A network theory for variable epicyclic gear trains
Citation for published version (APA):
Polder, J. W. (1969). A network theory for variable epicyclic gear trains. Technische Hogeschool Eindhoven.
https://doi.org/10.6100/IR104190
DOI:
10.6100/IR104190
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Published: 01/01/1969
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A NETWORK THEORY FOR
VARIABLE
EPICYCLIC
GEAR
TRAINS
PROEf'SCHAIFT
TER VERKRIJGING VAN OE GRAAD VAN OOCTOR IN DE TECHNISCHE WETENSCHAPPEN VAN DE TECHNISCHE HOGE;SCHOOl EINDHOVE;N, OP GE;ZAG VAN DE RE;CrOR
MAGNIFICUS, DR. IR. A. A. TH. M. VAN TRIER, HOOGlE·
RAAR IN DE AFDELING DER ELECTROTECHNIEK, vOOR
EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP
VR IJDAG 6 JUNI 1969, DES NAM I DDAGS TE 4 UU R DOOR
JAN WILLEM POLDER
GESOREN TE STOMPWIJK
Dit pro~fschrift is goedgekeurd door de promotor prof. ir H. BIClk,
CONTENTS 1.1, 1.2.
1.S.
1.4.1.5,
1. 6. 1.7.1.B.
2 2. l. 2.2.2.S.
2'.4.
2.5.2.
6.
2,7.2.8.
2.9.
3 3.1. 3.2. 3.3. 3.4.3.
5.
3.6, 4 4.1. 4,2.4.S,
4.4. 4.5. 4.6. lll,tX'oduction S<;:ope Rotating shaft Concept of a tht"e"'-pole NodeEp~cyclic gear train Tt"an6mi6s1on and vadator Power flow
Foundation of the mathematical model Performan<;e criteria Qf single
epicyclic g$a. trains
Angular velocities and tranf,n:l~6sion ratios Torques, powers and efficiencies
Dissipative powe., internal powers
Interrelationship", between binary effichmcies InequalitieS foX' angular velocities and tOl'ques Design of a s~ngle epicyclic gear train Power flow in a dngle epicyclic gea. train Sun X'atio and sun efficiency
Rapid calCUlation scheme
Introductory Consideration!;' on the n"dwork theory
Components of a variator network Junction and addition of transrni$$~ons Rapid determ~nation of torque ~'atio5 and efficiencies
Conversion
Inter'change of adjacent node6
Interchange of adjaoent epicyclic gear' trains Analy$i.6 of structure6 of variator networks Closed nGtwork Meshes ReHerative networks Inconsistent networks Reducible networks Parallel branches page 7 7 10 12 13 14 17 ,18 19 :;)2 22 24 26
27
27
33 37 3839
43 4.3 4.) 47 4951
52 55 55 56 57 59 62 675 5.1. 5,2. 5.3. 5.4. 5.5.
&:.~.
Clo,",,,ificatiun and synthesis of structures of variator networks Hesponsivity
DistribLition ()f poweI' Hackbon~ chain
ClassHication of variator network., Variator' netwO):'ks; with one variatal;: Variator networ'ks with two variators 5.7. Rec"-pitulaHo<l of variator networl(s w),th one
or two 'lariat","£; page
70
70 71 74 7476
30 1:156 Vax'iator network of the variable shunt
typ,:, 137
6. 1. Dc scription of a variable shunt B 7
6.2. Condition for the power flow 90
6.3.
S",lf-locldng vat'iable "hunt 92~.
I
RestI"ictions imposed by practic$.l reqllirementl3 937 Variable D",t:o,Ilork6 of the variable
['r'idg':' type 97
7. L Descript:i')ll of a vadable bridge 97
7,2.
Conditions for the powor flow 1007.3.
Power now throllgh the branches 1017.4. Roverse - sJillD1etI'lc vadable bridge 102
7.5. Doubl€ ,:,picyclic gear tr/lJn 106
8 Dynamic t'€sponse of a variator
netwClr'k 110
8.1. Schematic dynar:nk repre<lentation c)f an epicyclic
g", at" train 110
3. 2. Dynamic "lability condition» of an "'pi cyclic gcar
tr,,-in 115
Q.,~. Electri.cal a.nalogue of a va.r'iator network 119
Tel'm~"ology 122
Summary 129
Ref e '" i·~ [1 C e s 131
A.t tho top of each page in tho out"" margi.n appeal"' chapt~n' and section number. In addiHlm, each item (ddinition, theo:r'em, equatioll, "te.) i<l gi,vel1 a D\lInber that appear's in til", left-hand r:nargin. S"ch nurnbflI'S 'lre nurnber,,<:1 consecutively within ""eh secUon. When we refer in a section to all item within the sam" section, only tho item number is given.
1. 1, CHAPTER 1
INTRODUCTION
Thi" ",tudy deals with the transmission of power by rotating elements, ':;u"h as ",hafts, in certain combinations of e pic Y c I i c g ear t I" ai n s w{th con tin U 0 Ll " l y va ria b 1 e t I' an 10 m iss ion s .
Such combinations achieve in principle the same as variable transmission" themselves, viz. the transformation of an angular velocity into another and ",J.1:Ilultaneously of a torqu€> into another, whUe the ratio betweell the angular velocities Can be changed continuously. Tho product of the original <>.ng1.l1ar velocity and torque, called input power, decreased by a diss~pative power (absolute value) equals the output power (ab»olute valu0). An example i", provided by a variable ",hunt, also called split-power, divided-power, bifurcated, shunt, or differential tr'an",rn,ission; sce Fig .
.!..
! .
.1.
Fig. 1. 1. 1 A val'lable shunt. Speed increasing if
shaft end B is input shaft and .~haft end A is outPl)t shaft
2 A rotating shaft performs a powel' path .. Other elements allow a power path to branch into more power paths, arld two m- more power paths to combine into one. It is well known that an e pic Y c ~ ~ c g ear t I" a in performs such a b,anching or combination; see Fig. 1. 1. 2. A similar' branching or combination of power paths is performedby ~onnecting threo power transmitting shaft.:; to one transmission; $ee );.'ig . .
!:.!..
2.
In a systematic desc,iption this invohes that three !;lO-ax;i,al shaft ends are inter!;lonnected. A fixed interconnection of three shaft ends is called a 'n 0 de' .1.1.
Although nodes play;" part in th", description qf the systems to be consiciered her'e, they have 80 f<lr b""~rl oITerlook",d in liter';"turc to the Jctriwent of lac!, of g,.r",,·ality, N()(k" may be hLUld in sev",r'1l.1 ciesign,,;
o;ee Figs, ~,.!..,!, ~:.!::~, 1.1.5.
Fig, 1. 1.2 An epi"yelic g~'''u:' train coI""bined with a
tl·",no;rIli.~sion; tn actu<ll d""lgn (<l); ,,."pdrateo
from the trl\n~llli88ion tn dn equiv"lent cI(,~ign with co-axial shaft:s A, C, E (b);
sch",naticaUy (c)
t'W, 1. 1. ~l A ",.,dc: co[nt.Jinc:cl with;;. tr'ansrnission j,n actual oe>;ign (aL ",,,-parateo from th':! tr'ans!T)j,;;;:;ion i.n ;"ll cquivRh~nt design with pow"'r~ tr-an8mitling co-axial ~h"ft cncl~ 9, 0, E (b); :-;chematicctlly (c)
1.1. 3 This study is Gonr~nect to combinat~on6 of epicyclic gear trains with
continuously variable transmi$$~ons which a$ a whole act as a variable tr~6rnission and ther~fore are called I v a I' i a tor
net w ') r k!3 I . Later it will be proved that a variator I'HHwork has one
input >:!haft. one output shaft. and that the numbers of epicyclic gear trains and nodes must be equaL Combination" w~th different degrees of freedom like thos~ of Fig!3. 1. 1.4 and 1. 1.,~ are left out of consideration.
Fig, 1.1.4
Fig. 1, L 5
Combination of power path" through a node, Two ge",r drives, each with its own input shaft have a main gear and an outPl-lt shaft ~n common. Adua.l design (a);
schematically (b)
Split powel' gear dl:ive of the double-reduction locked-train type. Th" di6tl:ibution 'of power depend s on special propertie s of the lay" out. Actl-lal de sign (a); schematically (b)
4 A variator network has one or more of the following advanhges: - the range of the ratio of a variable tran5tni.~>:!ion in a variator
network may be I-ltilised to a larger extent;
- the variable transmission need not transmit all the network power; - the efficiency of a variator network may be better than th",t o£
it" variab~e transmission;
- in an emergency, a vadator network may transmit p<)w$r even H its variable transmission fails.
1.l.
1.2.
5 For deHcl'ibing and studying the pedormance of va)"'~ .. tor netwol'\(s witho11t undLle efi<)T'1: it is convenient to develop a consi>itent analytical formulatiorl. systematically applicablc to any design l,lf;ed in actual px·':',ctice. It is far 1",,,,, convenient to d€vclop :;m analytical formulation on the basi" of any gr()llp of par·ticular o;lesigns. The validity of such a sp,>dalised apPt'oach would be limited to the de"igns ch<.>r;cn and :i.t would be difficult to deduce a general theory fX',,,n it. The consL:<tcnt ~.pl'licll.tion of the analytic a I formulatiun, c all. f,,1 'm at hem at i cal mod c 1 " ~8tif;fies all requil"emcnts for basic design in a mO,'3t econOtI."J.ic waY;r v.iz~
- the D.nalysis of a varbtor networl, of given type;
- the 15ynthesis of a 'lD.ri.atll" network satisfying pre>icr'ibed r€qllireill€nts;
- the syntl1esia of a vat'iator network optimal in one or mOL'e I'cspect",.
1.2. Rot«ting shaft
1 A I'otating shaft is com,iderea, as uI:I\1a1, to b" one single ele:awnL Ith"", two shaft ends, eachmark"dwithaletter. '.I.'h€ ijllantiti.",;
w ang\llar' velodL,Y T tor quo
P sh"rt pow,:,,'
<I)"''' assignHI to <I "hal't end and further specified by one sing'le suff),)!:
corTe spaneling with Lhe lett",' of the Hha1t end.
2 A common engineering pNlctic" for simple problems i8 to worl,
'''XChl5~v''ly with positiv" value6 <ond to fit signs ~n fon~\lhe to til" case.
The lade of g,:,nf"r'al conventions and the limitations of the apPf'<.lach in (,xisti"g Hteratllt'C t'C6\lJtS in a mLlltitude of forrnulae, Such 00 pT'>lctice '" very ineffecUv'~ in mo"" complicated cases. It is neceHSat'y t<.' (Ievelop only one Het of f("'111Ul<l<:" in a mathematical mod"l operative f,w any de»ign con<.:civable,
U,:,,,cefo)"'th, 00 wi.d«r' Significance will be ""signed te, the v"lue of ~ quantity I:han it ,,,,mmonly has , while a widcr range of value s wj,)l be covered (rOt· inf>tl-l.nec, in chapter'
2.
imaginary v"lues ar"~ used).::\ 1"0" the inteT'jll'etati.<iIl 01' <I v"llle the ,-,oIlvenUuns to'" the sign,; are
very impo)."t,,-nt.
'rh"
sign "r a nurr,erieal v"lue ind~,-,ates a direction of th€'o qU<lntiLy,4 One llil'ectiort of rotaLion of "- shOon will be chosen a!:l the poHitive directioni
""0
Fig. l.:~.
.!..
A posihv<l directhm once cll,,8en is maintained in the ",-nalysis of tho systeITl iIldcp8n<lp.n'ly of th" posithlTl of the ()b8c~·ver.
5 An angl,l.lal' velocity is considered positive if the shaft rotates in the posHive direction.
Fig. 1. 2.1 PosHbe direction of l'otation. Directions for positive values of angular velocity 11.1
al)d torque T
6 A torque exerted on a rotating shaft is considel'ed positive if H would tend to drive th", shaft in the po",~Hve direction.
The above convention for the torque is the mo",t l'ecommendable one 1.2.
to a.vdd misunderstanding in the interpretation of torq\.\es and reaction torques. Linking the convention for angulal' velocities to that for torques is important to the interpretation of the sign of shaft powers.
7 Ash a 1 t power is the prod\.\ct of the angUlar velocity of a shaft end and the torque exerted on that shaft end. Hs sign is governed,
Hke in
algebra, by the signs ofHs
lactors. '8
A positive shaft power means an input of energy toward the "haft thro\lgh the :ohaft end consid<!!red. A negative shaft power means an output 01 enel'gy from the shaft through the shaft end consideI'ed, 9 The dynamk properties of a rotating shaft ar", described in accordance
wHh common engineering mechanics with the aid of ;\rl 0 ill en t s 0 f
in e r t i a lumped to a number of inertial element!), and s tiff n e sse s lumped to elastic elements of the "'haft; see F~g. 1. 2.2.
f - -
{}
(a) (b) (c) (d)
li'ig. 1. 2. 2 Schematic representation of a shaft ",nd (al. an inertial element (bl, and an elastic element (cl of a rotating shaft. An example of a rotating shaft (d) with shaft ends markedA,
e,
Illoments of inerti.a)\, )2' J" and stiffnesses ~, ~1. 3.
1.3.
In ch""ph;I'
Q.
it will be shown that moments of iner'tin anci stiffnE:!l:lscs H.>;signed t<, other e.l.,:,rnf,nts (plam!t ge[(r,~) may be h'atlsposed to the shafts. In a st(lHonary 'lituation [( 'r'otating' "haft ha'l one fixed v"luef<.H' the tocqU(' tran smilted [(no one for HH angula)"' velocity,
l,S, Conc':'l't of ,9. thre,:,-pole
Epicyclic gear tr·ains and nodes havc in common that th':'y interconnect tht'ce pow,:,r tr'an'lrnitting ,shaft end", thus constituting 't h r E:! ,,~ - pol E:! ,; , in H. va!'iatm' rletwor-k. No oth" r' entiti .. s than th,.ee~p('l«s aI'", rf''1l1ired
to p,~dorm ,,,Id seh..,rnatieD.Uy r'cpre"ent the inkr'connedions of rotating "hafts. If more thaYl three power p(l.th meet, they CC'Tl be descl'ibed by
d sequ"n<.:c of thl:'l'e-pol",;. See Hg. 1::~:
1:. '
Fig, 1. 3. 1 Rxampl,:, of 0. va r'i at or' !lHwork, Th':' tr-iangkfj
i,n (a) r",[,n; sent I:hr'ce-poles, the dott .. d lille"
c.ontain va J:'iablo tT'~n!-:;mis~;~f.")flH_
In (11) the three"pohe,; tlre diHtingul'lh."l in r,picy(,"U" gen)"' train~ anl! rlOdes, inuicnt'Hj by
dr'de:; ancl dotH, )'':'R!',;:ctively
2 In ITle),,!. oth"r' network" that c~,r'ry pow';,,', Kil"<;hhoj't'~ law,; 0'1'13
appliclIt)h:, for \[lstance or! the voltages and cun'':!llt,. of ",]"",Uical nE:!l:works, In a v[lr-iator llE;!twork Kj,r'chhoff'" laws. plltting i.t ,;impl,Y. lind with ;;o,(iHquate!:!ign conViientiODe, may b,:, interpr"~ted for' H node
the angular velocities of the: threE:! shalt E:!nds o.re rlllltun.lly equal (two lineal' equations);
- th<'. 'mm of the torq\l«'" is z~>r() (one lillear equation),
3 The "hove int,:,rpI'ctat:\(lll ot Ki.rdlhoff'f5 laws for a nod,:, docs not satisfy fot' [In "picycli<: gonr tr':iln. Contrary to nodes. «picycUc gear trains are ch"-I'aetedfiod by one linear equation for "ngular v<)locitief5, and two for' torqw?f;. Besiclc:s, thti!f;C equations contain c.-rtain paramet.-r,;. 4 As to both Kirchhoff'" laws, epicyclic gear trains and nodes have
different p1"(lpcrtie>:l, Hence. the conception ()f three"poles witho,lt
further >:Ip':'cifico.U(.>lI will llcaccely be used arId [In impodant dspect of
the network theory given hereafter' b;; the distinction of two types of three"poles, viz. epicycUc gear trains and nod~6. Therefo.e, no f"".reaching <;Ioaloguc to other spedalised nelwQ,k theorie", can be set up, neither' to the kinetic network lheo,y (lay-out of mcchan~'lms), n01' to the electrical network the()ry (except fo, the analogue to be put torward in chaptliI'
.!!.).
5 A" mentioIled above, a node satisfi"'6 two linear equation" for anglLlar velocities and One for torques, wher"~as an epicyclic gear' t"ain sati6fies one Unear equation for angular vclodties and two for tOI'ques. These sets of lineal' equations underlie the variator rletworj, theory in the sami£> way as tlli£> common Kirchhoff's laws do fOr' (.>ther netw()I'k thcod~s.
In the mathematical model a three,-pole will be specified by mean" of a set o£ three linea1"' equations. A further ,>peeificati(,m ~s obtaini£>d by
a few strikingly siITlple assurnpHoOF.'. given later on.
6 Fol' [h", .,ake of mathematical rigour and generality, in the mathematiC<;Il model the names of machine element,; should not be used. To avoid too ab",t,act a treatment, howevel', the term'l shaft, epj,cyclic gear train, and other tel'm., will be ui;~d for entitie., in the mathematical ITlodel. In an application any such term indicate" an element in the actual design. The double sense of a term if;! not Hl(ely
to cause erroneous interp1"'etations Or undesired limitations. Summarising,
a
three-pole with two linear equation., for anguhl' velocities and One for torque., is called a node. A three"polc w:tth one linear equation for angula.r velocitie s and two foJ' t01"'ques is called an ",p~cyclic gear train.1.4. Nod e
fin interconnectiun of three power t1"'ansmitting ~hafts is caUed a 'n 0 de'. The aynlbol tor a node is a dot with three line,>; see Fig. 1. 4. l. It can r",adily be accepted that th", two equation" for angul;;:-r-v.;iocities involve mutual equality of that three allgu~ar velucitiEOs, while the equation for torques formulates the cquilibrium
A~'
Fig.
l.i,.l.
Node2 3
1
I·i,
1. 5.
betw"on tb" torq\l"s. The set of oquaHoI\s foT' a noelE'! with ",haft ends
A, B, C is
Yr(,>rn~,
:!.,
and 1.2. i:l H follow,:; for the shaft powers tl1"t1 _ 5~ I,; pic Y eli c g" <l r' t);'" i n
.\ The cU1"lyGica.1 for'nwi",lion of ~u1 epicyclic goal' t1',,-in can be dedLwecl most ,:,,,,:;ily if done for a bl«.d<-box urtit with all th!'ee ~h"fts )·ot,,-ling. As )n"IlLion",d bofor";', an ",pic,yclj.<., Iie[(]' tr-ain will be rq'l'es",nlcd in the rnathematical model l' H a thr",,-polE:! with ,-HlC linear' equation 1"<."-'
.... ngubt' velocities and two line,~l" equations ror' torq\les. 'fhat
r<:!pl'es,:,nl.ation wi.ll be justifi<od by th'" fact that ony conedvable ,lfOsign saHMies !;l\H,h [( ,:wt of ,~qLlationH_ By ~c!ding " few <'tdldng'Iy simpl(, aSS\lIIlptionH, the (equation" will 1)(: specified completdy. 1<".1"'8t, W" ded\lc,' the '-,quation fo,' (lngubr v~,loeiti':'H and nt-,xt til':' two f<.>r· torques. Ttl':' symbol of (In epicyclic gear' tr(lin is [( dr'ele with thr'ec lin",,,: see Fig'. L G. 1 .
Fig. 1_ 5.1 £!:picydic gcm' I:,'ain
Let tIl'.' equ(lUCH\ fo" a"gulor' vclocitj,€s be wPitten in the g"~nero,l fo,'m
2 The only as,gumption th8n to be made for spedfying the cof:fficients
a,
b, c, d i'3 that in <;ase of intern~l blocking of the bl::\ck-box unit, the three (,o-axi~.l sll.\ft", will b~' allow"d to I:t"-ve the "",me lH'bitrary angular v<:!1ocity. Irr mo,thcmo,hcHl term,;, tile three «ngulo,)· vdoc~ti"'s. in c[(se (,f mutual cqu[I.lity, bave the !:lame I:wbitrm'y value. The tdentity11[1.R to be "Htisfi(~d (0)' "ny at'bitr'8.''Y vahLc of wA H",nce [a ~ b +~) =
a
£'end d = 0 .S~llce ii, band c are not simultaneously zero, one of them, say iI. may be taken
a."
O. Henceb b
WA-'-a,weH-'i-l)"t=
0 Instead of (_~) we w,He i and obtainwA - iWe + (i ~1)Wc~ 0
1.5.
in which i r€p,esents a par'ameter charactedstic of the design of the epicyclic geaJ.' train.
3 To avoid the chaotic treatment lmown from literature on the level customary in engineE!dng, the symmetry in the fOl'mula will be r€!etored. For this pU1"pose, the parameter is furthe~' I:lpecified by
two
suffixes separated by a virgule (/). The suffbe:e" determine the sequence of two rotating shafts. Moreover, for a reason explained h,,,,re below (1. 5. 71. the parametE!r is provided with a straight bar pla<;ed overh~ad. -So, wl"iting [AlB instead of i, we get4
~n. which the meaning of
i;.,il,l
beCOmes clear by blocking the shaft lind [Ale~ ~
fOr c.t = 0B
5 The parameter iNa called 1 b in a r y r' a ti 0 I stands for a gear ratio
in a situation with (In'" blocked (C) and two rotaUng shaft ends ( A and B ). Although equation.! may sugg",st a preference for a certain sequence of the shaft end", A, 8, C , yet such a preference does not exi.,t. Any eonsL,tent transposition of "ufiixes in.i yield" another binary ratio
depcnd~nt On (AlB' without di"turbance of the l"elation between WA •
We,
andl<ic.
6 A conei!)tent transposition of suffixes throughout the formJ..\lae is called a I per III uta t ion I . Each !)equence of suffh~eg is attainable. An
example of a permutation is the ~nterchangc of A and 8 in equation 5, which results in the relaUon of a parameter with a reciprocal one.
-- 1
'BJA"
;;"1137 The gtraight bar placed overhead distinguishes the binary ratio frOID
the quantity
tAlS
with a broken ba, overhead, The latter is calleds
It ern a r y !" a ti 0 I and is d",fined by
tA.l9-
~
in a situation with three rotating shafts For mo,e details, seE! chapter ~.1.5.
,),,, specify the "'lu.-.thms fo" torq\leS twv addHi"nal an~llmption'3 hdVe to h..:: ll.'L3.d" ,
10 7;;+18+7(; ,,0
11 The oth(Ol" [("HumpH",) i" that tll" lhl'ee torq1.le" may be !:l).mu1tancou'3.ly
ZE:H'O, H,,"ce, tfw second equation {or thp. lorq\](~n co." be wpitten
12
13
while 01. (J, 1 are gcnen,lly cli.fferent. Eliminativn of
Tc
yielo" 01.-7(fl_
11'1A+Ta ,,[)
ot-., ,
.
The rae.tor (0-1) 18 a COllst:\nt, leu-thel-. on tI-e~ted as '" para~~cte~'_ In
SP1Lt"! 01 the seerrllllgly C:Ull.11)"ULlS way oj
wI'ttlng
we )"'<l'lace (~)hY'Nei'iBJA
t..Joi'i~.)\7A +
19
~[)
In the pr'(JcI\<cI.iA
i13
i'i
BHI the par'arrwlC[' [NB is tho bino.ry nl.l:io d<:,finocl "hove, The rn..::aning ofil
BIA becomes clear by the ,!(,ductiolln.. 1 [ ii _
WB(~
Til),_~
"~/A
-tAtS
:AlB e",- %.1A
Ft.
for ~= 014 TIW paramC'ter' iie/A .1.0; call",! I bin~' l' y C 1 fi '.! lIOn c y '. The two eqlJ",tiO)lH.!:.Q and 1::> for tll", tOl"CI'Ws can be wy-itten
J G '1'11" >itl"o.ig'ht b",. p1ac,,'\ overh"ad cliHting\]iche~, tho bin,,-"y efficion<;y
tt'orl) Iho q1.1'lIltity ~BJA with a tn'oj,en bar ()verheacl. 1'11(, l'Ltt<'l' is callcel
I tel' n a r y co f r i (! i e n (! y' ane! is de fined by
1" f IIsrl"-p" ~
I } ,
l,n a Sltuatlon Wl.
.
.
t I t ' 11 t H"ee r'otu J.nl~ $1,\ jr
I:sA
1;'('H' mor'!} d\!Ll.ils, slOe Ghapt"y, ~.
1(;
Ii:
+'il
+Pc
= «)A~
+wsTs
+wcJ;: ,,{w
A- iA/Bii8/AWB + ([AiBiiljjll.-1)wc}J,;. 01.5. 1. 6. Generally, this sum is unequal to zero. Apart from the thr-ee shaft powers a power with another charactB!'istic has to be di,:,tingui"hed in the epicyclic gear train.
18 This power is Imown as the 'dissipative power', 0, in mOl'e popular terms the power loss, indicated by
Py.
20 According to the sign conventions a diss;'pat!.vE> pow'n' i6 never positive. Dissipative powers are n()t neglected th\"'oughout the study. A neglect would be unacceptable ... ,:, wHl be proved later on.
The epicyclic gear train will be discllssed extenSively in chapter ~.
1.6. Transmissio!l and variator
A common gear drive has two power t\"'ansmitting shaft end" connected to other units. The torques e)(erted on these two two shaft ends ar-e not in equilibrium. If the torque on the frame (gear box) is also considel'<:?d, we do get the equilibrium of three tOI'que",. ~rhe\"'efo\"'e, three elements will be considered. In this respect a COIYlI'><)TI gea, d1:"ive is similar to an cpi1:ydic gear t,ain.
In the mathematical model a c"Il:lIl:lon gear drive, or more generally, a transmission, will be defined as a particulai- case of an epicyclic gear' tr'air!, viz. an epicyclic gear h-ain with One "tationary shaft end
(angu~~r VE:'locity zero). So, all propert~e!l of a tran8mission to bc dealt with h, the theory will be taken into account automatically. A tI'ansr:niH6irm deduced from an epicyclic gear train need not have C()-i3)(i al 8haft'>, for it is not a requirement of the mathematical modeL The ,:r)-a::daUty of epicyclic gear trains is (>oly a p,actical aspect in the der::ign.
2 In6tead of 'gear drive' the mOre general name 't l' an s m iss ion' is uiOed. A transmi66ion may be any design wHh two shaft ends ensuring the desired ratio'> of angular velocities arId of torque". The above discussed equilibrium of torques need not in general be con8idered. 3 A contim,l()usly variable gear drive Llsually h.as a design substantially
different from that of common gear driVeS. In our theory the actual design will be disregarded, and the only kind of performance to be c(>n6idered in principle is the tr-an.,formation of an angUlar vdodty ),nto anothcr and of a tOl'que into another. Still one aspect remain", namely the onc.distinguishing a continuously variable gear ddve from a common gear ddve. 1. e. the variation of the transmtS8ion ratiO.
1. 6,
1.7.
4 The v"rlo.Uon of the trall.~ITli",f;ion ro.ti,) if; reo.li",(!d by c.;ontroJ. from outstd.., I.h(: val'i"ble gcal' ddve, ind':'p"Ildent of the ",ituation insidp..
AH to the eqU[1t\onH i'm' <1nguL<ll"' y"locities ["(no tOI'CjllOS Q conHnuously
VHr'iable geo.r dl"'iv<: iH simil(\r to d lransmission with fixed
"i:r'ansmission t'atio.
" A conti.nLJDusly \lo.ri["(bl .... gear drive will bc c8.11e<1 'y~. I' i a tor' , Irl<1inly to emphasise th" poosibl.: v<lY'i.HHc)rl of its trClnHll"lissiol1 r~H(l. A 'tn,n.,mission' is (;ollHidcJ'ed to hav« a fixed tl'",n"mission "atio.
6
7
8
TI'anSI11is<:;Jous and ""riat","" al'c special c,,-scs of an "'picyclic gea)" l["ain in Wllietl ono of the ShHn onds is con8icicl'cd pel'XX1ancntly blocb;ci,
A ciiHtinctioll LH;'[wcon bin"ry and tOl'l1o.ry P~ ,"'ameters lwed not be 111:.<3",. Ttl", aUg'ula!' ve.Lodty of tho block,," sh<1l't, s~'y
1I!c,
is zero, lknoo, eqUo.thHl .!.,~,i
bec(lTn<',,,The tor'quo of the 1)1""kc:d sh"ft "nci will be ignoI'Gc\ o.l1d ;;" is Gquo.tion
1, 5, 10. Eql\o.tion':l.!...~.
g
o.]ld,! ..
~. ~ becom<:,The ~ymbol for 1\ t.r·ansmissiorL is an oval. "flor the e;>rample of II
driving h",l1.. The mark of' the shaft end on tl1", 'pLllley' side is the fi)"';;l "uHi:>:: at th", tt'ansmissi<.Hl I'atio tAl8 and the second suffix ()f t.he
<:' ffic:iency 1)BIA' Tho inscriptiun 11fl,A~" bt'aciceted to enable II di.,tinction
from thc inscdplion
iwe,
E:''5p,;,cially wh<!n values ~l"e inscribecl instead of letter' ,:;ymbols; ;;p.e Fig,.!c •
.§".!...
'I'hp. 8Ylnbol ()f a vaci"tol" is that of II tI'ansmis,>ioIl, supplern"ntod with
,,)1 aY'['OW. It i!:\ oftoll PI',~ctl"al to simplHy the nota.tion of til,:, pa"arnete)''>
of a var'iatol' by intI'oducing such symbo18 as x fm' t...IB' o.nd 11. for 1)~IA;
sec' Fig, 1.6.2.
c~f...
'DJ(
+-(r;J
I''ig. 1.6, 1 'l.'.'a'lsmissioll Fig'. 1. fj,:': Vario.toY'
1.7. Power' flow
'j'".,
supply ")1d discho.rgE:' of "norgy may be rep):es"nlod by a Sankey-diagl·8.!ll. or TI1<)['O sin'ply by a p"ttern of al'row~,2 By convention, a positive pOw""r is repre~ented by an arrow towards the unit, a neg""Hve power by an arrow away from the unit.
The power flow in a va.dator network or in an cpicyclic gear train may include a closed loop; see Fig .
.!,-
2.
~-Fig. 1.7.1
BraI1Che,1 power flOW
Fig. 1.7.2
Blind-p;w;r flow. The arrow of the blind power is indicated hy a dot Fig . .!,<~: ~ Self-locking !;ituation. O\ltp\lt p<)Wf!I'
",ero
1. 7. L 8.3 DEFINITION A blind pOwer is the smallest power ~l()ng a closed path of cyclically equally directed powers.
-1 A branched power flow is a power flow that d()es nowhere include a dosed lc)()p with blind power. An example of a branched pOWCI~ flow is giVE'!n in Fig.
!.
2, .
.!..
5 A. "",markable situation aris:es when in ~pi.te of a power supply (on at least one shaft end no shaft power "an be wHhdrawn from any part
of the vl'l.l:'iator network. Such a situa.tion, caused by cxccss:ive blind
pOWer', iA <.:",ned self-locking_ See Fig. 1.7.3.
1.S. Foundation of the mathematical model
1n thO? previous $;ections ""venll definition$; were given and assumptiorls mQde which llllderlh, th~ variator netwoI~k theory. The definitions concern the significanct;, attached, and the name,,,, ghen, to entities in the mathematical model., "",e ~'emarl{
1:
~_ §. _ The assumptions a,"cself-ev~dent to such an cxtent that undoubtly any conceivable design is covcN:d by the theory developed her.,aft",r.
The definitions .. nd assumptions discus 8",d so far are concis.,ly
compiled below. For the sake of curhplet"'lless 7 (Il, (II), (III) and 9 (HI), (IV) a"e ac.lc.l"'d. The discussion of 7 and
?i
will follow iLl'Chapter ~.
-1 (I) DEF'INITION A t" 0 t
a.
tin g s haft in the generalised notion ill an element that t.ansmits power from one I5haft end to another shaft end by 110 other action than rotation about its axi$;. It may ~ndude at. B.
!:IhMt in the cor'llmon 81:Hl.'" and conll':'dillg elerntnu~ such (~'" '.!()Llpling'> ~.nd clutch':'8.
(TI) i\.SSUIVIPTION A .,haft end ~~ (,hat'acteJ"lf;(,d by (Ul >l.llIZle of
r () 1. a. t ion and a tor q \J (~ that gen':"'>Llly are tim" - dependent., (Ill) DEFINI.'l'lON The fir·"t derivative with re8pcct to tj.mH of th.~
",,[{Ie: of rol:a.tioll is th,·, angular vclocity.
(TV) ASSU IVI PTION A ,'otating sllaf[ will b,:, <:mt8ider,:,d ,~" alternattng ""'lllence of «lastic "-rid incdiD.l I:' I.<:,rrl<':nts.
(V) DEFINl'l'JON An eta",tic eloDlen! i" clE(r,~d('riscd by ~. c"llstrll1t quantity c:~l.l .. ,d s t i!"fn <:' S S .
(VO ASSUMPTION The t"r'qttc in an dastic et!:'mellt of [) ",haft is
I,qual to the pr'()cllld of th" "tiffneiO'l >l.nd the dJ.fkr'cnc':' of OI.lIg1es of
,'otiiliOl1 of thl! ,-,()nl1eoUng shart e.l.!:,Hl(!lltS.
(VU) DEFINITION AJl j",:J"tial e'':''''lont is chll.r'adcr).':l,'d by a confltant qUHntity cnl.lt:d mOrrl~'nt. ot in" "tia.
('/UT) 1\SSUlvlPTION Th" SLllll of \or'<jLLCS neting on nn in(,,'tial eJ.<:,mcnt
i~ eqllal to th,:, qllotient of Ih" second d,·!t'ivativ,:, or the angle of l'otatJorl with ",,'Hpeot to tim" ~Lnd tlw '''Lllnent <)f inc rtiD. ,
2 DEt'LNITTON
(I) A t h,' co c' - pol" is D.n elemont conn"cting "haft cncis of tht'cc ,'otatIng "hMt>:!:
(II) "shaft ,:,nd cannot he eOllnect,,,j to mOl'':' than onet.hr'cc-po.lE:!: UO) a thr"."'-pole rE:!H liHe:;; [L '>et of three lin"c,,' equations, either' two
f,,,' anguJ.:u' vclocitieH and onE:! for' torqu<oiO, or one for ang1JiH.r· velocities a[ld two fe",' t01'que6,
(TV) ASSUMPTION No other' (,lement~ l:han thr,,,,~-pole'5 >l.n:O necdt!ll 1:0
d,: sedbe the connecti()lls be\w,:, (,n L'Otating shntt s.
3 (1) DJ.~;FINTTION A Ih,'ee-poJ.I" with two '~quationH fo,,' <lng-ut""
vr:locitie>:l H.ilLI uno foT' i.Ell'ques i H ulllccl :, "" de,
(TT) 1\SSU IVlPTION 'I'he: two eq\J~.Lions for the angu.lrH' velociUeH invulv,:, TTlLli.u"l eqm\lHy or the'le <LngLll«r v"loc:iti",':l.
(ITT) ASSUIVlPTION '["'1.(: cqUD.hon rOJ:' torqllL'" fOI'l:t1UJ.:,.t"" the ,·:qui.libt'i\ml betwcen th': i.Or'que>;l,
4 (I) IH;lcINITI<)N 1\ thl":'''-polc' with one eqllll.lioll fo)' angul[Lr volo"ilk,s D. 11 (I two 8qun.tioll" 10).' torgue:> L~ canccl (\n
I:~ ~) i c
.v
c I, L (: g C 0. 1" .~ r' H in.PO i\.SSUNIl'T10N 'I'hr: ('qLLation for th,:, thn'<:' ""guldl' vdudtlcs
pl·Y'rnil.;; them, in th"
<.""".,
of llllltlial cqUD.1ity, \I.) hav,:, Ih" ~arn"nY'llil:l"H'y v"luo,
(III) /\SSUlVl PTTON 'l'ilL" two eq(lHlions for tOt'qU(;" pel'mit r:<ll. three of the,n to be ~imLlltD.n')')\J sly zero.
(TV) ASSlfMf'TION One of tho equations for' torque!:; formul"l(:;; the "quilibri,unl bctween thc toY'que8.
5 (ll DBF1N!1'lON A t ran s m iss ion is an epicyclic gear train with one stationa,y shaft end.
(II) ASSUIVIP'.('lON The angular velocity of a stationary shaft end
is
zero.[Ill) ASSUMPTION The to,que of a stationary shaft end, as well as th", I3haft end J.t"elf, ~"left out of consideration.
6 (l) DEFINITION A va, ~ at () 1." i6 a transmission of which the ratio between the an.gular velocities can be varied eontinuouslj by control from outside the tr'ani;mi",,~on, ~ndependent of the situation insic;le the tra"snlission.
(II) ASSUMPTION For every ratio between angular v<210<.:1[1"" th""", exists one equation for the torques.
7 DE:FmX'l'lON
1.8.
(I) A va ria t () r net w e) [" k is.9. cohe1."ent l3y6tem of rotating shafts,
epicycli.c gear tr«ins, n.ode", tran'3IDissions and variators; (m aU (~ngl,llaf" velocities ~n a variator networl{ are determined by "taHng ODe angUlar velocity.
(III) all tOI'ques in a variator network are determined by stating one torque.
3 (Il DEFI1'lrl'ION A shaft powel' is the pr'oduct Ilf Lh".angular velocity of a shaft end and tHe torque ,,:x;erted qn that .maft end. .
o
(I) DEFINITION Dis" i pat i v e power 10 OCCur in epi,cyclic gear trains, transmissions, as well a6 variators.(II) ASSUMPTION A dissipative power has a value l,:,s$ th",n qr
~qual to ;;;<;,ro.
(III) ASSUMPTION F<)r an arbHr-arily dissected part of a variatol' netwol'k til", Sllm of shaft p')w",r~ qf all shaft ends and the dissipativa powers is zero.
(IV) ASSUMPTION A dissipative power neWs not influence any relation between angulat' velocitie",.
2
4
2. 1. CHAPTER 2
PERFORMANCE CRITERIA OF SINGLE EPICYCLIC GEAR TRAINS
Although lh,· litel'atUJ'e of tho last half centm'y Jiseusse~ the sing'lt·, epicyclic g"'ll' h'ain if) <lNail, the1"'" iH still ~·e'H.;on to start all ove,' H.gdin. A big'111y c:onsist(!l1t rIlalhew<-IH<;al rnodcl h,~" to be dev,·d.opcd in o[',kr to inv"Hl.igat0 c01"nl'li""ted vadatm' netwo"'k,;, notro:'lctiV(·ly, nC,w D.SlwetH a,'c found ror' tho anl;.Iy"ls of 8impk :-;y.9tem~.
All w,dl-known j(H'rIlulac Ql'e deduccd with mInimum ,~rfod; a few ,,(rw fun<:l:i()ns CU'(' j.ntl"'(),luood. e~pc:cially fo'-' I:he dete)'minal.ion of tho< ciirediDll of tll'~ powCr' now, I\. complete fiLHlll1w.ry of !H)>;sible pow(,,'
now <lit'oebor,,; is given. Finally, H. VCt'Y eonvenicnt sch"me is
l)I"'oposocl f<.H' design ealcl.llations,
The c\ctnil.<,d '-Lnnlysi':l oi' the dogk opicycJ:ic ge£u' tro.).n
LII this ch"pu", may 11" "kippecl I)y the reClde r' w ItO i$ int,"'l;f;teci ill tlw 'VQJ'htor' nNwo)'i, th(",,'Y' P':'()[""', which
';['·J ... l" in cl1:'pl(:I'
1,
2, l, All g LL.l .. ~\ I' v co :1 0 e j t i
"fi
D. n d t ,. a 11 ~m
j f;" i <! n )' ,.1. l I () :.<P;qlJ:l.l.iolL
!, 2,
~ holds )"01'"V("'y
seqlll"''-'o of tlw fih<Lft el1dfi A, B. C,Ther'{:I\)J"C':o the c0l11plett: (~qLli.ltion for' angltlrtl"' vt~loait:i,(~~ i/o-;
(:OIWLLAI{
Y
tor'"-t=
0 A.B.C perlIlul"blei\ pe)'lnIJI':'ll.iL)n vX f;1J fi'ixc's i.'; il "'()Ilsi.~klll tJ"Cln,~pm·;llion of them
Ull'ougll()l.Ii: the i"(H'mlibe. 1·:i".:11 sequer",,., oj" ~\lfl'i.x(!., is "lhin.,ble.
A di.fr',.I·ont seq',,",llco of shaft onc\s pt'oo.iuces ,~ diffol'ent billCL!'Y l'oJi.<.) ,
hut doos not dbtul'b til<' T'clQtion h.-:lween w,,' "'8 ami
We,
There "" .. ,~"Ix binary ",Ltios
r
A/8 ,iiJ/C ..•• ,'
;;..11 d€tenYlineci if C)IlO of tlwll' is r;-iv,m, The tir",t r·l'lat;.o)1 ht>lwoen bin .. ",y j'CltiOi;l iH obt[)inecl by p,·,,'nlLlto.tir.m ()f lotter s ill ~, j'e Rulting inTh,·' H"c;ond 1'"LLtion llcLWC'('1l bin,u'y ,",,,!:iDS !'eslllLf; fro!n a tl'ansp()sition
5
6
and aubtr~o;;tbn of 1 fj:'OID 4
Sin"e therE:' i" but one ,elation for the angular vde)dth,,>, eqmltion
2
must be valid for arbi.tr~l"y values of wB and We, HenceA.B,C pl;!1"mutable
7 Fe)]" the iUust,atlon of the interdependency of the six binHY ratios Hnd 6evet"al other relations, diagrams with f;p<'idal 6cale6 will be \.\6ed. The scales contract the range of n\lJ;nbers il"Om -00 to + 00
to a finite segnl<'int. 'fh<'i semi~recip,ocal scale is well known for the purpose and will be used in this study later on. See Fig. ~ . .!. . .!.. 8 (1"\ the theory of epicyclic gear trains thre", interval8 of equal
importance at'e to be di6Hnguished, viz £r'om -00 to 0, from 0
to +1 and from +1 to +00 • Accordingly, we introduce a scale
2.1.
adapted to this division, here ca.lled the 't 1" i p 1 e • in t e r val s cal c '_ '('he positive part of the scale is identical with that of the
semi-redproc«l, 6c«1.,., The Dl;!gative part is condensed in one interval. See Pig. ~ .. .!.' ~.
9
10
Fig. 2. L 1 Semi-reciprocal scale
dista;-c';-t;; the origin l-2-~) for inscdbed value.; .11',,",.1
distance to the origin x for inscribed v"lues -1,,",x~+1 (lif;tanCe to the origin l+2-t) for inr,;cribed values .... ;;.+1
Fig. 2.1.2 Triple-intel'val dista;-c';-
t;;
the origin 1<~
distance to the origin
x
distance to the origin (+2~~) ,,<;:,,~#
fo)" ~nscl'ibecl values ... .;; 0 for inscribed values 00;;;.11'1\0+1
for inscribed value s
>;>
+ 1 The "ix bina,y ratios are represented in Fig. ~.1:,.
~ with tdple~ interval scales,In many considerations not the angular velocities thenH,elve!), but the differences betwecn them playa part, A )"elaHon for these difference", I>; equation.!. rewritten as
A.B.C permutable Or a 8 an extended proportion
11
12
3
4
ia.:.foa. ...
f·'j.g. 2,1. 'd Int<:'Y·,icl'encl<:nlcy ,)f the bins,T'Y ["atio'>
The product of L111~CC binm'Y "atios (par'amotcl's) with !l. "yclie
surux
':>(''1ucneen1ust ll,}[ be confu ~<.,d with that of thf"~c tcrnDJ'Y ",,-I:io,;; (illsts,nt;~nf,,)ug ratio~)
;:. 2. Tor q \( ,-H , ___
1'
0 W e r!:l ~ rl d e ff i eLl! II a i e ,'3J·.:q\J(ll.iOllS 1. 5.12 amI 1. S. "5 hold for overy ".;.qucnae of th~ lihaft
'«t,10; A, B.
C ,-
Thcrcfore-;-the complete <:'qualions areA,Il,C permutable
A.B.C pc,'muto,l)I."
S~TTlilady equati'H\ .2.-~. ~ is wdtton
A,8.C permutable
-
~
/)611>.=-R
A
COl{(JI ,.I Any for we = 0 A.B,C perrnlltable
Th<:'y'e ~,.c six binm'y (,f[iciencie!:l iiSIA,ijAIC' . . . . , all determined if
one of them is given, The first T'dation betwH8n binary .,ffidcncies is obtained by interchHrlging letter!; in
i,
resulting in5 6 7 8 9 10 II - 1 7! ~-"N8
ii!lN\
A.a.C permutabl ..The second rel:l.tion between binary efficiencies. deduced from a tran':lpo,,>i.tion of letters in
.!.
by comparisrJrt of
i
wHol
and by means of ~:1:
Q.
is A,e.G permutableThe product of three binary efficiencie':l (parameters) with a "y.:;lic lOuffix ""'quen.ce
- - - - 1-1;,,18 1-iEvAI'iAIB
'lBlA'lAII:7!CIB ~
7!eJA.,
r. ;; .
1 - ~ .1- -..IB'(AIA - tWA
must not be confused wHh that of three ternary eff~dencies (instantaneous quotient':l)
The p,oduct LAlBnSIA is a constant a!;j appears from
Fin. ally,
1.,
1,
7 and 2,1. t lead via ,osF/
p + -+...1<- ,,0
A ~ IlCIA
A,8,C perml<table to
12 LEMMA for
TiBIA
~ 1:;tnd to the inverse
13 LEMMA il8/A~ 1 for
P,
~ 0Proof Elimination of
'1.
in.!..!. and1:
:~, 19 forP,
~ 0 gives14
(ii
WA-1l(!iL
+ ~-) ~ 0 fOrp,
= 0'lBIA ~
Since.!.! must be valid for arbitrary values of ~ and
Pc.
(ilBlA~1)_D forR ;
0,15 The binary parameters lAJII and
119$,
can be addEd to the symbol for art epicyclic gear train in the same way as the parameters for atre.n!;jmi.~sion in Fig. l'~.l. An adapted 'oval' betw",en the shaft end .. 2.2.
C",,(,ct'ned il"Heat,,!:, t.hc din'dion for whieh th"f;\E, paran-l<·:I:':n. aI''::' (kl'incci; ~""': Fig. 2.2.1.
Fig. 2.2.1 A.n "l,ieyc\ic gear tr'ain
;], :l. I) i ~ ~ i pat i v C P (l W 0, r, i n t ~ I' n a.l power B
TIl<) "pp"",',m(:c of a pr'odud of 8, tor'que wHh "- cliff0rem:<> of angular vl'lociti",,,, in this tor'mula moti,v;l.I:<',,, the d<:>finilion of nn in t e ) ' ) l a I
pow C'r'.
2 DEl-'lN1T(ON •. A, b.B.o·C pcrmutahlc in pMr'o;
4
5
Thel'C: 0.1''' Hix. internal power's, ;.11 determined if n hin,,,'y rati,o, a
birlil"Y oj'fjdml(:Y and ("";' or the int~t'nal pow~"'s are g',i,V('IL The
nf'>"t
,",:hltiOl) 1'" ,. inte t'nn I powcr s re ~u I.l" from a. transp(>fiition of I"e \.1:(" , Sin 2 and [,-om 2, I. ,) aml 2,2, I .
..A,~·S.,.C per-lTH,llable in pa,irs
The ,~econd relati{)l'l r('sult~ frolll a tran"po"ition ()f lcttcr,~ in ~ "lllel ['['om ~". 1 0 "nd::: _ 1, 6
'·A.~·B,'·c pennlltable in pain"
'.A,b·e.o,C permutable in p"ir's .i\noth~"" important ,.-elation fc)llllW 8 front ~ and
£.
" b.o pcrmutabl",
REMAHK In an"act\lal design, only two of the six internal, powers can be inter'prete(i as planet powers (s",e ~.
2,.
~l. the others have n() actual rnell.ning. HOW<ivcr', detail" of design do not concern the formal2,4,
2.5.
analysis, whieh leads to a simplc and rapid calculation method of the moment and power distribution of a given epicyclic gear train. See 2.~.2.4. Interrelationship" between binary efficiencies
The positive product 2,2, B and equation 2,2,5 teach that from th,:, six b~nary efficiencili!S cither two or all eGe ::,:; positive, An inver'5i,O'l
of sign of a binary efficiency occurs if one of the thr-ee torques i,; zerO, In the diagrams 2.4,1 to 2.4,6 lncluaive this re~ult", in thr.;." transition lines. These transitionlfn0s and the lines
rA/e"
D,rA/B"',
andr""s,,((1
divid';' each diagram into ninE! fields. The interrelations of thes';' Held" are indil'atE!d by n\Ht,berf;l ~n the "'~x diagram s.Fig,>. 2. 4. 1 . .. 2. 4. 6 Interrelation"hip'" betwellll binary efficiencies. In the diagram,:; tdpl.e-intel"val scale.s are used.
The fields 2, 3, 5, 6, 8, 9 constitute the (shaded) area in which only
two bil1<1TY effici<!"ci<!", al"e positive. It now depends on the values of
the angular velocities whet.her th"ee shaft powers are non-negative or at least one of thc shaft pOWer!; i,,,, negative.
An epicyclic gear train is's.:: 1 f -lo c k i rl g' if one shaft power is positive and the others are non-negative.
The fields 1, 4, and 7 constitute an iiI'ea 10l' wl1ich the six binary efficiencies are simultaneously positiVE!. Becal.lse of 2. 2. 11 at least one of the shaft powers has to be negaHve. So the epicyclic gear Uairl cannot be 'Belf-locking' in this area.
2.5. Inequalities for angular velo{!itiee and torques H
wA=We=Wc
i", excll.lded, the p,oportion 2.1.10 implies that2 :\ 1 S 6 7 2. 5 ~
dthor ws<'wc< wA or WA<WC<% for lA/s< 0
dl.hol' wc<w,,:<:; wB or Ws<w,,<We for' 0< [A/8<1
'ilth"," w,,<wB<wc Or'" wc<%< wA for
r
A/S> 1Thu[;" fOI' a given de':lj,gn It is definl,tely establish~,d which ",,,gular
v(,lndty stand,; in the middlo of an in':'quality, j·'or' darity, th" shaft
,mcl c()llcern!;,cI ,nay be l"epl'(,,sented by " double Une in the symbolic r''''H·(~.sontatj,{)n; f;ee tabl" 2.5. 1,
Fig, J, S. 1 An opjcy"li,~ gear tnJrI with its lUoment:;; of jner'tia transp(),,~d to tho r'otating "hal'\<::
'l'h,:, h.>r·'lue with tl.,., largost ab""ll.ll.e value h~f; " ~ign diff,,,'cnt from
tl1,(' ol:h(,r.'f;. Hen<."", the incq',l:\]il:ies for torqllCf; can be written
T,., .lnd
78
,.qual ::5ign,1C
1 al:'g'cst absol,\lto, valuc fOl' lAIBfis",,< 0~
"nd ~ cqua,l "ign,l
lar'g'",,,t "b:=oolute v,I1IJe furo
<. [AIB~BIA< 17(
iJ,m.\ 1,;, Gq\J:,1 Hign, ~ larg",;l absolute v,,-lue tor iAll3ilslA :> 1Tlnr' the Clll(llYHi,; of the illl.'.,,'I'elation ,yf the inequaUlies 1, ;0, 3 ,,-nd
4, 5" U H 1 ~ flO!: sllfficient to eonsict"r only the qllHntitics m-entioncd
;-l)(~c:-'rh':' power' flow in arl cpicycH", gear trai.I' L>i not "eidolfl de:ter'mineci by I:h(, momonts of inortia ioherent to the d'~>iign. Thes'-, rnoments of im, dia will b" in(!"r'porated in auxilial'Y fu""lion", Th",
<lllltilia1"y fllnctions (I)'"" u"f,d in the in':'qu.'!.litic s which d<":ter'mine th.,
direction '.>f I:he intel'nnl powCr' !low. It will "llffice her'c to consid,.,r' th~ I'rwmcnts of h,,:,f'lia
4"
.Ie.
-!:
ill the configuration of .fig, 2.5.1(tran,,-[<:'1'[,(,<1 to the sha.fh; accol'ding to a method ~n chapter B), -The torques
I:lxeded on the epicyclic gear tr<ll,Tl :;lr'f.: ~,Te.7(, the to;::-quel; in the "hafts are ~,Me,Mc' '1'he angles of rotation of the shafts are ~,¥Ie,'A:;, Assuulpthm 2:,.~
.
.!.
(VIII) yieldsA,I3,C permutable
9 10 11 12 13 Thus Ms-Ja,s ~ -M J u; = T. • - 1"IBIl8 ", ,,- A TA "
fl'om which
9i.
'a'
4
are solved in view of a later elimination . .. _{r~eiiB8\-t+([Ala-1)(r"'lBiiB8\-ll-'e}MA
+"fAlij+"~
.. (1.l"IBlJeMc¥i -
1alc
+ l.0Bilsc.-tJ", t (lAlB-1)(ij,/BiiBlA-1J.{.lsIAleile,wtMA + WAlB~1)(!A18"iBlA-1)J",tkJMB + (IAle" l)lAlsilBPJ"Mc ~. ~~~~~~~-=~=--2~~~~~~~~~
~)c +
r.iJB
i!o.;...tJ...
+ ([AlB -1)(rAle'lBiA
.1LVe
Vic~
.(rAlBilBl.Cl-l)JeMA +'[AlB(fAl8ii~~l)JAMe+{ls+
f...)siiruo,J"}Mc
.laJc
+ r,0eiill.4\·tJA + ([A/Il-1)(iAlJ'iEllA-1l-VB
Now two ilwdUilry f~ncaOr)$ UA arid 9e&.arl'l ddined,14 DEFINITION UA=
t
-live";:
+ (lAle- 11-t
A.S,C perffilltable which, because of 7 and 1. 0 can also be written as 15 U"=
~
"
i'AfB~ +
(fAlB-1)~
A,B,C permutable1 ~1
-r
+~9i eb>,'" ;0. - C ~ A,\l.C pcrmu t a bl e
16 DEFINITION
17
18
19
L",le(~+T)
There arC six auxiliary fun <,:ti On '0 g~
iTA/C •.••. ,
between which the ['elation!';A,B,C permutable _
1"~ae.
~IA· 1- 18 A,a,C perffilltable
are similar to the relations 2.2.5 and 2,2.7.
'l'he tOJ;'q~e 1A via
7..
and.!l
expressl'ld in the auxiliary functions isUA
2,5,
2.5.
:10
T«bi.<:, 2. S, 1 The ineq\laUtle8 for epicyclic g<o!".r u"ains, eleter'mining the direction of th<i' powel' now
2.5.
The diS.:iipe.tive power e:Kpressed in these functions is
20 A,B,C per'mut".hle
A8"urnptiQn
l .. .1,1
(II) requiresp.:t<
0; hence21 CONDITION '(%-Wc) UA (firu,. -1) <!O 0 A,S.C peI'mut",ble
-
(~~-
1
,l ... iB
Ja
+ ..1cHIlB/A -
~This condition and the inequalities 1 to 6 inclusive have been wod;;ed out in table ~,,2
..
2.
and presented in dia,g;':ams ~.E.,
~ to 3..~. ~ inclusive. The only impossible combinatione arE! those for which simtl!taneOuo;ly1A
!a
E..
U ... <O, <.t<O, 4;<0.
iiSJt.
B
ii~
li2!l
{AC6} {BAC) {ABC) {.OCB} {CABl {CBA)
{SCAI {CAB} {CBA} (SCA) (aAC) (ABC)
0 1 0
{seA} {CAB} (eBA) {ACBI {CAB} {CBA}
lisp'
[E!!l
iiB,j!,{13CA\ SACl {ABC} 1181
1 (CAB}
It I 1
(Ace} {SAC} C8A) I~I
{ABCI
iAle
L,.,IB{ACB}
,
I~I 0(BAC) {AsCj 11(1
{BCAl ITeI
Figs. 2. 5. 2 . .. 2. 5. 4 lneq ualitie IS for angular' velod tie s - - - whe;;"
the
sign ofa
torque is known. Theabbreviation {ABC} means either
w
... < we
<
w~Or
wc<""a<
W ...Fig. 2.5.5 The torque with the largest absolute value
2.5,
Fig, 2. 5. G 'fh .. , auxiliary function
9ao>.
and Ih" sign ofYU,.,
~2 The internal powe)'", ".n: ",era in tl-l<,: unlikely <.:"'-He8
1)BI<l." CD i. c. pe1"rrHUlcntly
7;,
= 0TiBIA"
0 1. e. p<,rmarlently TB=
0ij8~=J-i.':. pe:rmamHltly 7(=0 tl\l9
i'isffI '" L". p01'mammtly WA=~="-'c'
2~l 'rh0 inter'"'' I p,)wers
,,,,(e
zCr'O intt,.,
compatibi<:' caseiiBlA"gelflfr(.HIl which follow,; UA=
Us=
UCO'
0 and hy vit'tue (.>1".22- •..G.,~: MA=JAYiA,Mg=JB¥is' Mc=Jcif!c' i,'o. eaohklrqu(: cal.I!;"H only tll'.' ang"ClL:u' acc"l';'ration of the: shaft "TId
COTI(!C ['ned ..
1A
=
78
=
Tc
= 0 "Tho i.""'lualitie» for ;.lngulal" veloe:ities and torque5 :in table ~.
2,.
2-WCl'Q ckdu<Jed by "lirninatio" ,)f the: angui.,u· acceler'fl,tinns, th1\!;
describJng ()nly ~\ lYl()[ne:ntary o;;ituationi thf:y dete~'lTIine: the cl~"',,<!lion
, .. l' the inhn'nal power' Hnw in any ,le .. ign. T() d e:ertain (:xtent they "rc dr:c:isive af:1 to which Cll.Hf."i; a1'e imp()HHiblc, b\.lt they n<'ed rlot
ncca ssaJ"il y imply thllt H ee:t'tain suppo sit ion ~f;l realisuble. A fllrther
limitation of thc possible <;aSQ S wiH he found in the eqUilibrium conditions cit';"!LLsscd in chapte:r ~.
2.6. 2.6. Design of <I single epicyclic gear tX"ain
1 The component parts of a single epicyclic gear train are two sun gea.X"", A, B, one planet carrier C, and planet gears 8, b
Th" ~haft ends of A, 8, and C arc co-axiaL Planet gear 8 meshes sun gear' A, likewise b meshes 8. A transmission in the planet carrier connects a and b. When the planet geaX"s bave a common ",hait, oX" when they ar.:. identical, their tranBmi""ion ratio is ialb = +1. A planet gear a, "- planet gear b and tho transmissic)1\ between them form 11
pIa net g r 0 UP. There <Ire II pbnet groups, if ~ 1.
2 All that will be ",aid about epicyclic gear trains with cylindrical gears on parallel shafts c<ln e<lsily be transferred to epicyclic ge<lr tr<lins with other types of gears. Therefore, the representation in Fig. 2.6. 1 with only cyl~ndrkal gears on parallel shafts will suffice here.
C
~B
Ze=+43 za'+2S zo=+38 z~.t25z: •
-250zr. ..
+247l,.A,= -
2.%s
k (lvisa' of {4h247-lo3..P!ill} =20877=3<6959 Fig. 2.6.1 General design of 11 single epicyclic geartrain with parallel $haftl;. FoI' dadty, one of the planet gr{)up" i" omitted and the planet carrier is indicated schematically. An Ilumerical example is added
If the design of the planet groups is completely identical, then there is a condition fat' the Ilumbet' of planet grOllp« whi.ch depend>on the numbe, of teeth z ... , Zs,
z..,
zb' and the raHo i"",'2. fi.
~, CONDT'I'ION i'm' ill"ntic,\l, p1;;,rlct gl"()\IP>;, "rjuo.lly dintt'ibuted ~n the
phu)<d; ""-L"l'iex'
4
f' J
{
l'tAZ;-Zgz:1 divisibl" hy Lhe lHlD1h"" or planet g,'oLlpS
za;rt, _ 2ii1
z:=
L.I~~z:cmd z;
mutmdl.y indivisiJ)kPro!) I" PlanGt gE:!Hr ", moving to the pln,ce of a in til<; !lext planet, gr·OLlP.
,'otate~' over'
ft
pitdWn and rn,\y b.:: shifted p.. pitches, With r'es))':)'.'t [0 the planetC,(I'T'i'~l',
iI n)tlll.cs.:;over' Hn anglet.<t
+p), .end b lil(ewise ,'o,ater, ov.,,' an anglet(j+'1)'
1T.,n.:o.::1
!A
.
Iz:
z.-I k +,0)= 1'lbl~(-?+q) wilich if< "'!'Iivalent i.()
p and q a.rbitrary inte:gex' ~
p ",nc! q IH'bitr'm'y inl0gC'l'S.
z:
(lnd l~ d0l'ine d by ~ anciQ,
Th.:: ten'"'(qz: -pztl
rnay ,'.::pre sent '"'"Y integ.·", hence "ondition±.
7 Condi.tio1l ~ is llnn""c;ssary [or planet ttr'OllP!:! nO llesi,gn(:d thO,t planetgcat":; <;«rl individually b'J ~lnfi(:mbler! fOr' corTe:.::t me~hing,
~ COJldilion
1
rn"y be om),I:l.ed wile)n the equO,l ,IiHtdbuti.on o[ the pl"-netgr'oup r, in the planet ""'Tier i.H r·cplac.'Od by a di5trilnttion ~n w hic.h tlH' angIE' hdwcen Lwo pln)")',>!, g"I'OUP5 i,H " multiple of .:lllO""
I '
I
ZAlb -ze z.
J Fig",~ ..S!,.
~ to ~.Q..
i!.
inc1 u:--;ive gi,vc >l rew exalYl.ples <)f :--;inglG(~picy(:li.c gC~0.r tl"aiI1:::i. ;l4 ZA
=
+17 ze _ -'79 Z. =Zb_ +31 k div~r of Iz,,-zel= =117-(-7911=96= 2sdFig,
i ..
Q.. ~ I'pkydic. gea)"' l"<.lin with Il.. 4.
Fig. ~.
§..!
Differ'ential gearA~
Z"A=100
ZB" 9:9
Z.;Zb
IZA-zel=1, ~e k_1
Fig. 2.6.4 fi:picycUc ge~)"' tr~dn with" l"rge transmission 1'atie>
Fig. 2.6.5 Epicyclic ge~r train with a degenerated sun g~ar
.A
.fl&.
¥IoZb
arctanLMail~)
Fig. 2.6.6 Epicyclic gear train with ~
non-linear degenerated sun gear. lOA, is the angle between a vertical plane and the plane of j.
2.6 .
10
11
2.0,
B
Fig, 2, G, 'I Harmonic JL'iv,",
c
AA.
ZA--2 za·+3
h ~1
Fig, ;), Ii, 8 Wn.llkd-engine, 'Sho.ft .<lllh;' A a,; well a" B
"Y','
Hl"lio[\",l'j, A th,..p.p.-i:()()th 'planet ge:>r.t iii )Tle>;h"H with a two~tooth in[(:rrl,,1 ',i;ll.ngenr",
.,"'C
L'eforence [121· Th,., combustionp"oces8 ,,\fib,s
R
>0. ThE:' ,,;h(,.l't power ofthe output bh"ft C is
Pc..:;
o.
'ChL' '-LIlj£lllar v<.jr.>dty of <L pl"net g""" iii relative to th" pland carri.er Cis
2.7. 2,7, Power flow i,l ii "irtg1e epicyclic gear train 1 A shaft power' is positive when "ner'gy is supplied to the epicyclic gear
t!"ain "",I ncgaliv« wh"n ';',,';'I'gy i", <:H",ch"rged from the epicyclic gear
[['",i!!, See Fig.
!2.,2'.!."
'I',~ble 2.7,1 R.epresentation of the p!)wer' flow when 0<
Ti
e ... < 1 The iir"r'{)\v':; of blind powers aI'€' marked with a dot. Tr'an"po':Oiti(ln of A and B gives thc pOw0.r' flowc
.A
0< ~18<1 0< ii6lA<1.A,
1< I <..J-AlB 11811; C.A
foriiru..
>1J. 7. <{.g,
j''ig. 2. 7, JJOWC'l"H in Lt ~i.l'lgh: u[Jic.'y(~l.ic g'('.'i..\L' tl"flln. indi.c~:\.lcd by ;H'I"Ow:!i in. po~il.ivC' dil:'U(~i.ions, wlwn C i,~ pl~l]()t ,,;uTie,)"'
~ H C it; pJ.:l.lld c,\l'd(T, the inh,,'nal. P()W(~I'c; lib i.llld Pb.'u'" (:;.,.11e(l· I) I. all" l pOW"" f i . i\cC()]"ditLg to ~,2'!i the eli ~Hi[lfltiv() POWCI'
R
c~n\.)c l'(:lal.{:d tu th(L'-;C pL.ttl(;t powcr':-L TIll:: dit-;t-jpD.UV(~ powcc it~ ncg.:-.llivc.
:j It' C is pUIlW!. c'U""i<.T, th" I'r-Odllct~ We"(;; allel We ~ ai"e c,l.ll.,'" c: at' 1':i " I' P (~ W c: I' I-i.. 'I,\clt\y n I'l.l not 1"(: I :ll.~·~d t() tI1€:l' d1 H Hipllti V(.:: pow 01.' ..
Tile :.,igl' ni":I. pOWl,."'~' "'i\\lC[-:-::C'S. W~l€·;n lh0 ~ign of the o.nglJbn' vc·loctt.y ·1'CV(~t·~.;e~:. TllC :-::igrL-: DI' I:hc' angul:1!' vl~l(}citif:=::-l ca.n be (i(~::;()l"ibocl by adding \1 noug'hl l-,(~twccn il'l(:q~lnl1.l:y ~ign.~; til I,h(: ir"J.eq1..\:J li.I'y or the n.ng:L~
Lar-vclucit:i.n.i...i (ill fuul' r .... la.t:c,~:::) i)t' hy ilclding nn (.!quali"~\"j1.ioil to ;.';(~l'O {in
iI,,'c'(' 1)1"",,:-;). 1,',", ;.n.~tanc,-" Ws< We":: WA 1,,,,,,()m(3.';;
'l'lli~ rrl~lk("'-:~ j"l'o.,-:,o..:,.ilJ],(,,1 th,' dLlgr:rtDlrnal.ic.:: L"Cp1"'CS(~ilt;J,tion or the ])()W(!t'" nUW"' i.1l t "Lil,<::; :2. i. 1,
2.tL Sur"! l'i.lt:i,) ;llld .";':Ill'l c'il"icic.rlcy
The (:o~~i'1'ich:lll;1-: ill thc' i~qu~.llil)1"l:-:1 for ~U1g1.lla"1'" vclo(:i.'til..',\.i .Q.,,~ .
.2.
~ull:1Lc.)I' hH'qUl~'8 2.~
.
.2.
~\lHl ~.~. ~ ~ll"C ucte"r~nli.iled by Ofll' lJin:n'y I'atio a.1'l~1Ol1e 1,l.ll\'I'.Y cfCic'i("·IH.:Y, l'.n pl"acl,i(:~~l \'l,ppLic~lti()nS it i~ c()nv\:Ilil~nt 1;0 ,.,dl'd til" 1,,,,",1)11"\'(:1":-; ~18 altd fiB"'- ['OJ" th", .~Ul\ geL\''''~ A, B, 0",:-;(,
1):ll'ill"(l(~t~'\I':-;' ;11'"(' CIlllt:(1 ~~\11' 1":'i.Lio ~u''J.(1 Hiltl efficir~ncy.
2 l)Er'lNTnON ThE,;; un"" f f i (: i '" II C Y is ii9IA if t;..1B i5 l;lln ratio.
2.8. 2. O.
3 The I'eciprocal quantity iSlA is a sun ratio as welL likewise
ii
AI8 h a sun efficiency, However, in actual app1ications on'ly0''''
sun r'atin andOflOC ;;un efflch,lH-'Y will !-!ufhce, 'rh", suffixes of the ,>un .,fficiency ar"
the interchanged suHixes of thE:! sun r':~tb,
4 A.ccording to 2.3.3 the sun efficiency '7B9>is the negative 1:'8.tio of tho:. pl(lnet powe\'8~
P.b
~tl.d Po.' ThE:!se planE:!t powers dE:!pend on the torques and on the relative angular velocities (~-Wc) and (we-We!' The sun efficiency is equal to the efficiency of the tt'an8mif;~ion between A ancia
while the planet earder is blocked and the "arne torque" and the same relative angulnr velocities are operative, Now, something can be ,>"iel about the pl:obablE:! valuE:!~ of th", sun effici"'ncy,5 SUPPOSITION FOr' a giv,~n ck;;ign unci",r' given cir'C;Urr15tanc",s. for "'ach "ornbindl:iof1 of I:or-quo:.
r,.,
dnd ro:.l".tiv<'! ~,llg'U("-r vo.(Qd.ty (fiJI-WC). if the pland. p"W""P.b
>0. the othel" planet power ,0 ... will have ()"ly on", v«\ue (.s\lffi.xp.~ •. A.b·B p<'!Y'rnutHblo:. in p'lirs),(i TI1e planet power,> cannot be both negative. at least one of them is positive. Hence, supposition.!!. lead,:; t,J
7 COROLLA.R.V Fc>r' ~ gi.v~n d"8~gn u'1det' given circumstances the sun efficiency has a value above I 01' negative for one direction of the internal power now, and a value below 1 (possibly negative) fOL" the other
direction of the internal power now,
[) Fa!' designs Qf prachcaJ. imp<,rt"'n<;c negative sun efficiencies are irnpr'obablc. Then the sun effideney ""'':''Y h':"ve two values, one 0..: iJB.«>.<O ilnd thf! "th<~T' llBIA>1.
9 For rno"t designs in cemml)tl. 0pE3raUvE3 ec)nditio"" the v,,-lu(.'~ of the sun efficiency vary between ,·~ther' nar'r'()w limits noar' to L In the following chJ.pters the sun <:'Hid",ney if; t:;c,)f\':;ideI'ed constant, either below 1, or (for reversed power flow) ~,b,)ve 1. The v;",riations of th" Hun efficiency, dependept on momentary condiUons 01' dll" to 1h(,
infl""nce of the design, should be taken into <\ccount in actual applications.
Of an .,picydic gear tr!.J.in are Mllppo,;;ed to be Imown: - a bina',:; ratio.
- which shaft cnd belongs to th(' planet carr'ier. - the sun cffidencie s fo!' both direction" of power flow, - one torque.
- two angular' velocities,
