2.007 –Design and Manufacturing I Gears: Strength, Gear Trains,… Today’s AgendaInterferenceBacklashGear SelectionWays Gears FailStress in GearsA Beam in BendingConcept QuestionStrength of GearsThe Lewis FormulaOr Use a Canned ToolDiscussion QuestionsContact Stress (Hertzian Stress) Contact Stress Quantitative CharacterizationSimple Gear TrainsCompound Gear TrainsCompound Helical GearsManual TransmissionsDiscussion QuestionsConcept QuestionDiscussion QuestionsDifferentialsPlanetary Gear TrainsAnalysis of Planetary Gear TrainsTypes of SpringsVolvo 1800ESRubber as a SpringSprings to CounterbalanceGears & Springs in MechanismsNext StepsUpdated ScheduleName That GearFollow upMIT OpenCourseWare http://ocw.mit.edu 2.007 Design and Manufacturing ISpring 2009For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.2.007 –Design and Manufacturing I Gears: Strength, Gear Trains,…Presented by Dan Frey on 19 MAR 2009Images removed due to copyright restrictions. Please see p. 1 inhttp://courses.washington.edu/mengr356/daly/Gear_stress.pdfAndhttp://content.answers.com/main/content/img/McGrawHill/Encyclopedia/images/CE064700FG0010.gifToday’s Agenda• Some discussion of homework #3•Gears– Selection of gears– Failure of gears– Strength– Gear trains• SpringsImage removed due to copyright restrictions. Please see http://www.hitecrcd.com/product_file/file/66/hs805.pdfInterference• Contact with the non-involutepart of the gear• In practice, avoided by under-cutting• Weakens the tooth, esp. on gears with few teeth (pinions) From Shigley and MischkeBase circleBase circleAddendumcirclePressurelineThis portion of profileis not an involuteThis portion of profileis not an involuteInterference ison flank of driverduring approachCADBDriven gear 2Driving gear 1O2O1ω2ω1Figure by MIT OpenCourseWare.Backlash• There is some space between the driving pair of gears and the next pair behind and ahead• When gears reverse direction, that space allows relative motion• Can be eliminated at some cost in parts and efficiency An “anti-backlash” gearfrom WM BergCourtesy of W. M. Berg, Inc. Used with permission.Gear Selection•Pitch• Face width• Material• Pressure angle• # of teeth• Hub style, bore, etc.Courtesy of W. M. Berg, Inc. Used with permission.• You call up the number 1-800-232-BERG and ask that, for a special application, you want a 48 pitch spur gear, but with a pitch dia of 0.32 inches. They will probably say:1. OK, no problem2. OK, but it will cost a lot3. No, this is not technically possible Courtesy of W. M. Berg, Inc. Used with permission.Ways Gears FailExceed static yield stress in bending“stripping”Exceed endurance limit in bendingExceed endurance limit in contact stress“pitting”Images removed due to copyright restrictions. Please see http://materials.open.ac.uk/mem/mem_mf6.htmhttp://www.hghouston.com/x/39_gearpit.htmlImage courtesy of deltaMike at Flickr.Stress in GearsImage removed due to copyright restrictions. Please see p. 1 in http://courses.washington.edu/mengr356/daly/Gear_stress.pdfA Beam in Bending26/ FtlWcIMt==σFWttlFigure by MIT OpenCourseWare.Concept Question• In selecting a gear of one inch pitch diameter, we are choosing between 48 and 24 pitch gear teeth. The effect on torque that can be transmitted before bendingfailure of the teeth is 1. Around a factor of 102. Around a factor of 43. Around a factor of 24. Much less than a factor of 2txaWrWWtrflFigure by MIT OpenCourseWare.Strength of Gears• Any good catalog will have a formula and tables• What factors must enter the equation?– Material properties– GeometryCourtesy of W. M. Berg, Inc. Used with permission.The Lewis FormulaFyPWt=σPoint of max stress due to bending“Lewis form factor”Low form factor → High stressDiametral pitch (teeth/inch)Face widthtxaWrWWtrflFigure by MIT OpenCourseWare.Or Use a Canned Toolhttp://www.wmberg.com/Tools/Please see “Spur Gear Tooth Strength” at http://www.wmberg.com/tools/Discussion Questions• I glued the third stage teeth of this servo together• Now I will apply a load to the output shaft (up to 10lbs)• What’s going to happen?Epoxy applied liberally hereContact Stress(Hertzian Stress) FFContact StressQuantitative Characterization0 0.5 1 1.5 2 2.5 300.20.40.60.81σzzd1, d2,()pmaxd1d2,()σyzd1, d2,()pmaxd1d2,()σxzd1, d2,()pmaxd1d2,()τyzzd1, d2,()pmaxd1d2,()τxzzd1, d2,()pmaxd1d2,()zbd1d2,()bd1d2,()2F⋅π l⋅1 ν12−⎛⎝⎞⎠E11 ν22−E2+1d11d2+⋅:=pmaxd1d2,()2F⋅π bd1d2,()⋅ l⋅:=FFSimple Gear Trains• A “simple” gear train has only one gear on each shaft• How does this arrangement behave?ω1Compound Gear Trains• A “compound”gear train has at least one shaft with multiple gears• How does this arrangement behave?From Norton “Design of Machinery”Image from Wikimedia Commons, http://commons.wikimedia.orgCompound Helical Gears• Note the difference in the handedness of the gears• Another book shows the gear with the same handedness•Which is better?N2N3N5N4Output shaftInputshaftAdapted from Norton "Design of Machinery"Adapted from Shigley "Mech. Eng. Design"Figure by MIT OpenCourseWare.Manual TransmissionsIf you find just two axles in a machine, does that mean there are just two stages?Image from Wikimedia Commons, http://commons.wikimedia.org. Please see http://mossmotors.com/Graphics/Products/Schematics/SPM-025.gifDiscussion Questions• Are there any disadvantage to a helical gear as compared to a spur gear?• How can the disadvantages be remedied?• Is a helical gear set stronger than a spur gear of the same diameter, pitch, face width, & material?Image from Wikimedia Commons, http://commons.wikimedia.orgConcept Question1. Pitch rises, face width rises2. Pitch rises, face width falls3. Pitch falls, face width rises4. Pitch falls, face width fallsA compound gear train is formed of eight gears. As we proceed from the pinion on the electric motor to the gear on the output shaft, how do the pitch and face width vary?Discussion Questions• How many stages in this device?• How do you suppose this number is chosen?• Are the reduction ratios typically all nearly the same in all successive stages?10 teeth on pinion gear72 teeth10 teeth36 teeth10 teeth48 teeth16 teeth42 teethDifferentials• Allows shafts to move at different speeds• Applies same torque to both• Slippage