1ME311 Machine DesignW Dornfeld07Nov2013Fairfield UniversitySchool of EngineeringLecture 9: Screws(Chapter 16)Thread GeometryHamrock Page 372DiametersMajorCrestPitchRootMinorThread HeightThread PitchThread AnglepinchThreadsn1)/( =The Pitch Diameter is midway between the Major and Minor Diameters.2Thread TypesHamrock Page 372Single-, double-, and triple-threaded screws. Also called single-, double-, and triple-start.Lead = 3 x PitchAcme ThreadSquare ThreadLead = 1 x PitchAcme threads are used in C-Clamps, vices, and cartoons.Details of Thread ProfilesHamrock Pages 372-373Relationships for M (metric) and UN (unified = US) screw threads.Example:UN: ¼-20, means 0.25in. Major diameter & 20 threads/inch.M: M8x1.25, means 8mm Major diameter & pitch of 1.25mmThread Heightppht8660.0)30tan(5.0=°=3Power ScrewsLooking at a square thread screw, we unwind one turn:αLead2π rmThis shows an inclined ramp with anglemrLeadπα2tan1−=WrmrcLeadµThread frictionCollar frictionMean thread radiusMean collar radiusLoad on nutµcThe Mean Radius is midway between the Crest and Root Radii.Square Thread Screw TorqueThe torque required to raise the load W isand to lower the load, we flip two signs:+−+=ccmraiserrWTµαµαµtan1tan++−=ccmlowerrrWTµαµαµtan1tanWrmrcLeadµµcHamrock Pages 376 & 3784If the thread form is not square but has an angle β, replace the thread friction µ with the effective frictionPower Screw Thread Angle)2/cos(βµµ=eThe effect:• Square: β= 0, β/2 = 0, 1/cos(0°) = 1.0• Acme: β= 29°, β/2 = 14.5°, 1/cos(14.5°)= 1.033• Unified: β= 60°, β/2 = 30°, 1/cos(30°)= 1.15The thread angle effectively increases surface friction between 3 and 15%βNote: Instead of β/2, Hamrock usesThe difference is negligible.)2tan(costan1βαθ−=nPower Screws - OverhaulingIf the collar friction is small (e.g., it may have a ball thrust bearing), too small a thread friction may let the weight screw down on its own.This can happen when(the numerator goes negative).This is the same case for a weight sliding down a ramp when the incline angle α exceeds tan-1µ.mrLeadπαµ2tan =<αLead2π rmµ++−=ccmlowerrrWTµαµαµtan1tan0αµtan−5Ball Screws Have Low FrictionRecirculating balls roll between ball screw and ball nut to minimize friction.These almost always overhaul.Our Scissors JackEnd with nutHandle End with ball thrust bearing1522 Lb Tension1015 Lb6Scissors Jack AnalysisThread ID = 0.398 in.Thread OD = 0.468 in.Estimate dp= (0.398+0.468)/2 = 0.433 in.Handle length = 135/25.4 = 5.31 in.What torque is required to raise the jack?What force is required on the handle?Lead = 0.10 in.Thread angle β = 29°Guess µ = 0.20 µc= 0 due to bearingW = 1522 Lb.C-Clamp AnalysisThread ID = 0.391 in.Thread OD = 0.480 in.Handle length = 3 in.N = 8 Threads/InchThread angle β = 60°Guess µ = 0.15 µc= 0 to simplify thingsW = 500 Lb.What torque is required to cause the 500 Lb. squeeze?Note: If Acme, could use Eqn. 16.5.4075.001.0)125.0)(5.0(48.001.05.0 inpddcp=−−=−−=But with a 60° thread angle, this is NOT an Acme.Estimate dp= (ID+OD)/2 = (0.390+0.480)/2 = 0.436 in.7Using Dornfeld Lecture Equationsdp= 0.436 in.N = 8 Threads/InchLead = 1/N = 0.125 in.°====−−−21.5)09126.0(tan)2/436.0(2125.0tan2tan111ππαmrLead..29.299842.026446.0)218.0)(500()09126.0)(1732.0(109126.01732.02436.0500tan1tanInLbrrWTccmraise==−+=+−+=µαµαµThread angle b = 60°µ = 0.15 W = 500 Lb.Because this is not a square thread, must use effective coefficient of friction = µ/cos(β/2) = 0.15/cos(30°) = 0.15/0.866 = 0.17320Using Hamrock Equationsdp= 0.436 in.N = 8 Threads/InchLead = 1/N = 0.125 in.°==×=°°===°==−−−−−897.29)57496.0(tan)57735.09959.0(tan)30tan21.5(costan)2tan(costan09126.0)tan(;21.52tan11111nnmrLeadθβαθαπα..29.2985231.022903.0)109()09126.0)(15.0(866.015.0)09126.0)(866.0()218.0)(500(21.5tan15.09.29cos)15.021.5tan9.29)(cos2/436.0(500tancos)tan)(cos2/(InLbrdWTccnnpraise==−+=°−°+°°=+−+=µαµθµαθThread angle b = 60°µ = 0.15 W = 500 Lb.[Eqn. 16.13]The equations are equivalent. Pick whichever one suits you best.How close is this to β/2 = 30°?08Overhauling Revisited++−=ccnnplowerrdWTµαµθαθµtancos)tancos)(2/(• Power screws can lower all by themselves if the friction becomes less than the tangent of the lead angle, α.• This corresponds to the numerator in the Tlowerequation going negative, with the transition being where the numerator is Zero.• You can use either Dornfeld or Hamrock equation, but remember that the Dornfeld equation is Effective friction, and you must multiply by cos(β/2) to get the actual friction.The equations are equivalent. Pick whichever one suits you best.++−=ccmlowerrrWTµαµαµtan1tanαθµtancosn=αββµµαµtan)2/cos()2/cos(tan===eeTransition when:Hamrock:Dornfeld:Failure Modes: Tensile OverloadWhen the tensile stress on a bolt exceeds the material’s Proof Strength, the bolt will permanently stretch. tAP=σWhere Atis the Tensile Stress Area for the bolt – the equivalent area of a section cut through the bolt.Hamrock Page 39129743.0)7854.0(−=ndActFor UN threads,2)9382.0)(7854.0( pdAct−=For M threads,dc= Crest Dia (in.)n = threads/in.dc= Crest Dia (mm)p = pitch (mm)9Failure Modes: Thread ShearShear of Nut ThreadsShear of Bolt ThreadsldAcrestshearπ= ldArootshearπ=lThe shear strength of the bolt and nut material might not be the same.Failure Modes: Shank Shear42shanksheardAπ=Bolts are not really intended to be used this way unless they are Shoulder Bolts:Typically the preload from tightening the bolt clamps the joint,and the friction between the members holds the joint.24222shankshankshearddAππ=×=10Bolt PreloadSo the bolt is really a spring that stretches and creates preload on the joint.JH Bickford explains : 'When we tighten a bolt, ( a) we apply torque to the nut, ( b) the nut turns, ( c) the bolt stretches, ( d) creating preload.'PKDTcrest=We use the Power Screw equations to determine how torque resultsin preload. This can be approximated simply by:Where T is torque, Dcrestis the bolt crest diameter, P is the preload, and K is a dimensionless constant. K = 0.20 for clean, dry threads and K = 0.15 for lubricated threads.Bolt StiffnessA bolt looks like two springs in series: one rod with the