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. [Phd Thesis 1 (Research TU/e /Graduation TU/e), Electrical Engineering]. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR104190

DOI:10.6100/IR104190

Document status and date: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

GREVE OFFSET N.v. IOINDHOVEN

Dit pro~fschrift is goedgekeurd door de promotor

prof. ir H. BIClk,

houglcr,lar aan de iechnische Hoge~chool Delft

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 Node Ep~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 38 39

43 4.3 4.)

47 49 51 52

55 55 56 57 59 62 67

5

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 74 76 30

1:15

6 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 93

7 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 100 7.3. Power now throllgh the branches 101 7.4. 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 Ii 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' .

7

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.

9

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.

10

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 of Hs 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 ~, ~

11

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"lue f<.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

12

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. Node

13

2

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"t

1 _ 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 tdentity

11[1.R to be "Htisfi(~d (0)' "ny at'bitr'8.''Y vahLc of wA H",nce [a ~ b +~) = a £'end d = 0 .

J4

S~llce ii, band c are not simultaneously zero, one of them, say iI.

may be taken a." O. Hence

b b WA-'-a,weH-'i-l)"t= 0

Instead of (_~) we w,He i and obtain

wA - 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 get

4

~n. which the meaning of i;.,il,l beCOmes clear by blocking the shaft lind

[Ale ~ ~ fOr c.t = 0 B

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, and l<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" ;;"113

7 The gtraight bar placed overhead distinguishes the binary ratio frOID

the quantity tAlS with a broken ba, overhead, The latter is called

s

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 ~.

15

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_1

1'1A+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'iBHI the par'arrwlC[' [NB is tho bino.ry nl.l:io d<:,finocl "hove, The rn..::aning of ilBIA becomes clear by the ,!(,ductioll

n.. 1 [ ii _ WB(~ Til) ,_~ "~/A -tAtS :AlB e",- %. 1A Ft. for ~= 0

14 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" ~ I}, . . . tIt' j r f IIsrl"-p" l,n a Sltuatlon Wl 11 t H"ee r'otu J.nl~ $1,\ I:s

A

1;'('H' mor'!} d\!Ll.ils, slOe Ghapt"y, ~.

1(;

Ii: + 'il +Pc = «)A ~ + wsTs + wcJ;: ,,{wA- iA/Bii8/AWB + ([AiBiiljjll.-1)wc}J,;. 0

" (1- tiB.¥J(wA -wcl'A

1.5. 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)(ia l 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.

17

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... 'D J( +-(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 Sankey­diagl·8.!ll. or TI1<)['O sin'ply by a p"ttern of al'row~,

lK

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,"c self-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 a

19

t. 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.

20

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",.

21

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"ble

i\ 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 rA/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 in

Th,·' H"c;ond 1'"LLtion llcLWC'('1l bin,u'y ,",,,!:iDS !'eslllLf; fro!n a tl'ansp()sition

Df the 1."'.I."'·s B ,,,,d C, and r'om'Y'IlTlgcment of l.el't'r1r:; in 1

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, Hence

A.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 semi­redproc«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 a8 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 ':>(''1ucnee

n1ust 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 ,'3

J·.: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 are

A,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 in

24

5

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 wHo l and by means of ~: 1: Q. is

A,e.G permutable

The 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

,os F/ 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, ~ 0

Proof Elimination of '1. in.!..!. and 1: :~, 19 for P, ~ 0 gives

14 (iiWA-1l(!iL + ~- ) ~ 0 fOr p, = 0 'lBIA ~

Since.!.! must be valid for arbitrary values of ~ and Pc. (ilBlA~1)_D for R ; 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 a tre.n!;jmi.~sion in Fig. l'~.l. An adapted 'oval' betw",en the shaft end ..

2.2.

25

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:(" , S in 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 formal

2,4,

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"', and r""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 that

27

2

:\

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 rA/S > 1

Thu[;" 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 fur o <. [AIB~BIA< 1

7( iJ,m.\ 1,;, Gq\J:,1 Hign, ~ larg",;l absolute v,,-lue tor iAll3ilslA :> 1

Tlnr' 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) yields

A,I3,C permutable

28

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.{.ls

IAleile,wtMA + WAlB~1)(!A18"iBlA-1)J",tkJMB + (IAle" l)lAlsilBPJ"Mc ~. ~~~~~~~-=~=--2~~~~~~~~~ ~)c + r.iJB i!o.;...tJ... + ([AlB -1)( rAle'lBiA .1 LVe

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 permutable

1 ~1 -r +~ 9i ;0. C t bl eb>,'" - ~ A,\l.C pcrmu a 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 is

UA

2,5,

2[l

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) requires p.:t< 0; hence

21 CONDITION '(%-Wc) UA (firu,. -1) <!O 0 A,S.C peI'mut",ble - (~~ - 1

,l ... iB Ja + ..1c HIlB/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;ly

1A !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 of a torque is known. The

abbreviation {ABC} means either w ... < we< w~ Or wc<""a< W ...

Fig. 2.5.5 The torque with the largest absolute value

31

2.5,

Fig, 2. 5. G 'fh .. , auxiliary function 9ao>. and Ih" sign of YU,.,

~2 The internal powe)'", ".n: ",era in tl-l<,: unlikely <.:"'-He8

1)BI<l." CD i. c. pe1"rrHUlcntly 7;, = 0

TiBIA" 0 1. e. p<,rmarlently TB = 0

ij8~=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 in tt,., compatibi<:' case

iiBlA"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 ~.

32

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~.t25

z: • -250 zr. .. +247 l,.A,= -2.%s k (lvisa' of {4h247-lo3..P!ill} =20877=3<6959

Fig. 2.6.1 General design of 11 single epicyclic gear train 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"",'

33

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)k

Pro!) 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 planet C,(I'T'i'~l', iI n)tlll.cs.:;over' Hn angle t.<t +p), .end b lil(ewise ,'o,ater, ov.,,' an angle t(j+'1)' 1T.,n.:o.::

1 !A . I z: 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 ~ anci Q,

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 planet

gcat":; <;«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"-net

gr'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= 2sd

Fig, i .. Q.. ~ I'pkydic. gea)"' l"<.lin with Il.

ll(:!g'aLlvc sun "·~·ll.io

. 4. Fig. ~. §..! Differ'ential gear

A~ 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 .

35

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.n

genr", .,"'C L'eforence [121· Th,., combustion p"oces8 ,,\fib,s R >0. ThE:' ,,;h(,.l't power of the 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

:w

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< Tie ... < 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' flow

c

.A 0< ~18<1 0< ii6lA<1

.A, 1< I <..J­

AlB 11811;

C

.A

for iiru.. >1

J. 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"" fi. 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:1

Lc.)I' hH'qUl~'8 2.~ . .2. ~\lHl ~.~. ~ ~ll"C ucte"r~nli.iled by Ofll' lJin:n'y I'atio a.1'l~1

Ol1e 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.

'\II

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 iiAI8 h a sun efficiency, However, in actual app1ications on'ly 0'''' sun r'atin and OflOC ;;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 anci a 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,

39

2. 'I'll(! 1.'<.:'Y1l:Ji [llllg <lng-\ltal'" v clocity n,,,ults fn)TIl eCj l\ C\tiQ 11 ~,~.l' '1'111'

"'ll,,"L\ti<.m of.' the l'<.:'ln: .•. i.ling tOl"q\J"" and thl:' powe!.'s i~ dorlc.' in c,

silllple' t!lllk, Vi~l int''''''''.11 pow,:,r~; ~ee tD.bJe 3.. '!!'·l,

Sl.IPP()bC til':' pl.anN c((1''l'i"l' i,s A and UK' known t"f'{-1lll3 i~ 7(. Then tlw

int\'l"'iI I I)OW"," In ~o" (Wc"WAJlC

\.1' ~">O, th,." , iiCts>l ; H ~b< 0, then iicIs< 1, which decj.d<''' til':' <.'.hoice­

iiU8 = 1/i'i, ur' 'be = lis, N coxt, IG.:" - 't18P'D ,

~l J.f I.he known 1.0l'qLI~ i~ th" 1.'H'q"e on tl1(·, pLlllC't c;al'l:i«." the n,r>;l:

intf~t'n;11. pt,)W\.~I'fl Lo l)lJ dl",)t€,:r'IHir"J.C'c.1 C~\l'1not be thf: pi,,!,ll't(.:.'l PI)\:V·~:~I".'-i, hC'l1c{"I

c>th,,,' l)l.LLI'.1 ,·J'I'i."i(-ndeC) i1:JV(~ 1.0 be ,,,,.IcLllated w'i,lh 2,2, 'Y I." find tl1':1

l'el,il.tiofl lJohV("I·nn 1. h c.' int(:~r'l1;'tl puWCl"s.

4 III th •. ' ALOUL (iO 1,,·uCe(ll.ll".' llc-low "haH end~ :.\'''(-, nun'hod by f;Llfiix rluml.)("'s in,stelld l.lJ' letters, ''('ht' procedlH'e opcn,l,·,,; in p""indl'lc in th(1

"'Hlle W:ly ".~ tll(~ h:Hld-WI'itt"1l (.,,,,lc:ul<ltion :-;(:hc:me,

10

'l'r.t.blc 2. :.l, J. Th,'('(' cxampl.l'''' ()f- [J. calcllJ.atbn started h'l)[[\

hi.";"-j l'atio [AlB::: -4 , sUIl efficiency 0.960 (". V0.960, !:oI'que 'C = 1200 N. rn •

:lllgula'!' vdodH.'H wB" +3 ,'ad!." wr;" +1. )'~d/s

r-' C----- ....

1r'-~' /I. _4 B

W r p f:.li. Nom W

A +'g -200 -1500 B + 3 -11:0) .3(0) C +4 +1M +4S00

be -5 -1X() '+5OOl cb - 4 + 1200 .~atXI c'i..,---- _. -200

liDD= 0.%0

C

APl:lnotcarr .... B

A~B W T P

'«\Is N-m w A + 8 _230 .18'3 B + 3 -969.6.2909 C .4 +1200 +'800

ac + S' -130 _1152 ca + 1 +1200 +1200 V -48

1iAlc ::: 0.960

III T p rOllis N-m W

A + a -23i4 ~1S59 8 + 3 - '167.6 -2903 C +4 +1200 +4800 ac +5 .2324 -1152 "a -+ 1 +1200 +1200 V "39 ilb +4 .:232.4 -930 ba -1 -967.6 +968 V -38

~~ 1lA!a= 0.960 fiAiC" 0.968

V 9<tO

--~:~~~~

2, ~,

pr'oceduI"c epicyclic g:c,ar train wil:h "h."-ft ,:,net., (hI, h2, h3) bin raHc>:(nl, n2, ratio) <;arrier:(n:l) sun off:(eff) t'\ng v,:,l:(kl, vell) ang vel:(k2, v(12) torquc:(k3, toY'S);

value kl, k2, k3, Ill, n2, n3; megor' hl, h2, 113, kl, k2, k,~. n1, n2, n3; !:.£Q! eff, r'atio, tor'3, vell, veL2;

eornmcnt fonnal pararnNero;: hI, h2, h3 sufixes of tho shaft cnd" of "n "pkycUc geClr tr ... in nl / n2 suffixes of a binary r,,,ti<! ratio value of this binar'Y ""ti() n3 suffix IA th", pl,,-net ca)'der df vl;l.lu':! d th':' eun efficiency, depending on the eli ,.·(",tion

kl, k2 v"ll, vel2 1(3

tOI'3

of the pOwel" fh)w, ~11 the calculation either maintained 01" l"':'ph.c,:,d by lieU $uffi:l\':'" o! l;l.ngular velocities v,~lue8 of these o.nguhr vclociU"s sufrix of a torqu,:, valu8 of thi" «I'"q u,:,

The procedure determines the quantiti<;oij bdow, the identifiers of which <lr':' flOt declared in the procedure b<.>dy:

sun!, sun2 suffixes of the Surl g",,,,r'l j.n the direction of th.:" phn«t power flow fr<.>rn 'l\ml to Bun2

omega[h] angulal: velocity of sho.ft end hl, h2, h.3 torquc[h] tr>l'que of shaft end h1, h2, h3 power [h] power On 'lhait end hI, h2, h3

"["he proQedun~ assigns the boole"n", below, the identin,)r", of which [(rc not decl",red in thE! procedure body:

8,:,cond solution ~ if intern:;,l power flow is possibh, ~n both d~rections for' th", ~ame input power on sh"ft ~nd k:l = n3. The quantitiEe'" of only one solution at'e determined with suffixes either n3 ' hl, ,>unl = h2, sun2 = h3, or n3 " h2, !;'I\ml = h3, sun2 = ttl, or n3 = h3, !:Ilml = hI, sun2 = h2 f()r a !;'Iun effiCiency "nter,:,d as a value less than 1. The suffix.:",. o;unl an~1 sun2 intereharlg" for a sun effich,n!;yabove 1. Thc other' solution can be obtained by repeating the whole I"'(),-,,:,dure for a redpr-r)cal value of the sun efficiency fal ~e in all other ca se s

no solution tr:.!!£. if no internal power flow is pc)!;",ible 1'0)." the given input power on .~haft end k3 = CIS. The s.ngular velocities [(rc tak',n ,,,,ro, the torques ar-e calculated (or sun efficiency = 1 false in all other' cases;

41

2. !1.

b,,'gi,n ~ntegel' j,j, ,)1, .12, j:l; ,.,,,,,1 disp()w(,,', lIU;

r-e"ll arT,\y u[ 1-2, ]::1, 1;:\], vel. todl:3l; ~)I)()l""n ciete,,"mincd;

~,'_h£E l'!rc)<:<'!dur-e N(n): Lntegcr n: N := !! 11 -, I'll lllen ) ,-,I",,, !J: rl h~ then 2 £.~ :l; jln":edllt',, -;upl'lNinl]-,

b"gini:l:.., (i - ,jl - .12; lll,U. ,il. j~l:= llll; ul,iJ, .)1" j:lJ :=- ) - till;

-~ ul.,i,j, j;>.. j 11 := lilt:;:;; 1! 110_'; lilij, ,p, j:lJ := lIU := I! uu;

ul.ij, fl, j~J :::; uu == ) / \111; IIfjj, fl, ,il] :::; 1/ lIll £.!!!.l; ull, 1.,11 :=- ,,[1,2, 2J :::: \Ill, :1, :lJ := u[:>., 1, lJ := \Il:!., :~, 2] := -J, lll2, :l.:l[:;;:; -1; Id:= N(kl); I,~:-;;; N(k2l; 1-;:1:= N(ld);

j) := [11 :::;; N(nl): _i2 :::: n2:= N'(n2), n:j:;;;; N(r,:)); ,i.i:"" ); uu:= I'OtiO:

~"Ipplt:lion; v1,'I,lklj := vcLl; v,-d[k2] ::::; v'-'.!.2: j:l := (J - 1<1 - 1<2; v"IIj.\]:= uIJ,j:l.ld].xvoLJ '-1l[1,.i:l.,k2JXv'·I.~. ,il:= n:;. + 1;

if,i1 4 th('n ,il := 1; j2 := ,d I 2; g j2 :l I.hC:1l j2 := p - :l;

,";,ml ::;; ,il: ":I,m2 := .12; ,i.i := 2; tl)l'U{:q := I.ol-:l, secund ~:~)ll.lt:i.{)n := tl'U(~; no ~()lutior"t := ~; detp.prnillcd ,_ j',:l.'l:.-ir~;

int<""Lt.I; Wl:= "ll • .iI, .i2J X (,ft; ,~lIppl('ti")l, kl := Ie', + 1:

out: (:ncl;

42

!l k1 ,1 th()11 Id :::: 1; 1;:2 := (; - ld - 1<:\; dio-;powe)' := - Lo,-:)'>; (vdlldl- v'('1[J<2)l X (1 - lll:!, k:l, kl]! ul.l. Ie:l, kl]); !l ,kl.e,'min,-,d ~her\ b"gin ~ r.ii,:powcr- ::- 0 then ,,,,,,,,nd f'ollltiOrl :=- ~; Y,,,to Ollt ,'rld; 1.!' disl'"wo)' ,() 1\ :';l:cunel ':olillion thc',., begin _,;c:(:nnc! :""!ULLOll ;:::; ~. "rr := J / dl'; /iiuto i,n\i,,'nal (mU; g dispnw(:r- ,> 0 A 1_<N:onci ,,1olution the)l1

begin no Ho'LUti()t1 := ~, .i.i := 1; vdl := vl,,12 :;;:; vel,:, ;= 0; df,=,pOWLT := ()

<:nd; tOI'[h'Hll] := • ()'<\ X ul,i.i. k:l, SlUlIJ;

i.,,,"1 "lm2] :~ - tc)l':l X "[j_l, ld, Slln:l]; ("ri,O " _~"nl - ,~lIl1:l] :-'" - tOl"l~111111 - tOl"l,:1ll!2], j,i ttll'jhl.llllj)( (v,'I[ hun1j - vel[!d]) ---, 0 til,on E~~~ ;1ual ;= :-,:lll"l:2; ~lln2 ~= (j - n.:~ - :-:':1.1I'11 (~nd;

:,ll,)l := !J :';lml 1 ~ ilJ c_d.~" U o;lml, ~ U1£!1 h:.l <,;,b£ 1\:,; ,-:1m;;; := g',:lIl'1~ -- 1!l10l h I ~ !1'-i\Hl2 - 2 ill21' 1r2 d,,~ I,:l: t)11",g;I,[lllj :::: vdll]; olX;<"g:,[h2J ;= vel(2]; omog'",iJ-dj :::: vel[:\]; tllrqudhlj := tn!'!:l]; tC1t·Q(.l,:,[h2] := tOl'[:';]; tOl'(jlldh:l] := ttjl'l :rl. IJ()wel'lhl] __ 0n1"g'11.[h1] >< tOl'qlH,[hlj;

[lowot:[h:!] ::;;- omcg<J.[h21X tOl'qud h2 j; !)()w(:t'll1:lj := Olltcgo.[h:ll X tOl'qut'[hJ]; doten"ined ::::;~; U "(,,,ond ;(lllll_ion {\ k:) " It:l ~ bq;in df:::; 1 I c-ff; got, .. intcl"11Dl ~; il_-iC'<:(')llll ::,o!l.Il_ion 1\ Id +- n:J then 'wcond :'_:<Jllrl.ion := '~;

3.1.

CHAPHR 3

INTRODUCTORY CONSIDERATIONS ON THE: NETWORK THEORY

In thi.~ dl().pter' the 'v[l.dalol' nelwor'k tbeor'yT proper startf] with ,'Ghemati~ing' th<:, ",i.errl<:,n\" menti(lYl",d in chapte)' ~, These element .. O.1"'e con,.,idered C\" int"'r('(lnn"ct\on~ of eHher tl1ree shaft ends or two ~haft enels. Tlleil' properties are governed by equation~ between Oong\tlal.' velocities an.d eqllations between torques, These equations a"e the basis of the variator network theory,

3,1, Components of a variat,,!" netW[)l"k

Special attention woo.~ given in chapter 1 to elements that interC(lfln(!ct three shaft ends, generally called t h r-e e - p () I e H (1. 8. 2), and

divided into nodes U::'!!'·~) and epicyclic g'",ar trOoins G. .~-i), In conu",:;t, element" inter'connecting' two shaft end5 were the rotOoting "hafts (1. 8.1), U",l1smi,;,;;ions (1. 8. 5), and vadato,--" (1,8, (J), A generalte-;-'l~ for' dements inte;:-,,(;u-;-ecLlng two Hhaft ",;;'d;- ~8 'b ran c h ' ,

DEFINITION A b l' a n chis an intcrconnel:l:i'.ln of two ~h"ft ,:,nd" of diffc,'ent thr-ee-pC)l"~'

A branch may be realised by a rotating shaft, a tram;mio:<>;ion, "­variator, or by a .~equence of these elemente; connected in ,;er-ios. The angular velocitie . ., of the two shaft ends have a cCI'tain ratio, and likewis", the lor''Iu",,, .

2 When three-polos arc interconnccted by br1l.rJ<.,h"", th"y constitute a 'network', Of th'" nUmer()UH conftgurl;ltions only thD"" will be

c;ondder<:d that act a8 a variator and, therefore:, will b~ call,~J 'va r' i a I: <) r' ne tw 0 r k '. See \<'ig, 3, J, 1, To achieve the limitatiDn, two r<:'~tdctil)ll,;;, already recorded in -1,-8,7 (II) and (III), are incol'porated in the ciefinition. - - -

:3 I)EFINITION A va ria to l' net w () r' Ie is a COh",""nt o;y"t",111 of tht'ee­poles and bl'anches, in which (Ill angul1u" v':'locitieo are determined by stating one angulal' velocity, and in which all tOl'que" are determined by stating one torque-,

The acting as "- vaI'ia.t()r', det",'min';ld by the two re~trictions in tho deflnition, yields two important thoorcms, ·0"" l'E;!gare1ing the input and output shafts, the oth';'l' tho. n\lmber of node,;.

4. '3

(11) A 0l)tP\~t <;;.hatt

lh ... :o ,r-E ..... polt:'

lhrce­pole

Vig, :L J. 1 An cx:arI~plj',~ or:t variatot' n(~tw~)I·k. in aC\(l,\i rk,;lgn ('l) ;1.nd schenlatiC:(llly (b).

See nho Pig". 2. . .2. .. 1., .!.J..~. 1. L :'.

8

4 TlrE()l~E:IVl !\ v:ll·ia\ol· ll':twork h"" twe) ~J1nn ,,"d~ nut JnC(.lJ·pol'aled in :1 bl-;ltlch.

'1\~H~:-iV twu :-.:h\1.rt Clldf-i j,d'('" known :lS ill put :'-: h a r L and (11.1 t put :::; haft. In OUI' t!l('<.)l'Y :1 <Ii ~l.irwti<")Il I.wtw'·"" input and '''' I. put ,~l1[ltt b lillt rOlev[lnt.

5 TJ:II·:()H.I':IVI in ,\ \1,\1"'1,,1.01' rllCtwor'k 1'.1,,; tlluniJer uf "picydic g',,;1" 1:I,,,,i11,.;

t~ (·~qU;ll. to til!..:' n~.lJl"lh(:l· of nodi~~,

.!,~.,I_~,~~:.~I.~)}!",(,~r'j.nL.-; "1"1]1(' 1) ltt"T"II)P"r" of urll(.nOwT"~ rrIU:-il. be! equD.l (.C) I.ril:

lHunb,:'!' "I' "quat-i.lm,.;. 'I'lle ,1l"11HIH,ration in the t"hl" be'low yi."lcl~ the two """(.Iition.·.: i.1I I.hc: lOW()"IIl"l.~1: level.. Ii. 1'(,sult8i,n

44

p:q i. (,. tlw tlWell.,,'!" nf epicyclic gear' tl'~.\ill:-; pi:; cqllccl to the I\llmll~'" 01' l"I()(Ic~ q.

fI;;; 2 i. (,. in 8 V:\r'lil.I.Ol' nctwo)"'k I.he,·o are two "ha.It ench rH,!.

.Ill'(,d]] (:olln"crccl to "thol' :,;h~.n (~ll(.l~, ill "the·,r· word". nolo il'H:or·pOj~ill.cd In, ~L bra,nch.

[(umber of terminology

shaft end"

Jp p cpicydk gear trains 3q q nodes 2r r transmi.$f;i.ons 2v v variators

(3p+3q+2r+2v) "total number of shaft ends, of which n shllft ends are not connected to other shaft ends, and (3p-l-Jq+2r+2v_n)

munber of equations ang. velocities torques

p 2p 2q q r r v v

3.1. 3.2.

connected in pairs ~JP+3q+2r+2v.n) t(3p+3q+2r+2v_n)

specified because of conditions 1. 8,2 (II). mr , ,

(Jp+3q+2r-f-2v) " 1+(tp +!Q+2r+2v-tn)

(Jp+ 3q+ 2r + 2v) = ",+(tP+fQ+2f+2v-tn)

3.2. Junction and addition of transmissions

In the classification and analysis of variator networks it is often practical to reduce the number of tr-an':lrrlissions, without disturbing the relations between angUlar 'l'elodUe'5 and between torquel; of the essential parts of th€ variator network. The most elementary ca>.;'" is the junction of two or more tran$ruissions in series.

ASSi'irtTION Two transrui"''lions in series connection e.re equivalent to a transmission with a transrui.!5!5ion ratio equal to th.:. prOduct of the original transmission ratios, and with an efficiency eq~al to the product of the odginal efficiencies. See Figs. 3.2. 1 and 3. 2.2 .

Fig, 3,2.1 Transmissions in series connection

F~g. 3.2.2 EquivalC'nt transmis .. ion

Froof The angular velocities and the torques Mtisfy

We ;:; iCt~ ivo %

To = -le/t;: LE!CIIJelCllM:l'/C

45

3.2.

If one of tho. tr',,-rlstnissions Dr' both al'e v8.ri",t<.)"s, then the eq\livalerlt tr-ansrrd.s~ion b:; aLso v~n:'i~t()r'_

2 COnOLLJ\RY Eoch hr'illlch tnO,y be ch",ra<otel'ised by only nrlO tro.ns:rni~sior~ or vLlriatoT'.

T·'o,· das siti """tiOt1 purpo",,,," tho branche f; are now cllstinguis h,:,c\ in 'h",..,! br'ancheoo' "wi '~oft b1""'rlchos'.: se", Fig. 3,2.3,

"'ig, ~,2. 3 '1'11':' varl;,_tor netwClr'k of Fig. 3.1.1 Sch,uIl,,-tical1y. Th., two tran8';'i;"i'ons onn tlw v;-H-iatol' arE:! joint to one variato1', for'ming " ",oft branch. The other brand, is a hard In'a.llch. '.l'b.e thr'.:oc-poles ar(. divided tntc) ."-ll epicyclic g,:,,,r' train (drc1,,) and a node (dot)

3 DE:FTNITION A h",. d b l' an" h is a br",'ld\ without a variato",

1 I.H: 1,'lNITION A,~ (J f\ bra n co h is a bl'anch with a v",riator.

The other' method of r"ducing the nutnber of tr'arlcmission,:\, or of ,;implUying the varint<:>,· network ~n another re!:lped, is the aclcling, in !O",de,; with an ,.,,,b,ting tranfimissiol1, of such Do traTI!Hnission th"t the br'(,rlch conc<!rrlcd is siJnpllfied to Do dir'oct connection. To maintaIn the r'('liition,~ f,>r' angular v",loeities D.nd torques it if; rlccessD.X'Y 1:0 add i,rlrCltical tr"n~mil-;8iOn'l in othor plD.c"'l, naroely ,-'ound a 'subnNw,,:tl-:';

~'''' Fig. ~.~'.i-

Fig. :L 2.1 AdditioIl of tr"nHrniH:,;ion~ to " Hubnetw(>J'k cr

" Dl:;YfNl'T'TON A" u b not W 0 r' k is a coher'"nl: part of H. var'btor netwo)·I, with at lon':lt olle three-p,.>l,,_

() i\SSEHTION A val'iDotOl' nHworl( maintains its pr·..,p"dies if ,:,quCl.l t""lT''lmlssions (1"" added t,) aU branohe" which connect: an D.rbitr'ar-y 'lubnctwork wi:th tho remaining part of the variator networ[<.

PI'nof J\11 rel~ti()ns betw<,en angular velocities or- between t,:n'qLlcs

46

3.2.

3.3.

in the subnetwork are homogeneou!; Unear equations. These equations remain valid if all angular velocities in the subnetworl, are multiplied by the same value i and all torque" are divide by the same value ilJ .

Th':! assertions! and 6 induce

7 'X'Hli;OREM F~ach variator network can be deo;(,db':!d as a networl, in which any three-pole is interconnected by any other three-pole along hard branches.

Proof S\1ppose there are two three-poles in a variator network not satisfying 2. Then, a subnetwork exists containing one of the throe­poles mentioned, and connected to the remaining part of the vari.atol" network by soft branches exclusively. To all these soft branches =d possibly to the input shaft or output shaft, equal variators may be atltled in such a way that one of the soft branches becomes a hard branch. This process may be continued until all three-poles are interconncctea ahlng hal"'d branches.

Theol"'em 2. is applied to simplify the t~' .. atment of variator networks by

8 CONDITION A variatol"' network will be described in .,uoh a way that any three-pole is interconnected to any other three-pole along hard branches.

3.3. Rapid determination of torque ratio., and efficiencie.a

In 1. 5.12 the torque ratio was written a~ a product of two parameter", ,~ar;h ;ith two suffixes. 1AIB~B"" . The laboul' of this complicated writing is rcwaraed by the theorem below expressing the close relationship bctW(!G:" t(Jrql.le ratios and transmission ratio ....

A" an i.nh·oductory l"'",mark it is mentioned here that a torque ratio ill equalS the corre.;p()ndi.ng transmiss'ion ratio i if th" <lfficiency fJ" 1. Now. assumption 1. 8. 9 (IV)' to the effect that a dissipative power does not influence-a;y -;:'elation between anguLar veloc;:i.ties, involve8 that transmission "aU,) .. i may be found if torque ratios and effidencies are given. Conversely, one can surmise that tOl'que ratios may be found by adding factors Il to the corresponding transmission ratios i, not only for a single unit, but a1,,0 for combinations of unH", "nd especially for the complete variatoI' networlL Illdeed, formulae of torque ratios may be identical, term by term, with the corresponding formulae of transmiS!ilion ratios.

The equations between the angular velocities of a three-pole in a variator network are of the types 1.4.2 01' 2. 1. 1. Those for the torques ate of the types .!..!. ~ or ~,~,.!.~ These a-;'e-all homogeneous linear equations.

47

2

3

4

5

(;

7

3

3.3.

The c<.,.,fficients a 1'(, either +1 or -1, ()r' depend <.>11 the panl,TIleter,~ iAJB a)ld ~. Such pail'.s of p(H'amoter.~ tA/9 and ilBBI. ~.r" eontri\)ut"cl to the rCH'Inulae by ~aeh epicyclie gear tr'ain, tnm"mi"sion 1\ncl vat'iato)", The po.t'amet"", I'fBBl.alwliYs appea,,, in the pro<iuct rA/B!'i~. III cOl1sequ<'mce ot the homogeneo\ls lineal'Hy, .,!..~. 2. (II) :;.nd (III) now read in fOl'nlulae

in whJch K, G, ",. sto.nd for a,'bitra):y ,:;haft end" in the v~r-iator netwoI'k. Tl' the nohlti(Jll of .!: 1· 2. and 1, :), 15 is ext\!nclcd to th\! "e arbitnu:')' shaft ('lIds, th",,,

THEOREM 'I'he function" J4:/G anci $G/K an: icientica1 if th\! h'l'nD.ry ,~ffidency flGlK b made ~G/K~ 1 by th<' a",,;umption that th", d;'",ipativ<:' powers O.l'(~ :,.';Cro in thl:::~ pat-t 01" th~~ vi:.l.L"io.to,~ nl,.~lwot·l( ~pccificd in th~ f,,['nllllae,

Proof Because (,f ~M<-,,1 :me.! \)ecmls~' c)f lemma.:'::~, ~ both functions have th" same VrI.I.1H" for the ~~.rnc indep",ndent pm:arnctcrs LAlB , (HIll,'"

lIenee th"y mll,~t tw idcnticnl,

The most imp()r'tant appli.<:i-ilion of this theo!:"ern concer'"" the efhci.:ncy "r (1 val'i,.cl,,,· n0twor-iL Let A be the input shc\ft, Z the OllLput sho.ft, <l.lld ~Z::!WA/WZ the: cr·nnr-:.rnission r'~lli0 of th(~ V 1:\r lo intOl" network ~~Xp(c~~p.d in Ute bin:ll',Y l'atio'>

" nA,/Z(ilvBilst .. , .. • iiwiiG'H,···l IlZlA= DA/Z(lA,/S"'" iHm .... J

This rn"allS thllt fcn' the c,a1clllo.ti.Qn of the effidency it suffice!:> 10 h[tve the clisp()sal of th", ,,()n.iug~V,d offici~"tdc$ and the equation of tfl(: t,'an srni,s r;ion 1',:,,liOh. It i ':\ rIOt nec!>.""ary to dc:duce B"'l'aI'nte ly a forrrnlia f'l" cffici<:"lldes.

fl Tho cODjllg<l.tion 0\ H-'1 dficien"y to the ,""'responding tnIl1Hl"lission ""tio is cXP'-" "kcd by th", ,., f i' i C i en" y tun co 1. ion

48

IO

11

If the transmissiun ratio ~s g~ven by a function like £. the efficiency f\lnction is defined by

3,3,

3,4,

12 Exampl!! Sllppose the transmission ratio>, of a variator network as a whole dcp(~nds on the transmission t'atio x ()f th" v"datol" lInd on the bina,"y ,'atio!; L, anel /2

(i,-1)1C' '" (i,-1)[1 y=-'----"------..:...

(r,_1)i?'l' + (1,-1)

The:' efHdency of the val"iator' i" '1., til", binary efficiency ii, is conjugated to r,. likcwlse fh to [2' Th", assumption that all di,,»ipative powers are zero makes the efficiency of the variator network lIy" 1. hence theorem 1 holds and the e:'ff:i.ciency lIy can directly b" written

{( r,iil-1)1C' 71. + (lib. _1)[, ii,}{(i, -1)12'" + ([2-1)} 1/ = eff(y)' .

y {([,-1)1C' + ((1-W,}{(i'IIl,-1)i;';hlC'1/.+(iii:l-1l}

13 EXAMPLE Eqllil,ti,on .~,~, 2. for the bina.ry efficiency of an epicyclic gear train

_ _ 1_ £AfBiiB/1I '1C/A - 1-i'i>.f8

"esults from oquation 3,-1 . .2. iAfC = 1 - [ .. .IE

by "ppUcation of theorem 5 .

0.4, Conversion

'C(lnv<:'r,>ion' is a mathematic1I.l artifice in the analy"i» and c;h""ifi<.'ation of variator nHwork", 1t is based lip"') th" di~tinct identity of t.he equations for allguLu' velodUe'l 8.nd torques, An lnt<:,,'(;h8.nge between ILl ,,-nd T produces a system with properties do",:,!.y .related to the origin8.~ one, Such all int<~"change will be called 'c.'(>Dv':'r8ion "

DEFINITION C <) tl v Eo " !, i 0 11 i8 the replacement of a variator network by 8. variator netwurk (.,r ~i.D1ilaI' structui'e {C)!' which in any part betw,:,en the angul".r velocities the ~amc': cqw,tion8 are operative Ol.ij odg~nally

between th<:' torques; and likf>w~o;<:l between the br'quei:l the 8 arnE' equations ~l""e operative ao; originally between the angula1" velocities,

49

'rhis definition dil·cc.tJ.y 1','1"1,, to

2 I'HOPER'J'Y 'i'h" c<)nver~i"'l of [t corwcded v(IT"L,[ol' nE'tw''''k yield~ th(~ or"iginnl v:u"iator \1~~tworlc

W" ,.;hall inv""l:ig:lte wi·,lli. ,",unver'>l(Hl mean,,; to .... ach ot th", ,,1onHmt.-J or il v;.).!'"i(,t'b)1" n(~lworJc

:) ASSI,:llTION A Ir·aIWDJ~~"j.ou conv',""!:" into D. tr'''nfimi".~·ion with D.

new tr·".nsmU";iml r·"tin and y,",t, Jcn' lh" ol'igin",1. ollr'ection, with t/1<.>

or'if,(in(\1 "rfidel1cy. Sc'" f·epre.,,,,n!atioD in diagram 2·~ . .!..

We = ;CtD%

TD = -iClD IlIJ1C 1C f. ~ -(-Ic/DI7D/C)'1c"Jo "'o=("'iC;;DIl~C

nEfVI.AIH': 'rhe: dir-"ction for whi(:h the pHr·"weters ar'e defj",,,, ("c'lllenC" of ~llfj\)(",,) is r<:'v("T"sc,d tor the, ~,1l{C of (1 ,.;impJ.o)'

'·"'I,,'e,;;el1tll!.ioll. Tlw [>""iliv<> di.r'cdion of ""[,,tiOI1 i.s llllch£\l1ged.

4 1\SSEf{'I'ION A i)I';lllCh WitllOlJt ,'1. tr[tl1:;miL~~iOl\ (,Ht'Cd conrwcliun)

collWcy'r:'J i.rl!() a ]):\l"'l.icular lr'l"\Tlfimis:siorl, i. c, orre which ~hllply inv~'r·l.~ lhe di,·"di()11 of l'ol.:r.ti()n, S,·", I"cpr"'~<:'I1I.,,tiOll in di:i.g'·ilrn :!.'.i .. ~.

;; J\::;::;I':WI'!ON A 110(ilo conv"l""ls into '·l. fiditio\1f; clement (:«lled "() II V ,. ,. ""' n (, cl", that il:1 (l hypothetic».l cpicyclh, geal' tl"'l1jl1 with llnlt'):\ iiy cl\lwl tl)l·'l'.l(";. ::it,,, r·"p,·C!s<3nt(l.tioll in cliagr'"m ~. i.]. .

~-.. _-_.

J c c

AA 1 ,-J._J--" A B

wA;WB=Wc 71.. To; 1(; .J L 1A+TB+7(;=O w",+wB+Wr.;=O

G ASSER'l'ION An epkyclic gear train C!onv~rt$ Lnto a node with two u''''",missions. See r'epresentation in diagram 1·.±, i,

WA -iAJsWe +(~-1)Wc=O

rB"-~~7,; 71; = (~IBiis;A-l)JA

~ -LNsTs+(r...-s-1l'c; ,,0

%= -iA;8~f.I}" We .. (ii.tBii8#\r 1)Uf,.

Fig, 3,4,1 Conversion of an epicycUc gear train, The symbol fOl' an epicyclic gear train is explaincd in ~. ~. 15 .

3.4. 3.5.

The relationship LNw'~"'n th", converse node and ep~cycUc gear train~ ik :wpUed in

7 ASSERTION An epicyclic gcar t",,-in C,,",11 be l'epl'eseIlteJ by "­conv"rs., node with two tr'''-n,;m~''Hi<:>ns, See representation in }'~g, 3.4. S.

Fig. ~.:!..~ All ep~cyclic geal' train r<'!pr'esented by a eonv.,r se node with two tI'>.n",rniss~ons

Proof The epicyclic gear h'ain is eonveX'ted; then, the node, thE:' two transmission", and the third shaft (branch without tr'an"mk!,sion) a,'" converted successively, after which to each shaft end a transmission with i, _1 and 1J~+1 is added.

3.5. lIICel"change of ad,jaccnt node!:'

The cl,:'\!i"ifiCCltion of variator networks is con",iderab~y 5implificd by th" d~v~ce 01" interchange, bcc<>u>;€: sf!v"rD.l configurations will p!'ove 1:0 be equivalent to each (>the",.

P£FINITION Ad j H." {, n t n () des are nodes NlIlIle(;ted by a hard branch, thuH f(H'rrdng a subnetwork.

51

3.5.

:~. H.

fly lllC"ans 01.' '\fl~"'·I.i{)n :J, 'l, r. lh" :;;ubn€"two,.'k of the ncijll,-,,,nt: lIodes ,'l'l'u::,entecl i r; Fig. 0. ~ 1 (;HI1 be replac,:,d by thnt of Flg". 3. S. 2 , Obvil..llJc,ly tlw Hulme,tw,,-;:'I:-oi' Fig. :~. ;i.:l {" l~qlliv'11':"11. 1.0 that 01."-Fig'"'·2·2·~alld :"~'~.'

Q~S Q-<::[' J L II) u

Figs. 1.:2,.2. : •. lId ,i.~. ~ !l.d.iLlC('llt 11<')(\(''''

Q~S ul ~i

Fig. 3, OJ.:~ Int",·"hanged 1<cljncent node,,,

2 !l.SSli;I~T()N /\. sul.11l'.,twol'k consi:'3Hng of two ncl,inceni: IWc\C:;; CQn I'll<

"('placed by a.11 cljllivnlent (.>nt:, ill which tWI) '"h",ft cnc«3 ol"iginallj COllll"ct.:'d i.() I.h(, S:cllH' nod" an: now G<.,nm,L'tcd to clHfc,'clIl nocic". 'ThL; .1'()I)I.(\(.",,"(,ntj.Kc'l.1.1('d iJlterchang"') of (\Cljil.e.,,.,nl. nude".

:1 H nod",; ~ll"f-' in\<"",.,onncc.teQ by " soft br'c,nch, intex'ch(\IlI;c will CQ,';;" .c:tn 1nc:r·(:i-I.~C of th(~ nurnbcr of var"ir.!,tors, ~n\~h)g()us to tJH~ inCl'cas\-=: of' t!';m':l)"nL;~i()n~ in I·'i.g'. ;l. 5. :;, 'l'h'~I"'CrOl'C. "m,h sll intel'chC1I"lg'C.' i.s not ;"U()w(.d.

:~, ti, 11'1 Le' [' c h :=1,ll gc {) f Z"I Ii j ace Il t e p j,(~ y eli c g\;.~ \1, I' ·(~ •. l- ai n. $

DI·:I;'TNTTIUN Adjacont "picyclic geat· tl'(ljnH <U"C' ()picy<.:Li<:

gC:lr tr';lirLo c())1n~'ci:<'d by a hGI"'d hr',"\ch, tlw;; fonning ,\ ,,1.1\mc:lwol'lc

2 Aool<:It'I'ION A ':llli>nc:lwoI'k cc.>m;i,;Lin;j ,)l" two iHlj;:u:;cnt crpi('ydiu gC'[(I' ITain>; can b,,' r'''plaoed by an ('ljlliv[(lent Oil'.', in which two shaft (,rlels Ol"'igill:llly cOlllvc·ct.·,d lu the' ""m" epicyclic g'",,-!. train Gl',; now connected l,j (1it'fet"(~nl ones. Thi~ t''-'":plD.cC'l-['"L(:~I'1'. i:; ci111ecl 1,nl,t":I:'ohr1.l!g'H oi'

Zt d j n c (~ 1'1 l C IJ icy (~ lie g (! ;:l. l' I: I:' a i II S. •

~ ,[,lv .• or'(.m i. J.. ~ C[(11 1 .. ,., ilPI)lied to the (binlF'Y) cfficiem:i"s, h"(,"LI~(' two of i:f", (Olll' ,;\][(fl: "nd,,; may he ~lIpp<.l5ed I.u be bloc.i<e.!d. 0U, ti,,,,",, is 110 <.>bj,·diun to >;I:n.l'ting the pr'()()!' to.ldng' all (bin[(l'Y) ,-,I'rl(,krlcie~ eq"(ll i.o unity, 'I'll.:: L'qufltiol1>l for tOl"q\H!H of th0 ~\lh-

4

networks in Figs. 3.6.1 and 3.6.2 are identical to those {o),"' angular v"lodUe!3 of the co-;:'v-;,ted ;;;ubn~t:;orks in Figs. 2·.2,·2 and 1.§'.~.

~ Q + 6 s

Fig. 3,6,1 Adjacent Zplcycllc g,:,ar h-a~n!3

- - iU'1l a I--..----<H S

[~1) 9rT4J~1) T U

Fig. 3.6.3 Converted adjacent epicyclic gcar trains with

binary "fficiencies put equal to unity

Fig. ~ . .§.. ~ Xnt",rch(mgec1 adjacent "picycHc ge-ar' tI'ain~

Fig. 3,6,4 Conver-ted int<!,ch.anged adjacent ",p~cycli.c gea, trains with bin8,r-y efhc~encics put equal to unity

The original subnetWQ)-k and the one with interch<>.ngec1 epicyclic gear train:; have to be equivalent. This leads viCl

to

{

l;L~i~ = 'r'u'R

L~iR(r;-1)" ((r-1)

(,~ -1) = £T ("Pu _1)

j 1+ (r (UiR - £T 1", 'r'viR - £T '''lr'uiR - £rLfl

Mel by vh-tue of theorem i..l- ~ to

_* (1 + LrTirrJiJRIlR -Lr il-r iRI'i~) 1lu = (1 + lr lUiR - lr 41) TiT''' TirTiuI'iR(l + £r LyiR - tr)

(1+ 'rilrll1!tfR'lR-'T'h)

7)~ = (1+ 'riir(JjuiR'lR-"i"ril-r)(l+ Ir(u(ll- 'riR)

[1 ... r,lhI~-rriir4tIlR)(1+lT[uiR- ir)

3.6,

53

3.6.

The Sllbnetwm'k of interch8.ngcd epicyclic g«al' t.l'ains is equivalent to the ol'igin"l one if' 4 nnd 5 aI'''' satisfied. See Fi.g. ~. 6.:; .

54

s s

~ Q Y R Y S

,. L.i

~ Q ~~ ~ S

U T

Iiig'. 3. fj. 5 I,,,,,.rnplc vf an inte1'"hange of ndja""nt '~l'""yclic ge[\l' l,'aiel" in act\lal ck,.;ign (a, b), anel ,;cbernt1li"ally (c, d)

4.1.

CHAPTER 4

ANALYSIS OF SiAUCTURES OF VARIATOR NETWORKS

"POJ:' th", af1aly:;;h, of a. vaI'iatoL' network we have the disposal of the concepts in tor a han g e of epicyclic g",ar traim; (3. e.) or nod",:;; (3. 5.), and con v e r s ion (3.4.). A few 'more concepts will be added, vh. closed network, mesh. prir'nitive subnetwork. Especially by the "tudy of meshes and subnetwork" ceL'tain ldnds of netw(lrks will prOve to be reducible to simpler' (IneS or will prove to have undesired properti.e". Thi) OInaly!;lif; c()TIceL'ns structures generally with more elements than would be used in practice. Therefore, examples of act\1a~ design" will scarcely be given.

4.1. Closed network

As appears from definition .!:.!!. . .I. (also ~ . .!..~). a variator network as a whole is equivalent to a variator. The intr'0ductinn (jf that equivalf!nt variator, callod 'reticulator', presents a convenient mathematical adifice to analyse seemingly diff"ren·t ~trudure" "t one go.

DEFINITION The ret i c ul ... tor' of ... v"ri"tor network L; a fic;titiou!'l val'iator for which the relations between the angular' velocitie!l a;rld

bctweerl the tOI'qllCB are identical with th""e "f thE'> variato,. nE'>t",ork "'!'l a whoI".

Apad fl'om being a substitute for the vaI'htor' rlE:twork, a I"eticulatClr may be introduced as a supplement to the variator network; sce Fig. 4. 1. 1 .

Fig. 4. 1. 1 A closed networl{ is a V:l.riator netw(lrk supplemented by a reticulatoI'

2 DEFINITION A c los e d net w 0 r k is a va.hator nctw,-,rk fictiti()usly supplemented with the l'etieulator.

If the reticulator is cOllsiciered a substitute for the variator networl{, its dissipative power is equal to the dissipative power of the variator network. If the reticulator is considered a supplement, the p0wer'

55

4. L '1. 2.

difl\:,,-cnce in th, .. r,:,ticlll"l:or' Inay be seen Q~ n ~upplcmentary p/)w'~r' wi.th Il. vil.lll(' OI.Jpo"it':' tc.) I:he, ,li,,_~ipiltive pow,:,r' ,,r the variator n':'twor"l" A r,(:,ttcllllll,or- (:,:m only in, thh p,u'ti,,"lal' re8p,:,d Ill< ,li:-;tinguishc'd h'ol'n

a va:ri..'J.t-or' 1.11 Ul(~ val-iat():r network,

:, C()I{ULI,AHY Tht· ,'om,o\f~tl of :1"Y v",r'iatol' of Q clo~"d nNwOl'k leaves a V,H'; ,:\1:"" IIctwc',,'k or whi ch th" 1"0tiC'Lllator to .i.cknti"Hl to the reDl()V('d V;-jrirli.ol'.

'1,:;, IVlc"hcH

DEI"lNrTJ:ON A me 0: h L; il d,,:;ec\ figur,:, 1"11""",,1 Ly :,l coll<:Kti .. >n or ~. numb,:,!" ''>1' lH':-l.rlchc:-; interc',lrJIl",d(,(] by the sam" [lumbcr' oi" thre;>­pole,,_

:; ,[)IIFTNITION A II c,,' d me s h is Q ptC!;)) including [10 vCtr'i(lY)L', but

p(l':\~ihly the ,'cticulatol",

:l DEVIN1'I'.\ON 1\ ~ 0 It Tn II H hi,. '" me:;;il jn,:Luciirlg a VQl'iatol' which is not the 1"(etic:ulH.t()[",

IVlc~hcs in a v~ri.:::tt.{}r' ru::twul'l( co.n hl.~ COLLI'J.tcd in 8eV(~f'(11 way::i, To .::\veJirl.

" chao!3 ~r) which the concept () f 'me:;h' lose~ ':lignific"-ilCQ, we (l.1.!:\o define th" nLl[nbe,' of 11),:,,,1,,,:;, Sec Fig, 4.~,,~,

I"ig_ 4_ 2_

B

A fiv"-m<,,sh variator' rwi:wock, with uight thl""'-polcs ( p" 4 ), tW() variat!))"h ( v,,2 ). five moshe,:, ( m ~5 ), of which lwo "-I"" hal'd Wf:l>;I-),e!; ( /)=2). j\II,,!;hn; number- 2

,'til (I 3 QI"" hard mo,;n",':L I'r A "nel B :\"c,HC:H.te th" input O:llld O\)tPl.1t o;harts, tIl'.' I.owermost 'v'll'i"l",-' is the l":,ticuliitm'

'l fH:PTNITION 'l'h" Cl u In bel.' '-' f Trl '" s he s in a var'i~tor network, closed by it,; 1'0ticulQto)", ifi eClual to th" ht'gcst nmn!,;" t' of meshe~ le-s-s lmity, by which aU I)1"IlIIC:hc:;, including the reticulo.VH', ,,"'0 ci<::;cl'ibl3c1 tWl(":',

;, TIIEOREIVI p",,, it val'iatol' ndw()rk with p "plcyclic- g<:'QI" h-'ains the nUI11bc:'T' of rn,,):;b,,~ IT]:(~ IT] = P + 1,

!)(;

4.2. 4.3.

PI'oof Ill'" do'Sed netwol'k the number of three.poles is .2p arld the number of branches including the reticulat!).· J.s 3p. Now the theorem of Euler' relating to the llLLmbel's of vertice", (here threc-pc)les). meshes and branches, reads 2p +m" 3p +1 (Eulers topol()gic theorem is tre,,-tcd in [39] and [38]. )

6 DEFINlTION The n uw bel' 0 f h "', d. me she r; ie the number' c)f

ITle.~hes in a clo,;;ed network H all vadators, excepting the reticulator, ar'" J'ernovecL

7 THEOREM For a variatOl' n<:>twork satisfying ::\, 3.8, with P <'picydic gear trcdns and v variator"s, the number' of hard meshe$ 1" h =p-v+l.

Prouf L<:>t the removal of v variaton; be coupled with the removal of 2v shaft erldij. TbG removal of each shaft end cau'Ses the removal of a three-pol., n.nd the junction of two branch"", into one branch. So (2p~2v) thrICe-poles and (3p-3vl bl'anc;he'> l'emain. If the variatoI' network satisfies 3.2. a, in each hard mesh at least one three-pole does not vanish. Thus,-the number of hard meshes rema~ns the same and the theo,ern of Euler rea.ds (2p-2vl+h= (Jp_ 3v)+'.

REMARK If the vRl'iator network, contrary to 3.2.8, corltains a hard mesh connected to the remaining pa.rt of the 11etwork by soft bI'anches exclusively, then this hard mesh may vanish in the proc~du,e mentioned ab<.>ve. In that case h ~ p-v+'.

4.3. R,dterative rletworlu;

The concept of 'reiterative networl{s' h",!:, hnpol'tancc to the classifkation of var-iatol' netwol'ks, and is especiaUy significant for Ihe analysi$ ",rlQ later on the !olynthesis of me"hes in variatoI' network.,.

DEF1NITION A variator netwOr]{ is a reiterative network if a subnetwork can be indicated that is a variator network in itself. An interchange of adjacent eptcyclic gea.r trains or a.n interchange of adjacent nodee may precede to the indica.tion of such a ~ubnetwork.

2 Such a subnetwork, considered aij a variator n<itwork, can be replaced by it" l'eticulator. SQ, the variator network is reduced to one with fewer' three-poles. See "xClmples in Figs. 4.3.1 ",nd 4. 3. 2. The reduction to a. variator network with few«r·th-;:ee-pole":;; justifies the

3 RESTRICTION B-dterative networi{s will be left out of consideratic>rl.

The e:>l:clueion of l'eiterative networb, has consequences for the COrl$totutiOl1 of me$hee in variator networks with two or mo!'e

epiCYCliC gear t,ains, with respect to the number of three-poles.

57

8(~ .. _--------- .. _--_ ....

Fig. ±-.:l . .!:. A reH"r·"tive network, in actual d~8ign (a), schematically (bl. "ud simplified (c)

Fig_ 1.3.2 A reite,..".tillc netwc)I"'k with (nt .. ,"change of Hc!'jacent ep,i(:yclic ge",,· train,~

4 ASSERTION III i,1 var"i,al.()r networ'k with two or YrlC"'C cpicyclic gea)" train>; the aTrlallo>:!t pO:-i"ible rn«"h, with respect to its n\Hnuer of llu-cc"p"]"", is (\ ;-",ft me,;h, Thi:. o;rr\alle5t 80ft trl<"!r;h or' il~ CO~lv~r:-;C (;O'(L!-7=iHl~ oJ t)n~~ L'picycli.c gear trairl; two nodes. and a.

Vo.r'ial:o.[" betW'I,"l thco nocho,,_ On" of the ,"erilaining brf-!.nche~ ,-n"y hay," a vo.T'LltO!~ :[1.::1 well.

P1"()"f '('h., vario\1~ cnnfigul"allvns Gonecivo.b]" or the !-;In,,lle,,t Hoft me"h HTe eQ>;i1j det<:>,'mined QTl<l vel'ifi,ed_ See \<'i1:>;- 4, :~,:l and 4, ~_ 4,

Tho exclIllplo in TCiA _ 1.:1. 5 shows" me>:!h which leJ.d; to ;- r',-,itcr'ati ;-, network, lind thu:--; do,l.,:\ -;,,;t SJ.ti~ fy i,

:ifl

Fig~. 4_ 3. :1 and 4. 3_ 4 S.mallest nl~shoe) wilh ,one v8.dator·

5 ASSERTION The 6mallest hard me'3h consists of two epicyclic gear tra.i.ns and two nodes in alternating sequence. See Fig. ~:~. ~.

~ The pos",ible confj,guration~ of the !i'mallest hard mco;;h al"'e eMily fOW1d out, and th1l6 the proof can rea.dUy be supplied.

F~g. 4.3. (; Smallest hard mesh

6 ASSERTION In every hard mcsh two epicyclic ge",r tr~ns and two nodes can be traced which constitute a o;;;J;;nallest hard me"h H all other epicyclic gear tra~ns and nodes are removed .. See Fig. 4.3.7.

Q-------{ }-------~

:Fig . .:!.~. 2. Hard mesh. The dotted linel> may contain an arbitrary number of epicyclic gear trains and node IS in any sequence

Proof .Il.djacent nodes in the ha1"d mesh can, according to a!5sertion 3.5.2, be replaced by one node ~n'3ide the mesh and the other outside; ;c;;- Fig, 4. :3. $, Adjacent epicyclic gear trains carl be tran'3po6ed similarly-:- Th;:;:", the remaining epicyclic gear trains alt€rnate with nodes, and constitute a me'3h with at lCast two epicyclic gear tra~n6 and two nodes satisfying a8serUon E..

Fig. ,i.,l . .§. Transposition Df adjacent nodes

7 ASSBB.TION A soft mesh with one variator has the (;onfiguration of a hard mesh in whiCh either one epicyclic gear train Or one node is replaced by il. vadator.

Proof The smallest soft metlh may be rc.lated to the sma,lle$t hard mesh in this way. Fut'ther' see the proof of ""l:l~ertion .§..

4.4. Ir\cQn~j,stent netw<;lrks

The two restrictions 1. 8. 7 (II) and (III) in the definition of a variator network, namely all angui;,r velocities to be determined by stating one

59

4.4.

Qng'ul'lI' velocity ,-,,"16 [,II torquo,; to h" dekI"mi,n<:,n by "tatil1g one t()l'qw;;, In,,,m a ~tr1.ng"nt c.ondition ['U!' the dCgl'OC:> of I\·.,,,dum i.n the con"titution

of" V<lriat,lk ""twOl'k, 'I'h(: i'ljst (O",,,:lu8i011,, w",'O theor"~1I1 3,1. 4 ('ollcc['nillg the input (\11\1 output ~h:J.f\"., ann tlteOl'ern .0., . .!.,.:!. eonc"l"'Tling

I.h(, rI"mb~'r' Dr epicyclic gear tr(\i"" and th\, numbol' of 110tlc':-;, Wo ,s;h<lU

"OW inve,~t;.g,,\l.e the (legNo"" u1" fl',:,,:"!,,!n of >;i\lbllctwork",

If a ~\Jh"Nwnrl{ """"Lits of "I.ldl unite; thCll it" set of (!quuti.ol1>;l fo!'

ang\dal' vclocihe!; <HId [UJ:' tor'quc" b; ill~'(Jn .. ;istent, [hen the v'-H'i"t()l'

nc:twoY'k will bo c"lkd <inc()n~islent<, ,for'in:;tanc<:', the only ~olLlti()n fo!" Lt r1IJulbcc' of nngulaI' v(:~lo(.:it.i<..'~-: IDDy h(~ w::: 0 ~ and the (OT'r'(!~pundj.ng tOl"'q\H:>; cfI<ty hav(,: a"..-l.litl'ary value", or in ",,,oth,:","' ">lo() til(e 1:0,''1"'''» nl"ly

h., ",or',-) aDcl the :'lngula)"' v", I.odtit:_',~ ",..hlt,.",ry,

DEFINITI()N A var'i"tm- l\C'lwod, i.'l arL inc 011 '" 1 f; ton t n <:' I; w" .: Ie if the a.,/£"lCll' v"l()dli(',~ (H' th" tOl'que5 ()f :iny subn"twol'k do ll()t have one

"rid only 01)<:, v,.lul! [0),' "",,;h p,'eSlIW('ri :ulgular' veJocil.y 0,' P"/=,,,,m.(,d tOl"'<'l'.L(' in the v"l'iato)' n(~ lw()['k.

2 In w,n",'"l, the inC(H'f;iHtUtlcy h not a p,'op<:,r-I.y of the _oubnctworl{ il.Helf, lmt ~ ,-'"lalioIl betWH,,, the; 8\I,l)n(~l\v()I'k (lnd I.h(: l'ClnD.j.ni,ng' pat't of th", vill'l:\tclf' network.

8 Tll a ~ubnl:1twol"k ,..'u'e l"otll'1d

P..,bopicy<.:!ll! g(:~'.t' tr':1il\H,

'T""b n(l(1/.'H,

b br',~""h('>;_

" ('onr:cding 'l111\ 1'1. (:nels to th(-, 1-('nEtiD~ng fldrt lOf th", VCl,-i(,tOl' n\·.~twol"-k_

5 The l'ic'.tn"lning P,11't 011:1\(' vaJ"l«I.()!" Ilotwo('k detcl'm~lIe" the (\ng'ula,' v(,lnoitieCl 01' '''' COIlIl"(:ling shaft ,,:mj,:; and th(" t.ol'que'l of CT c01w",,,l.ing

"I,an 0n("l.."

7 {

g '!'Ill£UllF,iV1 '1')1(' "ng-IIL'H' vclodti<)fi <end ti}f, to"que)s ,,[ il ~l,lJn"'tw()d,

\0

hn VIe one and (!Ylly ono V "l\H:' f".l' oach pr'" ':;umcci anguLLL' vc I ()(:i I.y U1"

PY',,' ~lI!l)od tOl"'ql.l(, in the vfll'i,,[or netw(lI'ic, if th.' "LLI,JIlctwo,'k fiati~Jie"

til" ,,()nditit)n,~

{ flO

C", " tk + P.Ob-q"'bl .. 1

c, "' -tk -Aut,+q"'bl ~,

6

')

II

12

which are equivalent to the condition!;

c",~1

4.4.

Proof 'rhe total number of shaft ",nd", in the subnetwork i!;(3,a.1II>+3q...b+2b), The (3Aub+3q..,b+2b)angular velocities of the shaft «rId", are determined by (,oSUb+ 2q,ub+3b) equation~; in the subnetwOf'k and by c., angular vdodties of the COrlrlectionfl. The numbet' of equation~ must be equal to 1:h.:, numb .. r of \1nknowns, so

Fr'Dm 4 <>.nd 13 we derive 9 and,hy c<)nversion,10. The $IJITI and the differ",-;;'ce of Q ancl .!2,lead tu the "q\J~valent conditiorls .!l and E.

11 'J.'HEOREM A v8.dator networl< is arl inconsistent networ~k if any subrwtw<)rk does not satisfy the C(mdiUons Q and .!.9..

Th(~ '5\lbnetworl{;3 in Figs. !.~: 1. and i. i.!:: do not satisfy!!. and 10.

~a T ~2J:L~." t<fi'l* 1

~(C'-Psut,+~l= 3

Fig. !.! . .!. A Hu'tmetw<)rk by which the shaft ends are blocked

Fig. 2:::'. ~ A subnetwork by which the shaft en<:18 are blocl,ed

The undesired prop"rtie'1 of an inCOllsistcnt Iletwork justify the

15 RESTRlCTION Inconsistent network$ wH!. be l«rt out of ""nddcration.

61

4.5.

DEFINn.'lON A ",,(\\,e.,il>lc network isav[lr1atorn<:>tw()l'kin which CI ';l.Il"lc:two)'l~ ,'.[1,[1 be I'cplaced by H. ~LLbnetwor'i< with 1cwer tin'~,,:,-p()lc:~ 0,' by a hl"R.n"'l, withnllt. ,li~tlll'bing th .. I'dations ror anf(Ul'H' y"locitiC:ll .\lld fell"' ["'-'lll"" in ti-,., l'e,-nMning pay't of the v~,r'i"lol' netwOl"k_

11' it val'i"tr)r rH:iwol'j( i~ \1i.,,-,-r-cducible. '"or"" of it~ ~ubrwl.w(H'k~ wUI by <kfinition b\, (!Cjllivalont to om, with few,:,y' l:h,-co-poJ.e~, One, can

~lJrrni"c tlwt ther(" acC <1 lin,iwo nUJube!' of ~1J{.:h subnetwol'ks ,,-nd it Illay bt' i.nlc:I-coting to c:xi"lline th(-d,' c:o!l1'iguraHorl;;_ They will be {!H.llecl TPl"j,THiLivc :;:UblH.:l.wor~k . .:; I.

2 DEFIN1'1'.ION A ,~lll)l1Hwor-i< withollt yadat()l'~ and without the rOticlll:lVH' i,; c:allC'd CI p r' i rn i t i v e ~ l.IL not W (> l' I, if tho eql1(1tiofLs of the' l"lngulat" vcluciti.e.i aud the torq\J~H of it,~ (;onl1{·~c:liI'Lg ~:i.hO:.ft (~rl(l.s

cannot lo,-, r'c,,"li~od by" SUL!lctW01'k wilh rOW\,r th""'L'-pu10'>,

i\ pdmitiv,:, "uhrLdwol'l~ C:lrlrI()t pO:.;sibly cmltain .~oft h'-,lllche"" "-i,,,,(:

'-""_'" vadato)' 'wtwOr-i( miglJt thr'rI be cOll'lid,','cd l'€,cjlldblc" vi,;, f'"du<!iI.Jlc to H" r'(:ticubt()l',

:1 A ";t'·p-by-:;top i.nYl',,,ti:;;,tiol1 of l'r'imitiv<:' ';;Ilbnotwori<s, indicateci ~n

[ilblu» 'J:.:), 1 lo '1. 3, :.l \rldLl"i"0, h[\!; beon Inn.de by means or [1-,," " . .,diic:icllt;'IJI.,l(!0,,--;Yf ti,,, coqLlatioll" for- angul:lr' vdocitiec), ro,,,- th"t

P''''I>O>-;<' , th,:, f,)llowing two th(:()t."oms W"r'''' prov,:,,-',

<1 'l'lll,:nnEM Two C!ubnotworkH wi.th identic:,l c()8ilio1,."t mat)'icE:!_o ['or :lnguL-u- v'elod.tlcH will Illno h"v(: identi<:"l ,-,oo:fj\ci,,,,l ",,-,t)'i,-'." ror­t'H-qlK't.;. rol.lowillg "d",pl-:~I:i()!l of the (binary) 8rfi.('.icnde~ in one or the Hllbnotw<ll' kH_

Pt-.~-'(!.£ Hy vir-tllo of :',,!-;1:r-ictioll ::._ ~_ is ;), _~\lhYlctwOl'k ,,,'l:i:-;fic.~ tlw <:ondltioll'> 4_ 4_ 9 "ltd 4:,4- 10 _ The HuhJlotW())"k detC:l'lJ'ilH:''' ,. !lund),-,,' of c -cw ;;; '1 -11Zi:o-;g(~nc"Z";Ll:: litv~~'" (tqua.tion:-) for' the anglllH.r" vt:locith':~ or the CO,,"(,ctillg ~h:lft end:; C).)1ri :l rlllmber' of '-', "So, efluation" foT' tho torqll\">; _ 'Iheo '~ql) <\ Lion" fot' the ~c "ng-"I "I' vC'loci_ti<' M 01.1'(: ):"p"""'(: ntocJ by :'1. (:()ortlch",1: lnatrix wi.th C Colllrnnf; dIld CT I'()W~, tl106'-' Eo!' th'~

torCjll<"s hy a ,-,,,,,fficient ITl(lll'ix with C (,olumn" [\nd c.., rows_

L'-'t the: >; llbn",twor' k:-; 0 :111<1 o~ h;n'8 ioe" I:i " ,,-1 coe)f",icnt n' D.tr'l'-:Q ,; fOl'

angular- vclociti<.'s, then both h,,-v'c thi? ,,~mo numher- ,)f collHnn~ " til" ":lInC' numh(·,,- oj" l'OW~ cT , and til", <;~"ne <iiff",y'cnee ,- 'T '" Cw of til(,,,,,, tW() llumlwy',,_ A '''It.lI'ldworl, (!, with (Gw -2) node" (l,ncl H. shaft ,-,nd A 1» connect".! to c.., ,:,h~-"ft onds of the 8ulm(->twor-l< 0, A >;LLbnetw()y'" ~ with (c, _2) "l'icycU" gear- tr:linH a.nd ., shaft: end B i,g <:onJlC'cted to

CT sh.,n ':'11('1" of the "IubnQtwori1: 0- Sot: Pig'_ i,E.,.!.- BocCI"H" of 4,1,12

f.2

G

7

10

11

. fig. !.~.! Compad£;on of tWe) subnetwor\{s

In thi", w~y. D. complete variator network is gerj(<r·~tf1d with D.n input "h", ft A ~nd all output shaft a.

4.5 .

Let the £;<im", oubnetworks a: and ~ now be connected to cor""csponding shaft end£; of the $ubnetwo1"'k G~. Let both variator netw,wb, h",ve th" ,.=e wA ,,-nd th" "arne 78. Because of theorem .:! . .:!. E.

The identical matrices of G and 0" and the equally cumplemented ~ubnetwork5 a: and i3 cause we = wg; whence

The d.daptation of the (binary) efficiencies i'i""". in the subnetwork a" if; c:l"Hn"c:l by

through which

The combined suhnetw,>r'k" a: ~Dc:l ~ must now ha>!", the same ';<.lEdficiellt matt'ix fOr' tor'que,. as the combi.ned subnetworks a: and a·. The suunctwuI'ks a and 1;1* IIlU»t then h~ve identioal coefficient md.tr~c",s for torques.

12 THEOREM Two 6ubnetworl{s have identical ooefficient mati'ice>; £0\' angular velocities it the ooefficient matrice'5 for torques can b", made identical by adaptation of th", (bin~ry) efficiencies in both subnetworks.

Proof Similal' to that of theOrem ± with [tn adaptation defined by ii " , andij"=1,

REMARK An eC(H\ornical adaptation of paramet«!r"6 tn only one >!ariat",· network wOlild surn""" viz. ~n ad~ptation of f* and If' \llldel' the condition

4. 5~

thClt r"ij" l'CroU)"'; tUlclwng'<d. Howt'v,·',-, the o.ciaptHtioll Ii =1, ij·o1 lS the rno;!-il_ (:()nVelli(~nl. one.

1:\ lI. :.;t\'p-hy->;tC)P nnd (up tl) fisc "()nn~,cti.n!~ ~haft 'mdH) CXhCl.llfilivc

invcdi.gati()ll or pr-iHlitiITC' ~llbr\Nw()I'I<8 I."." l.""'~l.lll."d in t",,,I('H i.~-..!. to !. Q.. ~ Lfu:lll~~;ivH, EXi.ll1"'1 p l,(:~:-; of r'C(1l1Ciblo ~u bn(-!~w o!-ks (~r(.~ given "in

Fig,~_ if, ri. ~ t\') -'I, fi, ~1 i!lc.lu;o;j,v(,~.

l.i4

'r"bk "_ :i.1 Co)"nplctc: .~lll'v"y of pr-i.mitiv,:, ~I.lbllctwor-k.~ with

three cOrHlc:ding ~h"n end.';

[)~' -

[-_. ~\j)[~

c;.,,2 cr ; 1

11[~1.o AyB [11 . ---- - '---".

-rabl." 4. S. 2 C:ompl"tc fiur·Vc.'y ()r pdmHivc: 8ubn(,tworkc; with four- i • .'O"),lH~C~LiI"Lg ~:=:hn ft_ end;,:,:

(1 -1 0 'lr~l'O It)O _11 I.~

A~..p+r) e Oi-()~l)

(+CI.+P-l)

C

.P'ig. 4.5.2 A r'cd.lLciul~: subnctwor'k iHld it,~ t:~qlJivalC't1t

pT'iIl.,iti. V(! >lUb"c:twork

Table 4_ 5_ 3 Complete ~urvey of primitive subnetwork,:; with five connGcting "haft ends

G...= 4'"-' --,,-- Cw=' --

t:'r =1

r'''m ti~Hi'l (1 1 , 1 ,)~" [ 1 I I ~ o 1 -1 00 -0 ABC D E A iii ~ D 001.10 -

000 1.1 c;,.3 "-'= 2 Cr- 2

f'" "I[~ rtT'?1 [i: \ 111~" [fill 010-10.0 D ACE o 0 1 0.1

~-----,.-""-"'"-""--~----

,,-,=3 "-'= 2 ~1

F' "I~ [<~{:n [" -" 'I ~-, l1""Tn A I l D E

00 1 .1 0 =0 00Ol·'wD

- E " C e A o 0 0 1-1

1;:l [~ ,~ <;,,=2

'"¢-- 111 0 0 ~ F' "I~ --- 1--'·"11 [0' i -1 .1] ~ =0 41' C E o i 1 1 0 =0

D E 0010-1 ~ D

,t c:;".3 il

2;:1 ~ 8~y-~ F"'I~" ,-A .. [! 1 ml~.' A~'y--T1 .. I OILllldp 0001.1% Y C D D E

E C

B c:;".3 D c;.,=2

,~, ~·;"Ol[~.,--<}$-' jrlmJl' """ O.1Pr 1 Wp , __ '2\ ~t3001

~~-'-j"-\ TO*", .~ •• -,~ p. H e C D _ _ co-; 1');D}-11: . • 12 ~"1 p. oJ, /'-'2 - ! ~,i2 /3,1 ~ ~ 2

~ ~= 1 /3=!.-d roil ~ o:~'i-1 fJ=-1

r:t~ \~--.~( J' - ,.~ A - '- ...... 8 B - C

L,,,

Fig. !.~.!

"

,0/ .. I

A reducible subnetwork and its equivalent prilniti ve subnetwork

4.5.

65

Fig, 4, ~.4 A, ,·"ducible ~lJbIl",tW01'k and its "quivalent primitive subn(,twork

"):;1;( A B

Fig. 1.5.5 A r<,ducibl" fiubnetwo,-k and its equivalent prirnilivc :;:ubn~!twork

Th .. , "voidance of lllHleCeSsnY'ily "CH1lpli.C~t",J varj.3.tor' network:!'! jUHtifies tl1",

14 l'LES'I.'l'tlCTION H.,;"lucibl'" n"twor'((s will h" left out of ,!<msid<:'rati.ofl,

A SUI,lrHHwOl'[{ with two conn"cUng shaft p.rld~ is ':'quiv'llent to " branch, in oth,~r' worc\s

lei COROL,LARY fh'iler",tive n!:'twor'](S (V'" r'cducibl ...

Restdction <1. :3. 3 ifi inchl<.l<;HI in restd,c[inn 14, Finally, (In important limit"t)ou ofthi::' -;."l\.Ilnber of ,:,picjelic gear train'> will be dil:l,;u88ed.

J f.i THEOREM V«r"iator netwol'l-;s in which th':' number. p, of epicyclic g<:'ar' t,'ains in mo,'e than twice th,:, number, v, of vQ)'i,ato"", are ,'(.dl,ldble, Proof Consid"r' the inpul shetH, the outPllt shaft, ~n(1 ehe 2v fihaft emln oj" tl1<:' I' vari(\t<.>r''; in "n a,'bitretry val'iatol"' netwo,k with p epicyclic g"'d" tl'''ill~, The equalions 1)E:![wccn the angular vdociUE:!s of the"" (2v+2)

~hatt "fHj:; con8i:;t of lin""" cqU<ltions which can be irnitated by those ,-,f o n"w variettor nctwork, At fir~t, four sh • ..rt c:nd8 rr\ay be cClllneet,>(l to " ,smallost h,,,'<1 me~h, Then, ~,ach paiy' of rem~.ini!lg shaft ends may b,,· connected with U", aid of two additional eph,ycIic ge"'r' tt'ain!:' and two etddHional nod,,:,;, H",l1{,''', p=2+(2v-2h2v "Ufl'iCE:!H [0 eon'3titute" vari"t()l' "dwoy'l< with II vadotol"'s, SeE:! Fig. 4.5.6.

Fig, 4.5. fi S'~qllenc", of hard nHH;hes (18 H.rl clCampl" of ~ v,l.I'iatoI' network for' which p = 2v

17 J.,JIVIfTATION p": 2v

Gfi

4.6.

4.(;. Parallel branches

Tho;, motive to define parallel branches is the presupposition that they may be characteristic of a variato!' network, e»pedaHy for the distribution of the power flow, The di!:ltdbution of ang\llar v«lociti(o", tOJ:'qu<:>s Or powar., Qver t p".r~.llel br"n"ha~ i,; specified by (f- 1) mutual !"'"tiD!:!. If it is d.;o';;il'ed to give each of these ratios its own value, (t_1)

indep.;ondent quantities have to be chosen, which implies that at least (1-1) v",-,'iator,;; are required.

This consid<:>ration is disputable, for it is not "ure that 1m inter­independency of the parallel b,an<;ha", is of tmportance. C<)rI':'''quently, the parallcl branches are mentioned here for the <;;al(e of completeness emly. They are in:Jignificant within the scope of tho present study.

2 DE}'XNl'1'10N Complementary ,;;ubnetw()rl<", are two ,;;ub­networks of a variator network, one containing th" inp\lt ",h::lJt, the other the output shaft, together induding aU thl'ae-pole!:l (,nIy ()nN,.

The brarlches b«tw«en the tw() complementary subnetwork,;; are called p "I' all" 1 b I' an c he" and are diatingLtish«d in har'd ami ,;;oft parallol brarlchc>;, morc or le,;;s similar to the pl'evious concept,;; of hard and soft branches. Sec Fig. 4.6.1.

Fig. 4.6.1 Complementar'Y $ubn",tw(."ki, and pa!'all~~ bran<;h""~

3 OE.l:<'tNn'ION Bard parJ.llel branches are the branchcs inter'­connecting two complementJ.ry 5ubnetworlH; for which, via each of the>;c branches, the connections between input shaft and output "haft at'C maintained if all variators are removed.

4 J)EFINITION Soft parallel branches are the branchcs inter'­connecting two complementary subnetworks for which, via each of tho"B branches, th<:> connections b<:>tween input shaft anel output shaft are interr'upted if (me e.lr more varhtor8 IU'" ,emoved,

5 THEOREM If pi.;; the number of ~picycl.~c gea, tr'ain8, v the number of val'iatol',;;, and t the llllmber' of hard p,~r'allel br'anch",~, then t 'Ep-v+l

P"oot' Let one of the complementar'Y subnetworks have /fA three-poles and Ii variatol'8, the other kethree-poleB and vavariators. There are I\, !-;oft parallel branche" with a variator in each of the branches

therrl;;«lves, and $ remaining '>oft parallel bnl.nches.

67

7

4. Ci.

If in the coroplBflle"t,,-,·y r;ubmltworl, A all V<lrl«l.o["S, a,'e removBd an.d all ])"1'"-11,,,1 br·,,,,,,hco~ arC disconn.ectBd, then thcre are (1+ 2VA +15 + I +s) unconnected Hl,aft cnds, including th", input ~haft. In the '5imp.l.e'5t case, the cornpl",,","la,'y s\.lbnetworl{ con"'nl:,; of Olle thl'ee-poll:' with thl'ee '511"ft ,,"dn, nO that

If kA increases to (11'"",1) the number of unconnecl.ed shaft end" Bither d,,,,,,e,,J'ics by 1, or inCJ:'eases by I, or increases by 3. '{'he latter CCltW C"ll always be ""vd,h,,d by applying theorem 2"~::i. Bence

F,'om G, 7, 11, 9 follows t"'5 .:;;p~v+T, and, after omission d s, thc theorem.

The number' p_v+1 is the numbe,' of hard Ul",,,,hes, h, accr.wding thl~o,..::rn '!. ~. 2.. The number of banl pa1'a11el br"nche:;, t , clepencln on the pl1\<;l~ of the di,.",ecU()n; ,;cc Fig. 4.6.2. If, instenn of thc probl",.",,-I:i\: tlulnbel' t, tho< lIlaxilnum -n':;-rnber <>f harcl par'a11el bl'anch",;, 1m.>. is c<'H1 ni,lc,"ocl, theoreIIl 2 may be wdl:lCn trna,::;h .

10 ASSErtTION A network for whkh

11

cont'lill~ "t least (HH:' hal'd mesh withollt the )'\1ticul ... tor. alJCl "t h:d.st thl'<:~e var"iators.

(-'I""oof The in",quality tm •• ~ 0 a.!ld SllpposHi,,,, Qleacl to h ~ 1. This DH:':lll>; that at JeaMt one bard me~h oxist8. Hthi~ hard IJl",,,,h (:ontain$ the l'c-ti.clIbt,,,,, a har'rl parallel bl"""rl"h cxists, tma.:a; 1, and by .!J,. h~ 2. FitHlHy, "- h:u·J mesh without the n~ti(:ul8.tor will be found. B",twl':c:n the hard m(:!;h with tlw !'oticulator' and the b,,)'d IIlcsh witho\lt it, thore if; one) ilo.r·d lJr·dllch to ~,~thfy c.onclHio" 2.~.!!., anc! IIO mOl'e than this

bil

4.6.

one llnrd branch to satisfy .!2:.. rn addition to the hard branch, at least two soft branches must interconnect the two hard me,>hes to avoid a reiterative networi{. Then, one interoonneotion must be added to avoid an inoonsistent network. Henoe, in addition to the hard branch, three soft branohes interconnect the two hard mesh",s. An example is given. in Fi.g. 4. e. 3.

Fig. 4.6.3 VariatQI' nHWQl,k with t,,;:p-II+1. The dissection presents t .... o soft parallel branches and one hard branch

69

4

S,1.

CHAPiER 5

CLASSIFICATION AND SYNTHESI£ OF STRUCTURES OF VARIATOR NETWORKS

5. l. I{ ~, 'l P () II '" i v i t Y

The 'r'esponfiivity' i:'! [( )n('(~~\.Ire rOt' the iDDI.I,-onee of <:t slight vidual v,lr'ation of the, I:r'ansmission ,-".tio .. of an )nclividual variat<.H' on the

ll"'1I1~missioD "I,tiD y of the vaT'iil.l.m' networJ<. It rnu::;t have a multiplicativo ch(lf'adCt' b~caus!:' of the multipliclltivc chD.l'[(ct!:'I' of thc t)'(1no;cnission ratio~, Thc deriv(\Hv,~ ~ would be in,,-ucquat<: becau",,~ of its I:\d,lil.ivc chal'(\ctE:'T', Thc: conc!:'pt of 'l·csponsivity' i~ idcntic(\l to "[h.!

concq)i: "f s c n sit ~ v i I. Y in oldol' cOntr"l system th!:'cwlef;_ Another' n(\rrlE;' iH intI'oducecl IH,,,,", 1.0 avoicl conlll!;ioll with nlodE:'T'1l interpretationH of i3el1,:dtLvity, and to ,"",-lphH8iso the signifloanco of thH CO!lCl3pt in th", PI'0'O<'rll. variator metwor-lc theol'Y, <'sp,,'cL<lly for thE:' distribution of power',

I)J:·:I·'J.NTTTON In ,'L clO~,td nCOlwor'j( the "hnfl: end" A and B Cln' i"lc:r­CI))1t)<'ch·"l hy S0111C v ... r.\[(t(.H' or' by tho l'eti.clllator'_ The Sil.l"l1e b I:r'ue or tllo .~IHlll","d" C ,-,-lid p, ij."., I.'ig_ ~_~,.!:., Wh.'n thc other- vari(\tOl'H lh".t (\"':' p(\.,~ibly lJllilt in m'.., ~llpp(Jf;Cd to hQve f(>,,,,(] values of tllej,r tr'~l.n:-::rllil-;~lor".l. .I"~l.tiO:-::r ~). l"c:-:p(:;.n:-:=-ivity it:

Vig-, ;j,1. 1 A clo!:'"d nClwork, Tlw r-c:ticuliltol' betwecn input ~hl1 rt A ,md outpul ",haft 8 h llidieatecl l)y it~ t1:·~,ns((liH:;ion l~nl:io y, ancl a V~'l r'iatu:r between .~h,:lfl. '-'tlds C Qnrl 0 h indic[(t"d by )(

A,B pel"TrlLllable ~,D pennL.ll.,-,blc

1\-8,C-D p,-,r-rr"lutable ~t1 palr-s

'['he g perl"1"llli.al:lon" L>[" A, 8, C, D produ(,e eight r'el:lpnnsivitie'O c.lf '" vaL-iuto)' 11,,"t.wo['k with .)11" vadator. (r (mc l'csponl:liv-ity is given, lho nl'hpJ·.'; or\'[' cieLen"ined by Lt." notations

'10

ACIA = -.i,_ A.",,,,;

AA/D"-il.,o,iC

A,B,r;,O permLltabl~

A,8,C,D p<:>r'rIlutable

5

6

It ~s cOllveuhmt to 1.1",e aooreviadon,,,, 101:" the transmission ratios

We x=wo (variator)

Y=~ (reHculat<)r) a

and to confine ourselves to one of the r'espon"iviti .. "" ",ay 2r

0.1.

~! ~:

'/ AA-t" _Y_ "K.~ dK Y dK x

!:l The number of mutually independent responsivities is equal to thE'! number of variators, For inBtanc':', the 24 r'espon-~ivities of a variator network with two vaI'iator$ hi:"ve mlltual ,elations like 3 and i,

IO

2

3

which reclu~,eQ the ,espons:tvitie.~ to be considered to

Tile product of these responsivitics 1$

and therMo"1i! only two al"e mutually independent.

5.2. Di..,tdbution of power

F'0l' p"l',chcl!.l. applications it is important to know the pOW (H' tran<lrnHted by an individual variator compared with the power :;uppHed to) the input shaft. ln a variator network with shaft ends denoted acc()l'ding to 5.1. 1 the distribution of power is represented by the ternary efficiency - - -

A.B permutable c,o permutable

'I·S,C·D permutable in PI:I';," s

"['here <Ire eight ternary efficiencies to repro sent the di",tdbution of power' hetw,:,,,,n a vadator and the reticulator. Their mutual relations are

fll\iC=J-'(CIA

A_B.C, D permutable

~DIA = FleIA 7lDIC A,8,C,D permutable

it ~s convenient to use abbreviations fOr the efficiencies

7l

4

5

5.2.

0.''','1 t<> conlin': our'selves to on., t)f the above t .. r'nar'y efficienci.e'3, s"'y 11[111.' Ii' ~Cli\' '/, ",I"1e1 I)y al'O uiven, an i:erIlar'Y efficj,<;md"'H arc dete1'J1li11 ed,

Th" di.st,'lbution of POW(-'" han" vel'y importa"t relation to th« r''''p()IlHivil.y, c'xpres5ed i,n the following thl',"" I:hc{)l"em-~,

(i TI-IEOI,n:IVI If the di'3!:!~F_"'llve poweJ:'5 i.n a variato)" n(,two:d( are o:er'o, then the r'''''jl()r\,sivity i8 equal to a ten1(l.,r'y el'ficiency which l'epre!:!,",",l,; the dintribllLlon or pow,.,-

'/

£FUur 1"01- tho ,~lJhndwO)'I\ wilh ,shaft ends A, B, C, 0, sc'e Fig, .:?" 3:,~.

Fig_ 5,6,1 A d()sed networ'k_ The l'eticulntor' betweel1

~npllL shaH A alld fJutp\lt ~haft B i.'5 indicated by i, t:fi tt, an8m i ~ ~hHl t- ,,-tio y, and <1 v ,"'i atol' hf.'I:wCC'll shaft enciH C and D i'3 :lnciicated by x

th.-! sum, of P()wc:(S is zfe,'()_ Hene!;!

d'1 + d~ + dPc + dfD + dJ; • "

Tdw -1l7,;. WAdWB + Tdw _rj~~dwo+ d~,O

A A Y '<'8 C [ , 0

]i'oll()willg M.bstitlJtioTl of 1),,,1, rjy~ 1, dp,. 0 •

%d~-~d~ w~~-~d% 1At.II8 2 + "cWo 1 ,0

WB Wo

~ T. d(UC 0 -r..%d(w ) + CWo (wI = ~ 0

R dy R~ rO A y + C x -

~ Pc JI dv , 1JUAo - T. -= y';r:; " "A/c

A

IIcll U' tilL lhc:ul'cm,

9

)0

11

on i>haft ends A and C in the form

T. t " . e.eff(E!)

" where @ff(6I) represent:; the effi<::hmcy function defined in 3.3.10, th", angulur velocities satisfy '-- -

'W €I :2.- -"'c' ~

and the di"tdbuhon of powe, "ati"fie"

flClA " AM:.@ff(E!)

Proof The "atio (Jf angular velocities

'" fA r i.f: • \- B H-t);; ~AIC·a~ff(e) -L C 'A

does not change if the dissipative power'''' are "upposed to be zero. Theorem.§. yields lQ, and then, .!..l.

12 THEOREM If a function !J expN;""",,, the relation between angular velocities on shaft end" A and C in the form

13

14

17

18

then the torque s satisfy

t'l.nd the distdbution of power satiafies

EXAMPLE C<;msid",r" vadator network with one variator und no other tI'arl:;mis,dons (nota ben",), $chematically represented by Fig. 5.2.1. The torques merlUOned ~n assumption 1. 8. 5 (III) ure torques -of the varia tor arid the ['eticulator. - - -

(1- "'I,Pc + (1- Y'l1)7;; = 0 which leads to

!c_ ~ 7A .- (1-x'lx)

.!:2. -'-.l!:ll we . ~AI( (1.xl

flC/A- AA/c (1-x)(1-YIJ,) (1-y)(l- x ll,)

5.2.

73

2'1,. B<:tckbone chain

The cr.HI(,cpt 'backhonc chain' will be il)tr(J(]L1ced <:t" [\ II"lathemat1cnl al'tinc~' to h;"lve a convenient (jcdudion D"l,el:ho(] 1'01' th" fO"mul<:te of ~.

vari.atr.H' nctworlc

DEVIN(TTON A. t.> ,'l C k bon e c h "i n is a collection of p nodes [In,1 p (·.,Hlven;o nod,:,,, "o[uteet,:,(:1 i:o each other' ill ono linE:', i""h,ding I1"Hhor'

.';"1 1.1'"ar)srnission HOI" ~.I. rnE-H~.h.

2 COI{()LLi\HY '1'[""-'(: '.Il'e (2p_l) intcrconll<!cti<"Jns betwf,en th", thn"'­polc~ and (2p+2) sh"lt (,"(if;. See example in Fig. :'\.~. 1.

Fig. ~.~.2. I\. bacl,f)()n'" (:!lain

:-) ASS F:n.1'TON A col.l,,'e:l.l(Jrl of p epic.y<:lh: ge<:t!' t ,., llim; and P Q()d.,,, int(""co'HIl,,:I:C'd to each othel:' in on,:, ]ine c,,-n be )""P'''(:Hcnted by n lJacklJon':' d'l\ill with t):(\'1,,,,,iHSions [\cldled to ec\cil of tl"1(: ,.ih,dl end" of I he bacl,bon<! chain.

:\:,1'oof '(,hl": epicyclic g(·',u· t,'<:tin':' (Ire' r'cplacecl by "ollvet'se nod",; a.nd tr,m:-;missions in """,)I'dance wi,l.h a.,;r;ertioll ~. 4.7. Til€' tr'1!.IlHlnissions bcl:w<:C'1l tho thr<:":-p()lc~s are "Iirninatcd by ni;j,l;iing: D..'>.~"r·ti()n,; 3.2.1 and :L 2.U. Sc(: C'xaD'plo 1" Fig. ~.~ . .:::.

Fig . .:2 . .2. J. IllustratiOl' of H.""edion ~,.i.}

1 A b:J(',kbone chain hi'S no po.'·'lInoter~, ndthel' (\ tr'ansmisd()!1 tatio nor an. {,-d'fi.cl(~I)(:Y.

2 .. !. Clo.:;;,~jfi""tioll of vD.rinlol' netw()"ks

S(:vc;t'al v[\r'i"tut' netwOI'ks h(\vl' ~\l>:'eady b,""n (:](clud..,c1, vi:<. - inc.:mlMistC'nt n .. l.w()r\(s (r"f;tr-iction 4.4.15) - 1",.'duciblc netw()T'k" (re':ltl'iction 4.5.14 anel

limitation 1.':""2:-17), among which reiten.tive netwol" k,; (corol,lar'Y~: 2- 15);

5.4.

while condition 3.2.8 protects the configuration of a variator networi{ aga~nst an unne;;-e;-s;-ry increase of the olllnber of var~ators,

Ni(>re()VeI" it is important to note that

a variator is a vaI'iatol:' network.

~ Consequently, variato!' networks with at least one epicyclic gear train, which contain a variator on the input shaft or on the output shaft, arc reiterative networks.

3 By virtuc of theorem .i, ~.l a variator network without hard parallel L,'anchcs is equivalent to one with a variato, on the ~npllt ",haft or' {m

the output shaft. Such a variator network i", 8. reiterative lletwor'k. See Fig. 5. '1.1,

Fig. 2, .1.1. A double variable transmission anel

its eqLlivalent reiterative network

4 '('he parameters applicd to the classification of variator networks are p number (Jf epicyc.lk geaI' tr"ins,

6

7

v number of variatoI's,

A third parameter may be used, viz. 1m", maximum number of hard parallel branches

for which in acco,dance with theorem 4.6.5

A fout'th parameter h number of hard meshliP3

depends on p and v according to theorem .1. ~.l,

FI"om 5 and 6 it follows that

75

5.1.

~~ 1 aH :;ificdtion of varialor' lie Lwot-ks

a ·l'QbJ.~· 5.4.1 li"ve,; Ll c.la8~:;f"'nl.ioll >;cheme for vad"tot· n~twQ).·k~" ~L)T(\ng:;;,(rlr~ o I"(le,. uf the POor:u",:lC'l'S V and n. Th", (,fily ''lariato)' n.-:twor'k' to b<:> IT"·,,l:ioned tOl' n"o h (1 vH.tiator itself. ~I"".,,)[·ding to 2. and ~. V'J.ti~tor netwo,.'kf; with p:> 2v a,e r'"dueible. Qcconling tu .:!: 2 . .!2. V'Ir-hlm' lletwo)'k:~ with one or two Vo.Y'i.,to"" satisfy tm»co h, o..~ ""lY be: concluded fro", ".;.;sertion 4. G. 10. Tho VariQtOl' networks in thi~ gr'()up, v~ 1. P:!O: 2 nnd v= 2, P-:!O: '4 W),1l hr~ dealt with bderly in thi." dlHptcr. In ::;edhm !:.. 2. ;,;, ,'ecQpit\11",t.i",' will be given, Dd.~cd on thi.~ chu ... ;siri(:~ttion ~cllp.ln(::.

" = 1 >,=2

v,,3

V: 4

h~p-v+1.

o 1

p= 0

t. 4 5

.... t~~.~

5

3. 5. Val' i '" l 0 I' net w () r' k H wit h 0 n e V" r' i a t 0 j'

The r\~rnova.l of the v(\"\';at(H' in a v[{f"intol"' nidwor-k with on~ variCttor­LcnV"'H " fillbnctwl))'I, with rOll!' connecUng ohaft ends. '['hi" Hul.Jnetwo).'k hOoS to h" il pI~imitive one,in 8.CCordJ,l1ce with table 4:. Ei. ~,8.nd must hOove equ:Jl l'\\.\mbor's of ':'pi,(;ydic geQr t""inH and no~le-;.-Fr-()m the pl'imitive 6\.tb",.two,'ks via th" dosed netWQr'k~ the operathmal Hl:h,~[:rH:8 of the v(\r'iator network~ ,,,'e made up. Vat'iator netw()['ks wi.th th['cc 01' mo).'e "I'i<:yelic gear' tr-ains are X''''(!Ll(Oible. Hut f01:' the val'io.t01' rll"l:wuI'i< mentic."H"d ill ~: i, . .!:.. (v~riatol' by itself) there 0.1":' two types of vari,,,l<.H· Iletworl{s w'll;h one 'lo.riato)·, uallcd 'var~~hh: shunt' .'lnd 'YOorio.bV, hr'irlgc', defil)!:'d below.

The v"rj,ah Ie ~hlult

2 IH!: f,'lNITION A V",' i,~ b 1 e s h \l n I: ifi a vQri<LtoY' IlClwol'k witil 1 "'pi<:,yclic gear t,."ill, 1. node, ~.nd 1 vaI'iOoto)', of which til" cl ClHed ni!l.wol'k is given in Fig. 5,5. l Hnd the openlt)..,,,,,l :>cheme i.n Fig. S.5.~.

:) It is of no int..,,·(:,;t which ,)f the "lwIt end", A and e is j,nput shaft or output shaft. TheI'otore, it fiuHices to Stll<ly only one v"riable shunt.

4

5

6

7

5.5.

r9.Tl 1. ......................... , . .1

A~B ~

:Fig. 5, 5. 1 Closed network of a variable shunt. The reticulato,; and tho v~riator are represented by dotted lines

Fig. 5.5. 2 V~riable shl,lnt

Especially for comparison with other variator networl(s it is prtlfitable to l"epreo;ent the variable shunt by a bac.kbone chain and two trartgmi!;$iorts. The quantities in the reprcscntation with a b~ckbC)n'"

chain ~re roarked by an asterisk; see Fig. 5.5.3, .The equations for the transmission ratios and efficiencie" <;an-be derived 'by one of the folluwing rn!:'thod ~.

The fil"!5t method is to write the equations in matrix notation, via

Fig. 5. 5. 3 The backbone chain ill the repr!:' ",entation of the va"iable "hunt

ane! elimination of wt, %, ut' w~.

[1 1 1 J 0.1,1(" =0 -lOy'

resulting in

x·"y"+1 =0

and, by virtue of theorem 1,.2.~'

77

5,5.

1\n()l:hl!r' method~" th" ,Ured r"ad~ng of Fig. 5, ~.4. The sum of angular velocities of 1:h', three shaft "nds of tile -;;o";;v<'!t',;C node is zen>. re'lulting in ('quation 2., 'I'h" sum of torq\l.)" of thc thn'~' "hRft ends of t1w nodo i,,; 7.<,ro, l-eslllting jn r'quation 2.,

Pig. ,,_ ,,_ 4 Angular vel()<;iti(,,; 'llld to\'qu",,, ill the

backbone cho.l,n elf,.. vadable sh\ml.

Th,~ respol1'ljvHy is

VOj' a mOL.',. "x:I:<'!n,;ive analY8i" (.If the vat'iable "huHt thc ,'eader i.3

l'c[erred to Ch~.ptCl' Q.

'l'he vat'iable bridge,

9 DEFINITION 11. va,' i 0. b I.", b rid g C is" variator network with 2 "picyclic ';""" tr'aillS, :l non.,,,, and 1 vo.1:i:1to,-', of which th" doscd ,,(1[wock is given in Fig. 5.5,5 and the (.>p",-atien0.1 ,,~:h'''r'l(: in "Jth,~f' Fig-_ 5.5. (i eH- S_ 5_ 7.

10Th." con fi.g'IH-ations of liigH_ ,,_ 5. (j Gnn S, S_ 7 are deduced from e",d, uthc!' by fl. tr',..nHp"sitiol1 of 1,-;(:1.(:1'8 or by (~);;-vcl'sion. 'l'her'dore, :i.t suffices to 'll:udy only one v,\.r·i"blc bridge, _~'"y in the conIig'ul.'D.tic)n of 1"'i.g, 2- 2- Q - The r-q>r.-csentn(.ion nf a VQ1'i"bh, bridge by 11l€:'D-11':l or ~.

bD.cl-::I)()n .. chain is giv<~n in l"ig. :2.~-Q-

Fi.g_ .s_ S_ 13 'I'h" bi-.ckbon<, <.;h,·\11\ in the j"erH'cficntation of the v al'i able br'i<lge

11 Thc h'ansmission with 9. tran8rni",,,,~on raHo rt and an efficiency ~ is ~ non-variable transmission !'csulting froIn the reduction of a hard mesh to a composing chain.

12

13

14

15

Th~ equathm" for the' t,ansmlBsion ratios and efficiencies can be derived by one of the following methods,

The fir!5t method is to write the equaHons in matrix notation, via

[

' 1 , 0 0 [w~" o 0 , ~ , o -1 0 ...... 0 OO.,0Y.: w5 -10001>"14

and elimination of wi: to wE

N~ ",ulting in

00, , , ,

[

' 1 , 0 0

o ., 0 ..... 0 ;;; 0 00 _10 ~ -10 0 Op"

= 0

and by v1,rtue of th",ol:em .:!·1· 2.

Another method is the direct reading of Fig. 5. 5. 9. whe,'e the angular velocities are expressed in ~ and the torque; i~ 1c~

16

Fig, 5.5.9 Angular vclocitie,;; and tOl'ques in the b,ackbone chain of a val'iable bridge

The J:esponsivity is

... ·0 Y*l y"+ p* AAIC= y*(1-x");;; x"- pM

For a more extensiv(;> analysie of the variable bridge the reader h, referred to chapter ].,

79

,~,6, Variat(J[' network,; with two variatoT'o;

Tho 1'"moval of tl", v"'r~btor5 leavcs a sUbnetwodc with !:1~X connoctiD-g shaft '>rItl8. Inste; ... d Df an inve~tigation into primitive Sllbm,[woeks wi,th six connccting .~h"Jt ends, IOV"n if restrid,hl to those with tht'ee epicyclic gear tntin« and three Il()dcs, it i8 peciera.hl .. to mal«> ~

deduction or the vaY'i"[,,,' netw<)rks direct.ly,

The double v""'ii·l.ble shunt

2 DEFINITION A double v,,-riable sh\lrl[ is a v[1r'iator' networJ, wilh 2 opi.cy"lic Iteat' train,;, 2 rloc\es, and 2 variators.

'['11("' four' [hr'eo-pol",,, ~.r·C intorconn"ded by haY'd bT'allches eith!:!y' in

in "Lain, see Fig. S. G. 1, or' in star, ",,€, Fig. 5. fi. 2. All "'r'r'angement~ ilL ,;tal'-c;n"titute l'eitel'!ltivc net;(;~:k:. So, th" ",TangeIYIl.'''!:'; of a cio111.>lc; val'iab!.", :.;hunt are dC'r'ived f)'(>Irl chree-pol,.,,; in chain. To avoid oth"." roiter"Hve notwork!;, the epicydic gem' tn,in:.; and no(les mU!it altern(lt", in lhis line. 'rh" dosed netwodc is g:i.v'~n ill I"ig. ~.£'2 ",nd the "pC:f"ationalschomos in l;·'ig:-;. 2,.§..,:! to 5. (i. 6 l,ldll si ve.

T Fig,s. 5. IJ. 1 iHLd 5. 0.;] Four' thrc::e-roks in <::h"in anci J.n star

r1L2n \ ......... ...... ::::::: __ ~::~: __ ... , .......... .J

Figs. 5.8.4 5. o. G Double val:'iable shunts

:3 Th<:! (,oni'igul'ation:.; of Figs. 5. l), 4 to G. 0. U ~ndusive C'lre d,'!duced from <:!,,-<::h othcl' by tr",r,spositio,~ ,;{ iettcrS:-·.rh<:!l"'efo!'0.it sufii<:"" to study onl.y onO variable bI'iLlge, say in the configuration of F'ig. £ . .§..:!. Th", ['o;'l)t'"csentatioll of a dQubl" variable 6h1)IIt by a backbon~' chain and thrce transID~.,si(}rl.s is given in Fig. ~.~. 2. The equation$ for the

!\O

4

5

transrnb;!;ion ratios and the efficiencies. re$1,llting from

1 1 1 DO o 0 1 1 1 o _10 .... 0 • 0 OO-'O/': _1000Z*

and theorem i.1. £. a1"(~

x"y*+ x" - y* -z· = 0

Fig. 5.6.7 The backbone chain in the representation of the double variable shunt

5.6.

Let the considered responsivities be AAiC' A=, AEIA' and the considered ternary efficiencies flC/A' ~EiC' ilAIC . The se quantities are represented in table 5.6.1.

Table 5. 6. 1 l)o\l.b~e vadable shunti cl06e;!O netWQr k, backbonE:' chain, responsivities, ternary efficiencies

The dO\l.ble variable;! bridge

"CJ<\' -\IC eft( 6) ttEIC -\t.: 11M AElAclf(l-A")

6 DEFINITION A double variable bric;lge i", avarJ.ator nl:'twork with 3 epicyclic gear trains, 3 nodes, and 2 variators.

It has one hard m~$h wHhollt the .eticulator. With the help of assertion i.2 . .Q all double variable bridges can be generated by fl.dding onl:' node and one epicyclic gear train to the smalle"t hard me",h. The working~ out in 10 closed networks or 24 variator networks is listE:'d in table 5.6.2.

The extended double variable bridge

7 DEFINITION An extended double variable bridge is a variator network with 4 epicyclic gear trains, 4 nodes, and 2 variators.

81

It has tw,> h"",d rneshe~ wilhoLlt the reticulator , Th",se hard lTI~Hhcs have two, ()!le, 01' no C(,llIlInOJl hard hnlnches. Variatol' network" based on two h",.,'d me she s wi I:hOllt COlTlrnon hard bnmch, shown in Figs, 5. f:L 3 «ml .5,6. 9, CODl:ltitute group 4 ill table 5,6,3, A non-ri!ducible netwOY'k ba:cd on two h,,-I'd mesh!:',:, with one COTI"lmOn hD.rd hranch conbin<:< "- Hubnetwork, an c:.:ampl", of which i.s "hown in Fig. 5.6, 10,

Figs. [). G, P, and 5, (;.9 Two hard m!:,,,he,,s without

COroID"" hard b)"andl

I"jg, S. 6.10 Tw', har'd meshes with one cornmon hard branch

The srnnllcst Iwrd ""!HII in this 8\lbnetworl{ Dl1lY be conn",,-,I:ed to th" remo.injllg' elwin in !Ol-'Vel'a} WD.YS, see g)",OllP '2 in table 5.6,3. A variate.H' network, "" its COI1ver'HC, based on two hard rneflhes \;:;hh ~o common he, ",I la'anch<;" "an be d!:' ~,'1:'ibcd by th" Hubnctwork in Fig, ~,~.l!.., to whid1." node 11<11:\ 1:0 bc added, Acldition of the node to the" places imlicatcd in Fig. ~.!:!:.!...!. 1'[" ,",uits in variato!' netw(.1r-ks of gro\lp 1 'Jr' 2 , Acidil iocl in all nth,,,' places Y'(ifiUltS in vad,H.lo,' networ'k" of group 3 ire

t~bl,~' S, G. 3. In th,· v<ldator netwO!'ks of eD.eh g"oup pdrnaivco sub­ndw()~~ks w:e n',")gr>1liable wHh "- o;tt'uct\l\"e "lOrCO OJ" I""H characteri",llo of (klt group,

J<'ig, ~. 0.11 Two hard In!:,,,hes with two comro(,m h",rcl t)r'"nche8 in "'qLCilfalent c()nl'igul"D.tiorlS. To g"n"'l·~.te an i!xtonded vll,riH.ble bridge, a node 11 .. ", to be added to eith"y' of the i.ndicated branches

'I'able 5_ O. 2 Double variable bridg",,,; closed networks, numbe!' of variable networks r",p1:'",,,ented by those closed networks, backbon'" chains. responsivities, ternary efficiencies

(~ ~ 3 r.et.....".

@ 3 r.oIv.orIo;

~----.--.--

--~-- ---'.,--,------

~~.------'

c ----'- __

'" 2....,,,,,,ks

® ' ....... _--_ .. 2f101WOr1d;

r~

[~~

ftCIA =\...c ~v:=-\~~ ~=.t",Aeff(.&=L) ftC/A~~z~ 1 f/OIC-~~z) 1WE-~A~ ft:.A~~

ftEft:~~f#) ~<VE ~"--EAeffl ~ )

~=~~ fj~-\:.,,~ ftvE=AE$,~ f!1;K>.~~

fk:.-\~~

Ik=-\A~ 16.-~ lk~~ I'fAlE=Ju. eff(l+1)

5.6.

83

5.6.

'I'abk 5.6.::1 J;:xtcndcd doubl" var-iable bridg,,;;; closed nNwor'ks, nUlnl.JCr' oJ: vo,Y'laJ:(JI" notwork" "'."d conve,,," nelworks "''Pr-c,;ented by th",;c; c10se(j I1ctwor'ks. Wi.nding line-~ indkatc a spl.it f'H' d b8.cJ(bone ch",ilL

grDUp 1

groLJp 2"

~""-'.-- / .~.---,,~.:.:/- 6W---;'ft-"" ~ ~ .. ------:~--i, __ ".-:, ... , .. ~;, .... -.,.;-- .. -. ( __ j !.,. __ ... _.. .. ..... i ,,,,).'

------ i;-j 313 310 ,.0 0.3 3.3

group 3

~ __ ,_. ~i -, ~,---:-,\

.... i " r'

.' .,' , , , '!' \ '--. ___________ . ----'l.l '-- ___________ - 2,1 \-________ --- 3,3

{$5 .. :--- ~,-.. --I ___ , , , . , .,............./ \, .... _--", .

........

2,2

5.7.

5.7. Recap!tl,IlaHon of variato!' networks with one 01' two variators

An outline of v8.l'iator networks with one ot" two val'iatorE, based upon the c1a",~ification scheme in table 5.4.1 is given in table 5.7.1. The n~ne given to the variator networlcs in a certain clas" has been­borrowed from one of the",,, variator networks which i" n~produced as [l stylised character.

Table 5.7. I Outline of variator n~twO\"k6 with v variator:>, v; 1 01'2, P epicyclic gear trains, and h hard meshes

,.---~.

!variabl.e !

p = 2 !variabll! !

p=~[;~ p,,' v,,, transmission ... ~ 1 II = 0 _ .... , m .• -----' h = 1 shunt \/,,' bridge 11_2

tl'1:!l":lC~ ,h;lroot.r

J[} <haraet.r V .. ,,""-, ..

b.,kboo, <",10 backbcml!: dlOllrl

r:;>n (::?j~:il do,,"" ""twol'k doSlJd net\o'w'Ort.c

r?::Jl \ ............... .! ~

oporat;O/\II1 5<1.", ... ~~ser..n-.. opor.llonal od\cmo.

~, ~ ~~ -- , ... , .. ,."-... ',.".,, p=2 [!EJ p= 3 I double

I p_4 extended

v. 2 var~ v = 2 variable v=2 daJbI.E, h", 1 5ho..tnt II ~ 2 bridge 11=3 variable

bridge

CharaoterJtl; <t'aracter~ cI"1;:1r'"iilltl!;:r

W l :

.b"'I;~boJ''lf: chall"l biiU:kbo"'l!: chain biekbom!: d1i;in:;

r?-r}n r-<irr1>T?11 r/r;:>r';'i;/il ,Io=j r<:!worl< rr:~::j?i{;;-{;'1l

@2J r'?j<;>j:rlt<h 10~ "0\1'1'0<1<. ~B cloOSed ncl'NOl"lci

operational scIlemes """stIMI~ ~, ""t=I<:; oon:;bh,lting 12, net ....... k.

~~ are ~ot.d ,,_~~~ ar! ('efIf~~ irl til~ ~§:~

85

2

3

4

5

7

8

G.1.

CHAPTER 6

VARIATOR NETWORK OF THE VARIABLE SHUNT TYPE

6.1. D""<;r'iptiOll of a variable shunt

The dennitiorl of a variable shunt was given in Q: Q.. ~. The most condensed representation was given in Fig. Q..Q..~ and in formula 5.5.6. Now, step by step, information about the epicyclic gem' ;;·;in-win be introduced, first thc parameters i and ii and Inter the

~ y Fig. ~ . .!. . .!. Variable shunt

position of the planet C<'Xdel'. The relation between Fi.g". 5.5.3 and §. ... !:.!. is given by the equ<ltion.~

Ii r}( )( =- 1_1

" 1 Y=-r-1 Y

b whkh tho;> parameters i and i'i are abbreviations

The <:'q"aHon f,)r the transmissIon ratios is ("ec Fig. 6.1.2)

and can be written explicitly

y + 4'-1 )(= --r--

_ (y-1J (" '(x~1)

The responsivity is "AiC' abbrevi<lted to A ($ee Fig. 2. . .!...:!)

87

G. 1.

all

x

Fig. G. 1. 2 1'h(, i:r'ansmission ~·[\ti,()s

()f'1 variable> shunt

-+--+-..... 1'

L ___ _

Fig"_ !!.. l . .:!. Th,~ Y"""ponsivity of <1 va '''Lab lc: shllnt

A _ IK_ >:"(y-1) , Y - 1'(.>(-1)

FiG, G. l. 4 /\llgulaJ' volocith'~ aml tOl'C[ues in " val'inble ~h\llli. ,,>xl't'CSS('c\ in Wa Hnd 1A

10

11

12

13

15

17

u3

~=+y.We7.;

~=-Ylly-%~

Pc ~ - [ijK-We1A

Po • + r~x'l.-We7.;

Tha c:hgg~paHve P<)WeI' of the epicyclic gear tr'ain ~" ("e~,' the an"ow marked -p'p in Fig. 2.]. . .!)

The dissipative power of the variator is

The efficiency of the variable shunt follows £r'om ~ and ~. i: ~

[ill/II, - Iff t 1 Fly" ix _ r + 1

The: distribution of powar is characterls<:!d by th<:! t<:!rnary ,dficiency

~ __ .'1: _ 1fu:.. - ~ - y

~C/A. ,",AIC -i'/

6.1.

The: efficiency fi of the epicyclic gear train depend:; on the value of th<:! sun ()fficieney If. and on the position of the sun gear" blltween which the sun efficiency is defined. In other words. i'j depends on fi. and on the

'----------_._----------- .. --T".ble 6. 1. 1 The binary efficiency ii

89

~ . .!.. 6.2.

p(Joiti()n of the plm1!d "<I[Tiel:' with re~ph;l t(J the shaft end5 twtwc",n whkh rand !j al'e defined. ln table .Q.~'l for' three positions of lb~, phn~l. car'de,' the bi1EH'y,dfldencj i'i is expr''''~fl(,d in the binary TalLo t and t!", "ull efficiency ii •.

tl.2. COlL<]ltlons for the pow"'" flow

Olily those power fl.ow~ fH'., I"HiGible the dissip::\Hv~ p,.>w(:n; of which <tH' neg::ttilf<'. Thill:', ~p<O "nd p'y<O. From ~:>O with Q:.!. . .!:.io.ml .!!: 1:.. 15 To llow,; "

CON Ill'I'.tON The power flow of H. var'lahle shunt ~"U"flc s

2 (1-II.H1-1)(1-}l<0 11

'rlli,; cOlH..lltion decid':'ij whi<..:h clir'eetion of th" power' flow in th" ('picyelic gear I:r~.ln (r'epre8':'\1t(,d by ij) belongs to '" giw,,, dil'ection of th .. power' How in th • .' val"iator' (represenled hj II,,). In ",. o;imllar way fO[1{)Wfl "­condition {()Y' II, [,'om Q. •. !: .!..Q ~md 2, . .!. . .!:! '

3 CON IllTION The power now of ~\ '1ario.bJ." flhunt ,satisfi<"",

4 (1-111)(1. ij)(1-J) > 0

')'h" l.n(,qllalitie.~ 2 and 4 aro relo.ted tl) each other' by tr'ans,losition of lctteJ"~ ,,11d revel-;-:"D.1. Orl:h0 inoquaUty ~ig[J. 1'h" evalclation of th,· I:wo condlt.1otl;; is given lit table Q.~ .. ~ ali a. math"lTIil Li<..:,,-l compl.et.,

T!lbl.~, 6.2.1 Ineqwd.il:i",; fur the POW<;' 1"' flow of 0. v".I"'.\>1l>lc shunt ......... " ..

12;> [<0 0<;'<1 1>1 if,;<0 O<7ls<1 ii,>1

---_. .. , .. --_ .. " .. ._--------

[<0 0<[<1 - r 1

~ Ik<r:r'L.

r- x [ >1 1l?f-r,J,. 1- is

..... ... . .. "'- /-.-.... i<O - i . 1 -

1;, 1/.?" -<.!';"i l1s<

'~ O<k1 O<ro.<I_i_! £:>1 t.=1-r<i1.<"~ .•

\x~.'L1- il<>1 i x:>1 y<Q y:»l y:>1

7J,.>t '1y<l { >1<1 Il(y~J [x< 0 y>l.;; . _. y<l-i

"-, . .. ,.

o

il,>1 1 r - 1 ,,=U<7I·< Ti,;<l 1 - [ 1 <;1I'<U=1;;

1 <'l.<1-'.r o <ii, .. ;::j

k=1-f<ii<1 fi.>l: !t<O

i'i,;",1 ;il,"",l-[ .. ~ o "'-7l"s<1

Tl,<l 1)<1 '1,.;> 1 1ly>1 y

._--_ ....

0<>1<1 0<x<1-1 11,<1 '1<1 1 <y.<)-f 1-1"0'.:0 y 7J,>1 l1y<l 71,<1 II >1 y

f..-. .,"-

k ... <1.:-~ l-r=x<O '. >/ O<y..;l 0-0/<1.[ / "

11,;> 1 1/>1 y If,<l lIy;> 1 11,;> 1 11<1 y

f--- .. ..... ...

0<>«1 1-f<l<<1 7J,.<1 1/y;>1

1-[<y<1 0<;1'<1 1'1,:> 1 1'1y>1 If,<l 1)y<1

....

'90

6.2.

relationship" In most pr9.ctic8.1 applications the values of 1/. a.Ild ij. ~re fairly close to 1. For such ca!je$ a conven~ently arranged extract (If table §..~ . ..!:. is pl'esfmted in table Q.~. ~. If, for instance, the transmission l'atio$ )( arId y and the direction of the power' nC)W d t"e v8.r-lable !:'''unt (r-epresented by 7ly) al'O given, aile can read from table ~. 3.. ~ whether 11. and '7. are below 1 (shaded are"5) or above 1 (bl9.nk ,,-,'ea,,) ,

'fable 0" 2. 2 Extract of table 6,2.1 for values of l1. and ii, (;h .. ~", to 1

91

G_ 3_

G_~_ S(::lf-loCI{'flit Vo.rii,ble shunt

'r'he vD.lll0 of 1/) cl~tOITnin<.'" the dh-ecti.,)1"\ of the pOW~I' now,

1/,>1 l!)e'U\.~ positive power on shaft end 8 and rlcg.:dive p,-,wer on sh'<.fl (Ond A

0< lIr< 1 mC·D.n::: po::;ilivo powe)"' <)" "haft €nd A .,nd u(Og ... ti'le pow,,,' un shaft ",nd B

l"], < 0 lIl(:[ulS pOSitive POw€, on ,;h",i't end A as well o.s pOSitiVE, power on "haft end B,

In th.., third on"", "ll inpllt pow<,r lr· ... nstOI'm fl i"to dissipativc power',

4 In o.ccol"'(\ .. ""e with .!: 2. 2. a vo.riabl", billmt with neg ... tive ",{Hd",ncy is calleel s" 1 f -1 0 0 t J. 1"1 g. FoX" clC)fleI investig~.tion the dfidelley of a var'i"ble 8hunl given in ~ . ..1. . .l§. mo.y b" "ewritten ~n variou8 ways_

7

L 0

lly = 1- A(1-l'j11.)

[ij( - - ) 1/ y ~ Y x1). -(Cle llslc

ThL' I',,:;.~ing of the: Vahle 1/y=1 h~~ already b<:,~r\ dealt with in table £-~.~. '1'11<:' limit~ 1/,,,0 "no 1),=0-:> will n~)w be inve8tig(1t"d.

The I.i.rnit lly" 0 "'''par-ales the ,,,,If-locldng o.r"H. hom the "lrE:a with Q

pow,:,r n.ow frO!"D input shoft A to ()utP1lt 6haft B. Thi,:, lirnlt COD8j.'lts "f the IJm:

'fh" other Umit 111 = 0-:> flopal'ates tho self-lockillg dr'ea h'om the ar,:,a with a power flnw frOI!I input shaft B to outP\lt. shaft A, Thi5 limit con.~i8t" uf the lim's

in which 10 stnl1d" for' the \H1illtoresti1lg' ''';.';;0 of a f,,,dy rotating "haft end C and a ulock,',j :-;haft ",n<l D. The ,'3elf-locking areas nt'<:' Hh()wn in tabh: 0,3 . .l .

92

6.3. 6.4.

Table 6.3.1 SeH-loddng areas of variable shuntB

)(7k ~ S::a~S<.(.=-? .. __ .. -+-,

A~B ~}

7l ... =.~ ... _ - - - - l-J=~~

_{-¢~<:=O -¢:>~¢::>

_ { ~ -(C;- <:::=> 11;-<0

D { c:;> r-(C:.-- c:;> 1ty<0

-{ 1--------------.-.. " .. ~ ........ --.--------____1

6.4. Reo;tdctions imposed by practical rcquir·ement5

Running through 7.erO ~$ \l~u"lly "llowed for the: transmbsion r,,,tio x

of the vll.l'iator, and sometirrle» for the tram;mission ratio y of tho variator network. ThiB doe6 not impose restrictions here. An unfav()ur'ahle rel"tion between I< and y may be caused by extremely ;;;mall or extremely high values of the binary ratio r. Ther(lfore, the only r'co;tr"iction for thE:! h'an6mi6sion ratios is

93

6.4.

A variable sh\lnt must not be self-loGklng for the principal direction r,f power 110w and CC.llnTll<lrtly not lor thp. ['cversed powcr Dow ,~ither.

2 )-tESTIUCTION

3

4

5

6

7

13

9

10

11

12

Anothcl' phw_~iblc ,'cstl'ictiort limits the maximum powcr through the v",riator, to" cecbin fr"dion eof the power' through th" input Shaft.

t{ i:';S'[,nICTION

~cU>.""e fOl' TI. <; 1 and '1y <; 1

~oU>.o;:e fo,' TIM >1 and lly < 1

~clao;:e tOt' '1M <; 1 and lIy > 1

~DlB~ e fOr' '1M >1 and Tly>1

while tile fra<.:tiorl e satisfie s

0< e.,; 1

The inequ,,-lities ± to 2. inclusive ,iI'C tran"formed into

A<;;+!-'1

for' I)M < 1 and lly < 1

Ao:;-!-..- for '1.>1 anel "11 <1 'I TIM

A';;;-? f'0)' TlM< 1 and '1y>1

A";+~ 1j1l-, for fl,>1 ... nd 1'(,>1

'I'he ,-c;,;tricHot,,, are "howll in table 6.4,1, The diagram,; would be morl;! completl;! if the CU).'Vf;>;; of Yig,-6:-1:-2 w!:!re entered in the unre;;l:dcted £IrEla,; _ The"" (:urves fo;'- trallsrrlisdon ratios, drawn I;3.S

far as there ",-I'e no t'e!'ltridions, w<.1uld show vlilues of the parameter f, the range of )( and the 1"allge of y t11 ... t are adc<.1uate fr'om the point of v!.!w of distdbution of power and restrictions.!. and ~.

94

6,4,

Table 6_ 4_ 1 Area!;> fe)r practical applications, calculated with ii.~ 0.96 Or 1/0_96, rV O.aO Or 1.25, e= 0.4.

The transmission ratio x is defined for' the direction represented by the oval. The transmission ratio y is defined y=~_ The diagr.~m5 in the right-hand column are repeated' with reciprocal x and y, to faci.litate a convenient comparison with those in the left-h1l.nd column.

El~A

_ reM~1"QI'TlI)

95

2

3

4

5

6

7

8

CHAPTER 7

VARIABLE NETWORKS OF THE VARIABL,.E BRIDGE TYPE

7_l_ Description of a var"iaol,a br~dge

The derinition of a variable bddge wa6 given in 5.5. (1 • The most conden6ed representation was given in Fig. 5.5:-8 and in formula

7.1.

S. S. 14. In£ormaUon about the epi<:yclic gea~ t~ai~s will now be int';oduced, first the parameters [" [2' i?" if, and latel' the pOf;itiorl of the planet carrier in each of the epicyclic gear' t1:'ao:n!;. The relation between Figs. Q.. Q.. ~ and 2.1::.!. is given by th0 equaHoTIe

Fig. 2.1:: 1:. Variable bJ."iclge

• (!,_1)_ x =- ((1-1) [2'"

Y~-T2Y

in which the parameters []I / 1 , il" til are abbJ"eviati.c>TI'5

L, = LACI8C

The eq\.\aHon lor the transmission ratios is

97

10

II

I. :~

1 ~\

[4

1(;

17

III

19

20

22

24

7.1.

""d CHIl be, wr'itten explicitly

x = (i,-1)(i, -y)

((,_1)([#-1)

([,-1)>< + ([~-1)[, y~

([,-1li0 + (r~-1)

(y - [')([1y-1)

y(i'(r 1)

A= Jy-IJ)~ ([,-1)(>I-1)y

A= _ (lzy-l)x(y-l) ((2-1)y(X- 1)

p. { -- ',i'j,.'} T A = Y 1 + II/Vih /1&-1 ,wB'At;

25

26

We (-- ~)T_ - 'MX1I.-- ,'AI:;

lib-'

Fig" 7. 1. 2 Anguhu' v~lneitic s and torque", in a vliriable bridge expt'I:' f:,,;ed in Wa and ~c

7. 1.

The dissipative power of the v aT'i lit or' between the epicyclic gear train., i~

OJ' )'ewritt"n

n Uli1- 1) (1 ")'" ) T. r.;,"- -_ - - ". ',-y ·WsAC

(1 , -1)

no

27

29

30

'1.1. 7.2.

I""l" a vnt'illhl.1.' bt'jdgp. with the vllr'j.D.tor' between the nOri<.'!O,F.'y i,>l th(e dlHHipilHv,== powel" of l:hH.l vllr·h.llul'"~ ~l1l.cl r:;"1. iH t.he.' ::::iu.pply r.H,)\.vC'llo.

The dish'ibuhon of p<.>w,·>r' j ~ ciul:"cteri:scd "Y th,~ I."",q\y'y <:'l'ficiclIcoy

ry :_ It = ([/11-1)([&-1)(1-[1[,)'" CiA Fi WTh-1)+XI)JlI)2(rlil~1)}{(!,-1)x ... ([,-1lq

rJ'hl: \:fflclt':r~t.:i(:~ 11M Ltlld 71y of the- v~rir..to:r ;t,11d thl~ v\,l1"iator' n~~'!:wol"k

rnay bc: dcd,H,(,d fnml Hand 10_

I) • L (Itil -1)XI). + (1!i?-l)'lti,

y- Y (1'll,-1l[ii1'%+ (iii-l-l)

'1,- }Ylb-1)«(lkYl)yL (ill)l .1) (i21)1YTly-l)

'1:"" "fl'lci"rll2ies ii, ,mel i'h of the 'JpkycH" It "".", tr'"in:;; depend on which l-i1" .. .!'1 (,,,d in erlch of the epicyclk gf:".'\1"' trai,,>; 1>; planet carrier, ,mu on I h" v,r.! u(, fi of the Sull eflieicncies 1l'1 Clnd 1l.z'

7, 2 _ C u n d i t i () n H f" l' I. h" pOW C ,- flow

Oldy 1,11,,:-;(: !lower Hows !I!'e pO~fiH)l(, Ihe dl>;sipcltive poWCI'~ of wlli,ch Ilt·" Tl(!g"ltivc,_ Tlw ~(lFl ()f til" inpul_ powe,' "nd output power iel p,-,~itiv("

J?o!" tl vttrinble bl"i.dg\~ wiLh tho VilL'into!"' b(::tW~~E~n the epicyclic gCi:.U"

tr"in<, follc.lWH [,-om P.1<O, 1;,<0, via ~ >0 with 7,1. 2:l 'lrl(j P,,, - - -'I, 1, 2:,

~ CONDITION if vR.ria,tol' b(:lwcc'r'l

<.'pi,,-'ydll, gear tr"ins

I<'il" II. val'i,"toI' l.Jctwooll til<' no(l"" follow>; from Py,..:;O,

R.=-P.I -F.'2-F.Jr >0 via ~>o "

:0 CON IHTlON i1' va!'iatol' IJ",twf:'f:'n nod""

In i1 8il11il.<11' wily 1:11.-, "ompal'i,son of F12 with p', ['(:sults in

100

4 CONDITION if val'iato,' between <i:pkycUc ge"l" tl"ain,'5

.5 CONDITION f~l{l-l}{U.} .:;:0 y-' 71. '1:1- 1 if' variator between node ~

Fro:m ~ ~.nd.1 ~.~ wen 11.'3 from 1. <tnd ~> with 7. L 30 follows

G CONDITION

"I CONDITION { !J..L}{1_Xl1'}{~} -<; 0 ([ -1Jy 1 -11. 1.lly

8 Suppose the epicyclic gear' tr',,-~n!:l :~l"e not of a self-locking type, viz.

neither !,flies between 1 and ii" nor' '!.i b"twef:''' 1 and ib and that the variator i~ not self-locking. Th" effkieAd"s of the f:'picyclic gear trains and the v"riator ,tlay eac;h ha,ve two value" in accord."nce with 2.8. 7 .

8 Then, because of 2, 4 and 7.1.29, a vsdll,ble bddge w:i.th the variator between the epicyclic-gear traiti:"S has t w" e qui 1 i b \" i uno 8.

10Th"", hecau"e of~, .1 and 2:.!.. 29, a variable bridge with the variat'H' between the node'3 ha'3 two e qui 1 i b j' i u m s if the ter'rYl

( Yl1y=~'il' l do~:;; not <,hange it", !;lign. Yl1y Yr2il2

7.3. Power now through the branches

There arc three meshes in a variabl" bddge. With regard to th".,e meshes foul' type" c>f power flow can be distingui>;hed. Power fl6ws are left out of cort,dderation if an epicyclic gear lr"-~I;\ dissipates all the power reeeived fr·om three branches, and/()r' if the variator di:;;!:lipate'> all the powe,' receiv<'!d. Power nbws which becoml£! id",ntical following trl>nsposition oj' suffi:xcs belong to the s~rne type. A type is char-<l.ct(Ori",ed by relations between thc dir<;cU011'3 of the branch pow"'r,;

IRe> ~D' 1&, Psr,. Formulae 2.1:.. 19 to 1,.!.22 indu'3ive are used to find the decisive inequalities given in table 1, .. ~: 1:..

2 If a power circulates in a mesh in unchanged direNion then this mc"h has what is krwwn ao; a b 1 i n d p o 'Iv e ,~ .

3 It is emphasised that a blind pOwer in a mesh containing the val'iator docs not imply that the val'iato)" i., highly loaded.

10l

7. :J. 7.4.

T"ble 7. :.l. 1 Four types of pow.,!' nuw with )"~g'ar'd to blind power in ;;, mesh. Til,:, numbers of pow,:,r' now schemes '"(,[Cr'

to the, sch8me" o'btained by trall':'po:;ition of letten:!. For' e~d\ type the df!ddve ineq\lali,ties are giV<:'ll

V1,OI'''''l'cI<>oo under 1 12 ro- llow 5C~ 6 powor flow _. 16 powor flaw ",t..rr.!.

fie,.,> 1, or ""'1 0-~1 6 p<m!:I now "',.,.,."... 12 _ flow SOI'\e"",.

7.4. 1{.,vCr80· "'Y"lmot1'i,,, variabl(e Dl'iclg(:,

Tho r<:'preHentation of thc vari~bl(: bddg<:' in Fig. 5.5. fj with a ~ompo,,~ng chain line! '1 minimmn l1\lxnber' of p[\)';~rn;tCi',:S has the cli"i".lvantag" that it i'l only a symbolic ['cpr'H"mtation. In H real desi.gn, the minirnum rnm'\b,n' of p[()"am"tcr.s, equal to the ll\l1"nbel' in th<:' "ePl'esenlalion with I:ho compodng chain, "dn be «chi<:,vc'd by p"tting

~, = 12 HIlJ i]" = 'l'2,

D),;I,'(NTTION A t'cver:;;.e-symrnei:ric v"r'iablc iJr'idge is 8:

var'iable br'idge with equal bin':"r'Y l'D.tiOb of the epkyclic get\)· (r'ains, cydicD.lly dir'ected .;,[1 the hard mc,sh, and with .~qll[(l sun efficiendes.

2 1,'(". <:omprll'is,,[! with (\ var·i",ble bridge with Y'kl.tios x ~'l1d y the CQ).'rm'l'ollding ,·al.i" 8 of th .. r'C \'€r8~ - !:'ymmetri<.: variable bridge (H"~

3

4

6

7

8

7.4,

written a>< and by, see Fig. 7.4.1 in comparison with Fig_ 7_ L L The equationB 2.1:: l to ~. !·1 inc1;:;:sive become

Fig. ]..:!:.!. Reverse-symmetric variable bridge

From 5 and 7.1. 3 via [2. ['[2 inste~d of concluding to i "I/f;f; we have the freedom to -w;':-He

by which

- ([,-1)\fF-' a_-- .... (l2-') ,~

b" Irr VIi 9 The particular form of ~ associates the values of I with the 8ign5 oft,

and '2' 'l'he value i: is iIT1ag~nary if r, and r~ have different Bigns.

10 Th", ratios a and b are independent of x and y. They are imaginary if X, and [2 have different ,,~gnf;. Tho areas of these rat~()!'; are speeified in Fi.gs. 7.4_ 2 and 7.4.3. The symbol j means i=\f-1. The tI'ansmission ratios !:latiSfJ - - -

b ,.,

br-------~~-----~-~

Figs. 7.4.2 and 7.4.3 Areas of a and b

103

II

12

'7.4.

a;«by + laby -1) by(al!+ 1)(.;1;<-1)

Tllf:'~ tr-.:.u)~r.Ili.~sif)n l~atio~ ancl the t"'e~~pon::;ivi,ty arc! 1.111J.'-i1.r'at:(~d In. Fi.g,::.;.

2· .:!:.:!:. tl) 2.'.::1:: 2, wh •. "·',' j rn<""ll;,; j=t/-1'

Ef j

00 ....•

/

.Qf>"

j~\lg's. 7.4.1 ;,.tl"H.I 7.;1.. 5 Tl~::J-n::-:rniHHiofl 1".1!.tLio:::; of

~:·c vO I' S~~' - ~YD ... rn(~t I'ie va l"' i~,lt),h.' br-lc..lp;('.,=-

.. !P -\'C

lii.g!;. 'i. 4. (; <tlll.1 7. 'I. 7 RCoIpon;,;ivlly 01" !'''VOI'~C"'lyl")"lrr\(\i:J'ic:

vdr·i"bh .. bt'idg,,~, '1'1", r'"U.l.liO!l bctwc'c,n "CIA

and ax is (kn"'iv~d l,}Y the pC l'tI"l1..'ltati(dl

by i)c'cOD1C-S ax, i 1J""o!UC!S-l

/..AICil(:c"m"" AUA

I(H

val·iable shunt, its responsivity does not pNig1'C$S morc favourably whn(~ tb, design i$ more complicated. See Figs.2.±.§. a.nd .§. • .l.!.

7.4.

l5 i\ val'i,,-bl,~ bridge with [2;.. 0, on thc contral"y, may havc a rem,,-r' kably t~vo\H'able r'~";p()n$ivity, see Fig. 7.4. 'I. Til", gr'adual change of the responsivHy A.o./C in a wide interval of. the r;>,ti(.l by is impot·tant becaus<:, th •. , t"',"nar'y efficiency 1lc/A tend s to "'ppr'Clximatc the value of )"AIC'

16 A ml,,'e detailed analysis of a vadable bridgc l!xceeds the !;leope of this study. The capricious lines found in tablc 6.4. :l" for the l'r'actical "-pplicatiorl of a vadable shunt are a warning against the complications inhcl'ent in a variable bridge. Suffic<:' it to obscrve a coincidence of a branched power now through the branches (table J.: i, .. !) with a [avolll'abl€, responsivity. A rough indication is made in Fig. 7.4.8.

~----+,---~~~---+.-----+~x

Fig. 7.4. B ROLigh incHcation of 8,r''''''''' f';\VQur~bl" t() C\ val'j.",ble bddge with the variator between the epicyclic gear tl"ains anel with rather low cii!;l!;lipaU.ve pow1,rs. 'I'h<:' c:lott<:>d lines illdicate minor impor·tant p",rt':l of the favo\lrable ~rea8. The Sa.r1lE! dia.gra.m 1101ds Jor ~ v(lri(lble bridge with the vari"tor between the nodes fOllowing th" perrnut(ltion

ax be(,,)rn.!~ by, i' becomes -L by he(,omE;! ,) il.J(, A.A/c become sAc""

105

2

7.5.

7,5. Double epicychc gear train

'rtw two epicyclic gear [rains of a vari",bl€> br'idge constitute a noteworthy design called 'double epicyclic ge"l' i.r'ain', Sf:!e Fig, 7, S, 1,

c

B

Fig, 2, :?:.!:. Double epicydic !te'at' train

DEFINITION A (l () \J h 1 €> 1C pic Y c 1 i, c,; g ear' t I' a i n c,-,nHi8ts of two singl,[" cplcsdic gen)" tr'ainH and two Dode", configurated "'~ " 8Inallest 11~U'd 111('''\',

Th(, (Lngll\n" vel()dtic~8 of the fo\lY- .. haft ends sat:i."fy tho proporti()n d(~ducC'd )'l'IHlt 2.1.1

:~ 'I'h" im.'ql[aliti(';-;; <.If the: "ng-ubr velociti;,8 ;;,,'0 summed lLl' in Fig:, 2: 2,. 3.'

106

j.'ig. 7 _ S, 2 In':'quahtie s for- anguhu' vel,odties_ The "bbr'1Cviation {ABeD} menDS eithel'

WA<%<WC<WD 0)' WD<WC<~<WA

7. O.

4 With ,e"p",d to the positions of the planet carriers four types of double epicyclic gear trains may be distinguished. M indicated ;in tabl", 2· ~-1,.

Table 7. 5.! Four typc~ of double epicyclic gear trains, not symmetrically ,elated to each other with respect to the positions of the planet carder"

1r,¢ *2~ {r3~ r--",.-

~~ 2 *~ , *~ 2

*~ 3 ~2~ *~ 1

A doubt'" epicydk gear train may be treated as a reiteration of a single epicyclic ge"'" train. A "pecial case, however, will be considered here, viz. the double epicyclic g~ar train with common sun gears and common planet carriers.

5 If the two epicyclic gear trains have a common "\HI gear, then their planet carriers must be coupled. Coupled planet ca,rier" a,e equivalent to a common planet carrier. A transformation of the design aft",r the '~xamplc of Fig. 1. E..-l demonstrates the equivalence of a double

A

Fig. 7.5.3 Transformation of a de.,~gn

",p~cydic gear train with couplod planet carricrs to one with a common planet carrier in which pla.net gears in a planet group havc a common shaft.

6 DEFINITION A Wolfrom~transmission is a double epicyclic gear train with a common planet carrier in which planet geare ~n a planet group have a common shaft.

107

7. :0.

n'(':l'refientation of W,)I..ft"',)[n-!['all::;miflfli()n~. ))[~Qig'ns

()bt:,jnn.hk by t,'anlwositiol1 of Ic.dt;'",,, h':IVe been ,-,rnil-ted

two pIonot goar> m~ be idcrl.:x:al

~ ~lQ1l}=5

'j A W"lfT'",n-tI"11nsmis~ii)n rnay bc: split into tWD ":pi"yclic gr,,,l' t"'''.ill'' i.n tin',·,,·, W'.l'y~, c"cil h:1ving " din"~['cnt syrnl)oli." r'(,pl'o.sent~tj()n, 'This 1'~:J.dfi I.u <l e.l.<H.~inc;:Jti.()n inli> two g!'oup':\, ,.,,.,,,, gn)llp wi.Hl two !"ll"·incil:.o.l

(1t's:i1(no; ".nd tht'oo oqllJ.Y~l\.'·'nl ~ymbolic J:epl"'~"nt"tions. 0e" I-'i.g. ,~ . .:2..1 :\[,,1 I.",b[" 7...:!.. ~. In )1"l""t: existing design,> the pL.lIlC>t cZlrri,'cl' of cl

Wol1'r'om-tr"n~)1"liHHioll is not conn0cted I.() a.ny element ,.,ul"ide the trallsmi~si'.m, 'I'hul-i, the transmi"siol"l i:,; e.qLliv(llent to H l-iinglo epicyc:iic": gcar tr:li.n \·)(·d:w""," the ~ho.ft (,[v;i", of lhe three. sun g"H.["l-i.

IOU

Fig. 7.5.4 A Wolfrom-tr·"T1~xrlis$ion split in three: diffcrent way"

8 Th~ efficiency of a WoHrom-tr"arlomission may be pOOr' fOr' the main pow!,!," flow and generally negative fQr the reversed power' flow (8elf­locking)_

7.5.

109

CHAPTER ~

DYNAMIC RESPONSE OF A VARIATOR NETWORK

H. I _ S (: h <.' ['rl. " I. Lcd .y. 11 " m L c_"'!::'_~j2 ... .!::_S >; 0 11 t (' I. i 0) n () f H. II

(' p.:~ cy (~'l i L g I,) C\,t.'_~~2

An (.~pic'ycJic g{!;.-u· train was n.!p,rC's(;~nh,~d In. 3. 'L '7 ;u-; a c()ny'I:~~":-;\'~ node h~,lving no tnli"J.:-;,.-ni:::1sio.n p;n··~Hn{~t(~I'· at aU .. l[lhl!_'tl:7ln~~ITlt::-;!-5ion parDI.net(-~l'·:-; Wt,,·(, tI·:\I\.,~r,,,,,,t:d ltl [11" ",11" fl.:;. If it) "dditioll t1l<.' dY'1,mlL<J PI"'op"rLi,,:,;

"f plMwt. g",,-n; ()'.Ilild 1.", I,,·,\.n~fol')',·d 10 tho SIJ:Jrtfi, the'll':lth"""<ll.iL'(ll IIl~)del ()i' iUI i'pic'yclic gc:l.I" tl'~'lin would be' ~\::..: (~nlp1.y tl:; th(~ rnodc'l (If n,

flock·. '!'hc~n .... t.hv :.:-:;ttldy of dyrlat:nic Pf'O{H",,'L'"tias of :~, vari"dor rH:lwo.r~k

wOllLIJ Iw ,'"clute,,,1 t<.1 tI'" f<l.ucly <)1" th" dyni\mic.~ of >;h~,ft,;, cuw"il""] int" " ~y~t('m ll'y p"'·'.\l\\"t'tl,,' ... I';·(l(: l'f>I;J.l.ion:. of thr'Hl ... ,l\llc~. 'i'hi~ "ppr'oncll will 1.>(, aJH:lw,,,',,d by til<' d,·dudions i:l,dow.

:,utfi1{"'"' A,S ~HHl g(~;J.l'"

1')1.\'1'1(~1. ci.."tl"r-iel"

plane'!. gClu'

l>ulJi'l C ;Hl 1"r, 1{" fi a, b

~ 'In;.1: VlO"«"·, P€;"'I""lndiculrtr' I." ;;L)(l~ a (pl>lnot g~'m'), (l\)P1QI1~H'

with (\xis a cel\cI axis C (pIn''''!: "'''TiC'r)

iffy lUlU VlOlO'''!:., 1l''''IH:llCliclll:u'lo <lxi,s a nnd '.l.-ci.s C ez llnit vC'ctut", nlo!.lg" t1.xi::: a

[./A,!oI6 binary ratio bctwe(;,n ,",un gCQX' ;~.,ld pLJ.net g"",r -1.-b,,z, -t.>JiJ t,.an:-;l\:"l"l'cd ,nOlllunt of tmlr·ti«

1, • ...10, moment or illodi:·' ,,[ pli.l.rlot g'.'"'" ubuut it" own md,.. jX?Jy~Jz II.:U .. )1':neni: of in(q'ti:~ of planot g(·~",.Lr a \l.bC,)lli: U C.u -ol"'C'linalO axi.;:.;'

m:.J,ml)

M,."Me M"b J;,/b s".rS~

:>;'0

rnas;-] of pi i'nil··t ge~) t"

t ,',.ul sf"1 '1·'" cI I.m·q LtC lorque 'in c!it,:-itic COI'1I'LI. .. 'ction ~.H~I:w(~'c:n pl:'lI1CL gco.r::J

,·alhlt,.. r,'mll centr'"l iLxis to C€.,,,I.'·C' o( gravlly of pl.,cnd uen.l'

tl'~\n,",I""·Tod tC.l1''"<iOl")'.I.l stifi'n(:>;s

t()l"Hiona,l 1:~ti.nllCMS uf connrt':'ction betw .... 'CIl pinnp( gCi:.lrS

~''B,I( tl':Jl1s['oJ'l'cd t01'qUe:

7,;, To to)'q'll< I.j·,·lllsnlHlcd by pl.:m(,l. gOCl!"'

~: 78~ t' ,,,mnocting' 1.00·qlW

J;r~'T,. ('''t!l[loJ1(;>nt nf the tot·que' (1\1<; 1:0 tim ~\c('."I.C:l'iltiol"1 of tho pl.i.,nN

g"a,· iiw.,l1B1b bjnH."Y oific.;"rl(!,Y between ~llD g'erll· o.l1ci pb.nc:t g'.~o.'" 3:."~ a.nglc: botw(~(·,rt plnnet ;,xi,; Clnl.i c"nl.t:·a:l <lxh,

'I'A'\?B'~ tr',·,.m,fen,.:".i I·<>lilting 'HIgh: 'PA.'Pg,'P( connecting' t'otCltJ.ng ilngle

i.:\{)

2

3

4

5

\'l,.1A;, rotating angle of pl,met gea, about its own axb wt,W;,w( connecting nngulnr velocity

"'.,wt, angular velocity of planet geD.r about its own axis w",wy.w, cornp<.>n,:,nt ()f anglll(l.J:' vr;d.odty

Acceleration of a planet gear

8. L

With respect to its dynamic performance, a planet gear should not be restricted to a ciesign with parallel axes, as was cione in~. Q . .;::, The mo:;;t general e",;e will be treated here.

The torque due to the acceleration of planet geal:' a (see Fig. !!,.,.!. .. !) is

where

Fig. 8.1.1 The mom':'nts of ine,tia J .. .It, of the planet gE:!"I r ", l~nd th<:> tors~on~J shrines'> ~b of an e l" stic connection between the plnnet gea, s sepal'ateci from the transmission J'ulletiolls. The omitted shalt end 8 is similar to A

V.+7A+r,:~,::: -..(.it,-)A-.J,~"

::: -J.Wxe, -J.w.f. - J1Wyt\ - JY~j - J,w-!, - J,wA

{ (1 m.1 J,. = J. + sin2.3;, )

(1 m.rl .iy= 2)· +~)

51n •

J,::: J,.

{ w" " - W~ sin-'.

wy " 0

W, = + W~Cc)5.'1. + W.

{ i. " iiic'~K " + W~CQ5.'1iY

ty ~ wC.ey " - w2Wli.'1.!. - w(5in-iJ..e.

i', = Wc'~, =+ wCsin..9-.e1

111

fl. 1.

G

( () Th,· I."'·que,s T,e, [\nd r.CDS.J.~, <lct on the b",,~, ... l.ng~, They <lre important

for' the: tC",lcuhtion of th". l)("I.l'ing load, but h",,..,, tlK'Y can be left (l\lt or cons'i(k"Hl.iotl if the fricti.nn irl the ue<ll'ing8 ~8 ,:,,,ffidently low.

HI 1tI,:flTHICTION Tlw fd,diorl lOr'quos dll" to Tyey and r,;~os.s.f, <1.1"", ig'rlOI'-CU.

11 The t01'([ll" T,sin.J.!', (\ct~ on the pl.anCl carTier

12

This torque is cquivalent to H\E:! tOI'que caused by ,m additiO\l(\l mon,.,nt

or inerthl (!J.sini .!\ + marJ) on trw planet c<lrrier', r\<"rlc,e

13 Jc = (1J.Sin~.J. + m.r.") + (~Jbsin?,,~v mbr~ )

14 The tOI'que 7;e, pH"l.ly ads on the HUrl Ite;:,u' A via tlw planet tr<lBS­rn; Hsion a-A I pm'tiy 'HI the sun gf:'(\J:' B via tho elastic (,,,,,,,cction i)("l.wcc" the' pl(\n':"t g<:""'H and the planet tJ'ansmissioB b - B.

J B

With

lrJ

17 .-A. b-B pel"mutabh, ill pairs

HI >-A,b-B p"I:'lnutablo in pail'S

19

,equation 12 yid<ls

20

a-A,b-B permutable in pairs

Elimination of ~ f['om the two equations 20 re!julta in

11,2

~: ,!..

21

22 When the configuration of Fig. ~.l.l is replaced by that of Fig. ~. 1. ~

23

24

25

26

27

28

30

the torques tot, 78" and th,e angle" f({" ~, 'PC must remain unaffected.

;"~..j" X~~'\ ~ .... ~ I;;;t- laJA i ~A1-U j' 16: \. {~.J j 'It Tc~

lOA I?o. "--_/ \Oc 'IE Fig. 8. 1. 2 The moments of inertia 01 the planet geaJ:'s

and the torsional stiffness of their' connection transferred to the shafts. 'l'he omitted shaft end 8 is similar to A

Hence, the two equations 20, and consequently equation 21, have to be .,atisfied. The configuration of Pig. f.,!.. ~ satisfies -

~-1A =..j"f+"

~~ s..,(I'}."-~)

Equations £ and 24 yield

Substitution of~, ~, ~ in II yields

(fAi;lii ..... J" - ~..(, Hi: +

+{rw..ii./A4 - tclAq,cos".- i.,I),J.+ -1,CQ!O\ - ~e-1,)}%+

+(ratbTjblBJ~ - [b/s"",I'B" +

+{rB/l)7Ib/e-'e - 'c/sC.!.cos.).- i.p.J.+ ..!t,CO$.)b - [bleJ~)}!D8 = 0

which prove!'; to be an identity for the following values of ..j., J~, ..j", le, to which the previous equation 12 ;1,6 (lupplemented_

f -h ~ r~ilw...r.

1 J~" f~ii~""

113

(..!:il

83

34

35

B.1.

{

-4. ~ fCli • ./II.iiAIII{(~QSJ..-fO/A»)' + (COS$b-ii,;e)Jb}

--le - 'C/B'b/BflBlb{(cos.9-.- '.",,>J. + (ws.9-b - [blBUo}

·~qu"-l:i'."lH 23, 25 and ~o rel:l1)lt in

a-A,D-B pet'lnutaule ill pail's

Substitution in 20 of 32 and 26 and elimination of ~ .. yields

{'L,i1.;ASA- S.b _ ;BlAS.b }(\o;-IOAl-('AI.cos-\.ll...(. (r6lfos~.llJb

_ {r~lI>lj",.eSB -S.b _ 'AISS.I> 1 (\lli'-'lb) _ 0 ('slows"o-l»)b «(A/;P"'~ _1»),-1

I'he eq\J,~UQn$ 30, 2.!., ..!2. ,mel 34 dct",'mine the complete transfer of dynamic P"op<:!1:'li<;>!:' to the shaft.s_ A special case occurs when between the pl(l.nret grearfj '10 dafjl:k d.<;>lnent exist.s, S.b;; en • .!\--!t. .

.7. +4 = [;./II.ijAli~Qd. + 'ClS)(J,tlJ

10 +18 '" ~8f1B10(~C05\ + iClt)(J,+Jbl " (b18

{ = t(~+1JSiJ4 + m •• / + mbtt,2

EXAMPLE An epicycUc gll,,-!' t!'ain with the d0sign of Yig. 2.6.2 l>nd one planet gear' a yi.,l ds

114

-\ = 0

z.= zb- -!(ZA+ZBl

- !t. :2 '0./11. ,,- z. ~ 1 + z~

z" ('til/AI, + relB l = z. + Ze "t

ZB-ZA

hence

--boo (1+ ZX)1 J. ZB

-k' m.r.1

8.2. Dynamic stability conditions of an epicydi<; gear train

The equilibrium conditions result £);"O);Il the dyn<lmic properties of an epkyclic gear train with three rotating shaft!; and particularly from the conljequences of a slight disturbance in the rotahon.

6.1. 8.2.

1 DEFINITION An epicyclic gear h:ahl h""" an una tab 1 e t~quilib:tium if after a slight dist1lrOl>11ce the angular velocities do not regain their original values.

:3 D.E ~'INXTrON An epicy-clic gear train has a s tab 1 e e q 11 iIi b r i u In

if after a slight disturbance the angular velocities tend to regain their odgin1J.1 values.

The equilibrium will be examined for an epicyclic gem' train of which the moments of inertia and the stiffnesse» are transferred to the shafts. See Fig. 8.2.1 for the model and for the notation.

¥'y~J,. ~~

\9c~-t ~

s~ A.0 ~~' ~~~

M, \9" -t Fig. 8.2.1 EpkycUc gear train of which the mO);Ilent8

of inertia and the 8tiffnesses are transferred to the shafts. The shafts are represented by an clastic ,:,lement between two inertial elements

115

8.2.

~~ The only supposition to be made for the torque8 M". ~. My. i,8 that they o.re consbnt. Theil' mutual raUos n<:!<:!o not b,:, tho"", <,d 7,;, 'lB. t, Even their Bum n':'eo not b':' zero.

4 The equilibrium is stable if IInO onl.y if in th", model a slight disturbance eauses a sinusoidal vib).'at;i.on wHh a n<m-increll,£illg amplitude Bup",rpQ"erl on the angulDx v':'[OCi.ti<:!H. fn ~, "'"",1. defligrl sLlch a vibration will be damped, A,~8umpU'm"..!.,.§. . .!. (VI) and (VIII) give the forronl",e

5 Mo. - f1..::!. )~,po:: A-tl,B-Il,C-y pe:r'!r'lutablc in pail's

6 ""e,( pel:'Inutll.Llc

7 A-o.,B-~,C-y pel"IIl11tablf, In pairs

Completed wHh <:!q11H.H()TI 3,,~: ~

(j q1", - ~i8 + (~18-1)¥>c " 0

9

10

11

12

and the o.bbr<:!viati()!ls

iiA = it-lih

H (1 .2 - le. _2 - ..!A.)~ 1\ = + 'Alil'lB1A.Is 'j. 'A/c'lC IA Jc SA

thC!y lead via

to lh" th,..:,,', equation>; of motion

A-",8-~,C-y permutable in pairs

A,e.C pennl,lt,ablG:

A-",Ol-~,C-y pen:out ... hlQ in pairs

4 H ., (K 1)' - ~!6...L - ~_:!&,~ -H M,,_O 1 ,"u," + A- "A t(A1RS 'Jr -sHA/C5 ), "C 'A' -

15

'" B :0. C "'"

"'-~,6.~.C-y p"'L'Lnutable in pail""

TIle8" dmLlltanoous diffe~'"mtial equations ne"d n()t be solved her", It suffk"" to consicler the quantity p in the gene>'al form of the 8())'utiorl

8.2.

Thi", quantity p satisfies the equation

1 uKA-Ht.P 2

16 1 -Ke-Hgpl " 0

17

18

1.9

20

22

1 1-Kc-Hcp~

or in another forID

p6+Q;p'+o.,p2+QO =()

Q ..&..:.l KB-l Kc- 1 2= H", + He + He

Q _ KAKe-KA-Ks KsKc-Ke-Kc I - HANs + HsH~

00 - KeKBKc-KAKe-KBKc-Kc& - HAHeH(

Odd powers of p are absent, so the 1'oot8 of 12 are either imaginary or real. Because of ~e8.ch root p ha., to be imaginary or :;;erO,

thu., p2..; 0 . According to the theorem of De.::cartes, 17,con.,;'dercd as an equation of the th~1'd degree in p2 haa either one orthree negative rooh; if all coefficients have the same .,ign,

.F'or a di.,tinction between one and thrce negative roots the left-hand "ide of 17 is comdd","ed a function tn p2, "eeo Fig. ~ . .:':.. ~.

f

Fig . .!!..~. ~ The left-hand sid", of G!quation ~,~, 1-.2

117

23

24

25

8.2,

''''06

There arc thre~ negative ,'oots p2 if f L, po"iti,v.;, fc)T' on", and negative for anothel' root oi' ,',,0 by whid. both "ootQ of ,',,0 are negative. The ,'oots of ,'= 0 aL'e uc'gH,I:ive if in ''',ldHiOl1 to 1 [J

0~-3Q1 :;. 0 SubHtitut.icln of ,'= a in (27f _9p1r_3Q/') yield s

2(1 27f=(2Q~-90P2+27~)±(O~-~O/'2 for (',:0

27

Th",.." a"'e "i,th.er cHHe"ent signs Jor f in 2G by virtue vi' 25 Or' three coin c'i.ciing' "om ~ pI if

4(Q~-3Q,?:<;> (201-901Q2+2700)2

Ct)ucliti,,,, 2,) L~ i,ncluded in ~"i, The complete condition fOl' a stable ",q\lnJ,l:>d\,~ formed by le,l!), 20 and 27 I and after admission of I'OOt,; pl"O, is - ~ - -

0, = 0 admitted if

0, ,,0 admitted if 21l CONOJTION

This condition may be written O,ol'!i; 0 0 ,-;: Q,ur> , in which O,n( and O,up are functions of Q\ ·md O2 , a,~ illustt'ated in Fig, 8.2.3,

lliJ

Q2 ........... -.-----======~~=::"':'l

.-'

'" I,

o

<oJ 0.D1 ,-5 'I

t­o ! r ! I

: I

Fig. fI, 2, 3

Orn~QO <: Q",p

Qio! Q""p

., •. < 0 1

Condition for >;l,,-1>le equilibrium

when 1'1", ~, My "'''''' ('OllSt<l.Ill

29 The eJ<:ceptional limit cases in w\:dch a binary efficiency changes its valuc (condition 2.5.21) during the disturbance, belong to unfavourable 8H\H!.tionf,l with ill-defined internal power flClW. They <J.r" not considered here.

30 If the torques M,., ""Il' M., are time-dependent, supposition 1. is not true and, cOJ;l»equently, functions for M,., ~, My have to be written in the equations of motion. Apart from being more ~abodou$ this pr'ocedure would not yield different viewpoints.

s.~. E~ectrical analogue of a variator network

AS mentioned in 1. 3. 4 epicyclic gear tI'ains and nodes do not sati6£Y Ki.r<;.hhoff'slaws inth-;; co:rrlIllon inh;'I'pretation. A. special type of electrical netwol"'ke ha" to be considered to make an analogue, viz. electrical network" wHh transformers

8.2. 8.3.

1 An epicyclic gear train oan be repree',Hlted in the analogue by a tran"forrner with three coils in '3 e r i e s, while a node is represented by a transformer with three CQils in par a 11 e 1 .

Evidently, conversion leads to a converse analogue. ln table 8.3.1 the correlations for the analogue and the converse analogue are g-;:;n;~ed up.

Table 8.3. 1 Co:rr",lation!;l for the el.,ctrical analogue

analogue <;.onver:;;e analogue

angular velocity voltage CUrrent

torque current voltage

power' power power

moment of inE!rti?- capacitor inductor

redprocs.l inductor capadtor

tor!;li()"",l stiffness

epicyclic gear train traJ'l/; former, trarh,for'mer. coil" in par alle 1

-------~------COil., in .,~ries 1------- - ---

>;;on ve rs e node direct connection

(transformer, t1."andormer, pode coils in p?-rallel) coils in se ries

diI'ect connection

transmission ratio transformation ratio

diSSipative power dissipative power dissipative power in transmission in parallel ~n ,,~l'it;!!>

resistor re!>istor

119

8. :~.

2 All pal'ameters of an I:'picycu'c It"'''''' tr'ain, hinary I'atio, binary efficiency, moments of inertia, stiffn<:!"ijl:''O, aNl I.ran!Of"r'I'ed to the "hafts. The l~cmilining converse node is reaU8eri in th" analogue by a ty'an"for'meI' wilh ,nutually equal coils in seri"'13.

:3 In p1:'J.ndpl,<:!. a n,)ol:' t" r'l~ ... liscJ in the analogue l)y a transformer wi,th th)""", mutually I:'qu»l coil" in pa,~allel, whieh in practice are r<:!pl.a(,(~d by,. oir"d \:')nrH~<:lh>Ll of the branches, See J!'ig, 8.3, l. ,

ie p;o,.II.1 m .,-:::--. rlP-911 -~~~ [- --

i~ rVf,

ITl j<'i.g. £!.' ~,l Electrical al1o.logu,:, of " compor;ing chain.

To bu.i.ld an ano.log\11:' of " ",ar'ialo,- network ideal. tr'ansformers. id",,,,l i"ductors and (dl:',,1 capacitors have to be added

4 1\ transmission is repl:I:'SI:c!nted by a (r'allsfol'mer with two COUll, "

vari" Un' by a val~iilblc transformer with tw,) (:oilr;,

5 The o.no.logu", d"scrih"d above ha'; idcal capacitors, i.d",,,l inductor-:.;, and d",,,.1 lr"-usforrner:,,. A circuit with non-id"al d",ncIlts needs cDr'!'ediolls, especially to approxi.m<lt" a""umption 1. 8. \) (IV), i. "', a dissipative power should. not i.nfluence any rolatio;;- b~t;I:''''n :mgllLu'

vcl.odtics.

6 The conception" 'in p".l"il.llcl' and 'in series'"!:<)'e rl(ll configuratiom; in the I:'l<?ctri'"d ".<1,,-logue only, They ~f'e ",l,,() recognis8ble in the me<;fiH.n;'Uns of cpicyclic gear tl'ains and nod'!"" Thl" may be

elucidated by the more Olimple analogue of forces of rectHine8.r' motions. See Figs. e. 3. 2 and 8.3.3, in which the analogue of a planet gear is "'- bogie.- - - - -

Fig-. 8.3.2 Forces in series Fig. B. 3, 3 Forces in pa:rallel

8,3,

121

TERMINOLOGY

1 . [!. 1 angular velocity h<.>~ ksnelheid W;lflke 19~ I3chwinctigkeit

w rad/s

L 2. 5 COr\v~nti"n foT' the sign of an ."ngular velocity

L 8.1 tOL'que

moment van een koppel Dr'ehrnoment

1. 2.6 convention for' tile sign ot 1\ torque

1. :3. a moment of incl'tia 1.8. 1 m.",;s."tt",,-aghoid8momenl

Tr1l.g-heitsmoment

1. 2. D 1. 8. J

torsionCtl stiffness tonie sti,j fheid Drehste ifigkei t

1. 2. 7 !:lh~ft powel' 'i,8.1i aevernlOgen

W",nenldstung

T N.m

2 J kg. In

PW 2 -3

lV1L T

The product uf the dIlgul8.r velucity Of "- :;h"ft end ,,,1<1

the [(lI'qL\<l e"e,·ted <>n. that shaft end

2 - 3 L 5_ \1:\ dissipative "owel' (lost powe!') P. W ML T 1. B. 9 dissipatievermogen (verliesvermogen)

Dissipationsleistung (Verlustlei stung) A p(JW(<r' that cannot be descI'ibod as a shaft powor

1. G. 6 tran!5IT1i6Skol1 ratio (J v«r'!Jrengv e rhouding libel' set "Ullg sver 111n tnis

1. 13.7 efficiency rendement Wirkungsgracl

1. 5. 5 binary ratio Z. T. 2" binaire overbrengverhouding

StandUber setzung sverh!Utni s

1. 5.14 binaJ;"Y efficiency 2.2."4 binair rem;ltiment

Standwi r kungsgr ad

The staH()Ilary shaft end is not necessarily the planet carrier

Other ratio6

1. 5. 8 ternary ratio ternair", overbrengverhouding LaufOber$etzungsverM.ltnil:>

.!: ~.l2. ternary efficiency 5.2. 1 tcrnair rendeJ;Ilent

Laufwirkungsgrad,

5. 1. 1

The shaft ends are a,bHr'ary shaft ends

rcsponsivity re oponsi teit Ite "ponsi HIt

Shaft ends A, S, r·espectively C, D, a,e shaft ends Of the reticulatm" "'.nel a variator

3. 3. 10 efficiency function eff(~/zl= ii'ZIA - - ~ rendementsfuncHe

Wirkungsgradfunktion

Energy flow

1. 7. 1

1. 7. 3 () 1. 7.4

L 7. 5

'l'he conjugaHon of !;l, lernary dficiency to the corresponding ter'nary ratio

powee flow veI'mogensstroom Lei$hmgsfiuss

blind power blind vermogen Blindleistung

Di"-g'rammatie representanon of the input and output of energy

Smallest power along a closed path of cydically equally dh:ected powers

branched power flow vert",lcte vtirmogensstroom ver~weigter Le~$tungsfiuss

A. power flow without a blind power

Concerns a tit1,latton in which self-locking zelireJ;Ilwing Selbsthemmung

no shaft power can be withdrawn

123

Elements

1. 2. 1 I.s.T

1. 2. 0 1. 8. 1

1. 2. !)

1. B. 1

L 2.1 1. 8. 1

L 6. 2 1. fl. 5

I--

-G

~;;-;;;;;;;=-

shaft end <l.seind Achsenende

illet'tial el<>ment rote rend lichaam I)n:'hungsk~rper

elastic element elastisch ded elastiseher Teil

!'otating ~haft draaionde as dr'Oh8rlde i\ehse

TerminD.l part of an element, only characterised by an angular velocity and [( torque

Element characterised by moment of inert.i,a

Element characterised by tor 6i onal stiffne B s

A unit with two shaft ends, oonsidered as a sequence of inertial and elastic element8

(fixed) transmission (va!;!:",) uvcrbrcnglng

(fostes) Gotricbo

A unit with two shaft eDd.~ having a fi:;.;:ed ratio for angular velocities and a fixed ratio for torques

1. D, 5 1. B. 6

--</:>--< variD.tor A transmission of whkh the re.tio between angUlar velocities can be changed continuously by arbitrD.ry interv<'!nl:l()r\ illrl"'pcndent of the r;H1)athm

3. 1. 1

3,2,3

3. <:.4

1. 3, 1 1.8,2

1,1,2 1.4.1 1. 3. ~l

124

~} :;:}

:+}

A

A

V"al'iator sture nlo se s G<:>td<.'l)e

br~.n<.:h

tal<-Zweig

hard branch hat'de ta" hax'ter' Zwe~g

!:'(lIt br'an<;:h z[(chte t<l." weioher Zweig

thi'ee-pole d 1"'1<='pc)(Jl Dreipol

node Imoop Knot .. "

A "'\ltaling >;h",n, "- tI'aIlsmission, ur' a var'iatoY'

A branch withQut variatol"'

A variD.tor

A unit with th,<:><:> Bhaft end"

A three - pole foY' which the 8.llg'uhr' velucities ar'C equD.l and the Stun oj" torques 16 z<:>ro

3.4.5 converse node conver se knoop konVeI'ser I'illoten

An imaginary element with three shaft ends fOr' which the sum of angub.r velocities i6 ~ero and the torques are mutually equal

l. 1. 1 1. 5.1 1. 8.4

epicyclic gear train planeetddjrwer k Planetengetriebe

A three-pole for which the angular velocities satisfy a Unear equation, permitting three equal ~rhitrary angular velocities, and for which the t(H'ques have fbl:ed mutual ratio ..

Design of an epicydk gear train

2. 6. I single epicyclic gear tr ain enkelvoudig planeetsteh,,,l cinfache6 Planetengetdebe

An epicyclic gear train with three and no more than three co-axial $hafts

2.6. I sun gear zonncwiel Sonnenrad

~. Q. 1. planet gear plancetwie\ Phnetenrl3.d

2. 6. 1 p\l3.net carrier pll3.neetdrager Steg

!.. 2,.,!, planet group planeetgroep Planetengruppe

A gear on on€ of the c;o - axial shafts

A gear meshing a sun gear. The 6\.lffix of a planet gear is a small letter corre,>ponding with the capital lette," suffb<: of the sun gear

.An element on one of the co-axial shafts in which the planet gears are incorporated. In the syx;nbolic I'cpresentathn a planet carrier may be indicated by a line extended in the circle

'X'wo planet gears, m",.,hing different ,>un gea1"", with their mutual transmis.,~on in the planet car·rier

~ • .§.. 2. degeneration of a sun gear on-taaI'ding'V= een- zonnewiel Entart1.ln-g eines Sonnen,ade ..

The replacement of I;L ,>un gear by an-y coupling between l3. planet gear and a <;o-a>l:ial shaft

2.8.1 sun ratio 2oonverhouding Standtlbersetzung

~. ~.1 sun efficiency zonrendement WlUzWit'kungsg,ad

The gear t'atio of the Bun gears if the planet carrier doe s not rotate

The efficiency of the power flow between the sun gears if the planet carrier does !;lot rotate

125

2 _ 3, 2 in_tc l'n':-l.l pC.)W~Y'

inw(!lldig v,'''rY1og':'n inncr'c L0.io;tl.lng

2.7.2 planet powCr'

plancctvc .'r~l"g"n W<1.lzlC,i;;l.ung'

2,7,3 cfl.rt'ier pOWCr'

dr-ilgcr- vC r'"f.I"H)g\:r)

Kupplung'fildfil:tlng'

The product of a torqu,~ 'lX<.lT't,:,<:l on a silo-it end and the diffcr'N1C\~ in ~ngular velocities of that shaft end with ~.noth<:'l'

Th .. pr'c)duct of (1. tOl'que exerted on a ",Jll ge ar and the difference in angular v,:,locities of that sun gear with thaI. of th" pll~n.et carrier

Thc p,'odud of the torq\i':' ""x,,,--t,,d QI1 a ,,;un g''''~r' ~l1d the anguLu' velocity of the pi" rH·!t c!lrrier

7, S, 1 dOLLblc cpi('ydic gca., I.r'ain

dul)bel p1Qneetclrijfwert ))oppe1pl<1D<.'t':'ngetriebe

Two flingh, ('picyclh, gcar' I:r'aLrli;

interconnected by two nodes

7. 5, G Wolfrom -trQtlsmission woUeon1 ddjfwe)- k

l)oub1e epicyclic gear t)-n.in with Q C01r1D"H)', plan':'t ,-,,,r',').<:,): j,n wh~ch planet gear::; fn a plan"l g'l"DUfl

have a CoLl:1rnon shaft W 01 fromg etde be

V~_t['ia1.or [lL'twork

1, 1. J variator netwO)'I, 1, 8, 7 v:u'iatornetw"r' k ::I. 1.::l VaL'iatnrennctz

A. cohe1:",l1t "'Yf:lt",m of thl" .... -pol"" ()f which the shaft ends, ",xc':'pt two of th«"" m'", interconnected by brQl1ches

1. 1,3 inpul shan, oulpLLt shaft :;. 1, -:I- illgaan,j(, "", ullg,\iUltk a" networl{ which does not connect

1':h,l:r'ill.>;w(.!llcl, !\.u,;l:dtt,;welle two three-poles

3.2. S HLlbnc:twol'\(

:; llbnctwer I, 'l'eilnet<;

A cohel'ent pal't ,.>\ :1 var'b.l.oJ:' network wtth :'It L""Ht Oil<: thr'cc-p()le

II. G, 2 compl.ern<:,nt",,..y Hubn(!l.wor-],,,

(":C)n-lpl.p.rnf:ntair'c:. cicclrlctwcl'kcn kon1p l.~rn~-nlt~,l't.~ '.1\:"'ilnc t~c

Subnetworj{.~, on':> with th", input: shalt, the oth,:,,., with th" output sllD.ft, together' ~ndll,Hng all three -pole s one':'

4. G. 2

4.5.3 4, G, 4

iJ., 1. 1

PQrQl.Lel br;mche" flQt'QHelle takken Pa.r-clllcbwcigc

",(,!tJ.,'lllal:o,·

)'ettc'L1 Ld.o,· Itehk"latOl'

Th,~ br'il.I1Chcs which intel'connect Cornpl.8rnenUll'y sUDnc:twol'ks

.An imQgimu'y tr"n5u,j~b~'.H\ betwcen thc inp ut shaft and the 0\\ tpllt "h,,-ft

4. L 2 closed network gesloten rtetwerk geschlossene$ Netz

4.2.1 mesh 4.2.4 maas

Masche

4. 2. 2 hard mesh 4.2.6 harde maas

harte Masche

4.2.3 Boft mesh zachte maa5 weiche Masche

5.3.1 backbone chain ruggegr aat~ keten R Uckg,at"ke tte

A variator network complemented with the r'eticulator

A collection of branche'>, irlterconnected by the same num.ber of three-poles, forming a. clo"ed figure

A mesh including no variato" but possibly the reticulator

A mesh with one or more variato!""

An equal numbe):' of node" and converse nodes, connected to each other' in one line, including neither,. transmission nor a mesh.

3. 4. 1 <; (HIve I' sian <;onveI'sie Konversion

The ,"'placement of a va!"iator rtetworl( by a sim:i.\ar one fo):' whJch between the angul"r velocities the same equations are operati.ve ar; originally between the tox'ques, and, lil{ewise, between the torques the "arne equations are operative as originally between the angUlar velodt"""

'I. 3. ~ reiterative network herhalingsnetwerk Wicderholungsnetz

4.1. I

4.5.1

incon$istent netwo!"k strijdi.g netwcrk streitig",':l Netz

rcducible network r'eciuceerbaar netwerk t'eduzierbares Netz

4, 5. 2 pl'irniti ve subnetwor]{ primitief deelnetwer]{ primitives Teilnet::.

A variator network in which a subnetwork is; a variator netwo)-k in itsclf, whether 0)"

not following intl:'r<;hange of adjacent epicyclic ge(1!' trains or interchange <:>f adjacent no.des

A variatoI' network in which the angulal" velocities Qj" the turques of any subnetwo]"k do not havl:' Olle and only one value for each p!"esumed angular velocity or pre6urned torque iII the variator network

A network of which a subni?twOl"k ean be replaced by one wHh t<i!wer three-poles

A subnl:'twork without soft branches and without the l'eticulator, which is not equivalent to a 5ubnetwQ);'!, with fcwer three-poles

127

S, S,:l var'iflblC! "hunt shunlJ,-l.ifwcl-k tiue ,-lagor-ung ~ga tdobo

S, ~. II variable bridge brugdd,j fw!:'rk t-l,-'Uck<:mg'!:'td!:'b",

5. O. 2 double v,ldnble -~hunt dub bel shuntdri,ifwerk

A variator networl{ with one VH.I'iH.toT' and or'lC (~plcycli(: glc:::tr train

A variatll!' rl8twod< with olle varia tor and two epicyclic gear trains

A var'iatol· n~two,'k with two var-iators and two epicyclic gear

Doppolllbol'lagel'ungsgotl'iobe I:r'ain!i

5, G, 6 d"lll,l" var';,a!:dt, bridg" dubbcl bnlgdd,ifwed,

DOI'Polucllckongctriebc

A var-i"-lor network wHh two variator;;; and three epicyclic gea.r trainH

5 _ G. 7 extended double v,,-ri,,-ble bridge L1itgebrci,] dubb81 brugdrijlwcrk (! r'wdte dc,s D')l'pclbrttckcngctl'iebe

7.4,1 r'cvcn;c-"ynllllotl'ic vm'iable bddge k" ",r' !-;ymnw t l'iseh [n'L1gdrijlwork

A volr'iator ll~tw(H'k,wil:h two vad,;l.tC)r!:l ,,-nd {(>lH'

epicyd~c genT' tr,~,)n!:l

oeykli "d,- "ymmc td" c he s Brttcl{cngetl'iebe

128

1\ variable bridge with equl'll lHu'arr",U,r" ()f the epicyclic gear trains cyclically dir-ected in the har'd mesh

SUMMARY

Thi,:; study deals with the tran,:;mission of PQwe," by means of 1;,)me COIrlbination of ~p1cyclic gear' trains and variator6, which as Il. whole ;:;tete a$ a variator. In SLlch combinations, called 'val' i a tor net W 0 r I~ is' , 'sets of three rotating shaft ends lire intereonnec;ted by epicyclic gear trains and by fixed int',H'connectivns, called 'n <) des' (1. 1. ). The e stablis\mlent of the netw<)rk theory $tarted with the admission that nodes are as sigflHicant as epicycli<: gear trains.

The properties of epicyclic gear trains as well as of nodes are governed by equation", between aD g u 1 a r vel 0 cit i e s and equations between tor q 1.1 e s. WhHe in other network theories thQse r'dations ar" determined by J(~rchhoff's law!;, in the variator nctwork theory the equations for epicyclic gear trains have to be distinguished from those fo,' node s (L 3. ) _ Therefore, rlO correspondences could be found to othe r specialised-network theories, and a complete new theory had til be established. For' analysing the structure and the power tran6mitting propo!!rties of va.r'iator networks a mat h e IT> at i cal mod e '1 h<,-s been developed, bascd upon linear equations betw~ell angular velocities and between torques. To specify these equations, a few surprisingly simple aHLLmption" have to be made (1:!!'- ). By virtue of <::onsistent conventions fClr' the signs of angular velociHe$, torques, afld other' q,uantities, the establishment and the ovaluaHon of the m,,,-thematical model p'rovel; to be very lucid_ In some re!3peC!t, the equatioIl':> mentioned above ar'e a substitute a" well as <1.11- extension of the commOll Kirchhoff'" law,:;_

It proves possible to intercltange epioyclic gea~' tr-ains and nodes in a variator netwOr' k. Such an interChange, oalled 'c 0 II V e r s ion', is defined as the i.nterohange of angular- v'elooiUes and t()r'que:;;. A convel"tcd variator networl{ is as significant in both theory and deSign as the original Oloe (l- i·)· The converse of a node i8 represented by a 'c 0 n v (0 r sen 0 de', being a fietitious epicyclic gear trai.n because of the spedal form of the relative equations. An actual epicyclic gear- train may be reprcsented with the aid of a conver-so node (3. *.), which is a useful tool in the theory, for' instance as a constit~e-;t par't of what i8 her'e called the 'b a c k bon e chain' (5.3,). The latter is used for' the oonvenient deduction of formulae fur-a given variator network.

Special attention is given to me she s in a variatc)" network (4. 2. alld

4.3.). and to subnetwork" (4,4. and4.5,). Illdningso, oertain kinds of networks prove to be inconsistent-or red\1cible and have, therefore, been e:.:cluded. l"or i.netance, a variato. network in which the number of epicyclic gear trains exceeds twice the number of variatol's, is reducible to one with less epicyclic gear trains (~.'~.').

129

'l'llt' pf'e:;ent outline of variator network,:; 1n confined to thone with <)[\(: ,-,r twr) V,~l"j~t()r,; (5, 7,). Tho~e with one variato!'. calloel 'val' i a b 1 e l'; h Ll ilL '

(ji.). ni1e! 'vtll'i.flhle bricige' (2,.). al'eelbcu:,;"edcxton"ivcl,Y. Thonc

will'! I.w" v''''i''tor'~ "'''e di.~C\\~sed only conci:>cly (:2,.Q,.).

D i " ~ i p" I. i v., P" w" Y·.S flY'e consistently conflidereel in all of the case ~tudie:-:: pcri'Ol:'1"I.1ClL To cach UIl.it with H. l r ;:1. n ~ In i H :.-:; ion r' ;'1. t i () ;'1,.rl

<:: r fj C' inC' y 1:,; conjLlg,.,tcd. When in "- vd.r·i",to,· Llclwodc " ,"ali" b(:lwC<:n

two angular' vclodtien in given as a fundio" or 1:l"al'l.~miH,do" l"f\tio", th~n

the- ratio bctwC(:rl lh4..:: cor'r·t~~pond.ing t()r·q1..1~::: (,::1n t-.~ w'!'"i,tt~n J.S ~ siD1iJ.~r

function of tr "nslnission n3.tio" and cfflci':'fl<:i(,,, (:1. :1.). C""",<'qu<ln I:ly, I:h .. cffidcncy of " va.l'lal:ol" ""hv()~'k ~6 ciet<:'1.'mineci Cls~s;on ClS thc rolation8hil' l.J(:1.w(:<:n :tnglJ'lru' V(,docities is ){llOwn.

!\ rn',!,1>;(1l'e of the iniluenoc 01' a slight virtud.l variation of ttw tl.'(1J1.S1:nj,88iOD

"·nti.o (,f D.ny in<iiviclual vadalo," 011 the trallsmis"ioll ""tio of tl1<.' VClri::ltor

networl, is its' r c:;; p on s i v i t,Y' (Q. . .!,.). Th" '·"'"pon!:iivUy 8t<mcl!3 for ",,)r.,! th:111 thi .. s m':.!flsure only, sinoe it has proved Lo be cl(od~iv,~ fOT' the cl i " I. l"i hili. i () n () f P () W" ,. in D. V::I"j,o.tor networl~ (1.~.).

An opicyclic goat' train in a variatoI' network may be ""PT'(,,,,,nted by a (".mv,-"·"e 110de .• lll1.ving no tl'Clnsmission j'atio r\(W cffiekllCY. Tho" par·an""!;.,,.,,, are tr·an.~j\!)Teci to units of the connecting ,:;hafts. It is p!'"vcd Lhal. "l,;() in" r·jj" I nJolYl<'nt" and 8tifl'nesses rn.l1.y be rcplClced by Imnpeel paramoter''; .U;S1plC:d to th., nhaft" (§.. 2..). 'l'h~6 f"dHtat(,F.I the clecluction of r'(:lilliollnhil'n d,ol:","mining I;h.·, dyna.mi,c performD.nce of ,=,picyclic geClt' 1.1',·,1n:-; (I>. ~. ).

LW

REFERENCES

Thl'> history of epicyclic gl'>ar trains was descdbed by Fllrster ['1]' and earlier by Wilson [2]. WolfrQ~n made the first pUblication about a double epicyclic geal' train with a common planet carrier, later caUed 'WolfroIn-tr'aIlsmission' (7.5.6). lIe I'eferred to the design by the tcrm "Planetcnd\.dergetdebe mit W11lzradantrieb" (epicyclic gear train with impelled planet gear), and in an examplp. he calculated a total efficiency of 0.55 for such a transmission with an overall ratio of 100 and a sun efficiency of O. 975. Terplim [4] made an interesting summary of th" calculation ml::thods of Kubbach, Willis, Swamp, Balogh and 8,,6ke, Poppinga, and Kosevnil{ov.

Merritt [5 J developed a use fut theory independent of the actual d",,:dgn, while Poppinga [6]. on the contrary, made tables with 50 cases (of which 8 were double-counted) transformed to unwOI'kable designs. Wolf [7J restored th", independence of the actual deBign, although his theory was not consistent. 'fhe ,;ymbol in Fig. 1. 5.1 and ","v;,,,al formulae in 2. 1. and 2.2. are siroHa, t(l those of WolL-Section 8. 1. is an extcIlsion-ofa part-of the wori{ of Str(5mblad [8), while sectioii-:-5. i~ an extension of Clr·nhagens theory [91. Kudr'javcev [10] eollectedfl{i"ssian knowledge on the subject.

Literature about te6Hng and about design is left out of <;Qn$ider'ation, with the el!ception of four references. Neussel[ll] tested a oO\1bl .. epicyclic gear train of a !3pecial dl'>sign. Meyer zur Capellen [12] proved the Wankel-en/l.'ine to be 8. special case of an epicyclic gear train (Fig. 2.6.8). Tuplin [13J and Hill [14] calculated the number of pland gr'oups (~.fi:--

The acceleration of epicyclic gear trains was studied by Tank r 15] with formulae reminisl)ent of the more complete eq\1ati<.m$ 2.5. 11 to 2.5. 1:) inclusive. The deductioIl of equilibrium conditions for epicyclic gear­trains (f.~. ) was inspired by a colloquium held by dl'> Beer [15J concerning a worm gear transmission. Denton [17] reported on an unfortunate applicaHon of an epicyclic gear train in combination with worm gear drives, but his conclusions were premature.

Rivin [18J proposed simpliHcl:l.HoIl!' in computing schemes of the dynamics of rotating shafts, while Zinov'iev and Umnov [19J warned against intolerable simplificatiom; in the equ"tiorl of motion for a mechanical system with a variator.

The oldest design Qf a variable transmission with split pOwer paths was patented in 1896 [20]. H concerns a hydraulic tran!1lmi!1lsion of whieh the common casing of pump and motor rotates and act8 a8 output shaft. Modern de"igns weI'", published by Sadler [22J, Himmler [23], and others. Since a single epicyclic gear train and a variable "h~nt have many

131

pl'incipl~s in common it is diHic,llt to ~ilY th~t ~llch d,~~igtj" ,;r-C

r.xdLH:;ivcly c:quivillcnt to either epicyclic gear h'ili.ns (Jr' v~,dab\" shunts. )J()w(,v,>r', a lI'min ch;H-acte:ristic in modern designs is an extension (If th(, slwft of the pump to th,; motor'. Tfw extended shaft hils the ilngular veloeity 01' the input shJ.ft, the pump h-an6fonn", th",~.')r·qll(' ('nodc plus tl'illlsmission'), The extend~d ~haft hafl the tin'que of the outPltt ,shaft, the motor h-andorrn'5 the ,'ng\)l~.T' vt,]odty ('q)i<::yelie geal- tI'ain'), Therefore, s\)ch V~\Y'i."l.ll" hydr'aulic transmiiOsions may be called 'shunt varjat,H'>;'.

The: que,stion whcn and whe).-e vadable "h\JI1tH ~.nd var-l>.blc bddges wer~ invented, will not b~' ~.n'lW(""'c.l h(,,'(, while references of the motor-cm' industr'y hav'c not been given_ A.ctu~l desig'l!" (,[ v".dable shunts were r'q)(ll'(ed by Wood l21J, Hic/{s 1271, Wahl [28], Jarchow ["fJ) , ,(fld Ol,k"".a[\ 1.:14). One may sUl'mise, that when M,~,-,mill".r\ [~('l mentioned the dh,-,ovr,,"y by D:1Vie" of a variable bridge with thi:! v",d".tol' bctwe0n the node~, In.trlill \,'r' 12.) I dIso knew sllch a design, although his di~.g"H.m wa" incon-ect rlllcl h,c<'JY1!;iHt(,nt with hi>; text. Actual design8 of vadl:l.bh, br'idgcs were de~cdbed by W(,!:'HlUY'Y, Smith and Glilze 1.2:01 anel by Wilson l:wl· Since about HHll p1.1blic(\UoY1~ ()n val'iable shunts and variable bX'idgeij h"v(, apf'ea,~ed l'~gulJ.r·ly, l24J to f.17J in(;luHive, commonly with elementar'y l.hco,'etical considerationI:', Th!:! 'Car'rlot-theorem' of French r:jfi1 way b" hupcdlt.lOUS by the intr'oci,lc;tion of "r-e>;pollsivity' (2 . .!.,). . A mathematiea} ba8is f<.ll"' nf'tw()!-k th""r-l"'; !Hay be fuumi ill >;everal books, fol' instance Ore r:18].l:)~J, S,,~hu and ne0d [40]. From the many bo()ks dealing with electricJ.l n",tW()~-1<:8 Cauey' (4lJ may be ,-dcrl'ed here, An interesting book; coveX'ing many other applicationB is Blaelnr(lell [42].

r1] Y5r.ster, l-\, J.: Zu,"' El\I:wkklung dar' Planetengetriebe H abi lHatit)n" H,~hrift

Ext,-ads in Automobil-Industrie 11(lf)66)4, 413-53 (8 r~f,) 12(19()'I)l, 37-44

l21 Wiboll, W. G,: Epicyclic geari.ng PI'OC, of the Inst, of J\.utunlobilc Engineers

~(1931/32) 21G-257

[~J WolfJ:'OlIl, U.: De,' WiI"kungsgr~d von Planetendld"rg"trlebcn W(:r-l,>;lat\Btechnik ~(U1l2)12, G15-617

[4] '!''',-pUi.ll, Z.: V"rschicdcnc Methoden f(\J":' die an(o.lyt~,,(;he Untcr­HLH:hung' d",,' cillfachsten Planetendlder'getdeb'" L\eta Tochn. Hung. 49(1964) 437-4S1 (17 !-efCl'cnees)

[~l Merritt, H.. I;:,: Gear- I. pain 0;

Pitman and Son!:', London, 1947

[61 Poppinga, n.: Stil'nrad Planetengetriebe Franck'sche Vedagshandlung, Stuttgart, 194\l (222 I'd, )

112

(7] Wolf, A.; Die Grundgesetze del' O;mllilufgetrieb~ SchriftEmreihe Antrie b$technik Heft 14 Vieweg \lnd Sohn, Braun$<::hwElig, 1958 (5 references)

[$] Strtlmblad, J.: Beschleunigungsver!a\1f und Gleichgewichtsdrehzahlen einfacher Planetengetriebe neb .. t Selbsthemmungsversuche Chalmers Univ. of Techn. Nr. 226, GNebor'g, 1960 (5 ref.)

(9] Qrnhagen, L.: On self-locking transmission .. Chalmers Univ. of Techn. Nr. 279, GNeborg, 1963 (10 ref.J

[lOJ Kudrjavcev, V. N.: Planetarnye pe);'edl;l.l$i ('EpicycHc gear trains') lsd. Ma~inostroenie, Moskva, 1966 (114 references)

[11] NElussel, P.: Untersuchung von rtlckkerenden Umlaufgetrieben mit und ohne SelbsthemIPung unter besonderer BerUcksichtigung von Koppelgetrieben Dissertation, Darmstadt, 1962

[12] Meyer Z1H' Capellen, W.: Kinematik de,.. UmlaufrlJ.dergetriebe und ih,e Anwendung a.uf den Wankelmotor Industr'ie-Anzeiger 83(1961)51, 1089-io92 (3 references)

[13] Tuplin, W. A.: Don't overlook the basic epicyclic gear MachlMry London, (1961)6, 122-126

[14] Hill, F. Einbaubedingungen bei Planetengetrieben Konstruktion ~(1967)10, 393-394 (2 references)

[15] l'ank, G.: Untersuchl,mg von Beschleunigungs- und VeI'zl'lgerul1gs­vorglmgen an Planetengetdeben VDI-Z ~(1954)10, 305~308

[16] de Beer, C.: lr.l6tabiliteit bij zelfrernmende woro7:fl- !in wormwieln ove rbroengingen Unpublished re?ort, Univ. of Techn. Eindhoven, 1960

[17] Denton, A. A.: Some disadvantages of the epicyclic gear' as a speed range amplifier Engineering Materials and Design (962) 577~57S (l r,:,L)

(18J Rivin, E. I.: lteducing the number of degrees of freedom in computing 6cheme .. RU6!;;ian Eng. Jou);'nal XLVI, 5, 39·43 (1 ,eference)

[19] Zinov'iev, V. A. and N. V. Omnov: The equatiml of motion for a mechanical SY6tl;!m with a variator JOurnal Mechani6m", Vol. 3, Pergamon Press, (1967),

269·274 (4 reference5)

133

[20J llall,.T. W.: Improvements in or relating to v(l.I"'·La.ble "peed appaI'atus for trnnsmitting powers nr-lH"h Patent Spec. Nr. 7479, (1890)

[21J WOI.,d, n, .I,; TIydl"aulic differential drive Machine DCllign ~(lY50)4, 129-1:32, 2313

[221 SCIcli,·,,·, C. L,: Differential type hydrosto.tic t!'an!:1rni,!:1,,~c'>flH

7 ttl Nat. Coll!. Industrinl HydrauUcs (1 flo 1) '6', 41-50

l2:l] IIimmlcr, C. I·e: Ce'3 vCl,rj,:'!h"IT'" de vil,~s"c moJertles LAl 'I'"chniquc MuJt,,'no 48( 1956)3, 186-1 \l5

[:H] Ma.<:miU"n, n. I·!.: Power flow o.nd 10,,'l in difter""I(ial. mechanisms ,jotH'""l Mcdl. Eng. Science !(I%1) 1. :17-41 (4,'p.fm'ence,;)

L:t.:'») Westblu'y, H .• C. P, Smith and S. G. Glaze: A doub1c-dHfeI'"nti",1. hydrostntie conrotant- speed "Her'nato,· dl:'ivo Pnlc. ConL Oil Hydrnulie Pow,>r Transmission and Co"!:,',,l (1961)'5', 50-61. Inst. lVIeeh, Eng. London (5 .<"eL)

Uti] I\!1C\ClnJ.I.I.'\tl, H. n.: Communication to 25, dHhl, Wl

[,~7] Hicks .. H. ,y.; v"d,,,,.t>i,,-r',,.tio lr';;.n"mission of high POW,":' "'ling "picydiu gears Atkn Engineering Review (1 fJ()~)43, 19-26

128] Wahl, C. C.: Gc-h'iobe Hi." g1'(>f:'~" L«i"lullgCll und hohc Dl'ehzo.hl<:,n mit vilt'iablel' Sel"mclit,.<l,.eh>:Clhl Alltriebsteehn~k :l(U164)3, 98-100 (r-efers to 27)

[20] ,fa.r'ch<>w, F.: Lc:istungsverzweigt", G",[ricbe VDI-Z 10u(l(lf.H:l6. 196-205 (3 ['"foI'oncos)

l30J Wi 1:-; <>Il, W. E.: Predicting th", ywrfo["mance of 0. hyclrodiff,er,,,,lial tl'o.nsmission Hydraulics and Pneumatic,:; .:t..2(1964) 11, 80-U5 (:~ ,.d.)

l31J M""millarl, n. H. o.ncl P. H. n"vi.;!H: Analytical study of <>y<>tem" fo]' bifu)'e",ted pow",. tr'an"'tli""ioll Journo.1 Meen. h;ng. fk:ic-ncc .:U1965)1, +'0-4'"/ (:;:0 r'~'f.)

[32J Gackstett€'l', G.: A.,,",wahl von Planetengetrieb'~n 7.1Jr' LCistungs­verzw<:,j,g1mg fu" negclgetriebe Konstruktion ..0.(l965)9, 349-355

[~3J Gackstetter, G.: Leistungsverzweigung bei der stuienlosen Drehzahlregelung mit vierwelligen Planetenget:ri.eben VDI-Z 108(1966)6, 210-214 (4 referonces)

t34

[34] Olderaan, W. F. T. C.: Rademakers~stoecki.(:ht hydrostatic variable ratio eph;:yclic gears Con!. A",s. TerIDotecnica Italian",. Se:!:. Lombarda (1966) Radelllll-kors, RotterdW:rl. 1966

[35] Wbite, G.: :Properties of differential transmissions The Engineer 224(1967)5818, 105-111 (6 refenmce!;l)

[36] FI'ench. M .. J.: A Carnot theorem for split torque variable spe"d gears Journal Mech. Eng. Science 10(1968)2, 198-201 (4 reL)

[37] 13eckmann, K.: Wirkungsgrad und Stellboreich von Planetengetrieben mit Uberlagerungszweig in Abh!lngigkeit VOID VerhlUtnis del' Uberlagerungsleistung ZUr Ge!>amtleistung Konstruktion 20(1968)9, 358-364 (29 referenceQ)

(38] Ore, 0.: Graphs and their uses Random House, New York, 1963 (9 refcI'ences)

[39] Ore. 0.: Theory of graphs American Mathematical Society, 1962

[40] Seshu, S. a.nd M, B. Reed: Linear graphs and electrical network!> Addison-Wesley, Reading (Mass.). 1961 (200 references)

[41l Cauer. W.: Synthesis of linear communication networlu; McGrawhill, New York, 1958

(42] Blackwell, W. A.: Mathematical modeling of physical networks Macmillan, New York, 1968

135

No. h<:'t d''>(H'l"p<:!'' V,on ('en lagere school te Voorburg bezocht ik de Gern<:'!,:mt<:'!Ujk" HBS tc Delft, wll.al' ik in 1943 het diploma HBS-B behaal<le Gedurende de hat'lte ooT'log;;jarc.r> volgdc. ik cen wis]mnde-opleiding bij prof ... h'. F, Schull. V~.r\ 1945 tot 1950 studeerde il{ werktuigbouwkun(lc. aan de Tn Delft. rn 1946 bc.haaldc. ik de MO-alcte Kl wisJrunde, aangev\lld met akt':' Q. G<:'!dur·"'nd,. ar\(lel'lt<l.lf j;;"1' was ik assistent bij de afdeling Wiskl,nd<:'! lIan de T If Dc'lft.

In rnijn lnat.~te !;ltl](H<:'!jaar' was lk wcrkzaam bij Van der Heem NV te '''I Gravenhage, "'n na mijr> militait'C dicnsttijd nag een kart", tijd bij d .. ""l£<1", NV 1n de UtI'cchtse fabriek. Van J 953 tot 1959 heb 1k mij bij Con,'"d Stol'l{ NV te Haat'l",m bedggehouden met theorie en pralctijk van tandwielen. In 1 !l5fl en 1960 veI'richne ik SpcL\l'werk op het gebied van autoDlaUf:\ch Wl:'rk"'nd<:! wrijvingsoverh,'cngingen bij Aug. bieren8 en Zn. te Tilb\\rg. Sindf:\ 1960 ben ik verbonden aan de Technische Hoge'>chool Eindhoven, thans nl'l w"'ten8c;ha.ppelijk hoofdmedewerl{er,

STELLINGEN

1. Bij de bewcrking van een inwendig cvolvent tandwiel ontl;ltaa.n geen valse ingrijpingsveX'!3cMjnllelen, indian djn topkrQ~t"'$tr ... ,",.l groter is dan de topkromtestraal van het steel{wiel. Is daaX'entegen de topkromtestraal van het Olteekwiel de grootste, dan ontstaat aH~jd 10 V e r s n i j d in g I, nog voordat f!r sprakc is van echte afwikkeUng. Door voldoend (Hep ~n!jnijden kan de".;: oversnijding woX'den overwonnen.

Polde).". ,f. W.: Over-cut, a new theory for tip interfeX'ence in internal gears (binnenkort te public",ren)

2. Voor een genormaliseerd vertandingssysteen zijn een betrekkeUjk klein aantal onclerling ortafhankclijkc geometrische gX'ooth.eclen aan te wijz.en, die geza.,Irllim1ijk een tandwicloverbrenging bepalen en d~e in I;;!en belastbaaX'he~d!'lberekening elk in slechts cen factor optreden, De berekeningsmcthoden kunnen daaX'om zo worden inger"icht, dat zander moeite een opt:i.male vormgeving wDrdt bercikt.

Voorlicht~ng!:lb1aden v(lOI' de Meta.alindustrie, verzorgcl door Werkgroep 'X'andwi.elen van de F ME: VM 09·03 Geornetrie Van de. uitwendige overbrenging, VM 09-04 Beh"tbaarheid van metalcn tandwielen. Voorstellen van commissie 5 van he t N N I in werkgroep 6 van ISO I T C 60.

3. Voor een eenpa):'~g :r-ote:r-ende beweging dient Hz (hertz) te worden gek:ozen a1s eenheid Van omwentelingsfrequentie ('toerental ').

4. Er is geen reclen om het oegrip 'specifieke glijding' 9 te handhaven. W"ar dat gewenst is kan beter het complement (1_ g) worden gebruikt, dat 'meesleeprendement' of 'sne lheio!3);",!TIdement' ZDU kunnen worden genoemd.

5. De uitdrukldng i~ voot' een gere(!uCeerd ma.ssa.traagheidsmoment is onjuist ( i overbX'engveJ:'ho\ld~ng, ) ma':;satraaghoidsmoment).

6. Het ontwerpen var\ eerl over'orenging$systeem met een vertakte vermogensstroom zonder het b~j de berekerling in acllt nemen van aIle dissipatievermogen!3 b, onve)"antw<)(lI'd.

7. De theoretische ond~rzoekingen van Poppinga aan dubbele planeet. ddjfwerken leiden niet tot eell bruikbaar criterium voor de keuze van zulke planeetdrijfwerken.

Popphlga, n.: StiI'rlI'ad-Planctengetriebe Frarlckh',:;che Verlagshandlung, Stuttgart, 1949

8. De lkh.a",~rll:;vur-m van kniptorren (Elateridae) mag ",1" uptima"l wOI'dE'n beschouwd V()OJ:" het effect van de 'sprong' of de lmipp<,ncic lJewcging die 7.ij maken indien zij in n(l()d vQr'ker-en. Dc nallW verwante goudgerande water-kevers (Dytiscidae) hebhen, ondanks .,en ovcr'cenkomstige bor-ststel{el, dit springvel'mogen n;i.et, hetgeen verb and kan 110uden met het geringere effect van een eventue.le knipbeweging in water en rnet d,~ krachtige ontwikkeUng van de zwempoterL

Oudem.ans, A. C.: Hct spt:ingen del' knip- of :;pringkevcI''' De lcvcnde natuur 25(}920)1, 9-)7

Doorman, G.: Die Mechanik de" SpI'unges del' Schnel1ki.Ue,. (Elateriden) Biol. Centralbl. 40(1920)2/3, 116-119 B;i.nagh;i., G,: Sulla meccanica del saIto degli Ebt"T'idi Boll. Soc. b:ntom. ltaliana .!!(1942)1, 1-6

9. Voor een gr()ep vall in bcdrijf :djnde gelijke machine" karl een schatting wor<:len gemaakt van de kans op storingen en de 'leven!:lduuI" op een belangdjk eenv()udigeI' wij 7,e dan vedal gcbruikelijk.

Polder, J. W.: Geven!;ldul,ll' van machines (nid gepllbliceerd)

10. Vaal' een groep van ;i,n bedrijf 1.ijnde gclijke machines kan een graf;i.ek van de verdeling van draaiuren (bedrijfsuren, logboektijd) worden gemaakt door langs de ab"ds draaiuren uiL te zetten en lang-s de ordi.naat de machines, die ni.et door ",en blijv0nde storing zijn uitgevallen, t .. rangschikken naar hun starld Van draaiuren op een bepaald Hjd"tip (kalen­dertijd). De verdelingslijn zal vrij spoedig een vrijwel vaste geb()gen vorm aannemen. De differentiaalvergelijldng van de verdelingaUjn vertoont verwant"chap met die van een trillende snaar, onder'wOr'pen (l,s,n demping.

Polder, ,J, W.: Levensduur van mach~ne!:l (n~et gepublieeerd)

11, E';'n me.;.r op de her-kenning van het wislrnndig mQdel geI'ichtc studie is o~et al,ken Van belang voor de ontwikkeling VaJ1 de jonge werktuigkundige ingen~el,lr, maar kan oak de studie aan de l' H beter' toegankelijk maken vOOr' icmand die op oudere ieMtijd met de studie begint.

12. Er is behoei'te aan de oDtwikkeUng en de toepassing van een filosoHe 11'001' de werktuigbouwkunde,

() juni 1969, J. W. Polder