Purdue University
Purdue e-Pubs
International Compressor Engineering Conference
School of Mechanical Engineering
1988
Epitrochoidal Versus Hypotrochoidal Gerotor
Type Pumps With Special Attention to Rubbing
Velocities
John E. Beard
Louisiana State University
Follow this and additional works at: http://docs.lib.purdue.edu/icec
Beard, John E., "Epitrochoidal Versus Hypotrochoidal Gerotor Type Pumps With Special Attention to Rubbing Velocities " (1988).
International Compressor Engineering Conference. Paper 656.
http://docs.lib.purdue.edu/icec/656
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for
additional information.
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Herrick/Events/orderlit.html
Mor eove r,
tl1e
refr iger atin g
beco mes fixe d afte r the
cap acit y con trol . As
for thes e resu lts, the
star tad a drop at N ~ torq ue Tr
1800 rpm,
and at N ~ 5000 rpmt
con stan t
to tha v<>lu<e ot N ~~n1800
rpm,
abou t
a
35%
redu ctio n
was
reac hed .
Con sequ entl y, wi tl1 a syst
em
smpl oy; ng
a
cap acit y con t<ol
com pres sor
ther e
is
no
atta chm ent and deta chm
elec t<om agn< >tic clut h ent of the
in
the
rang e o£ cap acit y cont
r.ol , so a
st<> ble valu e with no fluc
tuat ion
l. ~
rnai.nt~ine.d
thus
1
yiel din g
com fo< tabl e
tem
cegu l<>b on and driv ing pera ture
feel ing
(See Fig. 12)
capaci~y
st~rt
of
F~g.
12
Com pari son of
fluc tuat ions
CONCLUSIONS
the deve lopm ent a vane
rota ry type cap acit y con
com pres sor equi pped with
trol
u cap acit y con trol
enga ge0 the meth od of
mech anis
<>U
toma tical ly dela ying the m whic h
pos. i tion of suct ion
endi ng
pro cess , we stud ied dyna
mic beh avio r of
cap acit y
con trol
mec hani sm
and
came
to
con clus ions .
the
follo win g
l. The con trol mec hani
sm usin
with few add itio nal part g a arc- shap ed slid er is sim ple,
s and enab les cap acit
ovor the wide rang e from
y con trol
10 to 100% .
7..
1\t the refr ig<e ratin
g cyc le emp loye d the cap
com pres sor, with pres
acit
sure cont .t"ol valv e whic y con trol
suct ion pres sure con stan
h
t, refr iger atin g cap acit sta. bili za
cons t<>n t afte r the
y beoo mos
star
t
of
cap
acit
y con trol ,
redu ctio n of com pros sor
and
yiel d>n g com fort able temtorq ue is reac hed . Con sequ entl y
feel ing w.i th no fluc tuatpera ture regu latio n and driv ing
ion in the rang e of
con trol .
oap aci ty
3.
Thro ugh ana lysi s of cap
"
of theo reti cal ana lysi acit y con trol com pres sor, resu lts
one and the meth ods we s tend to agre e with e><p erirn anta l
ernp l.oyo d can esta blis h
con trol resp once and st.:l
outs tand ing
bili ty cha rac teri stic s.
In
/'1..1
A,
A,
C
d2
d,
Fi
Fli
F,
g
Ko
k1
k,
k;
M3
N
No
Pa
D,.
NOMENCLATURE
Pres sure rece ptio n area
Pres sure rece ptio n area of Diap hrag m
Pres sure rece ptio n area of Stea l bwl l
of Slid er
Flow rate coeffic~ent
Diam eter of out let hole
from
pres sure con trol valv
Diam eter of out let hole
from pres sure con trol chame
Sum of init ial spri ng stre
ngth s of spr.i ng (l) and ber
Init ial spri ng stre ngth
(2)
Side pres sure app lied to of spri ng (3)
Gra vita tion al acc eler atio slid er
Gain con stan t of Ref rige n
Spri ng con stan t ot spri rati ng cycl e
ng (l)
Spri ng con stan t of spri
Spri ng con stan t of spri ng (2)
ng (3)
Mas s of Slid er
Com pres sor rota tion spee
Co!n pres sor rota tion spee d
d at con trol star t
Atm osph eric pres sure
,...,....~ri,....r:~l
nl""~=>..:::,c:lJrP. rati
o
430
High pt"e:s sUt'e
rol cham ber
Pres sure of pres sure cont
press or
Suct ion pres sure of Com
Refr igera ting capa city
sure cont rol cham ber
In~flow rate to pres
cont rol cham ber
Out- flow rate from pres sure
Pin
l's
Qe
QJ
Q2
R'
To
'L'r
v
Vth
x,
X'
"Y11 1
Ym
w
K
11\)
nvo
1.
2.
3.
4.
5.
p~essure
Comp resso r disch arge
Pd
Pu
J.
side gap
Plow resis tanc e of slid er
Refr igmr ating cycl e
Repo nse time coeffici~nt of
e
torqu
Comp J:esso r requ ired
Volum e of cylin der cham ber
of cylin der cham ber
Maximum theo retic al volum e
ball
l
stee
of
Lift
Disp lacem ent of Slid er
Com press ion ratio
Gas spec ific wcigl 1t
Fric tion coef £1ci ent
Spec ific heat ratio
y of Com press or
Volu metr ic effic ienc
t
Com press or at cont rol star
Volu metr ic effic einc y of
Reic helt,
CONDI1;IONING
ct
OF
al.,
"REFRlGERATING
1')1\SS:CNGER
Cl\RS .
11
COMPRESSORS
Purdu e.
FOR AIR
Com press or
1986 , pp 537-5 49
Tech nolog y Conf erenc e, July ACITY CONTROJ, OF ROTARY TYPE
T. Maru yama , et al., "CAP
Purd ue
AIR-C ONDI TION ERS"
AUTOMOTIVE
L'OR
COMPRI:SSOI<S
91
eren ce, July 1982 , pp 284-2
Com press or Tech nolog y Conf
"V-5 AUTOMOTIVE VARIABLE
Skin ner, et al.,
Timo thy J.
RESSOR" SJ\E 8500 40, Feb.
COMP
NING
JITIO
CONl
DISPLACEMENT AIR
BLE
1985
LOPMENT OJ;' TWO-STAGE VARIA
Yoji Nish imur a, et al., "DEVE AU'rOMOTIVE 1\IR CONDITIONER"
DIS!?LJ\CEMCNT COM?RESSOR FOR
SAE8 50039 , Feb. 1985
NTINUOUS
al.., "CONTINUOUS AND DISCO
RS"
!!. llold ack- Jans sen, et
SPEED REFRIGERA1'ION COMPRESSO
lliGll
FOR
ROL
CONT
CITY
67-75
CAPA
y Conf eren ce, July 1986 pp
Purd ue Com press or Tech nolog
431
THE ANALYSIS OF DIMENSION
A!.. COMHONIZATl_ON Pl\OCEDUR
ES CONSIDERING
THE E~FICIENCY FOR '£HE ROL!.
.lNG PlSTON COMPl{ESSORS
Caio Fran co da Cost a
EMBRACO - Emp resa llras lleir
a de Com press ores S/A
P.O. Box D-27 - 8920 0 Join
vill e- SC- Braz il
ABSTRACT
The pres ent work dea1 s VJith
dime
nsion al t.offilliOni;;;ation proc
fere nt effe cts over the pedo
edur es and thei r difrman ce o£ tlle comp r.ess or.
The stud y is made theo J;eti cally with the use of a prev
ious ly valid ated simu latio
n program at a refr iger atio
pres sor seri es wlth capa city
n cornrang e from 650 to 1000 Btu/
hr.
Effi cien cy is anal ysed throu
gh
grap
hs show ing the vari atiu n of
cien cy rati o (EER ), capA -city
the ener gy effi and powe r consu rnptl on over
the swep t volum e doma in,
each of the c.om moni zatio n
for
alte rnat ives . The resu lts
of Lhe enG!rgc.tic and the volu
metr ic e£fic ic.nc .y over the
swep t volum e domain are also
shown.
Furthermore~ in orde
r to get a deta iled unde rstan
ding of the said proc esse s)
each spec ific ener gy and mass
loss caus ing the capa city
and cons ump tion vari atio ns
arQ. also disc lose d in grap
hs.
INTRODUCTION
Dime nsion al comm oniza tion
proc edur es are mean t pr.ir
n~rily as a way to atta
proc ess simp lific atio n and
in
cons eque ntly cost redu ction
. However such proc edur es
also be consi dG:r ed in term
must
s of the resu lting pc.r£orma
nce and G!ff icien cy of the
pres sor.
com-
The disp lacem ent of a rolli ng
pisto n comp resso r is give n
main dime nsio ns: cylin der diam
basi call y by ics thre e
eter, cylin der heia ht and roll
ther-e are thre e poss ible comm
er diam eter. Ther efore
oniza tion proc edur es for the
part s in the same comp ressor seri es; obta ined by chan
ging sing ly one dime nsion
keep ing the two othe rs constan t.
The. pres ent work deal s, with
the corn moni zatio n procedtir£
effe cts over the effic ienc
!S and thei r diffe rent
y of the comp resso r.
COMMONlZATXON PROCEDURES
As ment ioned abov e ther e
arc Lhree dire ct dime
nsio nal cornrnonization proc
for any grou p of pumping kits
edur es
of rolli ng pisto n comp resso
Also a four th proc edur e, a
rs of the same. seri es.
deri vati on, will be exam ined
(see tabl e 1).
The firs t, and most elem enta
ry proc edur e [,.1 II], is co ~han
cylin der (ll) keep ing ever y
ge the heig ht of the
othe r dirne nsiol l cons tant. 'the
seco nd type of proc edur e
[,.1 ¢r,~ 0e] is to chan ge che
roll er diam eter (0r) , and cons
of the shaf t (Ec) , keep ing
cons tant thQ cylin der heig ht, eque ntly the ecce ntric ity
part icul ary, the ecce ntric
cylin der diam eter (0c) and,
jour
thick ness ( tr) . The third proc nal diam eter (@e), henc e, shif ting also the roll er
edur e [,! 0r, ,f, 0e], a vari atio
achi eved chan ging the roll er
n of the seco nd one, is
diam ecer and the shaf t ecce
the ecce ntri c Jour nal diam
ntric ity yet alte ring also
eter, so that the roll er thick
ness is kept cons tant for
all kits in the g10oup. The
four th poss ible comm oniza tlon
[,! 0cJ, is co chan ge the
cylin der diam eter and the
ecce ntri
1he cons
city , keep ing cons tant ever
y othe r dime nsion .
eque nces of 12ac.h of thes e
comm oniza tion proc edur es
p10ocess can b" easi ly dedu ced
in the man ufac turin g
by thos e skil led in the art
tA1erc the cornr noniz ation o£
by exam ining tabl e 2,
a grou p of ''n'' comp resso r
kits is
factu
ring proc ess anal ysis woul
shwon. As the manud be out of the scop e o£
this work , v.re will not make
any furth er cons ider ation
on this matt er.
432
Y
AND COi1PRESSOR EFFICIENC
COMMONIZA.TION PROCEDURES
al cornmoni:zation proc ethe effe cts of the dime nsion
n
To anal yze theo retic ally
ious ly valid ated simu latio
prev
a
of
use
made
we
y
ienc
e
dure s on the comp resso r effic
disp lacem ent with in the rang
with
ors
press
com
of
es
cal seri
en
mod~l (1). A hypo theti
iger atio n mark et) were chos
stand ard rang e in the refr
the -23.3 /54.4 9C cond i-:
of 4.0 to 6.1 cc, the most.
comm oniza tion proc edur es, on
to be simu lated unde r the four
tion .
s foun d for the medium
cons ider ing t:.he dime nsion
kept
The conu noni zatio ns were made
nsio ns, i.e. the dime nsion s
dime
ing
lead
the
as
e
rang
swep t volum e (Vs) of the
t commoni;;:ation proc edur e were
r kit of each diff eren
cons tant for each comp resso
e (see exam ple in cabl e 3).
m swep t volum e of the rang
thos e calc ulat ed for the mediu
the four corn moni zatio n profor
foun d in the EER
with in
Figu re 1 pres ents the resu lts
e give s the most cons tant EER
edur
proc
H]
[,1
the
s
show
h
inte ris
It
e.
slop
EER
cedu res''· As the grap
est
high
0e] proc edur e give s the
the rang e, and the [f. 0r,,1
chos en betw een the rang e,
er wher e the lead ing Vs is
es will not be alt:e red.
rest ing to poin t that no matt
cdur
proc
line s for the four
chos en depe nds on if the
the rank of the slop es of the
be
to
e
edur
t:hat the bet.t er proc
a sma ller volum e than
or
Anot her impo rtan t issu e is
ter
grea
a
her
ing kit) has whet
ler than the deri ved
orig inal exis ting kit (lead
e~isting kit is smal
i£ the orig inal
the
the deri ved ones . That is,
s of effic ienc y, woul d be
the best procedure~ in term
a grea ter volum e
has
ones , stil l to be desig ned,
kit
inal
orig
the
if
,
rary
cont
[,1 0r, ,1 0e] proc edur e. On thebest chai se woul d be the [,1 H] proc edur e, for the high than thos e to be deriv ed the
er EER.
the ener getic effi volu metr ic effic ienc y (~v)and
It
Figu res 2a and Zb pres ent the
C'.ammoniza.tion proc edur es.
four
tbc
for
in
doma
e.
t volum
line s is the same of that
cien cy ( r1e) over the swep
the
of
es
slop
the
of
rank
the
can be note d that for the ~e nv thre e of the proc edur es show almo st the same line
rs. It means that the
of the EER line s, but for the
much lowe r slop e than the othe
a
and the [,1 H] proc edur e has
.
ance
of t.he ne than the nv vari
EER is much more a resu lt
vio~ it is
each diffe rent proc eclur e beha
for
ons
reas
the
il
deta
es. ln
loss
mass
To unde rstan d in
and
gy
ener
the
look on the vari atio ns of
nece ssary to take a clos e
front the medium (5. 09 cc) up
y the anal ysis is made only
hs the anal ysis shou ld be
orde r to simp lify this stud
grap
the
of
For the left side
to the grea cer swep t volu me.
made reve rsel y.
neen tbe roll er and the cyli
the. minimum clear ance . betw
its
The gas leak age thro ugh
cylin der heig ht, what e~plains
a func tion of the
der wall s (Fig 3a) is main ly proc edur e. On the othe r hand the gas leak age throu gh
H]
[#
the
for
ease
incr
shar p
Hj proc edur e, beca use this
almo st cons tant for the [#
[if:. H)
the vane edge s (Fig 3b) is
h does not change. for the
whic
city
ntri
ecce
i,;.he
of
ly
a almo st
leak age is a func tion main
es this leak age will have
othe r thre e proc edur
proc edur e. Whil e for the
ecce ntri city .
cons tant incr ease with the
tion of the cyli nder
(Fig 3c) is also main ly a func s of the grap h on
The vane tip fric tion lo"s
line
the
for
made
lar to that
is simi
heig ht, henc e, the anal ysis
Figu re 3a.
nal (Fig 3d) is a
roll er and the ecce ntric JOur
for the ['" H] proc eThe fric tion loss betw een the
ease
incr
t
grea
a
has
it
e
ht, henc
func tion of the cylin der heig [# 0, # ¢ej proc edur e this loss drop s beca use its also
case , decr ease s with
dure . On che conc rary for the
nal diam eter. whic h, for this
c JOur
no
a func tion of the ecce ntri
cons tant beca use ther e is
othe r two proc edur es it is
the volu me. Whil e for the
nal dian leter .
JOur
c
ntr:i
ecc.e
or
ht
heig
chan ge in neit her cylin der
vane insid e the cylin der slot
th~ sl.in ding of the
The fric tion loss caus ed by
the gas forc e alon g the
of the slid ing velo city and
city of
(Fig :; ) is mah1l.y a func tion
com binlt ion of the ecce ntri
s will respo nd for a
vane side s so that the
li~e
ht.
the shaf t and cylin der heig
the cylin der
adm issio n of the hot oil into
The ener gy loss caus ed by che
the refr iger ant gas temp erag
easin
incr
s,
side
vane
the
thro ugh the roll er face s and
ing loss (Fig 3f).
ion volu me, is calle d oil heat
ture , spec ially in the suct
rela tive valu es to thos e
figu res (exc ept Figs 2) a.:e
;, The valu es shown in the ne~t
4.
e
tabl
see
(Vs ~ 5.09 cc),
obta ined for the lead ing kit
433
I t suffe rs a great incre ase
for Lhe [# H] and the l# 0r, ~ 0e]
proce dures . In the
first bG:cause the oil leaka ge throug
h vane sides is propo rtiona l to
the cylind er
heigh t, in the latte r becau se the
oil admit ted throug h the rolle r faces
is inver sely
propo n:ion al to Lhe rolle r thick ness.
while for the [# i!Jr, fo 0e] proce dure
will be a small decre ase in c.he
there
oil heatin g loss due to the. decre
ase in the rolle r
medium perim eter, what is chara cteris
tic of this proce dure. For the [f:.
0c] proce dure
both heil]h t and rolle r medium perim
eter are kept const ant so that the
oil heati ng
loss is almos t const ant.
CONCLUSION
The effici ency of the rollin g pisto
n comp ressor , for each of Lhe £our
dimen sional cornrnon.ization proc~dures
possi ble
were exami ned. ThE> best proce dure
to be chose n
depen ds on the swept volum e o£ th~
origi nal exist lna klt relat ive to
the
volum
those kits still to be desig n. In
es of
case the new kits have smalle r. volum
es "Chan the
origi nal one the best choic e fer the
highc r effici ency would be the [~
H] proce dure.
If the new kits have great er volum
es than the origi nal one the best
proce dure is the.
[~ 0r, ~ 0e] proce dure. In
case ' whole ncw serie s of kits is
to be design ed from a
medium volum e leadin g kit the [# H]
proce dure would give the most const
ant effic iency slope . Those consi derat ions are
made only in terms of effici ency.
REFERENCES
1 -COST A, C. F. - "Use of a Simul
ation Hodel £or Theo retica l Optim izatio
n Analy sis
of a Rollin g Pisto n Type Comp ressor
", ICEC, Purdu e Unive rsity (1986 ).
2- COATES, D. A. - "Perfo rmanc e Impro
vement and Dimen sional Cornmonizat
ion with a
Comp ressor Simu lation ", PCTC, Punlu
e Unive rsity ( 1972) .
Types
H
0c
0r
Ec
0e
,;,
~
#
tr
[~ II]
[# 0r,
0e]
[~
0r, ,;, 0e]
[#
(icj
#
f.
~
Table 1: Types of Dimen sional Comm
onizat ion Proce dures .
Comrnonlza t i.on
Types
l~
H]
u
0r,
0e]
[! 0r, # 0e]
[# 0c]
I
Number of Par.ts
1Cylin d ·1 Rolle r
I
I
I
I
n
I
Sha~
I
I
" I
I
Number of Diffe rent Dimen sions
H
Qlc
Qlr
llc
0e
n
n
n
n
n
n
n
n
Table 2: Types of Dimen sional Cornm
onizat.ion and l1anu factur ed Pa:rts £or
a group
of "n'' kits.
434
Vs (cc)
I
I
) tr (mrn)
H (rom) l0c (rom) f/Jr (rom) Ec (rom) l0e (rom)
4.07
16
40
35.7
2.14
23
6.36
4.58
16
40
35.1
2.42
23
6.07
2. 72
23
5.78
34.0
3.00
23
5.49
33.4
3. 31
23
5.19
5.09
5.59
6.10
16
16
16
40
40
40
34.6
nizatio n for the [;k 0r,
Table 3: Example of Dimen sional Corno
proce dure (group of 5 kits) .
0e)
% of mass flow
Volum etric Losse s
Minimum cleara nce leaka ge
Vane edges leaka ge
4.28
1. 15
% of power input
Energ y Losse s
Vane tip fricti on
Rolle r/ecc entric fricti on
Vane /slot fricti on
1. 63
Oil heat:i ng
5.14
2.50
1.89
s for the Leadi ng Kit
Tabl<= 4: Volum etric and Energ y Losse
(Vs ~ 5.09 cc).
435
l. <l4
l. 029
l. 018
a - [;<\ H]
I. 007
b - l ;< ~r,
0e]
c - [ ,1 0r, ,1 0e]
d -
. 996
~
"-'
w
[ ,1 0c]
.985
.9?4
.963
.952
. 941
4.42
4.84
vs
5.25
5.68
S. I
ecce
95.5
95.2
u.:
94.6
lL
w 94.4
u
H
""
r-
94
w
5 93.6
_j
~
93.2
92.8
92.4
4. 42
4.84
5.68
VS [CCJ
F1g.2b
SWtPT VOLUME CCCJ *ENERGET IC EFF.
55.4
:::;
54.8
~
54.2
t:
53.G
u
53
w
H
t; 52.4
5 51.8
\!}
z
w5!.2
50.6
4.42
4.84
5.26
VS CCCJ
436
5. 68
6. l
6. l
*
SWEPT VOLUME LCCl
Fig.Ja
ROLLER/CYLINDER LEAKAGE
a
l. 15
w l.l
<;J
a: 1.05
a:
w
_j
"'
d
.95
_j
>-
u
\.
ll:
.9
_j
_j
.85
w
0
ll:
/
.8
.75
6. I
5. 68
5.26
4.84
4. 4 2
vs LCCJ
Fig.
3b
*
SWEPT VOLUME [CCl
VANE EDGES LEAKAGE
1. 2
l. 15
1.1
w l. 05
<;J
a:
"'
a:
w
_j
.95
.9
.85
.8
--L---~------
4.84
4.42
F'•g.3c
l. 16
z
*
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6. I
EPITROCHOIDAL VERSUS HYPOTROCHOIDAL
GERO TOR TYPE PUMPS WITH SPECIAL ATTE NTIO N
TO RUBBING VELOCITIES
John E. Beard, Assistan t Professor
Louisiana. State Universi ty
Departm ent of Mechani cal Engineer ing
Baton Rouge, Louisiana.
Abstra~t
ability can be used as a pump,
The pla.nar r.otary mechanism, by virtue of its volume changing
between the points of conta~t of
velocity
relat,ive
the
is
velocity
rubbing
The
.
compt~ssor
or
engine
rota.ry mechanism t !:ommonly known
the two working chamber$. The rubbing velodty of the planar
This pnper presents the effects of
a.s a ge:totor, influences the wear rates and heat transfet tates.
idal gerotors and the
design parameters on the rubbing velocity for epitrochoidal and hypotrocho
rubbing velodty versus displacement.
INTRO DUCTI ON
gerotor has been used sucThe lirst practical rotary mechanis m, commonl y known as the
as a pump although the
used
usually
is
gerotor
The
1920's.
the
since
cessfully for a pump
on engine and is
combusti
internal
an
as
success
limited
Wankel engine, a gerotor, has had
now obtaining increasin g usage as a compress or.
epitrocho idal generate d geroLeen1hius [1] presented the rubbing velocity (apex velocity) for
presented a method to detertors where the generatin g pin has zero diameter . Colbourn e [2]
motion. Hall [3] presente d
mine the evelopes of trochoids which are generate d by planetary
of curvature on the generate d
radius
minimum
the
e
determin
to
necessary
tics
mathema
the
theoretic al flow rate of epitroshape by the trochoida l motion. Colbourn e [4] presented the
g arc on the flow rate. Beard
choidal type gerotors and the effect of the shape of the generatin
to determin e its .-ffect on the
[5] varied the radius and location of the generatin g circular arc
of the design paramete rs
flow rate and pocket displacem ent. Beard [6] presented the effects
idal gerotors. Beard [7]
on the volume change ratio and size for hypotroc hoidal and epitrocho
of the epitrocho idal
velocity
rubbing
the
on
rs
paramete
design.
the
of
presented the effects
gerotor.
the epitrocho pidal versus
It is belived that the relations hips for the rubbing velocities for
the rubbing velocity (apex
hypotroc hoidal gerotors have not been previousl y presented . Since
when designing gerotors
velocity) for epitrocho idal and hypotroc hoidal gerotors is of concern
displacem ent for both
versus
velocity
rubbing
the
on
rs
paramete
design
the
the effect. of
.
presented
are
gerotors
epitrocho idal and hypotroc hoida.l
439
MATHEMA TICAL DEVELOP MENT
Epitrochoid al Motion of the Generating Arc
To generate a gerot.or, where the center of the path of the generating circular arc creates a.n
epitrochoid, the generating arc is placed oil the larger circle of radius r and this cirde is
2
rolled without slipping on the smaller circle of radius r 1 , Figure 1. Points q and qt are the
1
points nearest and furthest from the instant ce11ter and their paths generate the gerotor. To
-I)
Figure 1: Epitrochoidal path of the generating pin center plus the generated portion of inner and
outer
envelope
(a) External Contact
(b) lntetnal Contact
Figure 2: Epitrochoidal generation
generate complete closed curves the following ratio of radii is used.
r1
== rz(T- 1)/T
440
(I)
genera ting and genera ted shape respect ively.
Where T and T - 1 are lobes or teeth on the
can be contain ed in the same envelo pe, have
that
s
gerotor
of
les
examp
are
2b
and
2a
s
Figure
genera ting arc of r at a distanc e '• from the
a
with
ted
genera
were
ratio,
the same T /(T -1)
The rubbin g velocit y betwee n the moving
arc.
ting
genera
the
center of the circle contain ing
ns when the center of the genera ting
equatio
ng
and fixed gear is determ ined by the followi
path.
hoidal
epitroc
an
follows
arc
r
circula
(2)
= W*pq
v;,
v;,r o= w*pql
(3)
and is equal to:
where w is the angula r velocit y of the moving gear
w=t_T
(4)
is a linear functio n of the angula r velocit y of
Since the angula r velocit y of the moving gear
one for all calcula tions. The distanc e to the
to
equal
set
be
will
,),
the ann, for simplic ity
.between the fixed and 1noving gear is:
center
instant
center of the genera ting pin from the
pc =
Jrp' + r
therefo re
2
2
-
2.0r 2 rpcos( q)(1- 1/T))
(6)
pq = pc- r
and
(5)
(7)
+r
found
be
can
n1ent
develop
its
and
ion
derivat
t
The contain ed pocket area is presen ted withou
in referen ce 4.
pql = pc
dA
dq)
4r 2 r.,in( f)
T -1
T!
:n'(1'·~1!
±- •~-•
TT=r
(8)
Jrp
-Jrp 2 + r 2 2
-
2
+r 2 2
-
2.0r 2 rpcos (q)(l- l/T))
(9)
~- ¢(1 -1/T) ) dq)
(10)
2.0r 2rpcos(
and interna l contac t respect ively. Beard [6]
The ± are the displac ements for the externa l
pin, r , has little effect on the flow rate or
ting
genera
the
of
radius
the
that
ined
determ
l
r a direct compa rison betwee n the interna
displac el)lent. By using the following value of
made.
be
can
and extern al contac t geroto rs
(11)
Arc
Hypot rocho idal Motio n of the Gener ating
path of the genera ting circula r arc creates an
To genera te a gerotor , wh~re the center of the
smalle r circle of radius r 1 and this circle is
the
hypotr ochoid the genera ting arc is placed on
r 2 , Figure 3. Points q, and qr are th~
radius
of
circle
larger
the
on
g
slippin
t
rolled withou
their paths genera te the gerotor . To
and
center
instant
the
from
t
points neares t and furthes
hyptro choida l gerotor s the ratio
and
hoidal
epihoc
genera te comple te closed curves for both
y betwee n the moving and fixed gear is
velocit
g
rubbin
The
used.
is
one
n
equatio
in
given
gerotor .
determ ined in the same manne r as the epitroc hoidal
v;, = w;,pq
v.r = w;,pqt
441
(12)
(13)
wher~ w
is the angular velocity of the moving gear and is equal to:
(14)
y
-<P
T-1
Figure 3: Hypotrochoidal path of the generating pin center plus
the generated portion of inner and
outer envelope
As in the rase of the epitroch oidal gerotor the angular velocity
of the moving gear is a linear
function of the angular velocity of the arm and will be set
equal to one for all calculati ons.
The distance to the center of the generati ng pin from the
instant center between the fixed
and moving gear is:
(15)
with
pqo;pc -r
(16)
+r
(17)
and
pql '""pc
The containe d pocket area is presente d without derivatio n
and its develop ment can be found
in reference 5.
dA
d¢
4r 1 rpsin(~)
(18)
T
±~1ft
2
Jrp +r1 2 -2.0r 1 rpcos( ¢(T/(T -l))
2
-~rp'
V +r 1 -
2.0r 1rpcos( ~- ¢(T/(T -1)) dr/>
T-1
(19)
(20)
Tlu: ± are the displ~ceme~lts for the. external and internal
contact respectively. Since the
rad1~s of :he gen~ratmg pi~, r, h~s httle effect on the flow rate
or displace ment the same
relat.Ionslup used m the ep!troclw!dal generati on is used.
This allows a direct compari son
between the external and internal contact.
442
displ acem ent was scalo!:d
tors with different lobe ratio s the
For direc t comp ariso n betw een gero
insid e a unit circle. Figu res
fit
d
woul
tors
gero
l
oida
troch
hypo
such that both epitr ocho idal and
ing velocity versu s rv/r 2 •
displ acem ent and max imum rubb
4,5,6, and 7 are plots of the relat ive
....
'·""
= Rela tive Pock et Di8p lace ment
~
~V<IoO<,
Q•.IIOO
'·""
..,
'·"'
~~~-~
~O.:!tO
ij
:t
------.
. ,.
.::
g!l.2\)f ~.,
"(')"
~o::,,l$0 ..."~11.200
~
~
;:z: t.IOQ
g,,,,.
~
..
.....
..Ito
x r~ s.
+ r~
'·""
.
...... ....
,,~oo
6
CJ
r~
r~
4.
3.
2•
il.IIO
'·"'
and lnaxi mun1 rubb ing veloc ity
, exter nal conta ct, displ acem ent
Figur e 4: Epitr ocho idal gene ration
,,400
•• .11!1
'·"'
..,,.
''"GO
;::•·~
"'r~
= Rela tive Pock et Di8p lace ment
city
= Max imum Rubb ing Velo
~
'·"'
'i
~·-~
a: O.IM
~
""'•.m
i
~
"'
it
,..
i
loi!JO
....,
----
•• 290
..
...... "(')"-....
----------~
"'
-
...
~
..
~··'~
O.IH
'·""'
.,,0!.1
and maxh num rubbi ng veloc ity
, inter nal conta ct, displ acem ent
Figur e ·5: 'Epitr ocho idal gene ration
443
Relative Pocket Displace ment
=
= M ao;im.um Rubbing Velocity
•• - - - . - - - . -"(5" •• - - - . - - . - . - - - - . , . -
_______f _________ __ .•• ·--f
"'
----------f··-
«
----------x
:i! '"""b.--:~:::~::::::::::::::::::::::~----------~
'·""
II,!Of
"
'·""
"
'·"""
l •.iOO
"
<!> T= 5.
X T= 5.
"'
+ T,
,,Jot
4.
t; T= 3.
R~f~
OF'
~PW2
'·"'
Figure 6: Hypotroc hoidal generation , external contact, displacem
ent and maximum rubbing velocity
=
- - - - ""
Relative Pocket Displace ment
Ma:~;imum
Rubbing Velocity
'·"'
i'"~
~Q.XO
~
::< ,...,
"'
..
~11·1,.
----.-.-·-cr.
::::::::- ~ ;~--~ ~~ :~::: ::: ~;
. ----.---.-cr-.
:: :: :::~~:: :::;:- <!> T= 6.
X T=
o.ooo,+_,::,---:-,_..,c:---:-,.T..,::---:c:---Rft-:'"1 1r::'l'8-oF-R~,-~--2--,---,.r,.,-,-,T.,..--.,
+ T=
s.
4,
6 T= 3.
Figure 7: Hypotroc hoidal generation , internal contact, displacem
ent and mall:imum rubbing velocity
·444
CON CLU SION S
ation gerothe epitro choid al and hypot rocho idal gener
To make a direct comp arison betwe en
be comp ared. Since there arc
will
ts
pocke
ined
conta
of
er
numb
tors, geroto rs with the same
, the gear
and hypot rocho idal gener ation respectively
T and T - 1 pocke ts for epitro choid al
same.
be the
ratios for the equivalent geroto rs will not
the great est
gener ation, exten \al conta ct gcrot or has
It can be seen that the epitro choid al
al conta ct
intern
ation,
gener
al
choid
epitro
ratio. The
displa ceme nt for a given T and r./r 2
gener aidal
rocho
hypot
The
ct.
of the exter nal conta
(with T = 2) has a displa cemen t of 45%
choid al gener ation,
epitro
the
of
3)
==
withT
(
33%
of
nt
tion, extern al conta ct has a displa ceme
ceme nt of
gener ation, intern al conta ct has a displa
ext.ernal conta ct and the hypot rocho idal
ation,
gener
al
choid
epitro
the
al conta ct. Altho ugh
25% of the epitro choid al gener ation, extern
ined pocke ts and
conta
of
er
numb
given
a
for
nt,
ceme
extern al conta ct has the greate st displa
al generrubbi ng velocities than the epitro choid
r./r2 ratio, it also has significently highealr and
ation
gener
idal
rocho
hypot
ct,
conta
intern al
ation, intern al conta ct. Both the extern
extern al conta ct but the
ation,
gener
al
choid
epitro
the
than
have lower rubbi ng velocities
.
percent age is not significant.
epitro choid al
the
rates
wear rates and heat. transf er
Since the rubbi ng velocity influences the
ors with
gerot
of
types
other
over
tage
advan
an
have
gener ation, inte1·nal conta ct gerot or may
choid al
epitro
the
if
ine
determ
to
ed
s are plann
its lower nlbbi ng velocity. Furth er studie
than the exter nal conta ct for
stress
ct
conta
lower
a
has
or
gerot
ct
gener ation, intern al conta
a given displa ceme nt.
REF ERE NCE S
ressor s and
ector Loop Analysis of Wank el Comp
[1 ] Leem hius, R. S. and Soedel, W.,"V
Engin es", ASME, 78-DET-1978.
cation in
of Troch oidal Envelopes and their Appli
[2 ] Colbo urne, J. R., "The Geom etry
Vol 3, No. 4.
Rotar y Pump s", Trans action s, CSME ,
1968.
Proce edhtg s, Mech anism s Conference,
[3 ] Hall, A. S., "Pin Geari ng", ASME
Purnp s",
Gear
al
Intern
Theo retica l Flow Rate in
(4 ] Colbo urne, J .R. "Gea r Shape and
Trans action s CSME , Vol. 3, No. 4.
Comp resso rs",
of Gerot or Type Pump s, Engin es, and
(5 ] Beard , J. E., "Kine matic Analysis
1985.
rsity,
Ph.D . disser ation, Purdu e Unive
Gerot or Type
"Hyp otroc hoida l Versus Epitro choid al
[6 ] Beard , J. E., Hall, S. S., Soc del, W.
and Size", ASM E 1987 Design
Ratio
ge
Chan
e
Volum
to
tion
Atten
Pump s with Special
Autor nation Conference.
n Mech anics
Epitro choid al Gerot ors", 20th Midw estere
(7 ] Beard , J. E. "Rub bing Velocity of
Conference, Deve lopme nts in Mechanics.
445