EFFECT OF PITCH AND NOMINAL DIAMETER ON LOAD DISTRIBUTION AND EFFICIENCY IN MINIATURE LEAD SCREW DRIVES

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1 IJRE: Iteratoal Joural of Research Egeerg ad echology eissn: pissn: EFFEC OF PICH AND NOMINAL DIAMEER ON LOAD DISRIBUION AND EFFICIENCY IN MINIAURE LEAD SCREW DRIVES Vkrat Karmarkar 1, S.G.Jadhav 2, Rajesh Parab 3 1 PG Studet, Mechacal Egeerg Departmet VJI Mumba, Maharashtra, Ida 2 Assstat Professor, Mechacal Egeerg Departmet VJI Mumba, Maharashtra, Ida 3 Deputy Maager, Portescap Ida Pvt Ltd, Ida Abstract Lead screws are the devces whch are used for power trasmsso or to have lear moto. It s theoretcally assumed that appled load s evely dstrbuted amog the thread par cotact. However, practcally t s observed that load s ot uformly dstrbuted amog threads. he frst thread carres the maxmum load ad later the load o each thread reduces. Numerous studes have bee carred out for aalytcal calculato of the load dstrbuto usg sprg stffess method. But these studes are for screw ad ut combato. Not much study has bee doe to fd load dstrbuto o threads of a lead screw. he maxmum load actg o oe thread s a mportat parameter lead screw desg. he load decdes the fatgue lfe of the screw ad ut. o have better lfe of threads, the load dstrbuto should be uform to have fewer loads o sgle thread. he load s also mportat to kow the deflecto of thread whch affects the postoal accuracy of the lead screw drves. hs paper focuses o aalyzg mathematcally the varous thread parameters whch affects the load dstrbuto threads ad the correspodg effect o effcecy. he sprg model method proposed [1], [4] has dfferet costat coeffcet whch are depedg o thread geometry ad materal. If there are umbers of threads cotact, there wll be (-1) umber of equatos (-1) ukows. hese are lear dfferece equatos ad ca be solved by matrx elmato method. he results obtaed from aalytcal soluto are valdated wth the FEM (Fte Elemets Method) results. Keywords: Lead Screws, Load Dstrbuto, hread Parameters, Effcecy, Lear Drves *** INRODUCION Lead screw drves are smple, effcet, accurate ad less costly devces for force trasmsso ad lear moto applcatos. he basc prcple behd lead screw / power screw s smple as that of ut ad bolt mechasm. However, the thread agle used ut-bolt s such that t creases the frctoal cotact so that t ca be used deally for fasteg purposes. However, lead screws are devces for lear force trasmsso or for lear moto. he threads used lead screws are ether square threads or trapezodal threads. Square threads have best effcecy but are weaker at the root. Also they are costly for maufacturg. rapezodal threads o the other had are easy to maufacture ad thcker at the root of thread so they ca carry hgher load.. I ths paper, too, trapezodal threads are aalyzed. Varous studes have bee carred out to fd load dstrbuto the threads the screws. D. Mller [4] proposed sprg model for load dstrbuto threaded coectors. He developed mathematcal equatos whch were formulated assumg thread as a sprg ad calculated the stffess bedg. he mathematcs was verfed usg fte elemet aalyss. He formulated the equatos assumg o yeldg threads. W. Wag & K. M. Marshek [1] modfed the equatos formulated by D. Mller [4] ad proposed a model for yeldg thread codtos. Hua Zhao [3] proposed a ew vrtual cotact loadg method order to smulate the threaded coectos. he model was foud to be satsfactorly effcet ad accurate. Davd J Murphy [5] studed the load dstrbuto lead screw wearg uder varyg operatg codtos. W. Wag & K. M. Marshek [2] studed the load dstrbuto threaded coector havg dssmlar materal ad varyg thread stffess. Not much study has bee carred out to study the effect of varous thread parameters o load dstrbuto lead screw threads. hs paper focuses o how thread parameters affect load dstrbuto ad effcecy of a mature lead screw. he mathematcal data obtaed from aalytcal soluto has bee verfed wth Fte elemet methods. 2. MAHEMAICAL MODELLING he effcecy of a lead screw s a fucto of coeffcet of frcto. Lower the coeffcet better s the effcecy. Steel Broze combato s observed to have least frctoal coeffcet. Sce screw has to wthstad the bucklg alog wth the other body stresses as the ut, lead screw must be stroger tha ut. hs facltates the fact that whe ut wears out after log use, t s easy ad cheap to replace a ut tha a screw. I ths curret paper, steel for screw ad broze for ut s used. here are two cases of teso ad compresso as used [1] & [4]. Volume: 5 Issue: 4 Apr-216, 2

2 IJRE: Iteratoal Joural of Research Egeerg ad echology eissn: pissn: Case 1: Compresso I ths case, there wll be compresso of screw spdle. hs wll happe whe ut s movg away from the drvg ed of the motor or gearbox uder loadg codtos. For represetg screw ad ut a sprg model, varous sectos screw ad ut has to be represeted terms of sprgs. Uder loadg codtos, the thread of the screw acts as a catlever beam uder uformly dstrbuted load. he stud secto betwee two cosecutve helxes acts as smple crcular bar of costat cross secto (equal to mor dameter of screw) uder compresso. Smlar s the case for ut. Whe oe thread of a ut s cotact wth the respectve thread of screw, the axal deflecto for threads s same. So the resultg stffess s the combed stffess of ut & screw thread seres. hs s show fgure below. he body sectos betwee the threads have a sprg costatk s = L/δ s, where δ s represets axal deflecto due to load L. Smlarly, stffess of the stud secto s gve byk = L/δ, where δ represets elogato of the secto uder load S. he absolute axal deflecto of ay thread rg s the addto of the dvdual axal deflectos of all the prevous threads. So the absolute deflecto of k th thread s gve by, u s k = δ k = k δ s j j =1, 1 k (2a) he dfferece betwee axal deflectos of ay two cosecutve threads wll gve elogato of the stud secto betwee those threads. herefore δ = u s u s +1, 1 1 (3a) Puttg equato (2) equato (3), δ δ +1 δ s +1 = δ, 1 1 (4a) Now, the deflectos ca be expressed terms of appled loads ad sprg stffess. So equato (4) yelds, P K P +1 K +1 L +1 K s +1 = S K (5a) Fg -1: Equvalet stffess of threads bedg So the helxes of the screw are assumed to be lke thread rgs coected va the resultg sprg stffess. hs s better explaed the fgure. he load F appled o the stud s carred to threads by load P, where P represets load o th thread. S s the load o the secto betwee the threads & +1. L s the load o the body secto. Load o 1 st thread wll be the force appled F mus load o stud sectos betwee threads 1&2. he stud secto betwee th ad 1 st thread carres full load F. SoS =F. Sce there are threads cotact the body sectos betwee th & 9 th thread carres o load. So S =. hese are the costats whch are useful solvg the dfferece equato. he load o th thread ca be gve as P = S 1 S, 1 (1a) he load o body secto correspodg to k th thread s the load o the remag threads of the secto from thread rg k to last thread. So, Usg equato (1), L k = j =k L k = S k 1 S P j (6a) As dscussed earler, the stud secto betwee last actve thread ad the ext correspodg thread carres o load. So S =. he equato (6) gves, L k = S k 1 (7a) Usg equato (1) ad (7), equato (5) ca be modfed as S 1 S 1 + K K +1 + K K+1 + K s K K + S +1 K +1 = (a) Fg -2: Sprg model for compresso case Where,β = 1 + K S 1 β S + α S +1 = (9a) K +1 + K K+1 + K s K, Volume: 5 Issue: 4 Apr-216, 21 α = K K +1 For threads cotact, =. Puttg =1 to eght equato (9), we get correspodg equatos varable S. For =1, S β 1 + α 1 =.

3 IJRE: Iteratoal Joural of Research Egeerg ad echology eissn: pissn: SceS = F, the equato becomes, β 1 + α 1 = F () Smlarly, for other values of, β 2 + α 2 = () β 3 + α 3 = () β 4 + α 4 = (v) β 5 + α 5 = (v) β 6 + α 6 = (v) β 7 + α 7 S = Sce S =, the last equato ca be modfed as β 7 = (v) As the lead screw materal s uform throughout, K = K +1 he value of β ca be gve as β = 2 + K K+1 + K s K he seve equatos seve varables derved above ca be represeted matrx format β 1 α 1 1 β 2 α 2 1 β 3 α 3 1 β 4 α 4 1 β 5 α 5 1 β 6 α 6 F 1 β 7 = Also for uform bolt materal, α = 1. So the matrx becomes, hese matrces ca be solved by elmato method to get the values of S. 2.2 Case 2: eso he two cases of compresso ad teso are smlar. I teso case, the ut s comg travellg towards motor or gearbox ed, where the screw s attached, uder loadg codto. hs wll result teso the screw part hece the ame teso case. For mathematcal modelg of ths case, oly dfferece s wth the locato of fxg pot. I compresso case, the locato of fxg pot s ear the frst actve thread, whereas teso case, fxg s doe ear the last actve thread. It s worth otg that the drecto of appled force s same both the cases. he sprg model dagram for teso case s show the fgure 3. I teso case, load o body secto correspodg to k th thread s the sum of all the loads from thread 1 to k. So, From equato (1) L k = k j =1 P j L k = S S k he absolute deflecto of a thread rg s the combato of local thread rg deflecto ad sum of all the deflectos of the rgs from that partcular rg cosderato to last actve thread. hus, u s k = δ k + j =k Usg ths equato equato (3a), δ δ +1 + δ s = δ (1b) he deflectos ca be expressed terms of load ad stffess as P K P K + L K = S s K Usg equatos for P &L k, δ s j β β β β β β 6 1 F 1 β 7 = (1a) S 1 S 1 + K +1 K + K K + K s K + S +1 = S K K s K K +1 (2b) Followg the same approach as was doe for compresso case, S 1 β S + α S +1 = γ S Where γ = K K s (3b) Volume: 5 Issue: 4 Apr-216, 22

4 % of Load IJRE: Iteratoal Joural of Research Egeerg ad echology eissn: pissn: For all values of from 1 to, we have seve equatos seve varables as β 1 + α 1 = γ 1 F (v) β 2 + α 2 = γ 2 (v) β 3 + α 3 = γ 3 (v) β 4 + α 4 = γ 4 (x) β 5 + α 5 = γ 5 (x) β 6 + α 6 = γ 6 (x) β 7 + α 7 S = γ 7 β 7 = γ 7 (x) 3. LEAD SCREW DESIGN: For determg the dmesos of lead screw ad ut, ASME/ANSI B [7] stadards for acme screw were used. he accuracy class used was 2G. he omal dameter was kept costat at 5mm ad the ptch was vared from 1 mm to 1.5 mm. Due to toleraces gve o major ad mor dameters, there are two values (max & m) of each dameter. hus for calculatos, average of the dameters was take. he values of the obtaed dmesos are summarzed the table 1. I matrx form Fg-3: Sprg model for teso case β 1 α 1 1 β 2 α 2 1 β 3 α 3 1 β 4 α 4 1 β 5 α 5 1 β 6 α 6 1 β 7 = γ 1 F γ 2 γ 3 γ 4 γ 5 γ 6 γ 7 β β β β β β β 7 γ 1 F γ 2 γ 3 γ 4 γ 5 γ 6 γ 7 = (4b) Above equatos (1a & 4b) are smple dfferece equatos seve varables. hese were solved by a ole equato solver program [6], the solutos obtaed from the solver were verfed by Gaussa Elmato Method maually. able 1 : Major & mor dameters of screw & ut for dfferet ptch (All dmesos are mm) SCREW NU D p D maj D m D maj D m he sprg stffess costats are summarzed the table 2. able 2: Sprg stffess costat for screw & ut (K X*1 5 N/mm, p mm) No p K sb K sc K bb K bc K RESULS: 4.1: Effect of Ptch: 4.1.1: Effect o load dstrbuto: A load of 1 N was appled ad the load carred by each thread was plotted agast the thread umber for both teso ad compresso hread umber Fg-4: Effect of ptch o load dstrbuto compresso Case (sprg model) Volume: 5 Issue: 4 Apr-216,

5 % of Load % of Load Effcecy. % Effcecy % % of Load IJRE: Iteratoal Joural of Research Egeerg ad echology eissn: pissn: hread umber Fg-5: Effect of ptch o load dstrbuto teso case (Sprg model) For verfyg these results Fte Elemets Methods, Asys workbech (15.) was used. he model was 2D axsymmetrc, elemets sze was.5mm ear the cotact area. he results were plotted ad summarzed the fgure 6 & hread umber Fg-6: Effect of ptch o load dstrbuto compresso Case (FEM) hread umber Fg-7: Effect of ptch o load dstrbuto teso case (FEM) 4.1.2: Effect o Effcecy: he effcecy mproves as ptch creases. he mprovemet effcecy ca be see the fgure below. Fg-: Effect of ptch o Effcecy. Actually, the mprovemet effcecy wth ptch cotues wth a ptch value of 13mm, where effcecy reaches maxmum of 73.43% ad the t reduces. But for a ptch value of 1 mm to 1.5 mm, mprovemet effcecy s lear. hs ca be see fgure 9 below Ptch, mm Fg-9: Effect of ptch o Effcecy for dfferet coeffcet Of frcto o have more lfe of threads, the load carred by sgle thread should be as less as possble. hs ca be acheved by eve load dstrbuto by keepg the ptch mmum whch s evdet from fgures 4 & 5. But the reducto ptch hampers the effcecy whch ca be see fgure. hus to have best effcecy alog wth mmum load o a thread, the rato (Effcecy /Load) should be as maxmum as possble. hs s compared the table 3. able 3: Varato of Effcecy & Load wth ptch p(mm) ɳ (%) P(N) ɳ/P Volume: 5 Issue: 4 Apr-216, Ptch (mm) µ=.1 µ=.13 µ=.15

6 % of Load % of Load Effcecy % IJRE: Iteratoal Joural of Research Egeerg ad echology eissn: pissn: : Effect Of Nomal Dameter ( D ) : 4.2.1: Effect o Load dstrbuto: he omal dameter was vared from 5 mm to 6 mm wth a step of.2 mm for a costat ptch of 1 mm. he effect of ths o load dstrbuto s summarzed the fgures 9 & 1. It s evdet from these graphs that the crease dameter does ot have much effect o load dstrbuto. here s very lttle mprovemet load dstrbuto wth crease dameter. But the crease reduces the effcecy. hs ca be see from the fgure 11. Sce effect of dameter s ot sgfcat, the results from FEM model are ot cluded here Nomal dameter Fg-12: Effect of Nomal dameter o Effcecy hread umber D=5 D=5.2 D=5.4 D=5.6 D=5. D=6 Fg-1: Effect of Nomal dameter o load dstrbuto Compresso case (sprg model) hread umber D=5 D=5.2 D=5.4 D=5.6 D=5. D=6 Fg-11: Effect of Nomal dameter o load dstrbuto eso case (sprg model) 4.2.2: Effect o Effcecy: Effcecy reduces wth crease omal dameter. hs s due to the crease frctoal torque. he varato ca be see fgure CONCLUSION he load dstrbuto amogst thread was foud to be very ueve wth the frst thread carryg maxmum load. Icrease ptch causes more ueveess load dstrbuto ad load o frst thread creases. Chage omal dameter dd ot have much effect o the load dstrbuto. he sprg model preseted [1]& [4] was used for calculato of load dstrbuto amogst threads ad t gave satsfactory results compared wth Fte Elemet Methods. Amog teso ad compresso case, the load o frst thread was maxmum compresso case. Form table 3, optmum ptch ca be decded depedg o whether lfe of thread s mportat or effcecy s mportat for a partcular applcato. NOMENCLAURE p = ptch of thread D =Nomal dameter of thread P = Load carres by each thread F = exteral appled load K s = axal stffess of the screw secto betwee the Cosecutve threads K = axal stffess of the ut secto betwee the Cosecutve threads K sb = stffess of thread of screw bedg K b = stffess of thread of ut bedg K = equvalet sprg stffess bedg of thread par L = load o body secto = o of actve threads S = load o ut secto u = deflecto of ut thread rg δ s = axal deflecto of correspodg sprgs havg Sprg costat K s δ = axal deflecto of correspodg sprgs havg Sprg costat K δ = axal deflecto of correspodg sprgs havg Sprg costat K α, β & γ = coeffcet used fte dfferece equatos Volume: 5 Issue: 4 Apr-216, 25

7 IJRE: Iteratoal Joural of Research Egeerg ad echology eissn: pissn: REFERENCES [1]. W. Wag, K. M. Marshek, Determato of load dstrbuto a threaded coector wth yeldg threads, Mech. Mach. heory Vol. 31, No.2, pp (1996). [2]. W. Wag, K.M. Marshek, Determato of the load dstrbuto a threaded coector havg dssmlar materals ad varyg thread stffess, ASME, Joural of Egeerg for Idustry Vol 117 (1995), pp 1-. [3]. Hua Zhao, A umercal method for load dstrbuto threaded coectos, rasactos of ASME Vol. 11(1996) pp [4]. D. Mller, K. M. Marshek, Mohammad Naz, Determato of load dstrbuto a threaded coecto, Mechasm ad Mache heory Vol. 1, o. 6, pp [5]. Davd Murphy, Load dstrbuto o lead screw threads wearg uder varyg operatg codtos, ASME Joural of rbology, Vol. 13 (2). [6]. [7]. Machery s Hadbook 27 th Edto, pp Volume: 5 Issue: 4 Apr-216, 26