Gears: Volume 1: Geometric And Kinematic Design

880 Pages·2020·28.842 MB·English

by  Vincenzo Vullo

Springer Series in Solid and Structural Mechanics 10
Vincenzo Vullo
Gears
Volume 1: Geometric and Kinematic Design
Springer Series in Solid and Structural
Mechanics
Volume 10
Series Editors
Michel Frémond, Rome, Italy
Franco Maceri, Department of Civil Engineering and Computer Science,
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Vincenzo Vullo
Gears
Volume 1: Geometric and Kinematic Design
123
Vincenzo Vullo
University of Rome“Tor Vergata”
Rome, Italy
ISSN 2195-3511 ISSN 2195-352X (electronic)
SpringerSeries inSolid andStructural Mechanics
ISBN978-3-030-36501-1 ISBN978-3-030-36502-8 (eBook)
https://doi.org/10.1007/978-3-030-36502-8
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Aphorism
Oqῶm dέ re, jahάpeq έcx, rpotdaῖom jaὶ uikorouίa1 pqoersῶ sa ἀnio
kόcx1 jaὶ sὴm ἐm soῖ1 lahήlarim jasὰ sὸ ὑpopίpsom exqίam sesilηjόsa
ἐdojίlara cqάwai roi jaὶ eἰ1 sὸ aὐsὸ bibkίom ἐnoqίrai sqόpot simὸ1
ἰdiόsηsa, jah’ ὅm oi paqevόlemom ἔrsai kalbάmeim ἀuoqlὰ1 eἰ1 sὸ dύmarhaί
sima sῶm ἐm soῖ1 lahήlari hexqeῖm diὰ sῶm ηvamijῶm. soῦso dὲ pέpeirlai
vqήrilomeἶmaioὐdὲmἧrromjaὶeἰ1sὴmἀpόdeinimaὐsῶmsῶmhexqηlάsxm.aὶ
cάq sima sῶm pqόseqom loi uamέmsxm lηvamijῶ1 ὕrseqom cexlesqijῶ1
ἀpedeίvhη diὰ sὸ vxqὶ1 podeίnex1 eἶmai sὴm diὰ soύso soῦ sqόpot hexqίam·
ἑsoilόseqomcάqἐrsipqokabόmsadiὰsoῦsqόpotcmῶrίmsimasῶmηsηlάsxm
poqίrarhai sὴm ἀpόdeinim lᾶkkom ἢ lηdemὸ1 ἐcmxrlέmot fηseῖm.
Archimedes, The Method, Introduction
(Translation in Vol. 3, Sect. 3.2)
v
To my wife Maria Giovanna,
my sons Luca and Alberto,
my nephew Nicolò
and my students
Preface
Gears and gear power transmissions used in most mechanical applications and,
especially, in the automotive and aerospace industry constitute one of the classical
topics of Machine Design Theory and Methodology. In the opinion of many
scholars,includingtheauthorofthismonothematictextbook,thegears are,byfar,
oneofthemostcomplicatedbutfascinatingsubjectsofthemechanicalengineering,
because it is extremely polyhedral, as it calls into question a remarkable variety of
disciplines, of which the main ones are: plane and space geometry; various bran-
ches of theoretical and applied mechanics, such as kinematics, dynamics, elasto-
hydrodynamics, vibrations and noise; continuum mechanics and machine design
theory and methodology; static and dynamic material strength; structural opti-
mization; surface contact and lubrication theories and tribology; materials science
and metallurgy; cutting processes and, more generally, technological processes of
manufacture; maintenance, etc.
Despitethefactthat,fromahistoricalperspective(seeGears—Vol.3:AConcise
History), the gears constitute, after the potter’s wheel, the oldest mechanism
invented by Homo Sapiens Sapiens, we can say that a unified scientific theory
ofthegears,abletoconsideralltheaforementioneddisciplines,amalgamatingthem
in a “unicum” does not yet exist at present. Currently, according to the most
widespreadthinkingthatissharedbyhistoriansofscienceandscientists,ascience,
to be such, must have at least the following essential features:
(cid:129) All that science claims should not involve real objects, but specific theoretical
entities.
(cid:129) Thetheoryonwhichascienceisbasedmusthaveastrictlydeductivestructure,
i.e.,itmustbecharacterizedbyafewfundamentalstatements(calledaxiomsor
postulates or principles), and by a unified method, universally accepted, to
deduce from them an unlimited number of consequential properties.
(cid:129) Its applications to real objects must be based on rules of correspondence
between the theoretical entities and real objects.
ix
x Preface
Well, a unified and comprehensive science of the gears, which meets all these
requirementswithreferencetomultiplicityofthedisciplinesmentionedabove,does
notexistand,perhaps,itwillnotexistforalongtimeyet.Thewaytoachievethis
importantgoalisstilllongandfraughtwithdifficulties,sinceitisnecessarytoagree
the different scientific principles that form the bases of the various disciplines
involved. Based on the current state of knowledge, we can only say that there are
differentscientifictheoriesofthegears,asmanyasthedisciplinesthatcontributeto
defining a gear design.
Geardesignscholarsandexpertsarewellawarethat,withadesignbasedonthe
fundamental statements related to only one of the disciplines involved, there is a
reasonable certainty of not meeting the postulates related to other disciplines. The
difficultiestomakeatheoreticallyperfectdesignofagearpowertransmission,i.e.,
respectful of the countless design requirements related to the various disciplines
involved, are actually insurmountable, so the designer must fall back to a com-
promise design (the so-called design optimized, but often very far from the theo-
retical one), able to reconcile as best as possible the different and almost always
conflicting requirements, dictated by these disciplines.
Thescholars ofvauntedcredit,whoclaimtoknoweverythingandtobeableto
speak orwrite about any subject and whoboastorattributethemselves knowledge
inanyfield,considerthegearsasasynonymofobsolescence,asymbolofthepast
or, when they are benevolent, a nineteenth-century old stuff. This way of thinking
ofthosewehavebenevolentlyreferredtoasscholarsofvauntedcredit(noonecan
therefore accuse this author of not being kindly gentlemen) implies at least the
ignorance of the historical evidence that the theory of the gears, considered as
mechanisms, ismuch older, having it characterized thebirth ofscience,inthe first
Hellenism (see Gears—Vol. 3, Chap. 3). But the unjustified conviction of these
so-called scholars hides a far more serious ignorance. In fact, they prove not to
knowthatthemostsignificantcontributionsforthecalculationoftheloadcarrying
capacity of the gears were brought gradually in the entire twentieth century (with
theexceptionofLewis,1982—see references inGears—Vol.2,Chap. 3)and that
inthisbeginningofthethirdmillenniumnew andequallysignificantcontributions
of high scientific value appeared and continued to appear at the horizon (see, e.g.,
Vol. 2, Chaps. 10 and 11).
On the contrary, the true scholars and experts of gear power transmission sys-
tems are well aware that the gears were yesterday, continue to be today and for a
longtimeyetwillcontinuetobeanongoingscientificandtechnologicalchallenge.
Even today, surely it is worth investing significant financial resources, in terms of
manpower, tools and means, in R&S activities on this important area, as all the
technologicallyadvancedcountriescontinuetodo.Thisdependsonthefactthatthe
gearsareaverycomplexmultidisciplinaryfield,asfewinmechanicalengineering,
and knowledge still be acquired is numerous. We can affirm, without the fear of
beingdenied,thatgearsaretheresultofancientknowledgeincontinuousupdating.
Knowledge still to be acquired concerns not only those specific to each of the
disciplinesinvolved(geometryandkinematics;staticanddynamicloads,including
those due to impact; friction and efficiency; dynamic response and noise emission;
Preface xi
staticandfatiguetoothrootstrength;contactstressesandsurfacefatiguedurability;
nucleation offractures of any kind and their propagation until breakage; full film,
mixed and boundary lubrication; scuffing and wear; materials and heat treatments;
new materials; cutting processes and other manufacturing processes, etc.), but also
those arising from their mutual interactions. Other important challenges are those
relatedtothenewfieldsofapplication(see,e.g.,thoserelatedtothehelicopterand
aerospaceindustryandwindpowergenerators),whichrequiretheintroductionand
designofnewtypesofgeardrivesthatarenotreflectedinthecurrenttechnological
landscape.
Researchersworkinginthisfieldarewellawareofbeinginfrontofanimportant
goal:todevelopacomprehensiveandunifiedscientifictheoryofthegears.Really,
at present, a scientific theory of gears having a general horizon, capable of con-
sidering and balancing all the aspects involved in their design (i.e., geometric and
kinematic aspects, mechanical strength aspects and technological and production
aspects) does not yet exist. Only attempts of partial scientific theories of gearing
exist,whicharelimitedonlytothefirstofthethreeaforementionedaspects.Atthe
moment, partial scientific theories concerning the other two aspects mentioned
above cannot unfortunately be reported, as non-existent.
Among the partial scientific theories concerning the geometric and kinematic
aspects, it is however worth mentioning the laudable attempt by Radzevich (2013,
2018: see references in this volume, Chaps. 1 and 11) who, for the first time, tried
todevelopascientifictheoryofgearing.Inreality,Radzevichisheavilyindebtedto
Litvin(1994:seereferencesinthisvolume,Chap. 12)andhisfollowers,inasmuch
as he treasures the substantial contributions on the subject in terms of differential
geometry brought by them, presenting already known knowledge concerning the
aboveaspectsinthesystematicformofascientifictheory.However,Radzevichhas
thegreatmeritofhavingsetupageneralscientifictheoryofgearingthat,although
limitedtogeometricandkinematicaspects,iscapableoftreatingalltypesofgears,
including hypoid gears that include, as special cases, all other types of gears.
Notoriously,gearsconstituteaclassicmultidisciplinarymachinedesigntopic.In
this multidisciplinary framework, numerous and different areas can be identified,
whichhoweverarebetweentheminterdependent,sincethevariationofaparameter
or quantity relative to a given area, inexorably, affects a parameter or quantity of
otherareasinvolved.Usually,threeareas(itistobenotedthateachoftheseareasis
composed of two or more disciplines) are introduced, which identify three char-
acteristic aspects of the gears. They are as follows:
(cid:129) Geometric–kinematic area, which considers the gears as ideal theoretical enti-
ties, and that, in these ideal conditions, tries to optimize their geometric and
kinematic quantities.
(cid:129) Stressanalysisarea,whichaimstostretchasmuchaspossiblethelifetimeofthe
gear, considered as an actual structural machine element, working under real
operating conditions.