Novikov/Conformal Gearing: Scientific Theory and Practice

484 Pages·2022·18.068 MB·English

by  Stephen P. Radzevich

Stephen P. Radzevich
Novikov/
Conformal
Gearing
Scientific Theory and Practice
Novikov/Conformal Gearing

Stephen P. Radzevich
Novikov/Conformal Gearing
fi
Scienti c Theory and Practice
StephenP.Radzevich
SouthfieldInnovationCenter
EATONCorp.
SterlingHeights,MI,USA
ISBN978-3-031-10018-5 ISBN978-3-031-10019-2 (eBook)
https://doi.org/10.1007/978-3-031-10019-2
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In memoriam of Professor Mikhail L. Novikov,
the inventor of the Novikov Gear System,
whose110thanniversaryiscelebratedin2025
Introduction
Knowledgeispower(Scientiapotestasest)–FrancisBacon(1561–1626)
Historical Background
Gearsandgeardriveshavealong-lastinghistoryofevolution.Cogwheelsevolved
toahighprecisioninvolutegearcapableoftransmittingahugeamountofpowerat
highrotations.
Present-daygeardesignsarebasedonthelatestaccomplishmentsinthetheoryof
gearing.Ultimately,theseyieldaveryhighpredictabilityofthedesignedgear-sets.
Novikov/conformalandhigh-conformalgearingrepresentaspecialandrelatively
new kind of gearing. Gearing of this kind features a high-power-density gearing –
thisisapresent-daytrendofinthegeardesignandproduction.
Inventedinthemeadof1950s,gearsofthisdesignfeatures“convex-to-concave”
contact between the tooth flanks of a gear and a mating pinion. Favorable tooth
contactofthiskindenablesfavorablestressdistributionatcontactpointsofthetooth
flanks.
Despiteofenormousamountoftheresearchinthefieldhasbeencarriedoutinthe
past, the gear science and gear practice have been further evolved in the recent
decades.
Uniqueness of This Publication
This book is an invaluable source of information on the Novikov/conformal and
high-conformalgearing. Involute gears of conventional designare also considered
here. Gear experts all around the world have contributed their latest accomplish-
mentsinthefieldofgearing.
vii
viii Introduction
ThemainfeaturesofNovikov/conformalandhigh-conformalgearingarecovered
inthisvolume.Thisincludesbutisnotlimitedtothespecificdesignparametersof
gears of this particular design: Novikov boundary circle, one (or two) straight
pseudo-lines of action, pseudo-path of contact, instantaneous lines of action, and
soforth.
Intended Audience
High-powerdensity(or,inotherwords,highpower-to-massratio)isakeyfeatureof
Novikov/conformal gearing, especially of high-conformal gearing. Therefore, gear
experts those involved in the design, production, inspection, and in application of
advanceddesignofgeartransmissionsareamongthosewhomthisbookisaddressed
to.Mostofgearengineersandgearresearchersfromtheindustry,aswellasgraduate
students,willbebenefitedbythebook.
Organization of This Book
Thisbookiscomposedofelevenchapters.Thelistofthechapters,alongwithabrief
description of each chapter, can be found out immediately below. Also, several
appendicesareappendedattheendofthebook.
Chapter 1: Titled as “Novikov Gearing Is a Kind of Involute Gearing,” this
chapteriscontributedbyProf.StephenP.Radzevich(USA).
ThischapterofthebookdealswithNovikovgearing.Nowadays,gearingofthis
kind is commonly referred to as Novikov/conformal gearing. If properly designed,
manufactured, and applied, Novikov gearing features the higher power density
compared to that in conventional involute gearing of a similar size. Therefore, the
usageofNovikovgearingallowsfortransmittingmorepowerbymeansofgearboxes
ofa smaller size. Anincreaseof thepowerdensity isoneof thestrategictrendsof
evolutioninthegearindustry.
AbriefhistoricaloverviewofNovikovgearingispresentedinthischapterofthe
book. It is shown that Novikov gearing is a reduced case of involute gearing. The
conceptoftheboundaryNovikovcircle(introducedbytheauthor,andreferredtoas
theboundaryN-circle,forsimplicity)enablesoneclearanalysisofthegeometryand
kinematics of Novikov gearing. The necessary and sufficient conditions forperfect
operation of the conformal gearing (i.e., Novikov gearing) and high-conformal
gearing are outlined. Use of the concept of reversibly enveloping surfaces (or just
R -surfaces,forsimplicity)makesitpossibletostatethatneitherNovikovgearing
e
(moregenerally)norhigh-conformalgearingcanbefinish-machinedincontinuous-
indexing machining process; that is, the gears cannot be finish-cut by hobs, by
shapersandrack-cutters,bywormgrindingwheel,andsoforth.
Introduction ix
Chapter2:Titledas“MeshingTheoryforAbnormalNovikovHelicalGears,”this
chapteriscontributedbyProf.YapingZhaoandbySiyuLiu(bothofChina).
TheNovikovgearinthetraditionalsenseisenvelopedbyarack-cutterwithanarc
toothprofile,andthetoothsurfaceofthegearisverycomplex.Thischapterpresents
anewtypeofcirculararcgeardrivewithcirculartoothprofileinthenormalsection.
Thegearisdesignedasanewgearwhosenormalsectionisarc-shaped,andthetooth
surface equation of the arc-shaped gear is established by the spherical vector
function.Thetoothprofileisrelativelysimple.
Based on the transmission ratio condition, the augmented contact equations are
writtentodeterminethedatumpointwherethetransmissionratioerroriszero.The
instantaneous contact points are determined by solving nonlinear equations
established in the light of common point and common normal conditions, also
known as improved TCA method. The tooth width condition is used to determine
the endpoints of the contact paths of two tooth surfaces, and the internal contact
points are obtained. The calculation formula of relative principal curvature is also
derived.
The numerical cases show that the gears proposed are instantaneously in point
contact, tooth surface meshing belongs to point contact, and the contact trace is
roughly along the direction of tooth length. The gear pair can accomplish constant
ratio transmission, and its velocity ratio can completely be equal to the nominal
transmittingratio.
Chapter 3: Titled as “Helical Bevel Novikov Gears,” this chapter is contributed
byDr.MichałBatsch(Poland).
This chapter presents a mathematical model of convexo-concave helical bevel
Novikov gear meshing. A topological modification of this kind of gearing was
introduced. Results of tooth contact analyses aiming at the determination of the
effectofgearparametersoncontactpatternandtransmissionerrorwerepresented.In
addition, a comparison between the Novikov gear and a conventional spiral bevel
geargeneratedwiththeaidoftheduplexhelicalmethodwasprovided.Itwasfound
that the Novikov gear displayed larger instantaneous contact pattern than the
conventionalone.
Chapter 4: Titled as “Hyperboloidal-Type Normal Circular-Arc Gearing,” this
chapteriscontributedbyDr.HoujunChen,ZhilanJu,XiaopingZhang,andChang
Qu(all ofChina). Discussedinthis chapterhyperboloidal-type normal circular-arc
gearing (HNCGing) is a latest development of Novikov gearing to transmit the
motion and power between two orthogonally crossed axes. The basic principle of
molding-surfaceconjugation,which represents thekinematicsand thegeometry of
conjugation by a pair of directrices of conjugate surfaces rather than the conjugate
surfaces themselves, was presented, and both the conditional equation of conjuga-
tion and the structural condition of conjugation were established to provide a
theoretical foundation for constructing novel types of gearings. The generation of
thepinionandthematinggearinHNCGingwasanalyzed,mathematicalmodelsof
conjugatetoothsurfaceswereestablished,andtheinducedcurvaturecharacteristics
of conjugate surfaces were explored. This helps people get ready to acquire the
x Introduction
relationship of design parameters, calculate the Hertzian contact between the two
matingteeth,andcarryouttheinterfere-freetool-pathprogrammingforNCmachin-
ing of tooth surface. In order to meet the needs of digital manufacturing, an
integrated manufacturing software system of HNCGing was developed, which
connects computer-aided design, computer-aided manufacturing, and computer-
aided engineering in a single application platform. This software system supports
three-dimensionalmodeling,adaptivetool-pathprogrammingforfive-axisforming
millingoftoothsurface,andsimulationofmeshingandcontactforthetoothcontact
analysis.
Chapter5:Titledas“ModernMethodsofEstimatingandIncreasingoftheLoad-
Bearing Capacity of Novikov Gearing,” this chapter is contributed by Dr. Viktor
I.Korotkin(Russia).TheuseofNovikovgearingwithteethofdifferenthardnessfor
variouspurposesisconsidered.
Forgearboxesofgeneralmachine-buildingapplication,variantsoftransmissions
withhardenedteethareoffered,forwhichaspecialbasicprofilehasbeendeveloped,
whichispartofthestatestandard.
Afundamentalapproachtosolvingtheproblemofdeterminingthemainparam-
eters for calculating the strength of these gears, consisting of two stages, is
presented:
(a) Determination of both the main bending, contact stresses and stiffness of the
teeth,andthephase,takingintoaccountthecoefficientsofinfluenceoftheends
ofthegearrim
(b) Modeling of the process of real multi-pair engagement with finding criterial
stress in dangerous areas of the teeth that determine the load-bearing capacity
andservicelifeofthegearbox
Implementingthisapproach,generalizedresultsofsolvingthespatialproblemof
thestress-strainstateoftheteethatanypositionofthecontactpadalongtheirlength
areobtained,andthewayofobtainingpartialforcesatthecontactsitesandcriterial
stressareindicated.AnengineeringmethodforcalculatingthestrengthofNovikov
gearing is proposed, on the basis of which gearboxes of general mechanical engi-
neering with high-hard Novikov gears of relatively low degrees of accuracy are
created.Thismadeitpossibletoreplacehigh-precisionexpensive involutegearing
on the strained output stages of the gearboxes with less time-consuming in the
manufacture hardened Novikov gearing of coarser degrees of accuracy and reduce
thespecificgravityofthegearboxby1.5timescomparedtotheinvoluteanalogues.
Preliminarycalculationsofthecreatedpromisinghardenedhigh-precision Novikov
gearing are given, and their significant advantage in comparison with involute
analoguesisshown.
Knowndesignsof modifiedNovikovgearswithlongitudinalmodificationofthe
workingsurfacesofthermallyimprovedteethofdrivegearboxesofwidelyusedfor
areoilpumpingunitsdescribed.
Ontheexampleofoneofthemodelsofthegearbox,itisshownthattheuseofthe
proposed longitudinal modification of the working surfaces of the teeth in combi-
nationwithanincreaseinthegearmoduleprovidesasignificantimpactonreducing