1 fUNDAMENTALS OF MACHINE DESIGN P.ORlOV TRANSLATEDFROMTHERUSSIAN BYYU.TRAVNICHEV MIRPUBLISHERS.MOSCOW Aa B ~r'l' M Ee q Ht] Ie FIrstpublished1976 THERUSSIANALPHABETANDTRANSLITERATION AaaKKkXxkh B6bJIJII 1\1\ Is BevMMm 'I" ch rrgHRnillmsh lJ.11,d00'0' II\",shch EeeTInp1> .. " EeePprbI IiI Y lRlK.hCcsbb , 3.ZTTta.e IhrYyuIOroyu lUy 0.1nD,ea' With somemarginwetakethevalue ofdat 5mm. The seals are ma dein the form ofcy lin-dricalbossesontheimpellerdisksand fit,with clearance, intorings press-fitted intothepumpcasing.Toallowforpos-sibletouchingofsealingsurfacesthe ringsaremadeofanantifrictionmate-rial(softbronze,gradeBp.OIJ,C). Thesealscanalsobemadeoffluoro-carbonorsiliconeplastics,whichdonot swellinwater.However,thehighcost ofthesematerialsshouldbekeptin mindandtheirhighcoefficientoflinearexpansionmakesthe problemofringfixtureinthecasingmuchmorecomplex. (d)DurabilityofSupports Tentativelyassumethattheimpellershaftdiameterd=40mm andthatthesupportsare,ineffect,single-rowballbearings(light "eries)whoseworkcapacitycoefficientC=39000. Theworkcapacitycoefficientprovidingfortherequiredpump durability C =Rk. (nh)O.3 where R=loadonbearing(inthiscasethegreatestbearingload R=17kgf) k.=bearingdutyfactor(assumek. = 1.5) n=shaftspeed,rpm(n=2950rpm) h= givendurability(h= 40000h) 2.8.Design Example113 Consequently,C=17 1.5(2950.40 000)03.=6800.Thus,the selectedbearings willsatisfy(with an ample safety factor)thepump durabilityperiodandenablelargerloadsandincreasedspeedsto beusedintheevent. offorcedworking. Toconsiderbearingsfordurabilityaloneisnotsufficient,asthe calculationsassumednormalworkingconditionsforthebearings. Mistakes in assembly, too little ortoomuchlubricationmay bringallcalculationsto noughtandcausepremature wear or even failure long before theirservicelife hasexpired. (e)SpacingofSupports Whith the chosen ratioLll= 1.5thespacingbetweenthe supportswillentirelydepend ontheoverhangvalueIof thecentreofgravityofthe impellerrelativetothefront support. The latter value is de-terminedbythearrangement Fig.19.Arrangement01impellershalt supports ofthe seal betweenthe frontbearing and the pumphydraulic cavity. Onthebasisofthepreliminaryestimatesassumethattheseal lengthis45mmandthedistancebetweenthefrontsealfaceand theimpellercentre of gravity, 10mm.The bearingwidth is18mm. Hence,thetotallengthoftheoverhang /=45+10+9=64mm andthedistancebetweenthesupports L = 1.511 ~ iOOmm Thisstageofthepumpdesigneudswiththedrawingofasketch oftheimpellershaft,whichalsoillustratesthesupportspacing (Fig.19). (I)DischargeVolutes The cross sections ofvolutes can bepositioned sothat the extreme internalpointsofthesectionsareequispacedabouttheimpeller circumference. Thenthecentresofthecross-sectionsarearraugedalongaspiral describedbytheequation. Dimpdo,/''''+t-H+H1t-1+H 1.5VI . 1l.88Ul1l.8[l! I_+1 H.fHj-H--f+Ht I 10ccomp'kgf;mm2 Fig.236.Maximumstresses(Ymaxasfunctionofcompressivestresses(Jcompfor dillerentvaluesofa=Did(compressionofspheres) sharplydropwiththedecreaseina,Le.,whenthediameterofthe concavesphericalsurfaceapproachesclosertothespherediameter; tendingtozeroata=1,whentheconcavesphericaldiameteris equaltothespherediameter. 426Chapter 6.ContactStrength Thisdoesnot implythat thestressesvanish completely;but only meansthatHertz'sformulaisinapplicablewhena"" 1,because thisrendersinvalidoneoftheassumptionsunderlyingthebasic theory,namely,the assumption that thesizeof the compressionarea is infinitesimal in contrast to the dimensions of spheres.When a=1 (andevenwithvaluesveryclosetounity)thestressesshouldbe definedascrushing stresses. Now,let ustransformEq.(6.1)as follows.Substitute the radicand-%,by o"'"P' = 'where 0eomp' is the compressive stress kgfimm",_ produced by the action of force P in the central section of ad-diameter sphere(actualstressforsolidspheresandconditionalfortruncated spheresandbodieshavingalimitedsphericalsurface). Intheoverwhelmingmajorityofcasescontactjointsaremadeof steel(E=21000kgf/mm').SubstitutingthisvalueintoEq.(6.1) gives 3/"--umax =.430 y(Jcompr Vo whereoo-effectivestressEq.(6.2). (6.3) Thediagram,presentedinFig.236,embracesthethreekindsof loadings and shows 0max as a function of oeomp, forvarious magnitu-desofaandenablesanyproblemconnectedwithcalcuhtionsof sphericaljointstobeeasilysolved. Afewexamplesforillustration. Example1.Findthemaximumstressinaball,10mmindiameter,res-tinguponanat surfaceandloadedwithforceP=15kg!. Thecompressivestress(Jeomp'= 0.7i5d2 =1.27 "" 0.2.Startingfrom thepOintcreomEr=0.2ontheabscissamovingverticallyuntil- meetingline a=00,wefindontheordinateaxisthatcrms:x:=250kgf/mm.2 Example2.ThediameterofabaUandits loadarethe same.ThebaUrests inasphericalsocketwitha=1.02.Clearly,thecompressivestressinthehall isequaltothat in thepreceding case(acompr=0.2).Now,moving upwardfrom thispointuntilmeetinglinea=1,02,wefindthatO'max:=i8kgf!mm2 Example3.Tbegiven loadis 100 kg/;permissible stress (Jma. = 50 kg//mm'. Findthediameterofaballsuitablefortheseconditionswhentheballistobe mountedinasocketwithratioa=1.02. FollowfromthepointO'max=50kgf/mm2 alongthehorizontal,untilmee-tinglinea=1.02,wefindontheabscissathatGcompl'=4kgf/mm2.Hence, thebaUdiameter-d=' /1.27P=1/71".27"."10"'05.7mm IIO'compl'r4 Asthematerialinthecontactareaissubjectedtoomnilateral compression, high stresses (100-200 kgf/mm2)are generally permissible whendesigningcontact joints.With shockloadsthe permitted stres-sesarelowered2-3times. Contactjointmembersaremadeofthermallyhardenedsteels, mostlyofbearinggradesrnX15andrnX15Cr(hardnessafterhar-deningandlowtempering62-65Rc). o20'to5080fOOfZOd,mm Fig.237.:\laximuID stresses O"max(\vith P=1 kgf)as a function of sphere diame tel'd 2J Fig.238.Sphericalconnections 1and2- pointcontact;3and4- surfacecontact 428Chapter 6.ContactStrength Wenow explainin ageneralizedformthe influenceofspheredia-meteronstrength.FromEq.(6.3) (6.4) Consequently, the maximum stress is inversely proportional to d2/0 Thegraph,plottedinFig.237,is based onEq.(6.4).It is evident fromthegraph,thatincreaseinspherediametersharplylowers stressesin theareaofsmalldiameters butasdiameterincreasesthe stress lowering proceeds moreslowly.Beginning froma certain value of d,the lowering becomes hardly perceptible.In the given case when a=1.02-1.1,thisbeginsfromspherediametersof20-30mm.The rangeofadvisablediameterincreaseisratherlargerforspheres workingonspheres. In conclusion,let us comparethe strength under point andsurface contact(Fig.238).Assume() compT=0.1kgflmm2 forallcases.For cases3and4 having surfacecontactthis stress,obviously,isequal tothebearingstress()bonthebearingsurfaces)b=() compT= =0.1kgflmm2). Listedbelowarethevaluesof()m.x,forcases1and2,aswellas the ()max/()bratios,characterizing the comparative strength ofjoints having point and surface contact.Since the permissible stresses under contactloadingare,onaverage,5timeshigherthantheadmissible bearingstresses()b,thecomparisonisgivenintermsoftheratio ()max/ 5()b amax,kgf/mm2. (Jmax!crb..... crmax/5crb Case1} 200 2000 400 Ca.se2 15 150 30 Obviously,the strength ofajoint,whenasphererests upona flat surface,is400times,andwhenseatinginasphericalsocketat a=1.02,is30timeslessthanthestrengthobtainedundersurface contact. 6.2. Cylindrical Connections AccordingtoHertz,themaximalstress in cylindrical connections (}max = 0.6 V: V 1 +~ kg/lmm2(6.5) whereP=actingforce,ontheconnection,kgf E=Young's modulus for the material of the cy linders, kgf/mm2 d=diameterofsmallcylinder,mm I=lengthofcylinders,mm a=Did=ratiooflargeandsmallcylindersdiameters 6.2.CylindricalConnec#on,s429 Theminussignisforthecasewhenacylinderactsonaconcave cylindricalsurface. Denoting ]I (1+! )=Go; =G"mpr andassumingE=21000kgf/mm',wefindthat Gm., = 87 V aoomprGokgf/mm' whereGo=maximumeffectivecontactstress (6.6) aoompr=compressivestress,takenacrossthemeridionalsection ofacylinderhavingdiameterd Thevalues ofaDasafunctionofaareplottedin Fig.239,forthe threekinds ofloads:cylinder onacylinder,cylinder ina cylindrical seatan,\cylinderona flat surface(D=00;aD=1). ITIIITIr-..,.rrrrrm--r-rni Intheirgeneralappea- 1.6I'H--H-+-H-I+flI--l-t-Hi rancethe aDcurvesaresim- -PaO':r-'H-HittI---t--H-i1 ilartothecurvesofthe1.4\"/I sphericalconnections(seeP