Module 7 Design of Springs. Version 2 ME, IIT Kharagpur

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1 Modul 7 Dsign of Springs

2 Lsson Dsign of Hlical Springs for Variabl Load

3 Instructional Objctivs: At th nd of this lsson, th studnts should b abl to undrstand: Natur of varying load on springs Modification of Sodrbrg diagram Estimation of matrial proprtis for hlical spring Typs of hlical springs Dsign considrations for buckling and surg Dsign of hlical spring for variabl load In th arlir lctur, w hav larnd about dsign of hlical springs for static loads. In many applications, as for xampl in railway carriags or in automobil suspnsion systms th hlical springs usd ar constantly undr variabl load. Hnc, it is undrstood that whnvr thr is a variabl load on a spring th dsign procdur should includ th ffct of strss variation in th spring wir. Th mthodology usd is th modifid Sodrbrg mthod. w hav larnt about Sodrbrg mthod in arlir chaptr, hr, th ncssary modifications applicabl to hlical spring dsign will b discussd. In th cas of a spring, whthr it is a comprssion spring or an xtnsion spring, rvrs loading is not possibl. For xampl, lt us considr a comprssion spring placd btwn two plats. Th spring undr varying load can b comprssd to som maximum valu and at th most can rturn to zro comprssion stat (in practic, som amount of initial comprssion is always prsnt), othrwis, spring will loos contact with th plats and will gt displac from its sat. Similar rason holds good for an xtnsion spring, it will xprinc crtain amount of xtnsion and again rturn to at th most to zro xtnsion stat, but it will nvr go to comprssion zon. Du to varying load, th strss pattrn which occurs in a spring with rspct to tim is shown in Fig Th load which causs such strss pattrn is calld rpatd load. Th spring matrials, instad of tsting undr rvrsd bnding, ar tstd undr rpatd torsion. strss a max min = 0 tim a m Fig 7..1

4 From Fig.7..1 w s that, = = m a max (7..1) Whr, a is known as th strss amplitud and m is known as th man strss or th avrag strss. W know that for varying strss, th matrial can withstand strss not xcding nduranc limit valu. Hnc, for rpatd torsion xprimnt, th man strss and th strss amplitud bcom, Sodrbrg failur critrion max m = a = = Th modifid Sodrbrg diagram for rpatd strss is shown in th Fig 7... (7..) (, ) a Strss amplitud Strss amplitud m f A a c d Man strss Y FS b Sodrbrg failur critrion for springs Y Fig 7.. Th strss bing rpatd in natur, th co-ordinat of th point a is,. For saf dsign, th dsign data for th man and avrag strsss, a and m rspctivly, should b blow th lin a-b. If w choos a valu of factor of safty (FS), th lin a- b shifts to a nwr position as shown in th figur. This lin -f in th figur is calld a saf strss lin and th point A (, ) is a typical saf dsign point. m a

5 Considring two similar triangls, abc and Ad rspctivly, a rlationship btwn th strsss may b dvlopd and is givn as, a = (7..3) Y FS m Y whr Y is th shar yild point of th spring matrial. In simplifid form, th quation for Sodrbrg failur critrion for springs is 1 FS ( 1) (7..4) m a Y = + Y Y Th abov quation is furthr modifid by considring th shar corrction factor, K s and Wahl corrction factor, K w. It is a normal practic to multiply m by K s and to multiply a by K w. 1 FS K K (7..5) ( 1) s m w a Y = + Y Y Th abov quation for Sodrbrg failur critrion for will b utilizd for th dsigning of springs subjctd to variabl load Estimation of matrial strngth It is a vry important aspct in any dsign to obtain corrct matrial proprty. Th bst way is to prform an xprimnt with th spcimn of dsird matrial. Tnsil tst xprimnts as w know is rlativly simpl and lss tim consuming. This xprimnt is usd to obtain yild strngth and ultimat strngth of any givn matrial. Howvr, tsts to dtrmin nduranc limit is xtrmly tim consuming. Hnc, th ways to obtain matrial proprtis is to consult dsign data book or to us availabl rlationships, dvlopd through xprimnts, btwn various matrial proprtis. For th dsign of springs, w will discuss brifly, th stps normally usd to obtain th matrial proprtis. On of th rlationships to find out ultimat strngth of a spring wir of diamtr d is, As σ ut = (7..6) ms d For som slctd matrials, which ar commonly usd in spring dsign, th valus of A s and m s ar givn in th tabl blow. A s m s

6 Hard-drawn wir Oil-tmprd wir Chrom-vanadium wir Chrom-silicon wir Music wir Th abov formula givs th valu of ultimat strss in MPa for wir diamtr in mm. Onc th valu of ultimat strngth is stimatd, th shar yild strngth and shar nduranc limit can b obtaind from th following tabl dvlopd through xprimnts for rpatd load. Wir Typ σ ult σ y ult Hard-drawn wir Oil-tmprd wir Chrom-vanadium wir Chrom-silicon wir Music wir SS wir Hnc, as a rough guidlin and on a consrvativ sid, valus for shar yild point and shar nduranc limit for major typs of spring wirs can b obtaind from ultimat strngth as, y = 0.40 and = 0.0 (7..7) σ σ ult ult With th knowldg of matrial proprtis and load rquirmnts, on can asily utiliz Sodrbrg quation to obtain spring dsign paramtrs. 7.. Typs of springs Thr ar mainly two typs of hlical springs, comprssion springs and xtnsion springs. Hr w will hav a brif look at th typs of springs and thir nomnclatur Comprssion springs

.3 In th abov nomnclatur for th spring, N is th numbr of activ coils, i.., only ths coils tak part in th spring action.

7 Following ar th typs of comprssion springs usd in th dsign. (a) Plain nds Total coils, N T : N Solid lngth, L S : d ( N T + 1 ) +δ +δ Fr lngth, L : L S max allowan c Pitch, p : ( L d ) / N Plain nd spring Fig 7..3 In th abov nomnclatur for th spring, N is th numbr of activ coils, i.., only ths coils tak part in th spring action. Howvr, fw othr coils may b prsnt du to manufacturing considration, thus total numbr of coils, N T may vary from total numbr of activ coils. Solid lngth, L S is that lngth of th spring, whn prssd, all th spring coils will clash with ach othr and will appar as a solid cylindrical body. Th spring lngth undr no load condition is th fr lngth of a spring. Naturally, th lngth that w visualis in th abov diagram is th fr lngth. Maximum amount of comprssion th spring can hav is dnotd as δ max, which is calculatd from th dsign rquirmnt. Th addition of solid lngth and th δ max should b sufficint to gt th fr lngth of a spring. Howvr, dsignrs considr an additional lngth givn as δ allowanc. This allowanc is providd to avoid clash btwn to conscutiv spring coils. As a guidlin, th valu of δ allowanc is gnrally 15% of δ max. Th concpt of pitch in a spring is th sam as that in a scrw. (b) Plain and Ground nds Total coils, N T : N + 1 Solid lngth, L S : d ( N T ) Fr lngth, L L + δ +δ S : m ax allowanc Pitch, p : L / ( N + 1) Plain and Ground nd spring Fig 7..4 Th top and bottom of th spring is groundd as sn in th figur. Hr, du to grounding, on total coil is inactiv.

(d) Squard and ground nds Total coils, N T : N + Solid lngth, L S : d ( N T ) Fr lngth, L L +δ : +δ S max allowanc Pitch, p : ( L - d ) / N Squard and ground nd spring Fig 7.

8 (c) Squard or closd nds Total coils, N T : N + Solid lngth, L S : d ( N T + 1 ) Fr lngth, L L : S +δ max +δallowanc Pitch, p : ( L - 3d ) / N Squard or closd nd spring Fig 7..5 In th Fig 7..5 it is obsrvd that both th top as wll as th bottom spring is bing prssd to mak it paralll to th ground instad of having a hlix angl. Hr, it is sn that two full coils ar inactiv. (d) Squard and ground nds Total coils, N T : N + Solid lngth, L S : d ( N T ) Fr lngth, L L +δ : +δ S max allowanc Pitch, p : ( L - d ) / N Squard and ground nd spring Fig 7..6 It is obsrvd that both th top as wll as th bottom spring, as arlir on, is bing prssd to mak it paralll to th ground, furthr th facs ar groundd to allow for propr sat. Hr also two full coils ar inactiv Extnsion springs

9 Part of an xtnsion spring with a hook is shown in Fig Th nomnclatur for th xtnsion spring is givn blow. Body lngth, L B B : d ( N + 1 ) Fr lngth, L : L B B + hook diamtr. D/ hook hr, N stands for th numbr of activ coils. By putting th hook crtain amount of strss concntration coms in th bnt zon of th hook and ths ar substantially wakr zons than th othr part of th spring. On should tak up stps so that strss concntration in this rgion is rducd. For th rduction of strss concntration at th hook som of th modifications of spring ar shown in Fig Extnsion spring Fig 7..7 A complt loop is turnd up to a gradual swping curv A gradual rduction of nd turns from D/ D/ Extnsion springs with improvd nds Fig Buckling of comprssion spring Buckling is an instability that is normally shown up whn a long bar or a column is applid with comprssiv typ of load. Similar situation aris if a spring is too slndr and long thn it sways sidways and th failur is known as buckling failur. Buckling taks plac for a comprssiv typ of springs. Hnc, th stps to b followd in dsign to avoid buckling is givn blow. Fr lngth (L) should b lss than 4 tims th coil diamtr (D) to avoid buckling for most situations. For slndr springs cntral guid rod is ncssary.

10 A guidlin for fr lngth (L) of a spring to avoid buckling is as follows, πd (E G) L < C G+ E D L <.57, for stl blow. C (7..8), Whr, C is th nd condition and its valus ar givn C nd condition.0 fixd and fr nd 1.0 hingd at both nds hingd and fixd nd 0.5 fixd at both nds If th spring is placd btwn two rigid plats, thn nd condition may b takn as 0.5. If aftr calculation it is found that th spring is likly to buckl thn on has to us a guid rod passing through th cntr of th spring axis along which th comprssion action of th spring taks plac Spring surg (critical frquncy) If a load F act on a spring thr is a downward movmnt of th spring and du to this movmnt a wav travls along th spring in downward dirction and a to and fro motion continus. This phnomnon can also b obsrvd in closd watr body whr a disturbanc movs toward th wall and thn again rturns back to th starting of th disturbanc. This particular situation is calld surg of spring. If th frquncy of surging bcoms qual to th natural frquncy of th spring th rsonant frquncy will occur which may caus failur of th spring. Hnc, on has to calculat natural frquncy, known as th fundamntal frquncy of th spring and us a judgmnt to spcify th oprational frquncy of th spring. Th fundamntal frquncy can b obtaind from th rlationship givn blow. Fundamntal frquncy: f = 1 Kg W s f = 1 Kg 4 W s

11 Both nds within flat plats (7..9) (7..10) On nd fr and othr nd on flat plat. Whr, K W S : Spring rat : Spring wight =.47γd DN (7..11) and d is th wir diamtr, D is th coil diamtr, N is th numbr of activ coils and γ is th spcific wight of spring matrial. Th oprational frquncy of th spring should b at last 15-0 tims lss than its fundamntal frquncy. This will nsur that th spring surg will not occur and vn othr highr mods of frquncy can also b takn car of. A problm on spring dsign A hlical spring is actd upon by a varying load of 300 N to 900 N rspctivly as shown in th figur. Th spring dflction will b around 15 mm and outsid diamtr of th spring should b within mm. 300 N 900 N 15 mm Solution mm To dsign th spring for th givn data, th most important paramtr is th spring indx. Th spring indx dcids th dimnsion of th spring with rspct to chosn wir diamtr. Normally th spring indx varis ovr a wid rang from 3-1. For highr valu of th spring indx th curvatur ffct will b lss, but rlativly siz of th spring and strss in th spring wir will incras. Howvr, th ffcts will b som what opposit if th valu of spring indx is lowr. Hnc, it is bttr to start th itration procss with th spring indx of 6-7. Lt us start th problm with spring indx, C=6 and wir diamtr, d=7 mm. Th abov choic givs us a coil man diamtr, D =4 mm. Thrby, th outsid diamtr of th coil is 49 mm, which is within th givn limit. Computation of strsss: Th man load, Fm = = 600N

12 strss amplitud, F a = =300N Shar strss concntration factor, k s = = 4x Wahl corrction factor, k w = + = x So th valu of man shar strss, m = = 0.6MPa 3 π (7 ) and th valu of strss amplitud, a = 1.53 = 117.1MPa 3 π (7 ) Estimation of matrial proprtis: As no spcific us of th spring is mntiond in th problm, lt us tak Chrom Vanadium as th spring matrial. This alloy spring stl is usd for high strss conditions and at high tmpraturs, it is also good for fatigu rsistanc and long nduranc for shock and impact loads. Ultimat strngth of th matrial, 1790 σ ut = =134 MPa (7) From th rlationship of σ ult to y (yild point) and nduranc limit, w find that for chrom and y = σ ult 0.51 = 675.MPa = σ ult 0. = 64.8MPa vanadium, From Sodrbrg quation, y FS 1 FS a m y = m a y = + ( 1) y y = + 1 = FS FS 1.00 Factor of safty, FS=1.0 implis that th dsign do not considr any unforsn ffct that may caus xtra strsss in th spring. Normally in dsign of springs it is bttr to considr a factor of safty which should b in th vicinity of

13 In ordr to incras th valu of FS, in th nxt itration, natural choic for th spring indx, C is 5 and d = 8 mm. Bcaus C=7 and d = 6 mm will lad to mor strss on th wir and th valu of FS will not improv. With C=5 and d=8 mm and following th similar procdur as in prvious itration w hav, Thrfor, k = 1.1, k = s w m = = 131.3MPa 3 π a = = 78.4MPa 3 π 8 Matrial proprtis: Finally, 1790 σut = (8) = 197 MPa = MPa y = 59.4 MPa = + 1 = FS FS = 1.46 Th factor of safty obtaind is accptabl. Thrfor th valu of spring indx is 5 and corrsponding wir diamtr is 8mm. Hnc, man spring diamtr, D=40 mm. Outr diamtr of spring, D o =40+8=48 mm, This valu is within th prscribd limit. Innr diamtr of spring, D = 3mm Spring rat, k= = = 15 i 3 40N/mm N/m Onc th valu of stiffnss is known, thn th valu of numbr of activ turns, N of th spring is, 4 3 Gd N 3 3 8D N 8 (40) k 4 k = = = (40) 16 δ max = =.5mm In th abov quation, G = MPa.

14 Spring Nomnclatur: Lt us slct th typ of spring as squard and ground nds. For this typ of spring th valu of fr lngth is, L= L + δ + δ S max allowanc whr, L = dn = 8.0 (16 + ) = 144mm δallowanc S T = 15% δ max L = % δ 170mm max Chck for buckling: W know that for stl, L d Pitch, p = = = 9.65mm N 16 D L <.57 = 06mm C Hr, for th givn spring sat configuration, C = 0.5 Th fr lngth of th spring, 170 mm is lss than th critical lngth for buckling, 06mm. Thrfor th dsign is saf. Chck for critical frquncy: In ordr to find th critical frquncy of th spring, th wight of th spring is to b first computd, π d Ws = π DN 4 =.47γ Ws 3 3 WS =.47 ( 8X10 ) ( 40X10 ) = 7.74N ( )( γ ) d DN Thrfor,

15 Th fundamntal frquncy of th spring (for both nds within flat plats), f 3 1 Kg 1 40X10 X 9.81 = = 11.6Hz W 7.74 s Saf frquncy for dsign should b at last 0 tims lss than th fundamntal frquncy to tak car of mor numbr of harmonics. Thrfor, th spring frquncy for should b around 6 Hz. Qustions and Answrs Q1. Do th hlical spring xprinc rvrs loading? What is th loading typ calld whn varying load acts on a hlical spring? A1. Th hlical spring xprincs only rpatd load. It cannot xprinc rvrs loading, bcaus th spring will los contact with th nd supports. Q. What modification in Sodrbrg diagram is rquird whn it is usd for dsign of hlical springs? A. In th arlir Sodrbrg diagram, w hav usd in th dsign for varying loads on th machin mmbr, had only strss amplitud in th nduranc limit rprsntation, sinc, nduranc limit valu was for complt rvrsd loading. Hr, in spring dsign, w us nduranc limit valu for rpatd loads only. Hnc, w hav both strss amplitud and man strss valu of qual magnitud,. Thrfor, th nduranc limit rprsntation in Sodrbrg diagram changs to,. Q3. What should b th saf frquncy of a hlical spring? A3. Saf frquncy for dsign should b at last 0 tims lss than th fundamntal frquncy of th spring to tak car of mor numbr of harmonics. Rfrncs 1. V.Malv and Jams B. Hartman, Machin Dsign, CBS Publishrs And Distributors.3 rd Edition

16 . J.E Shigly and C.R Mischk, Mchanical Enginring Dsign, McGraw Hill Publication, 5 th Edition M.F Spotts, Dsign of Machin Elmnts, Prntic Hall India Pvt. Limitd, 6 th Edition, 1991.

4.2 Design of Sections for Flexure

4.2 Design of Sections for Flexure 4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt