calculo y diseño de resortes

8/6/2019 calculo y diseo de resortes

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MICROFICHEREFERENCELIBRARY

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R . H . WARRING

Spring Designand Calcula tion

Model & . Allied Publications Limited'3-35 Bridge Stre-Bt H - e m e t Hampstead, Herts., Eng~and


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Modil::l~& A IIL Sd Pu bll.,;.a~IO"~ ltdB o : : t Q k D ivisionS~al [on. Road, Ki I ')gs Lan.gl-ey" " e - r 1 f ordsl 'lH l! ! -" E f 'lQI andFII'~l PlJblL:!;i.he-d 1 973 :R H, Warring 1973~5BN 0 e5242 J'27 6Pnnted afld rnade in ng~afldby~


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----- - - .~ . . - ...SPRING MATERIALS 1T~H~so Ing insss " } f rnacals is .- ~Ia te d in a general way to thei r hardness.Lsac. for example. is r C sol: ,-newl, wit t o v i,'t~aUV no 'spring' orcpernes.Th same Wit r - a~,.HT.inlum.Lxtre~ll~ hardness, 'jl1 the other hand, aqainresults in luck. ot s:-:'iing ~'Iopertif'::;' necauce t",~ material is brittle ratherthan 'elestir.. The fdnGe of suitable sprinq materials are thus thosew;lich combine R ,! '~table. ila(dness wi~h 'el-.s.icitv'.

11~s ;)Iso im ~ J~t.ar.r, ;f spr in9 n~:torma rH:"e is to be co nsjsten t, th atthe material rstams tt ~ oftf-l:l1al p'opert~es, M~ny metal'. are subject to'work- hard.?ni~~,J'or ~ ch3ng~ O ! hardness when ~1 i"c 'S ,sed - .3nd allwork i- g S o p _ " . ngs ( lF ! SOUbjcc t ':"o r : : yc.les .~ st r 3:S"l . B -ass. for exampie. is a.neta I wi"! ~chi S re I.a~~....~y : 0 : :of L b d t ~-ep...~ued siessir t 4 J or WOf l s i ngc a l IS'S!S n s ha ~dness A 'I . l ~ r1 C r f~~se , w:~; til':' m t:ta I becom IPIg more spri n 9 yas a ccnsequence Thua wnilst soft brass is Quite useless as a springmatet i?t tu Ir y _ h t . . i r de I!~'d b .EltS ~f)S5esse : .~ reason abl y good sprin 9pr O ~ o E J o j nes.

The h.3rdne 5~of mar l V ru etals can ajs 0 be im proved l:l tIe at t reatme m ,End .~.s a result their spring ptGpertie:!' enhanced This is quite COmmo np. ~(;tic:eHI the preuaration co[basic spri!1g 'stock'. Heat. however. can.a Iso ~ .ud u ce ~he OPPD~ : : ~ res U It. Th us a hard, s I : J I in gy meta r Can orte nbe oermauerit! softer-e d b y heating and s r , " , ) w cooling (or annealinq).On the other ~.:tr;d. he~tillS and rapi('l cc..:,ling Q spririq materiel can~ncrease hs hcudness t ! : J \ he po in t of bnrt teness. Wi t noLIt co nsid fr8b~e~;:tp:r~~nce in the iechniq ues ::if heat treatment. therefore. sprin9rnaterlats should afways ba used :1S . s r _Wit h s u i t a bl e k n~wledg l. hcwever. processed sp r ing s rna 'I I often b ehea! treated 10 advantsqe - e.g. to remove ir i .ernal stresses remainingn the m~;!e,"j~. aft-3reo ld w or htrl~ to sh ape: c r fa rrn. Th e tempe re tu reand method of ne-at t reat ~I:e""t employed is depe nden 1 0n th e Com ~pcsitiou of the spri~g rnata.ial and rb e method of spring application,Another form of t~'3'atment which can produce ernbrittlernent in asp r i n g is else no rt latmg, T hi s a pplies partie u l a r Iy rn th I I case of c:aroo nstee I spr.ngs.. where plati ng may S O me t ~mes by t hou:ght des .rabIe toprovide resistance to corrosion. ff such springs are p:ated, regardless ofthe method used they require to be baked lrnmediarelv after platin-g todri ve Out h yd roge-l1 a bsor bed by the m ate r ia ~ du ring plating, Anyh'/drog en rerna inIJl9 ~n the pares of the spri ng mate r1ai wi t I causee rnbrittla me nt. S i r.)i ~ar COm rent a no Iies to plat d steel wirer used as aspfing materia I_The rc:nge of true spring materials is fairly Hmi t ed . Ordinary carbonsteel rendered in 'spriflg temper' form is the most common choice forge nara I pu rpose sc rin9S 0ia II types. In the cas e of wi re, the n ecessa ryr emp e r rn a y b e prod u c e d by the meth od of fabricatio n - e .g, cold

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SPRING MATERIALSdraw ing. The sorin9 tem pe J rn ay . howeve r, be fu rthe-r jmproved by heattreatment or oil tarnparinq. Such spring materials are suitable for useun de r OJdi nsrv ternpe ratu res, in norma I stress ranges - .e. wit hou t thelimit of proportionality of the material {see later} _For use under higherstresses. 0...hi 9 he r t empera tures, specie Ia110y stee Is may be need ed .Where cor ros ion rna V be a p robtem, the c ho i ce of sta In:es.s stee I or

non~e r rou Sspri ng mate r~als rnay be necessa ry - the termer wh e re highst resses h ave to be carri e d by th e s prin g . and the Ione J !or lower cost.easier workinq, wh-ere stresses are not so high- BefYU~um copper is anat rract ive ch0 ice where h ig h res ista nce to stress and ccrrosion arenecessary. .a nd 9a ode lectric a leo nd ucti vi ty is a lso req ...ired. If electricalco nductivitv is th e rnat n req uireme nt, phosp hor- bro tlze provides ach e a pe r a ttemauve: and brass even lower cos t (a ~thoug h bras sis a'margjnat spriflg material. even at full 'spring temper'). A nickel alloy(e -9 - mon el ) may be specifie d wh e re hig h tem peratures have to bea Ce O rn modata-d-

M echa n ic aI Consi de ratio nsHowever good the spring material, there are limits OV@J which it can beex pected to work consistent! y and sh ow a long 'spri ng Iife. Thecrit ica I faeta J invo Ived is the 03CttJ~ I Stress bor n b y the mate r i a1when

F i g _ t

,S ;h'oir'l __ ...

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SPRING MATE RIA L Sthe spring is . wotk~r'Jg. Up to a certain point with i(1creasing stress thecorrespondinq strein in the material fOflows a linear relauonsbip >F~g 1. 8 e yond thi50 limit Of propor tionillity t his. Iinear re I a t ion s h . p i10to nge r ~ppl i e s ijn d s u biecti ng the m a r 9 : ~ i a I to these hi 9 he r stress val ue smay pern1e ne nUy ch a nge ~he m ec han ica I properties 0 f th e mater iel.Tho lim it of propcrt ion alit 'I f lh us represe nts the uppe r st ress I rmi t forthe mate r.aI In pta c rice. a : lower Iimit s nor ma Ily em plo ved - 80 percent of the Jimit of prcportionalitv - to allow a safety factor in springdesign.Working within thiS limit will then ensure a consistent performancefrom a spri 1 19 rnatenai.This, however, 0nIy p r e - s e nts par~of th~ pictc r e . Th e strengt h of allYmater iel is d iffere n t fo r d ~fferen t ways in w hichi t is stressed. Maxim urnS it re rlg this uSU a IIV a 'riaU ab le when stre sse d in pu re compression, w.tha n almos t sl m . fa r va Iu e w he n stressed n pure tension. If su b j eG t totw isti rlg or torsion J the mater ia! stren gl h ava iIable is consldera b IVreduced.Basicallv, in fact the life of a ~pr.ng depends on four maio fr;ctors:

(i ) The manner in which the spring material is stresse-d.(i i) The mJximum worki ng Stress.( ,! i) The range of st. ess aver w h ic h the sprm 9 mater ial is wo rked.(iv) the n u m b e r 01 cvclr.s of stress or th e effects of fadg u eon m ate n a ~pro pert ies.Items (i) and (il) are directly related. Once the manner in which thernateri aI is stressed j S o establ ishad, a safe m a X I rnum work in9 stress can

be este b lishe d fOTa panic u Ia r rnateri al - see Ta b Ie r .The- stress range is more d l ff ic u11to esta b lis h. In genera t the h lg h e rthe range of st re s S ova r W hich th e $P ring is worked. the [ower sho 1J1dbe the maximu m permissi b Ie stress to ensu re long sp~iog life. However,this w j Il va r y with both d .Here ncesin mater ia I prope r t L e S and heattreatment and wi t h trequen cy of worki og. For si rnptic ity of desig n it isbest to ad opt rsafe' fi 9ures which err on the side of un deresurnatin g ,m a te ri al pe rforma nee, such as 9~ven in Table I.Wt1ilst mater ia t strength and stress determ ine the load whie 11e a n b eearned by a 5pd ng of given geometry. . a nd the Iife of the spdng,def lectio n char acterist ics are dete rmi ned by Ie m odu I i of th e material.Again this depends on the manner in which the spring material isdeflected or stretched. If r he spr. ng mate rial is under tens ion, then ~tisthe modulus of elasticity of Young's modu Iu s whie h is the parameterinvolved. Fo r a s prin 9 materia I su bject toto rs~on 1t is the roo duJus of(igiditr wh ich is .nvo lved in cal culatin 9 defle ct. on .Val ues of mod ulus of elasti city (E) and mod ulu:s of r igld it V (G) have,the refore, to be known for the sprin9 mate rials used before the full

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SPFUNG MATER IALSDt=rformanc c'f a sp'i ....~ can be evaluated. These are also given infa ble I. The mad u III S o of e Iasnc it y large Iy govern s the materia Iperfc rm -an ce of flat spr ings and torsio n s p ri ngs, T he mod u Iu s of rig id f ty gover nsth e mate ria t perform arlee in he I .ca I spri,.,g s, The actu ai SUess prod uced!n it spring. on th e other hand. is dependent only On the load cerr ied bythe spring and the spr.ng g~omatry, All these individual parametersap-pear in the spring desig n formulas in subsequent chapters.Formulas and Uoits5 prin g de sig n pro portions are not sorneth ~ng !at can be 'g uessf mated'wi than y degte e of accu rae V - and tri al- and -erro r desig n can prod ucea su ccess lon of faUu r e - sL T hV5 th is book 0 n Spri ng des ig 11 is fu U Offormu las, as th eon I y ac cu rate met hod of Pied ktlng sorin 9 perlo rman ce.However r all are sssentiallv orectice! working tormutes, and all areQU ite str a i9 htforwa rd to use. Eac h ce IClJ lati On is n oth ing more e Iaboratethan an a ,ithme t ical calculation - a ided by a slide rure or log t.ab le s ,No units are given with the formulas. si nee these foitow quite~og&ca'Iy depend&ng on wnettle r VOti i;lfe WOt kLOg to Eng ' .s h or metricst andards. M os t Q uan tit ies are linaa r dime n s i o n s, and it is on IV n e c e ssa ryto remembe r th at stress va Iues, etc, should be re I ld ered in th e sa meun its. Th us fo r worki n9 wit h a ltd imensio ns in in ches , stresses, etc,must be in pounds per square inch. Answers wilt then work out~o9icaU y in t he rig ht units.Fo! example, th e deflection pe' coi~of a heliC 81 comprssa ion spring

~S9~\ren by8PD3defle ction = ._-G d4where P is the loadD . $ the mean coil diameterd 1 S the wire diameterG is the moduIus of rigid ity of the spring materiaI.

I n Engl ish units, P would b e in pounds . Olmensions D and d wouldbe in inches, To be consistent. G must then be in pounds per squatsi1C h. The d eflect len. ca Icui ated fro m the fcrrnu la. is aneth ar linea rd imens ion and so woul d beg ~ven d1reedy in inches ..Using metric units the no int to wate his th at the mod u h .J S 0r stressv.alues used (0r calculated) are t n the same Lini ts as the linea, d imen-sions. The latter, for example, will usually be in millimetres. Moduli and5 1ress fig ure 5 rna v . h cweve r, b e Q Uote d ln k t1ogr6 m s :per sq ua re cem i-metre and wO uld ne e d ad i usti ng for cons i ste n'Cy whe n ~se d wit hmiiimetre linea r units.


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SIMPLE fLAT SPRINGSWe 1hen have b t" .-; X

From this point. either 'quesstirnate' a value of b and from thiscalculate th~ cortespcndinq va!u-9 ot t to satisfy the eouation: or'guesstimate' t and from this calculate b. The latter is the usual methodsjflee t hick ness is governed by th e stan dard sizes 0f materia Is avai lab~e,and thus there is a choice of specific values of t (e.g. 20 swg. '8 swg.etc.): Note: See Appendix B for tabular values of P.Any solutions derived by the above method will glV9' spring proper-lions s.at.sfyillg the- defl@chon under load requirements, It is nownecessa ry toe n te r t he s e val ue s j n the 5t resS form u ia togethe r \ A t ! t hload (P) and ca tcu late th e st ress resu lti rlg. Prov.d in 9 this is lower tha nth e rna X im um per rn:ss ibj e stress for the material used, then the $Pring9 eomeIty is sarisfacto tv . I f th e cal cu lated st reSsis highe' th ant hemaximum permissible stress, then the $pr.r tg geometry must berecelcu lated from the def!ection formula, using different values. This .ssimplest ~t the spri ng len g th is 1eft unalte red - 11 is then 0n~y necessaryto rer . ..HfI to the farmu~a

b t : : : ! ; ; ; : Xand use a greater thickness to calculate a new value for b. Check jfthis reduced the stress to below the maximum permissible value. I f nottry again.Square Wire SprjngsU~ the same Io r m utas and proced 1Ire. substitu t ing a::lfer bt2 in t he stressfo r rnu ta . a nd ,;.4 fOr bfl in the Deflecti on form u~a; w here a -::-d i mensio nof sn uars.Round Wire Flat SpringsE x ac t IV th e sa rna fo rrn utas (a nd desl 9 n procedure) fo Ilow in th e caseof a flat spring made trom round wire - Fig. 3 - except that band tare replaced by the wire diameter (d).

6PL . 4PL:JSuess ~ ~ Deflect ron : :5" Ed'"Flat Spring Supported at Each EndTh e stress and deflectic n formulas ar mod it ied when a t r at spr ing issupported a t both end s - F.g. 4 - a nd be come

3PLStress = ~2bt;iP L ; JDeflec t ion = 4 E . b t " ; !

De-s ig n proce d u re is th e sam e ag a i n .


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5IMPL fLAT SPRINGS

Fig. 3 p - - _ . ; ; . - , . . _ _ _\_ ---~.-

pFig_4

- - -..- \. .---

In the case of a frat sprjng made from round wire, suoported at bothends, bl2 in the 'S t ress ' formula is replaced by d 3; and bt~ in the'Deflection' formula is repla-ced by d"Des.ign of Contact SpringsA conrac t sp ri ng IS simpl y a flat sorin 9 desiqnad to appl y a cert ainpressure at .a particular point (conta-ct point) alonq its length - Fig. 5.~csn be derlveo from the stalidard JStress form uIa, few ritta n as aSO luno n fo r tead (P) Of actual co ntact pressu re prod I,J ced whendefleeted. vi:z

bt.2$P=~ 6Lp

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~ i g _ 6SIMPLE FLAT SPRINGS

. t o o . -- - - --

z o o

150 ~---

300 --~

~. 1 -.- _.._.~

, 50 t--------t- --~

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HElICA.L SPR INGSwhere- S is the des i9 r'I max ~mu~ workl ng 51rsss to r the rnateri a I to beused (..e. use 80 per cent of the limit of proportionalitv of the springmaterial from Table f)The s imples t way of lac k.11 1 9 desig n is to fi x suita b le value s ofsprinq length L and width (b). from which the requ ired materialth 'ckness can be calcula ted. IT th1$ y~eld5ia . not standard thickness. thenthe- neerest standard thickness ttl p or down} can be adopted, and thecorrespo nd in9 w idt h re -c ale u tated to provid e the reQ uired contactpressure.Itca n a ls0 be iostr uct ive. havin9 deeid ed 0n a suita ble spr in9 lengthand width. to calculate the maximum contact pressure available Over aran.ge of thicknesses for different spring materials. This is done inFig. 6 for a sprirlg l-ength of 2" and width i"~and clear~y mdicates (hes!Jperiorit y of bervr j ium coppe r as a co nract sprjn 9 mate r. a l.limitations to flat Spring Ca'cutationsWhilst the des.ign formulas provKie accurate theoretical soluuons, actualper to rmanc e may be modi fied sornew hat by the rnanne r in w hk h theend (Of ends) of a flat spring is (are) clamped.r n the case of contact spring s, perle rrnanee may be fu r ther mod f f~@dby the fact t ha t sue h spr.n gs are not necess ari Iy stm p Ie bea m shapes,but m ay be irregular in wkjth. Calculation applied to such shapes isted ious. ~ is best to design the s p l ; n 9 on the basis of a .meal1 O r'tvpical' width and check the performance by practical experiment----------- - ..-----~---.--~~- -~~-

HELICAL SPRINGS!n the case of he! ica! 5pritlg s whie hare e ft her compressed or extendedunder load, the spr ing rnateri a I is stressed intorsion a nd so the fo! lowi nghas ic f orm u Ias apply for rou nd wi re spr.n 9 s:T . I SPDorS~Qna s.tre ss = --.: r t d ~

8PD~DeUection - Gd4 x Nwhere P !!!!!! 1 0 ad

D :!!!! meand iameter of spri ngd !!!:! wire diameterG ;-::-mod u Ius of ,igid itv of sp rif]9 material.N :!!!! number of acti "Ie co i~sin the spri ng{see also Fig_ 7)

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H':;UCAl SPRINGS

F~g,7 p

. . . . . . . . . \\

IJ

The sti ffness of a heIicaI spnn g. theretore, rs pro po rt ion al to thelounh power 0 ' the wire d iameter, and varies Inversel vas the cube ofth e mea n d lamete r. Bot h d and D thu s have a marked effect 0n springpe rforrnance: Us in 9 a wire s i z e O n lyon e g.aug e o I J pea n apPffl crablyreduce the deflecti on, and \ 1 ' . ce versa, Sim flarly, on Iy a srna U increasein sprin9 diameter D ean consider abIyin crease the d ef lec t~on~ or asma II dec rsas8in D ca n rnak.eth88prin g muChstiffe J-The latta r ef fect p a rt ic u 1 a tiv , shOU Id be born e in mind whe n rnak.tnga heIica I spring by Wfap pin g arou n d a rnand ret. There wiIj be anmevita ble 'sp ring hac k . . resu Iti ng in a sp r~n9 ij noer dia meter SLIe greatertha n that of th e mandre LAn undersize rnand rei ~s thus requl red to forma sprinQ of ,equ ired d.anlete r. The dey ree of u .,d e rsi:ze c an on ly beestimated fro m ex pefie nee sil'lce It w.11 \l'ary wit h the qua lity of thes.pting m M e J i al used. a nd . a tao th e eol Iing tschn iqua.The nu~be ~of active coi Is is those act uaIly 'we rkF ng r a s a spring,Usua.l pracuce IS to allow! of a turn (or 1 complete turn] at each end

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f~g,8 lj!;N ~LON S~ ING l:NO~

1 . 0 1 ' 1 9 rQUn.t . : I 1 ! F \ dhook on {l!!rHI'I:!~~E " : . . I r!!nrj~ r ! : , r o ! : r;II'\i!'ither r :::o!nll 'e- Of "ide

~Cooed C ' f ' r d wi th~ha'1~; ' . ' ' ! ' I ~yr:

HEliCAL SPRINGS

Pl[] in ~nd'~

a l llQl'l !i l ~ q uC1 e ~dhor;j;: O t ' .r ce nt r C "= = i W$t:'D;7rt Q n . d . 'XU\o!rg loQ

IQ 0 1 ;1 .1 ~ form! 1 ' 1 9

~~~eo"Qr;I .r::nd w i~ I o r'S'fo' j " " e l h o -; J . ; .

1 "3

~r u l i l l

=GConed C ' l ' I r : : E to f r - ; ) i d1orr.g


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H{:LlCA L S pf:t INOSin the case of a plai n compression spring to produce para1lel ends. Thus.9eometrica I~ythe spring h as a tete I n IIrnber of col Is equal to N + l-t (orN + 2). the number of active COi[S being calculated for t~e requiredd e f lo ec tio n pe rforrnen ce. E;II ;W nsi on sp rin gS F 0n ttl " otber ha nd. co m rI"l"IlJny hav e aII th e co i Is Iactive', th e end s be iog mad e off at rig htang lesto the mai n coil. e.g. see Fig. 8.Allother irnportan t parameter is the sprin g rote (or 10ad rate). whi chis sim ply the toad divided by the de Uect ion.

Spring. Rate .= pd eflec no 1 1Gd4= 8ND3Where th e spr in9 is of const ant d . ameter and th e co.i s are eve n Iypitched. the spring rate is constant. A spring can be g.ven a variable

rate by 1aperin '9 the coi Lor u si ng a va r iebi e pitch. Con sta n t rate spr~ngsare the more usual, and much easier to work out.Basically..spring design mvolves caiculatinq the spring diameter andwire size rec u ired to Q ive a safe mate rial stress fo r the load to becarried. It is then simply it matter of dec~ing how many coits arereq l J i red ( i. e. how many active turns) to give the necessary spri ng rateor 'stiffness in pounds per inch of movement. This may also be affectedby th e amou n t of free move me nt availab] e f Q r the SDring.Th e same considera t i ons a ppl V to b ot h com pressionan d e)(tensionspri ngs, with on ~ d ifferen ce. Extensi 0n sp ring s may be wouJld withinitia I tsnsio Fl. which In some cases can be as h;g has 25 per cent ofthe safe load. To open the coils of the spring this ~oadmust be applied,an d 0111 y the remsinde 0 f the load is then avai lable for deflection. Thisdoes. not mod ify the spr.ng desig n forrnota - merely tha va ~ue of theapp lied load effecti V9 i n produc in9 def lecti on.Whi lst the worki ng for mu Ias are str aig htfolWard. spring desig n iscom pticated by ttl e fact tha t three variao las are involved in the springgeometry - d iamats r {D), wi re d iamete r (d) and nu-n b e r of activecoils (N). HOWEver, only D and d appear in tne Stress formula which isthe on 8 to sta rt with. So here it is a case of g uesstl mat i n g' one flgu n : :and caicu lati ng the other 0n that basis.DMig.... ProCedur'8

0) Either(a ) fix a va Iue fo r 0 and cale ulate d fo r the sere va lu e of work.ng stressfrom

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i-U:uCAl SPRINGS(Note: From the value of d3 so found the corresponding wirediameter can be found from the tables of Appendix E : i - there is noneed 10 work out the cube root of the answer to the formula)

or(b) fi)L the vatue of d (from an estimated suitable Or readily availableW, re ~ ~- ..:e. a fld from th is c ale u late the re qu ired value Of 0 from

n:Sd;Jo = 8P{Note: Ag.;_in you call ~oo-kup d3directly in the tables of Appendix B.)(ii) Check that the sizes are practical. 'For examote . if the value of D.s fixed the calculated value of d may be a non-standard wire size, lnthis C~S~, recalculate For the nearest standard size to yield an acceptablevalue of S. This can be avoided by ti>::ing the value of d to start with.but could vield an impractical value for D.

(j'i} Having arrived at suitable values for D and d, caieulate theflumbe r of active tu rns requ ired fo r the def lect ion to be accommodatsd.

Gd4 X deffectionN ~ SPO :)(Note: you can look up vetuss of d4directly in the Appendix tables).That in fact is all there featly . 5 to designing helicar compression orexten sio n springs, provid ed ex trerne i l CCU raey is not recu ired _Rem embe rto add on ior 1 turn to each ~nd for closed end compression springs_

Mo re Act::urate WorkingStress calculation by the above method assumes that the spring material~S stresse d ~r'I pure 1rsion - In fact fu r t he r stress is add e d beca use of(he- curvature ifl the w~re.Thus the true stress in the ma te ria l is higherthan predicted from simple calculation, vizTrue stress = K x Swhe re K IS a cone-clio n facto r fo r wire curvature

(n orrnai IV kn own a s th 8' Wah ~correctio n factor).Uoiortllnate~y .. the vatue of K depends on the spring geometry andthus the spri n 9 di emete r (0) a nd wire d iamete r (d) have to be deter-m . ned before itle correction factor ca n be found.

4c - I 0615K= +-~4c + 4 cwhere c = 0Id (which ratio is also knownas the spring ~ndex) .

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HELICAL SPRINGSHav'ng determined a suitable size of spring" therefore. the true stress

should b e calculated, using lhe Wan l correction factor calculated asabove. I f this true stress works out higher than the maximum perrniss: biemale ria r stress, the n the whole spr ing geo met rv must be reealc L J latedthrough.To save a lot of wor~ing, values of K are shown graphically againstspring index in Fig. 9, and also in Table IL

So'id Height of SpringTile solid O~ "closed' length of a helical spring follows by multiplyil1gwire diameter (d) by the total number of coils {N + dead" turn at eachend. where applicable). This. l-ength may be reduced somewhat bygrinding the 'dead' turns flat on a closed end sprinq see Fig. 10.

Helical Spri ngs ill Rectangular Wire SectionSimilar formulas apply, with wire width (b} and thickness (a) replacingd - F~g. 11. A tso add it ion el st ress facto rs a re ~ntrod ucad to take Intoaccou n t th e add i li0naI stresses ~rnoerted by bendi n9 recta n9u Iarsecti on W fre iI1tO a he lieal coi I.

PDKKStress ~ :z2a~bValues of K2 and K:I are gi\o"en in Table III. The spring index, for

determining the value of K the Wahl correction factor K , is 'found asfellows.For rectangular wire coiled on edge, C = D /aF or rectanqular wire coiled on flat C =D/b

For non - c ritica I ap plic at ion s th e dasig n of hel ica I coi I spring swou nd from recta ngul ar sectic n wi r8 can ignore t he correction s tostress, by adopti ng an appreci ably lower value of ma x i m u m pe rmissi blemateria I stress" Thi s wi ~ no t uti lise the full sprin 9 pote nti aI of themate ria I...but ccnside rab Iy simp lifies c a tc u tano n.PDSt reSS = 2a :lb

Hi


20/40

Fig_ 9

1.7

1.S

}. . : .J:OJ.~ 1.4.Jo.." ' E . . .-i1,3

rHeL .l CAL S .P r;lINGS

1----1-- T - - ~;

IW(lh,I"; -I;I rec Ion f C l ' l " ~ I'I!:~

I, ~,..H~1 i I: :u l GoH ~pr i l ' l ' 9 : : ' i .I, - i i'+i I ! . - - - - - - - I i

; I r . i I Ii : I I I .

I 1 : i : ! ! II I .: I: I' 1'-:--~-;--1 _ I I I_ J , - ' - + + - 1

t I .1.:2 t-----i---+---+--+---

t ,1 !t--------4- --+--+--

2 4 5 7 S 9 10 11 12 1 J 1.(

17

~ _ . -


21/40

HELICAl. SPRINGS

Fig. 10gfOO I " I d nO!

\II(II porcl lelj

J" r : N 3 i J d II ' I 1 . I r r ' l gt~IJ.f'Id fll;l t

Energy Stored in Helical SpringsThe energy stored in a compression or extension spring can easily beca I cut a ted from

P )( def~ectionEnergy =----~- 2

Fig. 11

wl.-e -coil ed on fl at

18


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TAPERED HELICAl SPRrNGS 4With the tapered or conical spring ..eae h coil is of different diameter.This gives the spring a variable rate. The stress imposed by any loadca us. nQ d e fle cn on is als0 va ria bl e from C oi i to C oi I. Far des.g n pu J posesit. s the maximum stress, w h ieh is most important. Thi s wi II occ uri nthe largsst active co iI - Fig, 12 - and the stress is larg 8 1y teosio n.

8PDM .a ) { , s t ress : r r c F J ' ) .. ' :Kwh&re K is the Wah 1correction factor

( Note: :sjnce stress IS proportion a r to spri n 9 dLameter D. t foUow s tha tthe stress In an y coil ca n be celcu latad b using the appropriate coi Idiameter: also that the maximum stress will occur when 0 is a ma)C.imom.i.e. equal to that 01 the ta rQes t coil).Fi-g. 12

-~ _ .- 01 ..------


23/40

1argluif g"'tL ..... ~Ql i~I ; J o 1 t ( 1 ( 1 " ! 5 . Hrst

TAPERED I-tELICAL SPRlNGS,

'70pI pI~ " - " l ~,--Io-------L.. sol id foIeight_ _ _ _ _ _ _ , . : : " " J

This formula can also be rewritten in terms of the maximum load toclose th e spr; ng so I i dp .::..:d '" x deflect. onmu 8D3Nz

The desig n of 18pere d sprinqs, therefo"e~ follows th e same lines asfor hel~caI coi ~sprin 9S (Section 3). usin:;i these mod ified form u las-So Iid Heig htThe s.olid heig ht of a taperad sprin9 is less tha n that of a hel ica I springsince the individual turns 'stack' to a certain e :de nt - Fig, 14_ Th@Fig. 14

20


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TQRS10N $p~It4GSeftective height (y I per cor~ can be determined f.rom the right L angledtr iang te show n, whe re

d2 !!!:. XZ r V~Or y = v:d~ - )( 2

The solid heijght of the sprinq tnen follows asSolid height = N ywhere N = number of active turns,Remember to add 2d to this to account for one 'dead' turn at each endin the case of springs. with closed ends.

TORS!QN SPRtNGS 5A he~,~altorsion spting is designed to provide an angular dettscticn 0",an arm at O ne end of the sprlnq _. se F~g_ 15 - the other end of thespri n9 be, ng anchored _The s l&ffness of sue h a spnn g {Of its reslstanceto d eflectio n is dire c tl y propo rtiona I to th e fourth powe r of the w ~fe

Fig. 15

Tt1R

p

diameter: a n d lnverselv propornonal to its diameter, The co~j diameter15 commonly fixed (e.g. the: spring has to fh over a shaft Or spindle):and thus choice of d i ffe re n t w~re sizes W I' I have a con sidarable effectOn spring performance.

21


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TO RS ION S PR ING SThe fo ~~owin 9 b aste fo rrnu las appiy;

32PAStress = )( K1 I : d3 4W her8 K4 is ths stress correction fact or "to r ro u nd wi resprinqs in torsion (see Table IV)

3665PRONA ng ular d ef lect ion (degrees) - ~~E-d-III--where E = Young's modulus

spring mate ria I ofDesig n calcu larion s are aga in based on work in g th e so r in 9 ma t e ria lwi t h. n acee pt able l im~ts of Stress. Th e force {P) acting 0n the spri n9 isap pi~ e d ove r a radius {R}r equal to the effective ~el"lgthof the free armott he SPfP 1 9 ' . Oes~gn c alcul ations ca P 1 proceed as taU ows:( i} Knowtng the force to b e accommodated and the spring armteverage required (R), use the stress torrnuta (w.thout correct-onfactor K4) to calculate a suitable Wire size:

32PR 10-18P Rd: ! : : : : ! ! !!!!!----1 tS Swhere S is the maxmum permiSSib lematerial stress,

(li) Adjust to a standard wire si:ZE L If nacassarv ,n i.) Ca tc u Iale the ang u I a 1 ' " deflectio n of such a s p ri n g. us ing aspee ified val ue 0f di amete r D" hom th e def leet;0n form ul a. and ~gno r.ngthe factor N_ This will give the deflection per coil. Then simply find outhow many coils are needed to produce the required deflection-Thi s stag e m ay .. of COIJ rse, be vaned _The load mornent P R may bethe critic a I factor ~ i.e. t hB sprinq ~s req u ired to axe rt (Q r resIst) acertain force (P) at a radius R with a specific deflection. In this case"hav~ngadopted a specific value for D r the deflection formuia can beused to find a solution for the number af turns required.

(iv) Having arrived at a possible spring geometrv" recalculate theI(ue SUesS as a check, using th e correctrcn factor K4If necessa ry, read] ust the spring geometr V to red uce the stress andrecatcu late the spri ng.If the spring .5 to be thted over s shaft Or spindle a check. shou1dalso be made that .n its bgl1tened position it does not bind on the shaft.

22


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TORSION SP~IN~S

Ftna~mean d,arneter :=; 0 ~ ~,where N, is the final number of turns when

tightened.Thi s ;S 5.mpis enouq h to w ork 0ut. A d e fle c uon of x degrees is e q 1 .1 iva lentto x/360 turns,

)(N, =N + 360Remember that the f;na~ inne.r djame~er of the coil w m be equal to

the final mean diamater minus d.

Thus

Torsion Sprio99 in Rectangular Section WireExactly the sam e procedure is involved, except that the bask formulasare modified sl~ghtly (see also Fig. 16).

6PRStrass > -- )( Ka : : i i : b "S. 2160PRDNAngular def~ecuan (degrees} = Ea3b

Square wire section & S sirnplv a special case of rectangular wire sectionwhere a .. b _

Fig. 16

23


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CLOCK SPRINGSEnergy Stored in Torsion SpringsThis is easily calculated from the deflection and moment.

PR x deflectlon (degrees)Stored energy ~ 115Th e s ame fotmu fa epp l ies to bot h rou n d and rectao 9 L J Ia r 'r/ 'J .re sactio n s.CLOCK SPRtNGS 6A clock spring is . a spedai type of spiral Or torsion 5pt'jng. wound fromfla t strip. M ain interest is in the tu rnlog mom e n t Of torque, and thepowet such a spring can develop.Th e stress deve loped jn the spri n9 materia I elln be ca lcul ated fro mthe sprin 9 dimen sions ina c tose w 0und and 1uUV released co nd itions.I f R....is the radius of a particular point; n the spring in a fully woundcondition and RoJhe -radius of the same point in a n unwound condition,a close a ppr ox imation to the stress is give 1 1 by:

Bending stress (So} = E ~ ( R :~J(se e F i g _ 1 7 for n otanons)

Fig. 17 fl.ll iy released

f t J . 1 1Y w o u t l . d

pointXPoir'lll X

Note that ;tis the materia~ stress in bend in9 (0r torsion) w h ich ap pliesn th is case, not the tensi Ie stress (whi ch is tower),24


28/40

ClOCl< SPRINGSThe .Denect~onI form ula can be rendered in te rms of ~hen umbe r of

turns (T) the spring can be-wound up.T or 6PRLnEeb

this can also be rewritten in terms of t h E ~ stress (Sb)T ::::;S tJ _nE t(see Fig. 18 fO f notation}

Fig. 18D-efll!o;: . l ioo -::..'IIu~r o Q f turm

. . . . .OVl' Id upUnwound

b!1IiI .I i~-R----1

This is by far the more convenient form .. but is not strictlv correctsine e it doe s not a~ow for the effe ct of curvatu re 0n stress (see TorslonSprings) _The complete form u1a fo r num o o r of turn sis thusT= LS~n:EtK~w here K~ is the curvat u re stressfactor (see Table V)

The le..,g th of spri ng (l) ca n be derived from basic geometry.L ~ nDNwhere N :;;;;;umber of active coils

25


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CLOCK SPR~NGSIn the fu~lywound condition

( Ru - R . .. .D ~ 2 R~'" -2~but tJ - R = N tI " " I I . J Y t

or R lI = N t + R . .. ..~ N t . ~ A , . - R o I )Thus 0 ~ 2 \ Ru + 2

= - 2Ru + NtSubstit uti ng in the f i rst form u:a

L x: 1 rN ( 2R~ + N t)These formulas can be used to determine the required springgeornetrv. with the mec han leal ou tpur glven b v

Turn. ng rno ment o.r torq ue 0= PR ~~f the applied torque is known. then the number of turns to wind upthe sprinq also fo ltows d irec t Iy as

6QLT=--1 tE t~yHorsepower CaIcutat ioIlTh e stored energ y j n a dock sprin 9 can be re~eased at va rious rates.according to the manner in whic h the movement is governed orrestrai ned.Note the relatlonshin between number of turns (T) and stress.No. of turns (T) to produce

S. LSstress n sp r i ng materi 8 1 ; : : -eIt tnEtTsuess (S) =lhus

also:. 1tSbt2Energ y per revo lution = - 6

To dererm ine the en erg V prod uceo by a ctcck springF proceed asfoltows:0 ) Care uiare fef1gth l from the .geemetry

.2 6


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CO . . . , STANT FORCE S'PRINGS(ji) Calcu late stress produced from the number of turns available towind up (this mus t not exceed th e maximum pefmis.sible bendingStress of the material} _(iir} From the stress cetculate the energy per revo~ution (EJIf the energy per revolution (Er) is determined in units of inch- pounds

(whic h will foUow using ir.cn units for tile spring ge-ometry aJ'ld stressin lb/sq.in).

' E t x rpmHorsepower ;;;;;: 96.000or say as a suitable approximation.

The time for which the spring w,~1develop power also follows asT/rpm, in minutes -- i , E ! ' . the number of turns which can be wound on.divided by the rate of unwindinq in revolutions per minute-The whole series. of calculations can, of course. be worked inreverse. That is. startinq with a horsepower output requirement arid aknown value of maximum permissible stress. suitable geometricproportions to r the sprin9 ce [lbed eterrni ned. togeth er wit h the" umbe rof w.nding turns available for the required rate of revolution andnumber of complete revolutions,Note: as a pracncai design feature the d i, a meter of the inne r coi I ofthe spring. in the fuHy wound condition, should not be less than 12times the spring str: o thickness. That is, the spr.ng should be woundon an arbor of this rninirnu m size, H wound up to a smaller diameter thespri ng 1s i.k ely to suffer fro m fat,g I J e effects.- - - - - - ~ - - ~ - - - ~ - - ~ - - - - - - - - - - - - - - - - - - - - - - - - ~~------CONSTANT FORCE SPR'NGS 7Constan t force springs are asp ec iaI tvp e 0f fjat stri p spri ng. presn essedto have a 1.10 i for m tenden C y to CU T I alon g its whol e - !engt n.' The V canbe used in two ways (see also Fig. 19),(i) Rolled ont Q a bush in9 to for m a constant force extenslon spr in9,

because the reststance to unrolling is the same at any extension.(ir) Reverse rwound a eo un d a second dru m to provfde a constanttorque spr ing, 0 r con st a11t to rqU e spr ing motor, S p,i ngs 01 t n i : ! i - t y p . . : ! ! ;;Iremad e b y Ten!ji i i rtGr l i mued, Acron Lane , H ar le5d'!:! .r ' I I , london,NW ' O . Thev are ikn ow n as : Tensstor' s .p rin gs ill" llthis COUfl1cy; and Ne.gate( s.pr ing9- in

A.rn:eti ca .2"1


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CONSTANT FORCE S PA ING SFig. 19

by ~II ing (11'\10 0 1 , : 1 1 ) 1 , , 1 ~ iI'I~ Gt ~QoOIpIOOUClI!!S 0- C flI'I"!;l-;]rl t fore .. :!! ~ dl!!n-5 i00 ~r ig

-~~" ~t\ j

b > , r-!V-E!1"54!' w i :l "l d i 1 " 1 9 O ( ) ta . 0 'afger iClrlllTllpro-dUo;A: :$ IJ c oo~tal'"ll torque ~ F 4 " in'9 motor

Constant torce springs of this type have the advantage of beinq moreco mpac t when re Ia xed, compared wit h hel ica I spri ngs. pi us the factth i;l t v@ry 1On9 e xten sion s are poss ible_ Eithe r end ca n be fj xe d toproduce an extension spring asshown diagrammatically iI)fig. 20- Thef i xed free - end config uration, for exampie, has proved panic ule rIyeffec tive for brush spr ingson electrlc motors.Fig. 20

,


32/40

CONST AN 1 ' FO l i CE SP RINO SThe constant to/que or spr~ng motor form is particularly interestingsince it offers a performance far supe rior to an ordinary ctockwcrkmotor, pertic ular~y in th e lenqt h of r u0 possi ble .a nd the greale rmechanical efficiency because 01 the absence of i ntercoil friction. Itsperformance can also be predicted quite accurately.

Extension Spring DesIgnFig_ 21 shows the static parameters of a 'Tensator' ex-tension sprjng_

F'g. 21 strip width::. b :. t

____.__obbin - .I i ' I I If-r---- l_

The toad to extend can be celcuteted directly from the load factor forthe material (see Table V) and the spring width and thickness-

p : ; ; : QbtThe work;ng extension of the spring (X ) will be specified. but can

also be determiner' from the actual iength of spring strip.X ' :::!:. L- 60:;:where L is the tota~ length of spring.

(Now' this formula allows for 1:} dead turns on the coil).The foHDwing formulas can a~sohe used to deterrnlne D'f and D2D1 = . , . , /1275(X + 4-750~1 t + D~

I" design this Should be increased by at laas t 10 p e T cent to be on thesale sid e . r 0 . a ' low ",0r a i r space betwee n the co iIs.O~ = 1 2 x natural free diameter of spring.. as made -

29


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34/40

MUlT1PLE lEAF SPRINGS 8Basicatlv a lamlnated spring consisting of a number of in d lvidua: t eavesis no different to a 51rgle ~eaf spring. except that the additional leavesinc tease the effective thickness and ~hus red uce bot h deflecno n 8ndstress i n the ind ivld u a i leave s for a 9iven loa d. S tre ss ca Icu ta t ions a r eu su a I 'y based 0n 1he ass u m ption of a propo rtioo.ate load on e a c . h Ie-af(l.e. proportionate to th e number 0 - 1 leaves).

Fig. 23 quorter - I ! ' J l:pll4;

~ " - - C - ' p " 4...... l --- - - ' 1 I- - - - L--It-.-I I- - . 1 : . . . . - _ . --...j

Three common configurations fer multiple leaf spr~ngs are shown inFig_ 23- The follow~ng deflection formulas apclv:Half elliptic;

Qua rtsr ell iptic: 4PL3Deflection = ---E b t : 3nD f~ . _ Pl3e secnon - 2Eben

where E ::::modulus ot elasticitvof so ring materi ain = number of leaves

Half elliptic C8l1t i leveJ:


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M U L T I P 'L E l E A F SPRINGSTh e cones pond.,9 stress for m u Ias . a r' (the rnaterl a j be in9 stressedin bending as with simple flat springs).

Half all iplic:15Plstress = = b;2t n

Quarter elliptic:6PLStress =--bt2nHalt eiiiptlc Cantilever:3PLStress =--bl~n

The re are sever aI possi hie des ign app roaches . lf the th ickness ofea-ch leaf (t) is decided. the spring w~dth (b) necessary to produce therequired deflectioo wit h 2 . . 3. 4, etc '. teaves can be ca leu latad, usin9the appro pri ate def lsctio n to rm I . J la_For axa mpie. 1n the csse 0f a quarterelliptic spring

4PLab=-~-----Et . J n x. del lect ionThis wit ~give suitabl e spr in9 9cornet ry wilh 2, 3. 4 leaves. et C O , f tomwhich the most attractive can be selected. This value of b can then beused in th e stress form u Ia to cnec k . tha 1 the max . mum perm isslb Ie

materia I stress is not exceed ad. ~ so. the nan altern ative sol urlo n mustbe ad opted (e.9_more leave sand srnaI~e width); Or the calcuIationsre-done starting with a different (higher) value of thickness (t).Sometimes it is simpler to work directlv from the load the spring willcarrv, which can be arrived at by rewritinq the stress formulas:load ca pa bi Iity:

H If 1 1 - . bt~nSIl-a e o p n c = 1.5la II" bt2nSpuarrer e ipnc = 6L

H If ~I' . C 'I bt ; : ; !nS~a e~ipnc anti ever = 3Lwhe re S j1 is th e max imum oermiss ib le materia Istress in bending,

A serie s of alternatlve spr iog desig"s ca n the n be worked 0ut in32


36/40

APPENDIX Aterms 01 different values of width (b). thickness (t) and number ofleaves, al~ of which would be capable of carrying the required load.' t j5 then a matte r of eslcu J ati n9 the de '~ectioo of eac h of these spr.ngsand deciding 011 the most suitable one. If none give a suitable value10r deflecti on, the n fu rther al te rnaties must be worked OUtbeari I1g inmind:spring stiffness i ncre asss i n dire c t pro porti 0n to spri ng width {b) andnumber of leaves (n):,nc rease S ; n dtreet pro po r t ion to th e cube of the teaf

thickness.

APPENDIX A S PR LNG TERMINOLOGY{and standard units)load (P) is the force in pounds (or kirograms) exerted on Or by 3spring producing or modify.ng motion, Of maintaining a force systemin equilibrium, Load 1$directly proportional to dsilecticn and is limitedby th e elastic l im it o f the spring mater ia l ,Deflectio n is the maximu m movement of a spring from its free lengthor free position to a specified operating position. I n the case of helicalco~~spdngS F deilect.on pe r co. I is equa l to the total deflection dividedby the number of active Coils,Rate or load rate is equal to load divided by deflechonF and is thusinversely proportional to the number of active coils in a coil spring.Free length ~s the true dimenslonal ~ength of a spring in ~tsun!oadedposition.Solid height is the geometric height (or length) of a col I spring whenit is ful~y compressed,Active coils - the number of coils in a coi I spring wh.ch deflect under-oad. End turns or p a r r - turns on 3 cornpression spring whiCh do nottake par t i0 d eflectio n are referre d to as 'dead I coi ls,P itc his the spac fn 9 or pite h dim e n s io n between adj acen t acnve coi Isin a coi Ispring _ Pitch determines the number of col Is per unIt length,Spring tat e is a lso de,oeode nton p iteh, be in9 substenn a II Y constan t ifthe pitc h is constan r .

33


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APPENDIX AStress is the ope rating stress 0n the spri n 9 materiaI unde r workingcond iti ons. It is important bot h to use the right stress value for themateria I (e.9_ de pend ing 0n whether the sprm9 material is beingsubject to te nsi0n or compressio n~ bendi n9 0r torslo naI loadi n9); . a ndaIs0 ensure that a maxim urn perm iss ible stress l.g u re is not exceed ad.The Iana r depend s on bot h load . a nd heq uenty of deflection.Mean diameter (0) The mean diameter of a helical coil spring isspec.ned as the diameter to the centreline of the coil. The overalldiameter of a coi l spring is thus equal to D + d: and the inner diameterof a coil spring to D - d. Note that diameters can vary with workingin the case of a torsion spring.Wire diameter (d) the actual diameter or wire size used in a spnngmade from round wire.Spring index. This IS the rat~0 Did and is used to determ ine stresscorrect ion faetors where th e stress load ing 0n a spring is not slmp:e(e.q: he! ica I COm pression and exte nsion so rings. and torsion Spri ngs) .

34


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APPENDIX B: WIRE SIZES AND VALUES OF d3AND d~--~~-- : - . - ----_.- ----oj

s : . . . . - ( I ' , , , d" ~4 l ~ :~ r" t J ' ' ' .,j.1~.---------- .- - - ---.l.l .0 01or. o 00 00 01 o . . ' : Q , ( ) . , 0 0 0 IX 'O ' I)()I)I) I} 0 6 - 1 ) .[) C lOO :!"'600 o 00001 2 .960032 001' . , 0 0 0 () (" l1 J:3' I: )OL~l.11~l o %1 e 0002:2699 o Q O o o , J84-~BJ.-O 00 I ;I~~ .0 ( l(IOCiO', j~~ 0 () 000: )0002"07:1-0 e I ; I B : : ! : o OOI; l~:: ' j~ . .t~ (I 0001)1 -4 111H(I (I rJ G ()( l{IOOl1 !l'~ 0 - OO(l{ l t . i "jO"'~~ 1 I'} (lB:i! D 00'.)2' 5005 o O O D D r 5 ~ : 5 : r o: ! "8 0014 : ;. OOOOOi7o l. 4 o O O O O O C h : l3 a 41 { : . 16 0064 O(;OD].a.~, '* o 0OOOllij.nne (I~5 o ",}OO(I C l3: !; 7 5 - o C()OO(l(lIJ~:""I6: i'5 (;IM6 o OOO;'"! :7ol6:."1 (: 0 0000 .. J.g'50~J : i I " CI(116: 1 1 eooo (1" " 0 !il!i , ';I O()(IoaOOfi!:.~re o (1-66 I) ( lOO2'.fI.j '~(I. (I- ( I ( I ( IOf ss 74 r

I) (I~-' -0 -:)'):)00 4.9 1 :3 - 0 ClGOOOOOEl: ! : 52 : I) (l;jI (J 0DI)0J 0 O ? 15 Q O O O O L O ' ! i ' 1. : : " '5 (1m!) () (K l Q OQ 5 6P 1~ OO(I I ;n ;!CI1 ( )4 - !j I7 . ti - o(). o e & ( l0M:; ! :14.4. ; :1 -I) ()(t(I(I/l ~~, 4 -

(I. 01-91 1;}~1 () 0iXl00I) 1 :) o - : : l ~ 1 (1 0 06 g 0 1 ) (0 1)3 2 8.S ' o 00002266701}o 02 0 o 0 0 00 0 81 X1 (1 o (01)0001 &001}0 {I ()'O o I)IJro4JOO - :; . 0 00 02 40 0' (10C o 02 ' o O OQ O O5 '2 '5 1 (I- 000000' ~4:11 ' IV " oJ , -0 . ; : : q o : : l - ~ 1 9 1 , o OOOOZ!}4 , " ' ! " 1

: : -4. I) ( I : : : - ~ o f) 00. ;: 1 0 ~ ()e,4:9 o OOUOOOi3 ; " ; ' ! >9 ,~ 0 () ()oj::;! ( I to :!n: ;!~ -: ) OQ(I( I 'l ' -a n o sI) C2l; I) ee co 1 :2 ~ E o 1) OOO{I (1(12")" 9 8 40 o ()orJ (I I)I](t3 89 P i Y o ! 0 ;) 00002S1"!182~J 1} O~ o QlJOO136H ~ O OO DO DJ;J;l 7 00 " 1J,4 1 , ; 1 'Il1J040~21 OI; lI; lW2"~1ioI) O! S (1 ) 0 0 (1 0 ~ 5 E o~ "f I IJ I (00000391)620 0 1 ) 30 C I . : ; w ; : x : : < I 2 / O CQ o OOOOlJ I; I8 ~ 0 0 0 1- 4 o o e o ( l -o - . . " "( I o ~ ' : O o o l)I):)I). to~CI ( ! - 0 3 1 o I')(}tJ(Io 2 9 ~ !il1 o 00.:)(100 g :n ~ : < - a - oa l . : )00 : . ~::H.;": n D O i :I D 4 " l 0 4 0 1 l : :'

21 1 :;. d.!:? i).C '< JO O ~ 7 6 0 1 - 0 1 } 1 (} (I (K I 0 'l 0 4 8 5 8 o 08:l o 0tXJ.~_'1 : : f , t o OOOO~Eo112200: '; ;3 1; 1 -I).'XI(J J~9 : 3 ; i' o 00000' , 8592: I}OBl o (1005 -7 )"'9 o 001):}d1451t3-0 D14 (I tOO 3 : N t 4 . : ) a o O O ( l r :} :3~ZI I) 084 G 0iJQ0I...I~2/1' (I oI):):Xt.G $1 7H1 ; 1005 C ~ :2 8 r! ; 1 J ( ; t ; " ) [ ] o : ) o J , ~"J(l6; ; : ' (1095 (1 ) ( lOO-&'~~: ! o OI)OO52ZI;>I)I ;-

20 000( ; o 0CIC6! I (j-:; Il o o 00tXI011111"i11132 o 1 ; 191 ! i -0 r l I J 0 Kl ~ 0 6 L) O C O Q ! ! : 0 4 J O O I : I ;0031 O~SJ o I(}I 'J(J(JO 181' 'I 6 o Ol!:7 o ( I( .1 t }I ! I5 8~0 C ~ :r 28 g s0 OJ8 1 ;1 -I)I)LIII)5-' 9 72 o 1!J00(I(I2(18~1" (1 0 0813 (1 ) 001" "168 '~1 . . . OOO l ) 5 "! 1g159Eo0-00::1 C I O DOO5~19 O~2:: l. 'J-44. o OOg o ( ' - .0 - 4 " 3 - r o l ' (I { ! I)62"7" : ':2 l

19 e 0.110 e (: I ( : I 1 } 1 } B 4 1 : ; 1 co 1 ;1 t';::01 ~ .. ~~~ - J ( ). , o s I) O(Il" g,!:i ( ) 0 0 0- ,4 1 U !L . . . . . . . . . . . _ - _ . _ .. - _._- --------_.

35


39/40

APPENDIX B: WIRE SIZES AND VALUES OF d3 AND d4~ - &. .... -_ .......r

oj d: 1 1 '1 '9 'II d" 1 ; 1 4 l..... in d"" ,j1._._-

0 1 '0 0 '0013310 o 0 0 6 1 . t@ . . 1I

:0 - () 1 11 10 o 004() :9&O l I ~~3;1! iCJo1n 0 OO1J~11S - (I. 0001 5 1 1 ! 1 l 01) 18-1 (I ~1733 : -D 0 I J 0 C > 6 J ~ 900112 {)001 4(I.t 9 o ( ,( 10 15 135 I 0111;1:2 00042$1 S o {I 0 0 < I I I I a9 15O11:J OOO1 . .. .. 2 ! 1 o (IQQ, 1S l0S 01"3oJ (I 0().t.J:l 0 ')" -0 . C( IIO ~G 5 511o 11" {) OO14B115 o (:01 )1)1!iE ls(! 0 1 S o l I () ~"1-(;r! ;11 (). 0091 23 3 ; 5 1011 !I - O O ( n $2(1~ Q OO


40/40

IIMIDiLl:.. :=!il"'l1ill'lllli MA I t:t'liIAL::::ii ~1J16I:h.IJ '> orI.-ql~' ...G i~;~o"j , r . : : ,

L , r n . 1 oJ~~lo:o~I~,(:o"" I~I '~" ,

I t o ~ Ir 1 "~ '; ;I ~o !; I . .. . _.p up ro 0 ( . ' 0 1-IfLlo0 1 1 '~I'fI~tod : i o l o : : ~ 1 ,"""oI 'f :Hw Q -!i1;N':n n~1 ' " 1 'N'I~> I - !o , n !0 : > 5 J J le l !! 18 :9 W I I ' '5r....,"'w ' II !!~ r.;!1.....,~I : : ; t .o \ O I "n A " ~ ~~ , ,, ,, ,,PI 'H).-! , I>"( li t'l'o}fI~.li3-u~""r.-co~1il::l.,.II'~ I;.Q~P\"r-P I i . ( 1 r o , o , o 1" ! i ,h O '

- C " o !I o '" I I J-!". I ~;!o'I"'9 o"i! , '~~I;o, l lr U o ; c - 6(:0 PO:: I O ! : ~ r " l i ffi ~tl l " , , ' u ' " 1 " 1 - !1. . . . . :r,:"I -:::. ' U o l r .O ~ ~ .' C ~ - ;1 1 II " "' H: pjTABLE u WAHtS COR-RECTION FACTOR K FoRROUND WIRE HELICAL COIL

SPRINGS .__-. Spring I n d F . : ! - x Old Kr . _ - _ . -----2 2 06II : 3 - , 584 ~40I 5 1)16 1251 8 1H3j 10 1 14~2 1 12! 10 10 :9I 20 10,l 25 1 06;

..J

Mool:"~ c o ,~ . ~ L.I~E \:~I:SL:l 1 " 1 1

10': '"~"II"~"' I. e o o w W . o - w'~:Jo.Q(D9 Q ~ 2 ( 1 1 X 1O~ 2 " ! ). O c - J

, 21).1XJ(I, 00. ro o100 IXlOI _ ! . O . tto QOO Iso C < I O

J:(I ( lOC I . I X I OJ.{I 000.. : : :00:!::) oeo IX lO2 "~ 00000)2 " 9 (I(lQ00 [)a c 1)00 e o o1 s IXIO.-:)oo~000.000:i " 5 OC(I (XII): 0 6 - - 1 B . 500.0DiJ ', 6.000 -IX:(

, : ! " .ow.1.Xl{ )~1 .500. CQ:)< 1 '5 IXI 000RJ:... ; ) ( I I )I 1 . ! !r O O VO!)1 1 ! Y : ' K I ~ : ' L 1 P .

o( j ~:)Q to,:.: ; S O : : : o - t O ug. (0-;)0 00t)

I) 7.1XI:} CIC(Is soo ooo

TABLE III CORRECTION FACTORS fORRECTANGULAR WIRE H'EliCAL COil SPRINGS---_. ..

Number S tress D~fle cliortof aei i l l 'E! Iacior factr.)fcons t : : ; 1 K :J10 4'SO 55519 4-2-B 40120 390 3322 5 371 3-0730 360 2924,0 345 27660 330 26110,0 315 2S0 j2Q 3'09 2-40 ,50 304 2-38 !100 302 237 _

TABtE IV STRESS CORRCnON TABLE V DESIGN VALUeS fORFACTORS FOR TORSION SPRINGS IEN~ATOR' SPRINGSI [ Dosig. ---,S t l 'C 'S 'S SW~::;~Noumber factor fu r factor fOr Carbon Stai"lessof activo tOul"'ld ri!'CC~l'Igtllar ~ I~ nQ. oi s . E e e - 1 stt::e~: e vcl es 0 S . Q S~coils wirE ! K . . wire l(~ 1- _ .. . _ ...2 16' 154 5000 521 0023 660 00273 1'33 1 2 : 9 - 10,000 418 0020 502 0 0234 123 12-0 20.000 271 0,015 350 0019~ 118 1-l5 40,000 H39 0-010 233 o - o i z6 114 1-12 70.000 123 0009 151 00098 1,10 1 09 100.000 101 0,008 87 000810 1-08 1007 ,000,000 e~ 7012 1-06 106-1 -5 1 05 , '0420 1-09 , .00.25 1 04 103

calculo y diseño de resortes

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