ME311 Machine Design

Transcription

1 ME11 Mchine Design Lecture 10: Springs (Chpter 17) W Dornfeld 9Nov018 Firfield University School of Engineering A Free Body Digrm of coil spring (cutting through nywhere on the coil) shows tht there must be torsion on the coil to blnce the lod. Compression Springs Coil springs hve these fetures: Wire dimeter, d or d wire Coil Men dimeter, D or D men Coil Inner Dimeter, ID D d Coil Outer Dimeter, OD D + d Spring Index, D C d You cn think of the OD s the Men Dimeter plus twice the wire rdius, so OD D + r D + d Hmrock Pge 495 1

2 Spring Index It is seldom prcticl to mke spring with n Index, C, less thn or greter thn 1. A smll Index mens lrge curvture, nd lrge index mens smll curvture. Springs with C in the rnge of 5 to 10 re preferred. Springs with smll C re hrd to mnufcture, nd hve lrge stress concentrtions due to the tight curvture. C C 1 Hmrock Eqn Stress In Springs P Torsionl Sher Tr Td () 8PD tors,mx 4 J + Direct Sher P 4P dir,mx A D/ P Totl Sher tot,mx 8DP d 1 + D Hmrock Pge 495

3 tot,mx 8DP d 1 + D Stress In Springs The term in prentheses is constnt, so this cn be rewritten: 8DK d P tot,mx [Eqn. 17.8] Where Kd, the Trnsverse Sher Fctor, is K d 1 C d D C C Torsion Direct Use this for STATIC loding. If the Spring Index, C, rnges from to 1, then the Direct Sher is from 1/6 th to 1/4 th of the Torsionl Sher. Hmrock Pge 495 Spring Stress Exercise A spring mde from 0.1 in. music wire hs n outside dimeter of 1 inch. If it hs lod of 5 Lbs pplied to it, wht is the mximum sher stress?

4 Curvture Effects Remember curved bems & their stress distribution? Adding the effect of curvture drives up the stress t the Inner Rdius. For cyclic (dynmic) loding, we include this curvture effect, nd write: 8D K P w tot,mx [Eqn ] where 4C K w + 4C 4 C Curvture Direct Use this for CYCLIC loding. Hmrock Pge 496 Sher Fctors Another Cyclic correction fctor is the Bergsträsser fctor, K b K b 4C + 4 C It is simpler nd very close to K w K Spring Index, C Clerly, the lower the Index, the higher the curvture, nd the higher the mx sher. Kw Kb Kd 4

5 Spring Mterils There re very limited number of mterils commonly used for mking springs, listed in Tble The llowble sher yield of these mterils Ssy 0.40 Sut, nd the Sut vries with wire size! The vrition is: A S ut d p m [Eqn. 17.] [Eqn. 17.] where Ap nd m come from Tble 17.. Cution: Use Ap in ksi with d in inches, nd Ap in MP with d in mm. A plot of Sut versus wire dimeter for the mteril in Tble 17. is shown on the next two slides, first in Metric nd then in English units. How much fctor of sfety did our spring hve? Hmrock Pge Min. Ultimte Tensile Strengths of Common Spring Wires From Hmrock, Tble 17. & Eqn Music Wire Sut (MP) Music Wire Chrome Silicon Chrome Vndium Oil-Tempered Hrd Drwn Wire Dimeter (mm) 5

6 Min. Ultimte Tensile Strengths of Common Spring Wires 400 From Hmrock, Tble 17. & Eqn Sut (KSI) Music Wire Chrome Silicon Chrome Vndium Oil-Tempered Hrd Drwn Wire Dimeter (in) Spring Deflection Cstiglino s theorem gives spring deflection s 8PC N Gd C δ [Eqn ] Becuse C is usully between nd 1, the second term would be between nd 0.005, nd the eqution is often shortened to 8PC N Gd δ [Eqn ] thereby ignoring between 0.5% nd 5.6% of the deflection. I suggest tht you generlly use Eqn , dropping the second term only if C is lrge. In both equtions, P is the pplied lod on the spring, G is the mteril Sher Modulus E, nd N is the number of ctive coils. (1 + ν ) 6

7 Active Coils The number of ctive coils depends on how the ends of the coils re finished. Squred Plin Plin & Ground Squred & Ground [~Tble 17.] Ends Plin Plin & Ground Squred Squred & Ground N end 0 1 N ctive N tot N tot - 1 N tot - N tot - L solid d(n tot + 1) d x N tot d(n tot + 1) d x N tot Pitch* (L free d)/n L free /(N +1) (L free d)/n (L free d)/n * Pitch is ONLY mesured when spring is UNLOADED! (L L free ) Spring Stiffness The stiffness, k, is the force per deflection. Gd k [Eqn ] 8C N (1 0.5 C ) + If C is lrge, this cn be reduced to Gd Gd 4 k 8C N 8D N Note: k, δ, nd re functions of d, D, N, G, nd P, but NOT of pitch nd therefore not L free. 7

8 Spring Exmple My Grge Door spring (yes, it s n extension spring, but it is close enough). d wire in. OD in. N tot 160 turns L free 6. in. L extended 6.5 in. G 11.5 x 10 6 psi D men OD d in. D 1.45 C 8.0 d Gd ( )(0.155) k.687lb / in 8C N 8(8.0) (160) Force k l (.687)(6.5 6.) 97.lb 4C K w C 4 C 8D K w P 8(1.45)(1.18)(97.) π (0.155) ksi C So we missed <1% Spring Force nd Deflection Two things to (lmost) lwys count on. P k x L P k x L ❶ At Free Length P 0 ❷ A chnge in spring length is lwys ccompnied by chnge in spring force k x L, up until the spring bottoms out. At Solid Length 8

9 Spring Buckling To mke it stble, you cn guide it on the inside or outside. Like ll long, skinny things with lod on them, springs cn buckle. Buckling is relted to the Free Length of the spring, nd to the End Conditions. Spring Vibrtion Springs cn vibrte longitudinlly (or surge) just like Slinky: The frequency is f n d G πn D ρ Hz [Eqn. 17.0] Here G is the Sher Modulus, nd ρ is the mss density (or weight density divided by g). To void resonnces, void cyclic loding spring ner integrl multiples of f n. For steel springs where G nd ρ re constnts, this cn be simplified to: For the grge door spring, f n 8.7Hz. For Informtion Only 1,900d f n Hz ( d nd D in inches) ND 5,000d f n Hz ( d nd D in mm) ND 9

10 Ftigue / Cyclic Loding of Helicl Springs Helicl springs re NEVER used s both compression nd extension springs (Hmrock, top of Section 17..7). Therefore, loding is never fully reversing, so we will use the modified Goodmn digrm insted of n S-N plot. 1. Get the stedy (men) nd lternting lods, P men nd P lt.. Compute the men nd lternting shers, using K Whl for BOTH: 8DK P W men men π d. FOS ginst yielding: SSy 0.4S n s + lt men 8DK P lt π d mx UT W lt 4. FOS ginst ftigue (Infinite life): n s S SE lt Design for Finite Life If finite life is specified, use the S-N digrm to compute the llowble sher stress for N cycles of life, to use in the Goodmn digrm (S se ): Use S L 0.7 S SU becuse this is Torsionl loding [Eq. 7.7] S SU is the sher ultimte strength: S SU 0.60 S UT [Eq. 17.9] Use S SE 45 KSI for unpeened springs, nd [Eq. 17.8] S SE 67.5 KSI for peened springs for mterils in Tble 17. with wire dimeter d < /8 (10mm). Note tht these S SE re corrected for ALL modifiction fctors EXCEPT relibility, k r. S-N Digrm S SU 0.6 S UT S L 0.7 S SU See Figures on next slide. S SF S SE k r x (45 or 67.5 ksi) N Life 10

11 Design for Finite Life S SU 0.6 S UT S-N Digrm S Sy 0.4 S UT Modified Goodmn Digrm S L 0.7 S SU lt Zero to Mx Loding S SF S SF Operting Are S SE k r x (45 or 67.5 ksi) N Life men S Sy S SU 0.6 S UT FOS ginst ftigue (Finite life): See Exmple 17.4, pge 500. n s S SF lt Helicl Extension Springs A. All coils re ctive. One coil is typiclly dded to the number of ctive coils to obtin the body length. B. The Free Length is mesured between the insides of the end loops. C. They re often close-wound with some initil prelod. P Prelod P δ x P k δ once P > prelod. D. Spring rte nd sher stress re the sme s for compression springs. E. Criticl stresses cn be in the end hooks. > See Eqns nd

12 Helicl Torsion Springs Similr to unwinding grden hose from reel, these springs work in bending. σ mx where KiMc K I K i i M 4C C 1 4C( C 1) M Moment C Index 1.4 Compre the mx stress to Bending yield S sy nd Bending endurnce S se Ki Spring Index, C Helicl Torsion Spring Deflection Torsionl spring stiffness hs different units from compression or extension springs: 4 d E Torque kθ DN Re v l 1 l k θ 4 d E 64 DN Torque Rdin D D men N body Active Turns: N N body ( l l ) π D Note: The ID chnges s the spring is loded: ID loded N N loded ID 1

13 d wire 0.0 in. ID 0.90 in. L 1 L in. Nb.4 turns of steel wire Exercise: Hnd Grip Wht is K θ? How much force does it tke to squeeze the hndles 1.5 together (mesured t in. rdius from the coil center)? in. Wht s the del with Lef Springs? Tringulr Plte loded t tip Side View Moment Repckged equivlent Moment of Inerti Mc σ I Mteril hs sme stress everywhere! Rubbing between pltes dmps out motion. 1