2.70/2.77 Week 2 - MITweb.mit.edu/2.70/Lecture Materials/Documents/Week 02/2.70 2017 lecture 2...

2.70/2.77Week2Spring2017

AlexanderSlocumPappalardoProfessorofMechanicalEngineering

slocum@mit.edu

1=

Week2Theme:FUNdaMENTALSofdeterminisIcdesign

•  Week2:•  Reading:FUNdaMENTALSTopics4,5,6,PMDChapter2•  Brainware:

–  DeterminisIcdesignbasedonsimplecheapelements(buildablefromstuffyoucanobtainyourself(thereisnobudgetormaterialssupplied):

•  Structure,bearings,andcarriageforsimple“precision”linearmoIonaxis.–  Makesuretoleaveroomfortheactuator!–  Usespreadsheetstoassignerrors(errorapporIonment)andcreatepreliminaryerrorbudgetsfor“best”

concepts…–  Usespreadsheetstoanalyzestructuralandmachineelementsyouplantousetoensuretheycanhandlethe

loads…•  Design(usethespreadsheet)athreegroovekinemaIccouplingtocoupleanythingyouwant(see

web.mit.edu/2.75forexamples)–  Youcannotjust“printit”:youhavetomakeit!

–  Seek&GeekExploraIon–  Updatewebsite

•  Hardware:–  MakesureyoureallyknowhowtocontroltheArduinocardfromyourlaptopandmakethe

steppermotormove–  MakeyourkinemaIccouplingandusealaserpointera]achedtoitthatprojectsdownthe

halltomeasurerepeatability.

NextWeek3Theme:•  Week3•  Reading:FUNdaMENTALSTopics7,8,PMDChapter3,4•  Brainware:•  A_erbuildingandtesIngyourkinemaIccouplingdesignedlastweek,evolveyour

iniIalspreadsheetstopredictperformance.–  ThisisclosingthelooponyourdesignsandhelpstobuilddesignintuiIon

•  EvolveyourlinearmoIonaxisdesignandmakepartdrawings.–  Designinwheretheactuatorwillgo

•  Seek&GeekExploraIon•  Updatewebsite•  •  Hardware:•  MakeandtestyourlinearmoIonsystem(doesnothavetoincludeactuator).

–  UseamountedlaserpointertotestandrecordchangeinposiIononpieceofpaperplacedfaraway

2/13/17 ©2016AlexanderSlocum 2-4

X

Y

Z

θ

θ

θ

Y

X

Z

Topics•  Accuracy,Repeatability,ResoluIon•  SensiIveDirecIons•  Occam’sRazor•  Newton’sLaws•  ConservaIonofEnergy•  Saint-Venant’sPrinciple•  GoldenRectangle•  Abbe’sPrinciple•  Maxwell&Reciprocity•  Self-Principles•  Stability•  Symmetry•  ParallelAxisTheorem•  StructuralLoops•  Preload•  CentersofAcIon•  ExactConstraintDesign•  ElasIcallyAveragedDesign•  SIckFigures

FUNdaMENTALPrinciplesThesearesoooooocriHcalsotheywillbereviewedincontextof2.70’sthememachine

2/13/17 ©2016AlexanderSlocum 2-5

Accuracy,Repeatability,&ResoluIon•  Anythingyoudesignandmanufactureismadefromparts

–  Partsmusthavethedesiredaccuracy,andtheirmanufacturehastoberepeatable•  Accuracy:theabilitytotellthetruth

–  Cantwomachinesmakeexactlythesamepart?–  Arethepartstheexactsizeshownonthedrawing?

•  Repeatability:theabilitytotellthesamestoryeachIme–  CanthemachinemaketheexactsamemoIoneachIme?–  Arethepartsallthesamesize?

•  ResoluHon:thedetailtowhichyoutellastory–  Howfinecanyouadjustamachine?–  Howsmallafeaturecanyoumake?

•  Howdotheseaffectthedesignprocess?

JosephBrownfromJ.RoeEnglishand

AmericanToolBuilders,©1916YaleUniversityPress

One-inchMicrometer(le_)madebyBrown&Sharpe,1868andPalmerMicrometer(right)broughtfromParisbyBrownin1867fromJ.RoeEnglishandAmericanToolBuilders,©1916YaleUniversityPress

DavidArguelliswins“MechEverest”withamachinethatrepeatseveryIme!

2/13/17 ©2016AlexanderSlocum 2-6

Accuracy,Repeatability,&ResoluIon:Mapping•  Itiso_enmostimportanttoobtainmechanicalrepeatability,because

accuracycano_enbeobtainedbythesensorandcontrolsystem–  WhentheerrormoIonsofamachinearemapped,the controller

multiplies the part height by the axis' pitch & roll to yield the sine error for which orthogonal axes must compensate

Y axis: Can be used to compensate for straightness errors in the X axis.

X axis: Can be used to compensate for straightness errors in the Y axis.

0 50 100 150 200 250 300-1.5

-1

-0.5

0

0.5

1

1.5

position [mm]

pitch

error

[arc

sec]

at 10

mm/

s

raw accuracy:2.44raw repeat:0.5

Crank-bore concentricity

-2.0-1.5-1.0-0.50.00.51.01.52.0

0 1 2 3 4 5 6 7 8

Trial #

δ c, m

icro

ns

JLJR

CrankBoreHalves

Block

Bedplate

AssemblyBolts

EliWhitneyfromJ.RoeEnglishand

AmericanToolBuilders,©1916YaleUniversityPress

2/13/17 ©2016AlexanderSlocum 2-7

SensiIveDirecIons&ReferenceFeatures

•  InaddiIontoaccuracy,repeatability,andresoluHon,wehavetoaskourselves,“whenisanerrorreallyimportantanyway?”–  PutalotofeffortintoaccuracyforthedirecIonsinwhichyouneedit

•  TheSensiHveDirecHons•  Alwaysbecarefultothinkaboutwhereyouneedprecision!

Workpieceinalathe

Tool

SensiIveDirecIon

Non-sensiIvedirecIon

2/13/17 ©2016AlexanderSlocum 2-8

Occam’sRazor•  WilliamofOccam(Ockham)(1284-1347)wasanEnglishphilosopherand

theologian–  OckhamstressedtheAristotelianprinciplethatenHHesmustnotbemulHplied

beyondwhatisnecessary(seeMaudslay’smaximsonpage1-4)•  “Themedievalruleofparsimony,orprincipleofeconomy,frequentlyusedbyOckhamcametobeknownasOckham'srazor.The

rule,whichsaidthatpluralityshouldnotbeassumedwithoutnecessity(or,inmodernEnglish,keepitsimple,stupid),wasusedto

eliminatemanypseudo-explanatoryenIIes” (h]p://wotug.ukc.ac.uk/parallel/www/occam/occam-bio.html)

•  Aproblemshouldbestatedinitsmostbasicandsimplestterms•  Thesimplesttheorythatfitsthefactsofaproblemistheonethatshouldbeselected•  LimitAnalysiscanbeusedtocheckideas

•  Usefundamentalprinciplesascatalyststohelpyou–  KeepItSuperSimple(KISS)–  MakeItSuperSimple(MISS)–  “Siliconischeaperthancastiron” (DonBlomquist)

2/13/17 ©2016AlexanderSlocum 2-9

Axis Move # Force (N)Velocity

(m/s) Distance (m)Efficiency, net system Move

Battery dissipation

Σ power for move #

Energy for move Σ Energy

Drive to pucks 1 3 0.2 1 29% 2.10 8.30 52.0 52.0Lower arm 1 0.5 0.5 0.04 29% 0.88 8.30 11.28 0.7 52.8

Scoop 2 3 0.2 0.02 29% 2.10 3.00 5.10 0.5 53.3Raise arm 3 3 0.2 0.05 29% 2.10 3.00 5.10 1.3 54.5

Drive to goal 4 2 0.2 0.5 29% 1.40 3.00 4.40 11.0 65.6Dump pucks 5 0.1 0.5 0.05 29% 0.18 3.00 3.18 0.3 65.9

Power & energy budget for individual moves, total (S) for simultaneous moves, and cumulativePower_budget_estimate.xls

Last modified 9/01/03 by Alex SlocumEnters numbers in BOLD, Results in RED Power (Watts) Energy (N-m)

Newton’sLaws•  1st,2nd,&3rd“Laws”areinvaluabledesign

catalyststhatcanhelplaunchmanyanidea!–  (Theonlyreal“law”,perhaps,is300,000km/second!)

•  ConservaIonoflinearmomentum–  Ifnoforceisapplied,thenmomentumis

constant

•  ConservaIonofangularmomentum–  Ifnotorqueisappliedtoabodyaboutanaxis,

angularmomentumisconstantaboutthataxis

•  Aforcecoincidentwithanaxisdoesnotapplytorqueaboutthataxis

Table Speed to Free Shot-puts

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50

Table hole diameter

RPM

to fr

ee S

hot-p

ut

Table_rotate_dislodge_ball.xls

Projectile trajectory

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

Time (s)

Dist

ance

(m)

X position (m)Y position (m)

projecHle.xls

2/13/17 ©2016AlexanderSlocum 2-10

Newton:FreeBodyDiagrams&SuperposiHon•  FreebodydiagramsareagraphicalrepresentaIonofNewton’sthirdlaw

–  TheyallowadesignertoshowcomponentsandtheirrelaIonshiptoeachotherwithrespecttoforcestransmi]edbetweenthem

•  Invaluableforproperlyvisualizingloadsoncomponents•  Inordertoproperlyconstrainacomponent,onehastounderstandhowitis

loadedandconstrained

•  SuperposiHonallowsacomplexloadtobebrokenupintocomponentseachofwhichcanbeappliedoneataIme,andthentheirneteffectsadded

Whatsupportstheotherendofthesha_towhichthegearisa]ached?Howwillthegear-toothradialforcesberesisted?AsimpleFBDofeverycomponentcanbeacriIcaldesignsynthesiscatalyst.FDBsarecriIcaltohelpingidenIfyhowtoproperlysupportcomponents!(inafewpages,Saint-Venantwill…)

JustbecausethisdesignSUCKSdoesnotmeanitisavacuumcleaner!

Sureitmightworkforthecontest,butitislikelytofailunderload.Designlikethisinindustryandyourjobwillbesentoffshorepronto!

A B

C

E D

F

A B

C

E

AxAxAxAx

NyE NyD

D

F

C

Cy

Cx Cx

Cy

A B

2/13/17 ©2016AlexanderSlocum 2-11

ConservaIonofEnergy•  Whatgoesinmustcomeout:

efficiency energy outenergy in

force out (N) distance out (m)

torque (or moment) (N-m) distance (radians)

En EE dFE α

× =

= ×

×= Γ

Fin

Fout

a b

d in d out

Ffulcrum

Ødin

Ødout

AB

C

GIn, a in

Gout, a out

in outin out

in outin out

in out

out in

out in

Assume and are known inputs; find and :F1 equation and 2 unknowns

geometric compatability (small angles)

use in the first equation

d dFd dF F

d da b

ad d b

aF F b

× = × ⇒

⇒ = ⇒

⇒ = ⇒

⇒ =

in out fulcrum

in out fulcrum

in out

o

Force and Moment Equilibrium (Newton's First Law):Assume is a known input; find and F

0 0 1 equation and 2 unknowns

0 1 equation and 1 unknowny

A

F FF F F F

a bM F F

= ⇒ − + = ⇒−

= ⇒ × = × ⇒

⇒

∑∑

ut in

fulcrum in out in

use in the first equation

1

aF F b

aFF F F b

= ⇒

⎛ ⎞⇒ = ++ = ⎜ ⎟⎝ ⎠

in outin in out

in outin out

in outout

inoutin

out

outout in

in

Assume and are known inputs ( = ); find and :1 equation, 2 unknowns

geometric compatability2 2d d

dd

dd

πα α α

α α

π α

π π

Γ Γ× = × ⇒Γ Γ

⇒ = ⇒

× = ×⇒ Γ Γ

⇒ =Γ Γ

in in

in outefficiency

inout

Assume torque applied to screw is over one revolution ( =2 )Lead is defined as distance nut travels in one screw revolution

2 1 equation, 1 unknown

2F

F

πα

π η

πη

Γ

× × = × ⇒Γ

Γ⇒ =

ll

l

For a machine of mass m to move a distance x under constant acceleration in time t: 2

2 efficiency

a Fvtx v at F ma Pη

= = = =

Solving for the power consumed 2

2 2 3

2 2 2 4

efficiency

x x xm mxa v F Ptt t tη

= = = =

If the percent weight of the vehicle over the drive wheels is β, then the minimum coefficient of friction between the drive wheels and the ground is:

minimumFmg

µβ

= SeePower_to_Move.xls

SeeScrewforce.xls

SeeSpurgears.xls

t

a

t

v

t

x

x

m

2/13/17 ©2016AlexanderSlocum 2-12

Saint-Venant’sPrinciple•  Saint-VenantdidresearchinthetheoryofelasIcity,ando_enhereliedon

theassumpIonthatlocaleffectsofloadingdonotaffectglobalstrains–  e.g., bending strains at the root of a canIlever are not influenced by the local

deformaIonsofapointloadappliedtotheendofacanIlever•  TheengineeringapplicaIonofhisgeneralobservaIonsareprofound for

thedevelopmentofconceptualideasandiniIallayoutsofdesigns:–  ToNOTbeaffectedbylocaldeformaIonsofaforce,beseveralcharacterisIc

dimensionsaway•  Howmanyseatsawayfromthesweatydudedoyouwanttobe?•  Severalcanbeinterpretedas3-5

–  Tohavecontrolofanobject,applyconstraintsoverseveralcharacterisIcdimensions

•  ThesearejustiniIallayoutguidelines,anddesignsmustbeopImizedusingclosed-formorfiniteelementanalysis

•  OneofthemostpowerfulprinciplesinyourdrawerofFUNdaMENTALSBarrédeSaint-Venant

1797-1886

2/13/17 ©2016AlexanderSlocum 2-13

Saint-Venant’sPrinciple:Structures•  ToNOTfeelsomething’seffects,beseveralcharacterisIcdimensionsaway!

–  Ifaplateis5mmthickandaboltpassesthroughit,youshouldbe3platethicknessesawayfromtheboltforcetonotcauseanywarpingoftheplate!

•  Manybearingsystemsfailbecauseboltsaretooclosetothebearings

–  Bewarethestrainconeunderaboltthatdeformsduetoboltpressure!•  StrainconesshouldoverlapinthevicinitybearingstopreventwavydeformaIons•  BUTcheckthedesign'sfuncIonalrequirements,andonlyuseasmanyboltsasareneeded!

•  ToDOMINATEandCONTROLsomething,controlseveralcharacterisIcdimensions–  IfacolumnistobecanIlevered,theanchorregionshouldbe3Imesthecolumnbase

area•  Toocompliantmachines(lawnfurnituresyndrome)o_enhavepoorproporIons•  DiagonalbracescanbemosteffecIveatsIffeningastructure

2/13/17 ©2016AlexanderSlocum 2-14

Saint-Venant’sPrinciple:Bearings•  Saint-Venant:LinearBearings:

–  MakefricIon(µ)lowandL/D>1,1.6:1verygood,3:1awesome–  EveryyearsomestudentstryL/D<1andtheirmachinesjam!

•  Widedrawersguidedattheoutsideedgescanjamb•  Widedrawersguidedbyacentralrunnerdonot!•  IfL/D<1,actuatebothsidesoftheslide!

•  Saint-Venant:RotaryBearings:–  L/D>3ifthebearingsaretoacttoconstrainthesha_likeacanIlever–  IF L/D < 3, BE careful that slope from sha_ bending does edge-load the

bearingsandcauseprematurefailure–  Forslidingcontactbearings,angulardeformaIonscancauseasha_tomake

edgecontactatbothendsofabearing•  Thiscancausethebearingtotwist,seize,andfail

•  Somesha_-tobearingboreclearancemustalwaysexist

Bad

Good

Stable: will not jamb

Stable: will not jamb with modest eccentric loads

Marginally stable: jambing occurs with small offset loads

2/13/17 ©2016AlexanderSlocum 2-15

Saint-Venant’sPrinciple:Bearings•  Modelofasha_supportedbybearings:MinimizethedeflecIon

oftheendsofthebeam–  SeeBearings_rotary_spacing.xls)

•  Modeloftheeffectofbearingwidth,fricIon,andlengthspacingontheactuaIonforce(drawerjamming)•  SeeBearings_linear_spacing.xls

c2a

mW

Fpull

mFB

mFB

FB

FB

?Y

X

2b

Ouch!

Ouch!0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Pull force offset c/a

Pull

forc

e/w

eigh

t

Coefficient of friction, mu 0.3Drawer weight, W (N) 1Slide bearing spacing, 2a (mm) 300Slide bearing spacing, 2b (mm) 100Pull force offset, c (mm) 50Pull force, Fpull (N) 0.43Bearing force, FB (N) 0.21

Design Parameters Valuesa (mm) 50b (mm) 20c (mm) 250d (mm) 100Diameter, D_1 (mm) 15Diameter, D_2 (mm) 10Bearing radial spring constant, KA (N/mm) 2.00E+02Bearing radial spring constant, KB (N/mm) 2.00E+02Modulus, E (N/mm^2) 6.70E+04Tip force, F (N) 10.00Moment of inertia, I_1 (mm^4) 2.49E+03Moment of inertia, I_2 (mm^4) 4.91E+02Spring force, FA (N) -110.00Spring force, FB (N) 100.00End deflection of just D_2 segment (mm) 1.01E-01End slope of just D_2 segment (rad) 1.52E-03Ratio (deflection left end)/(deflection right end) -0.103Position along beam: 0, a, (a+b)/2, b, c, (c+d) deflection (mm) slope (rad)

0 -1.20E+00 3.51E-0250 5.53E-01 3.54E-0235 2.39E-02 3.52E-0220 -5.03E-01 3.51E-02

250 7.91E+00 3.78E-02350 1.17E+01 3.93E-02

Bearing width, wb (mm) 5.00Diametral clearance loss at first bearing (a) (mm) 0.177Diametral clearance loss at first bearing (b) (mm) 0.176

Bearing gap closure (for sliding contact bearing supports)

Saint-Venant’sPrinciple:Bearings

•  MaximizeAquaIcAvianLinearity:–  Week2BrainwareAssignmentfordeterminisIcdesign,canitbe

somethingyoualsolatertaketoaninterview…

•  ReverseengineertheFRDPARRCforthesliderbelow–  ProfSlocum’sweekendassignment—totalbuildImeabout1.5hours)

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

5.0

4.8

4.6

4.4

4.2

4.0

3.8

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

width/height

roll

angl

e (d

eg)

2/13/17 ©2016AlexanderSlocum 22-17

TheGoldenRectangle

1.618:1 1:1

α

100

162

262

Seeh]

p://lady-macbeth.m

it.edu/2.007/geekdate/

h]p://w

ww-gap.dcs.st-

and.ac.uk/~history/Mathem

aIcians/Pythagoras.htm

l

PythagorasofSamos569BC-475BC

h]p://encarta.msn.com/media_461550503_761579463_-1_1/Leonardo_Fibonacci.html LeonardoFibonacci(1170?-1240?)

•  The proportions of the Golden Rectangle are a natural starting point for preliminary sizing of structures and elements

–  Golden Rectangle: A rectangle where when a square is cut from the rectangle, the remaining rectangle has the same proportions as the original rectangle: a/1 = 1/(a-1)

•  See and study Donald in Mathmagic Land!–  Try a Golden Solid: 1: 1.618: 2.618, & the diagonal has length 2a = 3.236–  Example: Bearings:–  The greater the ratio of the longitudinal to latitudinal (length to width) spacing:

•  The smoother the motion will be and the less the chance of walking (yaw error)•  First try to design the system so the ratio of the longitudinal to latitudinal spacing of bearing

elements is about 2:1•  For the space conscious, the bearing elements can lie on the perimeter of a golden rectangle (ratio

about 1.618:1)•  The minimum length to width ratio should be 1:1 •  To minimize yaw error•  Depends on friction too

2/13/17 ©2016AlexanderSlocum 2-18

Abbe’sPrinciple•  Inthelate1800s,Dr.ErnstAbbe(1840-1905)andDr.CarlZeiss

(1816-1888)workedtogethertocreateoneoftheworld’sforemostprecisionopIcscompanies:CarlZeiss,GmbH(h]p://www.zeiss.com/us/about/history.shtml)

•  TheAbbePrinciple(Abbeerrors)resultedfromobservaIonsaboutmeasurementerrorsinthemanufactureofmicroscopes:–  Iferrorsinparallaxaretobeavoided,themeasuringsystemmustbe

placedcoaxiallywiththeaxisalongwhichthedisplacementistobemeasuredontheworkpiece

•  Strictlyspeaking,thetermAbbeerroronlyappliestomeasurementerrors•  Whenanangularerrorisamplifiedbyadistance,e.g.,tocreatean

errorinamachine’sposiIon,thestrictdefiniIonoftheerrorisasineorcosineerror

εL

L(1-cos(ε)) ≈ Lε2/2

Lsin(ε)

L

2/13/17 ©2016AlexanderSlocum 2-19

Pitch

Roll

Yaw

Abbe’sPrinciple:LocaHngComponents•  Geometric:Angularerrorsareamplifiedbythedistancefromthesource

–  Measurenearthesource,andmovethebearingsandactuatornearthework!•  Thermal:Temperaturesarehardertomeasurefurtherfromthesource

–  Measurenearthesource!

•  ThinkingofAbbeerrors,andthesystemFRsisapowerfulcatalysttohelpdevelopDPs,wherelocaIonofmoIonaxesisdepictedschemaIcally

–  Example:SIckfigureswitharrowsindicaIngmoIonsareapowerfulsimplemeansofdepicIngstrategyorconcepts

OnBrown&Sharpe’sverniercaliper:“ItwasthefirstpracIcaltoolforexactmeasurementswhichcouldbesoldinanycountryatapricewithinthereachoftheordinarymachinist,anditsimportanceinthea]ainmentofaccuracyforfineworkcanhardlybeoveresImated”

a d

( )( )

2

arcsin

1 cos2

H

HH

δθ

δθ

=

Δ = − ≈d

H

q

D

2/13/17 ©2016AlexanderSlocum 2-20

Abbe’sPrinciple:CascadingErrors•  AsmallangulardeflecIoninonepartofamachinequicklygrowsas

subsequentlayersofmachinearestackeduponit…–  AcomponentthatIpsontopofacomponentthatIps…–  IfYouGiveaMouseaCookie…(greatkid’sbookforadults!)

•  ErrorbudgeIngkeepstracksoferrorsincascadedcomponents–  DesignsmustconsidernotonlylineardeflecIons,butangulardeflecIons

andtheirresulIngsineerrors…

MoIonofacolumnasitmovesanddeflectstheaxisuponwhichitrides

R

Tool

WorkError

L

h

a F

Fdh

dv

Beam: modulus E, section I

q

3 2 33 2 2

32

3"transmission ratio"2

vv

vh

h

v

F F dL Ld EI EI Lhdhd L

hdLd

θ

θ

= = =

= =

⇒ = =

2/13/17 ©2016AlexanderSlocum 2-21

Maxwell&Reciprocity

•  Maxwell’stheoryofReciprocity–  LetAandBbeanytwopointsofanelasIcsystem.LetthedisplacementofBin

anydirecIonUduetoaforcePacInginanydirecIonVatAbeu;andthedisplacementofAinthedirecIonVduetoaforceQacInginthedirecIonUatBbev.ThenPv=Qu(fromRoarkandYoungFormulasforStressandStrain)

•  Theprincipleofreciprocitycanbeextendedinphilosophicaltermstohaveaprofoundeffectonmeasurementanddevelopmentofconcepts–  Reversal–  CriIcalThinking

JamesClerkMaxwell1831-1879

Drive point measurement

1 2 3 4 5 6 7 8 9 11 12

10

10. 15. 20. 30. 50. 70. 100.

0.00001

0.000050.00010

0.000500.00100

0.00500

Hz

H(w)

Node for 2 modes All modes visible

ZYX

FRF

1 1! !!

opportunity Ahhhhhproblem Ow

= =

2/13/17 ©2016AlexanderSlocum 2-22

Maxwell&Reciprocity:Reversal

δCMM(x)δpart (x) before reversal

after reversal

•  Reversal is a method used to remove repeatable measuring instrument errors–  A principal method for continual advances in the accuracy of mechanical components

•  There are many applications for measurement and manufacturing–  Two bearings rails ground side-by-side can be installed end-to-end

•  A carriage whose bearings are spaced one rail segment apart will not pitch or roll–  Scrapingthreeplatesflat

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

probe before reversal

probe after reversal

probe before reversal probe after reversal

2

CMM part

CMM part

part

x x xZx x xZ

x xZ Zx

δ δ

δ δ

δ

= −

= +

+−=

Abbe&Maxwell:ElasHcExcepHons

•  1/Rigid=ElasIc….–  ThinkLegos™

2/13/17 ©2016AlexanderSlocum 2-23

2/13/17 ©2016AlexanderSlocum 2-24

Self-Principles•  Themannerinwhichadesignreactstoinputsdeterminesits

output–  ReciprocitywouldphilosophicallytellustolookforasoluIonwherea

potenIallydetrimentalresultcanbeusedtocanceltheeffect–  MarIalarIstspracIcethisprinciplealltheIme!

•  Self-Help:Adesignthatusestheinputstoassistinachievingthedesiredoutput

–  AniniIaleffectisusedtomakethedevicereadyforinputs•  Thesupplementaryeffectisthatwhichisinducedbytheinputs,

anditenhancestheoutput–  Example:Airplanedoorsactliketaperedplugs

•  Whenthedoorisshut,latchessqueezetheseal,makingthecabinairIght

•  Astheplaneascendsandoutsideairpressuredecreases,thehigherinnerairpressurecausesthedoortosealevenIghter

–  Example:Back-to-backangularcontactbearingsarethermallystable–  Example:Icetongs–  Example:Abe]ermousetrap!–  Example:BalancedforcesonhydrostaIcbearings:A.M.vander

Wielen,P.H.J.Schellekens,F.T.M.Jaartsveld,AccurateToolHeightControlbyBearinggapAdjustment,AnnalsoftheCIRP,51(1/200),351-354,(2002)

•  Otherself-principlessimilarlyexist:–  SelfBalancing,Self-Reinforcing,Self-ProtecHng,Self-LimiHng,Self-

Damaging,Self-Braking,Self-StarHng….

2/13/17 ©2016AlexanderSlocum 2-25

Stability

face-to-face mounting can accommodate shaft misalignment

but cannot tolerate thermal expansion at high speeds

Back-to-back mounting cannot accommodate shaft misalignment

but can tolerate thermal expansion at high speeds

•  Allsystemsareeitherstable,neutral,orunstable–  Saint-Venant’sprinciplewasappliedtobearingdesigntoreducethechanceofsliding

instability(e.g.,adrawerjamming)–  Asnap-fitusesanappliedforcetomovefromastable,toaneutrallystable,toanunstable

toafinalnewstableposiIon–  Wheelsallowasystemtorollalongaflatsurface–  Astheloadonatallcolumnincreases,infinitesimallateraldeflecIonsareactedonbythe

axialforcetobecomebendingmoments,whichincreasethedeflecIons….•  Reciprocitysaysthisdetrimentaleffectcanbeuseful:firesprinklersareacIvatedbya

columnthatbuckleswhenitbecomesso_…–  Back-to-backmountedbearingsareintolerantofmisalignment,butuseaxialthermal

growthtocancelradialthermalgrowthforconstantpreloadandthermalstabilityathighspeeds

–  Face-to-facemountedbearingsaretolerantofmisalignment,butaxialthermalgrowthaddstoradialthermalgrowthandcausesthebearingstobecomeoverloadedandseizeathighspeeds

?L

J?

J

mode n k c k c k c k c1 1.875 2.47 3.142 9.87 3.927 20.2 4.730 39.52 4.694 6.283 7.069 7.8533 7.855 9.425 10.210 10.9964 10.996 12.566 13.352 14.137n (2n-1)π/2 nπ (4n+1)π/4 (2n+1)π/2

Cantilevered Simply Supported Fixed-Simple Fixed-Fixed

24 2bucklen

EI cEIk F

L Lω

ρ= =

2/13/17 ©2016AlexanderSlocum 2-26

Symmetry•  Symmetrycanbeapowerfuldesigntooltominimizeerrors

–  Thermalgradienterrorscausedbybi-materialstructurescanminimizewarpingerrors

•  Steelrailscanbea]achedtoanaluminumstructureontheplaneoftheneutralaxis•  Steelrailsonanaluminumstructurecanbebalancedbysteelboltedtotheopposite

side–  AngularerrormoIonscanbereducedbysymmetricsupportofelements

•  Symmetrycanbedetrimental(Maxwellappliedtosymmetry)–  DifferenIaltemperatureminimizedbyaddingaheatsourcecancausethe

enIrestructuretoheatup•  Onlya]emptwithextremecare•  Be]ertoisolatetheheatsource,temperaturecontrolit,usethermalbreaks,and

insulatethestructure–  Alongsha_axiallyrestrainedbybearingsatbothendscanbuckle–  Remember-whenyougeneralize,youareo_enwrong

•  ThequesIontoask,therefore,is“Cansymmetryhelporhurtthisdesign?”

Blocks to push components against precision ground reference surfaces

2/13/17 ©2016AlexanderSlocum 2-27

ParallelAxisTheorem

•  TheParallelAxisTheoremisusefulforcalculaIngthemomentsofinerIaforcomplexobjects

–  ThesIffnessofadesignisproporIonaltothesquareofthedistanceofthecomponentstructuralmembers’neutralaxesfromtheassembly’sneutralaxis

•  The assembly’s neutral axis is found in the same manner as the center of gravity, and it is located a distance yNA from an arbitrary plane

( )21 1

N N

i NAi ii i

I y yI A= =

= + −∑ ∑∑

∑

=

== N

ii

N

iii

NA

A

Ayy

1

1

1.0

1.5

2.0

2.5

0.0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.2 9.1 10.0

Center Element Gap/Beam Height

Nor

mal

ized

Def

lect

ion

Series1

JohnMcBeangoestotheextreme!

y top

y NA

Y

ht

y bottom

w

y i H

y NA

ØDY

2/13/17 ©2016AlexanderSlocum 2-28

StructuralLoops•  TheStructuralLoopisthepaththataloadtakesfromthetooltothework

–  Itcontainsjointsandstructuralelementsthatlocatethetoolwithrespecttotheworkpiece

–  ItcanberepresentedasasIck-figuretoenableadesignengineertocreateaconcept

–  SubtledifferencescanhaveaHUGEeffectontheperformanceofamachine–  ThestructuralloopgivesanindicaIonofmachinesIffnessandaccuracy

•  TheproductofthelengthofthestructuralloopandthecharacterisHcmanufacturingandcomponentaccuracy(e.g.,partspermillion)isindicaHveofmachineaccuracy(ppm)

•  Long-openstructuralloopshavelesssIffnessandlessaccuracy

2/13/17 ©2016AlexanderSlocum 2-29

Preload•  ComponentsthatmoverelaIvetooneanothergenerallyhavetolerancesthatleaveclearances

betweentheirmaIngfeatures–  Theseclearancesresultinbacklashorwobblewhichisdifficulttocontrol

•  AnexampleistheLegorollercoasteronpage3-10

•  Becausemachineelementso_enhavesuchsmallcompliance,andtoaccountforwear,backlashiso_enremovedwiththeuseofpreload

–  Preloadinvolvesusingaspring,orcomplianceinthemechanismitself,toforcecomponentstogethersothereisnoclearancebetweenelements

•  However,thecomplianceinthepreloadmethoditselfmustbechosensuchthatitlocallycandeformtoaccommodatecomponenterrorswithoutcausinglargeincreasesintheforcesbetweencomponents

–  Linearandrotarybearings,gears,leadscrews,andballscrewsareo_enpreloaded»  Onemustbecarefulwhenpreloadingtonottoooverconstrainthesystem!

–  Structuraljointsarealsoo_enpreloadedbybolts

Internal clearance Zero clearance Light preload Medium preload

Load distribution on rolling elements due to radial load applied to bearings with various preload conditions

2/13/17 ©2016AlexanderSlocum 2-30

Centers-of-AcIon•  TheCenters-of-AcHonarepointsatwhichwhenaforce

isapplied,nomomentsarecreated:–  Center-of-Mass–  Center-of-SHffness–  Center-of-FricHon

•  TheCenter-of-ThermalExpansionisthepointaboutwhichthestructureappearstoexpandequallyinalldirecHonswhenheated.

VeeandFlat

DoubleVee

Boxway

0.25

1.00

4.75

2.00

2.50

5.00

ck cg

Force for no tilt due todynamic loading (if

there were no springs)Force for no tilt due

to static loadingK K K

CarriageBearing blocksBearing rails

Center of stiffness (ideal locationfor attaching actuator)

Funnyimagefoundonh]p://zeeb.at/oops/Nothing_Changes.jpg,photographernotcredited,wouldliketo,emailslocum@mit.edu

θ

Lw

Lcg

X

Y

mg

FTr

FTf

FNr

FNf

hmg

2/13/17 ©2016AlexanderSlocum 2-31

ExactConstraintDesign

•  Everyrigidbodyhas6DegreesofFreedom(DOF)–  Anexactlyconstraineddesignhasnochanceofdeformingor

havingitsfuncIonimpairedbeitbyassembly,fastenerIghtening,thermalexpansion,orexternalloads

•  Makesureyouhaveconstrainedwhatyouwanttoconstrain!

–  ForabodytohaveNdegreesoffreedomfreetomove,theremustbe6-NbearingreacIonpoints!

–  ToresisttranslaIon,aforceisrequired.–  ToresistrotaIon,amoment,ortwoforcesacIngasacouple,isrequired!

•  Saint-Venantrules!Donotconstrainasha_withmorethan2bearings,unlessitisverylong…

Overconstrained Properly constrained

Whippletree (aka wiffletree)

2/13/17 ©2016AlexanderSlocum 2-32

ElasIcallyAveragedDesign

Curvic coupling

Bulkhead

Disk spring washers

Turret index gear

Stepper motor

Piston

Sleeve bearing

Turret

Error(µm)

•  ApplyingReciprocitytoExactConstraintDesignimpliesthatinsteadofhavinganexactnumberofconstraints,havean“infinite”numberofconstraints,sotheerrorinanyonewillbeaveragedout!

–  Legos™,fiveleggedchairs,windshieldwipers,andGeckosarethemostcommonexamples,andmanymachinecomponentsachieveaccuracybyelasIcaveraging

K.Autumn,Y.Liang,W.P.Chan,T.Hsieh,R.Fearing,T.W.Kenny,andR.Full,DryAdhesiveForceofaSingleGeckoFoot-Hair,Nature.405:681-685(2000)

2/13/17 ©2016AlexanderSlocum 2-33

SIckFigures•  Useoffundamentalprinciplesallowsa

designertosketchamachineanderrormoIonsandcoordinatesystemsjustintermsofasHckfigure:

–  ThesIcksjoinatcentersofsIffness,mass,fricIon,andhelpto:

•  Define the sensitive directions in a machine

•  Locate coordinate systems•  Set the stage for error budgeting

–  ThedesignerisnolongerencumberedbycrosssecIonsizeorbearingsize

•  It helps to prevent the designer from locking in too early

•  ErrorbudgetandpreliminaryloadanalysiscanthenindicatetherequiredsIffness/loadcapacityrequiredforeach“sIck”and“joint”

–  AppropriatecrosssecIonsandbearingscanthenbedeterminisIcallyselected

•  Itisa“backwardstasking”soluIonmethodthatisveryverypowerful!

ExamplesofDeterminisIcDesign

•  IdenIfyCosts,Physics,Risks,CMTHENcreateFRs….ANDTHENASALASTstepDesignParameters

•  WindPower•  DirecIonalDrilling

2/13/17 ©2016AlexanderSlocum 35

HydrocarbonswithSNFstorage•  Oilgotusintothismessanditcangetusout…

–  Cantheoilindustrycanbethesavioroftheplanet?•  DeepgeographicalformaIonmappinganddeepdrillingtechnologyleaders

–  DeepBoreholeDisposal•  Boredeephorizontalholesneareachreactor•  Dropspentfuelin,curvedholetoslowitdown…•  Newdrillingtechnologymakeitpossible

2/13/17 36©2015AlexanderSlocum

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