whatwallthicknessdidyourteammateschooseforyourcooler?Whenthecoolingsystemisturnedon,whatisthevalueforqin? Pleaseuseappropriateunits.

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Control System Off with Cooling System On

0 2 4 6 8 10

Time (s) x 104

Control System On

0 2 4 6 8 10

Time (s) x 104

Figure17.25.2: PrototypeCoolerTests. Thetopplotshowsthecoolingsystembeingruncontinuously. Thebottomplotshowsthecoolingsystembeingrunwiththenon-linearcontroller.

-20

-10

0

10

20

30

T vac

( C)

0

5

10

15

20

25

T vac

( C)

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17.26 VoltageDrivenRRCCircuitConsiderthecircuitshownbelow.

R

R C1

+

_V

in

+

Vo

Figure17.26.1: Voltagedividerwithcapacitor inparallel

(a) DerivethedifferentialequationdescribingVo(t) in termsofVi(t).(b) TheoutputVo(t) is givenby

Vo(t) = 1;t 0

Thiswaveform issketchedbelow

Vo(t)

t (sec)

t =-ln(1/2)/10=0.0693 sec

1

-1

Figure17.26.2: SketchofVo

What isthe inputvi(t)forallt? Explainyourreasoning,andmakeasketchofvi(t).

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17.27 LinearMechanicalSystemwithPositionInputConsider the mechanical systemshown in Figure17.27.1, where w(t) is an input position source,andx(t) isthemassposition.

w(t)

m

k1 k2

b1 b2

x(t)

Figure 17.27.1: Linear mechanical system with two springs, two dampers, and a position sourceinput,w(t).

(a) Drawa freebodydiagram forthemassm,showing forcesactingonthemass,asa functionofthedisplacementsw(t) andx(t)and intermsofthesystemparameters.

(b) Writeadifferentialequationforthesystem, intermsof inputw(t) andoutputx(t).(c) Letm = 10kg,k1 = 30 N/m,k2 = 10 N/m,b1 = 2 Ns/m,b2 = 6 Ns/m,withw(t) = 1us(t) [m]

(a unit step). Plot the system poles in the s-plane. Solve for x(t), t > 0, from rest initialconditions.

(d) Plotx(t), using accurate dimensions, and units. Also indicate the overshoot value, with itsassociatedpeaktime,aswellasthe1090%risetime.

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17.28 TankwithPumpInletLowerthanOutletThisproblemconsidersthesystemshown inFigure17.28.1.

Figure17.28.1: Diagramoftankconfiguration.

Asshowninthefigure,apumpactsasasourceofflowqp(t)intothecylinder;thatisthepumpflowisaspecifiedconstantindependentofloadpressure. Thecylinderisofcross-sectionalareaA[m2].Ataheightho [m] fromthebottomofthecylinder,weconnectafluidresistanceR [Pasec/m3].TheentiresystemisexposedtoatmosphericpressurePatm.At t = 0, the cylinder is empty (h = 0), and the pump is turned on at constant flow rate qo =106 m3/sec,andsothecylinderbeginstofillup. Thefluidheightabovethebottomofthecylinderisdefinedash(t) [m]. NotethatnoflowenterstheresistanceRuntilh(t)ho. The liquid inthesystemiswaterwith= 103 kg/m3. Theheightofthefluidisrecordedasafunctionoftime,andthe data is plotted as shown below.

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@VUHWHP>WK

hf

ho

t1

W>VHFRQGV@

Figure17.28.2: Fluidheightasafunctionoftime.Note that the fluid height reaches theheight of the resistance att =t1 = 30sec;thatish(t1) =ho = 0.1 m. Fort > t1,theheightexponentiallyapproachesafinalvalueofhf = 0.2 m.

(a) Usingthedatafor0t t1, what is thevalue ofA in[m2].(b) Using the data for t > t1,what is the value ofR in [Pasec/m3]? (This can be answered

relativelyeasilyifyouthinkaboutitcorrectly.) Whatisthetimeconstant [sec]associatedwiththeresponse fort > t1? (It isacceptabletodeterminethiseithergraphicallyorviaanappropriatecalculation.)

(c) A longtime lateratt =t2 >> t1,thepumpflow issettozero,qp =0. Makeadimensionedsketchofh(t),t > t2.

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17.29 ThermalPowerChipAnalysisApowerelectronicchipisattachedtoasubstrateandalsoexposedtoairasshowninFigure17.29.1.HeatislosttotheairthroughathermalresistanceR1 [K/W]andtotheboardthroughathermalresistanceR2 [K/W]. Theair and circuitboardareat ambienttemperatureTA [K]. The chiphas a thermal capacitance of C [J/K] and is at temperature Tc [K]. Power dissipation in thechipismodeledasheatflow in,qin [W].

Figure17.29.1:

(a) Writethegoverningdifferentialequationforthissystem intermsofTc andqin.(b) For R1 = 90 K/W, R2 = 10 K/W, and C = 1 J/K, solve for Tc(t) and plot Tc as a

function of time for qin = 10W from rest initial conditions where Tc(0) = TA. Be sure to labelanddimensionyourtimeandtemperaturescales.

(c) If the maximum allowable steady-state temperature of the chip is 100 K above ambient,what isthemaximumallowablepowerthechipcandissipate?

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18 MathTechniques18.1 ComplexExpressionReduction

(a) Reducethefollowingexpressionstoasinglecomplexnumber inrectangularform:1.

(2 + 3i)

(45i)2. (2 + 3i)/(45i)3. (6 + 7i)(3+7i)4. (6 + 7i)/(3+7i)

(b) Expressallabovecomplexnumbersinpolarform,thencalculatethefinalresultinthepolarformats.

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18.2 ComplexExpressions(a) Reducethefollowingexpressionstoasinglecomplexnumber inrectangularform:

(i) (1 + 2i)(34i)(ii) (1 + 2i)/(34i)

(iii) (6 + 4i)(5+7i)(iv) (6 + 4i)/(5+7i)

(b) Expressallabovecomplexnumbersinpolarform,thencalculatethefinalresultinthepolarformats.

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18.3 MatrixOperationPracticeAssumeX= [123],Y = [980] andZ= [s2 s1], where denotestransposeoperation. Calculatethefollowingexpressions:

(a) 5X;(b) XY;(c) XY;(d) FindtherootsoftheequationXZ= 0

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19 RecitationProblems19.1 Recitation1ProblemJusthoweffectivearesnowbanksatstoppingaslidingcar? Wecanmodelthesystemandstudyitsdynamicandstaticbehavior.

xb

m F

x0

Figure19.1.1: SlidingCarModel

Consider the model shown in Figure 19.1.1. A mass m, initially at rest, is subject to a constantapplied force F. After the mass has travelled a distance xo it impacts a damper with dampingcoefficientb. Assumethesurfaceisfrictionless.

(a)

Formulatethe

first

order

differential

equation

describing

the

velocity

of

the

car

as

afunction

oftime,beforethecollision.

(b) What isthevelocityatthecollision?(c) Formulatethefirstorderdifferentialequationdescribingthesystemafterthecollision.(d) What isthetimeconstantforthesystem? How longwill ittaketostopthecar?(e) What isthepeakforce? Asafunctionofxo? What ifxo is0? Whatifb is?(f) Solvetheproblemagainusingenergymethods.

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19.2 Recitation2Problem

bm

k

x

Figure19.2.1: AutomobileSuspensionModel

Thesuspensionforanautomobilecanbemodeledbythespring-mass-dampersystemshowninthefigurewithmassm=500[kg],springconstantk= 4103 [N/m],anddamperb= 2103 [Ns/m].

(a) Determinethecharacteristicequationforthesystem. What istheformofthehomogeneousresponse to an initial condition x0 = 0 and x0 = 1? To an initial condition x0 = 1 andx0 = 0?

(b) Find the damping ratio , undamped natural frequency n, damped natural frequency d,attenuation,andthetimeconstantoftheexponentialenvelope.Hint:

b k= and n =2 km m

(c) Isthesystemundamped,underdamped,critically-damped,overdamped(orjustright)?(d) From the homogeneous response determine if the system is stable, marginally stable, or

unstable. Doesthismatchyourphysical intuition?(e) What iftheshockabsorber is leaky? Discusswhathappenstothesystemasbchanges.(f) WhathappenswhenyoudrivedownMassachusettsAvenueandhitapothole? Thinkabout

the forced response.(g) Ifyouweredesigningthesuspension foracar,whichdampingbehavior(undamped,under-

damped,critically-damped,overdamped)wouldyouchooseandwhy? Forasportscar? ForgrandmasCadillac? Forapickuptruck?

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19.3 Recitation3ProblemConsideraswingingdoor,suchasyoumightfindattheentrancetothekitcheninarestaurantoratthefrontdoortoasaloonintheWildWest. Thesetypesofdoorsdontslam;theycanovershootandoscillate. Theschematicrepresentationofthedoor isshown intheFigure19.3.1.

bJ

k

T T(t)Figure19.3.1: Swingingdoormodel

Thesystemrotatesaboutacentralshaftwithangulardisplacement, moment of inertiaJ, springconstantk,anddampingconstantb. Whensomeonepushesonthedoor,theyapplyatorqueT.(a) Determinethecharacteristicequationforthesystem.(b) Find the damping ratio , undamped natural frequency n, and damped natural frequency

d. Rememberthestandardform:f(t) =

1n2+

2n +

(c) SketchtheresponseofthesystemtoasteptorqueoftheformT(t) =T0us(t)

withinitialconditions0 = 0,0 = 0. Usen =1[rad/sec].(d) Definethemaximumvalueoftheangulardisplacement. Addittothesketch.(e) Definerisetime. Showrisetimeonthesketch.(f) Definesettlingtime. Showsettlingtimeonthesketch.(g) How does the above forced response ofa second-order system comparetothehomogeneous

response?

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19.4 Recitation4ProblemTwo identical tanks with cross sectional area A are partially filled with a volume V of incompressible fluid of density . The fluid flows between the two tanks through a short tube with asmalldiameterwhichcanbemodeledasalumpedresistance,R. Thetopoftank2 isopentotheatmosphere(withpressurePa).Initially,thetopoftank1 isclosedandthepressureatthetopoftank1issetsothatthelevelofthefluid inthefirsttank,h1 issubstantiallyhigherthanthe levelofthe fluid in thesecondtank,h2. At time t =0, the top of tank 1 is opened to the atmosphere. 19.4.1 shows the system justafterthetop isopened.

P

h1

h2RA,U

A,U

Pa

a

Tank 1 Tank 2

Figure19.4.1: TankSetup

(a) How many independent variables as a function of time do you need to completely describethestateofthesystem?

(b) Whatdoyouexpectthesystemtodophysically? Beforemakinganycalculations,sketchtheheightofthefluid intank1andtheheightofthefluidintank2asfunctionsoftime.

(c) Findarelationshipbetweentheheightofthefluidintank1totheheightofthefluidintank2.

(d) What isthepressureP1 atthebottomoftank1? What isthepressureP2 atthebottomoftank2?

(e) Findaconstitutiverelationshipforthe lumpedresistance inthetube.(f)

Find

the

differential

equation

in

h1

forthe

system.

(g) Sketch the height of the fluid in the tank 1 and tank 2 as a function of time, given initial

conditionsh1(0)andh2(0). Willthissystemoscillate?(h) Doesthetimeconstantofthesystem dependontheamountoffluid? Whatcouldyoudo

tomakethesystemrespondtwiceasquickly?(i) This system is first-order, but the demonstration we saw in class on Wednesday of water

flowingthroughatubewasasecond-ordersystem. Whyarethesesystemsdifferent?