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/ h t t p : / / w w w . h
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Mechanical springs, by A.M. Wahl ...
Wahl, A. M. (Arthur M.), 1901-
Cleveland, O., Penton Pub. Co., 1944.
http://hdl.handle.net/2027/uc1.$b76475
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/ h t t p : / / w w w . h
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/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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PUB I H IN GCO .,C E V E L A N D13,O H IO ,U.S .A .
F S T E L . MA C H IN E D S I GN . TH E
W E Q U IPM N T DI G S T • R E V I TA I N D U T R IA L
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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ManufacturingCompany
H I N G CO M PA N Y
D ,O H I O
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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PUB I H IN GCO MPA N Y
D ,O H I O
t . , W e st m in s te r , S . W . 1
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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undamentalprinciplesunderlying
pringsand bringstogetherincon-
erofmachinesthe moreimportant
eoryandtestingwhichhaveta en
lthoughmechanicalspringsoften
ponentsofmodernmachinesandde-
ngstoooftenhavebeendesignedon
ruleof thumb methodsw hichdo
e limitationsofthematerialsused
esset upduringoperation.Thisis
tionswherefatigueorrepeated
conse uence,unsatisfactoryoperation
hasresultedin manycases.
thatthepresentboo maycontribute
idanceofsuchconditionsbyhelping
springsonamorerational basis.
esearchescarriedoutunderthe
esearchCommitteeonMechanical
canS ocietyofMechanica lEngineersand,
heWarE ngineeringBoardSpringC om-
utomotiveEngineers, havebeenf reely
aterialin theboo hasalsobeenbased
heTransactionsof theA . . M. . , the
anics,theS .A . .Journalandtheauthor s
n MachineDesignduringthepast
eofthehelical compressionor
argeamountof spacehasbeende-
nlyhavethetheoreticalaspectsof
pe ofspringbeentreatedin con-
emphasisalsohasbeen laidupon
chsprings,as wellasonthe fatigue
singeneral.This hasbeendone
periencethatthe lim itationsduetothe
aterialsareaptto beoverloo edby
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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antaspectsofthehelical springdesign
chaptersincludecreep effectsunder
lateralloading,vibrationandsurging.
thefundamentalsofdesignof
esincludingdis ,Belleville,flat,leaf,
ngspringshave beentreated.Be-
ance,thevolute typeofspringhas
abledetail.
dinarilythoughtof asaspring
applicationofrubberspringsandmount-
adetheinclusionofa chapteronthis
ble. S incethesub ectofv ibrationisin-
plicationof rubbermountings,some
rationabsorptionandisolationalso
thatinaboo o f thisnature it
thefactorswhich enterintothe
enapplication.Conse uentlythe
esultsinanyparticulardesign,close
ntainedbetweenthedesignerandthe
dertobenef itf romthelatter se per-
ow ever, ak now ledgeof thefunda-
udgingthefeasibilityofadesign.
wouldnothavebeenpossible
estinghouseE lectricandManu-
ctiontheauthor sthan saredue,
hubb, Director, R esearchL aboratories,
, Manager, MechanicsDepartment. Than s
e llso f theL aboratories, andtoTore
y , B. S t e r n e an d H . O . F u c h s o f th e S . A . .
dS pringCommittee.Theencouragement
or sresearchbyS . T imoshen oand
ppreciated.ToL .E .Jermy,E ditor,
W. Greve, A ssociateE ditor, thew riter
ablesuggestionsconcerningthepres-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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pringDesign1
S pringMaterials—TyposofLoading—
S urfaceC ondit ionsandDecarburization—
ariationsinDimensions—F actoro fS afety
ompressionandTensionS prings25
ppro imateTheory—E x actTheory , Ef fect
pringsw ithLargeDef lection50
eetoR otate S tress, Def lection, Unw inding
p r in g s wi t h E n d s F i e d A g a i n st R o t a t io n D e -
tress
stsonHelica lSpringsandSpringMateria ls. . . 69
DeflectionTests—V ariationsinModulus
a ining, SurfaceDecarburization—Determina-
ty DeflectionMethod,DirectMethod,
od—ModulusofR igidityV alues Car-
l loy , Sta inless, MonelandPhosphorB ronze,
atigueTests S mall-sizeS prings,S hot
prings, F ew S tressCycles
rStaticL oading95
glectingC urvature—L oadforC omplete
nofF ormulastoSpringTables C urvature
e la ationTests—A naly tica lMethodsof
Creep, R e la ation, S hearStresses
oadingofHelica lS prings119
S ensitiv ity toStressC oncentration, A p-
tationsofMethods—ComparisonofTheo-
A lternativeMethodofCalculation
DesignofHelica lC ompressionSprings134
sUsedinPrarice—S pringTables—Design
ation
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tionsforH elicalCompressionS prings.157
rns—E ccentricityo fLoading, A llow able
ngTolerances,Deflection—E ffectofMod-
ssatS o lidC ompression O ver-stressing,
tresses
ompressionSprings169
o a d , H i n g ed E n d s, F l e u r al R i g i d i ty , S h e a ri n g
ds—Def lectionUnderC ombinedL oading—
imumSpaceE f f iciency183
o l id a n d F r e e -H e i g h t V o l u me , I nf r e u e nt L o a d-
, Ma imumEnergyStorage—S pringNests
taticLoading
gs StressandDeHectioninEndL oops,
fE ndC oils, Wor ingStresses—C om-
ssionS prings
nular-WireC ompressionS prings203
, S mallP itchA ngle MembraneA nalogy
r i n g s of S m a ll I n de S m al l P it c h A n g l es ,
x actTheory—R ectangular-WireS prings
hartsforCalculatingS tress,Calculationof
hA ngles—TestsonSq uare-WireS prings
mulas
gofHelica lS prings222
esonance, Principa lF re uencies, Surge
onforV ibratingSpring A cce leratingF orces,
t u ra l F r e u e nc y S p r in g E n d s F i e d , O n e
n e S p r in g E n d W e i g h te d — S u r g i ng o f E n g i ne
ignE x pedients
leville)S prings238
S implifiedDesignforConstantL oad
heory—W or ingStresses—F atigue
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prings263
gs—S pringsofC onstantThic ness—L arge
dCalculation
gs28C
apezo ida lP rof i leSprings—S impleLeaf
oading—PlateSpring—S tressC oncentration
tches, S harpB ends, C lampedE nds
gs314
dtions—B inding—B uc ling—WireS ection—
arTorsionS prings—CircularW ireTor-
i n g S t r e ss e s
sWithoutC ontact C lampedOuterE nd,
pringswithF ew Turns—L argeDef lections,
nerR ing, OuterR ing—Def lection—Design
le B ottomingL oads, Def lection, Stress—V ari-
L oad, Def lection, TypeC urves
Mountings378
impleShearS pring—C y lindrica lS hear
ight, C onstantStress—C y lindrica lTorsion
c ness, ConstantS tress—A llow ableStresses
S teady - tate , Shoc
cityofV ariousS prings399
prings—C antileverSprings R ectangular
eafSpring—H elicalTorsionS pring
ularW ire—S pira lSprings—R oundB ar
lCompressionSprings—TensionS prings
eCapacities
operties—E nduranceL imits—Descrip-
dMateria ls.
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BO L S
ec/in.2
rala istocentero fgrav ity inch
nginch
city lb/ s in.
ationcycles/ sec
tyin./(sec)
b/s in.
nch
rea in.
ertiain.4
—
/ in.
ionfactor—
tationfactorduetocurvature—
threductionfactor—
concentrationfactor—
cientofmodulusdegrees '
on sratio—
wistingmomentsperinchin.-lb/in.
mentsperinchin. - lb/ in.
omentin.-lb
mberofleaves,etc —
frubberslab —
oils(he lica lspring)—
ieldinglb
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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BO L S
—
area lb/ s in.
l loadlb
engthlb/in.
s
omentin.-lb
springperunitvolumein.-lb/in.
runitvolumein.-lb/in.
rdinates—
aminch
matanypoint inch
edegreesorradians
radius. —
ctiondegreesorradians
ardegreesorradians
melb/in.
ngleradians
#/dt)radians/sec
-
n—
in.
riablecomponentsofnormalstresslb/ s in.
t re s s lb / s i n .
ionlb/ s in.
mtinbendinglb/ s in.
asedonstrengththeory . lb/s in.
l st r es s es l b /s i n .
in.
riablecomponentsofshearstresslb/s in.
hearlb/ s in.
on,angulardeflectionradiansordegrees
y , circularf re uency radians/ sec
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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/ h t t p : / / w w w . h
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S P R IN G
N S I D R A T IO N S
I GN
ybedefinedasan elasticbody
odeflector distortunderloadand
shapewhenreleasedafter being
stmaterialbodiesareelasticandwill
not allconsideredassprings.Thus
willdeflectslightlywhenaweightis
tisnotconsideredasa springbecause
todeflectunderloadbut ratherto
elicalspringusedinthe ordinary
esignedso astodeflectby arelatively
d.Conse uently,thedeflectionand
ned.This,therefore,functionsasa
tstressedbeyondthe elastic
ingwillhaveastraight-lineload-
wninF ig.2.Thismeansthatthe de-
e load,i.e.,iftheloadis doubled
ed.Therelationwill holdtrueeven
ueormoment,providedlineardeflec-
rdeflection.
r load-deflectiondiagrams,how-
eflectiondiagramsasshownin F ig.
maybeobtainedwithathin flatcir-
gedeflection.CurveBmayresultfrom
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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S P R IX G
orBelleville)spring.Thesetwospring
aptersX V andX IV , respectively . B e-
ct betweenturns,theordinarycloc
a lineartor ue-anglecharacteristic,
Y III.
S P R I N G
tionsofspringsthe followingare
nt:
andMitigateS hoc : Inordertoabsorb
epea loadsthespringmustdef lectby
fie S ons
calesprings
n e ampleof theuseofaspringtoab-
arspringshowninF ig. 4. A nother
vespringsforindependentsuspension
hichmustbe abletoabsorbtheenergy
esoverabump.Thesealsofunctionas
evehicle.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
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S I GN CO N S I D R A TI O N S
orceohTor ue: A n automotiveva lve
hichholds thevalvefolloweragainst
gsuppliesthetor uenecessary toover-
vingmechanism.S ometimesspringsare
-def lec-
as etpressureasin thehigh-voltage
etspringsshowninF ig.6.Thefunction
tainanoil-tightgas etsealregardless
emperaturechange. Loc w asherscommonly
headsalsofunctionessentiallyassprings
oad-
f
gardlessofslightchangesinthe bolt
mperaturechanges,etc.Thisforce
romunwindingeventhoughvibration
assesonToIsolateV ibration:The
edto supportmovingorvibratingmasses
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
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S P R IN G
evibrationorimpact.Thuselectric mo-
gsupportedtopreventthetransmission
tontothefoundation. Li ew ise , thesprings
sionsnotonlytendto mitigateshoc
roadsurface(Point1)but alsopre-
rStee lC o.
ew of ringspringindraftgear
ecar bodyofob ectionablevibration
waves(wash-board)intheroadcon-
gsona railwaycar,F ig.7,tendtoprevent
tshoc sf romtruc irregularit ies. A n
heuseof springstosupportavibrating
anautomaticwashingmachine,F ig.
iblyonsprings,transmissionof
dmassesinthe eccentrically-loaded
oadorTor ue: Oneof themostim-
sforindependentsuspensionoffrontwheels
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
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S I GN CO N S I D R A TI O N S
sis thatoffurnishingafle iblemember
siderableamountwhensub ecttoa
eof suitablemechanismsthisdeflection
whichindicatestheamountofloador
e isthesca lespringinF ig. 1.
P ivotorGuide: Sometimesoneormore
ncombinationtofunctionasan elastic
nalfrictionsuch pivotsoftenhavereal
sorantifrictionbearings.Thusthefle ible
untingsfortheMt. Palomartelescope,
formainta ininggas etpressure
agroupofstraightbars radiallydisposed
iscanbe consideredessentiallyasaspring
cpivotis re uired.A napplicationwhere
isinthebalancing machine,F ig.10.
I A L S
allydeflectbya considerablede-
owsthatarelativelylarge amountof
enthespring isinthe deflectedposition.
andload formostspringsarepropor-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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S P R IN G
energyisproportionaltodeflectiontimes
eraltheamountofenergywhichmay
othes uareof thestress. H encefor
wor ingstressesmustbeused.This
gmaterialshavehightensile strengths
higherstresses thaninotherfields.
erialforsprings atpresentiscar-
zesofwire,foran .8to.9per cent
gand patenting,ultimatetensile
0,000to400,000poundspers uare
ofthewire)maybeobtained.This
nasmusicwire.In thelargersections,
bonsteelandheattreatingafterform-
mayreach200,000to 240,000pounds
larva luesmaybeobtainedf romchrome-
Propertiesofvariousspringmate-
springsonN ew Y or C entra l" Mercury
erX X III.Inthe largersizes,helical
atedafterforming,the latterbeing
fhelicalsprings, ontheotherhand,are
etemperedmaterialormusicwire.
lievinglow-temperatureheattreat-
ordifferentspringmaterials,includingdataonheat treat-
rX X III.
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S I GN CO N S I D R A TI O N S
arepresent,stainlesssteelsprings
8percentnic e lcomposit ionaref re-
rial(whichisalso usedforhigh-tem-
yhaveatensilestrengthvaryingfrom
ortforautomaticw asher
areinchfor3 16-inchwireto280,000
or1/32-inchwire.Phosphorbronzealso
present.Thismaterial,however,has
thanstainlesssteel,valuesofultimate
0,000poundspers uareinchare
A DIN G
anycases, springsaresub ecttoa load
stantor whichisrepeatedbuta few
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
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S P R IN G
espring. Suchspringsarek now nas
x amplesaresafetyvalvesprings
ctedtopopof fbutafew timesduring
ucinggas etpressure , typif iedby thecon-
/ g. 6 springsincircuit-brea ermech-
eroperatesbutafewtimes initslife.
loadedspringsitisfre uentlyim-
ntainits calibrationtoasufficient
faspringcompressedbya given
time goeson,theloadshouldnot drop
mount(usuallyafewpercent). This
sk now nasrela ation. F ore ample,
hereis somelossinloaddue tore-
ftera periodoftime,thevalvewill pop
hat forwhichitwasdesigned.S imilar-
ducegas etpressure , re la ationof the
lementsforgimbalo fMt. Palomartelescope
ofpressure.W hilesomelossinpres-
ted,toomuchrenders thespringin-
asealingdevice.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
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S I GN CO N S I D R A TI O N S
toaconstantloadratherthanacon-
stressistoohigh,there willbeaslow
susuallyk nownassettageandis due
susedinbalancingmachine
hematerial.Inmanycases thisisun-
spring isusedtosupporta givenload
-actioncar,settageofthespringwill allow
elativeto thecar.S uchadeflection
thesteeringmechanism.A nother
asespringsshow ninF ig. 11. If settageof
,ob ectionablelossinpressurebetween
dwirewouldresult.Thereforethe
othat thissetisk eptsmall.
s, if thepea stressinthespringis
iclimit oryieldpointofthe material,
a tionwillse ldomoccur. H ence indesign,
arenotinvolved,thema imumstress
y ieldpo intdiv idedby thefactoro f safety .
o f safety ista ene ua lto1. 5a lthough
n manycases.S omeofthefactors
ofa factorofsafetyarediscussed
estheef fectsofcreeporrela ationare
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
e designstressshouldbebasedon
ontests.Unfortunately,notmuchdata
todesigners,althoughsometests were
P.Z immerli-.Thesetestsindicatedthat
prings,theeffectsoftemperatures
ahr.werenotverypronounced.
ahr.theuseofstainless-steelspringsis in-
furtherdiscussedin ChapterV .
b ecttostaticloadingit issug-
pringmaterialwhichhassomeduc-
tas muchasstructuralmaterials),
tssuchasthosedue tocurvaturemay
mple , inasimpleplatew ithasmallhole
nload,theoreticalanalysisbasedonthe
hat,forelasH cconditions,thepea
oleisaroundthree timesthestress
orfatigueorrepeatedloadingthis pea
ever,forstaticloadstheusualpractice
glectsuchstressconcentrationeffects4,
esarelocalizedandmaybe relievedby
uenceofthematerialductility.A vailable
fact thatsimilarstressconcentration
tocurvatureinhelicalsprings maybe
dsareinvolved.
manyspringapplicationsthe load
butvariesw ithtime. F ore ample , an
initiallycompressedbya givenamount
goperation,itis compressedperiodic-
nt.It may,therefore,beconsidered
mumandama imumloadorstress.
issaidtobesub ecttovariableorfatigue
chstress cyclesforsucha springsub-
c stressbetweentheminimumstress
mvaluea „ ^ . Thisise uiva lenttoastaticor
atureonC oiledS tee lSpringsatV ariousL oadings , Transac-
1 9 41 . P ag e 3 63 . A l n " R e l a a t io n R e s i st a nc e o f N i c e l A l l o v
ta l. T ransactionsA . . M. . , In ly, 1942. Page465.
engthofMateria ls, PartII. SecondE dit ion. 1941. V anN ostrand,
scussionof stressconcentration alsoTheoryofE lasticity.
75.
ionarticlebvC . R . S oderhergon" W or ingStresses ,
..1933,A PM55-16isrecommended.A lsoTimoshcn o— S trength
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S I GN CO N S I D R A TI O N S
altoha lf thesumofma imumandminimum
mposeda variablestress< t,.Thevari-
a lf thedif ferencebetw eena, „ x andami ,
eingconsidered.Thediscussion con-
thischapteralsoapplies,in general,
oadingconditionismuchmore
icatedinF ig. 12. F ore ample , an
springs
spring. F ig. 5, issub ecttopractica lly
ar istravelingoversmoothpavement.
roads,however,thespringmaybe
oadingconditionwithstresscyclesof
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
atedinF ig.13.Thesamethingis also
motivesprings.Insuchcasesthe de-
tressesismoredifficult,particularly
ngresults forcaseswherethevariable
geswithtimearealmost whollylac ing5.
gsaresub ecttoloadsor stresses
ontinuouslybetweenaminimumand
dif ferencebetw eenthema imumand
ow nasthestressrange thisisa lsotw ice
stressar.This stressrangeisof
nfatigueorrepeatedloadingisin-
erialstheendurancerangeis prac-
heyieldpointis note ceeded.
s avwhichthespringwill j ust
emeanstressa onw hicho> issuper-
agramsuchasthatshownin F ig.14
pointP onthiscurvemeansthat,if
variablestresslargerthana, if super-
willeventuallycausefatiguefailure
ana,willnotcause fatiguefailure
twhenthestaticcomponentof
onditionofcompletelyreversedstress
ressac forzeroa shouldtherefore
atigueF a iluref romStressC yclesofV ary ingA mplitude , Jour-
s,December,1937,givesafurtherdiscussionofthis problem.
vcrstressintheF atigueofF errousMeta ls , byR usse lland
. . T . M. , 1936, Part2, page118.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S I GN CO N S I D R A TI O N S
durancelimita> forcompletelyre-
on orbending(dependingonwhether
sareconsidered).O ntheotherhand,
ow, testsshowthatthecurvetends
trengtha . O fcourse, forhighvalues
ecreepmaybe e pectedtooccurso
esofvariableamplitude
oneofavoidinge cessivesettageor
enduranceorfatiguetest results
onsisshowninF ig. 15. Heretw o
ngo- „ uanda, ( respectivelyareplotted
anddash linecwhichrepresentsthe
peri-
C T0
d bcurvesrepresentupperandlower
stre uiredtocausefatiguefailurefor
ntsonthesamevertical line.Itis clear
nateofthemeanstress linecrepre-
omponentofstressa whilethevertical
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
nstresslineand eithertheupperor
blestresscomponento> .Theupper
atw hilethe low ercurvebrepresents
mumandminimumvaluesofstresso-ma
toanycombinationofstatic andvari-
maybereaddirectly fromthecurves
fdeterminingw or ingstressistore-
ecurveby astraightlineconnecting
heendurance lim ita , asshow nin
wever,beconsiderablybelowthe
thatwor ingstressesdeterminedin
whattooconservative.H owever,the
manycaseson accountofitssim-
nationofstatic andvariable
nypointPon thelineA Bconnecting
vnanda / nisdef inedashavinga
isthestaticcomponentof the
, thevariablecomponent, f romthege-
ybeshownthatthefactorof safetymay
methodof rep-
results
s
dbyS oderberg—" F actorofS afetyandW or ing
onsA . S . M. E . , 1930. A PM52-2.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S I GN CO N S I D R A TI O N S
owingrelation:
onas wellasforbending,pro-
, areta enasthey ieldpo intand
respectively.Torsionstresses,how-
ytheCree lettert.
ostspring steelswillusuallybe
calrelationshipbetweenthe values
sesnecessaryto causefatiguefailure.
ndurancecurvewill berepresentedby
pse, A C B inF ig. 17andintersecting
Tvandt, . If itbeassumedthatno
eyieldstrength,thedottedportionof
acedby the lineC B e tendingatan
abscissae.Thisisdonesince, when
ponentsofstress representedby
hema imumstressw ill j uste ua l
asisof thisellipticallaw,thefactor
onof
a -
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
esamemeaningasthose usedin
tests areplottedinF ig.17to
madebetweenthe ellipticalandthe
nthe caseoftorsionalfatiguestress-
ttheresultsobtainedby Weibel7
eelwiretestedbothin pulsating(0to
edtorsion,thesurfaceof thewires
d condit ion. Whiletheseresultsshow
eamongthe fewavailableforsuch
n F ig.17representtestresults by
sofsilico-manganesespringsteelwithma-
seen thatinthesecasesthe elliptical
test results.Itshouldbenoted,
N S T R E S S L B ./ Q . IN .
T IGU T S T B W E IB L O N T MP R E D
E L W IR E . ( U R F A C -A S R E C IV E D )
A T IGU T S T B H A N K I N S O N S I -MN
. ( U R F A C MA C H IN E D )
verepresentingfatiguefailureandcom-
ancetestresults
inelaw,F ig.16,issaferto usein
ecompletetestdataarelac ing.
meanstressa ,theconsensusof
pringWireB endingandTorsionTests —E . E . Weibe l,
.,N ovember,1935,page501.
e T es t s on S p r i n g S t e e ls — G . A . H a n i n s, D e pt . o f S c i en t if i c &
it ish)Specia lR eportNo. 9.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S I GN CO N S I D R A TI O N S
stress-concentrationeffectsmaybe
rials9.Thisisconsistent withneg-
oneffectswherestaticloadsonlyare
pea sduetocurvature inhelica l-com-
ctinfatigue
n,.7per
lundEisen,
S S
sarelocalized,it isbelievedthatfor
beconsideredasdue tostresscon-
cemaybeneglectedin computingo- .
eeffectofdirectshear inhelical-
dbeconsidered,becausethisis not
g thevariablecomponenta> ,how-
maynotbeneglected.
rtofthis methodliesincertain
rsunder combinedstaticandvariable
ch test1 on.7percent carbonsteel
,thefull linesbeingresultsforspeci-
ntrationandthedashedlines for
ybeseenthat themeanorstatic
efinedbyS oderbergasthosehavingelongationsover5
ostspring materials.
oudremontandB enne , S tahlundEisen, V o l. 52, page660.
. PetersonofR eporto fR esearchC ommitteeonF atigueof
. T . M. , 1937, V o l. 37, Part1, Page162.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
otand dashlineisnot diminished
neffect,whilethevariablestressrep-
tancebetweeneitherthe fulllines
nishedinamoreor lessconstant
trationeffectofthenotch.W hileit
lablefatiguetestdata madeforthe
ss-concentrationeffectsundercom-
ressare rathermeagre,itisbelieved
areathand,stressincreasesdue to
gsmaybe treatedinthismanner.
ethodtothedeterminationofw or ing
giveninChapterV I.
ingtheproblemofcombinedstatic
alculatethestressrangebyta ing
cts(duetocurvature,fore ample)into
wherestressconcentrationispres-
elocationof stressconcentrationwill
ntsince thematerialwillmerelyyield
ema imumpointo f thestressrange
tstressofthematerial.To determine
durancerangeof thematerialiscom-
culatedinthismanner . A further
thatthestressat ma imumload,
ressconcentration,mustnote ceed
erial,sinceotherwisee cessiveyield-
onofthismethodinthe designof
tratedinChapterV I.
T O P R A TIO N
ecttore lative ly few cyclesof load-
ingstressmaybeconsiderablyin-
eforaninfinite number.E x amples
usedincertaincontrol mechanisms.
raphforhelicalcompressionspringsof
mzerotoama imumisshow nin
e factthatthematerialmaynot befullysensitiveto
thattheactualstress rangeasfoundbytestmay begreater
sinutheoreticalstress concentrationfactors),areduction
emade,providedtestdataareavailable.F urtherdiscussion
Two-andThree-DimensionalCasesofS tressConcentration
tigueTests PetersonandWahl, Journa lo fA ppliedMechanics,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S I GN CO N S I D R A TI O N S
orthistypeof stressapplication,the
failurein tenthousandcyclesofstress
wiceasgreat asthatre uiredto
cycles.Providedsomepermanent
ionable,thissuggeststhataconsiderably
ouldbeusedif thespringistobesub ect
C E S T O BM UR E
-cyclecurveforhelicalsprings
es.It mustnotbeinferredhowever,
ewor ingstressisusuallypossible
tivelyfew.Inmost casestheincreased
ablyinterferewiththeoperationof
pringsforman integralpart.
D ITI O N S A N D D CA R B UR I Z A T IO N
surfaceconditioninspring steels
hefatiguestrength ofthematerial.The
emanufactureofsprings,becauseof
eatingandformingoperations,the
edtosomee tent.Thusthereis,in
carbonsteel(whichisrelatively
thespringw hichiscomposedof the
bonoralloysteel.Underfatigueor
nsthewea erlow-carbonsteelonthe
c whichthenspreadsacrossthe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ceofthehighstress concentrationatthe
tua ltestshaveshow nthatavery thinlayer
alissufficientto greatlywea enthe
ntheef fecto f surfacecondit ionson
ngshas beencarriedoutbythe
oratory inEngland. Theresultso f this
used inleafspringsshowconclusively
ronthe surfacecombinedwiththe
tofsurfaceirregularitiesproducedby
smayreducetheactualendurance
lessof thattobee pectedonthe
dorgroundspecimens. F ore ample ,
ch barsofheattreated.61per cent
steel(asusedin leafsprings)made
howedanendurancerangewiththe
undof0 to128,000poundspers uare
asleftuntouched,the endurancerange
spers uareinch,areductionofabout
ecurvesofF ig.20areplottedfrom
perimentersandshowthetremendous
tionsin thisparticularcase.A lthough
nunusuallygreatreductionin strength,
owimportantarethesurfaceconditions.
ta inedbyH an insandF ord tw ho
nesesteela± 60,000poundspers uare
ersedbendingon specimenswhich
rgrindingto size.Inthiscase there
elayerleftthere bytheheat-treating
weremadeonspecimensof thesame
eat treatmentbuthavingathinlayer
offafterheat treatment,theendur-
103, 000poundspers uare inch. F ur-
ecimenswhich hadbeenheattreated
such awayastopreventthe forma-
r inthiscasethe endurancelimitwas
gthofC arbonandA lloyS tee lP latesasUsedforL aminated
g InstituteofMechanica lEngineers, 1931, Page301.
tallurgica lP ropertiesofS pringStee lsasR evea ledbyLab-
insandF ord- JournalIronandStee lInstitute , 1929, N o. 1,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S 1 GN C O N S ID R A T IO N S 21
uare inchorpractica lly thesameasfor
rheat-treatment.Thisindicatesthat
bythe usualheattreatmentwasto
bleforthelowerendurancelimits found
otbeenmachinedafter heattreatment.
elyinterestinginthattheyaffordani n-
neby meansofspecialheattreat-
tiguestrengthof actualsprings.
estrengthbecauseof surface
rvedinreversedtorsion fatiguetests
4onchrome-vanadiumspringstee lw ires.
that,wherethewireshadbeenma-
C E S O F S T R E S S
I CA T IO N S F R O M Z E R O T O MA X I MU M)
ecursesfor. 61percentcarboncom-
romtestsbyBatsonand
.,1931,Page301)
rsionalendurancelimitswerein-
evalueobtainedfrom specimensin
n. Sw an, Sutton, andDouglas1 1a lso
umsteelunderpulsatingtorsional
o m eS t e el W i r e s U nd e r R e p ea t ed T o rs i on a l S t r es s es — L e a
ngsInstituteofMechanicalE ngineers,1927,Page403.
nofStee lsforA ircra ftE ngineV alveS prings , P roceedingsInstitute
s,1931,V ol.120,Page261.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
imumtothema imum), anincreaseof
urancerangewherethespecimens
ed.Theresultsofthese varioustests
esurfaceconditionsof springsteels
estressing.O ntheotherhand,it
orsionfatiguetestson S wedishvalve
showedpracticallynodifferenceinthe
etweenspecimenswiththesurface
ngthesurfacelayergroundoff. Prob-
nedby thefactthatthismateria lhasa
nsothatthe effectofdecarburization
t-blastinghelicalsprings hasbeen
stheendurancerangeby50per cent
rings16.Thisprocessconsistsofpro-
highvelocityagainstthespringsur-
centrifugaltypeof machine.A ppar-
steel shotpropelledagainstthesur-
oco ldwor andthusincreasethe
decarburizedsurfacelayer.Thismethod
calwayofobtainingsatisfactory
utthe e penseofgrindingthesur-
owever,shotblastingorshotpeen-
nnotbee pectedtogivesatisfactory
decarburizationorsurfacedefectsare
ussionof thisisgiveninC hapterIV .
E F F E C T
sub ecttoevenmildlycorrosive
stressing,theendurancelimit for
reducedgreatly.In suchcases,fa-
outformanymorethanthe usual
argenumberofcorrosionfatiguetests
doutby McA dam18,showthetre-
H ow ShotB lastingIncreasesF atigueL ife , MachineDesign,
lsoL essellsandMurray—" TheEf fecto fS hotB lastingandIts
P roceedingsA . . T . M. V o l. 41, 1941, Page659.
ueofMeta ls —H . J. C ough, E ngineer, V o l. 154, 1932, Page284.
sionF atigueofSpringMateria ls . —D. J. McA dam, Jr. ,
.,1929,A PM51-5.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S I GN CO N S I D R A TI O N S
ndurancelimitfor springmaterials
orsa lt-watercorrosionfatigue. F orspring
watercorrosionfatigue,thevalueof en-
sbutone-fourthtoone-ninththat ob-
nsin air.H ighervaluesofendurance
tionswereobtainedoncorrosion-
mium-platedspringsshowedmuch
dersuchconditions,i.e.,abouttwice
a springsteelwithaplating than
sshowthatthespringdesignermust
omcorrosiveaction, orelseusee
sses.E venthen,ifcorrosionis present,
eventualfatiguefailurewillnotoccur,
erof stressrepetitionsofstressta e
I N D IM N S I O N S
espringdesignershouldk eepin
sanunavoidablevariationinthesize
a ingsprings. Theef fecto f these
ge,especiallywhenitcomes toob-
tioncharacteristics.F ore ample,in
,acumulativevariationinbothcoil
percentwillresult ina7 percent
oncharacteristicofthe spring.Thus
entvariationwouldcorrespondto
nly.001-inch.S uchvariationsareeasily
ctice.H ence,itmaybenecessary
cturersomeleewayinchoosing the
compensateforunavoidablevaria-
ia lw irestoc . F ore ample, if thew ire
shappensto beslightlyundersize,the
beabletocompensateforthis by
ameter.In mostcases,thisslightre-
uldnot bedetrimentaltotheopera-
mercialsprings,actualstressesand
sticsmayeasilydeviateby5to10 per
luesandthis mustbeconsideredin
onsaretobeta enby thespring
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
maybereduced, butatincreasede -
cussionof thisisgiveninC hapterV III.
A F E T
safety , n(asdef inedinE q uation1)
dby manyconsiderations.Ifthe
areserious,thena higherfactormust
enspringcausesbutlitt le inconvenience,
designertolowerthe factorofsafety.
ofuniformandhigh-gradematerial
he manufacturingprocessismain-
etymaybeused.If, inaddition,
particularspringmaterialsemployed
rethetest conditionsappro imate
designfactorof safetymayagainbe
and, ignoranceof thepea loadsacting
wnfactorssuchascorrosionor tempera-
increasein thisfactor.
o f thisboo toac ua intthede-
alsunderlyingspringdesigninorder
nintelligentselectionofsprings for
heless,itisadvisable,incaseswhere
onsareconcernedtohavethe design
withthespringmanufacturer,inorder
se perience.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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U N D -W I R E CO M PR E S S I O N
S P R IN G
nceinmachinedesign arehelical
rtensiontypes.Theyaremadein
dusedintremendousq uantities.
ideacceptanceandgeneraluse are
ringsarerelativelycheapto manu-
eenoughq uantitiesarere uired
maticspring-windingmachinery.
relativelycompact,aconsiderable
ueezedintoasmallspace.
erialisstressedfairlyefficientlyunless
ofcoildiametertowirediameter)
discussedinChapterX X II.
e helicalspringisas broad
self. S everalofthemoreimportant
licalspringshavealreadybeen men-
omotivefieldthese includein-
rontwheels,F ig.5,suspensionofrear
R ailroadsarelargeusersof helical
ghtandpassengercar suspensions.
ricale uipment,springsareused
sinswitchgearandcontrole uipment,
sms,etc.Innumerableotherapplica-
ned.A typicalapplicationofaheli-
ermechanismisshow ninF ig. 21.
portanceofthis typeofspring,
o f space isdevotedto it inthisboo .
ssesthetheoryfor stressanddeflec-
rings, theapplicationofthistheory
coveredinsubse uentchapters. The
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
pterwillbelimitedto springswhere
nottoolarge(notmore thanhalf
of largepitchangles,however,
desmostpracticalsprings.
ulatinghelicaltensionsprings
hat usedforcompressionsprings
eeffectsoftheend loopswhichare
ngs,additionalconcentrationsof
. F orthisreasona low erw or ingstress
aspecial typeofendfasteningis
r,effectsdue toendturnsboth
nspringsare e cludedfromthe
theseeffectsareconsideredlaterin
Y TH E O R Y - L A R G IN D X A N D S MA L L
E
rings,theelementarytheoryas
boo sonstrengthofmateria lsorma-
heassumptionthatthespringmay
asastraight barundertorsion.This
ate ly truewherethespringinde is
angle issmall. Sincethee lementary
oaccountthedifferencein fiberlength
tsideofthe coilwhicharisesbecause
ingbarorwire,considerableerror
oryisusedforsprings withsmall
eflytheelementarytheoryis as
inde underana ia lloadPas
ressedbetweentwoparallelplates as
esultantload ingeneralwillbeslight-
sshown.Thiseccentricityis neglected,
scussion.R eferringtoF ig.22,each
pringcoil maybeconsideredto
emomentPrw herer= meancoilradius.
e lementsof lengthtl iscutf romthe
pitch anglescombinedwithlargedeflectionsareconsidered
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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IN G 27
diculartothebara is. A ssumingthat
r distortduringdeformation,itfol-
rmationsandhencethe shearing
stributionalongaradiusas shown.
tressdistributioninastraight bar
tadistancep fromthecenterO
nofhelica lspringinacircuitbrea er
t— 2ptm/d(fromsimilartriangles)
umshearingstressatthesurfaceof thebar
hemomentta enupby theshaded
uspw illbedM= 2nrp-dp 2ptm/ d)
mentPrw illbe
- - C 3,
mumstressT ,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
for calculatingstressinhelical
nte tboo sorhandboo s. A sstated
erableerrorforspringswithsmall
(1)The effectofdirectshearstress
sneglected and(2)The increase in
ceinfiberlengthbetweentheinside
n g o l a rg e i nd e ,
producedbywirecurvatureis not
willbemorefullydiscussedlater.
Tocalculatedeflectionofthe
duremaybeemployed.Consider-
urfaceofthebar andparallelto
se lement, a fterdeformation, w illro tate
thepositionac.F romelastictheory
a ltot, div idedby theshearingmodulus
Eq uation4.
7d , forsmallanglessuchasare
mentaryangledctthroughwhichone
respecttotheotherw illbee ua lto
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
umingthatthespringmaybeconsideredas
2irnrwheren— numberofactivecoils,
tingthe angulardeflectionofoneendof
eotherw illbe , usingEq uation5,
mentarmof the loadP ise ua ltor,
willbe
formulafor springdeflections.In
stressformula,whichmaybe incon-
c-
tionformulaisq uiteaccurateeven
esandforlargeheli angles. Tests
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
eaccuracyof thise uationarediscussed
TH E O R Y — S MA L L O R M O D R A T IN D X C O N -
eofaheavyhelical springwhich
gisshownin F ig.25.Itwill benoted
afatiguecrac neartheinsideofthe
angleofabout45degreestothea is
ailuresare typicalofheavyhelical
erathersmallinde es, itmaybee -
umstressoccursatthe insideof thecoil
6a. Thereasonsforthee istenceof thema i-
re:F irst,thefiberlengthalong the
sswhich meansthatahighershear-
ivenangularrotationof ad acent
g. 26b, if theradia lsectionsbb and
allanglewith respecttoeachotherand
inside(andshorter)f ibera b w illbe
ershearingstressbecauseofits short
erah whichislonger.S econd,the
a b isincreasedbecausetheshear
ia lloadP isaddedtothatduetothe
ispo int. Intheoutsidef iberabthisstress
tothetor uemoment.Theresult
nsideofthe coilreachvaluesaround
outsideforspringsof inde 3, asmaybe
perIV ) andtheory forlargerinde es
A T UR E E F F E C T
nalele-
n
e fracturedsurfacema inganangleof 45degrees
fatiguefractureofastraightcylindricalbar underalter-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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IN G
,not sopronounced.
ee actso lutionof theproblemof
gsofsmall inde iscomplicated
ro imatesolutionwhichissufficiently
withinabout 2percentfor practical
follows3:
isassumedsincethisassumptionisvalid
gs. Consideringanelementofan
thmeanradiusofcurv aturercutby two
ectionsaa andbb asshowninF ig.
efailure
hiselementareresolvedintoa twisting
aradia lplaneandadirecta ia lshear-
etupbythis twistingmomentare
differsinseveralrespects fromanappro imatesolu-
cr, " B eanspruchungZ y lindrischeSchraubenfedemmitK re is-
913.Page1907,butthefinal numericalresultsareonlyslightly
hor spaper" S tressesinH eavvClose lyC o iledH elicalS prings ,
.,1929paperA .P.M.,51-17.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
resuperimposedonthe stressesdue
omentM= Prthetwocross-
w illro tatew ithrespecttoeachotherthrough
ntionedbefore,thiswillresult inmuch
e fiber
springsof
tress^ act-
nF ig. 27f
edinto
, para lle lto
ndatrans-
ndicular
o
saa and
achotherand
rpendicularto
through
butionof
thestress
erper-
a isw ill
the
uchadis-
toa
siblesince
center
othe le f tand
ecouldbe
brium.If,however,rotationoccursabout
29, whichisdisplacedtow ardthea iso f the
ntO ,adistributionofstress isobtained
actionofapuremomentM. F rom
etransversestresscomponentst will
mwhenrotationoccursaboutanypoint
. 27b. Po intO' maybefoundasfollows:
rotationaboutO' , thestressract-
withcoordinates.vandy maybefound.
andbb haverotatedthroughasmallangle
calspring
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
her,therelativemovementoftheends
e sp o nd i ng t o d A w i ll b e d /8 ( 2 + { / 2 ) ' 4
co ilo fhe lica lspring
d is(r-y - )dtl, theshearingstress7
be
ofthisstresswill be,F ig.27,
ade,thisdistributionofstress is
vedbar4andthedistributionof the
cn o, S (rmcf/ infMateria ls, V anNostrand, Part2, S econd
en o, Mrenf if /infMateria ls, va
discussionof curvedbartheory
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
cinform, F ig. 29. Thisdistancey isde-
onthatthe integra lo f r dA (w here
ea)mustbezerow henta enoverthe
edbartheorythedistance * maybe
at e ly a s r :
tedin thedenominatorsincein
mgreaterthan1/3 andhencet/2/16r-
y. PuttingE q uation10in9,
at theordinaryformulaforangleof
uation6)w illapplyw ithsuf f icient
ressdistributiona longa
tionaboutthecenter
nofdfi/dd (borneoutbyactualtests,
).Thus
hisinE q uation11,
ge7-1.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
sclearthatthema imumvalueof t
2—d2/ 16r, i. e ., a tpo inta inF ig. 27b.
butionalongatransversediameter,
hepointO '
uation13andalsoputtingthespring
essata inF ig. 29becomes
tpo inta( ig. 29)ontheoutsideof
d/ 2—d2/ 16rinE q uation13anddropping
a loadPactsa longthea is, as
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ee ternalforcesactingoverthecross
toa twistingmomentM= Pranda
,assumingthat thepitchangleis small.
ng momentmaybefoundbysubstitut-
ons14and15. Onthesestressesthe
( ig. 27b)duetothedirecta ia lload
tappearsreasonableto ta eforthis
eoryofelasticityatthe outeredges
cantileverofcircularcross section
heory6givesavalueof stresse ual
isstressto thatduetothemomentPr
m a i m um s t re s s t, u . a t a m ay b e e p r es s ed
pressionforthema imumstress
g,a iallyloaded.Comparingthe
his formulawiththoseofamore
Goehner(seePage42),itmaybe
pringswherethe inde cise ua lto3
ithin2percentof themoree act
esarenegligiblefromapractical stand-
oagreesw ellw ithe perimenta lresults
mentsonactualsprings(ChapterIV ).
fthecoilat a,( ig.28b),may
ation15andsubtractingthestressdue
actsintheoppositedirection,giving
imenta lresultsindicatesthatthise ua-
eycorrect.
thema imumshearingstressinahelica l
oryofE lasticity , McGraw-Hill , Page290.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
nfactorK is
a imumstressissimply thestress
mulaofEq uation4multipliedbya
thanunity andwhichdependsonthe
nvenience inca lculationva luesofK are
springinde cinF ig. 30. It isseen
3thefactorK = 1. 58, w hichmeansthat
X
ectionfactorforhe lica lround
nsprings
dinaryformulamustbemultipliedby
imumstress.
ethat thestressesderivedhold
ditionsprevail,i.e.,aslongasthe yield
ematerialisnote ceeded. If thisisnot
stressmaybelessthan thatcalculated.
caseswhereyieldingoccurs,the formula
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R 1N G
tresswhichis ofmostimportance
urtherdiscussionoftheuse of
e loadingisgiveninC hapterV I.
Y
developedintheprecedingsec-
helicalspringsofsmallor moderate
reviously,sufficientlyaccurateformost
beingaccurateto within2percent
erthanthree).W heregreateraccuracy
thodofcalculationdevelopedby
ismethodwillbe brieflyoutlined8.
eferringtoF ig. 31, if t9-andrroare
gstressactingon theelementA ofa
springasshown,thecoordinatesofA
ch angleissmallsothat theelements
sideredunderpuretorsion,allshear
eptt : andrromaybeassumedzero.
mthetheoryofelasticity,the condi-
indricalcoordinatesgivethe follow-
uation
ore uires• thatthefollowing
w hatarek now nasthe" compatibil ity
satisf ied:
0( 21 )
spannungsverte ilungimQuerschnitte inerSchraubenfeder ,
0, Page619 " S chubspamiinigsverteilungimQuerschnitte ines
A nw endungaufS chraubenfedern , Ing. -A rch. V o l. 2, 1931,
gsverte ilungine inemandenE nd uerschnittenbelastetenR ingstab-
o l. 2, 1931, Page381 and" DieB erechnungZ y lindrischeS chraub-
rch12, 1932, Page269.
S. T imoshen o, McGraw -Hill , Page355, givesamore
method.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G 39
,21and22 astressfunction< /> is in-
I d< h \
d. B ysubstitutionofEq uation23in21
uationsareobtained
= 0 ( 24 )
d t > \
pressioninparenthesismustbea
enotedby—2c . Thus,
0 ( 26 )
eoustointroducenewcoordinates
omes
0 ( 27 )
beconsideredsmall,
)
solveE q uation27bymeansof a
imations.F romtheconditionthat
ssatthe boundaryofthecrosssection
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
undary,thefunction< f> maybeshown
boundary . Withthiscondit ionthee -
s:
( 29 )
0
, 3 d < t > o
3 $ d $ o
0 e t c.
, us i ng P ~ r - - i ,
actingoverthespring crosssec-
f r(3D
t/ r)- inEq uations30maybee -
ollows:
3 2 )
ndi, bymeansofsuccessive
ncarriedoutin thisgeneralmannerby
ngresultsforround-wire,helical
ssforacircular ringsectorwithzero
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
1 6 )
withinonepercentevenfor
scomponentsincross-
e acttheory )
esnot applyaccurately,however,
ppreciable.
rethe pitchangleais notzero,
a forma imumshearstressbecomes
actualradiusofcurvatureof theheli ta -
count.Thefirsttermof thise uation
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S P R IN G
on33wheretheactualradius ofcurvature
econdtermof thise uationarisesbe-
he shearloadPcosa. Thetermin
econdparto f therightsideofEq ua-
heseries whicharisesinthecalcula-
ation34isconsideredveryaccurate , it is
calculations forsuchcasesthefollow-
with anaccuracywithinonepercent:
reaterthan3andforpitchanglesa
a imumshearingstressis
/ j _ y v
2 P / J
rthan4anda< 20degrees.
s(whichincludesmostpractical
ula fort, „ . , e pressedinterms
onvenientandisto bepreferred:
c T + - cd ( 37 i
osa= l. E q uation37dif fersby
q uation16derivedbyappro imate
esof threeormore.
vetheshearingstressduetothetwist-
andthedirectshearPcosa. How ever, there
mu presentduetothebending moment
tensionorcompressionstressduetothe
tthema imume uiva lentstressinthe
o- , „ . rmustbecombinedw iththetor-
basisofa strengththeory(Page44).
umbendingstress(T, „ x theresults
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
ybeused.A somewhatmoreac-
he generale uationsofthetheory
Goehner .Thisinvolvesessentially
uil ibriume uationsandthecompatibil ity
nding<pin
ulafor
and
A N D IA M T R
T R
oordinatesandtheirsolution bya
ro imations,similartothatpreviously
torsionstresst, , , . F ina lresultsare:
stress:
m + 4 / d
" 2p
onstant= reciproca lo fPo isson sratio .
. 3thisformulasimplifiesto
Thelastterminthebrac etsd/ 8ry ie ldsthe
ace381. A lsoTheoryofElasticity—Timoshen o, Page361.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R I N G
nsionorcompressionwhichis
nderyieldsthestressdueto thebending
thedenominator1—d/ 2p represents
swhicharisesfromthe methodofsuc-
s.
needonlybeusedforre latively large
de es. F ortheusua lcasethefo llowing
sufficientaccuracy:
4
m 8
rrespondingtoPo isson sratio= . 3
tressinthespring,the shearing
dingstresso- , u . w hichactatagiven
dbefore,becombinedaccordingtoa
strengththeorywhichiswidelyused at
m-sheartheory,whichstatesthatfailure
imumshearingstressatanypointin a
shownfromelastic theorythatifa
dashearing stresstm< uactata givenpoint,
resst, basedonthema imumshear
mu f 1 +
whichiscomingmoreand moreinto
heory1:(alsok nownasthevonMises-
heorystatesthatfa ilurew illoccurw hen
yof distortion)ofthehigheststressed
oshen o—S trengthufMateria ls, Purt2, SecondE dit ion, Page
ussionol strengththrones.
, Part1, Page122.
art 2,Page479givesafurther discussionofthistheory.
. X uda i, Eng. S oc. Monographs, McGraw -H ill , 1931.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
shearenergyofane lementinana ia lly
eldpoint (orattheendurancelimit
atiguefailure).Ifo- a2,ando-3are
ure,a mathematicalstatementofthe
r ) + ( < r l — < r 3 )- = 2 oc = C on st an t (4 3)
thertotheyield pointortheendur-
ftside ofthise uationcanbe
othe shear-energyorenergyofdis-
al.This energyise ualtothetotal
nergyofthree-dimensionaltensionor
nabendingstresscrl nranda
simultaneouslyasinthiscase, thee uiva-
gto theshear-energytheoryis
bydeterminingtheprincipal
clefor acaseofcombinedtensionand
seprincipalstressesare
( r , ) ( 4 5)
t ^ Y ( 46 )
ncenonormalstressactsonthesurface
ingE q uations45and46inEq uation42
a lentshearstressiy= av / 1. 73(w hichis
lenttoasimpletensionstressa- , ) , Eq ua-
nusuallylargeamax willbesmall
formulasforprincipalstress forcombinedtensionand
ngthofMaterials,loc.cit.,Part1, Page48.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
sothatingenera lformostpractica lsprings
romt , x .
eformulasgiveninthischapter
asthedeflectionsperturn aresmall
er)so thatthecoilradiusand pitch
onstant.Theseconditionsusually
acyinpractical springswherethe
eforsmallormoderate inde ese cessive
hedeflectionsperturnapproachthe
ampleof theapplicationof themore
hischapter,assuminga springwithan
anglea= 12degrees, sothatEq uation
s< x = . 978andtana= . 2126. Using
a
tion41
44 4 s in a
imum-sheartheoryapplies,bysub-
2,
beta enasabasis, bysubstitu-
mthe valueobtainedbyusingthe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
.Italsodiffersbyonlyabout1^ percent
viouslybyappro imatemethodsand
( q uation16). Thisindicatesthat
disaccurateenoughformostpurposes.
ssumingthatthespringdeflec-
1relativetothecoil radiusandthatthe
redverysmall,the springdeflec-
. It isseenthatthisissimply theor-
q uation7multipliedbva termin
sonthespringinde . The largerthe
rw illthistermapproachunity . How -
ptiona lly smallinde o f3, theterminthe
on48ise ua lto . 977w hichindicatesthat
2.3 percentsmallerthanthat cal-
mula.H owever,forbestaccuracy
considered.
ingis considered,theprocedure
tictheory1 itmaybeshow nthatthe
elicalspringis
naland initialpitchangles,respectively.
ectionis smallsothatthecoil radius
ant.Multiplyingthis bythetorsional
ngmomentM, = Prcosoc. Li ew ise
notw illbee ua ltothef le ura lrigidity
acte pressionforthechangeincurva-
tions(whichmay occurwithoute cessivestressonlyfor
dinC hapterIII.
2, Page272.
e—TheoryofE lasticity , third. edit ion, CambridgeUniv . P ress,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R I N G
ngwireorbar willbe
sandtheresultsof elastictheory,amore
thedef lectionofhelica lspringshasbeen
rateformula,whichassumessmall
n
a t a n a (5 2 J
figuredbyusualformula,E q uation7,
dingonthespringinde c= 2r/ dandon
. 32va luesof theconstant<phavebeen
pring inde cforvariouspitch angles
ulations, aratioG/ - . 38w asta en(cor-
tely toPo isson sratio= . 3), w hichho lds
rmostspringmaterials.H owever,
nthisratioG/ w il lhavebuta
ueof i^ .Thedashedcurveforzero
zedin practicalspringsbecauseof
coils alsoapartof thecurvefor5
wndashedsince itcannotberealized.
cticalspringswherethepitch angle
greesandforthesmallerinde esthe
y,whichmeansthatthe actualdeflec-
thenominaldeflection,E q uation7.
st sinceonewoulde pectthatthe
ncreasethedeflectionoverthatgiven
tobedescribedlater(C hapterIV ), how -
conclusion.Itshouldbe notedthat
tisusua llyverysmall thusforinde es
15degrees,thedeviationsbetween
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
aandtheordinary formulaEq uation7
nt. F ormostspringsw herea<10de-
enceisunder onepercent,a figurewhich
tice sinceotherfactorssuchas the
andwiresize,shapeofend turns,
ordinarilybe greaterthanthis.In
sfore ampleincertaininstrument
mumaccuracy,itmaybedesirableto
uation52inma ingdeflectionca lculations.
gastee lspringof thefo llowingdimen-
in.,meancoilradiusr= .286in.,wire
tveturns, springinde (2r/ d)=3. 28,
grees,wor ingload140lb,the deflec-
nalformulaforthese dimensionsis
tchangleof7% degreesandinde c= 3. 2
985. Thedef lectionca lculatedby themore
= . / , „ = . 985(. 0745)= -0733in.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S PR I N G
W ITH L A R G D F L E CTIO N S -
y isre uiredthetheorydeveloped
edhelicalspringsissatisfactoryfor
spring deflectionsandstresseswhere
der10degreesand thedeflectionper
coilradius.H owever,forcases
e islargeorwherethe deflectionper
ntheuseof theusualformula , Eq ua-
g deflectionswillresult.Thiserroi
initialpitchanglesaround20 de-
urne ualtotheinitial coilradius.The
usualformulaispartly thatthe
eroin thepreviousderivationand
us changeswithdeflection.Thus
giscompressedfromthe initialposi-
thatshowninF ig. 33b, themeancoil
tor. S incethespringdeflection, other
oportionaltothecube ofthecoilradius,
becomesmoref le ibleasit iscom-
ct,ofcourse,occursin tensionsprings.
he effectofpitchanglemaybe
reaccurateformula,E q uation51,
gle intoaccount.Ifthespring deflec-
owever,thisformulawillalsobesome-
deflectionssincethechangein coil
neglectedinthe derivation.
pterwillbelimitedto springs
effectsdueto changesinthepitch
mostpronouncedinsuchsprings.
moderate inde highstressesaresetup
meslargeenoughsothatchanges in
areof importance. H enceEq uation51
s.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S PR I N G
eica lspringissub ecttoana ia l
ction,thereis atendencyforthe
rwords,oneendof thespringtendsto
eotheraboutthespringa is. If this
reelyandwithoutrestraint,we have
lly- loadedspringasindicatedinF ig. 34.
econditionintensionspringswith
ehoo sarenotrigidlyhe ldbuthavesome
espringa is.If,however,theends
nbyfriction(as isusuallythecase
by clamping,endmomentsacting
ila resetupw hichtendtopreventthis
itis necessarytodistinguishtwocases:
ringa ia lly loadedandw ithendsf reeto
isand
aendsaref i edsothatrotationaboutthe
d.
N D F R E E TO R O T A T
nopen-coiledhelicalspringof large
ig. 34issub ecttoatensionloadP ,
f reetorotateabouttheco ila is. If
ngleandrtheactua lco ilradius, the
gontheelementA oflengthdswill
momentPrsin a,atwistingmoment
sa andatensionforceP sina.The
w istingmomentPrcosaw illbee ua l
he torsionalsectionmodulusfora
Thus
stressaduetothebendingmoment
nde c
isassumedlarge, stressesduetothe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ndthe tensionPsina willbeneglected
elementofthesurfaceofthe coil
reactingasindicatedinF ig. 34c. A s
hapterIIthesestressesmaybe com-
imum-sheartheoryand,therefore
used. Hencethee uiva lentshearstress
tgivenbyEq uationsS 3and54
ee pressionfore uiva lentshear
a = — — — ( 56 )
under theradicalisunity.
heoryapplies, thise uationshow s
uiva lentshearstressise ua ltothatgiven
Pr/W ' regardlessofthepitchangle
m- hearandS hear- nergyTheories
/ t c
nddirect shearbeingneglected),pro-
a enasthatactua llye ist ingwhenthe
w ninF ig. 33, thisw illbedif ferentf rom
atzero load.
rgytheoryofstrength(asdiscussed
andtgivenbyE q uationsS 3and54
ation43. If thisisdoneane uiva lent
ua lto
n -a ( 5 7)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S PR I N G
enE q uations56and57show sthat,
spring,thedifferencebetweenthe
ories (ma imum-shearandshear-
tforpitchanglesunder 20degrees
nglesbelow30degrees,( TableI).
encebetweenthetwotheories,
springwithlargedef lection
uation56willbeused inthefollowing:
simplerforthedesigner touse
o loadasabasisforca lculationsince it
sured.Itisshown,Page57,that foran
eactua lco ilradiusisgivenby r= K 2r
f theratio8, , / nr betw eennomina lde-
coilradius,andofthe initialpitch
eatzero load). Thenomina ldeflectionS „
ydeflectionformulausing theinitial
, is
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
essedasafunction8 /nr andofa and
comes, f romE q uation56,
)
ssedsimplyastheordinaryformula
multipliedbyafactorK 2w hichmaybe
s of F i g . 35 i f < x a n d $ / n r a r e k n o wn .
aybeca lculatedf romEq uation58foragiven
thatnegativeva luesof8 / nr cor-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S P R I N G 55
positivevaluestoe tensions.The
unityforopen-coiledcompressionsprings
asesasthespringis compressed the
nsionsprings.
. 35itmaybeseenthat, if the init ia l
greesandthecalculateddeflection
rethanthe init ia lcoilradiusr , the
aduetopitchanglechanges areunder
reach15per centiftheangle a
edeflectionper turne ceedsthe
t actualapplicationswherex ,<
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
< . 5, theerrorisunder3percentand,
edformostpracticalpurposes.
der todeterminetheeffectof
reasein stressduetobar curvature
enneglected. A sanappro imationto
ess,thestressesfiguredinthis way
ythe curvaturecorrectionfactorK
9. E venforspringinde esbetween10
yfrom1.07to1.14andis thusofim-
ate,butmorecomplicated,methodis
, niandama givenbyE q uations34and38,
s A ssumingahelica lspringa ia l-
eetorotateasthe springdeflects,F ig.
ryofelasticity1that thechangeinbar
pringdeflectsfromaninitial pitch
hangleais
hecoilradiicorrespondingtothe init ia land
eThise uationissim ilartoE q uation49
nco ilradiusisconsidered. F romF ig.
usingthischange incurvatureis
stbee ua ltothef le ura lrigidityEI
incurvatureA k . Thus
felasticity,/= momentofinertiaofcross-
t ion60:
mayalsobeshow nthatthetw istA f
asthespring deflectsfromapitch
ea is
lasticity , C ambridgeUniversityP ress, ThirdEdit ion, Page421.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S PR I N G
.C O S a c o
etorsionalrigidity GI (for
wistingmomentmt.Thislatteris,
cosa . Thus
o s a \
dityandIp= polarmomentofinertia
nd63,
- a
-
elengthI ofthespringremainscon-
pringlengtliwith
forsprings oflargeinde )thespring
ontheheli cylinderasindicatedin
ometryof thisf igureastheanglechanges
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R I N G
ringdef lectionS becomes
rnr , w herenisthetota lnumberofactiveco ils,
ritten:
* „ ) ( 66 )
nd66thetota ldeflection8of thespring
enomina ldeflection8 (ca lculatedf rom
edbyafactorThus
D F L E C TI O N P R t UR N
R A D IU
ndingdeflectioncorrectionfactorV ' , ,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S PR I N G
( 6 7)
ntheinitial pitchanglex .,theratio E /G
armoduli, andontheratio8 / nr be-
perturnandinitial coilradius.V alues
ensiondiagramsforopen-co iledhelicaltension
rotate
tedforaratioE/ G— 2. 6(whichapplies
springsteels)and theresultsplottedin
esofaandSc/ nr . A lthoughava lueof
gtoPo isson sratio= . 3hasbeenassumed,
erablechangemaybehadin thisratio
hangein thefinalresults.
o fF ig. 37show sthatforpitchangles
ectionsperturn lessthanhalfthecoil
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
theerrorintheusua lspringdef lectionformula
ent,i.e.,doesnotdifferfrom unity
percent.F orpitchanglesaround20 de-
urne ualtothecoil radius,however,
rcent.
alcalculationsof open-coiledhelical
sarytodeterminethedeflection8 for
ordinaryspringformula,E q uation58,
esorcharts. F romthisva lueof8 the
T N S I O N P R T UR N
D IU
culatingtwistofspringends.
reeto rotate
ound. Thenk now ingthisandthe init ialpitch
/ < , maybereadf romF ig. 37. Thema imum
Pw illthenbee ualto^ 18 .
ectiondiagramsdeviatefroma
al pitchanglesa andforvarious
curvesofF ig.38havebeenplotted,
tensionspringswherethe endsarefastened
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S PR I N G
ationaboutthe springa isissmall.
representvalues ofPX 64r,r/Gd and
theload,while theabscissasrepresent
rndividedbyinitial coilradius.These
d,whichmeansanincreasein spring
eflection)withload.Thiswouldbe
ngssincethe coilradiusrdecreaseswith
hlinerepresentsthedeflectionas fig-
la.In thiscaseitmaybe seenthat
tchangles thereisaconsiderablede-
erepresentingthe ordinaryformula.
nds—Whenatensionspringise -
n,thecoilstendto unwind,atleastat
ountofthisunwindingmaybe cal-
mF ig.36,theanglein radianssubtended
totalspring lengthintheunloadedposi-
ulartothea isw illbe
anglewillchangeto
rtherelativerotationofone endwith
ethedifferenceinthesevalues.
2 w nr / c os ( „ ,
ssedinradians.
nd67thisanglemaybee pressedin
r asbefore . Ex pressing$indegrees
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
rturnin degreesandmaybereadfrom
/ nr arek now n. Itw il lbenotedf romthisf ig-
tlynegativefor smallvaluesof8 /nr,
greaterthanzero.Thismeansthat
ringhas aslighttendencytowind up.
thedistortionsof theelementsofthe
insuchadirectionastocausethiswind-
ngdeflectionincreases,this tendencyis
geinpitchangle whichcausesanun-
N D F I X E D A G A I N S T R O TA T IO N
Wherethespringendsaref i ed,
ngaboutthea isofthespring during
realizedin manycompressionsprings
he licalspring
ntsare
ntheendsand thesupportingplate
,asimilaranalysismaybemadeto that
occurs withoutrestraint.Inthiscase,
a e intoaccountthemomentactingat
entsthe coilsfromunwinding.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S PR I N G
famomentM anda loadPareacting
slyasindicated,thebending andtwist-
ting onthewirewillbe
(72)
(73)
saandM siniofthe a ialmoment
ctorsinF ig. 40b.
the wireA k duetothemoment
entdiv idedby thef le ura lrigidityEI.
0becomes, f romE q uation62,
o s a .
ons72, and73inEq uations74and75
btained,fromwhichthefollowing
maybefound:
i n a, C O S a '
CO S a \
\
aocosa. \
ofthespringarepreventedfrom
f> 2asgivenbyEq uations68and69aree ua l
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
D F L E C TI O N P R TU R N
R A D IU
ngdeflectioncorrectionfactor
aga instrotation
q uations76and77andsimplify ing,
i n a H — 7 T 7~ ' a n a( co s a ~ c os a )
espondingto< / < forthecasew hereno
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S PR I N G
, Eq uation67. B yusingE q uations66
maybee pressedintermsof init ialpitch
r asbefore , andtheresultsaregivenonthe
arisonofF igs. 37and41indicatesthat
etwocases,i.e.,endsfi edorfree,is not
esof8 / niv• A tlargervaluesthereare
msasdeterminedforcompression
sbyusingEq uation79aregiveninF ig. 42.
pitch anglesanddeflectionsthereis
MP R E S S D N P R T UR N
I R A D U
pressiondiagramsforopen-coiled
dsf i edaga instrotation
mthestraightlinecalculatedfromthe
bnotedthatthequestionofbuc ling
notconsideredhere1 . Wherethebuc -
hecurvesofF igs. 41and42maystil lbe
to preventlateralmovement.
smethodsofdeterminingbuc lingloadsincompressionsprings.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R I N G
urvesfortensionspringswithends
boutthespringa isaregiveninF ig. 43.
pro imatelyfortensionspringshaving
o le inaplate , sothat, w henthespringis
cannotrotateappreciably. Ita lsoho ldsw here
T N S IO N P R TUR N
R A D IU
ensiondiagramsforopen-co iled
ndsf i edaga instrotation
ringendswhicharein turnfastenedin
nyrotation.
entStress—F ora ia lly- loadedhelica l
thestressis modifiedbythepresenceof
attheendsof thespring. F romE q uations
dtwistingmomentsactingonthewire
ated.A ssuming,asbefore,thatthe
ofstrengthisva lid, thee uivalentshear
55,foracircular wirecrosssectionbecomes
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H E L I CA L S PR I N G
& andm, givenbyE q uations72and73in
andPrgivenbyEq uations76and
ereducedto
w hichtheusua lformulat= 16Pr / w d
aintheactua lstress. V a luesofK . , a re
L E C tI O N P R t UR N
D IU
ectionfactorK / , endsf i ed
„ andofS „ / nr inF ig. 44. Itshouldbe
reviouslyadditionalstressesdueto
ndthesemaybe ta enintoaccount
apterII.
thischapterindicatesthatwhere
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
nt,theerrorin theusualdeflection
houldbeconsidered.Thiserror mayap-
alpitchanglesnear 20degreesandde-
tothecoilradius. F orusua lapplications
e isunder10degreesand thedeflec-
thecoil radius,theresultsindicate
ulaoflessthan3% percent.H ence,
acyisdesired,theseeffectsdueto pitch
ncoil diametermayusuallybe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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T IG U T S T
S P R I N G A N D S P R I N G MA T R I A L S
validityofthe formulasderivedin
ndwirehelicalsprings,a seriesof
gH uggenbergere tensometerswas
springsofthetypeused inrailway
oilscutfrom thesesprings .Thesprings
nd3withan outsidediameterof6
ro f IV 2inches. Thislow inde w as
sesvaluesas lowasthisare usedin
more, theuseofa low inde springinthe
encebetweentheresults calculated
ndthosecalculatedby moree act
ble.H ence,abettere perimental
d.
U R E M N T
hefull-sizedsprings weresuchas
oplaceane tensometeronthe insideof
mumstressoccurs),semicoilswere
loadedinsucha wayastosimulate
springunderana ia lload. Todothis,
dtothe semicoilasshowninF ig.45.
edbyspecial eyeboltshavingspheri-
a ia lloads. A photographof thesemi-
ngmachinewiththee tensometer
co ila tthepo into fma imumstress
easurethetorsion stressinthecoil, the
replacedatobanda tob at45degrees
thebar2. F romthesestra inmeasure-
ressesinHeavyC lose lyC o iledH elicalS prings, T ransactions
P . M. 51-17givesfurtherdeta ils.
sistsessentiallyofa tensionstresscombinedwithan
ssatrightanglesthereto bothofthesestressesbeingat 45degrees
trainmeasurementsta enat45degreesto theshearstress
nofthelatter.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
basedonelastictheory,it ispossible
s,providedthe modulusofelasticity
ek now n . B yusingarelative ly shortgage
hepea stresscanbefound withsuffi-
gementofF ig. 46thusma espos-
pea stressontheinside ofthecoil
ico iltestarrangement
a ial-loadingconditioninacomplete
sF ormulas—L oad-stresscurvesob-
surementsontheoutsideand thein-
w nby thefull l inesinF ig. 47. S im ilar
bytests onadifferentsemicoil.F or
sultsthe dashedlinesrepresenting
valuesta enf romEq uation18forthe
moshcn o— StrengthofMaterials, SecondE dit ion, part1,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G 71
co ilandf romEq uation17forthe
lsoshown.Itwillbenoted thatthe
dfromE q uations17and18agree
thee perimentalresults.Thedashed
helica lspring
tensometer
stresson
scalculatedfromthe ordinary
eeffectsofcurvatureand directshear
thetwoe perimentalcurvesandis
asthema imumstressisconcerned.
that themeasuredstressontheinside
imesthatontheoutside . F orspringsof
nce,ofcourse,wouldbeconsiderably
semicoilswererepresentative
sa iallyloaded,acompletespring
nasshow ninF ig. 48. A sbefore , e -
centimetergagelengthwereapplied
omeasurestrainsat 45degreestothe
d-stresscurvesobta inedinthismanneron
esofthespringare showninF ig.49.
stresson oneside,thefullcircles
ppositesideof thespring.Itmaybe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
esideisabout10 percenthigherthan
atthehigherloads.Thereason forthis
at, becauseofthepresenceofthe end
tlyeccentrictothespring a is(further
PU T D B
E L I CA L
U A
0
T R E S S - B ./ Q . IN .
scurvesforsemicoil
I).Thedashedlineon thisfigurerepre-
tsideofthespringas calculatedfrom
st curve(whichgivesthestressduo
showntogetherwiththecalculated
ula.Itis ofinteresttonotethat this
deswiththat obtainedontheout-
.47,upto aloadof3000pounds,thus
testsdo simulatetheloadingofa
mparisonthestresscurvecomputed
rmula.E q uation4,whichneglects
reffectsisalsoshown.
atfor smallinde esthesimple
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G
pringsmaybe inconsiderableerror.
appro imateformulaofE q uation18
alculatingstressesin helicalsprings,
sumed.
T S T
flectionformula,E q uation7,for
ual springswerecarriedoutsome
r sdirection1.Thesetestsalsoserve
e actdef lectionformulaofE q uation51
nteffectsduetospringinde andpitch
odwastowind threetension
9. 5, 4. 7and2. 7fromasinglebarofcar-
hdiameter.A totalofninespringscut
eretested.Themethodoftesting
eloadbeingapplied throughaneye-
meterusedtomeasurestress
nsweremeasuredbetweenthe
db-b inthebodyof thespringtoe lim inate
eend turns.Byta inganaverage
onHelica lS pringsofR oundandS q uareWire , Transactions
e 2 17 .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
voidableeffectofslight eccentricities
.Itwasfoundthat thetestload-
moste actlystraightlines.A typical
arge inde isshowninF ig. 52, the
own.O nthiscurvethe circlesrepre-
eofthespring,the crossesrepresent
oppositeside.Becauseofthe un-
tyofloading,thesedonot coincide.
arge inde madef romagiven
almodulusof rigiditycouldbede-
on7forthatparticularbar. A veragew ire
bymeasuringdt, d2, d3andd F ig.
ch coilafterthetest.To dothisit
igidity
hespring.Coildiameterswerefound
,andD.,,correctionbeingmadefor
thedeterminationofmodulusof
testedareshowninTableII.
themodulusvaried bylessthan
ifferentbarsand, hence,thatthemate-
25000
T R E S S — L B ./ Q . IN .
scurvesoncompletespring
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G
uniform.
pared—F orthespringsof smaller
s indicatedbythetheoreticalcurves
ectionwas inmostcasesslightlyless
rdinarydeflectionformula,i.e.,^ was
ltestcurvefora springofsmallinde
0 2500030000
T R E S S — L B ./ Q . IN .
scurvesforcompletespringundera ia lloading
ig.53,thiscurverepresentingthe
dondiametricallyoppositesides.The
narydeflectionformula,E q uation7,
ltsobta inedonsi springshav ing
iveninTable III, w hichshowstheper-
nthetestcurvesandthecurves cal-
ormula,E q uation7,forthevarious
vedeviationmeansthatthedeflection
calculatedbymeansoftheordinary
t theaveragetestdeviation(for
ars1, 2,and3)was— 1.7percent
and—1. 0percentforspringsof inde
tedbyusingthe factoripofF ig.32
leandspringinde w ere—2. 4percent
husseenthat theaveragetestvalues
ecorrespondingcalculatedvalues
ethod.Thisindicatesthat slightly
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
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S P R IN G
ringdeflectionsmaybe obtainedby
sfiguredbytheordinaryformulaby
ation52orF ig. 32. Itshouldbementioned,
Deviations
pringF ormula
TestCa lculated
eviation
usingfactor&
2 . 4
0— .7
ve,i.e.,deectionswereslightlylessthancalculatedfrom
ation7. E f fecto fpitchangleconsidered.
flectionformulaforhelicalsprings
mostpractical purposes.
I N MO D U U O F R I GI DI T
ofdeflectionsin helicalsprings,
ofmodulusofrigidity ortorsional
def lectionformulaEq uation7isneces-
s,theeffectivetorsionalmodulusofa
dbe usedinthespringformula)may
tobee pectedonthebasisof torsion
esamematerialwithgroundand
ngthereasonsforthisdifferenceare
hematerial,presenceofa decar-
e,and residualstressesresultingfrom
ng.
actors willbediscussedincon-
ablein theliteraturerelativetothe
lsprings andspringmaterials.Un-
wsthattheeffectivetorsionmodulus
varyfroman averagefigurebysev-
cases.
ng—Thevalueofmodulusof rigidity
ntbyoverstra iningthemateria l. How -
r apartofthis reductiontobelost
dfor sometime.A dams5foundthat
arsofhigh-carbonspringsteel in
pMemoirs, Iron& S tee lInstitute , 1937, Page1.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G
ralper centinthemodulusof rigid-
thisdecreasecouldbe eliminatedby
heattreatmentaftertheoverstrain-
eobta inedbyP letta , Smith, andHarri-
ngsmade from%-inch diameterbar.
decreasesinthetorsionalmodulus
percentdependingonthe amountof
ncyofthemodulusto partiallyre-
thespringhasstoodfor aconsider-
carburization—A factorw hichiso f
nfi ingtheeffectivetorsionalmodulus
zationofthewiresurfaceinthe com-
A PP Y
N G .
lspringfordeflection
at,ifthereis adecarburizedlayer
whenthe springisstressedthis ma-
rbonsteelandwillyieldat arelatively
verstra inonC lose lyC oiledHelica lS pringsandtheV ariationof
eCo ilsw ithL oad byP letta , SmithandH arrison, Eng. E x p.
, V irginiaPo ly technicInst.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S P R IN G
the springwill,therefore,acttosome
decarburizedmaterialwerenotpresent
flectionratewillbe roughlythatcor-
ehavingadiametere ualtotheactual
hic nessofthedecarburizedlayer.
ris usedinthespring formula,this
effectivemodulusofrigidity werede-
, aspringof% - inchhot-woundstoc may -
rburizedlayer e tending.01-inchinto
erloads,since thedecarburizedlayer
ngth,the load-deflectionratecorre-
of.5—2(.01)— .48-inchor4percent
d-deflectionratevariesasthe fourth
uation7, thismeansareductionof
entintheformer, oradecrease inef-
of16 percent.Inmanycases,spring
I N G, L B .
O N O N E S I D O F S PR I N G.
N D IA M TR I CA L L Y O PPO S I T S I D
deflectiondiagramforhelicalspring
)
modulusfigureofabout10per cent
pringsteel thanforhard-drawnma-
pleshowsthatadecarburizedlayer
e- inchdiameterstoc w ouldbesuf f icient
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G
modulusvalue.S uchalayermayeasily
materials.
Ingeneral themodulusofrigidity
ithincreasein temperature.This
ofa springunderagivenload willbe
I N G, L B .
N O N E S I D O F S PR I N G.
O N D IA M TR I CA L L Y O PPO S I T S I D .
deflectiondiagramforspringof
otethatmeandeflectionof tw o
theoreticalvalue
res.H owever,availabletestdata
urearelimited.O neofthefewin-
hislineis thatcarriedoutbyK eulegan
vestigatedthechangeinmodulusof
o f temperature(f rom—50 C . to
ingmaterials.Theseinvestigatorsfound
mostmaterialsthemodulusofrigidity
o imatelyby
gdityat0 C , m— temperaturecoeff i-
temperature,C.
es. . V o l. 10. 1933, Page305. S eea lsoB rombacherk
ch. R eportNo. 358, 1930.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ainedbytheseinvestigatorsfor
regiveninTable IV . F ore ample ,
intemperaturefrom0to50 degrees
thetorsionalmoduluse ualto
sofModulusofR igidityC
C.00025
98% Cr..24% V a.00026
8% Ni . 00040
0
c anandHauseman, B ureauofS tandardsJourna lo fR esearch.
.
ed,
0G- . 013Gor1. 3percentinthisrange.
cyisdesired(as ininstrumentsprings)
tant.
modulusofrigiditydropswith
—100 degreesF ahr.to600or 800
btainedfromthecurves ofF ig.54.These
smoothcurvesthroughtestdata
ndhiscollaborators8.S incetheactual
considerableirregularitiesandscatter,
sideredas givingonlyaroughindi-
uluschangewithtemperature.
N O F M O D U U O F R I GI DI T
methodswhichhavebeenused
ofrigidityfor springmaterials:
surementsofdeflectioninhelical
pression
mentsoftwistofa straightbarin
thod:Measurementsoftheperiod
mwhichby k nownformulasthe
determined.
ng thefirstmethod,deflections
. T . M. , 1930, PartII, Page356.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G
icalsprings loadedintestingma-
eviouslyitisadvisableto measure
ncoilsinthebody ofthespringto
etotheeffectsof endcoils.A lsoto
avoidableeccentricitiesofloadingit
flectionsondiametricallyopposite
rageof theseva luesthenista en. If
ennturnsofthe springisfound,the
e foundfrom
dbysolvingE q uation7forG,the other
meaningsasbefore.S lightlyhigher
- F
effectonmodulusof
ls
cularlyforsmall inde es,bydividing
t a e n f ro m F i g . 3 2. H o w e ve r , fo r i nd e e s
aybeneglectedinmost cases.It
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
t,foraccurateresults,carefulmeasure-
sionsat manypointsarenecessary.
spring mustbecutup afterthetest
rediameter.
nsomereluctanceonthepartof
flectionmethodonthe groundthat
asmayintroduceun nownerrorsin
r sopinionthatthequestionsre-
ormulahavebeensettled,both e -
cally,andthattheresultsarereliable
edarecarriedout. Inaddition,the
advantagethatthemodulusofrigidity
tespring and,hence,maybemore
wounduptoformactualsprings.I t
besomedifferencebetweenthema-
gandheattreatedand straightbars
uiredbytheothermethodsdiscussed.
hedirectmethodfor findingthe
ht,roundbarof thespringmaterial
ngmachine.Toeliminatedisturb-
dsit isadvisabletomeasuretwist
hof thematerialundertorsionby
torsion measuringdevice.O nemethod
tachmirrorsonthe baracertain
deflectionsofthesemirrorsaremeas-
cale.If6is theangleoftwistin
nthe gagelengthIthemodulusof
ducingthetwist, Z = gagelength, d=bar
eoverallangularmovementof
achinearemeasuredaresub ectto
nindefiniteamountoftwistin the
endsof thespecimenandthismayintro-
e results.
lyderivedf romthek now nformulaforangulartw isto fa
eeS trengthofMateria ls—Timoshen o, PartI, Page261.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G
od—Inthethird ortorsional
htissupportedbythe springwireand
uencyofoscillation/in cyclesper
omthisthetorsionalmodulusG maybe
ation
tofinertiaofpendulumbob,1= effective
rediameterasbefore.Thise uation
mthek now ne uation.
pingconstantof thew ire1 .
ylong toreduce,asmuchas
teeffectsduetoclampingatthe ends
est, o fcourse , themateria lissub ect
and tensionstress,thelatterbeing
endulumbob.
A summaryofava ilabletestdata
arbonspringsteelisgiven inTable
ereobtainedby variousinvestigators,
tmethodsdescribedpreviously,i.e.,
nalpendulummethod.Ineachcase
etherwiththemethodusedis indi-
hetable.
sentsanaveragefigurecalculated
rementsbetweencoilsofhelicalsprings
hemodulusfigurebeinganaveragefor
usingthetestdata,deflectionsbeyond
imitofthe springmaterialwerenot
sare seldomloadedtosuchhigh
orsiontestofa %-inch diameterbar
testedbyA dams,whileItem3refers
overstrainingat ator ueabout50
enHartog—Mechanica lV ibrations, McGraw -Hill , 1940, Page43.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
uecorrespondingtotheproportional
at treatmentat228degreesCent.
frigidity from11.82to11.14X 10
by thistreatmentgivesanideaof there-
omstressinghelical springsfarbeyond
material.This investigatorshowed
igidityGforC arbonS pringStee ls
hearthQ & T
dandtempered.
a ined, mildheattreatment.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G
ct thatnodecarburizedlayerwas
wasnotoverstrained.Thelower
thercasesareprobablydue largely
woeffects.Theselatterfiguresare,
ativeofthosetobee pected.
e landPhosphorB ronze summary
e modulusofrigidityofspring mate-
ls isgiveninTableV I.
roughaveragevaluesofthe
eta enas: 11.5X 10 poundsper
lloyS tee ls,
ndPhosphorBronze
igidityG
.in.y10 )Investigator
T .f % 1 1 .7 5 A d a m s
T. f . 14811. 2Z immerli, Wood
. 37511. 45B erry1
. . A . S rC . D. t. 14810. 5Z immerli, e ta l
D. t. 04- . 16210. 8Wahl
. D. t. 1259. 1Z immerli, eta l
rC . D. t. 096. 7Z immerli, e ta l
ayre
6.2BrombacherorMelton
dandtempered.
edandco lddraw n.
a , .2 1 % N i .
a .
.
pMemoirs,Iron& S teelInstitute,1937,Pages1-55.Directmethod
ained.
. M. , 1930, PartII, Page357. Directmethodused.
.E ngrs.,1938,Page460.Directmethodused.
Deflectionmethod.A verageof14springs.
M. . , 1934, Page556. Torsiona lpendulumtests.
ica lR eportN o. 358, 1930, Page568. Directmethodused.
vanadiumstee ls, 10. 6X 10 forsta inless
, 9X 10 forMonelmetal, 6. 4X 10 for
itshouldbe notedthatindividualtest
heseaveragesbyseveralpercent.
nyinvestigationshavebeen
durancepropertiesofhelicalsprings
oading.Theusualmethodof fatigue
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
testsofautomotivek nee-action
.Theresultsof suchtestsareof direct
ngineerswhoare responsibleforthe
ngunderfatigueloading.A nother
evalvespringused ininternalcombus-
eshowsthatthere areconsider-
urancelimitsor limitingstressranges
rtedbythevariousinvestigators.The
inthefactthat theendurancelimit
fspringis verymuchdependenton
spring wireorbar.S lightsurface
edecarburizationresultingfromthe
ayresultinaconsiderablereduction
nge.Thelow valuesreportedin
es maybeduetothis. A further
esults obtainedliesinthefact that
maybeused. Sincethesensitiv ityof
sconcentrationeffectsdueto bar
ferenceinresultswouldbee pected.
ChapterV I.
A mongthemore importantinvestiga-
calsprings,thetestsconductedby
mentioned.Thesetestsconsistedofen-
stressrangeson small-sizedhelical
tivevalvesprings.A typicalendur-
onchrome-vanadiumsteelspringsin
nF ig.56,thestressranges actually
theverticallines betweenthecircles
tress -.Thecircleswiththe arrows
imitsoftherange whichdidnot
ioncycles,while theplaincircles
d.O nthebasisof theseteststhees-
erangeisrepresentedbythe upper
diagramit maybeseenthatthe
areaboutasfollowsfor thismaterial:
00,40,000to 98,000,60,000to108,000
ssR angeforSmallH e licalS prings F . P . Z immerli, Engineering
niversityofMichigan,July1934.
agramhasbeenpreviouslydiscussedin ChapterI.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G
. Thismeansthatthespringcouldbee -
elywithinanyofthese ranges.S imilar
notherspringmaterials.
sobtainedbyZ immerlitogether
erinvestigatorsisincluded inTable
ingk indofmaterial,heattreatment,
rs
tso fk neeactionhelica lsprings
sofrupture,yieldstrengthofthe ma-
wiresize,coil diameter,numberof
aregiventogetherwithvaluesofthe
hereseveralvaluesoflimitingen-
iagramssimilartothose ofF ig.56
fora givenmaterial.Inallcasesthe
q uation18w asusedtocalculatethe
atiguefailureofalarge helicalspringis
1.
ations,theresultsofwhichare
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 108/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IX G
arethoseby Johnson13atWrightF ie ld
directionof thespecialresearchcom-
ringsof theA . . M. . andreportedby
seinvestigationscoveredtestsonthe
around% -inchbardiameter)and
ange fromzerotoma imum.
ryof e pectedlimitingstress
edataof TableV IIandassuminga
elow10,000-20,000poundspers uare
0100000
, L B . / IN '
amofendurancetestsonhelical
umsteel.F romtestsbyZ immerli
smallerwiresizes.Thusthelimiting
old-woundcarbonsteel wiremeansthat
rangeof60,000poundspers uareinch
s,i.e..rangessuch as0to60,000,5,000
000pounds pers uareinch.Thefig-
matedfromtestsreportedbyvariousin-
rist icso fHelica lSprings , IronA ge, March15, 1934,
ressR eportN o. 3onHeavyH elicalS prings , Transactions
r1 9 37 , P ag e 6 09 .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G
en thereareconsiderabledifferences
edbydifferentinvestigatorsforthe
Thereasonfor thisisthatthe q uality
gesSmall- izeHelica lS prings
mMinimumStressN earZ ero)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
orthy.Bythisprocessit appearspos-
cerangetovalueswhichmaybee -
hedbars testedintorsion.Thus,
shot-blastedhelicalspringsof chrome-
nendurancerangein zerotoma i-
undspers uareinch.Thiscompares
ndby Johnsonongroundandpol-
diumsteelforarange fromzeroto
' andw ithava lueofonly70, 000pounds
gs withouttheshotblastingtreatment.
propersize ofshotandpeening
Manufacturersf re uentlychec the
ardA -typespecimen,3incheslongby
hthic treatedtoR oc w ellC 44-50.
aheavyplateandsub ectedtothe
tas thespring.A fterpeeningthe
easuredona1.25-inchchord.F rom
gesforLarger- izedHelica lC ompressionS prings
)
bleV III.
lmenthefollowingvaluesaresatis-
wrediametersprings, shotsize . 040and
25-inchchordwiththe standardA
sand torsionbarsof1.25-inchdiam-
e.060,deflection.012to.015-inchon
thic nessof.0938-inchbutis other-
ecimen. F orf la tsprings. 020- inchthic ,
003- inchonanA specimen1 .
hisisgiveninthearticles: Z immerli—MachineDesign, N ov .
Tra ilsinS urfaceF inishing , S tee l, July5, 1943, Page102.
nedS urfacesImproveE nduranceofMachineParts ,MetalProgress.
. " Improv ing F atigueS trengthofMachineParts , Mechanica l
43, Page553.
ivesmoredataonendurance limitso f springmateria ls(asdis-
springs).
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L ICA L S PR IN G 93
ehighendurancerangeswhichare
gcannotordinarilybeutilized inde-
000100000
. in.
mateendurancediagramsforgood
imitingstressrangeread
minimumstressandlines
tingma imumstress.Curva-
used
tressisused, e cessivecreeporload
ThisisdiscussedinC hapterV ). How -
blasttreatmentgreatlyreducesthe
that theproblembecomesmainlyone
etorloadloss.
E stimatedlimitingstressrangesfor
helarger-sizedspringsare summarized
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
agiveninTablesV II andIX ,
wingthevalueof enduranceranges
f romgood- ua lityhe licalcompression
Thesecurvesholdroughlyfor
between5and10. F orlargerinde es,
aybee pectedandviceversaforthe
ethesecurvesrepresentroughaverage
ationinindividualinstancesmaybe
ssconcentrationnearthehoo ends
hatlowervaluesofenduranceranges
57mayalsobe found.
O N S P R IN G W ITH F E W S T R E S S C C E S
obe availableintheliteraturefor
obut asmallnumberofstresscycles.
mentioned.Z immerli11foundon
springw ire( A E 6150, inde 7. 4)a lif e
esfor astressrangeof86,000,the
0poundspers uareinch(curvature
nd followingdata).E dgerton
0,000to 250,000cyclesforhot-
sofinde 5atastress rangeof100,000
(minimumstresszero). H. O . F uchs
fcenterlessgroundwire(446Brinell)
648-inchdiameterwire,inde 7.3.
m 170,000to409,000cyclesata stress
nimumto138,000poundspers uare
im ilarspringsof . 628- inchdiameterw ire,
to 178,000cyclesatastress range
00poundspers uareinch.
in thischapterapplyonlyfor
turewithnocorrosionpresent.. In
ginstolose itseffectivenessattem-
eesF ahr. 1\ Itshouldbeemphasized
ssrangesfoundinfatiguetests should
sdiscussedinChapterI, amarginof
ountunavo idableuncerta inties, isre uired.
.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G UN D R S T A T IC L O A D I N G
cal springsbasedonelastic
tensivelyinChapterII.It shouldbe
calculationsbasedonproportionality
donot applyrigidlyaftertheelastic
materia lhasbeene ceeded. A lthough
terII areofbasicimportancein
rationofwhathappenswhenthe
lhasbeene ceedediso fva lue in
oadedhelica l
ablestressforhelicalsprings.In the
basisfor thechoiceofwor ingstress
dingbasedonsuchconsiderations
C hapterV Itheq uestionof fatigue
discussed.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
alspringmaybedefinedas one
oadortoa loadrepeatedbutare lative ly
fthespring.A springloadedless
ngits lifewouldusuallybecon-
dincontrast tofatigueloadingin-
fcycles.
tantapplicationsofstatically-loaded
ybeenmentionedinChapterI.
esprings,springstoprovidegas et
ngs inmechanismswhichoperateonly
eotherapplicationsmightalsobe
sinlightningarrestersF ig. 58. Herethe
maintainadefinitespacerelation-
fthearrester,regardlessof temperature
ectto fatigueorrepeatedloading,
evelopmentofafatiguecrac which
of thespring( ig. 25). F ractureof
veroccurs,however,wheresprings
ads insuchcasesthedesignermustguard
porlossinload(whichta esplace if
sareused).Theseeffectsare particu-
edtemperatures.Ifasmallamount
eto lerated, ahigherw or ingstress
thecaseotherwise.
icularimportanceinthedesign
cttostaticloadingisthespringinde ,
diameterandwire diameter.Where
,thehigheststress isconcentratednear
enthe loadisca lculatedby ta ing
willbeseenlater,a highervalueis
springsthanwouldbethe case
es.
C U A TI O N S
stressisbe low theelasticlim it, in
sub ecttoastaticloadandnormaloper-
essdistributionalongatransverse
imatelyby the linebe inF ig. 59a. The
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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O A D D H E L I CA L S P R IN G 97
binthiscase isca lculatedbyEq uation
ticconditions.
enthat, foraspringofsmallinde , the
hecoil ismuchlargerthanthe stress
eco il, i . e ., mosto f thehighstressiscon-
fstressalongtransversediameterofbarof helical
S mallinde ata , la rge inde atb
smeansthat aconditionofstress
sisshow ngraphicallybyF ig. 60aw here
ntratedinthe relativelysmallshaded
islarge,conditionsareconsiderably
g. 59b. Herethestressabonthe in-
A K
stributionof regionsofpea stressincross-sectionofbar
e ataandlarge inde atb(forthespringof
tressis concentratedneartheinsideofthe coilata)
ttle largerthanthestressa conthe
ea stressisgivenappro imate lyby
uation18. Thisfollowsf romEq uation19
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
forvery largeva lueof the inde c. F or
tstressesare locatedinthering-shaped
g.60binsteadofbeing concentratedin
arthe insideofthecoil asisthecase
ll( ig. 60b). Inotherw ords, inthecase
onlyarelativelysmallportionof
sub ecttostressesnearthepea , w hile
ndia -
m
- IN C H E S P R I N C H
f large inde , are lative ly largepart
ecttosuchstresses. Hence if the
ieldingoccursovertheentirecross-
tis clearthatthespringof small
rryamuchlargerloadthanw ouldbe
f thema imumstressca lculatedf rom
umespurelyelasticconditions.Thereason
asticlimitis passedandyieldingbe-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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O A D D H E L I CA L S P R IN G
tionwill beeffectiveincarryingload
sinceagoodshareof thecross-section
isa lreadysub ecttostressesnear
loadnecessarytoproducecomplete
tionwill notbesogreat asinthe
spring, w hereonlyasmallparto f the
cttostressesnearthepea .
lshaveconsiderableductility
hlessthanhavestructural materials).
nstress-straindiagramofatypical chrome-
orsmallhe lica lsprings isshow ninF ig.
shapeoftensilestress-straindiagram
aterialwith afairlysharplydefined
fthe usefulspringmaterialshave
percent in2inches,and stress-strain
ig.61,(althoughtheymayhavea
dpoint hasbeenpassed)itappears
sductile materials.Insuchcases,
ved,asbroughtout inChapterI,it
tstressconcentrationeffectsindesign .
augmenttobarcurvature(which
sconcentrationeffect)maybe
estressunder static-loadconditions.
Curvature—Tocalculatethe stress
gstressconcentrationduetobaror
ureisas follows:A ssumingahelical
ringundera loadPandneglecting
nsandpitchangle,the torsionmoment
w illbee ua ltoPrw hilethedirect
The distributionoftorsionstressalong
tothe momentPrwillbeasshown
a torsionstresst, duetothismoment
y theusua lformula( q uation4). Thus
uperimposedtheshearstressr.,due to
archBulletinN o. 26, UniversityofMichigan, R ivesother
ss e s — C . S . S o d er b er g , Tr a ns a ct i on s A . . M . . , 1 9 3 3, A P M 5 5 - 16 .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
hichforourpurposesmaybe consid-
overthecross-section3.Thisstress
dasshownin F ig.62bandis
btainedby superpositionofthe
andbgiving aresultantdistribution
ma imumstress, T . , thusobtainedby
ationeffects,is
8
itten
illbecalled ashear-stressmultiplication
tionofspringinde cinF ig. 63.
pearslogicaltouseE q uation89
nthatstressconcentrationeffectsmay
formanideaofthe marginofsafety
ng,the stresscomputedbyE q uation
hthe yieldpointofthematerialin
ngmaterialsmaybe ta enasabout
yieldpoint.
M P E T Y I L D IN G
ent(andperhapsmorelogical)
fdesigninga springforstaticload
menon-uniformityinthe distributionoftheshear
veasimilareffect tothatofstressconcentrationdue
eglected.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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O A D D H E L I CA L S P R I N G 1 01
ionofstressesinhe lica lspring—stresscon-
tureneglected.A tais shownstress
stressduetodirect shear,andcsu-
wnata andb
basedonthe considerationofthe
completeyieldingofthe materialin
loadbeingthenta enasacertainper-
uiredtoproducecompleteyielding.If
tress-straincurvesimilartothatof
ectedthataf tere ceedingthey ieldpo int
ross atransversediameterwillbe
owninF ig. 64aforaspringofsmallinde
of large inde . A ctua lly itmaybee -
rialssomerise inthestress-strain
C - ^ f-
ngshearstressmultiplication
esintoaccounteffectsdueto
obarcurvature
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ldpo intw illta eplaceduetothe
hatactua lly thecurvesofF ig. 64w illbe
dalinform.H owever,theassumption
n,whichlendsitself tosimplicityin
yaccurate.Becauseofthenecessity
direct-shearload,particularlywhere
stributionoftorsionstressunderplastic con-
rentinde es. A t a isshow nsmall
b. F orthesmallerinde estheareaA , is
ta ecareof thedirectshearload
ll, itmaybee pectedthatinsuchcases
terthanA , inF ig. 64a. O ntheother
e,thesetwoareaswill beaboutthe
tothiseffect, thepo intO' w herethestress
mounte f romthegeometrica lcenterO.
P atwhichcompleteyielding
n occursrepresentsaproblemin
remelycomplicatedforlowinde
adeterminationofthedirectionsof
t allpointsofthe crosssectionunder
aluatetheshear forceandmoment.
matesolutionbasedonreasonableas-
dasfollows:
ctionsoftheresultantshear stress
ofthecrosssection mayberepre-
sasshownin F itf.65a.Thecenters
displaced sothateachcircle intersects
ate ua linterva lsbetw eenPC andO' B
sthepo into f zerostress(the lineB B rep-
rsetothespringa is). Ifnostra inhard-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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O A D D H E L I CA L S P R I N G 1 03
heresultantshearstressista ene ua l
t atallpointsand thedirection
t ispossibletocalculatethe resultant
ra givendisplacementt ofthepoint
. F romthisthespringinde maybefound.
thecirclew ithcenteratA represents
65a. Onthebasiso f theassumption
oints ofintersectionofthecircles be-
B , t h e ra d iu s p o f t hi s c ir c le w i ll b e
po intDa longO ' B , aseriesofcircles
ig. 65a.
ntdA atradiuspand angle6,
lbe actedonbya shearforcerydA
ththeradiusOC . Thearea is
orce aboutthecenterO willbe
) pd A
his,
93)
diagramofresultantshearstressdirectionover cross
icalspringunderyielding.Pointof zerostressO '
icalcenterO aw ay fromspringa is
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
illbe theintegraloftheseelementary
rosssection.Thusthe momentMyfor
sectionbecomes
O (94)
aedfunctionofbothc , pand6and
ofEq uation94ingenera ltermsis
ssumingagivene anddandbydraw -
nF ig. 65a, theva lueofsinipcanbe
andradiusp.Drawinge ually
r ofthecrosssectionO asindicated
theva lueof t p2sin, palongeachradius
p, the integra lo fE q uation94maybe
hedeterminationoftheareaunderthe
plyingbya constantdependingon
radiiandaddingthetotal.In this
ora givene anddmaybeobtained.
overthecrosssectionmaybe
er.R eferringtoF ig.65cthevertical
rceactingon theelementdA is
9 ) dA
sincetheshearforceis considered
thee pressionfordA givenby
ppde(95)
ceP actingoverthesectionfor
eintegralof theseelementaryforces
ence
dO (96)
atedbydraw ingcirclesasinF ig.
pacedradia ll ines, measuringifz— Q , and
sin(i/ < — 0)a longeachradius, andf ind-
urve.Byaddingthesewithproper
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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O A D D H E L I CA L S P R I N G 1 05
ingbya constant,theresultantshear
ndbardiameterdisfound.
er e r ec o il r a di u s,
M andP obtainedbygraphicalor
q uations94and96arek now n, the
foragivene andd. F romthisthe
a ined.
ringisessentiallya straightbar
mentM, = P ' r, directshearbeing
forconstantyield stresst is
onw ithEq uation3, it isseenthatthe
' )forcompletey ieldingofa large inde
thigherthanthat atwhichyielding
uation3by ta ingrm= ry ). Inactua l
ardeningandother effects,higher
e pected,however.
,/ ( q uation97)fora large inde
uation94)forasmallinde spring, anesti-
irectshearloadis possible.Theratio
comparabletothefactorK H ( q ua-
byneglectingstressc oncentration
pletesolutionofthisproblemobtained
4and97forvariousspringinde esisnot
eindicationsat presentarethatthe
co n si d er a bl y l es s t ha n K „ a s g iv e n by E q u a -
useofpossibleinaccuraciesintheas-
gtheshearstressdirections,F ig.65a,
eningeffectsnot considered,themore
givenbyE q uation90w illbereta ined
ntuis k nown,theloadatwhich
ringoccurs (wheretheloaddeflec-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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rTurnfor
e licalS prings0
ofSprings(inches)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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rings(inches)
enin. i
stressT,
ectionpe
1OO , OO 0
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
hatbelowtheactual loadobtained
enbyE q uation90isprobablysomew hat
eningeffectscomeintothepicture.
F F O R MU A S T O S P R I N GTA B E S
nofE q uations89and7 inthe
helicalspringsTable X ,hasbeen
sloadsanddeflectionsperturn ata
ers uareinchanda torsionmodulus
pers uare inchascomputedf romEq ua-
standardoutsidecoildiametersandwire
s usedforsizesup to.090andthe
rsizesf rom. 106to% - inch.
00, 000poundspers uare inchmayac-
cticalcases,itshouldbe notedthat
blefor convenienceonlyandisnot
ndedw or ingstress. Theactua lw or -
oadedspringshouldbee ualtothe
materialdividedbyafactorofsafety
IIIgivedataonthey ieldpo intsof spring
ssin tensionisk nown,theyield
a enasabout57percentof thatin
basedontheshear-energytheory
aluesoffactorofsafetyas usedin
to aslowas1.25in somecases.
alueofstressother than100,000
theloadsanddeflectionsgiven inthe
nthesameratio . A sane ample, as-
houtsidediameterand.135-inch
adanddeflectionperturn are103
pectively,at100,000poundsper
edf romEq uations89and7. If the
withanultimatestrengthof260,000
nthissizeanday ie ldpointintension
0poundspers uare inch, they ie ld
percent ofthisorabout 120,000.
etyof 1.5basedontheyieldpoint is
ingstressof120000/1.5= 80,000
Thismeansthatthe permissibleloads
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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O A D D H E L I CA L S P R I N G 1 09
ndertheseconditionswould be80
hetable.In thecasecited,the
rnwouldbe.80(.141)= .113-inch
ouldbe.80(103)= 82.5pounds.
orintermediatecoilandwire
bleX , thechartso fF igs. 66and67
g.66theordinaterepresentsload al
R , IN C H E S
. 20
R , IN C H E S
000lb./s .in.Tofind loadatanyotherstress r,loadsgiven
0,000.N ottobeused forfatigueloading
ulatingloadsinstatically-loadedhelicalsprings
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
areinchtorsionstress(calculatedby neg-
due tocurvature)whiletheabscissa
achcurverepresentsagivenoutside
sfor awiresizeof .090-inchand
-inchtheloadat 100,000poundsper
bout61pounds.
per turnrepresentedbytheor-
twirediameterforvariousoutside
ire sizeof.106-inchanda coilout-
thedeflectionperturnis .035-inch.
mallerrorwillresult inreadingthe
igs.66and67 andforbestaccuracy
uld beused.Thesecharts,however,
rmostpracticalpurposes.
re uently , springtablesorchartsbased
chyie ldthepea stressincludingcurvature
pterV II).Thesetablesmayalsobe
sson whichthetableorchart isbased
w h er e K C = K / „ v al u es o f K a n d K „
uations19and90. Theva lueofK , .
entrationeffectdueprimarilyto wire
,representstheincreasedstressdue
a ia lload. Thisfo llow ssinceK — K C „ .
culation, va luesofK , -aregiveninF ig. 68
c. Thus, if a tableorchartisbased
undspers uare inchandif thespring
1. 15, thestressca lculatedbyneglecting
0/1.15or87000poundspers uare
be comparedwiththeyieldpoint.
e ampleof thisdesignprocedurefor
aspringmaybeconsideredofinde
lhavingatensionyieldstressofaround
areinchandayield stressintorsion
r110, 000poundspers uare inch.
f safetyof1.5basedonthe yieldstress
ewor ingstressforthe static-load
sngEq uation89w ouldthenbe
oundspers uare inch. F oraninde 3,
ig. 68) hencethea llowablestressas
tion18(whichincludescurvatureeffects)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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O A D D H E L I CA L S P R I N G 1 11
aluesshouldbomultipliedbyT/100,000.A lso,ifthemodulus
0 ,valuesshouldbemultipliedby 11,400,000/G
ulatingdeflectionsinstatically-loadedhelicalsprings
orsionmodulus, G= 11. 4x 10 Ib. /s . in.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
98000poundspers uare inch. If
uationareused(suchasthosegivenin
ring proportionssuchasactiveturns,
eandsolid heights,aredetermined.
: A springhasaninde o f15w ith
asintheprev iouse ample . F oran
1. 06f romF ig. 68. A ga inassuming
byneglectingstressc oncentration,
ressintorsionor73000poundsper
blestressf iguredf romEq uation18w ould
oundsper s uareinch.Thisstress
valuein thepreviouse ample.This
pea calculatedstress(withcurva-
rywiththespring inde ,assuming
eingmaintainedbetweenthewor -
uiredtocausecompleteyielding.
L A X A T IO N U N D R E L E V A T D T M P R A T UR E S
he determinationofwor ing
helicalspringswasbasedon theyield
altemperaturesbeingassumed.A s
essisk eptw ellbe low thispo int, no
shouldbee perienced,providedthe
otmorethanabout200 degreesF ahr.
mperaturesitis notusuallysuffi-
ntheyieldpointor elasticlimitof
hodofdeterminingwor ingstresses
actua lcreeporre la ationtestsat
fortunately,thereisnot agreatdeal
amountoflossofloadwhich maybe
mprehensiveseriesoftestssofar carried
tionorlossofload inhelicalspringshave
immerli4.Thesetestsweremadeby
sbyagiven amountinaspecialtest
dspringwasthenput intoafurnace
evaryingfromthreedays forthe
tureonC oiledS tee lSpringsatV ariousL oadings —
clioniA . . M. . , May , 1941, Page363.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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O A D D H E L I CA L S P R I N G 1 1J
springstoten daysforthestainless
eating,thespringswereremovedfrom
1 1
13 1 4 1 5
C=
ngstress-concentrationfactorK c
ossinf reeheightdetermined. F romthis
permanentset)thepercentageloss
d.
eresultsobtainedbyZ immerlia re
orvariousspring materials.Thevalues
ntpercentagelossinloadin aperiod
eraturelisted,e ceptforthestainless
tswere runtendays.Thestresseswere
ection,E q uation18.Iffiguredwith-
esevalueswouldbeabout 10per
sweremadewith springswhichhad
usbluingor stressrelievingtempera-
minutes.Inthetablethe valuesof
ptimumbluingtemperaturesare
lowertemperatures,thevaluesof
iderablygreater.Itappearsthat at
uresnotallthe coilingstressesare
tterarecombinedwiththe loadstresses
ethanwouldbe thecaseotherwise.
rsthat,atstressesof100,000pounds
90,000figuredwithoutcurvaturecor-
loadlossmaybee pectedwithinthree
.6percent carbonsteelwire,when
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
sof350degreesF ahr. Somew hatlow er
dforthechrome-vanadiumsteel.S tain-
eshowedaloadlossof onlyabout
esF ahr.whichincreasedto11.5per
hr.atthesamestress(100,000poundsper
tertestsw ererunfortendays. F orvery
oad lossesmaybee pected.
Z immerliconcludedthattheusual
henstressedto notmorethan80,000
(ortoabout72, 000poundspers uare
turecorrection)attemperaturesupto
ovethistemperatureandupto 400degrees
aybee pected, w hileordinaryspring
temperaturesabove400degreesF ahr.
stainlesssteelsofthe18-8 typeresist
tterthanothers,e cepthigh-speed.
wnfromthisseriesof testswasthat,
t-treatedaftercoilingshowedno
undfrompretemperedwireproperly
g temperaturewasfoundtobeinthe
adforHelica lSpringsatE levatedTemperatures
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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O A D D H E L I CA L S P R IN G
withoutob ectionableloweringof
rties.F orfurtherdetailsonthis the
riginalarticle1.
culations—S omeanalyticalmeth-
byN adai forca lculatingcreepand
resumeof thesemethodsw illbegiven:
ppro imationthatthespringis
etorsion.This willbeappro imately
ngs. L ettingP= loadonspring, r=
rediameter,and6theangleof twist
wire,thetwistingmomentwillbe
shearatadistancepf romthecentero f
n
problemstheunitshearstrain i
to fanelasticpartf andaplastic
q uation98,
(9 9)
followingrelationholds:
tradiusp( ig-24, C hapterII)and
ofthematerial.
on99withrespectto timet,
erentiationwithrespecttotime.
t= g(r)w hereg(r) isa functionof
q uation100andbysubstitutionin
UnderV ariousS tressC ondit ions * —-A . N adai, Th. ton
o lume, 1941. A lsobibliographygiveninthisre ference.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
by
wire.
latethe steadycreepofthespring
willbeassumedthattheshear stress
sisgovernedbya powerfunctionlaw.
reewithtestsoverlimitedrangesof
singEq uation101,
( i o4 )
isane ponentw hichdependsonthe
terminedbyactualcreep tests.
f t intoEq uation103, andinte-
ufficienttimehas elapsedsothat
becomeconstant,thee pression
on102. So lv ingfortheva lueof6byusing
bstitutinginEq uation104,
of stressoverthecross-section
oadPisindicatedby thecurvedline
calculatingt asafunctionofp from
ightlinewhichrepresentsthe initial
ing oflargeinde )isalso shown.
edistributionofstressgivenby
ursaftera considerabletimehaselapsed.
ibutionfor theintermediateperiod,
ow s: F romE q uation105sinceP is
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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O A D D H E L I CA L S P R I N G 1 17
dr/ dtgivenbyEq uation102,
E q uation102andrearrangingterms
rentiale uationresults:
d f i- g (r ) ( 10 8 )
ationfort= 0istheusual formula:
isgivenbyE q uation106.
ulatethere la ationorlossinload
essed(ore tended)toagivenlength,
S
eadyF ig. 70—S tressdistribution
ring rela ationofspring
howingtime effect
s:S incethelengthofthe springdoes
ngleof twistperunitlength6
e6— 0. UsingE q uation102thefo llow -
isobtained:
unctionis againassumed:
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
nt.
uation109,
ctto timeanddeterminingthe
theconditionthatfort = 0,alinear
cross sectionoccurs,
he init ia lstressdistribution. So lv ingfort,
distributionof shearingstressesover
ustimest isillustratedbyF ig.70.
epea stressdropsconsiderablyas
essdistributiontendstoflattenout
rmdistribution.A fteraconsiderable
tressesapproachthevalue:
( 1 13
inga rectangulardistributionof
orloadPat largevaluesoftbecomes
obtainedfromrela ationtests.
nsforloadmaybeobta inedbyusing
uation112inE q uation103and
mericallyorgraphically.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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A R I A B E L O A D IN G O F H E L ICA L
ationalbasisfordetermining
ca lspringssub ecttostaticorinf re-
gwasdiscussed.Incaseswheresprings
orrepeatedloading, asfore ample inau-
omewhatdifferentapproachtothe
or ingstressisnecessary.In the
problemiscomplicatedbythefact
sub ecttoa load(orstress)w hich
auetoama imum. A sshow ninChap-
oaconstantorasteady loadonwhich
oralternatingload.Thus inthecase
ring,theconstantcomponentofthe
itial compressionofthespringwhile
determinedbythevalvelift.
lculatedbymeansofthe curva-
q uation18, conservativedesignw ill
pearsj ustifiedif thespringissub ect
tress infatigue.W herethereisa
mponentonwhichis superimposed
t,however,itappearslogicaltoneglect
tsduetocurvaturein figuringstresses
ofthe load.Inthisconnectionthere
amongspring engineersthatthe
esult intoo low valuesofw or ing
tthe stressescomputedthiswayaretoo
medtoa certaine tentbytheresults
defatiguetestsonsmallhelical
escarriedoutbyZ immerli1. These
ressrangein fatigue,whenfigured
ashigherforthesprings ofsmaller
ltsw erereportedbyE dgerton- inconnec-
M. . , January , 1938, Page43.
M. . , October, 1937, Page609.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
r onheavyhelicalspringsbythe
esearchC ommitteeonMechanica lSprings.
dlater.
efullstress-concentrationeffectcor-
ecorrectionfactorK doesnotalways
ding)lies inthefactthat somema-
eto stressconcentration.Inother
lsaretestedbymeansof specimens
llets,thefatiguestrengthreduction
ofsuch" stressra isers isnotasgreat
basedontheoreticalstress-concentration
eoreticalstress-concentrationfactors
byanalyticalmeansusingthe theory
asurements,orbyphotoelastictests4.
stressconcentrationeffectsis in
inthesmallersized specimensandfor
eontheotherhand thefine-grained,
areverysensitivetosuch effects5.In
,thedecarburizationofthesurface
eattreatmentand theeffectsofshot-
epresentfurtherfactorswhichtend to
-concentration.
C A L C U A T IO N
viousdiscussionand thatof
luatingwor ingstressinhelical
dingbasedonthe followingassump-
effectsdueprincipallytobar orwire
maybe neglectedinfiguringthe
imitingvalueofthe staticandvariable
refollowsalinear law
ofE lasticity—Timoshen o, McGraw -Hill . A lsoNeuber
pringer,Berlin formethodsofdeterminingstressconcentration
ods.
toe lasticity, V o l. I, Wiley .
onon" CorrelatingDatafromF atigueTestsofS tress
ns , T imoshen oA nniversaryV o lume, Macmillan, 1938, and
ssC oncentrationF actorsinDesign , P roceedingsSociety forEx peri-
is, V o l. 1, No. 1, Page118, discussthis. A lsoarticlebyPeterson
ppliedMechanics,March,1938,PageA -15anddiscussionDe-
46.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D IN G O F H E L I CA L S P R IN G 1 21
ertiesofthe materialarethesame
es,assumingthesamesizewire
ofloadingdue toendturnsare neg-
ducedbyheat-treatmentoroverstress-
glected.
sedmorefully later.F orthepresent,
alto stressconcentrationwillalso
ffectsofvariationsin thesensitivityin-
duetosurfacedecarburization, shotblast-
.
ressC oncentration—R eferringtoF ig.
typicale perimentalcurveoffailure
inationofstatic andvariablestress.
aluesofvariablestresswhichwill j ust
mposedinthestatic stressesshownby
gthatfatiguetestsaremadeon aspring
) s o t ha t K c = 1 , a nd l e tt i ng r ' d e no t e
erotoma imumstressrangeobtained
T ,
nofstra ightline law , tohe lical
mitt eforpulsatingloadappli-
m)andy ie ldstresstyarek now n
intPonthediagramis determined.
imumstress)boththestaticandvari-
ua ltore / 2. A s anappro imation, thee -
eplacedbythe straightlinePA drawn
bscissasatt , thetorsiona ly ie ldpo int
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
one sinceingeneralnostress should
Toapplythis diagraminactualdesign,
springisoperatingundera fatigue
totma w herethesestressesaref iguredby
orrectionfactorK = K , 3(seePage
mponentofstresst, is
esfullsensitivitytostress concentra-
tofstresst ,whenfiguredbyneg-
neffectsduetobar curvature,then
uationisdividedbyK c= K / „
adybeenusedinf iguringrma andt, , ;
ermining
for
1
ermining
rC trfor
1
arplydef ined, asanappro imationitmaybeta en
asticstrainis. 2percent. S ee" C oncerningtheY ie ldPoint
e lls, P roceedingsA . . T . M. , 1928, Page387.
rsioncould besubstitutedforj yifdesirable.In
results incloseragreementwithtests,butthe results
point will,ingeneral,beon thesafeside.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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D IN G O F H E L I CA L S P R IN G 1 23
sconcentrationeffectsduetoc urvature
ary. A nana lytica le pressionforthe line
f t a n d t i s
tionofastraightline passingthrough
titutingtheva luesof tvandt givenby
ermining
f o r t / t <
16inthise uation,
ma imumstress3tma intermsofK ,,
/ t c a n d ma y be w r it t en
f un c ti o n of K C , tu , t , i / t , u r a nd t u /t , ,
thevariablecomponentofstressisnotH rea Tthanthestatic
stressconditionscorrespondingtotheline PA inF ig.71an
alwaysthecase.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ofspringinde c, va luesofC w may
hartsforvariousspring inde es,and
' .
evalueof t, o givenbyE q uation119
fetyN sothatthewor ingstress
amountstoassumingaline CD,
nePA andintersectingthea iso fab-
f romO. A nycombinationofvariable
lls onthelineCDis thusassumed
.
lation,chartsshowingtherela-
n/ tma forvariousspringinde esat
aregiveninF igs. 72, 73and74. These
dusingthee pressionforCwgiven
assumingadef initeva lueof tu/ t, ' . Thus
o ista ene ua lto2. Thesechartsshow
mpermissiblestressincreaseswithin-
dthatthisincreaseis greaterforthe
.H owever,thisincreaseinallowable
eepandrela ationeffectsasdiscussed
ensitivetoS tressConcentration
maybeusedfor caseswherethespring
eto stressconcentration.Itwillbe
it ivity inde of thematerial(w hichisa
itivitytostress concentration)has
venmaterialandwire sizebyactual
it ivity inde q isdef inedas
ote5giveafurtherdiscussionof " sensit iv ity inde .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D IN G O F H E L I CA L S P R IN G 1 25
trengthreductionfactor,i.e.,theratioof
ressconcentrationtoendurancelimit
present.H ereK cisagainthe theo-
nfactordueto barorwirecurvature.
yinsensitivetostressconcentration
ation122, q= 0. F ormateria lsfully sensi-
nceq = l. Thusthemoresensit ivemate-
nedalloysteelsinthelargersizes,
fq thanwouldmaterialsnotsensi-
n.A smentionedpreviously,thesur-
arwouldalsohaveaneffecton the
heinde .
effect oflac ofsensitivitytostress
rangetimu: — i- , i shouldbeca lculated
threductionfactor K /ratherthanthe
isk now n, byso lv ingforKt inE q ua-
att, u andtmi havea lreadybeenca l-
yusingthecurvaturecorrectionfactor
ore, if theva lueofmultipliedby
ueofvariablestresst w il lbeobta ined
countthesensit ivityef fect. Hence,
this,
e -l )
ons116and125inE q uation117,
l( fullsensit ivity )thise uation
18. F orq = 0(nostressconcentrationef -
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
hise uationisinef fectdiv idedbyK ,
ncentrationeffectsdueto curvature
eneglectedentirelybothfor thestatic
nents.Thisise uivalenttousingthe
ti o n 90 .
isobtained:
uation120whenq = l.
nctionofspringinde cfora
V i a n d fo r t / t , e u a l to 1 . 5, 2 . 0, a n d 2. 5 a re
77, respectively.Thesechartsare
estoshowtheeffectofa reductionin
ialas measuredbytheinde con
romthesef igures, itmaybeseenthat
tostress concentrationtheallowable
rablyhigherfor thespringsofsmaller
he largerinde springs.
To illustratetheapplicationofthe
inpractica lw or , thefo llow ingcondi-
springofV 4-inchwirediameterand
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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D IN G O F H E L I CA L S P R IN G 1 27
er,i.e.,c= 3.F atiguetestsonsprings
hesamesizew iresub ectedtoapulsating
e ldava lueofendurance lim itt/ =
reinch,whiletorsiontestsshowa yield
0poundspers uare inch. F urtherthe
stressrangef romt, i to r w w here
hstressesbeingcomputedbyusingthe factor
sensitivityofthe materialisas-
t / T , / = 2, thecharto fF ig. 73applies.
n de c = 3 , C , = 1 . 53 w h en t m in / tm a — - ^ > -
ailure maybee pectedattma — Cw
0= 92, 000poundspers uare inch( q uation
rof safetyN — l. 5, thewor ingstress
tion121, t = C T/ / N= 92, 000/ 1.5-=
reinch(thestressbeing figuredbyusing
ginde w ere10insteadof3, thefac-
g. 73, andthew or ingstressrw —
6, 000poundspers uare inch, assuming
ductioninthesensitivityinde q ,
sofconsiderablysmallerinde inthis
eassumedtoshow avalueofq — ^k .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ig.76(t /2>' = 2, q = V z)C= 1.65
= % . Inthiscase, thea llow ablew or ing
afetyN= 1. 5, wouldbetw = C w tc / N=
6, 000poundspers uare inch.
fundamentallimitationinthe
discussedforspringsundervariable
ptionthatthestressconcentration
edin figuringthestaticcomponentof
under fatigueloading.A sbrought
theloadis purelystatic,thisappears
er, furthertestsw illbere uiredtoestablish
ionwhenappliedto combinations
s.The alternativemethoddiscussed
ge131) doesnot,however,involve
emethodis theassumptionofa
estaticandvariablestress components
failure,i.e.,astraightl inePA inF ig.
ctuallimitingcurve.Usuallythee -
omewhatabovethisline asindi-
ueof thezerotoma imumendurance
ctualtestsona springoflargeinde
nderconsideration,itappearsthat
basis,thecalculatedresultswillbe on
ectionitisnecessaryto determine
heactualwire sizeused,sincethe
angeconsiderablybetweensmalland
thatthelineoffailure PA tends
n torsion.Inmanycasesit will
proachestheultimatestrengthintor-
e lattercouldbeuser1insteadof t inap-
ulas.If thisisdone,however,a
uldbe usedthanotherwise.
Cv,it hasbeenassumedthat
ofthematerialdo notchangebetween
einde assumingagivenw iresize. A l-
areasonableassumption,thereare
ganE ngineeringR esearchBulletinN o. 26mentionedprev iously,
her.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D IN G O F H E L I CA L S P R IN G
estrict ly true. F ore ample, inco ld-
re size,springscoiledtosmaller
ctedtothegreatestamountofco ldwor -
s maypossesssomewhatdifferent
nwouldotherwisebee pected.Ifa
tisgiven,thisdifferencemaybeslight.
ee pectedw herespringsareq uenched
ctof theheattreatmentmaybedif-
ntinde es.Theshot-blasttreatment
(ChapterIV )willprobablyalsointro-
tivityinde .
hemanufactureofcompression
pringsolid, thusgivingita permanent
ntroduceresidualstressesof op-
hespringisunderthew or ingload,
ducedby theamountofsuchresidual
thetendencywill beforthefatigue
causeof thepresenceofthesestresses.
vatureofthe springshavingsmaller
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
rtointroducesuchresidualstresses in
happenthat thefatiguestrength
de maybeincreasedbysuchtreat-
re tentthanisthe case,forsprings
effectsofeccentricityofloading
beenneglected.Theseeffectsmayin-
ressfrom4to 30percent,dependingon
endturns,andonthe totalnumberof
nof thisw illbegiveninC hapterV III.
F T H E O R E T ICA L A N D T S T R E S U T
ledgethemostcomprehensiveseries
theef fect. o f springinde onendur-
ngs werethosecarriedoutby
easeriesof testsonspringsof . 148-inch
wedishvalve-springwire,havingin-
. 5to12. Itw il lbeof interesttocompare
eobtainedby theapplicationofthe
d74 whichassumefullsensitivityto
l) . V a luesof theminimumandma i-
J-)of thelimitingstressrangesas found
ousinde esaregiveninthesecondand
I.
a imumendurancelimitj > ' ,
, thetestresultsforc= 11. 9areused
19, 000poundspers uare inch, tma —
are inch, andrmi / tm(U= . 21. A ssuming
1 . 5, f r om E q u a ti o n 11 9 , rm u — L l l t f o r
ing, r, = 91, 000/ 1. 1-= 82, 700poundsper
eyie ldpo intintorsionforthematerialw il l
spers uareinch(orsomewhatabove
, itmaybeassumedthatt / t ' =1. 5
owingtheuseof F ig.72.Usingthe
ndassumingthe minimumvaluesof
sgiven, the lim itingva luesofma imum
edusingthecharto fF ig. 72andE q ua-
luesoftma thusobtainedaregiven
M . . , J a n ua r y, 1 9 38 , V a s 4 3 .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D IN G O F H E L I CA L S P R IN G 1 31
eX II. F orcomparison, va luesof the
—rmiH asfoundby testandasdeter-
venin thelasttwocolumns.
ntheselast twocolumnsindi-
culatedvaluesoflimitingstressrange
nt.Thisofferssomeindicationthat
uesofL imitingS tresses
im itingR angeinS tress
ests9C alculatedtmo —tmin
m l n t in n B y t es t C al c ul a te d
. )( lb. / s . in.)( lb. / s . in.)( lb. / s . in. )Ob. / s . in.)
095,50086.00081,500
096,00075,00077,000
93,50074,00074,500
92,00071,00073,000
091,00072,00072,000
usingthecurvaturecorrectionfactor K .
gw or ingstress, usingE q uation118
vitytostressconcentration(< 7= 1),
eementwithactualfatiguetests,at
dwiresizes.
ionedpreviouslyweremadeon
ledf rom% - inchdiameterbarstoc , one
eotherof inde 5. Theendurance lim its
econventionalformula,E q uation4,
orthe twogroupsofsprings,while
atedbyusingthe K factordiffered
ndicatethatforthesesprings neither
directshearstresshaveanyeffect on
ontrasttothepreviouslydiscussed
doshowthatthewire curvaturedoes
cerange eveninsmallsizesprings.
atawill bere uiredbeforedefinite
.
E M TH O D O F C A L C U A T IO N
methodofevaluatingwor ing
dervariableloadingisbasedonthe
ncentrationeffectsduetocurvature
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
atingthe staticcomponentofthe
hemethodproposedbyS oderberg
or ingstresses.A nalternative,how-
hatsimplermethodisthefollowing:
operatingbetw eenma imumand
ndP , , , thentherangeinloadwillbe
ntorsionstressrristhencomputed
ngthefullcurvaturecorrectionfactor
ataare availableregardingsensitivity
erials, thevalueofK maybereducedto
ndsonthesensitiv ity inde q andisgivenby
absenceofactua ltestdataore perience,
hatasensit ivity inde q e ua ltounity
henca lculatedf romtheloadPma using
actorK . If thispea stressisabovethe
atterva lue ista enasthema imum
nnearlyallcaseslocalized yielding
stothisvalue.Then theactuallimiting
sthetestva luew iththepea stresse ua l
n. H ow ever, if thepea stresstma is
oint,thenthe actualendurancerange
sta enasabasis. Thismaybefound
ofthetypeshownin F ig.56orfrom
nTableV II,Page88.
at, if thepea stressdoesnot
torsion,therewillnotbe muchvaria-
urancerangeforvariouspea stresses.
ses,anappro imatefigureoflimiting
ew iththepea stresse ua ltothey ie ld
c onservativefigure.Theallowablestress
factor K ,wouldthenbethis limiting
bythefactorof safety12.
essivepermanentsetthestressat
W, ca lculatedbyneglectingcurvaturecor-
hapterV , shouldnote ceedtheallow -
g.This alternativemethodofde-
pearstobe promisingandissome-
of thismethodw asgiveninapaperon" H elicalS pring
tandardC ode TransactionsA . . M. . , Ju ly1942, Page476.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 152/462
D IN G O F H E L I CA L S P R IN G
ussedpreviously.F urthertestdata,
bletodifferentiatebetweenthetwo
either wouldbesufficientlygoodfor
ampleof theuseof thismethod: A car-
soutsidecoil diameter,%-inchbar
hree, sub ecttocontinuousa lternating
mof1700poundsanda minimumof
uation18thestressatthepea load
correctionwillbe82,000pounds
somewhatbelowthetorsionalyield
dof1700—1200= 500poundsthe
00poundspers uare inch. F rom
heat-treated,carbon-steelspringthe
oma imumloadapplicationwith
poundspers uare inch, maybees-
oundspers uareinch.Thisgives
0/24,100= 2.9onthestress range
tressfiguredwithoutcurvaturecorrection,
0poundspers uare inch. S incethee -
tofthis materialshouldbeabout
are inch, TableV IIPage88, thefactor
lding wouldbe110,000/61,000=1.8.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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L E CT IO N A N D D S I GN O F H E L I CA L
S P R I N G
fthischapterto presentdataon
as chartsandtables,whichmaybe
cilitatethepracticalselectionofhelical
ns.A lthoughthemethodsofevaluat-
cribedinthe precedingtwochapters
htothe designproblem,inmany
involvedbytheuseof thesemore
rranted.Thisis particularlytrueif
incharacteristicsarere uired,for
venmechanism,plentyofspacebeing
use ofthespringtablesgivenhere
ry.
maybecaseswhentheproper
ng isvitaltothe successfulopera-
hileatthesametime,theavailable
se,aconsiderableamountoftime
g re uirementsonthebasisof the
andV I w ouldprobablybe j ustified. Even
used,however,thechoiceofthe proper
useofthechartsand tablesgivenin
E S S E S U E D I N PR A C TI C
eswhereaq uic selectionof
ewof suggestedwor ingstressvalues,
rces,isdesirable1.Inutilizing these
hatfor bestresults,aconsiderable
re uiredandforthisreasoninimportant
withaspringmanufacturerusually
arcdiscussedfurtherinauthor sarticle , TransactionsA . . M. . ,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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F H E L I CA L CO M PR E S S I O N S P R I N G 1 35
nofw or ingstressesusedasa
esignbyWestinghouseE lec. & Mfg.
ingstressvalues,whichshouldbe used
ngselection,applytospringsmade
chasmusic oroil-temperedwire,hot-
edafterforming.Inmostcases the
h e a r -H e l i c al
tee l
eServ iceA verageServ iceL ightServ ice
. in.)( lb. / s . in.)
3,000
085,000
074,000
065,000
056,000
50,000
uality springstee l. A llstressesbasedontheuseofacurva-
tabledoesnot holdwherecorrosioneffectsorhightem-
rphosphorbronzesprings50per centandforrust-resisting
valuesareused.
TableX IIIwillbe foundtobecon-
eincreasedaftera carefulstudyof
nbasedon thestresseslisted
Pages138to149are given.
beseenthatlow erstressesareused
ndforsevereserviceinaccordance
ce.Theclassificationofparticularap-
ge,orlightservicedependsto acon-
udgmentof thedesigner. Ingenera l,
cttocontinuousfatiguestressingin pul-
eretheratioof minimumtoma i-
ss, asinvalvesprings,for e ample,
reservice.O ntheotherhand,a spring
plicationsof loadduringitslif eortoprac-
ormaltemperaturewouldbelight
nswherespaceisat apremium,
ressesaresuggestedby theS. A . . War
ingC ommittee(Manualon" Designand
andS pira lSpringsforOrdnance ). S ug-
gstressincompressionspringsof music
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 155/462
S P R IN G
or.015-diameterto140,000for.15-
erablelowervaluesfortensionsprings.
essionsprings,hotwoundandheat
estedvaluesofstressvaryfrom 116,-
to80,000for1-inchdiameterbar. These
rHelica lS prings
Ma i m um S o l i d
ss,
0
00100,000
50,00080,000
00080.000
0
0,000
houtcurvaturecorrectionandapplytc
ofthesestressesassumesthat coldsetting
nsareusedto obtainma imumstrength.
houldnot beusedwherelonglifeor
uired.
fw or ingstressesusedinpractice ,
IV aresuggestedinapamphlet
on-RaymondDivisionofA ssociated
uesrefertoma imumw or ingstress
tress. Wherepossible it isfurthersug-
thespringbelimitedto % to2/3the
ress.
Wire CompanyintheirManualof
ggestforspringsofplain carbonsteels
mumTorsiona lDesignS tressesforHelica l
derA verageS erviceConditions
3inches
3inches
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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F H E L I CA L CO M PR E S S I O N S P R I N G 1 37
ditionsvaluesofma imumdesign
.
herspringmaterials, sa fewor ing
estedby thiscompanyasfollows:
uare Inch)
0
esesuggestedstressesandthose
erageserviceconditionsdefinedasnon-
rmaltemperatures,andwithslowly
ndividualcases,wherefatigueor
nt,lowervaluesofstress willbere-
caseshighervaluesmaybepossible.
S
fsprings foragivenpurpose,
mputed,basedonthe stressesof
ce.TableX V Iappliestocarbon-
alitysuchasmusicor oil-temperedwire,
gs.TheallowableloadsPare based
hilethe deflectionsperturnywere
dsusingamodulusof rigidityof
rs uare inch. Thislatterva lueappliesto
fficientaccuracy.Thetotal deflection
ebee ualtothedef lectionyper
berof activeturns.F oradifferent
sG thedeflectionsgiveninthetable
1. 4X 108/ G. F orw or ingstressesother
loadsand deflectionsmaybeta en
IIIsim ilartabulationsaregivenfor
orbronzehelicalcompressionsprings.
steel springsisbasedon stressese ual
fTableX V Iw hileTableX V IIIf o r
basedon stressese ualto50per
eel.Theshearmodulususedinc om-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 157/462
erIurny forC arbon
are inch
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 158/462
elica lS prings
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 159/462
S P R IN G
nperturny forC arbon
are ineh
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 160/462
F H E L I CA L CO M PR E S S I O N S P R I N G 1 41
elica lS pringsf
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 161/462
S P R IN G
urn yforS tainless
are inch
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 162/462
F H E L I CA L CO M PR E S S I O N S P R I N G
elica lS pringsf
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 163/462
S P R IN G
erturn_v forS ta inless
are ineh
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 164/462
F H E L I CA L CO M PR E S S I O N S P R I N G 1 45
elica lS pringsf
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 165/462
S P R IN G
rturn yforPhosphor
are inch
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 166/462
F H E L I CA L CO M PR E S S I O N S P R I N G 1 47
elicalS pringsf
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 167/462
S P R IN G
nperturny forPhospho i
u a re i n ch
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 168/462
F H E L I CA L CO M PR E S S I O N S P R I N G
elicalS pringsf
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 169/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 170/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 171/462
S P R IN G
sta inlessstee lw as10. 5X 10 w hilethat
V IIIforphosphorbronzesprings
ers uare inch. F orotherva luesof
sperturn giveninTableX V IIshould
lO' / G, those inTableX V IIIby
erageorlightserv icethese loadsanddeflec-
proportiontowor ingstress(see
useof thespringtables: A stee lcom-
edforamechanismto give160poundsat
spaceavailableis suchthatanout-
maybeused. If thespringissub ectto
X V Ifor.263-inchwirediameterand
allowableloadis161poundsandthe
rn.124-inch.Toobtain.8-inchdeflec-
24= 6. 45, say&k , activeturnsorabout
apterV IIIdiscusseseva luationofend
aspringofabout1. 1(8. 5)(.263)4-
ngth, a llowing10percente tra lengthfor
henthespring iscompressedataload
at aloadof160 poundswouldbe2.46
about 10percentless orabout2.2
olidlengthwouldbeabout10 percent
he tableisbasedor52,800pounds
tve ly low stress. A furtherdiscussionof
imumstresswhenthespringis com-
inChapterV III.
T
preparedwiththecurvature
dareshownonF igs. 78and79. F or
sarebasedona valueof100,000
w or ingstressandatorsiona lmodulus
ndspers uare inch. Itshouldbeempha-
essisusedmainlyfor convenience
recommendedwor ingstress.The
edbyW allaceBarnesCo.inTheMainspringforJune
rereproducedthroughthecourtesyof thiscompany.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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F H E L I CA L CO M PR E S S I O N S P R I N G 1 53
erange inloadbetweenoneand
79,between100and10,000pounds.
tesrepresentloadat100,000
orsionstress,theabscissas,inches de-
er activecoil.Thustheabscissa,
rofactiveturns willyieldtherecip-
ntinpoundsperinch.The setoflines
sto thehorizontalinthesecharts
s,whilethesetinclinedat about30
ecoildiameters.Theintersectionof
ato f theothersetf i esthe loadat
areinchstressandthe deflectionper
n. Thus,fore ample,ifthewire
tsidecoildiameter' /i-inch,theload
uareinchstresswill be9.3poundsand
load perturnwillbe .0025-inch.If
hespringconstantwillbe 1/.025= 40
t anyotherstresstdifferentfrom
are inchw illbe indirectratio tothe
alow ablestressof60, 000poundspers uare
e amplebecomes9.3(60,000/100,000)
stressarek now nthere uiredspring
m thechartsofF igs.78and79.Thus,
essof60, 000poundspers uare inchis
ngloadof30pounds, thew or ingload
uare inchw illbedirectratio tothe
00)= 50pounds.F romthechartit is
sizeswillyieldthis valueofload.
izeof .100- inchandanoutsidecoildiam-
loseto it.Inthis sizethespringwill
002-inchper poundofloadperactive
t30poundload,assuminga steel
10 poundspers uare inch. If , say,
uiredat30poundsloadthenumberof
uldbe.25/.06orslightlymore thanfour.
dif ferentf rom11. 5X 10 poundsper
nindeflectionmaybemadetota ethis
gthedeflectionby 11.5X10V G.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 173/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 174/462
oil— Courtesy,WallaceBarnesCo.
bpers in. )
ge100to10, 000pounds)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 175/462
S P R IN G
78and79aseriesofdashedlinesat
orizontalareshown.Theserepresent
dastheratioofthe deflectionat
areinchstresstothe netsolidheightof
g.Itis clearthatspringswitha large
ealargedeflectioncomparedto the
Thus aspringof.100-inchwireand
f% - inch(asusedintheprev iouse -
ationratioofalmost 100percentat
are inch, F ig. 79. Thismeansthatthe
ndspers uareinchstresswill beabout
t. A t 60, 000poundspers uare inchthe
bout60per cent.
erewillalwaysbeasmall in-
esigncharts.This errorshouldnot,
ent andwillusuallybewithin2 percent.
dbenotedthatbecauseofmanufac-
nsinwiresize,and incoildiameter,
entestandcalculatedresultswill
rcent,unlessspecialprecautionsin
ta en.F orafurtherdiscussionof this,
esevariables.Thismeansthat thecharts
dbesufficientlyaccurateformostprac-
r,incaseswherema imumaccuracyis
emadeusingE q uations7and18,or
rX V IIImaybeused.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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G CO N S I D R A T IO N S — H E L I CA L
S P R I N G
siderations, otherthanwor ing
tindesigninghelical compression
ssedin thischapter.Theseinclude
cesforend coils,effectsofeccentric-
iationin springdimensions,variation
sat solidcompression.Theeffect
atera lloadingtogetherw ithbuc ling
in thefollowingchapter.
T O E N D T UR N S
mployedinhelicalcompression
.80.Themostcommontype— endsset
dicated inF ig.80a—has theadvan-
entricityofloading(andhencea lower
anwouldbethecase wheretheends
ig. 80fo , cord. InF ig. 806theends
closed, w hile inF ig. 80c, theendsare
ng.Thistype ofspringwouldgive
ntricityofloading.The springof
thatatce ceptthattheendshavebeen
2turnateachendisf la t. InF ig. 80e, a
ssetupisshow n.
onofdeflectioninhelical compres-
hattheef fecto f theendturnsbeesti-
curacy.S omee perimentalandan-
1indicatesthatfortheusua ldesignofend
andground, F ig. 80a, thenumberofactive
mberofcomplete ly f reeco ilsplus% .
thiscaseis determinedbythe
pcontactpoints.)Thus ifacom-
iv e C oi l s in H e l i c al S p r in g s , T r an s ac t io n s A . . M . . , J u n e, 1 9 34 ,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
coilsand12total coils(tiptotip of
numberofactivecoils wouldbe10J/2,
einactiveateachend.H owever,when
is someprogressiveseatingofthe
berofcompletelyfreecoilsdecreases
reasesthenumberofinactiveturns.
rison2madeaseriesof carefultests
ngaspecial setuptodeterminethe
softhesetests indicatethatatzero
eturnsw ase ua lton - f -% w heren
yfreec oilsatzeroload.A stheload
umberofinactivecoilswasfoundto
the endcoils,theamountofincrease
rnatusua lw or ingloads, w ithan
inceforcalculationpurposesthe
turnsinthe rangefromnoloadto
ary interest(thisw ouldbeusedinthe
earsreasonabletosubtractabouthalf
umber ofactiveturns.Thisgivesa
urnsvary ingf romn ton + V t. B e-
f turnsisn + 2fortheusua ltypeof
eansthatthe inactiveturnsfoundin
out1% to2withanaverageof1.85.
. C . K eysor3indicatesthatthetotal
thespringisappro imatelye ua lto
so lidturns basedonthecommonly
thenumberof " so lidturns e ua ltothe
orwirediameter.S incefortheusual
0a , the" so lidturns aree ua ltothe
edf romtiptotipofbar, m inusV - iturn,
e uivalenttoadeductionof1.7 turn
inactiveturnswasgiven by
eresearchof theS pecialR esearchC om-
pringsofA . . M. . Theaveragevalue
as1.15asadeductionfrom " solidturns
rstra inonC lose lyCo iledHelica lSpringsandtheV ariation
eC oilsw ithL oad , V irginiaPo ly technicInstitute , Engineering
onB ulletinNo. 24.
lasticC urveofaHelica lC ompressionSpring —H . C . K eysor,
.,May,1940,Page319.
Desig , December, 1939, Page53.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I D R A T IO N S — H E L I CA L S P R IN G 1 59
mtotalturns.
eseinvestigationsasabasis,it ap-
signofendcoil thenumberofin-
about1.65 to2consideredasa deduc-
blya meanvalueofP/iinactivecoils
asany touse inpractice . F orthe
resomewhathighermaybe j ustified,
usedforlowerloads.Theseating
easesalsotendsto produceaslight
ctiondiagram.F orfurtherdetails,
einvestigationbyPlettaandhis
asbeenconcernedonlywith the
stresultsconcerningtheothertypes
N D G R O UN D O R F O R C D (b )S Q U A R E D O R C O S E D E N D
T P ) N O TGR O U N D
( d) P A I N E N D GR O U N D
E T U P
urnsasused inhelicalcompressionsprings
. 80are lac ing, butappro imateva lues
low s: F orpla inends, F ig. 80c, active
totalturns forpla inendsground,
ren— 1. / / 2% turnsateachendareset
80etheactiveturnsmaybeta en
F L O A D IN G
usualdesigniscompressedbe-
sin atestingmachine( ig.23),it
raltheresultantloadisdisplaced from
allamounteasindicatedinthisf igure .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
loadingistoincreasethe stresson
meteranddecreaseitonthe other
ple, by the load-stressdiagramofF ig. 49
son onesideofthespring thanon
of thiseccentricityofloadingbased
salsobeencarried outbyK eysor
tyoftheanalysis,it willnotbegiven
resultsofthe analysisaregiveninthe
natesrepresentingratioe/r between
usrandthe abscissasbeingthenum-
tipcontactpoints.The totalnumberof
gnw illbee ua lton + 2. It isseen
uctuatesbetweenzeroand ma i-
esoccuringappro imatelyatn =
Theoretically itshouldbepossibletoget
eccentricity)bychoosingn toconform
ver,becauseofvariationsinactual
ecauseof errorsintheassumptions
nnotingenera lberea lizedinpractice .
refore,theenvelopeofthec urveasin-
beemployed.
oe/rthefo llow inge pressionsgiven
d
- ( 1 29 )
so lidco ils. Thisw illbeappro imately
nthenumberofco ilsn betw eentipcontact
1. 5. B yusingthesee uationstheratio
sanappro imationitmaybeassumed
e isfa irly largethestressw illbe in-
e/rascomparedwiththestress for
adebythe writerwhichgivea
ula.Thesewerecarriedout incon-
perimentsmadebyPlettaandMaher— " H eli Warping
prings , TransactionsA . . M. . , May1940, Page327.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I D R A T IO N S — H E L I CA L S P R IN G 1 61
whereit wasdesiredtoobtainas
loadonahelicalcompressionspring.
all helicalspringsusingaspecial
resoarrangedthatthe eccentricityof
ed.E ssentiallythisconsistedofaflat
achingdeadweights120 degrees
hene ualloadsw ereappliedate ua l
T UR N S B T W E E N T I P CO N T A CT PO I N T
tw eeneccentricityandco ilradiusasafuntionofn
ysor
ndthattheloadingplanesat each
tparallel.Theloadswerethenad-
smof these loadingplanes f romthe
edloadstheeccentricityofloadingc ould
aresummarizedinTableX IX ,
r,wirediameter,numberofturnsn ,
pringstestedhadgroundendcoils of
columnthevaluesofthe ratioe/r
coilradiusas calculatedfromE q ua-
en.F orcomparisonthetestvaluesof
arioussprings arealsogiveninthe
.
cases theagreementbetween
s sufficientlygoodforpracticaluse,
edthatthetestspringswerehand made
sta eninformingtheendturns.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
entricityof
prings
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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I D R A T IO N S — H E L I CA L S P R IN G 1 63
ctioncharacteristicmaybebrought
.Ifsprings aretobeheld torelatively
oallowsomeleewayonthecoil diam-
rofturns,sinceotherwisethe cost
s—Inwindingspringscold,there
bac " . Inotherw ords, the insidediam-
inC ommercia lSpringWireS izes
-227-39T\ i
wire. .A -229-39T> <
31-39T I(
gwire....A -230-39T^ i
pringwireA -232-39T/I
ndingwillbeslightlygreaterthan the
sa conse uenceoftheelasticand
terial.A lthoughthiseffectmaybe
aslightlysmaller mandrel,fordiffer-
pectedthatsomevariationsincoil
ualvaria tionsinco ildiameterstobe
tolerancesgivenby onespringmanu-
ableX X I. Itmaybeseenthatthese
thespringinde D/dandonthe
mall variationsincoildiameter
estimatedq uantitativelyasfollows:
mulaforhelicalsprings( q uation7)
he nominalmeancoilradius
tively.S upposingthatthetruemean
ngineeringpublishedA mericanS tee landWireC o., Page97.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
t er a r e r = r ( l + « ) a n d d = d ( l- f A )
uantities,relativetounity,thetruede-
medthateandA aresmallre la tiveto
igherpow ersmaybeneglected. Hence
ttenwithsufficientaccuracy(since
/( l + A ) , sr l — 4 A ) :
0)
ectionS 1ismerelythe nominal
term l+ 3e—4awhich dependsone
w thattheactua lmeanco ildiameteror
terthanthenominal,i.e.,e= .01,
truewirediameterisoneper cent
= —. 01. Puttingtheseva luesofc
13 0 ,
c onditionswithaonepercent
andwirediameterfrom thenominal
nwillbe 1.07timesthenominalde-
ter.
acticale ample , anactua lcasee amined
ssed.Thisspringwasmadeof
= .5625inch.A ftercuttingupthe
imensions,itwasfound thatthe
s.551-inchwhichwouldcorrespond
of(.5625— .551)/.5625= 2.04per
ssumingthetruemeanco ildiameterof this
enominal, i. e ., thatc= 0, thenf rom
edef lection8tw ouldbe1—4A =1. 08times
ctualcoil diameter,however,had
t lessthanthenominal,whichmeant
hisvalue inE q uation130thetruede-
esthenominaldeflection.Thus,the
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I D R A T IO N S — H E L I CA L S P R IN G 1 65
enmadeslightly smallerthanthe
gma ertocompensateforthede-
reused.
beshownthatoneper centvari-
rmeansappro imatelya3percent
aonepercent changeinthecoildiameter,
tress.Usually,thestressdoes nothave
ilDiameters*
icalS prings)
ationsinDiameter
1 2
5 ± 00 5
0105
0130
18
8
S tee l& WireC o.
itsas thedeflection however,acon-
ommercialvariationsin wiresize
for certainhighlystressedsprings.
O DU U O F R I GI DI T
ofthe deflectionofactualsprings
eef fectiveturnsbek now n, buta lso
of thespringmaterialbek nown
ndicatedinthe discussionofChapter
reportedintheliteraturevaryconsider-
ctof adecarburizedlayeronlya
emodulusbyseveralper cent.
ultsgiveninTablesV andV I agood
sofrigidityfor carbonandalloysteel
ers uare inch. How ever, forhot-wound,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
-rolledmaterialinthelarger sizes,
mmendamodulusfigureof10.5X 10
. On thebasiso f thetestdatagivenin
odulusofR igidity
1.5X 10
0. 5X 10a
averagef iguresgiveninTableX X II
ring materials.Itshouldbenoted,
ofseveralpercentmayoccur.
O L I DCO M PR E S S I O N
ionspringsitisdesirableto choose
enthespringis compressedsolid,no
twilloccur.Thereasonfor thisis
espring mayattimesbecompressed
condit ionsitta esaset, the loadat
bechanged.Thusthe springwillno
cteristics.
mpressionspringisinitiallywound
ygreatsothattheelastic limitofthe
henthespringiscompressedsolid,the
a diameterofthecrosssectionis
pringof large inde 7. A tlow loadsbe-
chedthedistributionisappro imate-
. 82a. A f terthe loadisre leased, the
w illbe li e thatinF ig. 82c.
de theseresidua lstressesmaybe
lyfromtheconditionthatthemoment
ythe triangleobcaboutpointo must
ofthe stressesrepresentedbythearea
sagainappliedthe resultantstress
hodsof calculationofloadsforcompleteyielding,see
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I D R A T IO N S — H E L I CA L S P R IN G 1 67
g. 82d. It isclearthatthema imum
nreducedbytheoverstressing,since
tesignare inducedandthesesub-
uetothew or ingload. H ow ever, in
or overstressing,thefreelengthhas
initial freelengthofthespring ismade
reelengthby theproperamount,
heldtothespecifiedvalue.A tthe
s overstressingprocess,ahighercal-
pressionmaybe permissible.
dacertainlimit,there willbe
this process.Inotherwords,be-
ength,thefinallengthafterthe set-
same.Thereasonforthisis thatthe
oflattenout ( ig.61)sothat ahigher
fstressesovercross
f large inde (a),
alloadbeforecold-
aboveelasticlim it (c)
etlingwithloadre-
mal loadaftersetting
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
essesatSo lidC ompressionfor
cluded.
preciablygreatertorsionmoment.If
co ldw or andlossofductil itymayoccur.
f fectwhichoccursw henthistypeof
w hatisk now nas" recovery . Thusim-
eoperationonacompressionspring,
ngwillbea certainvalue onstanding
thesettagestress istoohigh,thefree
.Thisagainwill changetheload-
ofthespringand isob ectionablein
ample,instrumentsprings).
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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R A L A N D A X I A L L O A D IN G B UC L I N G
CO M PR E S S I O N S P R I N G
madetoolongrelativetoits
at atacertainload,a suddenside-
This phenomenonisessentiallysimi-
ongslenderco lumnwhenthe loade -
hedesignofhelicalcompression
guardagainstthislateralbuc lingby
tionsinsuch awaythatthecritical
ysbe greaterthananyloadencoun-
otdone,somesort oflateralsup-
eforaguide)must beprovided.
ngloadforhelicalsprings maybe
e samemannerasthatusedin
er,theanalysisinthecaseof thespring
rdinarycolumntheoryinthatit is
ecreasein lengthunderload.In
column,ontheother hand,thisde-
eneglected.Thereasonforthislies in
tyof moststructuralmaterials,which
engthfromnoload tofullloadis
nt.Thisisnot true,however,fora
a verylowmodulusofelasticity.Be-
underload,itis alsonecessaryto
softhespringdue tolateralshearing
eassumedthatthespringis close
glemaybeconsideredas small.
sionofco lumntheorysecTheoryofE lasticS tabil ityby
ill, 1936.A discussionofbuc lingofhelicalspringsis
it- ona lre ferencesonthebuc lingofspringsseearticles
i ( . V r r . d . In . . V . 5 4 , P ae e 1 38 , 1 91 0 b y R . G r a mm e l. Z e i t A n e ru .
e384, 1924 andbyB iezenoandK och, Z ettA ngew Math. Mcch.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
ngthofspring / —lengthofspringaf ter
berofactiveco ils r= meanco ilradius x ,
ompressive, f le ura l, andshearingrigidit iesof
dcondition-anda,fi," arethesame
ssionofthespring.
eshownfromcolumntheorythat
nsina columnwithhingedendsare
dis
earingrigidityof theco lumnandPristhe
I/ t2, EIbe ingthef le ura lrigidityof the
lloadis thatfiguredbyconsidering
andneglectingshearingdeformations.
stoaspring withshearrigidity7
e criticalloadbecomes
gthecompressive, f le ura land
inverselyproportionaltothenum-
h( q uations135, 141, and144). Hence
7 . .- , - ' 1 3 3 )
on133inE q uation132,
pterII,compressiverigiditybecomes
yisme(inttheratio ofloadtodeflectionper unitoflength
direct compression.F orabarof cross-sectionalareaA
thecompressiverigidity ise ua ltoA E . L i ew isethe
tioofbendingmomenttocurvaturefor abeaminpure
modulusofelasticitytimes momentofinertiaofthecross-
tyise ualtotheratio ofshearingforcetoshearing
hand forabeamise ualtomodulusofrigiditytimes cross-
pliedbyaconstantdependingon theshapeofthesection.
age140.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D I N G— C O MP R E S S I O N S P R I N G 1 71
w henthespringiscompressedtothecrit ical
uationholds:
E q uation135,
rgivenbyE q uations134and137, thefo l-
nthise uationmaybereducedto
0 ( 1 3 8)
C a lculationof thef le ura lrigidity / o f
ishedbydeterminingthe angular
spring undertheactionofa moment
he coilasindicatedin F ig.83.This
ingaq uarterco ilsub ecttoamoment
ig.84.Themomentishere repre-
crosssectionatanangle< f> thebend-
os< f> whilethetwistingmomentM,
sideringa lengthds= rd< f> , thetota lcom-
outthea isofthemomentwillbe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 191/462
S P R IN G
rethef le ura landtorsionalrigiditiesof
ectively.Thetwistdue tothebending
pliedbycos < f> toobtainthecomponent
hemoment thatduetoM< mustbemulti-
ectedtotransversemoment
momentof inertia inbendingof thesection
sionasIp. S ubstitutingMb= Mcos< £ ,
d, t= rd< f> inE q uation139gives
< t > + s m < p d < t >
oracompleteturnwill befour
betw een< f> = 0and4> =ir/ 2. Thus:
henumberof turnsperincha ia l
meansthatthe angulardeflectionin
llben6/ l . Thisw illbea lsoe ua ltothe
ng1 = 2Iforacircularcrosssectionof
lrigidity / „ , w hichistheratioofbend-
sseento be
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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D I N G— C O MP R E S S I O N S P R I N G
calculate theshearingrigidity,the
g(or coil)underashearforce Q is
nsideringthedeformationoftheq uar-
b,thebendingmomentatan angle$
lundermomenttrans-
thisdividedbyEIandmultipliedbydsw ill
d6in alengthds.H ence
isy— ywillbethis anglemulti-
ngturnundershearforce
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
gives, usingEq uation142,
, integratingbetween0andr/2,and
shearingdeflectionyfora complete
w O r
turnsperincha ia llength, thetota l
cha ia llengthw illbe
gidity,or ratioofshearingforceto de-
comes
onsinEq uations135,141, and144fora„
ti o n 13 8 a nd t a i n g G= E / 2 (1 + " ) w h er e » = P oi s -
0(145)
uationhasonerealposit iveroo l
calvalueofz atwhichbuc lingoccurs.
n,thecorrespondingcriticalloadis,using
( 14 7 )
uation145,maybee pressed:
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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D I N G— C O MP R E S S I O N S P R I N G 1 75
stantofspringor loadperinchdeflection,
ndingontheratio l / rbetweenf ree
r= crit ica lorbuc lingload.
dendsaswasassumedinthe deriva-
nby the low ercurvea inF ig. 86. A
opivotsasindicatedin F ig.87bmight
— I 1— I 1 — I — I — I— I — I I
L E N G TH
A D IU
dngbuc lingloadfactorC i> . C urvea
ds curvebforhingedends
atelyasaspringwithhinged endspro-
ssmallcomparedtothe freelength.
s—W heretheendsof thespringmaybe
milaranalysismaybe carriedout.The
thesameformasE q uation148e cept
actorC a isnow tobeta enf romtheupper
seof built-inendsissimulatedwhena
dbetweenparallelplatesas indicated
eof incompletef i ityo f theendsandof
ngloadmaybelowerthancalculated .
cularw irethespringconstantC is
noandKoch, loc. cit . fo ra furtherdiscussionof thisproblem.
investigatorsshowgoodagreementwiththeanalysis
ofcoils isnottoosmall andthatthecoilsdo nottouch
lsocommentsinMachineDesign,July1943,Page 144.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 195/462
S P R IN G
uation7, E q uation148maybow rittenas
consideredastheratioof thecritical
lingoccurs)tothe freelength.Thus,
aybee pectedatadef lectione ua lto . 4I.
testsshow agreementw ithEq uation
meinaccuracymaybe e pecteddue
nsionsandthe effectofendturns.
e a m pl e o f th e u se o f t he b u c l i ng l o ad
buc lingload,asteelhelical
efollowingdimensions:F reelength
diusr= .75-inch,outsidediameter==
hf i edandhingedends
rd= .25-inch,activeturnsn— 12.
79(C hapterV II)fo rthesedimensions
142poundsperinch. F romF ig. 86the
unde ualto . 64forZ „ / V = 6/ . 75=8. It
ring iscompressedbetweenparallel
arecompletelyrestrainedfromrota-
ig. 86forf i edendsmaybeused. Then
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D I N G— C O MP R E S S I O N S P R I N G 1 77
eca lculatedbuc lingloadPcrisC B Z cC =
nds. A ssumingama imumw or ing
ers uare inchtheactua lloadonthe
echarto fF ig. 79, C hapterV II)P=
ercent ofthevaluefor100,000pounds
heseconditionsthereis aconsider-
w or ingloadandthebuc lingload.
spring werehingedasindicatedin
raintduetorotationoccurs,thenusing
onstantC M= . 2. InthiscasePcr~
170pounds. Hence, w iththistypeof
bedangerof suchaspringbuc ling
of190 poundswasreached.
L A N D L A T R A L L O A D IN G
ngsaresometimescalledon towith-
s,butalsotransverseloadsas indicated
e Q representsthetransverseload.
icationsarecertaintypes ofrailway
ngunder
ia lload
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
calspringsmusttransmitlateralloads
ds.Incertainrefrigeratormechanisms
pportedonhelical springs,these
bsorblateralforcesduetothe un-
echanismaswellasa ialloadsdueto
ral deflectionsofaspringunder
nedeffectofthe a ialloadPand the
nsidered.Ingeneralthe largerthe
ebuc lingloadthe largertheef fecto f
n.
dbotha iallyandtransverselyas
beconsideredas acolumnundercom-
seloads5.Itis alsoessentiallythesame
r combineda ialandtransverseload-
I. Theprocedure inca lculatingsucha
tthelateraldeflectionof thespringis
ew erenoa ia lload. A ssumingthe
swayis 8 ,thecriticalload Prrisfound
rthecaseofbuilt- inendsusingcurveb
ia lloadactingonthespring, theratio
il lbeshow nlaterinC hapterX V Ithe
tionduetothe a ialloadwillbe given
nofP / P, , aregiveninF ig. 156of
actuallateraldeflectionwill be
8 whichwouldoccurifno a ial
sof beamtheorymaybeused.The
oadedbya latera lloadQ ( ig. 147
onsideredastwo cantileversoflength/,2.
to bendingof
lasticS tability,Page4.
methodofdeterminingthisfactorisgiveninthereferenceof
thattheappro imatee pressionissuf ficientlyaccuratefor
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L A N D L A T R A L L O A D IN G
ura lrigidityof thecantilever.
lyloadedhelicalspringofF ig.
/ 3givenbyEq uation141isused, ta ing
ssedlengthIunderthe loadP.Thus
deflectionduetodirect shear
videdbytheshearingrigidity7 and
. Tof ind7E q uation144isusedta ing
shearingdeflectionbecomes
sthesumof 8,and8 .Thus,usingE q ua-
15 4)
ngs, E = 30X 10 poundspers uare
poundspers uare inch. Usingthese
4andsimplif ying, thee pressionforthe
ia lloadbecomes:
1 . 0 6r = ) ( 1 5 5)
sedin E q uation151tocalculatethe
auseofthis lateraldeflectionthere
stress.A naccuratecalculationofthis
sandwould beverycomplicted.A s
smay becomputedasfollows:The
a ialloadPwill bePr.Theeffective
byanamount8/2dueto theeccentric
sionmomentdueto Pbecomes
dit ionthe latera lforceQ producesa
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
sresultsina totaltorsionmomentof
themomentMtwillbeobtained
tion(4c—l)/ (4c—4)fortheef fecto f
pringinde , asindicatedinEq uation
6
l \
eto thedirectshearloade ualto
mthe secondterminthebrac etsof
ectshearstressatthe insideof theco ildue
lbezero . Hencethema imumshear
onsideredas veryrough.S ince
uchgreaterthant , itmaybee pected
adingistoincreasethe stressinthe
9
ceP ismuchlargerthanQ, andfor
esareofprimaryimportance.Ina
ressesmaybecalculated.
ample: A stee lspringhasthefo llow -
diameter— 5inches,meancoilradius
meterd= ? 4- inch, f ree lengthl0=9 A
ringof% - inchw ireand5inches
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L A N D L A T R A L L O A D IN G 181
of 1100poundsthedeflectionper
chor1.5inches for8activecoils.
= 1100/1.5=732poundsper inch.
3= -4.48andfromF ig.86thebuc ling
gtheseva luesinE q uation148the
(9 . 5) 7 32 = 4 8 60 l b
a la ia lloadPonthisspringis2400
00/4860= . 494. F romEq uation150, the
actorC,is2.Tocalculate thedeflec-
5isused. Theva lueof Iusedinthise ua-
minusthedef lectionduetoa loadof
i l lbe2400/ C = 2400/ 732= 3. 28
3. 28=6. 22inches. A ssuminga latera l
bysubstitutioninEq uation155,
04 ( 6- 2 2) 2 + 1 -0 6 (2 ) , -r " 2 7 4
thedeflectionata latera lloadof200pounds
. 5 48 i n ch e s
casetheactualdeflectionwitha ial
calculatedbyneglecting-theeffectof
ssatana ia lloadof3050poundsis
areinchwithcurvaturecorrectioncon-
sa ia lloadthestressw ouldbe
8, 500poundspers uare inch. F rom
orC fordeterminingthe increase instress
2 5 f or « = . 5 4 8 ,Q = 2 0 0 lb
ein stress.to1.25(78,500)=98,000
maybee pectedduetothe latera lload.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
eatmanydifferentspringswas
C haplin, andSheppard7. Thetests
asteelplate ortableonfour springs.
a ialloadwhilethetransverseload
le.Inthismanneressentiallythe
.88wereobtained.R esultsofthese
lescatterbetweentestvaluesof lateral
atedbyusingE q uations151and
nin thetestpointswasfrom about
tof thetheoreticalvalues,mostofthe
gwithin20percent ofthecalculated
tbyLehrandGrossinGermany
d20percentor morebetweencalcu-
Byta ingspecialprecautionsin
springandbyaccuratelydetermining
mberofturns,these investigators
goodagreementbetweentheoryand
loadedcompressionsprings.F orthis
chiefcausesofthediscrepancybe-
n (1)imperfectclampingoftheend
ghtrotationunderloading whileno
theory,and(2)inaccuracyinde-
mberofturnsandlength ofthespring.
ompressionspringswiththe
dturnsthedesignermay,therefore,
nsasmuchas20percentandevenmore
oflateraldeflectionas obtainedfrom
calSpringsunderTransverseLoading byB urdic , C haplin
cionsA . . M. . , October, 1939, Page623.
edbyV . D. I. , Berlin, 1938, Page100.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G F O R
E F F I CI N C
uentlyarisesinhe licalspringdesign
gwithgivenloadand deflectionchar-
ablebeinglimited.S inceloadtimes
oenergy,thismeansthat acertain
storedwithinthe givenspace.A con-
bleenergystoragevarieswith spring
erest.
G
sproblemis tocalculatetheenergy
ringwhen thecoilsj usttouchand
ssatsolid compressionbeingassumed
wablevalue.Theamountofenergy
calculatedforvariousspring inde es.
ace occupiedbythespringthe amount
oredw illbeama imumatadef inite
ow ever, thisoptimumvaluewillde-
ghas variableorstaticloading1.
htV o lume—F orstatica lly loadedcom-
springsare compressednearlysolid
e occupiedbythespringwill ap-
htvolume,i.e.,thevolumeoccupiedby
ualtotheoutsidespring diameterand
e heightofthespring(this neglects
endturns).H ence,forapplications
springs,theuseofsolid-heightvol-
s acriterionofefficiencyof space
and,wherethe springisundervari-
ma imumstressrange,thespace
hissimilar lothatusedby J.Jennings( ngineering,
34).H owever,themethodusedbytheauthordiffersfrom
atadistinctionis madebetweenstaticandvariable
eflectionformulais usedinsteadoftheW oodformulaused
eris q uiteaccurate(seeChapterIV ).
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
nunloadedmaybe muchlarger
me.Inthiscase thefree-heightvol-
edbythe activepartofthespring
samorerepresentativecriterionthan
owever,itshouldbementionedthat
on offree-heightvolumeiscompli-
lueofallowablestressat solidcom-
.Ifa lowstressisused,the difference
edbyusingthe free-heightandthose
eightvolumewouldbemuchless than
igh stress.
nswherespringsare sub ect
finitial compression,acriterionof
atebetweenthefreeandsolid-height
mostrepresentative.Thisvolume
mountofinitialcompressionandon
dheight.I naddition,forbestac-
dbe considered.Toavoidallthese
criterionofsolid-heightvolumewill
marilyas aconvenientguidefor
fspaceutilization.
Toapply thiscriterion, thepotentia l
duetoaload Pandadeflection8
storedenergyis
ingisgivenbyE q uation7.S ub-
q uation160thestoredenergybe-
eusualmeanings.
the loadactingonthespringisvari-
anautomotivevalvespring,the
hespring asfiguredusingthecurvature
uation19, may, asaf irstappro imation,
sub ectbyE. L atshaw , MachineDesign, March, 1942.
nanumberofcurvesbasedon free-heightvolumeanda
ers uareincharegiven.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- F F I CI N C
heloadcarryingability .Thisstress
venbyE q uation19.
PandsubstitutinginE q uation161
denergybecomes
rion ofsolid-heightvolume
energymustbestoredin avolume
derw ithadiametere ua ltotheoutside
nda lengthe ua ltotheactive length
atterisfully compressed.This
angle,whichissmallforpractical
olumeisthus
- = — — ( c+ 1 )
.
on164in163ane pressionfortota l
pendingonthespringinde candis
t,foragivenvolumeof spaceoc-
stress,theenergystoreddependsonly
twhichin turndependsonlyonthe
morecompletediscussionof this.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
fC , areplottedagainstspringinde c
.89.Thiscurveshowsthat forvariable
ngthema imumenergy isstoredin
tagivenpea stressif thespringinde is
C ~ J
of f icients forvariable loads
er,itshouldbenotedthatif thefree-
enasabasis thisoptimumvalueof
eensomew hatless . Thus, if astressof
areinchisassumed,theoptimuminde
hefree-heightvolumeisusedas the
aceutilization.
orage—Wherethe loadisstaticorre-
ringthe servicelifeofthespring as
. L atshaw , loc. cit .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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- F F I CI N C
indicationsarethatcurvatureeffects(but
ear)maybeneglectedincalculating
stresst0 shouldbecalculatedby
\ . ,
q uation90andta esintoaccountthe
ct shearload.
eadofE q uation162andproceeding
,theenergystoredbecomes
C = * r
f f icientsforstaticloads
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
thespringinde c,thelower
ined. Thiscurve indicatesama imum
alestpracticalspringinde . Thismeans
nvolved,ma imumenergystorage
beobtainedinagiven spacebyusing
eofthe inde (whichwillusuallybe
creasingtheamountofenergy
ivenspaceisto useaspringnest,
rmore springstelescopedonewithin
nestof
ig. 91. A practica le ampleof the
wninF ig. 92w hichrepresentsanend
for alocomotivetendertruc .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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- F F I CI N C
ystoragetheso lidlengthsofa llthe
tshouldbe thesame.A ssuminga
ngs,thismeansthat
sefollowingthesubscripts1 and2refer
ngsofthenest,respectively.In addi-
atthefreelengthsof thespringscom-
ocomotiveWor s)
nestforlocomotivetender
esame.Thismeansthatthe total
thesameatanygivenload,i.e.,that
orvariable loadingthedef lectionis
ions7and18.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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L S P R I N G
r t ni ( 1 71 )
e esclandc, thesee uationsmay
-- * * * * . ( 17 2)
stressineachspringisassumed,
d1= n2dif romE q uation169thismeans
c,andc-. (andhencealsothecurvature
K . , )shouldbethesameifS l= S „ . If
adethesame,theenergy coefficients
heinde es)willbethe samefor
uation165thismeansthatthe total
by
thevo lumesenclosedby theouterand
whencompressedsolid.S ncc
ationmaybew ritten
)
uch,theouterdiameterof the
a ltothe innerdiameterof theouter
2+ d2. UsingEq uation164thisgives
= — n , 2 22 , J (c , - 1) 2
d n1 < 2 1 = n d 2 , fr o m th e se e u a ti o ns i s o b-
uation173, foratwo-springnest,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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- F F I CI N C
osedbyouterspring.
e f ficientC , areplottedagainst
9. F romthisit isseenthatforatw o-spring
g,thema imumenergystorageisob-
saround5to 7.Thesevaluesaresome-
ainedfora singlespring however,
wouldbethecaseif thefree-height
asabasis. Thusfore ampletheana lysis
previously2indicatesanoptimumvalue
nneste ua ltoabout4, basedonanal-
poundspers uare inch. F ora low er
ofspring inde wouldbehigher.
donsolid-heightvolumemaybe
st.Thisgives,for energystored,
nbyEq uation166.
aga instcinF ig. 89show thatfora
eassumptionsofe ualdeflectionand
ma imumenergy isstoredinagiven
shav inginde esaround6to8are
swillbe lowerifthefree-heightvolume
oraspringnestsub ecttostaticloading
n e actlythesamewayasbefore,
isf iguredf romEq uation89w hich
ntdueto curvature,andinsteadofC,
on168)isused. R esultso f thisana ly sisare:
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
aticallyloaded,thestored energy
1 (1 80 )
taticallyloaded,storedenergyis
( : ) i ( >
" areplottedaga instspringinde cin
tappearsthatforstaticloadsthema imum
pate willbehadbyusing springswith
ratw o-springnestandw ithinde esof4to
Theseoptimumvaluesaresomewhat
forspringsunder variableloading,
olid-heightvolume.However,if the
umeisusedfor springsundervariable
weentheoptimumvaluesofinde for
willbe small.Inanycase,theresults
imumenergystoragewithina
onsideration,aratherlowvalueof
yaround3 forasinglespring andabout
ree-springnests.
A ssumingthedesignerre uiresa
anddeflectionforagivenapplication,
edis fi ed.Byusingtheformulasof
amountofspacere uiredforthespring
ea stress. A ctua lly , o therpractica l
elargerspacere uirementsthanthose
sshouldgivea roughindicationof
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
dcombinationtension-com-
mthatof compressionspringsinthat
inreducingallowablestressshould be
S I O N S P R IN G
eninmanufactureafairlysharp
aratthepo intw herethehoo j o ins
, F ig. 93b, mayoccur. Thiscurvature
ss concentrationwhichisnotcon-
dof stresscalculationforhelical
ahalfendturnis bentupsharplyso
elysmall,whichtendstoresult ina
s.F orthisreason,mostfailuresof
ch pointsandthisis onereasonwhy
ngstressis usuallyrecommendedfor
edwithcompressionsprings.This
ctmaybereduced,andthestrength
endturn sothattheminimumradius
possible.
stbeconsideredin tensionsprings
on.Bycertainmethodsofcoiling
bringthecoils togetherinsucha way
eappliedbeforethecoils willbeginto
hisinitialload islimitedtoa value
round6000 to25,000poundsper
neglectingcurvatureef fects), thee act
pringinde . A f terseparationof the
heload-deflectiondiagramisthesame
tainedforaspringwithno initial
smstheinitialtensionis important.
s A lthoughane actca lculationof
elicalspringsis complicated,arough
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
nin F ig.93(whichhasa sharpbend
maybemadeasfo llow s: Thebendingmoment
harpbendbegins)duetothe loadP isP rap-
albendingstressatthispoint willbe
hesectionmodulusof acircularsec-
gwithhalf-loopcoilend
s greatasthenominaltorsionstress
momentPr).Thema imumbend-
emultipliedbyafactorK , w hereK,
actordependingontheratio2r../d,the
atthestartof thebendinthe plane
dinF ig. 93. A nestimateofK , maybe
fF ig. 180, ChapterX V II, f o rtorsion
undwire inbending.Tothisbend-
dedthedirecttensionstress4P /V d- .
bendingstressatpointA ' e ualto
93bnearthepo intw herethebendj o ins
pringthestressconditionis prin-
maybemadeasfo llow s: F romE q ua-
caseofpuretorsion actingona
matee pressionforstressconcentration
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
)w here inthiscasec, istobeta en
gtheradiusofcurvatureof thebend,
umstressduetothetorsionmomentPr
rstresspresentat pointA dueto
tshear,however,doesnot actatthe
thetorsionstressgivenbyEq uation
uently , E q uation184maybeta enas
essionforthema imumstressatpo intA .
small,andtheq uantityintheparen-
gwithfullloopturnedup.
eappro imatelye ua ltothe
ng, E = B / 3appro imately ,
n = numberof turnsindimension/
psbegin. Wor ingturns
maybelarge.Thismeansa highstress
showstheadvisabilityofk eepingr,
ndA therewillbeacombinationof
i ma te ly
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
eswhichdependonthe shapeofthe
diir, andr2. Sincethepea sof these
ssesdonotoccur atthesamepoint,
presentsconsiderablecomplication
here . F orpractica lpurposesE q ua-
bablysufficient.
ypeoftension springwithafull
inF ig.94,theminimumradiusof
blylarger,andthe stressconcentra-
he casewiththespringshownin
pisturnedup.Dimensionsas commonly
alsoindicatedinF ig.94.
Def lectionofTensionSprings—To
ctiveturnsina tensionspring,the
pointswheretheloop beginsisdeter-
edthedeflectionduetothe endcoils.
dicatethatahalf coilturnedupto form
g. 93ise uivalentto . 1full- coilasfar
Thus,if aspringhadn turnsbe-
psstart,the totalactiveturnswould
a. 2turnbeingthee uiva lento f two loops.
wnanalyticallyasfollows.Thehalf
oopise uiva lenttothequarter- turn
loadedasshow n. UsingEq uation143
arter-turnbecomes
emeanradiusof the loop(ta ene ua l
inceformostspringmaterials
2. 6G, appro imately , w hereG—
uationmaybewritten
ately , w here$ isthedeflectionperturn
up, F ig. 94, thee perimenta lwor
f lectione ua lto , 5-turn. Inthiscasethe
M. . , 1934, Pngc558.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
sw ouldben + lw heren isaga in
tswheretheloopsbegin.
ntof initialtensionwhichcan
dsprimarilyonthespringinde
thelowertheinitialtensionvalues.
pondingtopractical valuesofinitial
IV w erepublishedbyWallaceB arnes
ca lculatedfromEq uation4, curvature
encetheinitialtensionloadmay be
ofstressusing theformula
stress.
lationforf indinginit ia ltensionload:
nch outsidediameterandy4-inchwire
2r/ d= 7. Whenw oundwithma i-
pondingtoInitialTension
T e « ° " S t re ss
u a re I n ch
tess, fromTableX X IV , dueto init ia l
00poundspers uare inch. F romEq ua-
loadbecomes
of winding,valuesofinitialtension
, 1941.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
obtained.
Usua lly theendturnsof tension
mpleformsindicatedinF igs.93and
fullturn isbentupto formaloop.
ons,however,awidevarietyofend
hareshownin F ig.95,maybeused.
particularlywheretheloopor hoo
considerablygreaterstressinthe
donthebasiso fana ia lload, E q ua-
E
D N D
E
H O L D
L E Y E
I TH
T
pesofendloopsfortensionsprings
atthe side,themomentarmofthe
mumtorsionstressinthe springdepends
ichw oulde istif apure lya ia lload
pring.This meansadoublingofthe
oadingoftension springs,evenif
isused,theline ofactionoftheload
considerableamountfromthea is
considerableincreasein stressmay
onsideredbythedesigner.
madewithplain ends,F ig.95d.
d" springends areattachedtotheseasin-
enusingthese,thespring isclosewound
arespreadapartbyscrewingthe
iscaseaninitial stresscorrespondingto
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
urns neartheendsis setup.This
ndtoa loade ua ltothe init ia lten-
oadcorrespondingtothe distance
secondtypeofspringendis shown
wedintotheendsofthe springcoil.
educestressintheendco ilsare indi-
8. InF ig. 97thediameterofendco ils
etheend loopisformed.Then,when
emomentarmofthe loadatthepoint
wire isthesharpestwillbe small.
hespring isreducedaccordingly.S uch
pensivethantheusualformofend loop,
tresses areunavoidable.
cingthestressin thespringisto
nghoo sateachendtofitoverthe
is arrangementasharpcurvature
pringendsfor tensionsprings
rreducedstressintensionspring
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
pointsof highstressisavoidedand
hespring reduced.Inadditionthis
uentlyofadvantageinmechanism
ecttoaw hippingactionas, fore ample ,
gisattachedtoa bell-cran which
andstopssud-
ofthespringis
selfhas avelocity
orsuchapplications
bythedesign
e inreducingthe
d,a springfast-
aplug wouldbe
stressatthesepo ints
.
F ortensionsprings
ndloop,F igs.93
remaybe rather
tedthatthestrength
anfor compres-
materialandheat
rlytruewhere
fatigueorrepeated
hestressconcen-
harpcurvature
lsoitis difficulttopresettension
ertheinitial tensionwillbelostor
sultbetw eenturns. B ecauseof lac o f
smorecreeporloadloss may,there-
orproperlypresetcompressionsprings.
ductionofw or ingstressto75or80per
onspringsisfre uentlymade.
E S S IO N S PR IN G
aspringtoe ertbothtensionand
ase inpointisacran - typefatiguetest-
gfull-sized impulseturbineblades .
nof thisseeTheMainspring, A ugust, 1939.
roon—" TurbineB ladeF atigueTesting , Mechanica lEngi-
31.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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IN G
pressionspring.Thespringis clampedatboth
andcompressionloadsmaybee erted
avingtheshapeshowninF ig.99are
heryoftwocircular platesasshownin
platesismovedbac andforthbyacran
oacrosshead.Bythismeansan al-
weentensionandcompressionisap-
specimenwhichisheatedatthe same
emperatureandvibrationconditions
ingsuchsprings,careshould be-
ilshavea gradualtransitionbetweenthe
straightportionofthe end.Inthis
eend turnsmaybereducedandthe
tionminimizedasmuchas possible.
ngementofF ig.100isthat adefinite
st specimenevenifdeflectionsof
riouscauses.
pressionsprings,itshouldbe
ressiscompletelyreversed,thestress
faspringsub ecttopulsating(zero
thesamepea va lue. Thusif theendur-
o f
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
dspers uare inchforspringstestedina
(thisisan averagevalueforspringsof
ee pectedendurance limitw ouldbe
ers uare inchinthecaseofatension-
samematerial.A dditionalstresscon-
endturns wouldtendtoreducethe
svalue.O ntheotherhand,the en-
stressmaybe somwhatgreaterthan
zerotoma imumstressrange. In
arsthatusuallyawor ingstressabout
ra zerotoma imumrangemaybe
.
g.100,whichhaveawire diameter
of3. 5, wor ingstressesof±25, 000
figuredwithcurvaturecorrection,have
nservice.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D R E CT A N GU A R - W IR E CO M PR E S S I O N
-wirecompressionortensionsprings
applications.F ore ample,inthede-
es, arectangularcross-sectionen-
na morenearlylinearload-deflection
cationofthis typeutilizingrectangular
utysca lesofF ig. 1C hapterI. A nother
chspringsisthe setofinterchange-
ig.101madefora testingmachine.
eof thes uareorrectangular-barsec-
maybepac edintoagivenspacefor
epossible forroundwire.H owever,
nullifiedbythe factthattheefficiency
lfor therectangularsectionisnot as
areinvolvedandsprings arecold-set,
butionoccurs andthisdifference
dandrectangularwiremaybe small.
A R G IN D X , S MA L L PITCH A N G E
arge,a helicaltensionorcom-
reorrectangularw ireandof large inde ' -
allyasabar ofs uareorrectangular
oatorsionmomentPr, F ig. 102w hereP is
meancoilradius. Toca lculatethetor-
arectangularbarundertorsion,
e a na l og y m a y be u s ed ' .
hisanalogymaybebriefly de-
etchedmembranehavingarectangular
formtension atitsedges,combinedwith
onof thispointtogetherwiththeoreticalresultssee
orest—" N ew SpringF ormulasandNew MaterialsinPrecision
presentedattheA nnualA . . M. . Meeting, December, 1934.
maybeconsideredastheratio 2r/abetweenmeandi-
wirecross-sectionbtii.102.
ofE lasticity,McGraw-Hill, 1934,Page239,givesamore
sanalogy.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
causingit tobulgeout.ThePrandtl
imumslopeofthemembraneat
earingstressat thecorresponding
andthatthevolumeenclosed
ami S ons
changeablerectangularbarspringsmadefor atest-
y f rom1/ 10to1 poundsperinchdef lection
neofitsedges representstor ue.
hatthedeflectionsofaloaded
epartialdifferentiale uation:
nofthemembraneatanypointhaving
, q isthepressureperunito fareaof the
nsionforceinthe membraneitselfper
.In additionthedeflectionzmust
hichrepresentsarectangularcross
w here(b> « ), thedeflectionzof the
essedinseriesformas follows:
, Page240.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- W IR E CO M PR E S S I O N S P R IN G 20 5
nofyonly. B ypropercho iceofY„ , the
186)maybesatisfied.
ation187inE q uation186ande -
q uation186intheformofaF ourier s
orY„ maybedetermined. Thef ina lre-
alogy,theshearingstressatany
slopeofthemembraneatthe cor-
mq/ be ingreplacedby2Gi9w here
pringof
y loaded
nd6 theangulartwistperunit length.
ation188andta ingy= 0, x =a/ 2, the
he mid-pointsofthelongsides)becomes
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
, f romTableX X V .
terminedin termsoftheangular
hevolumeunderthe deflectedmem-
ectangularBarsinTorsion
e ingaga inreplacedby2GH . S incetw ice
8 f or z a n d ta i n g q / = 2 G h th e e p r es -
s
t an h \
itten
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- W IR E CO M PR E S S I O N S P R IN G 20 7
atio b/aandmaybeobtainedfrom
of6givenbyE q uation191in
imumstressrmmaybedeterminedin
enfromTableX X V .
helica lspringof large inde and
ngmomentMt = Prwherer= mean
ma imumshearingstressthenbecomes
se uationassumesa large inde ,
eto curvatureanddirectshearis neg-
arbarsection
swillbe consideredlater.
gsw herea=b, thise uationre-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
theangulartwist6perunit length
t= Pr. Thetota langulartw istw illbe
,thisvaluemultipliedbythe coil
nr-9, orusingEq uation191
T ab l e X X \ r .
gsw herea= bthise uationre-
sdef lectione uationa large inde
e actcalculationbasedonelastic
nde greaterthanfourthe errorwill
sin contrasttothestressformula
yinerror evenforinde esgreater
largedeflectionsatheorymaybe
irespringssimilartothatdescribedin
springs.
E S PR I N G O F S MA L L IN D X
simplyasas uarebarundera
of themembraneanalogyyieldsan
umshearstressw hichisgivenbyEq uation
de esthisvalueofstressw illbeappro i-
llor moderateinde estheerrorwill
ses,as forroundwire,amoree act
inthe ma imumstress,theordinary
ult ipliedbyafactorK dependingon
ountforcurvatureanddirectshear.
ThefactorK' maybecomputedf rom
erechnungZ y lindrischerS chraubenfedernV . D. I. V o l. 76,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- W IR E CO M PR E S S I O N S P R IN G 20 9
wayasforthecase ofroundwire,
analysis showsthatforsmallpitch
aterthanthreeane pressionforthefactor
percent, is
ginde .
onstressfors uarewirebecomes
ottedasfunctionsofspringinde 2r/ a
ofthisfigurewith thecorresponding
g.30ChapterII) showsthatthevalues
C= Z %
utiplicationfactorK' f o rs uare-w ire
r/ o> 3)
undertheK va luesforroundw ire , thedif fer-
ntforaninde o f3, and4percentfor
re-w irespringisw ound, asacon-
272.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
rmationduringcoilingthe sectionbe-
atedinF ig.105.Insuch casesan
hadby ta inganaveragevalueofa
nde e ua lto2r/ a, forf indingK ' . The
ectangular-wirespringstobediscussed
hodis sufficientlyaccurate.
s uare-wirespringsofsmall
6. derivedonthebasiso fastra ightbarin
ect towithin4per centforspring
applicationof themoree actca lcula -
oy y ie ldsthefo llow inge pression
ndsmallpitchangles:
thesesrepresentsdeflection8 based
tbar, E q uation196, w hilethef rac-
stheeffecto f thespringinde . F oran
s.963whichmeansa deflectionabout3.7
uredf romtheusua lformula inEq ua-
of4thedeflectionw illbeabout2per
d79 whichapplytoroundwire
usedforanappro imatecalculation
uarewirespringsatgivenstresses.
alculatetheloadanddeflectionat
sinthe correspondinground-wirespring,
utsidecoil diameter,numberofturns
a ltotheaveragesideof thes uare
usfound aremultipliedbythefactor
.738to findthoseforthes uarewire
s orbestaccuracy, how ever, Eq ua-
beusedinsteadofthe chartsmentioned.
E x actTheory—A pplyingthemore
rmannerasoutlinedinC hapterIIto
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- W IR E CO M PR E S S I O N S P R IN G
thefo llow ingmoree actformulashave
a eintoaccountthe effectofpitch
s
2 p/
rethe inde c=2r/ a> 3anda< 10
maybee pressedas
Eq uation197orF ig. 104.
egreesandc< 3,thefollowing
umtorsionstressho lds:
fhe lica l
rew ire
alin
ingcoiled
wire.
m-
on
derivedbyGoehner,loc.cit..Page271,using methods
n ChapterII.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
choccurs asaresultof the
tedusing curved-bartheory,ta ing
ua ltoPrsinaandusingasim ilarpro-
apterII.Thebendingandtorsion
en becombinedinasimilar way
uiva lentshearstress. Ingenera l, f or
uivalentstresswillnotbe greatlydif-
umtorsionstresst, .
pressionforcalculatingtheef fecto f
aybe derivedinasimilarwayas
rings,E q uation51.Theanalysis
is givenby
4)
onofs uare-w irespringf iguredf rom
uation196, and
tana(205)
uationissim ilartoE q uation51
re ample, if c= 3 anda=10de-
eoffromE q uation205is.97,which
willbe about3percentl owerthanthat
epitchangle andcurvatureeffect.
- W IR E S P R IN G
f rectangular-w irespringsiscompli-
urvatureinincreasingthestress is
arlytrueif thespringiscoiled flat-
herethe longsideof thecrosssec-
nga isandw heretheratiob/ a is
ig. 102, anappro imatee pression
llpitch angles)is:
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- W IR E CO M PR E S S I O N S P R IN G 21 3
06 )
f romE q uation197orthecurveofF ig.
reb/ a isbetween2. 5and3this
saccurateto withinafewpercent forin-
cisbetw een3and4theerrormaybeas
herectangleisparallelto the
andZ > >3a, anappro imationforstress8is
ictheoryhavealsobeen developed
angular-barsprings .Itshouldbe
mtorsionstressina rectangularbarunder
midpointofthelongsidesof the
rue wherethebariscoiled inthe
nde . F orsmallerinde es, however,
ndstooccuratthe insideof theco ilbe-
ctshear effects.Thustwoopposing
nto play.W herethespringisc oiled
ea stressmayoccure itherontheshort
ndingonthespringinde andthe
ress—F orpracticalcalculationsof
rings thecurvesofF ig.107,based
iesec e f romGoehner se uationsfor
rings maybeused.
sonsshow nonthes etchesinF ig.
ringstresst, inthespringisgivenby
erpendicularandparalleltospring
meancoilradius, / J= a factorto
7dependingontheratioa / borb/ aand
. D. I., 1933, V o l. 77, Page892.
Page425.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L S P R I N G
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- W IR E CO M PR E S S I O N S P R IN G 21 5
2r/ a . E achcurverepresentsagiven
onisusedfor intermediatevaluesofc.
hiscaseb alwaysrepresents
ralle lto thea iso f thespring hence,
e.
e a mp l e of t h e us e o f th e c ha r t of F i g .
imumstressina rectangular-wire
atwiseasindicatedonthe figure.
ctorC,forrectangularbarsprings
% - inch, a / b= 2, r= 1. 5inch, c— 2r/ a
nds, f romF ig. 107fora /b= 2andc= 6,
ema imumstresst, becomes
Tocalculatedeflectionsin rec-
glargeinde es(saygreaterthan8),
edontorsionofa straightbarof
eldresultsaccurateto withinafew
ngleisnotlarge:
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
sthe longsideandathe short
laforrectangularwirespringsof
ing1
2 0 9a ( ta n h + - 00 4 ) ] ( 21 1 )
ducesto
herectangleisparallelto the
102, Eq uation210w illy ie ldresultsac-
centevenforinde esaslow as3,
ththespring inde .Ifhigherac-
ofF ig.109shouldbe used.
edf la tw iseasinF ig. 106, Eq uation
ccuratetowithinafew percent,for
hanfive.In thecaseofsuchsprings
andforlargerinde esw herehigher
olow inge uationmaybeused1 :
ndingonb/ aandisgivenby the
trmC isgivenbyE q uation211or
enthattherightsideofEq uation213
ndingterminEq uation210div idedby
c= 3andb/ a= 4, C 2= 1. 18, i. e .,
Page271. It isassumedthatthepitchangle isunder12
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- W IR E CO M PR E S S I O N S P R IN G 21 7
uation210forsuchcasesmaybearound
argeva luesofcthefactorC be-
dE q uations210and213become
ninrectangular-wiresprings
be simplycarriedoutbythe useof
Inthiscasethema imumdef lection
.1933, Page892.
alculatingdeflectionsinrectangular-wirehelical
byLiesee e, V DI, 1933P. 892)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ndson theratioa/borb/a,F ig.109.
sta enasthesidepara lle lto thespring
pendicularthereto . If thespringis
f sc uare-w irehelica lspringinposit ion
bis ta en,whileifitis coilededge-
.
sco iledf la tw isew itha= % - inch,
h, inde c= 2r/ a=6, P=300pounds,
5, G= 11. 5X 10 poundspers uare
. 109theconstant7= 6. 7fora / b= 2,
ti o n 21 4 ,
3 . 3 7X 5 . ,
i n ch .
1 0
beenthepracticeinspring
odulusofrigidityG forrectangular-
hatused incircular-barspringsof
ethereisno goodreasonwhythe
bedifferentfor springsofthesame
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- W IR E CO M PR E S S I O N S P R IN G 21 9
sbeendoneto compensateforinac-
yusedempiricaldeflectionand stress
arsprings.Comparisons12ofsomeof
cticewith themoree acttheoryfor
showconsiderableerrorsupto100 per
ob/a.It istheauthor sopinion
rewill yieldmoresatisfactoryresults
springsthanwilltheempirical form-
beenused inthepast.
ne actca lculationofstressinrec-
largepitchanglesis complicatedand
owever,anappro imationsuffi-
20000
T R E S S , L B / Q U A R E IN C H
scurvesforsemico il, c= 3. 07
racticalpurposesmaybe obtained
.107andneglectingthepitch angle.
nsforrectangular-wirespringsthe
pressionta espitchangle intoac-
e inMachineDesign, July , 1930forfurtherdeta ilso f thiscom-
e271.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
by
T R E S S , L B / Q U A R E IN C H
sscurvesforsemicoil, c— 4. 14
tion211or212andEI= f le ura lrigidityof
outana ispara lle lto thespringa is.
UA R E -WIR E S PR IN G
ula,E q uation198,fors uare-wire
rementsweremadeonsemicoilscut
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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- W IR E CO M PR E S S I O N S P R IN G 22 1
springs.Thestrainmeasurementswere
as thosecarriedoutonround-wire
semico ilw ascutf romanactua ls uare-
armswereweldedonasindicated
entsthesemicoilin positioninatesting
ownhavesphericalpointssothatthe
a lloadasisthecase inacomplete
heHuggenbergere tensometerusedto
wnin positionontheinsideof the
mstressoccurs.S hearingstressesplotted
edbythefull linesinF igs.Illand
of twotestsontwocoils,one ofin-
f inde 4. 14. F orcomparison, dashed
resscalculatedbythe moree act
8,arealsoshown.It maybeseenthatthe
ismoree acte pressionagreew ell
rcomparisonacurverepresentingthe
inarytorsionformulafor s uare
hichneglectscurvatureeffects)isalso
ttercurveshowsconsiderabledevia-
F F O R MU A S T O S T A TI C
O A D IN G
eformulasforstressgivenin
ndrectangularbarspringsare based
erefatigueloadingisinvolvedthe
a imumstressrangewhichis of
er,forstaticloadingtheseformulas
cflow withresultantincreased
ryload.Insuchcases ananalysis
V forroundw irew ouldbere uired
escopeofthis boo .Intheabsence
on,useofrectangular-barformulas
nddirectshear( q uation206ta ing
ablybe j ustifiedw herestaticloadingonly
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D S U R G IN G O F H E L I CA L S P R IN G
stressanddeflectioninhelical
edthattheloadis applied(orthe
wrate1 sothatadditionaldynamic
vibrationdonotoccur.In mostprac-
sassumptionis probablyrealized
herearealarge numberofapplica-
namiceffectsduetosurgeorvibration
p.
iveva lve
headditionalvibratorystressesthus
toaccountby thedesignerif fa tigue
hemostimportante ampleofsuch
neinwhichthe timeofapplicationoftheload is
ralperiodof vibrationofthespring.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D S U R G I N G
r automotiveenginevalvespring.
picalautomotivevalvespringand
3, whileas etchofatypica lva lve-gear
mentisshowninF ig.114a.
alve-liftcurveshowingvalveliftplotted
ig.114b.Thislattercurvealso rep-
ftheendof thespring(beyonda
cva lvespringandgearfor
e
againsttime.It isclearthat,if
asuddencompressionoftheendof
mpressionwavetotravelalongthe
edfrom theend,thetimefor the
dof thespringtotheother being
re uencyofthespring.Thisphe-
ongaspringmayeasily bedemon-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
gf le iblespring, suchasacurta inrod
hedbetweenthe twohands.Ifone
movingonehand,a compressionor
seentotravelbac andfortha longthe
esame assurgingofvalvesprings.
icationofdynamicallyloaded
pefatiguetestingmachineshowninF ig.
emachinesofthistypeare tooperateat
onshouldbeconsidered.
I D R A T IO N S
signofspringssub ecttoarapid
hasvalvesprings),it isimportant
ble,resonancebetweenthefre uency
fthe endofthespringand oneof
ofvibrationofthespring. Usuallythe
cy isof themostimportance. F oraspring
allelplatesthefirstmodeofvibration
stnaturalfre uency)willconsistof
iddleportion ofthespringwiththe
.Thesecondmodeofvibration(cor-
uency)willhaveanode(or point
)inthemiddleof thespring,while
co ilsoccursatpo intsV \ and% of the
end ofthespring.Thenatural
ngtothesemodesofvibrationmaybe
cussedlater.
F ore ample, if aspringissub ect
ymeansof asimplecran arrange-
.115,providedtheratior/l between
ectingrodlengthisnottoo large, thee -
acementfromits positionattop
sufficientaccuracybythee uation2:
c o s 2 wt J ( 2 17 )
hecran inradianspersecond
ica lV ibrations, SecondEdit ion, McGraw -Hill , 1940, Page
on.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D S U R G I N G
h er e N = s p ee d i n re v ol u ti o ns
tforspringssub-
eansofasimple
cipalf re uencies
:
encyof rotationasrep-
ermofE q uation217.
hisva luerepresentedby
q uation217.
resonance,
noughsothat
ncy,calculated
sconsiderablyhigher
cyof rotationof the
tedbyacamas
ft curvey—f(t)
mplesinefunction
ortwo termsas
insteadacompli-
beassumedto
ofsinusoidal
ries). Thusthee -
aybewrittenas
n (U - - < t > i ) + C i s i n ( 2w t + < t > : ) +
) + * ( 2 18 )
dofthespringmaybe considered
v ingacircularf re uencyw e ual
speedi nrevolutionspersecond,on
monicsof2,3, 4....timesthis fre-
desof theseharmonicsarec,, c3, c> . . . .
eachof theseharmonicsmaybeas-
y.Inpracticeharmonicsas highas
beconsidered.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
enera l, itshouldbenotedthatthe
esehigherharmonics,representedby
creaseastheorder oftheharmonic
efounddifficult toavoidresonance
drangesbetweenoneofthe higher
e uency(usuallythelowest)ofthe
splace,severevibrationorsurgingoccurs
ndthis mayincreasethestressrange
tor more.Thisistrueeventhough
ularharmonicin resonancemaybe
eisa largemagnificationofthe
ons.
suchresonantvibrationsinvalve
reopen.Inthe firstplacethenatural
maybeashighaspossiblesothat
orthe higherorderharmonics(which
ude).H enceanimprovementisob-
t upbyresonancewiththesehigher
as thosesetupby resonancewith
ramplitude.
cingsurgestressesin valvesprings
ursoasto reducetheamplitudesof
importancein thespeedrangewith-
eused. F ore ample , itm ightbe
th anoperatingrangefrom2000to
tethetenth,eleventh,andtwelfth
ewiththel owestnaturalfre uency
speedrange.H enceacamcontour
forthese harmonicswouldbe
n manycasesitispossible bya
toreducethemagnitudesofthe
thincertainspeedranges4.
pitch ofthecoilsnear theends
entoftenmaybeobtained.The
onanceoccurswithone harmonic,
pthuschangingthe naturalfre uency
of Iheamp ; tudesof thevariousharmonicsmaybede-
lysisforanygivenvalvelift curve.
" S chwinffungenS chraubenfoermigenVentilfedem
ertation,T.H .Berlin,publishedbvDeutschenV ersuchsanstalt
dlershof.1938.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D S U R G I N G
othrow itouto f resonance . F rict ion
ree-prongeddevicewiththeprongs
rcoils ofthespringhavealsobeen
oscillations .
R V I BR A T IN G S P R I N G
uencyandvibratorycharacteristics
arytoderive thedifferentiale uation
lementA (showncross-hatched)
ingcom-
plates
g.116compressedbetweenthetwo
dered.Itis assumedthatwhenthe
elementA oflengthds,isat adis-
of thespring.Theactivelengthof
f thespringista enas/ , w hile the
eneglected.Thedeflectionofthe
meanposition(or positionwhenthe
et willbedesignatedy.Ifwd:/4
ofthe wire,7theweightofthe spring
and gtheaccelerationofgravity,
is- rrdrtds/ A gandtheforcere uired
willbe
ndL ehr,Page115.publishedbyV .D.I.,Berlin,1938.
ineV a lveSprings, bySw anandS avage, S p. R ep. N o. 10,
esearch, London.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
uationforcee ualsmasstimesaccelera-
of thiselementisd2ydt2.The par-
yisa functionofbothsand t.
cedswill be(dy/ds)ds.F or
e willbeA (/=27i-r dy/dswherer—
otalforcePactingat adistancex
pring deflectionformula,E q uation7,
gidity.
na lengthdswillbe(dP/ds)ds,
ceF (,actingtoacceleratethe ele-
ntia tionofEq uation220, thisforcebe-
* y .d ( 22 1)
dditiontherearedampingforces
sesincluding:
springmaterial
intheendturns
energyinthesupports.
a inga llthesesourcesofdamping
elesslycomplicated.F ormathemati-
ngforceisassumedproportionalto
Thismeansthat,ifc isthedamping
ewireperunitof velocity,thedamp-
ticforce F i H ence,frome uilibrium,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D S U R G I N G
ions219, 221and222inthisanddiv id-
- y dy
ere I= active lengthofspring, and
ns
r2 n2 d -
sevaluesin E q uation223andre-
wingdifferentiale uationisobtained:
2 2 4)
w eightofactivepartcf spring(225)
t(226)
asureof thee uiva lentdamping
willvarywith suchfactorsask ind
motion,designofendturns, rigidity
determinedbyactualtestsonvi-
mpingiszero , b= 0andE q uation
ewell- nowne uationforlongi-
rt iclebyC . H . K ent, MachineDesign, October, 1935, fora
oreferencesoffootnotes4and6.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
inprismaticalbars,abeing theveloc-
a longthebar .
E Q U N C
uencyofaspring,it isper-
gsincethe smallamountofdamping
es notaffectthenaturalfre uency
ispurposethe simplerdifferential
used. Toso lvethise uation, the in-
tanypointof thespringisassumed
nctions,oneafunctionof x only,
ftonly.Thus,
(2 30
/ < (f )arefunctionsofx andtrespective ly . Then
• d > ( ) — • l ( t)
uation229,
) d -
esatisfiedif bothmembersofthis
aconstant, say— w 2. Then
ationsare
t c os a t ( 23 3 ) -
- B 2 c os ( 2 34 )
ionProblemsinE ngineering,S econdE dition,Page309,V an
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D S U R G I N G
arearbitraryconstantsdependingonthe
venproblem.
tions233and234inE q uation230a
satisfiesthedifferentialE q uation
d—Ifbothendsof thespringareas-
ed,thismeansthatregardlessof the
0 f or x = l .
sre uiresthatB.,= 0,whilethe
wl/a= 0.Thismeansthatthefollowing
• e tc .
/ isanaturalf re uencyof thespring,
hatthenatura lf re uenciesare inthe
nbyEq uation227, thee pression
yofthespring(in cyclespersecond)
orderofthevibration,i.e.,m= l
esecondmode,etc.
orthespringconstantk andspring
uations226and225, the lowestnatura l
omes
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
usuallyofmostimportancein practice.
tforagiven materialthenatural
ng isproportionaltothewire diameter
totheproductof thecoildiameter
oils. F ortheusualsteelsprings where
spers uare inchandt= . 285poundsper
lowestnaturalfre uencyreducesto
ree—F oraspringwithoneendf reeand
westnatura lf re uencywouldbee ua l
wiceaslong butwithbothends
ingEq uation237maybeusedif the
nastw icetheactua lnumberinthe
ampleof theuseof thesee uationsin
ency,assumingasteelspringclamped
nch, r— 1- inch, n= 6andusingEq ua-
lfre uencybecomes
rationthespringfre uencywillbe
persecond.
e ighted—F oraspringw ithaw eight
wninF ig.117,thelowestnaturalfre-
aybecalculatedasfo llow s: It isk now n
ctedbyacertainamount8 underits
e springbeingsmallcomparedto
ural fre uencyincyclespersecond
.
anica lV ibrations, Page45.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D S U R G I N G
ssissupportedbyahelical spring
dicatedinF ig.117ithas beenfound
hirdofthespring isaddedtothatof
heca lculateddef lectionmaybeusedforf iguring
W isthespringw eight, E q uation225,
tin poundsperinchdeflection,thefre-
nd becomes
N G IN E V A L V E S P R IN G
utomobileenginesrunat variable
viously,itispracticallyimpossibleto
oneofthehigher
urve anda
espringatsome
nthisoccursthe
dtheresulting
dprimarilyon
onicwhichis in
untofdamping
bythe damping
4).
essin the
itisassumed
inF ig. 116, say
an amplitude
n
eamplitudeoftheparticularharmonic
chisinresonancewith anaturalfre-
g, usua lly the low est. Thecircularf re -
aharmonicista enasw = 2w f . The
ressdueto thisharmonicmaybe
edif ferentia lE q uation224incon unc-
aryconditions.Thisstress isthen
cstressdue tothevalveliftas indicated
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
dot-dashlinerepresentsthestressdueto
imumvaluebeingt . O nthisis
uencyvibrationrepresentedbythe
sonancewitha givenharmonicofthe
4forthesteadystate condition,the
fromtheendofthe springisas-
4 1)
ef u nc t io n F ( ) i s a f un c ti o n of x o n ly . Th i s
entoscillationsdueto suddenspeed
oncernhere.Theassumptionrepre-
41isj ustifiedsince, fora forcedv ibration
partso f thespringmustv ibrateatthis
re:F orx = 0F ig.116,y= 0re-
neendof thespringisassumedf i ed
c sinw tsincetheotherendof thespring
monicmotionofamplitudec produced
uation218w hichisinresonancew iththe
ferentia lEq uation224satisfy ingthese
eenobtainedbyH ussmann10.
pingsuch asoccurinpractical
sto therelativelysimpleform:
U + 4 > ) ( 24 2)
mumamplitudeofmotioninthespringand
angle . F romthisso lution, forsmalldamp-
blestressobtainedis
staticstressinducedbycompress-
ntc . T in se uationindicatesthat
otnote4.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D S U R G I N G
g.118isinverselyproportionaltothe
tlyproportionalto thefre uencyf
atter, inturn, isproportiona ltothe
ticularharmonicinresonance.
dampingfactorh inactualsprings
magnificationof100to300times occurs,
o 300timest8cIt hasalsobeen
ngfactorbvaries withtheamountof
TR E S S D U
I T.
tionofvibrationstressesonstresses
orvalvespring
pringand thatitincreaseswiththe
monicin resonance.Thisisreasonable
internal dampingofthespringma-
lbe lower. A lso fore tremelyhigh
ervaluesofbare found,resultingfrom
tbetweenturns.A tmediuminitial
edampingfactorarelowerwhile,at
esevaluesagainrisebecauseofdamp-
ingoftheend turnsfromthesup-
about 1sec-1atloweramplitudesof
ehigheramplitudeshavebeenob-
stva luesbe ingbetween2and4.
useofEq uation243assumingthat
ency / o f thespringise ua lto200
thecamshaftspeedis1200revolu-
nsthat resonancebetweenthisnatural
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
nd thetenthharmonicofthevalvelift
mingalsothattestson springsunder
ownadampingfactorb = 5sec-1,
mpressionofthespringbyan amount
ndif thea lternatingstressduetothe
curve is, fore ample, . 002t (asfound
011001200
P E D , R .P.M .
peolresonancecurveforvalvespring.
ateach resonancepea
healternatingstressdueto resonance
e251X . 002t, < orabout. 5t . This
e stressrangewillbeincreasedto 2
ration.
a lstressrangeinthespringis
tialcompressionofthespringwith
sition,therangeinstresswill befrom
t 1 — t t o a m a i m u m v al u e tm * = t 1 +
nwithendurancediagramssuchasthose
relativemarginofsafetyofthe spring
ybeestimated.
esimilarto thoseobtainedbyactual
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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D S U R G I N G
howninF ig.119.Inthis curvethe
hemiddlecoil ofavalvespringis
peed.Itisseen thatthiscurvecon-
pacedatinterva ls, eachpea be ingdue
hevalveliftcurve(indicatedby the
pea mar ed10isduetothetenth
rve,i.e.,to avibrationfre uency
spersecondforacamshaf tspeedof1200
eamplitudesofthesepea svarysince
thec sofE q uation218)of thevarious
DI N T
b ecttorapidreciprocatingmo-
the followinge pedientsareoften
hnatura lf re uency
snearend ofspring
ssibleofresonancebetweennaturalfre-
dane cit ingf re uency
soastoreduce themagnitudeofthe
urve withincertainspeedranges
n,incon unctionwithtestre-
ngesin actualspringsundervibra-
eand inthiswaythe marginof
redetermined.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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E D DI K (B L L E V I L E ) S P R IN G
thedirectionofloadapplication,
dis springsisf re uentlyofadvantage.
lso k nownasBellevillesprings,consist
so fconstantthic nessandhavean
suitable variationoftheratioh/t
htanddis thic ness, it ispossibleto
veshavingawidevarietyofshapesas
fF ig. 121. F ore ample, re ferringto
ncharacteristicfora ratioh/t— 2.75
dbyC urveA . S uchashapemaybe
ngdeviceisbeing designed.Byre-
aload-deflectioncurvesimilarto
ypeofspring, k now nasthe" con-
aconsiderablerangeofdeflectionwithin
lyconstant.S uchacharacteristicis
pplications,suchas fore ample,
eddis (B e llev il le )spring
iedto agas etisnecessary.Byvary-
ediateshapes,F ig.122,ispossible.
f theapplicationofsuchspringsisthe
ero iderectifierstoprovidea con-
astac o f rectif ie rdis stogether.
espringsforproducinggas etpres-
pacitorsandincondenserbushings
t.Insuchcases,springswiththe char-
BF ig.121havebeenfoundadvantage-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ndingdeflectionfactorC,forinitially-coneddis
ncharacteristiccurveswillbesimilar
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
llsupplyappro imate lyconstantgas et
evariationofdeflectiondue totem-
rcauses.S uchdeflectionchanges
amplebytemperaturechangesasacon-
R A L L E L
R I S
stac inginit ia lly -coneddis springs
nce ine pansioncoeff icientsbetweenthe
ecopperpartsof electricale uipment
ringsmaybestac edinseriesorin
. 123. Bystac ingthespringsinpara l-
edforagivendeflection stac ingthe
argerdeflectionatthegivenload.
stac edinseries, ratiosofh/ tbetween
ssgreaterthanabout1.3shouldnot
tabilityor snapactionisaptto occur,
ctioncharacteristicwillthenresult.
hematicaltheoryofelasticityto
coneddis springsise tremelycom-
practicalsolutionoftheproblemmay
heassumptionthatduringdeflection,
eoryofP latesandS hells—S . T imoshen o, McGraw Hill,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G
dis srotatewithoutdistortionas
ine inF ig. 124c. If theratior / r, be-
erradius isnottoolarge,tests show
llyield sufficientlyaccurateresults
atleastas farasdeflectionsarecon-
tionsonflatcircular plateswith
nmadeon thebasisofthis assump-
sectionsrotatewithoutdistortion,
odresultswhencomparedwiththose
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ory2.F ore ample,comparisonbe-
ppro imateso lutions1show sthatwhere
outerandinnerradii isnotover3(w hich
es)theerror indeflectionmadeby
verabout5percent,while theerrorin
coneddis springwhichfol-
mptionofrotationofradialcrosssec-
ndisduetoA lmenandL aszlo . These
medeflectiontestswhichindicate
sfactoryforpractical use.
initially coneddis springcut
btendinga smallangled6,F ig.124a,
ternalload,aradialcross sectionro-
sindicatedby thedashedoutlineofF ig.
entarystripoflengthd atadistance
tionundertheseassumptionsconsists
acementdrand arotation< f> ,thelat-
nofthecrosssectionabout pointO
lacementdrcausesauniformtan-
entd (thesmalldifferencesipradial
fthespring betweentheupperand
entd arehereneglected).Thero-
tangentialbendingstrainwhichis zeroat
ma imumattheupperand lower
stresses duetothemovementsdr
ducemomentsaboutpointO whichresist
aldisplacementdrmaybecalcu-
circumferentiallengthofthe ele-
onis
ngle.A fterdeflectionthislength
sdescribedinC hapterX V .
obo, Jr.—" S tressesandDeflectionsinF latC ircularP lates
ransactionsA . . M. . , 1930. A . P . M. 52-3. A lsoS . T iinoshen o—
anN oslrand, Part2, 1941, Page179.
t io n D is S p r in g , T r an s ac t io n s A . . M. . . 1 93 6 , Pa g e 30 5 . A
taperedspringsis givenbyW .A .Brechtandthewriter
dDiscS pring , TransactionsA . . M. . , 1930, A . P . M. 52-4.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G
) d 9
) - co s 0
s i n < t > — x c o s 0 ( 1 — c os 4 > ) ( 2 45 )
thattheanglesfiand < f> aresmall
springs)sothat
s in 4 > = < t > ; l — c os 4 > — ~ ^
cos< f> is obtainedbyusingthecosine
sabovetheseconddegree.
on245,
(2 46 )
tangentialdirectionwillbe,
heradial stresseswhichareas-
ss isobtainedbymultiplyingthisby
modulusofe lasticityand/ =Po isson s
onlytotheradialmotiondr becomes
gnifiescompression.Thefactor1— > r
tractionor e pansionofelementsof
thise pressionmayalsobeobtained
aforcalculatingstressesfromstrainsin
stress5.
T imoshen o—S trengtho Materials, 1941, Part1, Page52.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
duetothetangentialstresses
w illbe
s i n( 0- < t > )
t > ) ~ / 3 — < t > f o r sm a ll a n gl e s an d u si n g E q u a -
' d
c — r t o x — c — r , ,
. ,
0 -r . )+ r f o g, — | ( 2 49 )
stressdue tothebendingstrain
o f thee lementd , F ig. 12A b, it isneces-
in curvatureinthetangentialdirec-
w hichis
uralrigidityinthe caseofbeams.L et-
curvatureoftheelementduringdeflec-
mentactingon elementd duetothis
— - c x ( 25 1)
tureoftheelementin theun-
mate ly sinf i/(c—x )w hilethecurvature
ssin(/ J—< £ )/ c—x . Thechangeincur-
t >
251,
,PartII, Page120,givesafurtherdiscussionof platerigidity.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G 2 45
atthesurfacewillbe thisvalue
cionmodulusof thee lementd w hichis
q uation252,
eneutra la is(ta enposit ive inan
sswillbe
used tosignifycompression.
mentdM2whichacts inaradial
uation252,
c—r tox = c—n, themoment
ar motionoftheelementsofthesec-
= E t W e l og i n. / r. )
2 (1 - ,. )
ointO thusbecomes,usingE q ua-
25 6)
pointO tothea isofthedis is
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
atthe sumofallforcesover the
becausethereisnonet e ternal
o f thedis . S incethebendingstresses
nt,onlythestress a/duetothe radial
nsidered.Thus
8
gforc
on257inE q uation256,
r ,) 2
2 < : s ( 25 8)
ualtothee ternalmomentonthe
hichis
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G
nof thespringandusingE q uation
2 -> + < H . • ( 26 )
(- ) < 2 62 >
tC , = C . fo rpractica lpurposes.
+ ' ]
asfunctionsof theratioa= r / ri
uation263itmaybeseenthatthe loadP
deflection8.Byusingthise uation,
sticsmaybedeterminedforvarious
22. TheapplicationofE q uation263
4. 550
terminingfactorC2
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
facilitatedbythemethoddescribed
ressata pointatadistancex
of thestressesa, anda . H ence, using
54
6 4)
tthe uppersurfaceofthespring
r, andt/ = t/ 2. Ta ingtheseva luesto-
r , ) , < f > = S / ( r — r < ) a n d s u bs t it u ti n g th e
uation257inE q uation264thisma imum
inneredgeofthe springisob-
1/ 2inE q uation264. Thisy ie lds
( 26 8)
ativevaluesignifiescompression,a
I G N
ofE q uations263and264in
lly coneddis springsthefollowing
ordeterminingtheload-deflection
.E q uation263maybewritten:
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P II IN G
1
on theratiosh/tand S /twhile
— r / T ionlyandmaybeta enf rom
rf romE q uation261.
putations,valuesofC,havebeen
tforvariousva luesofh/ t inF igs. 122
S S
rdeflectionfactorC , forB e llev il lesprings
e ight: thic ness
g.126maybeused toobtaingreater
luesof h/t.
ecurvesofF igs.122and126also
characteristicsforspringshavingvari-
alcone heightandthic ness.Thisis
tlyproportionaltotheconstant Cr
ndependentof theratior r, (andhence
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
apeoftheload-deflectioncharac-
teriallyonlyby alteringtheratioh/i
htanddis thic ness. A th/ t— 1. 414,
6,thecurvehasa horizontaltangent
determiningstressfactorK , for
/ r(= 1. 5
ethe springrateisvery low.F or
vengreaterrangeof low springratebut
slightlyafterreachinga ma imum.
of about2.8theloaddropsbelowzero
that permanentbuc lingofthespring
orintermediatevaluesofh/tmaybe
cyformostpractical purposes.
ofstress a,E q uations265and
ows
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G
gvalue:
beforetheconstantC.. thestressin
espringisobtained,whileusing the
essat thelowerinneredge.Itis
safunctionof r / r( , h/tandh/ t.
omputations, va luesofK thavebeen
atioh/tforvariousvaluesof h/tin
ratiosa= r / r(e ua lto1. 5, thecurves
vevalue ofK 1representingtension
determiningstressfactorK , for
/ r, = 2to2. 5
presentingcompression.Itwasfound
nby thecurves,forvaluesofh/t
imumstresswillbecompressionat the
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
8< 2/ . F ordef lection8e ua lto2h, the
edgee ualsthecompressioninthe
tensioninthe loweredgebecomesthe
sshownby theuppercurveforh/ t= l,
2tthenthecompressionandtension
sbecomee ual, andfor8/ <> 2theupper
F ormostpracticalcaseswhereh/t
a imumstressesw illbeobta inedby
urves. Interpolationmaybeused
h/t.Indoubtfulcases where8> 2fi,
edbyusingE q uation271. A further
nofstressin thesespringsisgivenon
sdistributionofstress inatypicalcase.
E x ample1: To il lustratetheuse
6to128 inpracticaldesigna spring
llowingconditions:Thespringis to
plicationw herethe loadistobeheld
at6000poundssothat thetypeofload-
thatfor h/t— 1.5,F ig.121.The
usingan 8%-inchoutside diameter
espring mayvarybetweenand
adandthema imumcompressionstres
on271istobe lim itedto200, 000pounds
auehasbeenfoundbye periencetobe
taticorrepeatedbuta fewtimes7.
ches,ri= 2V & inches,a= r /ri—2h from
K 1 ^ l / r , w he r e K , = — 6 . 7 f r om F i g . 12 8 f or
ma imumvalue. So lv ingthisfortand
undspers uare inchcompression,
1. 5, C , = 1. 68onthef la tparto f the
fora= 2, C 2= 1. 45. F romE q uation269
e
wor ingstressseePage25U.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G
ired,it willbenecessarytouse 5
l giveappro imatelytherightload.
enthatforh/ t= l. o , C , isappro imately
to5/ f=2. 1. S incef= . 134thismeans
mate lyconstantf romS ~ . 75(. 134)= . l-
ichisaboutw hatisre uired. If thema i-
i -
A C
ch, thema imumvalueofS/ fw il lbe
mF ig. 128for8/ f= 2thefactorKtw ill
6.7whichmeansthatthecalculated
lightlyless than200,000poundsper
e calculatedloadfrom6500to6000
f thedis maybereducedabout2percent.
esuchasthatshow ninF ig. 122for
snapactiondevicetooperatein such
reachesacertainpointrepresented
,thesystembecomesunstableand a
hresultingsnap-action.A lso,a
520poundsis desired,spaceisavail-
rdis , andastopisprov idedsothat
-inchbeforecomingagainstthestop.
eflectionwhenthespringis againstthe
dbythepointonthe curveofF ig.126
ingto8/f= 3. S inceatma imumde-
smeansthatt= . 25/3= . 0833- inchand
8- inch. A ssuminga= 2, f romF ig. 128,
3, h/ t= 2. 5. F romE q uation271onPage250,
efollowingrelation:
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
0poundsper s uareinch(compression),
g, r = 3. 8- inchsay3% inches. F rom
1. 45andf romF ig. 122thema imum
gtoma imumload)forh/t=2. 5is
269thepea loadis
5 l b .
520poundsweredesired.To geta
helatterf romE q uation269increases
ua l)thethic nessmaybereduced
• = . 935. Thusf= . 0833X . 935= . 078-
eofcurveh/ tmustbek eptthesame
078=. 195- inch. A tY 4- inchdef lection
2whichwillbesomewhatbeyondthe
3, whichinthiscaseis permissible.
tressw illnotbechangedappreciably since
2. 5and8/ f=3. 2, thefactorK, is practic-
. F romF ig. 122for8/ f=3. 2, h/t=
4. 6atthepea load, the loadwhenthe
willbereducedto 1.3/4.6X 520=147 _
I GN F O R C O N S TA N T - O A D
5,aload-deflectioncharacteristic
type isobta inedasindicatedbyC urveB
vefor7i/ f= 1. 5ofF ig. 126suchthatthe
5percentfrom adeflection8= .8fto
whichareof particularvalueinmany
ybedesignedina simplemanner
mallow ablestressisgiven. L ettingDbe
spring,then theconstantloadPand
ssf toobta inthisloadaregivenby
stedbyR .C.BergvalloftheW estinghouseCompany.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G
dK dependonthema imumallow able
or(r /r()betweenouterandinner
eseconstantsmaybeta enf romthe
000180000200000
L By Q I N A T D F L E C TI O N < T = 2 .2 5t -
131. Itshouldbenotedthatthethic -
dtotheva luegivenbyEq uation274
dcharacteristic.Inall casesthema i-
medas2.25timesthethic ness.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
esofconstantloadP , thic nesst, and
25faretabulatedforma imumstresses
100,000poundspers uareinch,and
gf r m1. 25to2.5. It isassumedthat
oadB ellev il leS prings
a e n a s 30 x 1 0 l b ./ s . i n .
hichthemodulusofelasticityE may
p o un d s pe r s u a re i n ch .
fE q uations273and274as wellas
V Imaybeca lculatedasfo llow s:
it ispossibletoso lveforthethic nesst
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G
q uation274inthis,
uation273.
dagiven stress,thefactorC.may
25w hileK w illbefoundf romE q uation
T IO N £ = 2 .2 5t -
= S P R IN G TH I C N E S S .
M T R
80000200000
, a / O J N . A T D F L E CT IO N < 5 = 2 .2 5t
determiningthic nessforconstant-
s
ppro imatelythatgivenbytheflat
1.5,F ig.126.Usingthesevaluesthe
igs. 130and131andtheconstantsin
mputed.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
E DW ITH TH E O R Y
estshavebeencarriedoutbyA lmen
yconeddis springs. Thesetests, made
rtions,showcurvessimilar tothose
hat themethodofcalculationdevel-
eformost practicalpurposes.H ow-
sarenote actand, inpractice , devia -
uchas10to15percentmaybee -
alculatedvalues.F orhighestac-
testsshould thereforebecarriedout.
,aload-deflectioncharacteristic
ourdis sinpara lle lisshown. These
about1.45andshowclearlythe constant-
tiallargedeflectionwasdue to
I N IN C H E S
stof stac o f fourstee lB e llev il lesprings, r = 4V 4.
48
esbetweenthedis s.Inspiteof the
eindividualspringswereslightly
esisloopbetweentheloadingand
ned,indicatingconsiderablefriction
singagroup ofspringsinparallel the
imatelyinproportiontothenumber
eappro imatemethodofca lculation
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G
results asfarasdeflectionsare con-
ctedthatdeviationsbetweentestand
stressthanfordeflection.Compari-
atedbythetheoryweremadeby
ltso f sometestscarriedoutbyL ehrand
lculatedS tresses
thic Bellevillesprings.Theresults
ents,obtainedwithaspeciale tensom-
elength,arecomparedwiththere-
eX X V II.
stedbythese writersthedeflec-
hgoodagreementasTableX X V IIindi-
ectedforlargedeflectionsrelativetothe
pected, how ever, thateveninsuch
ultsmaybeobtained.
E S S E S
ere init ially coneddis springsare
or toaloadrepeateda relativelyfew
e perienceshowsthatstressesof
spers uareinchascalculatedby
eusedeventhoughthey ie ldpointo f the
is madeisonly120,000poundsper
. A lthoughthisstressseemse tremely
esign,May,1939,Page47.
, 1936, Page66.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
eredthatit islocalizedneartheupper
ig.129.Conse uently,anyyielding
ributethe stressandallowthere-
ngtota eagreatershareof the load.
ssiscompressioninthe usualapplica-
oramorefavorablecondit ion. A lsodue
themanufactureofthe spring,
tesignare inducedandthesereduce
owthatcalculated.
t ingw or ingloadsforstaticloading
t-loadtypeofspringis tofigurethe
A ssumingthatthespringis flattened
ameterofthe verticalreactionacting
enceofthespring willbe(P/2)2r /ir
yo fasemicircleof radiusr isata
hediameter).Themomentofthevertical
lf-circumferenceaboutadiameter
. ThenetmomentMactingonadi-
gwill bethedifferencebetweenthese
78 )
endingstressesacrossthedi-
efiguredfrom theordinarybeam
hestress-concentrationeffectsofthe
pressionforma imumstress>rbecomes
ivenbyE q uation278)
onmodulusof adiametralsectionis
a m pl e o f th e u se o f E q u a ti o n 27 9 : A
calculatedstressof 200,000pounds
f romEq uation271andhasanoutside
romTableX X V Itheconstantload
ndsforD/ d—1. 5. Thethic nessw ill
. 0297- inches. UsingP= 74poundsand
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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P R IN G
ation279thecalculatedstress(neglecting
mes
0 0 lb . /s . i n .
guredinthis way(i.e.,stress
cted)thestressisonly about80,000
ndspers uareinchfigureobtained
oy . Thise pla insw hyspringsdesigned
sfactorilyunderstaticloadssincethe
reinchfigureis wellbelowthetension
.In thecaseofthe constant-load
ded,it maywellbethatthe stress
tion279w illy ieldabetterpictureof the
rry load. How ever, themoree act
sconcentrationintoaccountshould
peatedloadingis involved.
herefatigue loadingof init ially -
olved,considerablylowerstressesthan
cloadingmayhaveto beused.S o
e fatiguetestdataavailableby
strengthof suchspringsmaybe
mateofthefatiguestrengthofthis type
wever,asfollows:
goperateswithina givenrangeof
essin theupperandlowersurfacesof
byusingE q uation271. A ssumingno
ntheusual casetherangein theupper
intermediatevaluetoama imumin
thelowerinner edgewillbefrom
sion,orfrom anintermediatevalueto
esidualstressespresentinactualsprings
s.
gtensionstress willbemore
ingcompressionstressonly.H owever,
latedcompressionstresswillbebe-
dpointofthe material.H ence,yield-
uallyoccureitheronfirst loadingorin
withtheresultthat tensionstresseswill
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
range.In practice,therefore,itappears
sisfordesignthema imumrangeof
willusuallybea rangeincompres-
uation271.Theactualstressrangein
comparedwiththeendurancerange
eddis springsareheattreatedaf ter
ationofthesurfacematerialwillno
etheendurancerangetobe usedfor
obtainedonspecimenswithunma-
icatedinC hapterX X IIIthisla tterva lue
s ofthevalueobtainedonmachined
thefatiguestrengthof Belleville
edasvery roughandthefinalanswer
ual fatiguetests.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A T DI K S PR I N G
conedor Bellevillesprings,initial-
ntagewherespaceislimitedin thedi-
whileat thesametimehighloadsare
e Bellevillespring(whichmaybede-
of load-deflectioncharacteristics)the
aload-deflectiondiagramwhichis
andconcaveupwardfor largeones.
gbecomesstifferasthe loadincreases.
dewith acrosssectionof constant
ig. 133orw itharadia lly -taperedcross-
1341. A s-w illbeshownlatertheuseof
ethic nessisproportionaltothe
niformstressdistributionandhence
ofthe springmaterial.S uchsprings,
nesstypeorofthe radially-tapered
sshowninF ig. 135. B y thismeansthe
ncreasedwithoutreducingthe load-
metime,the springassemblyisen-
w ellasvertica lloads.
A P R E D S P R IN G
noftheradially-tapereddis spring
chrepresentsasectionalview ofacom-
ailwaymotors.Thefunctionofthe
se istosupplyaconstantpressureforho ld-
gether,whileatthe sametimeallow-
othate pansionsduetotemperature
ew ithoutproducinge cessivestressin
rbars. F orthispurpose, thedis spring
F oranappro imateca lculationof the
B rechtandtheauthor, " TheR adia llyTaperedDis S pring ,
.,1930A .P.M.52-4.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
dis springs,asinthecase oftheap-
viouslydiscussed,itwillbeassumed
thespringrotateduringdeflection
c ness, init ia lly - f latdis spring
risonwithamoree actsolutionas
this appro imationissatisfactoryfor
videdtheratior /r(betweenouter
3.F orstress,theagreementbetween
apereddis spring
tetheory isnotsogoodwheretheratio
usingtheappro imatemethod, how ever, it
nonlinearload-deflectioncharacteristic
thisisfarmorecomplicatedif thee act
temethod,anelementofaradially-
ially-taperedspringismeantaspring havingathic ness
t anypoint.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R I N G 2 65
cutout bytwoneighboringradial
angled6 asshowninF ig.137,the
wninF ig.134.UndertheloadP the
anangle < f> asshownbythedashed
tac ingradia lly -tapereddis springs
erotationisassumedtota eplaceabout
cecf romthespringa is.
nitiallyata distancex fromO
heneutralsurface(ormiddle plane)
ngthofthiselementis
anangle< f> thefinallengthbecomes
y s i n < t > ) d 9
iselementduetodeflectionof the
weenthesee pressions.Thus
— x ( l — c o s$ )
ssmall, sinand1—cos< f> = < f> - / 2,
bethisdifferencedividedbythe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L S P R I N G
altotheunit elongationmultiplied
yE (effectsduetolateralcontraction
ngthise uation,thestressbecomes
ngusedinra ilw aymotorcommutator
ifthe springdeflectionissmall,theterm
glectedandthestresswillbegivenbyE <f> y / r.
mwillthenoccur wheny= t/2=k r.Thus
ade,itisseen thatforsmalldeflections
onstantalonga radiusforthistypeof
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R IN G
ctingon theelementG,F ig.137,
perpendiculartothepaperw illbe
edbyta ingtheradialcomponentof
multiplyingbythe leverarmy.
nthe slicecutoutof thedis will
ementaryforcesoverthecross-section,
nthe limitsc—r andc—uandybeing
t s — k ( c — x ) a n d - - ( c — x ) .
essionforagivenbyEq uation280in
lmomentbecomes
ee ualtothemomentdueto the
testhatthemomentactingon the
oplanesatanangle d6willbed6/2ir
section,radially-taperecldis spring
uetothe loadPw hichisP (r —r().
fM givenbyE q uation283in
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L S P R I N G
tingandso lvingforP ,
+ ^ - W + r ^ + r . ) ( 28 5)
n8isgivenby
2 86 )
w oe uationsdeterminePasa
q uation286forandsubstitutingin
+ r < 1 ) ( 28 7)
ns, E q uations281and287maybereduced
ms
s, Eq uation287maybewritten
„ .
l ) J
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R IN G
rac etsisusuallynotgreatlydifferent
imationfor8willbe obtainedbyusing
hisva lue inE q uation293, acorrectedva lue
ained.Thiscorrectedvaluemayagain
93forasecondappro imation, ifdesired.
toconvergerapidly.A nothermethod
avalueof8and calculatethecorre-
q uation292.Byassumingseveral
tioncurvemaybe plottedreadily.
hec theaccuracyof theresultsob-
atemethoddevelopedpreviously,itis
ee actflat-platetheorytothisprob-
ations3whichmustbesatisfiedat any
atewhichis loadedbyaloadP uni-
sedges,are
+ - -f = 0 (2 94 )
5)
96)
naradialdirectionat radiusr,inch-pounds
tangentialdirectionat radiusr,inch-pounds
teatradiusr
lope inradia ldirectionatradiusr
usr
2 k 7 3 ( 1 - A » ' ) f o r a ra d ia l ly - ta p er e d di s w h er e
on295withrespectto rgives
u b \ / d d > d > \
differentia le uationsisgivenbyT imoshen o—S trengthof
o l. 2, Page135, V anN ostrand a lsobyA . N adai—Ktastische
ger, Page52.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L S P R I N G
onsform2anddm drgivenby
97inEq uation294y ie ldsthefo llow ing
t > P
ise uation4is
- i " « > - — — — ( 29 8)
re:F orr= r ,m^ O sinceno
theedge hencefromEq uation295,
a n d
ablethedeterminationofthetwo
uation298. Theseva luesare
l, ,r - l ( 29 9)
~ * - r rw r. -
s + M )
)
-
oDr. A . N ada io f theWestinghouseR esearchL abora-
rdingthemethodofintegrationof thise uation.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R IN G
heplate willoccuratthe inneredge
tinneredge(tb= 2 ri) .
298,299,and300,andthe derivative
u a ti o n 30 1 , an d t a i n g r= r f t h e m a i m u m
) r( 1- ) 1
(M-8-_)
torK dependsprimarilyonthe
outerandinnerradiusandonPo isson s
sofK areplottedasfunctionsof the
ioe ua lto . 3, whichisappro imately true
nsiderablechangeinPoisson sratio
butslightly.
mthee uation
givenbyE q uation298andintegrating
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
3 - ) P
sisdeterminedfromthe condition
roattheouter edgeoftheplatewhere
tion305thiscondit iongives
Cr „ (- " - > P
306)
q uation305andta ingr— r the
ecomes
deflectionconstantChavebeen
r / rif o rPoisson sratio^= . 3.
sC , C, K , K ' f iguredby thee actand
q uations308, 291, 303, and290, are
II.
istableshow sthatforva luesof the
uter diameterslessthan3thereis
dCwithinabout10per cent.This
ctions(say,lessthan abouthalfthe
greementwithinthis percentagebe-
redby thee actandby theappro imate
nt accuracyisusuallysufficientfor
appro imatemethodmaybeusedin
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R IN G
tiosa lessthan3.H owever,theagree-
od,sincethe differencebetweenK
0percentforva luesofagreaterthanabout
e e acttheorybeingsomewhathigher
pro imatetheory.
q uations302and307w erederived
deflections.H owever,whenthede-
ndK ' fo rV ariousV a luesofa
ee actf la t-platetheorybecomese -
nimprovementinaccuracyforcalcu-
btainedbymultiplyingE q uation307
mthemoree actplatetheorybythe
uation293. Thisgives
1 )-
eativetothic nessEq uation309re-
7sincetheterminthebrac etsbecomes
ns,thee uationcorrectsfortheeffect
rt ionasisdoneintheappro imate
sedontheappro imatetheoryshow s
dediameterandstressthedeflection
byA . N adai, Pade284presentsafurtherdiscussionof this.
ote1givesadditionaldetails.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
mesama imumforva luesofdiameter
proportionsalsoresultin bettercondi-
forgingthandis soflargerratios.
etterinpracticeto usevaluesofaaround
dictateotherwise.
as—Inthepracticaluseof radially -
appliedasmalldistanceinsidethe
.140.Insuch casesitisadvisableto
ostressconstantK forra -
gs
theloadwereappliede actlyatthe
sthen multipliedbytheratio
dial distancebetweenpointsofcontact,
eductionofstresscausedbythis effect.
ationsincethemomentperinch ofcir-
hisproportionaltostress)has been
hedeflectionofthe inneredgeofthe
outerwillalsobe reducedintheratio
flectionof thepo intsof loadapplicationw ill
rethanthisor appro imatelyinthe
seofthesecorrections willimprovethe
.
applicationinpractica lca lculationof
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R IN G
ischapter:It isdesiredtodetermine
f lectionforaradia lly- tapereddis spring
mutatorapplicationinF ig.136.The
: r = 9inches, r(= 6inches, a= 1. 5
A D IU
E R R A D IU
Thema imumloadP is90, 000pounds.
39 o r T ab l e X X V I I I C = . 2 71 a n d K -
omE q uations302and307,
s . i n .
rectedbecausein theactualde-
cationisV fe-inchinsidetheedgeasin-
splacedin-
spring
sthedistanced=23A inchesandr —r4=
ueof stresswillbe110,000X 2.75/3=
areinchandthedeflectionat thepoints
ll be.157(2.75/3)2= .132-inch.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
aremuchsmallerthanha lf thethic -
dthattheload-deflectioncharacteristicof
imatelylinear,beingmodifiedonlyby
ysupportingthe springattheneutral
ppededge( ig. 136)thisf rictionmaybe
W I TH T H E O R Y
eorygiveninthischapter, sometests
rrangementshownschematicallyin
ment,theloadwasappliedinatest-
cylindersto aheavyringwhichap-
oundtheouter circumferenceofthe
springw assupportedbyacylinderrest-
H E A D O F T S T IN G MA C H IN E
A D O F T S T IN G MA C H IN E
ent
rn g,
t e n-
etestingmachine.S ufficientclear-
ntheedgesofthe dis spring,thecylin-
ely,to ma ecertaintherewouldbe
onofthe load.Inmostofthe tests
ring wasbeveledasshown,greatly
epointofload applicationwasdefinite.
ometers,placedalongtheinneredgeofthe
measurementofma imumstress.A t
ibletoreadthee tensometerswhilethe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R IN G
asesstrainmeasurementsweremade
sidesofthespring todeterminewhether
l.F ormeasuringdeflectionsadialgage
atotalofsevenradiallytapered
erdiametersvary ingf rom3.8to4 inches,
om1% to2% inches,andminimum
% inches.
dload-stressdiagramsasobtained
by thefulllinesof thecurvesof
thetheoreticallydeterminedcurves
and309(correctionbeingmadeforthe
epointofload application)areshown
tresseswere determinedfromthe
ofthespringwherethe ma imum
thepointwherefailurestarts as
sts).A modulusofelasticityE —
uare inchw asusedinconvertingthe
ntwasfoundbetweentest andcal-
tthelowerloadsfor deflectioninall
oundtobe q uitegood,butatthe
therewasasmalldeviationbetween
blytothe factthatthepointof appli-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
t on d i s s p ri n g A '
moveinwardsothatthedistanced
thusma ingthespringslightly stif f er
r,theagreementinallcasesbetween
entlygoodfor mostpracticalpurposes.
CO N S T A N T TH I C N E S S
ingofconstantthic ness,F ig.133a,
r manneraswasdonein thecase
spring.
F oranappro imatetheory7theas-
atradial cross-sectionsrotatewithout
e< f> asindicatedbythedottedoutline
ecaseof theradia lly - taperedspringfor
sin anelementGata radiusrfromthe
f romthemiddlesurfaceof thedis w il l
uation280). Themomentof theforces
boutana isthroughO w illbeasbefore
attheelementGiscutoutby tw oslicesat
ThetotalmomentM w illbethe integra l
ngthofMaterials,PartII,S econdE dition,Pago179,V an
00100000 120000
0 IN A T IN N E R E D G
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R I N G 2 79
entsta enovertheareaof thesection.
glimits
ngonthespringise ua ltoP , the
ontheelementwillbe thesameasthat
4fortheradia lly - taperedspring. Eq uating
yEq uation312tothatgivenbyEq ua-
f> ,
a)
nisthen
it isclearthatthestressw illbeama i-
dr= r, . Usingthesevaluesandtheva lueof
ation312ainEq uation311andsimplif ying, the
mes
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
e acttheory forca lculatinginit ia lly -
stantthic nessisbasedonthek now n
f> betheslopeatany radiusro facir-
oaded,thenfromplatetheorythefol-
tionmustbesatisfied:
Q
t712(l-M:)
ceperunitcircumferentia llengthatany radiusr.
ig. 133w herethe loadP isdistrib-
dges
uation318andintegrating
ationconstantsto bedeterminedlater.
y radiusr, theslope< / > = — duy f / r. Inte-
w ithrespecttor,
~ C J og . r+ C , ( 3 20 )
Cl,C2,C3arefound fromthe
heouterand inneredgeswherer= r
ndingmomentsm, mustbezero. F rom
eansthat
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R IN G
0
nablethedeterminationofthethree
q uation320.
thetangentia lbendingmomentm2per
momentisama imumw hen
mstressisthen
on319andta ingr= r(intheresult-
andd< t> / dr, substitutinginEq uation321the
mes(for^ = . 3):
23)
n8isobtainedfromE q uation320
henbecomes(for/ i= . 3)
e pressionsforstressanddeflec-
masthoseobtainedbythe appro imate
4and316).Comparisonsofthenumer-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
onstantsforDis S pringsV a luesofa
' obtainedby thee actandappro i-
TableX X IX .
esshowsthatupto aratiofora of
IU
D I U
terminingconstantsCand
ofconstantthic ness
ntbetw eenthee actandappro imate
lation,valuesofC andK areplotted
ain F ig.144.
ssesnndDeflectionsinF latC ircularP latesw ithCentralH o les
w riter, TransactionsA S M , 1930, A . P . M. 52-3givesafur-
ar plateswithvariousloadingandedgeconditions.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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R IN G
E C TIO N S
uslydiscussedforcalculatingini-
basedon theassumptionthatdeflec-
er halfthespringthic ness,forrea-
heredeflectionsarelarge,thee act
orpractical use however,theap-
nit ially -coneddis springsC hapterX IV
taccuracyfor mostpurposes.Itisonly
foraninit ia lly - f la tspring)inE q uations
efollowinge pressionfortheloadP:
ation261orF ig. 125.
safunctionof theratioS / tbetween
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
sinthecurveofF ig. 145. Thiscurve
urvedeviatesfromastraight lineafter
eaterthanabouthalfthe thic ness.
aybeobtainedfromE q uations271
Thisgives:
0 (3 29 )
venasfunctionsof< x —ro/ rt inE q ua-
esofK, area lsogivenby thecurvesfor
128fordif ferentratiosof r / r< .
C U A T IO N
d-deflectioncharacteristicofthe
isdesired, theca lculationof re uired
ametersbecomessimple . B yproceeding
neforthecase ofinitially-coneddis -
ectiondia -
/ t= 1. 75
N
N E S S
maybecalculatedfor steelsprings
dbyK .C.BergvalloftheW estinghouseCompany.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R IN G
30X 10 lb. / s . in. )asinTableX X X .
load lessthanthema imumload
foundbyusingthecurveofF ig. 146.
are lative ly simplemethodofdesignfor
swithagivenload-deflectioncharac-
pro imatemethodinvolvestheas-
atS tee lDis S prings9
urveofF ig. 146w henma imumdef lectionJ= 1. 75t.
sectionsrotatewithoutdistortion,but
ttheresultsare sufficientlyaccurate
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
trm" f la tsprings isagenericterm
striporbarstoc madeinaw ideva-
heshapeswhich arepossibleforthis
discussionis beyondthescopeof
ronlythefundamentalprinciplesunder-
simplerformsofflatspringssuch as
igs.147and148)and theirapplica-
lbeconsidered.Inadditionthe ef-
tressconcentration,andcombined
will alsobetreated.Morecomplicated
analyzedbysimilarmethods.
cantileverspring overthehelical
espring maybeguidedalongadefi-
usthespringmayfunctionas a
asan energyabsorbingdevice.By
particularshapeitmaybepossibleto
s,thussimplifyingdesign.F ore ample,
maybedesignednot onlytoabsorb
carry latera lloadsand, insomecasesto
u e a s we l l.
S P R IN G
ngisa flatstriporplate ofrectangular
ectionas showninF ig.147.A ssuming
ndandloadedattheother, thema i-
the well- nowncantileverformula:
g,E = modulusofelasticityofthema-
nertiaofspringcross-section.(J--
w idthofspring, h= thic ness.)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
antilever
e
nsofthedeflectionshowthati fb
sforspringsofcloc -springsteel)the
ratio. S inceformostmateria lsnisaround
his e uationisabout10per centless
uation350.Thereasonforthis difference
ring ofrelativelygreatwidthcompared
pansionorcontractionofe lementsnearthe
evented,whichresultsinaslightlystiffer
amtheory.Thisstiffeningis ta en
1—n2)inEq uation331. Inmany
tionsthedeflectionwill probablybe
alculatingdeflections
pezoidalprofile
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
tedbyEq uation331thantothatca l-
301.
built-inedgeO ,F ig.147,isgiven
eseformulasarebasedonusual
essmalldeflections.Thecasewhere
consideredlater. Where variable
areinvolved,thestressesmustbe mul-
tionfactorswhichdependon thecon-
e.A discussionofthiswillalso be
gs—In manycases,leafsprings
wninF ig.149may,forpracticalpur-
deredascantileversprings oftrapez-
ig. 148. S uchaprof ilema esamore
lthan doestherectangularprofileof
loadPthema imumstressisagaingiven
re inthiscaseb isthew idthatthebuilt-
nordinarybeamtheory,however,shows
reasedoverthose obtainedinthe
rectangularprofileby anamount
betweenwidthatthefree andbuilt-
analysisshowsthatin thiscasethe
givenby
M
-, , :)
ert iaatbuilt- inends. ThefactorKl de-
eta enf romthecurveofF ig, 148. It
IV givesafurtherdiscussionof thiscorrection.
ainedbydividingthebendingmomentbythe section
rnet section.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
ctionofatrapezoidalprofilespring is
ngularpro f i lespringP l:< / E I mult iplied
om1forfo / fo = l (rectangularpro f i le )to
rprofile).Theoreticallythemosteffi-
whereb/b 0sinceotherthingsbe ing
imumdef lectionforagivenva lueof load.
owever,usuallydictateavalueofb/bn
sesw hereb islargecomparedtothe
arytomultiplythedeflectiongivenby
ctor(1—/ 2)ase pla inedpreviously.
smentionedpreviously,thebeam
tions331to333are basedassumessmall
gise uivalenttoacantilever
e
pringlength.In somepracticalcases,
ctionscannotbeconsideredsmall.This
0. Whenthespringisdef lectedbyan
heorywillhold. H owever,whenthede-
,S 2itmaybe seenthatthemoment
isconsiderably lessthanthe lengthIo f the
creaseinbothstressand deflection
fromE q uations332and333.
the deflectionislargecompared
oreaccuratemathematica le pressionfor
ofthebeamisused.I fx isthe
endO of thebeam( ig. 150)and8
ce,then bye uatingthecurvature
rnalbendingmomentdiv idedbyE IX ,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
esults:
inertiaatdistancex .
widthisalinear functionofthe
nessthemomentofinertiaat adistance
appro imate lyby
essioninE q uation334, thefollow ing
tion1utilizingtheboundarycondi-
atx —0, y= 0, anddy /d = 0, thereduc-
E CTI O N
ring withdeflections
onbelowthosecalculatedfromE q ua-
e pressedasfunctionsofthedimen-
/ I andtheratiob/ b betweenw idth
atbuilt-inedge.In F ig.151aregiven
uation335forestimatingthepercentage
rossandL ehr,publishedbyV .D.I.,Berlin,1938,Page133,
gration.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
erspringsof trapezoidalprofilefor
va luesofc=P l2/ Iaascomparedw ith
eusingE q uation332. Whereb/ b ~0
dand,in thiscase,thestressreduc-
stimatingstressreduction
antileversprings
12percentfor valuesofcbetween0
ore ample , if c= landthestressis
t ion332, theactua lstressw illbe12per
pringsofrectangularprofilethe varia-
tw ithinarangec—0toc= l. InF ig.
o estimatethepercentagereductionin
ca lculatedbyusingEq uation333. F or
hiscorrectionvariesf rom0to8percent
(b/b = l)andf rom0to18percent
b/ b 0). It isclearthatthecorrec-
becomessmaller.A lthoughinmost
secorrectionsmaybeneglected,for
rlywhereb/b issmall,theyshouldbe
ample, to il lustratethepractica lutil iza -
,a cantileverspringoftrapezoidalpro-
efollowingdimensions( ig.148):1=
b = . 2, b = 6in. , P= 200lb. Themateria l
0 poundspers uare inch. F orthis
cbecomes
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
M ) 3
thenomina lstressis
0 l b. / s . i n .
1, forc= . 77, b/ b = . 2, there isa6V 2
ssasaconse uenceofthelargedeflec-
R
iecalculationoflarge
prings
is96000(1-.065)= 89,700pounds
m F i g . 1 4 8 fo r b /b . 2 , K , = 1 . 3 1 an d t he
q uation333,
orrectedby thepercentagegiven
2forc= . 77andfo / fo = . 2w hichindicates
estimatedlO MtpercentifEq uation
eflectioniscorrectedby afactor(1—
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
.105)= 9.05inches.Inaddition
hb islargecomparedw iththethic ness
eductionbymultiplyingby (1—/ i2)= . 91
cussedpreviously).Thisgives afinal
1)= 8.25inchesisconsiderablyless
uredby thesimpleformulaofEq ua-
S P R IN G
etchofasimple leafspringloaded
dsupportedbya force2Patthe bot-
ffrictionasafirst appro imation,this
sa simpletrapezoidalspring.Ifthere
andwidthb1andif thereareatotal
ocomotiveWor s
aspringforlocomotive
g. 149, b= nib1andb =nbi-Theratio
hecurvesofF igs. 148, 151, and152may
falargeleaf springasusedin locomo-
ig. 153.
S P R IN G UN D R C O MH I N E D L O A DI N G
ngw hichf re uentlyoccursinprac-
gwithoneendrigidlybuilt inandthe
rallybut restrainedfromrotation
ype shownschematicallyinF ig.154.
eightin theverticaldirection,this
by thea ia lforceP . S uchcasesoccur
edcasesofe ll ipticleaf springsandthosesupportedby lin s
risreferredtotheboo byGrossandL ehr, loc. cit .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
supportedbyspringsofthistype,the
byacran arrangement. A n e ample
wninF ig.155whichshowsanappli-
apermachine.Theverticalstripsvisible
sofMicarta( now nassha esprings)
fthe table.
cttoessentially
wnschematic-
ia lloadF is
hebuc ling
essmayeasily
eamtheory.
are
m( ig. 154),
moment
w idth, h— thic -
ton, a= nomina l
ss concentrationneglected).When
tigueloadingas mentionedpreviously
be multipliedbyafatiguestrength
ntoaccountthe stressconcentrationat
altestdatarelativeto thevaluesofsuch
P, F ig. 154isnotsmallcomparedtothe
tions336and337no longerapplyaccurately .
rateanalysisshowsthatthestressand
ymultiplyingtheresults calculated
y factorsC , andK , w hichdependonthe
nthisPcr= E Iw 2/l2istheEulercrit i-
ngedends.These factorsare:
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S PR I N G 2 95
husbecome:
luesofC 1andK 2areplottedaga inst
ppro imatemethodofca lculating5thefac-
Y -
espringsofcantilevertypeonF ourdrinierpaper
combinedlatera landa ia lloading
heactionofthe loadsPandQ the
e curveoftheform
dinthe beamis(usingthise ua-
ethodisappliedtothecaseofasimplysupportedbeaminTheory
T imoshen o, McGraw -Hill , 1936, Page28.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
eflectionS increasesbyasmall
uation342thepotentialenergyV changes
)A 8/ 16/ 3, neglectingsmallquantit iesof
orceQ doesthew or @A 8anditmay
lmovementof thea ia lforceP is
8/ 8Z w hichmeansthatthew or doneis
culatingthedeflectionofacantileverspring under
ia ltypeof load
atingthechangeinpotentia lenergy tothew or
, thefo llow inge uationresults
.2 I
ameasE q uation341. Ma imumstressis
modulus(Z = foh2/ 6).
essionforQ obta inedfromEq uation
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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A F S P R IN G
ectionfactorforcantileverspringsunder combined
ottedagainstloadratios
q uation340isobtained.
owsthatwherethe a ialloadis half
ad, thefactorC , = 2, i. e ., tw iceasmuchde-
edthanifEq uation336basedonordinary
ntheotherhandforratiosP / P r, e ua l
ig.157indicatesthatthe stressformula
ssthan10percentinerror. Inaddit ion,
casewherea cran arrangementis
thestressesfiguredfromE q uation
higherthantheactualstresses.
twhenthespringwidthis large
ss(asitusuallyis) thecalculateddeflec-
t ion341shouldbemultipliedby1—n2as
stressasgivenbyEq uation340di-
Po isson sratio . F ormostmateria ls
percentincreasein stress.
G
noftheuse oflargeflatspringsin
chematicallyinF ig.158whichrepre-
ringssupportingtheframeof a60,000
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
phasealternating-currentturbo-generator.
ementisto absorbtheperiodictor ue
gle-phasegeneratorofthis typewith-
ectionablevibrationtothefoundation.
T R A M :
carrangementofplatespringassemblyfor
ting-currentgenerator
his typeofspringis shownbythe
romthebeamtheory,thedeflection
edas
ectionofaplatespring
imensionsshow nonF ig. 159andP is
rverywidespringsin relationtothe
nshouldbemultipliedby1—/ <» ase -
a imumstressisgivenby
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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A F S P R IN G
normally sub ecttore lative ly low
evereshortcircuitconditions(which
thedesignstressesare reached.
methodsforcalculating nominal
variousflatand leafspringapplications
ses,however,theeffectofthese nom-
bystress concentration,whichmaybe
mpededges,sharpbends,sudden
thespringisunder apurelystatic
epeatedarelativelyfewtimes,such
tsmayusuallybeneglected,inprac-
erialsof goodq uality.H owever,
ho leunderbending, ho le
omparedto strip
euppercurveofF ig.
d tofindtheoretical
r
oadingoccurs,carefulconsideration
stressra isers by thedesigner, since
artat localizedpointsofstressconcentra-
etefailureofthe spring.
tionintoaccount,it isnecessaryto
alstressto whichthespringis sub-
ththediscussionof ChapterIthe
ameanstressa andavariablestress
ethema imumandminimumnomina l
(f fm< w + amm)andac=^ (o-ma —amin)-
C N TR A T IO N E F F E CT
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
ss a,afatiguestrengthreduction
To beonthesafeside, intheabsence
ueofK fmaybeta ene ualtothetheo-
nfactorK t(seePage125).In some
ctorK fmaybeappreciablybelowK t,
factorsmaybenearlye ua l. F atigue
articularlytrueofthefine-grained,
sedforsprings.F orthisreasonthe
rationfactorsgiveninthe following
tisfactoryforuse untilmoretestdata
it isdesirableornecessary toprov idea
dingthespring inplaceorfor manu-
S S
R
R
ress concentrationfactorsfor
g.N otethatthelowercurve
circularnotches
mmone ampleisthesemiellipticauto-
uallyis providedwithaholein the
isholeholdsthe leavestogetherand
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
etweentheleaves.
lessmall indiameterrelativeto
sshow ninF ig. 160(thismayoccurin
gs),it appearsreasonabletoapply
estson tensionbarswithholes 5.The
gw ithhole large indiametercompared
whered/hislargeconsiderably
factorsmaybee pected.F orsmall
3comparedto1.85forsmall d/w
howsvaluesoftheoreticalstress con-
rminedphotoelasticallyasafunction
iameterandplate width.Thuswhere
orK tapproaches3, awellk now nresult
wever,forsmallhofesitis possible
ngthreductionfactor K Iwillbecon-
the" sizeeffect maybepronounced.
hinsprings(fore ample, thosemade
wheretheholediameterd islargecom-
esshasshow ninF ig. 162, bothtests
derably low erva luesofKt w ille istthan
all.A mathematicalanalysisby
mall holeinawidestrip underpure
hered/h islargethisfactor K (— 1.85.
mallerthanthe valueof3foundfor
/ hissmall. Sincew hentheholediam-
idtha factorof1maybe e pected,
, A ug. , 1934, Page617, andMechanica lEngineering, A ug. ,
scriptionsofsuch tests,togetherwiththeoreticalstresscon-
, V . 22, 1936, Page69. Thisw or hasa lsobeenchec ed
montusingstrain-measurementsonalargealuminumplate.
hnica lN oteNo. 740. V a luesofK(= 1. 59w erefoundforrf / t = . 145
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
urve isthatshowndashedin F ig.161
stressconcentrationfactorsisil-
ig.163whichshowsanestimateden-
high q ualityspring-steelstripina
tion(thic ness.006-inch).Theen-
alwas foundbytests(carriedout
of theWestinghouseR esearchL abora-
poundspers uare inchinreversedbend-
gthwas275,000.It maybee pected
ub ecttoastressrangebetw eena , i
acharacterist icappro imatelyasshownby
Thusthestress rangeforcompletere-
andBorbetween-(-130,000and
uare inch. F orarangef rom0toma i-
aeGandH f rom0to180, 000. N ow , ifa
trip, assumingthatd/ w issmalland
actorK, — 1. 85. Sincethismaterial
tiveto stressconcentration,itmay
thatthe fatiguestrengthreduction
atelye ua ltothisvalue. Onthisbasisa
owadiagramasindicatedby the
thepointC andDrepresentingnominal
±70,000.Theordinatesbetweenthe
andeithertheupperorlow erfull l inearea lso
o toma imumstressrangeforthe
esentedby the lineEF or0to112, 000
Inthis case,therefore,thestrength
cationhasbeenreducedbythe pres-
,000to112,000poundspers uareinch
112,000= 1.6.Thisisconsiderablyless
eductionfactorassumedfor completely
1.85.Thisdifferenceis aconse uence
ssconcentrationeffectsmaybe
ticcomponentoftheappliedstress
tualtestswouldprobablyshowa
ncelimitforthestripwith aholethan
thisw ay , dueto" sizeef fect forsuch
se,however,themethodofcalculation
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
fordesign.A lsotheassumptionthat
tsmaybeneglectedincalculatingthe
maynotbe entirelycorrect( ee
roducea furtherdeviationbetween
ults.
presentsthevaluesto bee pected
ecurvesforahigh-gradespring-steelstrip,
facegroundandpolished
h-gradestripmaterialinavery thinsize
acein goodcondition,i.e.,ground
hic ersections, suchasareusedinleaf
surface intheconditionleftby rolling
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L S P R I N G
0000OO C OO
S
urancecurvesonleafspringmaterial
reatmenton thebasisofavailabletest
esofendurancelimit maybee -
nF ig.163.Theresultsofendurance
sedinleafspringsare indicatedin
low ercurvesA andA ' representthe
steelplatesasusedin leafspringswith
ethedecarburizedlayerleft byheat
eshadedarea representtestresults
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 324/462
A F S P R IN G
B atsonandB radley , andHoudremont
springsw iththesurfacesuntouchedafterheat-
pectedthat,forleafspringsthe results
somewherewithintheshadedarea
weringoftheendurancerangetofrom
rmachinedorgroundspecimensistobe
uenceof thedecarburizedsurface layer
hnotchesinbending. Where
urveofF ig. 166. Where
wercurveofF ig. 161
owever,forhighq ualitymaterialsand
eatmentsitispossible thatimproved
erthevalues indicatedonF ig.164.
tressconcentrationeffects(holes,
thevaluesoflimiting stressrangeas
ucedstill lower.
ificandIndustrialR esearch, (B rit ish)Spec. R eportN o. 5,
onofNfechanica lEngineers, 1931, Page301.
, publishedinSta ll \ indEiscn, July7, 1932, Page660.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
sitisnecessaryforvariouspractical
si nthesidesof flatsprings.W hen
rcularform,the stressconcentration
sfollows:Ifthe stripisrelatively
/hbetweennotchdiameterandspring
dicatedinF ig. 165a, itappearsreasonable
toelastictestsonnotchedbarsin ten-
esofthetheoreticalstress concentration
R
IP
ress concentrationfactors
thic strips,smalld/h
swayareplottedas afunctionofthe
ameterandplatewidth.
sinthin stripmaterialsasshownin
ratiod/hislarge, it isreasonabletoe pect
bepracticallythe sameasthatfora strip
ameterdandthesamewidth w.Itis,
thelowercurveofF ig.161mayalso
ationforthiscase.Thedashedcurve
note6discussresultsof suchphotoelastictests.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
ore,representtheendurancediagramfor
hsemicircularnotchesunderbending
all).
ngflatsprings,sharpbendsarefre-
ampleof thisisthebendatA inthespring
pshow ing
tdue
at A
ecauseoftheirsharp curvature,these
tressconcentrationeffectwhich may
rrepeatedloadingby usingastresscon-
vedfromcurvedbartheory1-.V alues
ousvaluesoftheratior/h betweenmean
nessofmaterialaregiveninF ig. 168.
ctorK tincreasesrapidlyasr/h ap-
tanceofavoidingverysharpbendsin
whensub ecttorepeatedloadingis
nwasmadepreviouslyof theeffect
atespringsincertainapplications.
uentlynecessarybutbecauseof theclamp-
ountof stressconcentrationissetup
erfatigue loadingthisw illresult inare-
belowthatobtainedwhenthe spring
nbythedashedlineb inF ig.169which
ssconcentration.Inaddition,under
soa certainamountofrubbingat
,whichresultsin thisso-calledrub-
t.This latterresultsina further
ngthwhichmaybe particularlygreat
erials.
tobelittle dataavailableonthe
ppliedtospring steels,someresultsof
edbyH an ins . Thesetestsw eremade
edmorefully inChapterX V II.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
springsteel(% - inchthic )w iththe
erheattreatment,theloadingcon-
oseofF ig.169.W henthe,specimens
rmwidththeendurancelimit incom-
tressfora.48 percentcarbonsteel
O F B N D
N E S S
centrationfactorsK tfor
the specimenswithoutstressconcen-
evalueof± 47,000.Thefatiguestrength
fore,was47,000/33,600=1.4.S imilar
onspringsteel showedareductionin
00to45,000anda fatiguefactor
eregivenastotheactualclampingpres-
uldbenotedthatfor higherstrength
cehasbeengroundafter heattreat-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
D TH
gw ithclampedend. Duetoclamp-
ntrationoccursatA
ofK f thanthesearepossible . F or
usedinrollerand ballbearingswhere
ts, va luesofKfashighas3to4havebeen
aftsareused.F oramedium-carbon-
-inchdiametercarbon-steelshaft
m1.4to 2,dependingonfitpressure,
O F F L A T S PR I N G
aratherunusua lapplicationw herethe
edoutadvantageously isthe latera le -
ig.170.Thepurposeofthis instrument
)isto measureminutelateralcon-
modelofBa elitewhenstressed.
001-inchtotal toobtainsayoneper
arytomeasurethesemovementsto
ownthattheselateralcontractionsare
sumof theprincipalstresses15atany
nderload.Bythusdeterminingthe
nypoint,anddeterminingtheirdif-
photoelasticmethodsthecompletestress
sonandWahl, " F atigueofS haf tsatF ittedMembers , Journal
5,PageA -lgivesfurtherdetails.
,1939,presentsamorecompletedescriptionofthesee -
simaginedascutoutofa f la tspecimenunderloadand
dtoberotateduntilno shearingstressesactonits sides,the
pendicularplanesare calledprincipalstresses.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L S P R I N G
tensometerutil izesf la tsprings
menmay befound.
ale tensometerisshow ninF ig. 171.
nd P arepressedlightlyagainstthe
thepressuree ertedby thef la tspring.
holdthepointsat adefinitelocation
wholeassemblyis supportedbythe
ig.170,astatic balanceoftheinstru-
ansofthe weightsshown.Becauseof
tyofthesupportinghelicalsprings, slight
ofthetestspecimenmayoccur without
of latera le tensometer
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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A F S P R IN G
f thepointsPandP ( ig. 171)resting
beseenthatany latera lcontractionof
ea relativemotionofpointsPand P
otionofthe barBwithrespectto the
cordedontheH uggenbergere ten-
areheldagainstthe instrumentbya
ereflatspringshavebeenusedto form
theshortgage-lengthe tensometershow nsche-
cs etchforarrangementof
someter
sinstrumentisused inthedetermina-
nwherea shortgage-lengthisneces-
rumentconsistsoftwolight, hollow
feedgesB andB . Thesek nifeedgesarc
ngstee lstripsS ( ig. 172a)w hichare
ung, " A nInvestigationof theC ross- pringPivot , presented
. M. . meetinggivesdesignformulasfore lastichinges. Thesemay
esas highas45degrees.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ee tensometerpointsarepressed
eansofa specialclampingarrange-
R eferringtoF ig. 172aw hendeforma-
urspo intB movestosayB . Thiscauses
erarmT asindicatedbythedashed
mationofthespringsteel strips.The
semblypivotsaboutthepointO ,and
e leverarmsaconsiderablemagnifica-
5in thiscase).Thusanyrelativemo-
sthuscommunicatedtothetargetsA a f ter
erratio.The relativemotionofthe
bymeansofa microscopewithameasur-
odahigh magnificationisobtained.
( ig. 172a)itw asnecessary to
sothatbuc lingduetotheclampingload
sametimesufficientfle ibilitymust
reciablerestrainttothedeformationof
Byusingflat springsinthisway,the
heirdisadvantageswasavoided.
tspringsis showninthespecial
orholdingthee tensometerpo ints
Theclampingpressureis suppliedby
wn.TheflatspringsDand D allow
amp whenthescrewE isturned.
maybelocatedaccurately.A definite
tensometerbycompressingthehelical
t.Itis importanttomaintainadefinite
hightopreventslippageof thepoints
causebuc lingof thef la tsprings. A
o f
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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A F S PR I N G 3 13
eclampingpointP (notshown)isalso
isessentia llya f la tspringw hileA isa
eof thisarrangementistoallow
tensometerrelativetotheclamp,
tensometerandmagnetclamping
sitiononshaftfillet
test specimen,bothlaterallyand
ultinginlateralforceswhichmay
umentor slippageofthegagepoints.
ctswould produceerrorsintheresults.
espossibleathree-pointsupportforthe
e sametimeslightdistortionsofthe
BandCmaybe ta enupbydeflection
gappreciablelateralload onthe
atsprings itwasagainnecessaryto
ndertheactionof theclampingforce.
someterclampedontoa largeshaf t
4. ThemicroscopeM, thetargetA , the
ntsB , andthemagnetN are indicated.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I
S I O N S P R IN G
veessentiallythesameshape as
sionspringse ceptthattheends are
thespringmaybesub ecttotor ue
useofthemodeof stressingsuchsprings
ural,incontrastto thehelicalcompression
primarystress istorsional.S ome
orsion springsareshowninF igs.175
ng endismadeprimarilyfromthe
nge ternaltor uetothespring. Tor-
arnesCo.
poftorsionsprings
widevarietyofapplicationsamong
oorhingesprings,springsfor starters
sforbrushholdersin electricmotors.
methodof loadingatorsionspringisin-
erethespringissupposedtobew ound
beingfastenedtothe rodwhilethe
pro ectingoutward.Ifthisarmis
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S I O N S P R IN G
adiusR f romthea isinsuchamanner
nthemomenttendingtotwistthe spring
the figure.Becauseoffrictionbetween
actualmomentmaydecreasealong
actcalculationbecomesinvolved.
gsare woundcold,itisadvisableto
hatthespringtendsto windupasthe
y lesofendsusedintorsionsprings
for thisisthatthe residualstresses
ceof theco ldw indingare insuchadirec-
epea stressduetotheloading,pro-
esamedirectionas thatinwhichthe
ngsub ectto forcePatradiusR
ection ofloadingissuchas toun-
abletoheattreatbymeansof ablu-
uchresidualstresses.
loadingthe springinthisman-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S P R IN G
177fora loadtendingtow indupthe
againstthearborandthepea bend-
w illbePR . H ow ever, if the loadisin
atshown,the pea momentwillbe
meancoilradius. Thismeansaconsider-
ticularlyif risaboutas largeasR .
manycasesw heretheendsof the
pecialendsareused,somestressc on-
ctedneartheends.Thesestressconcen-
yconsideredparticularlyifthe spring
ading, orif it istobesub ecttoa large
duringits life.O ntheotherhand,
cationsissmallduringthe springlife,
smayprobablybeneglected.
nspring (forusualapplications)
,itsdiameterdecreases.Indesignit
clearancebeallowedbetweenthe
e springiswound,andtheinner
s isnotdone,thespring maybind
andhighstressesmaybeset up.The
beestimatedfromthecalculatedde-
springasgivenbyE q uation367. Thus,
90degreesorV 4-turnandthespringhas
changeintheratioofV * to8orabout
wedforin design.
beandis loadedsoastounwind,
beallowedbetweentheoutsidediam-
nsidediameterofthe tube.This
edinthesame mannerasbefore.
esquite longtorsionspringsareused.
s alwaysthepossibilityoftorsional
oidedby providingsuchspringswith
es.Byproperlyclampingtheendsor
nload,it ispossibletoavoidbuc ling
se ofguides.
formanufacturingreasons,torsion
wire. H owever,wherema imum
redinagivenspace, theuseofs uare
advisable.Thereasonfor thisisthat
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S I O N S P R IN G
rrectangularsectionhasa largerpropor-
tedtostressesnearthepea va luethanis
. Conse uently , forthesamepea stress,
aybeobtainedfora givenvolumeof
ctangularwirethanfor circularwire.
nde , morematerialmaybecompressed
meterin thecaseofrectangularwire
mountof energywhichmaybestored
X IIdiscussesthisfurther.
signed,mosttorsionspringsmay
cttoa purebendingmomentaboutthe
eusua lsmallpitchangles, thespringmay
ngclementactedonbybendingmomentM
vedbarsub ecttoabendingmomentM,
bartheorymaybeapplied1.
bed,F ig.178,cutfromthe spring
passingthroughthecenter ofcurva-
andincludingasmallangled > , the
f thebar, V > , themeanco ildiameter, is
ngthofMateria ls. Part11. SecondE dit ion, page65givesa
eory.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
rta inradiusr therewillbenostressin
cecorrespondingtothisradiusisk now nas
tMactson thiselement,ifitis as-
tionsnormalto thecenterlineremain
nbc deflectsthroughasmallangle
e s u p th e p os i ti o n ef .
alelementshownshadedata dis-
surface is(r —y)d< f> beforebending
helengthincreasesbyanamounty (A d< / >).
sbe
willbethis elongationtimesthemodulus
material.Thus,
tdiestressdistributionacross thesec-
sindicatedinthe diagramofF ig.179.
wnradiusr oftheneutralsurface,
ardef lectionA d< f> , tw oe uationsart-
edfromtwoconditions:
orcesactingoveraradial crosssection
e ternalforce(butonlyane ternalmo-
tsoftheelementaryforcesabouttheneutral
thee ternalmomentM.
areaoverw hichthestresso> acts
ularcrosssectionw herehisthewidthof
mthefirst conditionmentioned,
on,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S I O N S P R IN G
ybewrittenas
- ( A d < t> )
_ M
-yI
hisgives
eofthecentroidofthe sectionfrom
uation347,
345, thee pressionforstressaa
ndrectangularcrosssectionsthe
urat theinsidesurfaceofthe spring
y= ruthe insideradiusof thebar, F ig. 178.
sevaluesinEq uation348, e pressionfor
. ratthe insideof thecoil, F ig. 179, becomes
utsideofthe coilisobtainedbyta ing
r 2 i n E q u a ti o n 34 8 . Th u s,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
gnifiescompressionforthedirec-
ated inF ig.178.
B A R T O R S I O N S P R IN G
sectionofw idthb, F ig. 178, dA = bdy
uation346,
tion, thevalueof r maybedetermined.
nisfacilitatedbythefollowingmethod
o2. L etting / , = / - (-<? w here(/, isthe
crosssectionfromthe centroid,then,
a n d e~ t — r , f r o m E q u a ti o n 34 9
_ f M_ r -d A - = Q ( 3 50 )
,
on351,
- dA = A a + » > ( 3 52 )
tions351and352inE q uation350,
forarectangularcrosssection,the
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S I O N S P R IN G 3 21
ation351maybee pressedinseriesform
. . + ( ^ y , + . . )
/
uation351, ta ingdA = bdy l, andso lving
y , + _ ,
1 r 3 /
2r/h,thevalueofmbecomes
h ( 35 4
2)—eforrectangularsection, bysubstitution
q uation353,
ubstitutingE q uations353and355in
i n g r , = r — h 2 a n d A = " b h , th e ma i m um
tion becomes
thespringinde c= 2r/ 7iforarectangu-
rec isoverthree,theseriesof
vergesveryrapidly andsufficientaccuracy
beobtainedby ta ingtwoterms.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S P R IN G
thise uationisverysmallcompared
springs,thismaybewritten:
orminE q uation356, thema imumstress
ssimplythe ordinaryformulafor
sub ecttoabendingmomentM (i.e.,
f actorK, greaterthanunityw hichde-
c
candw hichmaybeconsideredasa
r.V aluesofK .,areplottedasfunctions
nthat, asw ouldbee pected, thevalue
ninde , sincethespringbarthenap-
straight barunderabendingmoment.
de o f3, thecorrectionfactorK. , ~ 1. 30.
stressisabout30per centgreaterin
dbytheusualformulain whichthe
cted.
he deflectionofatorsionspring
e-sectionw ire , theusua lbeame uation
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S I O N S P R IN G
ctingeffectsoffrictionandassumingthat
constant momentMasbefore,theangu-
tof thespringoflengthds willbe
nertiaofthewire section.Thetotal
radians)willbetheintegral ofthis,or
elength ofthespringwireand nthe
R -
c - T- O R ^ -
ssconcentrationfactors
andrectangularw ire
ngthis valueof/,theangulardeflec-
ualbeame uationw ithmoree actresultsca lculatedfrom
hatthedifferenceisnegligiblefor practicalpurposes.Thisis
ressionsprings,wheretheusuale uationfordeflectionis
practicalpurposes,althoughderivedin anelementaryway.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
pringindegreeswillbe 57.3timesthis
ngissub ecttoaforcePattheend
sindicatedinF ig. 177, themomentM—
ialdeflection8atthe endofthisarmwill
uation358thedeflectionatradiusR is
T O R S I O N S P R IN G
lthoughtherectangular-wiretorsion
cientuse ofmaterial,springsofciicular
yusedforreasonsofeconomy.Thestress
uredinasimilarwayas before.Ifd
singE q uation348thema imumstress
ssedasfo llow s:
oceduremaybeusedasin thecase
rng. F romE q uation360, ta ing
c= = 2r,dandA -* • (/-4forcircular wire,
ritten
uation352isused.Ta ing
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S I O N S P R IN G
s
/
pressionundertheradicalmaybe written
uation362, ta ingA = (ir/ 4)d-andsolv -
rarectangularcross-section,theinde
sectionc= 2r/ d)
herethe inde c> 3, thisseriescon-
msaresufficient forpracticaluse.Thus
hebrac etsof thise pressionissmall
ttenwithsufficientaccuracyas
q uation361,andsimplifying,the
umstressbecomes
o llow inge pression
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
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S P R IN G
uation364representsthestressfig-
afor astraightcircularbar,i.e.,stress
noversectionmodulus. ThefactorK,
asedueto thehyperbolicstressdistri-
esofK, areplottedasfunctionsof thespring
tisseenthatforaninde o f3thestress
about1.33.It mayalsobenotedfrom
sofK, donotdiffermuchf romthoseofK . ,
hesameinde .
he deflectionofatorsionspringof
pressionmaybeusedasthat usedfor
tthatthemomentof inertia / ista enas
vesfortheangulardeflection
ghl= 2irrn, thise pressionmaybewritten
berof turns, sinceoneturn= 27rradians,
(3 6 7)
toamomentsetupbyaforcePat
radiusR , F ig. 177, thenM= PR andthe
atthisradiusw illbee ua ltoR < f> ,
radians. H ence, usingEq uation366, the
mes
A sane ample , thedesignofabrush-
catedin F ig.177willbe considered.
dtoapplypressureonthe carbon
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S I O N S P R IN G
ssumingthefollowingdimensions:
meancoil- radiusr=3/ 16- inch w irediam-
ginde c= 2r/ d—9. 4, the load' ? atthe
* pounds. F romF ig. 180thefactorK ,
q uation364thema imumstressis,
25X . 875, BB nMt. . .
lb . /s . i n .
s,fromE q uation367thedeflection
10 poundspers uare inch,
X . 187X 10
5 4 tu r ns
)
ctionof slightlymorethan180degrees.
tressesforTorsionSprings
n H i - in c h W i r e S i z es % t o V i - i n ch
M a i m um M a i m um
al W o r i n g To t al
sStressS tress
00180,000215,000
00215,000140,000180,000
00180,000120,000155,000
,000180,000115,000145,000
5,00048,00070,000
0060,000
or ingstressesfortorsionspringsof
diameteraresuggestedbyW allace
1.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
oodguidein designingspringsofrea-
neraluse.Theseareshownin Table
Wire Company(ManualofS pring
vesvaluesofma imumdesignstress
asindicatedinTableX X X II. These
foraverageserviceconditions,defined
ere,temperaturesnote ceeding150
velyslowlyvaryingorstaticloads. The
mumDesignStresses
tions)
ucedin generalforthelargersizes of
esarealsogivenforstainless steel,monel
ybesatisfactorywhencomputed
or364whichta e intoaccountthestress
andprovidedthespring issub ectto
ssduringits life.
ecttorepeatedorfatiguestresses
ge,ingeneralit willbenecessaryto
sesthanthosesuggestedinTablesX X X I
lsotrue if thespringissub ecttoe levated
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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II
G
stingessentiallyofflat stripwound
nyadvantagesfromthestandpointof
itedspace,particularlyif thespring
r ue. Inaddit ion, suchspringsarere la -
re.Becauseoftheseadvantages,
edin cloc s,watches,electricalin-
ces.O therapplicationsinclude
g.182,phonographmotors,etc.A nun-
sho lderspringtormotor
ringasan energystoringdeviceis
ta lcircuit-brea ermechanismofF ig. 183.
undthatindividualturns donot
sisfor thespringmaybecarried out
cy. S uchane ample isprov idedby
ntheotherhand,if theturnsofthe
gether,asis trueofaphonograph
rtof analysismustbemadebecairse
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
menta lmeehanismutilizesspira lsprings
hesecaseswill thereforebetreated
rposeofthischapterbeinga discussion
alspringcalculations.
M A N Y T UR N S W I TH O U T CO N T A C T
hefirstanalysisit willbeassumed
pringisclampedby amomentM,as
so willbeassumedthatthespring has
hichare,however,separatedsuffi-
tturnsdonotcome incontactduringde-
he springisfastenedtoan arbor
andisactedonbyator ueM .
, attheouterendA o f thespring
alforceR (passingthroughO )and
ee ternaltor ueM is
ntatanypointof thespiralhaving
y , thenf romthestaticalcondit ionsofe ui-
( 3 70 )
TheoryofE lasticity , Ox ford, ClarendonPress, 1936, Page66.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
ation369andsubstitutinginEq ua-
ortlengthdsofthe springacted
mordinarybeamtheory2
felasticity andIthemomentof inertia
erestripmaterialisused asindicatedin
urateresultswillbe obtainedbyreplacing
/ = P o i ss o n s r a ti o .
hespringis
enoverthetota llengthof thespira l.
statesthatthepartialderivative
hrespecttoastatically indeterminate
trengthofMaterials,PartI,S econdE dition,V anN ostrand
ng
e
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
oesnow or , mustbezero . Sinceneither
entM, dowor asthespringdef lects,
ddifferentiatingundertheintegral
hofthe spiral.
dconstant, f romthesee uationsthefo l-
,
ogetherw ithE q uation371inE q ua-
M, - R x ] ^ - = 0 ( 37 7)
ations371and375,
M, - R x ] x d s = 0 (3 78 )
alsostatesthatthepartial derivative
hrespecttoane ternalmomentgives
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
tothis moment.Thustheangularde-
rnalmomentM becomes(usingE q ua-
on371withrespectto M,
and380in379angulardef lection< f> becomes
T) ~ M r ~ H ( + 7 ) ( 38 1)
ew ritten
—-ds- fA f,—ds-C R x 2ds=0 (382)
largenumberof turns,thefol-
o ldw ithsuff iciente actitudeforprac-
0 Cx yds=0
hreeintegralsofE q uation382are
tbezero itfo llowsf romthise uationthat
attheouterendA o f suchaspring, F ig.
i l lbezero . F ora smallnumberof turns,
ver.
,
s - f M ^ d s - C R - - ds = Q
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
O a n d / ( y / r ) ch = 0 f o r a la r ge
uationreducesto
annotbezero , Eq uation383showsthat
= M X . U s in g t hi s c on d it i on i n E q u a ti o n 36 9 ,
hetangentia lforceattheendA , F ig.
onditionassumed.S inceR isalsozero,
ation371showsthatM= M w hichmeansthat
ngthe lengthofthespring.
= 0, E q uation381reducesto
erof turns/ (y / r)ds= -0and/ ds~ l
ection< f> becomes
dians(or degreesdividedby57.3).
sthattheangular deflectionofaspiral
of turnsandalengthI withbuilt-in
hatofa straightbeamoflengthIbuilt
yamomentat theother.
stantalong thelengthofthespiral
ctingcurvatureeffects)for thecase
nby
crosssectionandli= thic nessofstrip.
sconcentrationat theclamped
ueorrepeatedloadingispresent(as
h),inaccordancewiththediscussion
ouldbeta enintoaccountbymultiply ing
q uation386bya stressconcentration
ionswherethenumberof repetitions
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
espringis small,stressconcentration
ever.
f turnsisinvo lved, Eq uations385
das discussedlater.
re uently inpractice , formanufac-
dof aspiralspringmaybe heldwith
mped.N eglectingfrictionnomoment
andtheloadingconditionswillbe
. Inthiscasethee ternalmomentMn
o nottoucheachother,the mo-
piralhavingthecoordinatesx andy
)
hise pressionmaybewritten
orem,asbefore,
uenceofthefactthat theradialforceR
def lectionof thespring. ThereforeEq ua-
on388w ithrespecttoR andsubsti-
3, thefo llowinge pressionisobtained:
mberofturns,thefirst twointegrals
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
encethise uationgives
ntf romzerothismeansthattheradia l
ozeroforthepin-endedcase. F ig. 185.
otation< f> isgivenbyEq uation379,
byE q uation389.Differentiatingthe
ttoM andsubstitutingtheresultto-
389inE q uation379,
( 1+ 7 )
bezerothissimplif iesto
t/ yds=0 fora largenumberof turns,
o f turns
te.UsingitinE q uation391,thee -
sto
uation385it isseenthat, forthe
M,,aspiral springwithahingedouter end
tmoreangulardeflectionthanthecor-
pedouterend,providedad acentturns
inthespringwill occurwheny= r
ngy= rinE q uation389, sinceR —0, this
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
eM= 2M . Thema imumstressisthen
momentM thisstressistw icethatfor
uteredge( q uation386). How ever,
ccursata pointoppositetothepinned
ssconcentration.Ifthearbordiam-
r,themomentattheinner clampedend
ameasthatforaspringw ithclamped
the stressatthisend willalsobethe
lwayssomestressconcentrationatthis
hatinsomecasesthis isthelimiting
hecoils, asmayeasilyoccurin practice,
essgivenbyEq uation393. F oramore
spiralspringswithlarge numbersofturns
hearticlebyV andenB roe 1.
rsionspringhav ingapinnedendA F ig.
t e rn a l to r u e M e u a l to 2 5 i nc h -p o un d s.
ches,thebarsectionis " 2by.06-inch,
ches. R e uiredthestressandthede-
odulusE = 30X 10 poundspers uare
forapinnedend
, J .
1. 73radians
0 -
ularrotationof 1.73(57.3)=99 de-
thema imumstressis
endO wherestressconcentration
r oc — " S p i ra l S p r i n gs T r an s ac t io n s, A . . M . . , 1 9 3 1, 5 3 -1 8 .
yTheory, Wiley , 1942.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
hisor84, 000poundspers uare inch
ersmallcomparedtor. H owever,as
tterstresswill beaugmentedbystress
toclampingofthe end.
erelarge tor uesareinvolvedit
ivelyheavycrosssectionforthespiral
oflargerdiameter.Thismeansthat
springmayberelativelysmallso
edtheory(basedona largenumber
owever,ananalysisofthiscase
lar methodstothosedescribedpre-
bebrieflyoutlined.
small numberofturnsasshown
edthatthespringisclampedorbuilt in
hile theouterendA maymoveinan
ngulardef lectionof theendA ( in
movementofA a longthearcdiv ided
t onthespiralhavingthe co-
egivenbyEq uation370, usingr, forr,
x ( 3 94 )
uantitiesM PandR inthise uation
hreee uationsobtainedbyusingthe
cethepointA isassumedconstrained
rcaboutO, thew or doneby theforce
ansdU/ dR — 0andE q uation375ho lds.
ation394partia llyw ithrespecttoR ,
on394inE q uation375, thefollow inge -
" S pira lSpringsw ithSmallN umberofTurns , Journal,
o l. 225, 1938, Page171.
E W T UR N S
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
dof thespringthroughanangle
w illa lsomovethroughthesameangle . F rom
isconditiongives
, dM/dM^ l. UsingthisinE q uation396,
on394,
M ds = ^ j J ^ P (r t+ y ) + M 1- R x ] d s (3 97 )
soobta inedfromtheC astigliano
thetotaldeflectionin thedirectionof
a ltodU/dP . SinceP isa lw aysassumed
rcofmotionof theendA , F ig. 186, this
H ence
4, dM/dP~ r2- -y . Usingthistogether
q uation398,
) + M, -R x ] ( r + 3 ) rf s( 3 )
ringista enintheformofaspira l,
nd399may
llength
eous
f rom
titiesmay
e,the
point
uation394.
tionof
themoment
btained
a lueofF igise—S pira lspringw ith
M, . IfM smallnumberof rums
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
thisisa lsothema imummomentfora
nda largenumberofturns)the ratio
mummomentande ternalmomentmaybe
entrationfactor.V aluesofthisratio
8409201000 1080
G E O F CO L ( IN D GR E E S )
centrationfactoraforspringwithfew turns
87asfunctionsof thetotalspiralangletI
functionA =(r2— r1)/ rihavebeenob-
venport1. Theva lueof6ista enasthe
us vectorintravelingfromoneend
enthatthestressconcentrationef fect
urnsissomewhatgreaterforthe larger
hestress concentrationvaluesare
angle 6isnear 360,720or1080de-
turns.This isshownbythedips inthe
isofadvantagewhendesigningspiral
awholeratherthana fractional,num-
omF ig. 187it isa lsoseenthatasthe
s,thema imumvaluesoftheratio
.e.,thestressconcentrationeffectde-
for0= 1080degrees(3turns)andA —
stil le ua ltoa lmost1. 2whichmeansthat
be almost20percenthigher thanthat
6, derivedontheassumptionofa large
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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G
ress concentrationeffectmaybe
ectsofimperfectclampingat the
thatbecauseofthe smallnum-
omewhatstiff erthanwouldbee -
simpleformula( q uation385)de-
fturns.Themoreaccurateanalysis
ectionata momentM isgivenby
sgreater thanunity)depends
spira landontheratioA . V a luesof this
188. H ow ever, forratiosA between. 4
0000O f lC
E O F CO L C I N D G R E E S )
actorpforspira lspringw ithsmallnumberof turns
woturnsofthe spiralthisfactordiffers
ercentand mayusuallybeneg-
es,i.e.,theusual formula( q uation
w ever, fromF ig. 18 itmaybeseen
nvolvedin theusualformulawhere
honlyoneturn.
ctionat variouspointsalongthe
ingeneralthedesignershouldtry to
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
560 640720800880960 1040
O N G CO I C IN D GR E E S )
ndingtheradialdeflectionofcoils ofspiral
suredf romouterend
hduring deflection.Theseq uantities
asfunctionsof theanglealongthespiral
sand forvariousvaluesoftheratio A ;
ig.189.Theordinatesrepresentradial
f> r. , . Inthis< f> istheangulardeflectionof
ternalmomentw hile \ p istheanglemeas-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
theouterend.Byusingthesecurves
weencoilsmaybewor edoutfor
ersofturns.F orfurtherdetailsthe
aperbyK roonandDavenport4.
ngsw herethethic nessisfa irly
entrationentersdue tothefactthat
rvedbar.Thisstress concentrationis
eterminedfor agiventhic nessand
gcurves givenfortorsionspringsin
II.
nd188apply onlytospiralsprings
heretheouterendis pinconnected
,ananalysismay becarriedoutusing
ee acte pressionsfordef lectionand
pressionsarerathercumbersomeand
eris referredtothepublicationby
E S S E S
ssesinspiral springsmayrunas
0poundsper s uareinchormorewhere
afactor. F ore ample, anordinary
femaybesub ectto lessthan5000cycles
dmuchhigher thanwouldbethecase
einvolved.W herefatigueconditions
mpleinthespiral springforthebalance
s rangeshouldbek eptwellbelow
material,stress concentrationcondi-
being considered.S omedataonen-
forspring materialsaregivenin
E CTIO N — CO I S I N CO N T A CT
asbeenbasedonthe assumption
toucheachother.Thiscondition,
manycases,as fore ampleinthe
wer springofaphonographwherethe
eF edern, Page73, 1938, V . D. I. , Berlin.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN C
de ahollowcaseasindicatedin F ig.
how nwounduponthearbor. Whenthe
againsttheinsideofthecase asindi-
redbysuch aspringmaybe
yasfo llow s : If Iisto ta llengthofspring
ess,thetotalsectionalareaof thewound
F ig. 190thisisa lsoe ua lto
ngthatthe coilsarewoundtightlysothat
dneglectingthe turnsconnectingthe
Thus,
acentco ilstouch, thenumberof
fd2givenbyE q uation401, thee -
hespringiso iled, ad acentturns
hic nessof theo ilf i lmandthisw ill in-
e uation.
whenthespringis unwoundas
n bethenumberof turnsof theunwound
ngtheturns connectingtheinsideof
arbor,
sDeliveredbyF latC o iledSprings , TheMainspring, S pring7,
publishedbyWallaceBarnesC o.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
area ofthewoundportionmustbe
on406in404,
deliveredbythespringin un-
ositionofF ig.190tothe unwound
ng
bethedifferencebetweennandn .
nsdeliveredbecomes:
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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C A L S P R I N G
ngthe woundpartofthespring
neglectedinthis derivationtheresults
8aresomewhathigh.Toobtainmoreac-
shouldbemultipliedwithacorrec-
ity.V aluesofthiscorrectionfactor are
f drumareaminusarborareadivided
usva luesofmassuggestedbyWallace
TableX X X III.
m between5and1.5areduction
thatca lculatedf romEq uation408of
aybee pected.
ssconcentrationduetocurvature
risusuallymadearound15 to25times
pringisw oundf romastrip% - inch
nd100incheslong.Thearbordiameter
ousV a luesofm
85
iameterof thecase2V 4inches. The
mberofturnsdeliveredfromthesolid
sh= M5, Z = 100, D,= 2. 25, ^ = . 375.
,
I, by interpo lationform= 2. 58, f c= . 76.
h, I, D., anddlinE q uation408
becomes19.6turns.Thismustbe mul-
tor fc— .76whichyieldsavalue
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
forthenumberofturnsdelivered.
sa largenumberof turnsthee ua-
and deflectionbecomesimplepro-
eincontactduring deflection.Where
allandtheouterendis clampedthe
ctionmaybecalculatedbymeansof the
dtightly,as inthecaseof the
relativelysimplee pressionsgiven
turnsdelivered.Becauseoffriction
thiscase adeterminationoftor ue
e angleornumberofturnsis rather
iscussedhere.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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da relativelylargeamountof
, a typeofspringk now nasthe" ring
onsiderationbythe designer1.Thisis
cationis onewherealargeamount
e,such asfore ample,draftgear
ringspringconsistsessentiallyofa
al surfacesandassembledasindicated
. Whenana ia lloadisapplied, slidingoc-
c-
ces withtheresultthatthe inner
heouter ringse tended.Inthisman-
niformdistributionofcircumferential
nnerandouterrings.Becauseof this
ofstressdistribution,thering springis
" T il R ingS pring , Mechanica lEngineering, F eb. , 1926, Page
csof theR ingS pring , A mericanMachinist. F eb. 14, 1924.
ra ftGearS prings—PastandPresent , R a ilua9Mechanica l
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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essentiallyasabar insimpletension
glyhighefficiency(consideredon the
torageperpoundof metal).A ctual-
nstressesatthe conicalsurfacesof
be aslightnonuniformityine uiva-
hereatensionandcompressionstress
iscase, thee uivalentstress—onthe
sheartheoryofstrength—w illbethesum
esofthesetensionand compression
a lthic nessof theringsisap-
nuniformityincircumferentialstress
eathic cy linderunderinterna lore -
stsprings,however,thisnonuniformity
ributionwillnotbe largeandhencethis
elativelyhighefficiency.O ntheother
at thedampinginthisspring isob-
ofacerta inamountofwearonthesliding
accordingto usualpractice.
gineeringandF oundryC o.
gandrotatingbloc for
springisutilized
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
C U A TI O N S
hysteresisloopforthering spring
romthisitmaybeseenthatonthecom-
igherspring constant(intermsofpounds
nedthan forthereturnstro e.This
sontheconical facesoftheringswhich,
dded totheelasticforcescaused by
foradecreasingloadaresubtracted
IN C H E S
-deflectiondiagramofringspring
sloopduringloadingandunloading
sa largehysteresisloopisobtained
energyabsorptionpercycle.
orpracticalpurposesof analysiseach
springmaybeconsideredassub ect
distributeduniformlyaroundthecir-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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f o rceF = / JV (w hennisthecoef ficient
actsinthedirectionshownwhen the
d,andin anoppositedirectionwhen
nded. TheseforcesN andF produce
cingonelementof ringspring
thering,althoughthere isatthe same
gtobendli eabaronelasticfounda-
owever,maybeneglectedfor practical
be assumedthattheringthic ness
meandiametersothatthenonuniform
nduetothethic cylindereffectmay
ngthe innerringofF ig. 195andas-
compressedsothatthefrictionforce
n,thetotalradialforceactingwill be
sina)w herea istheangleof taperof the
lloadp perinchofthe circumferen-
willbe thetotalradialforcedivided
eanradiusofthe ring.H encethisload
se uationbecomes
trengthofMateria ls, V anNostrand, SecondE dit ion, Part2,
cit . , Pnge236.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
pressivestresswillbe
reaof the innerring. S ubstitutingE q ua-
thespringduringthecompression
ngthecomponentsofN andF a longthe
c e
a = N ( s i n a + n co s a )
substitutinginE q uation412, thecir-
estressa,-intheinnerrings becomes
a
ducedtothe simplerform:
1 4)
f E q uation414,valuesofK are
for variousfrictioncoefficientsnin
sofF ig. 196. Wherema imumac-
tionshouldbemadeby usingE q ua-
calculatingthecircumferentialten-
ingsis used.Thisgives
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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sectionalareaofouterringandK isgiven
enotedthatEq uations414and416give
asafunctionof loadforincreasing
dicatedpreviously , toobtainthee uiva-
gs,thecompressivestressesdueto the
95shouldbeaddedto thecircumferential
dfromE q uation416.Thesecompres-
utedasfollows:F ortheouterring
fmeancircumferenceis obtainedfrom
insteadof r, . Thus
vs
tensionstress,fromthering
ation417.
ro ecteda ia llengthofcontactareaatthe
.192(whichlengthmaybe obtainedfrom
fthespring anditsgeometricalpropor-
ompressionstress< » inthecontactre-
)/2= meanradiusof innerandouterrings.
areaoverwhichtheforceN actsis
utinginthisthevalueofN givenbyEq ua-
becomes
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
ctionof thespring,theradialde-
firstbefound.F ortheinnerring the
ppro imatelye ua ltoacrm/ w hereE
The a ialdeflectionofthespring
betwotimes theradialvalueacrm/
r twoisusedsince therearetwo
enceifn,is thetotalnumberofin-
ngof halfthefullsectionbeing con-
deflection8< duetotheseringsis
a ldef lection8 duetoouterringsis
ro fouterrings. Thisw illalsoe ua l
to ta lnumberof " e lements inthe
stingofahalf innerandahalfouter
ations421and422,
° < + » c ) (4 23 )
givenbyE q uations414and416
ngthecompressionstro emaybee -
adP.
uation415orF ig. 196.
ccuringduringthe unloading
eanalyzedinasim ilarmannerbycon-
thedirectionof thef rict ionforcesF , F ig.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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3 55
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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L S P R IN G
stheload and3,thecorresponding
rnstro e,
edforconvenience inthe low ergroup
P,(return stro e)andtheload
tanygivendeflectionis obtainedby
uations424and425. Thisgives
fthespringconstantsfor thereturnand
spectively it isonlynecessary tota ethe
ivenva luesof / ianda. Thisistruesincethe
rtionaltothe respectiveloadsatany
A T IO N
applicationof theseformulasinprac-
ofthefollowingdimensionsastested
considered: A , = A , = . 584in.2, r(=4. 42
= 4 . 58 i n . ta n a = . 2 5 , a — - 14 ( a pp r o . ) ,
n , = 9 , n = 1 8 . T e st s o n th i s ri n g sp r in g i nd i -
on^ = . 12. F romF ig. 196, forx = 14
onK = . 095andK , = . 031.
dP , = 100, 000pounds, f romEq uation
thedeflection8is
. 0 95
tro eforthissamedeflection
pliedby theratioK , K . Thisgives
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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32, 700lb
compressionstro eis
eis:
00X -^ -= 10, 000lb/ in.
eouterringat P,= 100,000pounds
tion416,
iscase, f romEq uation414, theforego ing
ecompressionstressinthe innerring.
er,theinnerring areaA ,ismade
hasbeenfoundf rome periencethat
maybeused incompressionthaninten-
aftgearspringshavebeendesignedfor
tressof125,000poundsper s uareinch
aredwithacompressionstressof 210,-
inchininnerringsw henspringisso lid .
nproportionsof(he springareso
tedcontactlengthb, F ig. 192, is. 79- inch
ds, thenf romEq uation420thecom-
thecontactareais
onstressat= 143,000poundsper
valentstressintheouterringbecomes
900poundspers uare inch. Thisisa
wouldbeobtainedif thecontactcom-
glected.
dicationof loadsanddeflectionspos-
g. TableX X X IV , isuseful5.
w aterS tee lC o.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
tings
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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I G
essentiallyofa relativelywide
blade,whichhasbeenwoundtoform
sectioninF ig.197.Beforewinding,
howninF ig. 199. A f terw indingbut
springwill haveeitheraconstant
eandavariableco ilradiusasshownin
ngoperationusuallyemployedwill,
ngeintheheli angledistribution
scondition.W henloadeda ially
behavesessentiallyas acurved
ipalstresses beingtorsional.
t
ring
cation
yin
gthese
ng:
nufac-
yfric-
pring
h de-
otect
d.
tially
ble
e
rthe
gth.
n
the
o ilsabovea
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
atbeyonda certainloadsomeofthe
portingplate, increasingthestiffness.
stressdistributionthe thic ness
aperednearthe innerendof theco il .
L I X A N G E
calculatestresses anddeflectionsin
entofthecoil may,forpracticalpur-
ntiallyasaportionof ana iallyloaded
coilradiusandhavinga rectangular
dthusneglectsfrictionbetweenad-
pingsuspensionforM-5tan
certainsecondarystresseswhicharediffi-
fthesestressesarisefromthe factthat
197 ingenera lw il lnotbea ia lasas-
ut willbedisplacedfromthea isof
toadditionalstressescausedbythis
certainstressesk nownasconeandarch
maymodifytheresults1.To simplify
iscussionofvolutespringcalculationsincluding theeffect
thic nesssecarticlebyB. S terne, " C haracterist icso f the
rna lS . A . . , June1942, Page221. Seea lsopaperbyH. O . F uchs,
ryS tressesinV o luteSprings, T ransactionsA S M , July1943, Page
gnMethodforV o luteSprings , JournalS . A . . , S ept. 1943, Page317.
arcgiveninarticlebyB. S terne, TransactionsA . . M. . , Ju ly
P roceedingsS ociety forE x perimentalS tressA na lysis.
ults ofstrainmeasurementsandeccentricitydeterminations
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I G
lutespring
eeheli anglewillfirstbeassumed.
ehe li anglewillbetreated.
eferringtoF ig. 200w hichrepresents
fthe bladeofavolutespring,the
atedbytheline A B,abeingthefree
onstant).Inthistheordinaterepresents
nterline,andtheabscissathe distance
moderateloadsbeforetheouterend starts
engthwillberepresentedby the
eatheavy loadsw henaportionA C of the
developedlengthisrepresentedby
chwillbe calledtheinitialbot-
uter coilj uststartstobottom,the
sticwillbeastraightline asindicated
sload, astheco ilsbottom, thespringbe-
ndthe load-deflectioncharacteristic
ttomingloaditwillbe assumed
O N G CO I —
tofcenter lineofvolutespringfor constant
theavy loadsspringbottomsbetweentoandr
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
angle 6fromthebuilt-inouter endA
pesentedappro imate lybyaspira l. This
eformost practicalpurposes.Thus
)
beginningandend,respectively,ofthe
g( ig. 197)andnisthenumberof
ahelicalspringofnarrowr ec-
erethelongside ofthesectionis paral-
dw herethew idthb( ig. 197a)isgreater
ecaseinvolutesprings,is givenwith
fo llow inge uation :
g/ i= thic nessofblade, b= bladewidth,
coilradius.
rementofthedeflectiond will
pliedbydd/2w . Hence, usingE q uation430
usatangle6( ig. 197/7).
m ingof theouterco ilmaybee pected
heli anglewhenthesloped / dsatthe
ig. 197)ise ua ltothetangentof the
npracticalvolutesprings thetangentof
8.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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I G
entaccuracybeta ene ualtothe
gds—rd6andtanoc= a , thiscondit ion
uation431, andsubstitutinginEq uation
oadP,for constantfreeheli angle
eacise pressedinradians(degreesdivided
fdeflectionwillbe discussedfor
heretheloadsarelessthan initialbot-
f lection
ng
eyaregreater.
ethedeflection8for loadsPthat
ottomingloadP E q uation428and
itutingthevalueof rgivenbyE q uation
tofdeflectioninsmallangled6 becomes
) ^ ' -
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
helimits0= 0and6 — 2irnwhere
oils,thetotal deflection(forloadsun-
ad)maybe e pressedas
, ) ,w h en P < P , ( 43 5
uation433, and
tedasfunctionsof fi- (r — ri) / r in
andr, inthise pressiondependonthe
ig. 197). Wherethese latteraretapered
three-fourthsturnateachend isfre-
ndingfactorK, asfunctionof$
tive,butthisfiguremaybe changedas
vailable.
natany loadPlessthanP, the
ation435maybeused, thevalueK 1being
ig. 202. ItshouldbenotedthatE q uation
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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C
caseofa variablefreeheli anglepro-
oils occurs.
he loadP isabovethebottomingload
constantf reeheli anglemaybecon-
woparts, e. g. , apart8 ( ig. 200)dueto
tomedportionA Cofthespring and
lectionof thef reeportionC D. A ssum-
ttomedtoaradiusr andangle6 asin-
nf romthecondit iond / ds— aatthe
oceedingasbeforethefo llow ingisob-
s p ri ng i nd e a t r~ r .
a n d 6 = 6 i n E q u a ti o n 42 8 , th e a ng l e
asfo llow s:
s p r i ng i n de a t r = r , ,.
ttana= a , thedeflection8 isgiven
thisandintegrating,
40)
33, 437and438thise uationmaybe
eratioP/ P , asfo llow s:
obtainedbysummingupthe ele-
tweenthe limits6= 6 and6=2w n. Thus,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R 1N G
/b)
gandaddingtothe valueof8
1, thetota ldeflection8becomesforP>P1:
2 n r a ( - l - — )( 44 3)
f theratioP / P ,.
>
enasfunctionsofP /P inF ig. 203. B y
ofF ig.202,thedeflectionatany load
In thismannerthecompleteload-
fthespringfor aconstantinitialheli
hallcoils bottomtheprocedureis
uation437, byusingthee pressionforP ,
3,
swill occurwhenr,= r(and
o n 44 5 a nd t a i n g / = ( r — r , ) / > „ t h e fi n al
es
oad P,isobtainedbyta ing6 —2n-n
ntegrating, usingtheva lueof rgivenby
ogivesthe differencebetweenfreeand
,m U ( - -r ) (4 47 )
ei angle intermsof is
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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I G
he ightsof thespringarek now n,
ians)maybeca lculatedf romthise ua-
hich initialbottomingoccursisob-
435ta ingP= P1. Thisgives
mateload-deflectioncurveforany
/ t>
indingfactorK: f romloadratioP/ P ,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
tialheli angleitisonlynecessary
nd8, f romEq uations433, 446, 447and449.
awnbetweentheoriginandpointA
ig.201).PointB(representingP,and
yasmoothcurveconcaveupward.F or
d,additionalpointsonthis curvemay-
uation443.Thustheload-deflectiondia-
simply.
calculatethe stress,theformulasfor
ngswillbe used,modifiedtoapply
mentionedpreviously,thesestresses
asfirst appro imationsbecauseaddi-
uallywillbepresent.
he loadP islessthanthe initia lbot-
stressw illoccuratthema imumradius
imatee uationforarectangularbar
hthelong sideoftherectangleparal-
discussedprev iously4, thema imumshear
ecomes
s p r in g i nd e a t r = r ( l. W h e r e h/ b i s sm al l ,
ecomparedto thethic ness,theterm
nasunit) . Thisgives, appro imately ,
he loadisgreaterthanthe init ialbot-
umshearstresstw illoccuratr= r
tw hichbottomingoccurs. ThusEq ua-
tt ingcv= cfw herec ~ 2r / h=spring
g i ve s
P i
.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I G
n 44 5 , c = c \ / P JP , b y su b st i tu t io n o f
thestressatany loadP(forP>P, )becomes
curs,theloadP = P,.Usingthevalue
tion446inE q uation453thepea stress
oadP , maybee pressedby
ginde atr= r, .
valueofP . , givenbyEq uation446
IN C H E S
000
ectionandload-stresscurves,
tomingloads
byE q uation433, thestressr2forf ina l
mplee pression
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
eeffectofbar curvature.Ifitis
ct wherestaticloadingispresent,the
singthesame e uations(450to455),
sionintheparenthesisofthe numerator
ation450, todothisc ista eninsteadof
or). F ormostvolutespringsthisw illnot
ference,however.
Design—A sane ampleof theuse
cticaldesigna volutespringwitha
gleandwiththefollowingdimensions
2% - inch, r, = lV 4- inch, h~V 4- inch, b=5-
e ight—7Mi- inch, c — 2r / 7i= 20,
numberofactiveco ils, / 3= (r —r()/ r =
bethedifferencebetweenthe
hus8. , = 2% - inch. F romE q uation448
forsteel,thepea shearstresswith
f romEq uation455,
5 3 1X 1 1 1 , .
P,is, fromE q uation433,
3X . 0531X . 969
a lbottomingloadP , fromEq uation450is
s i n .
thef ina lloadP , isfound:
nitial bottomingloadP1isgivenby
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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distributionofhe li angleas
usz values
heva lueofK , = . 47givenbyF ig. 202for
2 Jr X 4 X 2 . 5X . 0 5 3 1X . 4 7 = 1 . 57 i n .
bottomingloadP.,willbethe difference
height,i.e.,8.,= 7.5— 5= 2.5-inch.
dP , a loaddef lectioncurvesim ilartoF ig.
sparticularspring.A similarload-
d,since thestresswillvarylinearlywith
gloadP,.Thestressatany loadbe-
lculatedf romE q uation453. Inthis
-stressandload-deflectiondiagrams
eobta inedforthiscase. F romthesedia-
deflectioncurvemayalsobe plotted.
E L I X A N C E
y,theprocessofmanufacturein
anglewhichincreasesfrominsideto
mountof thisvariationinheli
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 391/462
itionsobtainingduringthe cold-setting
of winding.A nanalysisbasedon
riationoffree heli angleislinear
rradiushasbeencarriedout by
nmaybee pressedby
theheli anglesatradiir, r, andrrespec-
lativevaria tionof theheli anglemay
berzw here
ept. 1943, Puge317. Thisa lsodiscussesdesignofpresetting
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I G
constantf reeheli angle isobta ined
matelyconstantbottomingstresswille ist
ocurvature). R atiosof ( / a areplotted
. 205, forthevariousvaluesofz. Thisgivesan
nin freeheli anglewithradius,for
sumingelasticconditions,similarload-
obtainedforall springswithgivenval-
actualload-deflectiondiagrammaybe
ven" typecurve bycertainsca leratios.
rvesforva luesofz= 0, V * , % and1have
hs andaregiveninF igs. 206, 207, 208, 209,
curvesaredraw nforrj r e ua lto . 3, . 4,
gtosprings withsmall,mediumorlarge
diameters.Initialandfinalbottoming
circlesoneachcurve.The abscissas
R A T IO f
forz—% ,
R A t IO j
forz=
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
in termsofthema imumpossible
atesintermsofa loadP . , w here
d z is g i ve n b y E q u a ti o n 45 7 . V a l u es o f K 3
curveofF ig. 211forvariousvaluesofz.
o tt e d in F i g . 21 2 a ga i ns t < 7 = r , /r . F o r
referredtothearticleby F uchs5.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I G
d-deflectioncurveforanyspring
dq theordinatesofthecorresponding
pliedbyP , andtheabscissasbyS b. In-
necessary.
einitialbottomingloadP,maybe
,
s
on433usinga= a . F ortheusualspring
arstresst atwhichbottomingstarts
ation450ta ingP= P1. Thisgives
/ h a t r= r . I f t he v al u e of P , g iv e n by
utedin thistheinitialbottomingstress
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 395/462
S P R IN G
lectedtheterm unityinthe
ressionisdropped,obtainingthesimple
albottomingloadisgivenby
ga= a(andusingr, insteadof r,. . Thisgives
t )
rend(r= r, )a tf inalbottomingload
55usinga, inthiscase insteadofa :
e atr= r(,
lectedthis reducestothesimple
sane ampleof theuseof these
ngr = 3. 75, r, = 1. 93, fo= 7. 50, h—
10 lb, s . in. , a - - . 076, a , -= . 060,
mEq uation457. F romF ig. 211, forq — . 515,
, andf romF ig. 212, ^ = . 053. UsingE q uations
)3 X . 0 6 0X . 0 5 3 X 5 . 5
0 X 5 . 5 = 4 . 95
nobtainedbyusingthe curvesfor
i g s. 2 0 7 an d 2 08 ) f or q = . 5 d oe s n ot a m ou n t
yload,whichisverysmall compared
enceeithercurvemaybeusedforcon-
teload-deflectiondiagram.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I G
ngstresst ,neglectingcurvature,
. Thisgives
rvaturethisstressismultiplied
4asafunctionof rl/ r
w he r e c = 2 r / h = l 8 . 7. T h is g i ve s
= 94, 000poundspers uare inch.
mingstresst,,neglectingcurvature,
. Thisgives
urethe stressthusfoundismul-
reC i= inde 2rf /7i=9. 7. Thestress
137,000(10.7/9.7)=151,000poundsper s uare
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G A N D M O UN T I N G
bberspringsincludehighenergy
dthe possibilityofformingincom-
sereasonssuchsprings havefound
brationisolatorsformachinery,
tomobileandaircraftengines,mount-
drichC o.
ringsappliedtocompressormounting
ibleelementsincouplings,andmany
of theapplicationof rubberspringstoa
ountingisshowninF ig.213.
t isso largethatane tensivetreatment
ntingsis beyondthescopeofthis boo ,
hatatleast thefundamentalprinciplesof
should beemphasizedthatthee -
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
fthebehaviorofrubberunderstress is
he moreusualspringmaterialssuchas
horbronzeandthe li e . C onse uently ,
ringsby availablemethodsisatbest
mongthereasonsforvariationsbetween
haviorofsuch springsarethefol-
shearmodulimayoccuramongdifferent
houghofthesamehardnessreading.
ionspringsofrubber,frictionbetween
varythroughwidelimitsthusaffecting
hererubberpadsarcbonded to
swill notoccur,however.
micmoduliof elasticitywilldiffer.
sare deflectedbyrelativelylargeamounts,
oredifficult tocalculateaccurately
ssanddeflectionof varioustypes
stbe treatedafterwhichthefunda-
inthechoiceof fle ibilitiesforsuch
d.
lyusedtypes ofrubberspringsis
oc showninF ig.214whichrepresents
r compressedbetweentwosteelplates.
esdevelopedatthesurfacesof thebloc
iffnessofsucha springisfarlarger
dtotheothertwodimensions)than
S P R I N G
n
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
berisnot bondedon,thestiffness
differentamountsoffrictionbetween
usif theplatesarelubricatedwith
muchlargerdeflectionsmaybee -
wouldbethecaseif drysurfaces
L A S TI CI T I N C O MPR E S S I O N ( L B / O , IN .
ourna lo fA ppliedMechanics. 193 .
lasticityofrubberincompressionas
dnessnumber
uncertainamountoffriction present,
lculations ofdeflectioninsuchcases
yappro imateonly.
ethodwasdevelopedbyS mith1,
rageofaconsiderablenumberof tests.
percentagedeflectionofslabofrubber
sectiona lareaofslab, / ff—ratioof
thic ness, £ —modulusofelasticity
veragecurveofthevariationof modulus
e" R ubberMountings —J. F . Dow nieS mith, Journalo f
ch, 1938, PageA 13 and" R ubberS prings—S hearLoading
urnalo fA ppliedMcch., Dec. , 1939, PageA 159. Otherarticles
" R ubberSprings —W . O . K eys, Mechanica lE ngineering. May ,
lasticB ehavioro fV ulcanizedR ubber —H . H enc y , Transactions
45: " R ubberCushioningDevices —H irschf ie ldandP iron, Trans-
g. 1937, Page471 " TheMechanica lC haracteristicso fR ubber —
ransactionsA S M . F eb. , 1939, Page149. " UseofR ubberinV i-
. H . H ull, Journa lo fA ppliedMechanics, S ept. 1937, Page109.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
nwithdurometerhardnessis givenin
ulusofelasticityo f rubberhaving55durometer
ectionofa1-inchcubeof 55durom-
maybeta enf romtheaveragecurveof
k now n. Thenanempiricale pressionfor
ofrubberslabsis
ampleof theuseof thise uation, assum-
ometerhardness,asectionalarea
thic ness= linch, andload=10, 000pounds,
40 3 0 6 0 60 1 0 0
F H N CH C UB P R C N T
eflectioncurvefor55durorubber
2. F romF ig. 215themodulusofe las-
bberisE= 310poundspers uare inch.
erE= 430poundspers uare inch. The
00/ 32=312poundspers uare inch.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
f lectionn o fa1- inchcubeof55durometer
be57percent.Using thesevaluesin
centagecompressionat10,000poundsload
therloads,aload-deflectiondiagram
R S P R IN G
earinvolvesnovolumechange,fric-
contact suchasmayoccurin com-
entand betteraccuracyincalculation
bbershearspringswhichhaveessentially
17consistingoftwo rubberpadsbonded
sed forvibrationisolationandmachine
icationisshow ninF ig. 213.
calculatedasfollows:A ssuming
diansisproportionalto shearstress
ubbershearspring
tothemodulusofrigidity G(accord-
ptiongivesthebetteragreementbetween
theshearstress t=P/2A and
areaof eachpad.Thefactor2is used
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
etw opadssi/ b ectedtothe loadP .
greestheva luegivenbyE q uation469
).Themodulusof rigidityGdepends
0
R I GI DI T , L F A / Q . IN .
gidityasfunctionofhardness
softherubber andmaybeestimated
18.
8of thespringofF ig.217,ifthe
f thepadand7=theanglefiguredf rom
e (saybelow20degrees),for
ngentmaybeta ene ualtotheangle .
nd470the deflection8thenbecomes
prings aremadeinitiallyobli ue
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
esofF ig.219,thepositionof therub-
ingindicatedby thedashedlines.By
anner,itispossibleto introducead-
sesduringdeflection.Thesestresses
ubberspring
ds.
om-
rload
ndpointofthe adhesionofthebond
esteel.If thefinaldeflectionleaves
lyflatsothatc/his notlarge,asisusually
ttoogreat, itmaybeshownthatEq ua-
enoughaccuracyforpracticalpurposes2.
S H E A R S P R IN G
ype ofshearspringconsistsessen-
dedtoasteel ringontheoutside and
sideasindicatedinF ig. 220a. A load
isasshown. Theshearstresstatany
sameradius( ig. 2206)theslope
negativevalueof theshearangle.The
inceydecreasesw ithincrease inr. Since
a ltor/ G(appro imate ly )andthederiva-
entof theshearangle , usingEq uation472,
nalo fA ppliedMechanics, December, 1939, PageA -159.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 404/462
G 3 85
4 7 3)
henusingthek now nseriese pression
e,
= r(andr=r , thetotaldef lection
s(w hereb/ r < . 4)thetermsof this
be neglectedwithoutseriouserror.
imationthefollowingformula:
bber springofconstantheight
inga longa is
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
thic nesshofacy lindricalrubber
onaltothe radiusr,asindicatedin
twillbeconstant andbetterutilization
ubber
stress,
verse ly
ained.Thisfollowsfromthee uation
tutioninthisformula , PPP
ssatouterradiusr , F ig. 221. S incethestress
heshearanglev = t/Gwillalsobe con-
e
fort
essthan20degrees)wherethe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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G
ppro imatelye ua ltotheangle , thede-
tlyaccurateformostpractical uses
T O R S I O N S P R IN G
nthiscasethethic nesshof thespring
222w hileamomentMisassumedtoact
sindicated. Theshearstresstatradiusr
umshearstressw illoccurw henr=r(and
gulardef lectionaboutthespringa is
sion
essh
ressacting ontheelementalring
2b,then
entialdeformationoftheouter cir-
twithrespecttothe innerisdrtan 7
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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L S P R I N G
t/G.Dividingthisby ryieldstheele-
nd(). S ince i= t/ G, byusingE q ua-
hG, thise pressionbecomes
( 48 3 )
ntseriesasbefore,theangularde-
utinglimitsthise uationreducesto
-77)l^+ - - - J- (484)
hatthis seriesconvergesrapidly
lywillbesufficient.This gives
cylindricalrubberspringis madeso
g. 223variesinverse lyasthes uareof the
uation480theshearstressw illbecon-
thedepthista ene ua lto
f rubberatthe insideradiusr=r(, F ig.
uation480thestressbecomes
e,thentheangular deflectiondueto
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G 3 89
hadedinF ig.223bwillbe,as before,
andr= r , thetotalangulardef lection
4 8 8)
sion springwithconstantstress
of theanglee ua ltotheangle(w hichis
ectionis nottoolarge),
ingacy lindrica lrubberspring(60durom-
oadedasinF ig. 222underator ueof
onsasfo llows: r = 3 inches, r(=2
mE q uation481thema imumstressat
dspers uare inch, is
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
i n.
^
er, fromF ig. 218themodulusof rigid-
uare inch. Usingthef irsttermof the
4theangulardeflectionbecomes:
10.2degrees
uation484isused, thisresultw il l
nt.
onstantstresstype( ig. 223)with
2inchesf romE q uation487thestressisthe
ously . F romE q uation489assuming
ulardeflectionat10,000inch-pounds
ees
og, ,. 1. 5—. 405.
88wouldgivearesult e ualto17.2
entgreaterthanthatfoundfrom
t asmentionedpreviouslythe
houldbeconsideredappro imateonly
ndstressesarelarge.
S TR E S S E S
fdatain theliteratureonallow-
rubbersprings. Hirschf ie ldandP ironJ
gstressesinrubbershear springs(and
nds)belimitedto25 to30pounds
re ceptincasesw hichhavebeen
sostatethatunderfavorablecondi-
hstandconsiderablyhigherunitloads,
ndspers uare inchhav ingbeenused.
, A ug. , 1937, Page489.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
ywiththoseofK eys4whostatesthat
dspers uare inchareusedonmeta lto
suggestedthatthethic nessofashear
anone-fourthofthe smallerofthe
esultsoflong-timecreeptests on
terhardness.W hentestedinthe form
g. 217, at50poundspers uare inchshear
reep ofabout.017-inchfor5/16-inch
gtoashearangleof .055-radian)after
rature. A t140degreesF ahr. , a tthe
epwas.15-inchi n5/16-inchor.48-radian
thertestson1- inchthic rubbersand-
ound(38durometer)at 40pounds
essand atnormaltemperatureshowed
hearangleof .055-radian)after500
p-timerelationwasappro imately
showedthatthecreepmaybe greatly
peratureandthatwidevariationsmay
dsareused.The necessityfork eep-
sesife cessivecreepistobeavo ided
these tests.
ainsonrubbercompressionsprings,
andP iron1suggestama imumcompression
ntof thef reethic ness, andanupper
se ua lto700poundspers uare inch
aushalter alsosuggeststhatthede-
ringsbelimitedto 15to20per cent
p.Where fatigueloadingisinvolved(as
gsforcouplingsin electricmotorap-
usstartingandstoppingisinvolved)
ntagecompressionsshouldbeallowed^
chesforusein shearorcompres-
esirabletok eepthema imumthic -
possiblebelow1inch.This isdonein
curingduringvulcanization.Ifmore
stac o f rubbersandw ichesinseries
ring,May,1937,Page347.
, F eb. , 1939, Page157.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
s,caremustbeta entoavoidinstabil-
hiscanhappennot onlyincompression
prings.
D S H O C I O L A TI O N
sbeenconcernedprimarilywith
stressesanddeflectionsof rubber
s.H owever,thedesignerisalsofaced
ingwhatfle ibility,ordeflection,is
venmounting.In doingthishemustbe
plesforvibrationandshoc isolation.
tantofthese principlesparticularlyas
ntingsformachinery,andmilitary
brie fly considered6. F orpurposesofdis-
dedinto twoparts:
n.
shoc .
redas avibrationwhichlastscontinu-
onsideredasamotionwhichdies
me.
ion—Thisk indofv ibrationisencoun-
atingelectricalmachineryasa conse-
herotatingparts. Itmayalsobeset up
lternatingcurrents.A furthere ample
aircraftstructuresduetoengineun-
plosionforces7.Inmachinerymount-
nbalancedreciprocatingmassesis
e.
mountingforsteady-statevibra-
siderthevariouspossiblemodesof
anycasesthesystemcanbe simplified
sasingledegree-of-freedomsystem,con-
e spring-mountedmassonavibrat-
nF ig. 117ofC hapterX III) . Thisisthe
scussionofthegeneralvibrationproblemis giveninMechani-
enHartog, McGraw-Hill , S econdEdit ion1940, andV ibration
—S . T imoshen o, V anNostrand. SecondE dit ion1937.
oche, Mechanica lEngineering, A ugust1943, Page581presents
cussionofrubbermountingsforaircraftand militarye uip-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
taininstrumentmountingsinaircraft.
anatura lf re uencygivenby
ofthe massundergravity,inches,
cyofsystemincyclespersecond. Inprac-
stantoftherubbermountingshould
t / willbeconsiderablylowerthanthe
ationofthesupport.F ormachinery
e ua lto1/ 3to1/10thenormaloperating
condare usedinpractice .
ninvibrationrealizedby arubber
atthesupport( ig. 117, ChapterX III)
namplitudegivenbya sinw tw heret=
f re uencyof thee ternalv ibrationin
ectingdampingthedifferentiale ua-
y betweenthemassandthesupport
..
= m a t i na t( 49 1)
springconstantof spring. F orasteady -
ntothise uationis
fthespringsupportedmasswill be
q u a ti o n 49 2 f or y a n d ta i n g < , > / < » „ = / / / n
dR ubberC ompoundsinIsolatingMachineryV ibration -—B . C .
M , A ugust1943, Page619.
Page41.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
f/ f approachesunity , thise uation
chcasestheeffectof damping(which
ation)is veryimportant.H owever,
danappro imationforf re uenciescon-
sonance,andinsuchcases,it maybe
motionof themasshasbeenreduced
scomparedwiththeamplitudee peri-
mounted.Theaccelerationstowhich
willa lsobereducedinthesameratio . Thus
ngineering, A ug. 1943.
ficiency intcimsof f re uencyandstaticdef lection
tionf re uency, f—30cyclespersecond,
so chosenthatthenaturalfre uency
d, f / /n= 3andf romE q uation493, the
emassw illbea / (32—1)— aa/ 8. This
vibrationamplitudeandacceleration
obtainedwitha fle iblemounting.
sedonE q uations490and493,and
beusedtoestimatethe percentagere-
edbythe useofrubbermountings
ction8 .Theordinatesofthis diagram
e uencyincyclesper minute,while
estaticdeflectionof themounting.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 414/462
G
toa definiteamountofreductionin
theshadedarearepresentsaregion
plitude.Thus,fore ample,ifthe
say,800cyclesper minuteandthestatic
eweightofthe mountedapparatus,
f80percentnormallyw ouldbee -
mphasizedthatEq uation493and
basedon theassumptionthatdamping
mostpracticalpurposesthis willprob-
lycloseto actualconditions.H ow-
mpingmustbeta enintoaccount.The
isproblemis toassumethatthedamp-
onstantctimestheve locity . Thisis
ermcdy / dttothe le f tsideofEq uation
enbesolvedfor steady-statecondi-
ctualtests,however,showthatthein-
s afunctionoffre uencyandinmany
pro imatelyasinverse lyproportionalto
ercompounds1 . Inaddition, ithasalso
usofelasticity ofthematerialand
asusedinEq uation491may insome
.Thisresults,forsomerubber
bledeviationbetweencalculatedand
vesbasedonconstantvaluesofc andk .
oseagreementbetweentheoretical
cand k values,andtestcurveshave
range offre uencies8.
onsanalternatingload P=Pn sin
tedmass,thesupportorfoundationbe-
previouscaseconsideredwherethe
vbrate . A ne ampleofsuchanapplica-
ountedonrubberbushings,theforce
alternatingload inq uestion.Inthis
wsthat ifdampingisneglectedtheload
ionisgivenby
F uchs, TransactionsA S M , A ugust1943, Page623 paper
tion byS . D. Gehman, Journa lA ppliedPhysics, June1942, Page
ynamicalP ropertiesofR ubber byC . O . H arris, JournalA pplied
-132giveadditionaldata.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
tthereductionin vibrationfortheno-
thesamee pression(/ V / nl)—1asthat
v ibratingsupportinE q uation493. If
naltovelocityis assumed,ananalysis
sthattheratioof theforceP (transmitted
ssedforceP isgivenby
tngfactor, cc= 2V n fcdef inedas
. Insomecasesva luesofc/ ccaround. 02
butthis mayvaryconsiderablyfordif-
8.A curveshowingtheeffectofdamp-
(i. e. , ra tioP , / P ) isgiveninF ig. 2257.
ffectwithno dampingpresent,
calculatedontheassumptionofacon-
hiscurve alsoindicatesthat,forcon-
sesthetransmissibilityabovef/fn=
wthis value.Becauseofvariationinc
dtrue inallcasesparticularlywhere
cyisinvolved12.Thusforaneffective
ibilityshouldbechosenso astoobtain
above1.4.Ifthemountingis madetoo
ructuremaynotfunctionsatisfactorily.
ustusuallybereached.V aluesofratios
igherhavebeenusedintan anda ircra f t
notedthatastructuresuch asanair-
fferentfre uenciesofvibration(dueto
or ue, fore ample)w hilethesemay
ngeatvarious speeds.Consideration
,loc.cit..Page87.
: u c hs , F o o t no t e 10 .
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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G
nciesisnecessary if resonanceande -
avoided.
hisproblemisparticularly important
mountingsforprotectinge uipmentin
omthesuddenmotionsresultingfrom
epth charges,orenemyaction.
trumentina shipmaybefle ibly
n spiteofthetransientmotionscaused
T UR I I N G F H O U i N C TT O N A I U A l F M O U N C T ( : h > )
ngineering, A ug. 1943.
ampingontransmissibilityatvariousratios f/fn
uencyandnatura lf re uency
Ingeneraltheimpacts ore plosions
scillationscombinedwithhighfre-
uchloweramplitudes,thefre uencies
dbythestructuralcharacteristicsof
ghf re uencymotionsmayresult in
uipmentrigidlyattachedtothe
iblemountingsthedamagingeffect
escanbeentirelyeliminated.H owever,
alsothelowfre uencylargeampli-
ountingssothatthemotionacrossthe
duringtherimeofthe transientto
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
emotionacrossthe mountingise -
achcasemust beconsideredwith
tionthatise pectedandthedesign
ble,forinstanceatlow speeds,two
beemployed.Thefirstis tointroduce
rmeans,whilethe secondistouse a
d-deflectioncharacteristicsothatthe
withincreaseddeflection.S uchmount-
anbe obtainedbyusingstopsorother
cillationsoccuratagiven fre-
nancew iththenaturalf re uency for
tudewilltendtobuild up.A sitbuilds
hecurvedload-deflectioncharacteristic,
ncreasesandwithit thenaturalfre-
throwthesystemoutofresonanceand
hemountingmust,of course,bede-
re uiredcharacteristicsatthegiven
untingsforagivenmachineit isde-
ethemin oneplanecoincidingwith
ismannerthetendencyofdifferent
mecoupledis reducedwhileatthe
yisobtained.
ibil ityhasbeendeterminedforagiven
ngsmaybedesignedbyusing the
eofthemountingmust besochosen
ntherubberare avoidedundertheanti-
tudeofmotion,whileatthesametime,
tismainta ined.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I
R A G CA P A CIT O F V A R IO U S P R IN G
factorsmustbeta enintoaccount
efora givenapplication,totheprac-
mountof energywhichcanbestored
ofprimaryimportance.Thisis true
anddeflectionaregiven,whichmeans
agivenamountofenergy.Thisis the
edesignoflandinggear springsforair-
hespringsmustbeable toabsorbthe
ssof theplanefallingthroughacertain
presentchapter istocomparevari-
ashelical,leaf,cantilever,etc.,fromthe
agecapacityperunitvolumeofmate-
imumstress.Thiswillgivethe de-
mumvolumeofspaceneededforagiven
lumemaybemuchgreaterdepend-
tness).Itmaybenotedthatthis ap-
omewhatdifferentfromthatofChapter
fspring(i.e.,thehelical)wasdiscussed
lspaceoccupied.
O N - BA R S PR I N G
edas anidealspring,consisting
funiformsectionsub ecttoana ia l
thebarisloadeda ia lly, thestressdis-
is uniformandforthisreasonthis
umconditionfromtheviewpointof
geperunitvolumeofmaterial.If Iis
ss-sectionarea,thestressa willbeP/A .
a/ ,whereE isthemodulusofelas-
e totalelongation8willbeal/ .
ualtothearea undertheload-deflec-
ension-bar springtodistinguishitf romthehelica lten-
nownasatensionspring.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
nergy17= %P8.This conditiongives:
vo lumeofmateria l.
staticloads, thetensiony ie ldpo int<rv
ngstress. Thecriterionofenergy
thereforebethevalueofU when
theotherhand, if thespringissub ect
ng,thestress, attheendurancelimit
ningenergy-storagecapacity.Because
chisusually presentneartheendsof
ped,itisals necessarytointroduce
nfactor2.Ifthis fatiguefactorisK t,
atfor variableloadstheenergystored
be
tbecauseofthepresenceof stresscon-
ragecapacityforrepeatedloadingwill
dealvalue(whichwoulde istifno
present).
S P R IN G
Thisspringconsistsessential-
frectangularprofileandconstantthic -
V I). F orsmalldef lections, thedeflec-
sing thenotationofChapterX V I,
by thefollowinge uation:
factorshavebeendiscussedinChapterV I.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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R A G CA P A CIT
heenergystoredmaybee pressedas
this, andta ingthevolumeV of thema-
hl, theenergyUbecomes
staticloads, thema imumenergystored
e ua ltothey ie ldstresso- w il lbe , from
uation496, it isseenthatthisva lue
ergywhichmaybestoredin theideal
samevolumeatthe yieldpoint.H ow-
t, whenthee tremefiberisstressed
tileverspringwillstillhavea consider-
lyieldingoverthecross sectionoccurs.
d energybyafactorofabout twoto
ethetension-barspringdoesnot have
9mentionedpreviouslyispessimistic
gecapacityofthesimplecantilever
thisreason,probablyabetter basisof
sis theenergystoredata loadproduc-
thesectionat thebuilt-inendofthe
aybemadeasfollows:
gulardistributionofstresse ists
completeyielding.Thismeansthat,
tressisaconstanttensione ua ltothe
nthecompressionside, thestress3ise ua l
angularsection, thee ternalbendingmoment
ngsteelsthe stresswilltendtor iseaftertheyieldpoint
agramofF ig.61).H owever,theassumptionofaconstant
presentpurpose.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
followingrelation:
of load50percentabovethe value
chisbasedona linearstressdistribution
oad-deflectioncurveup tothisload
rbecauseofyieldingeffects.H owever,
lbe appro imatelylinearandfor
onedirection,the energystoredwill
fromE q uation500,usingthehigher
msreasonabletousethe lattere uation,
condition,asabasis forcomparison.
ulatedfromE q uation503andsubsti-
0thee pressionforenergystoredbecomes
for theidealcase.( q uation496).
erethe loadisvariable , thestressgiven
bemultipliedby afatiguestrengthreduc-
e intoaccountthestressconcentrationatthe
r.The storedenergyattheendurance
threductionisassumedfor the
t inasimpletension-barspring,the
ntheformerforfatigueconditionswill
er1.Thereasonfor" thismaybefound
verspringof rectangularprofileonly
totalvolumeofmaterial(i.e.,that
ub ecttoanythingapproachingthema i-
ss,inspiteofitsrelative inefficiencyin
springstill findsafieldof useparticu-
pringmustfunctionasa guideinaddi-
rengthreductionfactormaybeconsiderablylessin the
alvaluedependingonthe design.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R A G CA P A CIT
onfunction.
eafS pring—Theefficiencyof utiliz-
antileverspringmaybeincreasedby
ngularor trapezoidalformsothatthe
ngthof thespringwillbeappro imate-
nditionintheusualtypeof leafspring
ngularprofile,thedeflectionattheend
henotationofC hapterX V I,
hesameas thatgivenbyE q uation499.
hestoredenergymaybee pressedas
E q uation507andta ingthevolumeV = =
mes
staticloads, theenergystoredw henthe
y ie ldpo inta is
is aboutone-thirdthatforthe
o int, Eq uation496. F orcompletey ie ld-
spring,theloadwill beabout50per
enbyE q uation509andforthiscondi-
springofrectangularprofile)the
maybeassumedtoincreaseroughly
Thisgives
gecapacityisabout25 percentbelow
lcase.
fatigue loadingtheenergystoredinthe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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L S P R I N G
larprofileis
strengthreductionfactorattheclamped
va luesofK f , thisvalue isappro imately1/ 3
case.
S I O N S P R IN G
ssumingahelica ltorsionspringof rec-
oaconstantmomentM( ig. 178), the
uation357,
ependsonthespringinde tota e into
edueto curvatureofthebar.F rom
gulardeflectionof theendof thespringis
scasewillbe halftheproductof
.Usingthis e uationtheenergybe-
nbyEq uation512inthisandta -
ae ua lto2w rnbh,
staticloads, thecurvaturecorrection
eredasa stress-concentrationfactorand
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R A G CA P A CIT
Theenergystoredat theyieldpointay
uation509foracantileverspring of
mpleteyielding,conditionswillbethe
antileverspring henceEq uation510
ase.
variable loadingtheenergystoredw ill
ngattheends inatorsionspring,
entrationeffectwhichmayberepre-
hreductionfactor K f.Ifthisfactor
refactorK 2,F ig.180,theformershould
15insteadof the latter.
orsionspringofcircularw irethe
tion364)is
correction factor.
diansis, fromE q uation366,
efore)
ir-rnd2/2 andusingE q uation516
edenergybecomes
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
astaticload, theenergystoredw henthe
nt< tvis,neglectingthecurvaturefactor
theenergystoredinthe ideal
yieldpoint.H owever,anincrease
elyintheratioof 1:1.7isnecessaryto
verthecrosssection foraround-wire
shownas beforebyassumingyield-
integratingoverthecircularcross
pondtoanincrease inenergytoabout
q uation519.Thus,forcomplete
centbelowthatcorrespondingto the
orvariable loads, theenergystoredw ill
ion518,
ctionfactor K 1attheclampedend is
mershouldbeusedinthise pression.
ueofKf forbothcases. E q uation521thus
oadingtheround-wiretorsionspring
sthe ideal(tension-bar)spring.
G
iporrectangularbarmaterialwith
llhavethesameenergy-storageca-
aterialas thetorsionspringofrec-
attheendsare clampedandthatthere
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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H A G CA P A CIT 407
ntheturns sothattheydonot come
V III) . Insuchacaseasshow npre-
ro imatelyconstantalongthelengthof
aseofahelical torsionspringunder
fatigueloading,however,thestresscon-
different,dependingonthemethod
eends.
D R T O R S I O N
sub ecttoatorsionmomentMt
onstresstisgivenbyE q uation4, ta ing
f> ,inradians,ofa roundbaroflength
odulusofelasticity.
ne-halfthe productofthemoment
ence
valueofMtgivenbyE q uation522
ofmateria lV =w d- l/ 4,
staticloads, whentheshearstresstj ust
torsiont ,theenergystoredbecomes
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
atio . F ormostmaterialsmaybeta enas
/ 2. 6. A lso f romtheshear-energy
earingy ie ldpo intt maybeta enas
y ie ldpo int»> . Thus
uation524,forstaticloadingthe energy
ustreachestheyieldpointis
hanthat storedineithertheround
icaltorsionsprings( q uations514and
benotedthattheselatter springshave
weentheloadatwhich yieldingstarts
ingoverthe section.Thismargin
f thetorsionallyloadedround-bar
ofthesectionisstressed tovalues
henceforstaticloading comparisonon
s519and526isprobably toofavorable for
fairercomparisonmaybemadeby
ngoverthesectionasfollows:
utionofstress overthecrosssec-
ouldoccuraftercompleteyielding
sta enplace), themomentMtisgivenby
uation522thismeansthat,for com-
tise ualto4/3timesthe valuewhen
gthofMateria ls, PartI, Page57, S econdEdit ion.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R A G CA P A CIT 409
rj ustreachestheyieldpoint intorsion.
ent-anglecurve(whichwillberealized
firstloadapplication)the energystored
na ltothes uareof themoment. Thus,
uation526w illbe increasedintheratio
3percentlowerthan thestored
ndshowsthatthehelicalc ompression
inde (w hichappro imatesacondi-
tivelyefficientasfarasenergystorage
is concerned.
orastra ightbarintorsionunc erfatigue
uation522maybeused. How ever, in
e somestressconcentration(at
e),andhencethevalue ofMfgivenby
videdbya fatiguestrengthreduction
isintoaccount. Ta ingtcastheendurance
uation523theenergystoredthus be-
droughlythat theendurancelimit
tothatintensionaediv idedbyV 3. Using
ingG= E / 2. 6asbefore, thefo llow inge -
andassuminge ualfatigue-strength
gy-storagecapacityofastraightbar
hatlessthan halfthaiofa simpleten-
er,aspreviouslyindicatedthiscompari-
differencesin theactualfatigue
ondesignand material.
ear-energytheory.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
PR E S S I O N O R T N S I O N S P R I N G
spring,a iallyloaded(from
edbecomes
staticloads, thestressisca lculatedf rom
earstressmultiplicationfactor, F ig. 63.
uation531, ta ingthevolumeofma-
= ir2d2rn/ 2, andthestresse ua ltothe
hisgives
torfora straightbarintorsion,
e factorK0. A ga inta ingt, e ua ltothe
dby \ / 3andta ingE = G/ 2. 6this
33 )
or completeyieldingoverthesec-
multipliedby (4/ 3) asinthecaseofa
. Thisgives
I giveamorecompletediscussionofhe licalspringsunderstatic
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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R A G CA P A CIT
fatigueorrepeatedloading, forarough
hodmaybeappliede ceptthatinstead
ctorK ( ig. 30, C hapterII)which
ffects duetocurvatureandtodirect
hisgives(f romEq uation533), ta ing
eadof ay
3and535it isseenthatthe largerthe
he energystorageperunitvolumeof
decreasew ithincrease inthe inde .
sonismadeonthe basisoftotalvolume
efoundthattheuse ofmoderateinde
imumeff iciency (C hapterX ).
uationsforspringcapacityasprev ious-
edinTableX X X havebeencom-
ergy-storagecapacityasfractionsof
nsion-bar)spring.Thusforstatic
ergystoragewhenfirstyieldingoccurs
gwillabsorb anamountofenergy
hatof the idea lspring. If comparisonismade
rageatfirstyielding,it appearsthat
ngisfar moreefficientthanother
,if comparisonismadeonthemore
agefor completeyieldingoverthe
deal springisnota greatdealmore
Thus,fore ampleonthisbasis(from
e),thetriangularcantileverspring,
gularsection,and theroundbarin
percentofthec apacityoftheideal
ding. F ora he lica lspringof inde 5,
astco lumnofTableX X X V , theenergy -
S T O R A C C A PA C I TI S
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
fvolumeon thisbasiswillbe
atoftheideal spring.H owever,when
basis ofenergy-storagecapacityfor
efatiguestrengthreductionfactors
ngis farmoreefficientthantheother
icethesefatiguefactorsmayvarycon-
torageCapacityforDifferentTypesofS prings
nsof thatinthe idea lcaseofasimpletension-barspring )
angu-TorsionStra ightC ompres-
pringR oundsionor
al(R oundBarinTension
gSpringB ar)TorsionSpring
11. 33. 33. 25. 43. 43/ , 2
. 77/ ,
-
. 33 / , 1 3 3/ , . 25 / , . 43 / , . 43 /1
torcompletey ie lding(staticloads)foridea lcasea2V / 2 -
ance lim it(variable loads)foridealcasece V / 2J .
gthreductionfactorsK finthis rowwillvaryamongthe
. Endurance lim itintorsionta enasf fe / y3.
nthe figuresgivenshouldbeconsid-
ationofthecapacitiesof thevarious
atin thechoiceofspringtype by
factors areinvolvedbesidesenergy
husabilityto fitintoamachineor
mountimportance.H owever,the
sasgiveninTableX X X maybehelp-
toformsomej udgementastothe
undergivenconditions.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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II
I A L S
concernedprimarily withadis-
antspring materials,theirproperties,
ticularreferencewillbemadeto
chmaybere uiredasaconse uence
eemphasiswill,however,beplaced
aterialitselfratherthanonthose of
terials,itshouldbeborne inmind
rictions,whereverpossibleplain
c wire,oil-temperedwire,hot-rolled
beusedinsteadof alloysteels,
rousmaterials,allofwhichutilize
als.Inmanypracticalapplications
sedwithoutlossof essentialproperties.
P R T I S O F M A T R I A L S
importantpropertiesofthedifferent
n T ab l es X X X V I , X X X V I I a nd
X X V Igivesatabulationof thecomposit ion
ordingto specificationsoftheA meri-
Materia ls. TablesX X X V IIandX X X V III
calproperties,includingultimate
sion,modulusof rigidity,modulusof
orferrousandnonferrousspringma-
asizedthatin individualcases,spring
mewhatfromthevaluesshown.
summaryofavailabledataonendur-
alsas foundintheliteratureis given
ndX L , theformerapplyingtotorsionand
propertiesofhelicalsprings werediscussedinChapterIV .
in, IronA ge, A ugust16, 1934, Page18, " Mechanica l
byWallaceBarnesC o. , 1944, and" ManualonDesignandA p-
pira lS pringsforO rdnance , publishedbySA . . WarE n-
eadditionaldataonspringmaterials.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
nentinformation,includingk indof
surfacecondition(i.e.,whetherground
d),ultimateandyieldstrengthsin ten-
ndyieldstrength intorsion,andlitera-
ogetherwiththelimiting endurance
urancerangefrom0-110,000pounds
pringMaterials
pecifications)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I A L S
endurancediagramofF ig.226has
sentswhatmaybe e pectedforgood
gmateria linthic nessesaroundV * toV i-
en thatforgroundandpolishedspeci-
endurancelimitsmaybee pected
erva luesmayalsobee pectedforhigh-
als.
causeofstressconcentrationeffects
mpededges,etc.,theactual endurance
atspringsingeneralare considerably
wninF ig. 226. Thisisshow nby thetests
s reportedbyBatsonandBradley
L II. F orthesetests, asshow nonthis
stress inthemasterleafof thespring
40, 000poundspers uare inch. F or
alS pringS teels
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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S P R IN G
ncentrationeffectspresentintheactual
nC hapterX V Iasmallho le inaf la t
ancerangeabout 50percent.
F S P R IN G W IR E S A N D M A T R I A L S
tiesanduses ofthemoreim-
re andspringmaterialswillbebriefly
eferencetothedata giveninTables
I I , an d X X X V I I I. P e rt i ne n t da t a of i m po r ta n ce i n
mateendurancediagramsforgood
ngmaterials
ationanduseof thevariousmaterials
.
ua lity carbonstee l, thisw ire isw ide ly
springs,particularlythosesub ectto
hehighstrengthofthe materialis
ofabout.70to1.00per centcarbon,
gtosize.The compositionasspecified
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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I A L S
sgiveninTableX X X V Iandtypica lphysica l
V II. Thisspecif icationa lsoca llsfor
mtensilestrengthvaluesasshownby
esofF ig. 227. A sw illbeseenf romthese
sofS tainlessS teelandN on- errousMetals
stionanceModulus
of
nchesinhendingrigidity
< ■ * < * > l T
0. 000toTO . 000. 026^ 10.
mPer 160. 000to110. 000to26 10 3-850.0009. 5 10
40,000
" V " ' 1 80 ,0 00 t o 13 0, 00 0 to 3 0x 1 0 5 -1 0 11 ' » > '
0170,0006to
O t o 6 t o
8 0 .0 0 0 to I , 1 0 5 - O ^ , .
ependingonheattreatment.
ofmusic wiremayvaryfrom255,000
orthelarger wiresizesto440,000pounds
smaller.Thecarboncontentofthisma-
the wiresize,thesmallersizes hav-
omespecificationscallfora limited
carboncontentfora givenwiresize.
sedforspringslargerthan about%-
canbesuppliedinlarger sizeson
f musicwire,thewindingisdone
erwinding,itis advisabletogivethe
eheattreatmenttorelievecoiling
mentmaycallfor heatingthesprings
500 degreesF ahr.foronehourfor
to30minutesfor thesmallersizes.
smadelargelyfromS wedishsteels
tyandhighqua lity . A t present, A meri-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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I A L S
toma ethismaterialwithsatisfactory
btainedwithcadmium-platedsur-
recorrosionisa factor(intheapplica-
ustbeta entoguardagainsthydrogen
happlications,cadmium-platedsprings
bstitutefor18-8stainless steelwire.
e—Thisisagood- ua lity , high-
theopen hearthorelectricfurnace
080910IIJ2. 13J41516
R , IN O C
mandminimumtensilestrengthcharacter-
rioussizes, f romA S TMA 228-41
co ld-woundsprings. A S TMA 229-41
istedinTableX X X V I, C omposit ionA
er7/32-inch.
iscold drawntosizeandthen
werlimitsfor tensileultimate
wireasgivenbythe foregoingspeci-
stwiresize inF ig.228.F urtherlimita-
h callforavariationin tensilestrength
poundspers uare inchinasingle lo tin
notmorethan25,000pounds per
ve. 120- inch. Itw il lbenotedthatthe
e aresomewhatbelowthevaluesfor
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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PhysicalPropertiesofS pringMaterialsinBending
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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I A L S
sicwire,F ig.227.O therphysical
ableX X X V II.
wire,springsmadefrom oil-tempered
dand thengivenathermaltreatment
Thismaybe donebyheatingat500-
-hour.
O f low erqua lity thanmusicor
terialisutilizedin caseswherethe
highdegreeofuniformityisnot es-
posit ionasspecif iedbyA S TMA 227-
T R , [ N O E S
mandminimumvaluesofult imatetensilestrength
mperedwire,A S TMA 221-41and229-41
X V I. Thisspecif icationfurtherre uires
lotofmaterialshall notvarybymore
manganesebynotmorethan.30per
rboncontentsare usedforthelarger
valuesoftensilestrengthinthese
mvaluesofultimatetensilestrengths
eninA S TMA 227-41areshownby
228asa functionofwirediameter.A
hisspecificationisthat theultimate
llnotvarymorethan 40,000pounds
below.072-inch,norbymorethan30,-
inchforsizesabove. 072- inch. Windingis
rmaltreatment.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
wire—W ithhighductilityin thean-
sutilized incaseswheresevereform-
ryinthe manufactureofthespring
torsionspringswithcertainshapesof end
dPhysica lP ropertiesofL eafand
nB ending
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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I A L S
gs,heattreatedafterforming—F or
prings(overabout% to% -inchwire
altowindthesprings cold.Insuch
woundhot fromeithercarbonoralloy-
eated.F orcarbonsteelbars,the com-
T M A 6 8 - 39 i s g iv e n in T a bl e X X X V I .
rings, A S TMA 125-39ca llsforheat-
00degreesF ahr.andcoilingona pre-
gsarethenallowedto cooluniformly
ll ipticLeafS prings
e llStressR ange
Ib/ s in. )
4453, 000to32, 000t
3492.500to46,000
3424,100to43,000
dB radlev , Dept. o fSci. & Ind. R esearch(Brit ish)S pecial
neffectsacttoreduce strength.Theseareduetoclampsused
.
hichtheyareheat-treatedtoatempera-
greesF ahr. andq uenchedino il. A f ter
retemperedbyheatingto 800degrees
willgivea hardnessaround375to425
opertiesofthismaterialare shown
re— Inthepastthis alloy-steel
ly specifiedw hereahigh- ua litymateria l
eraturesaresomewhathigherthan
eforautomotivevalvesprings.Because
loysteels,however,its useshouldbe
thisconnectionit shouldbenoted
Z immerli4didnotshow amar edsuperior-
eelcomparedtocarbonsteelsas faras
a ationatelevatedtemperaturewas
e-vanadiumvalvespringq uality
tureonC oiledS tee lSpringsUnderV ariousL oadings , Trans-
1941, Page363.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
M A 2 3 2 -4 1 i s li s te d i n Ta b le X X X V I . F o r
msteelspringwire(asdistinguished
w ire), somewhathigheramountsofphos-
ur(.05% )areallowed.
btainedeitherintheannealedor
on.W henwoundfromannealedwire,
eatedaftercoiling.A fterwinding
e-vanadiumwire,alowtemperature
00-700degreesF ahr.shouldbegiven,
turesbeingpreferredforapplications
ratures.Othertensile propertiesofthis
eX X X V II.
ingwire— S tainlesssteelshavinga
rcent chromiumand8percent nic el
sub ecttocorrosionconditions.They
ted-temperatureconditions.A typical
ialcallsfor thefollowingcomposition:
wireisdevelopedbycolddraw-
,000to 320,000poundspers uare
zeasshow ninTableX L III.
lpropertiesofthistype ofsteelare
I I I .
steelwire arewoundcoldandmay
heattreatmentata temperatureof
minutestoan hour,theshortertimebe-
resizes.
dingitsgreatestusein caseswherea
conductivityis desired,phosphor
plicationswherecorrosionresistanceis
presentbecauseofhightin content(5to
elyrestricted.A possiblesubstitute
desiredis berylliumcopper.Typical
veninTableX X X V III.
, m in.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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I A L S
analloyconsistingessentiallyof
mandtherestcoppertogetherwith
ys5.Ithas theadvantageofhavinga
ywhilenotre uiringanytinin its
pringWire
wiremadefromthismaterialis
egreesF ahr. andthencolddraw nto in-
tercoiling,itis heattreatedtoincrease
isheattreatmentmayalsobe varied
asticity ortheamountofdrift orcreep.
ertiesof thismaterialaregivenin Table
nalloycomposedof about70percent
ncwhichiscold rolledtogiveit high
esare listedinTableX X X V III. B e-
stressesmustbek eptmoderateifthis
ver, ithastheadvantageofnotre uiring
,whileatthesametimepossessing
yandcorrosionresistance.
pper-nic e la lloy tow hich2to4per
added.A typicalcompositionis:
c e l66percent a luminum2.75percent.
en asolutionheattreatmentandthen
g,springsaregivena finalheattreat-
essand strength.Bythismeans,ulti-
C arson, " S pringsofB ery ll iumC opper, A eroDigest, July
oysforS prings, P roductE ngineering, June1938, giveaddit iona l
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
0,000to200,000poundspers uareinch
rdataonphysicalpropertiesare given
pringsof thisa lloyareusedforcorrosion
cetoelevatedtemperatures0 .
ion-resistanta lloycontainingabout98
erialalsohasgood mechanicalproper-
ub ecttoelevatedtemperatures.Be-
fterheattreatment,springsofthis
ndfromheattreatedwire.F urther
X V III.
onsforcriticalspring materials—
eredfor useasasubstitutematerial
itionsandstaticloadingare involved
springuse,this materialconsistses-
steelhavinga thincoatingofcopper
osion.Tensilepropertiesinthe heat
btainedwhichapproachthoseof hot-
ted.W herefatigueloadingisin-
otbeadvisabletouse thismaterial
uefailureof therelativelywea surface
tpresentfor staticloading.
mglass rodsandtemperedhave
undercorrosionconditions7.The
cingsurfacecompressionstressesby
herebygreatlyincreasingthetensile
wtensilestrengthof glassrelative
naftertempering),muchlowerwor -
acticaldesign.S incetheenergy-storage
esas thes uareofthestress,other
meansthattheglass springwillusually
anacorrespondingonemadeofspring
ughthemodulusofelasticityofglass
owever,wherespaceisavailablefor
smaybek epttolowvalues,this ma-
foruseparticularlywhere corrosion
. . T ransactionsA S M , July1942, Page465givesdata
ofthisandother nic elalloysatelevatedtemperatures.
hael,MachineDesign,A ugust1942,Page85,givesfur-
ula s, asw ellasarticlebvT. J. Thompson, P roductEngineer-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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or ingstress)
rome-vana-
el,etc.)
85, 165
18,420
, 341
wire
he licalsprings. . 30
gs(seeV alvesprings)
ation.260
5b
ulations.261
6
239,258
0
59
60
glected260
262
6
rings361,375
s
s316
soF latS prings)
93
(seeE nergy-
modulus80
8,420,422
18,420,422
ar-barsprings214,217
ded..151,154
111
4
4
re
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
ory,helical
0
berspring. . . 379
cal(seeH eli-
lutespring.... 360
springs(see
thic ness)
4
materials....22
ncipa lf re uen-
5
culation..115
calsprings
callyloaded
mplitude....12
bber384
ubber.... 387
5
8
ntings...395
y397
t20,304,420, 422
idity77
springs...151,154
,254,256
1
ring.. .386
prings388,390
thic ness. 279, 285
gs196
94
ndlatera l
gs
ralloading178
6
changle , e -
changle . . . 47
49
springs
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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3
ation,modu-
ned(seeBelle-
t
27b
m284
given
84
ement.274
73
theory276
elicalsprings
60
tings,inreduc-
ation,helical
calsprings...26
diagram....16
gs193
springs157
5
seea lsoEndur-
terials..304,416
s13,14
91
ued)
2
2
7,420,423
alsprings..88,89,91
ts)
sspringfunction2
es399,412
410 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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S P R IN G
tra lloading. 293
3
ects299
calsprings171
terionforenergy
3
pe
gs
68
ttheory43
5
111
vaturein-
ralloading.. 177
y44
4
59
ation183
o157, 196
y183,410
gs(continued)
alculation131
28
22
n414
ature113
163,165
65, 166 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
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springs(continued)
prings
210
0
10
rgystorage189,191
eeTorsionsprings)
sconcentration
gs
steel
ngs350
frontwheels4
210
42
s...213,219
208
elicalsprings 96
rings18,-94
ngs193,197
gs(seeBelle-
s(seeDis
83
lsprings
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 451/462
S P R IN G
5
415
ation....82
n77
165
6
,417
f80
odofde-
6,415,417
bbersprings)
19
7,420
rings335
ction
s219
48,51,56
212
s
s211
48,54
211
2
ran arrange-
licalsprings
rgystorage
ting,.7% car-
sts
blestress17
essconcentra-
21
gs50
tation62
rings166
modulusof
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 452/462
84 386
7
andcalcu-
0
ocentero f
24
entration,fa-
ngs120
urface22
9, 91, 420
ntensity...92
ngpeening
ot-blasting)
82
onfactor100
springs173
erionforen-
mounting.... 398
rings199
41
ncontact. .343
7,418,420,423
417,418,420
de , spring)
ngs P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 453/462
S P R IN G
ia landlat-
berofturns
193
ging234
ncerange, En-
oading)
8
375
or o ingstress)
8,121,124
tors,flatsprings
rvature
24
25
eeStra inmeas-
1
e(seeE ndurance
)
gs
gs,methodsol
nt,frontwheels
ancediagram
t20,304,420
idity77
ertiesanden-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 454/462
tionstress
vibration226,234
ress
5
o lutespring. .. 371
lsoF atigue load-
ndur-
sF atiguestress,
urance
s
rigidity76
ofreducing
antages..359
0
nued)
0
371
onfortan . . 360
lf re uency. . . 232
a inlessstee l
35
ngs202
helica lsprings,
129
100
eModulusofelas-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 455/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 456/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 457/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 458/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 459/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 460/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
8/21/2019 Mechanical Springs Wahl
http://slidepdf.com/reader/full/mechanical-springs-wahl 461/462 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e