2.70/2.77 week 2 - mitweb.mit.edu/2.70/lecture materials/documents/week 02/2.70 2017 lecture 2...
TRANSCRIPT
2.70/2.77Week2Spring2017
AlexanderSlocumPappalardoProfessorofMechanicalEngineering
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
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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
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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)
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60.0
70.0
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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,[email protected]
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
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HydrocarbonswithSNFstorage• Oilgotusintothismessanditcangetusout…
– Cantheoilindustrycanbethesavioroftheplanet?• DeepgeographicalformaIonmappinganddeepdrillingtechnologyleaders
– DeepBoreholeDisposal• Boredeephorizontalholesneareachreactor• Dropspentfuelin,curvedholetoslowitdown…• Newdrillingtechnologymakeitpossible
2/13/17 36©2015AlexanderSlocum
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