Reciprocating Masses 1
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Transcript of Reciprocating Masses 1
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8/9/2019 Reciprocating Masses 1
1/11
512
BALANCE
OF MACHINERY
lhe speed
at
which the
rotor
is
to operaie A
snall
unbalance
of
mass may
be
tolerable at
lop
speed
srne
the
jnenia
force represenling
unbalance
ma)
be
small.
bur
becau\e
rhe
unb.lan.ed
lorce
increasi'
a'
rhe
square ol Lhe speed.
lhe
unbalance kansrnitted
to
the beaflngs
may be Large at
high speed.
The
rotor
of
a
jet
engne
operalLng
atgrealer
than
10,000 rpm, for
example,
nust be balanced
to
a
high
degree.
Fo.
such
roto :,
the
individual
compressor
and
turbine
blades
a.e
balanced
i. pairs at
opposite locations
on the
rotor Ln such a way
that,
ifone
blade is danaged,
the
pair is replaced
to
restore balance
\.The
p.incipl€ on wbich
dlnamic
balancing machines are
based N
sho*n in
Fir
I0.
10.
_De
rotor
to
be dynamically baldced
is supponed on
flexible
springs
roror is
ro
lbe
oscinatio;
rhe
is idro
d the
€ mo
sPrin
oscrllat ilh
wlnch
of
tes
lhe
idtant
ial
and the
ampltudi is
mardmum
After
the
amounl
and angllar posilLon of
the
unbalan@
is read.
the.otor 6 lenoved from
the machine
and
mateiialis
added
by
solderLng
masses to
i( o.
is removed by
d.illing
holes As shovn
in
Fig r0
10,
nraterial is
usually
added
or
removed
at lwo
specific localions
where
il
is not
injur'ous
to the
rotor
sudace- In
electric notor
armaiur€s, fo example, il is
not
llqals
possible
to add or
renove
malerialin the
region
of
the
electrical windings-
Long
rotors such
as
amarures,
c.ankshafts, and
jet
engine.oto6 are balanced
in machines
of this
kind
It
is
often
necessary
to
baladce
a
rotor that is too lalge to be handled in a
balancing
machine
or to
balance
a
rotor
assernbled
in
its
own
beanngs
Also,
when
a unii
js
being
rebuilt,
it may be rhpraclical to transpon
lhe
rotor back
Io
rhe
.nop
ior
rcbalancin8.
in ttch
cjses
ldd
bala4148 nun
be
used
Tne ilirial
a."t"p*-,
"r
this.mthod
or barancns_was
tr::t,^11
:Lri::.1:t]
y-"j:
BALANCE
OF RECIPROCATING
MASSES
51II
'iecentiescriptions
of ttris
nethod
are
gived bv Hirschhorn:
and
bJ Shisley:and
10.5
BALANCE
OF
RECIPROCATING
MASSES
FICURE
1O.II
Et
Theane,
DynrnicBalancinginlheFicld
lda
lJr,E /oattuloJ
EnSk"tihslot
Innaq
rr
Hi^chhom,
Dlanid
o/Mc.[ii.4
Bah6
and Noble Ne*
York' 1963'p
34
rr
J
ShrgleylndJ
I
Urcke\Thea"oIitaAj'6orl
M'd@irj-
lLc€q
+lill N'c'lqk
lgm
FIGURE
IO.1O
-
8/9/2019 Reciprocating Masses 1
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"*.,(-.'.:-.,,)
I
I
I
514
BAL"{NC€
OF
MACHTNERY
BAI_ANCE
OF
REcrlRocAm{c
MAssEs
511
conne.ting
tbe
engine
btock
to lhe
supporting
frane,
the
oscillations
duo
shaking
couple may be
isolated
from
the
frane
for
cerrain
shaking
couph
isu e
10.12
show
a
tygicat arregemenr
of cran*.
in
a six{ylDder
in-tino
ln this engjne,
(he
crlnkr
are
fD(ed
ar
120. ro
eactr
otber as
showD
and
cqank
pans ale the
same
as to
size, shape, aod
weight.
As
shopn
in tbo
of
Fig
10
12,
the
inertia force
F
of
the individuar
reciprocating
masses
aro
al-.j
::.= tic lclowing
equation:
which
M
is the
combined dailes
Mr ald M4
for a
siDde
g.liDder,
a
is
th
*
I€ntth,
, is
rhe
@mecting{od
tength, @
is rhe
q€Dk
algltar
velcity,
and
€
is
the
craDt
dgle
trom l.D.c. (rop dead
cf,Drer)
Ar rbe
table
io
FiB.
10. U sho*r,
rbe
arratrtede
otcrark
for
a sir_cltjrdcr
.Dgine
i5
such
rhar
rhe reulranr or
$e
sn iDrnia
for6
is
zro
for tte plxition
1::
of
the
qekshalt
CiveD
by
0r
=
20e. n
may
be
shom
thar
de
rEuhani
is zro
for
a'l
positions
ol
lhe
crankshaft- Thus,
no shaung foEe
is
transnined
ro
ftrj
'r=oa777b
*J,i
€,'
t20
22$
I
t4C
s
260
620
+0
904
0 766
-0_766
-o
t74
+0
?66
-0.940
+0
t7,1
+0,174
-0
940
+0_19t
-0.235
+0
044
-Q
735
+0191
)=0
+l
l3l
-0
409
-0
721
-0.722
-o
409
+r
l3l
>=0
+1450
52s
-925
-925
-525
+t450
t=0
FIGURE
10.12
>=0
t=0
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516
BAIANCE
OF MACHINERY
10.6 ANALYIICAL
DETERMINATION
"
oF
UNBALANCE
Analyticdl
merhods
are arai
aole
lor
the
deler,nrnallon
ot
r}le unbalance
or rhe
\haking
fo .c
of
a multicylind€r enginer'lhe
nethod l€ad. r. sinple
algebmic
expressions
which
Bive
the
magnrlude
and
sense oi unbalan@
as
a
tunction of
crank
positior
8,-
Cr€nk
posirroD
of
a
multicylinder ensine
in
rhe
ensine cycle
h
given
by tbe crank angle 6,
of the fi61cyli.der
as shown in Frg. 10.13.
In
au-
tomotive engines,
the
first
cylinder
is.at
thefrontend,
aDd
0,
is
measured
clock'ri*
in
the
direction
of
rolation
when
lie\{ed
fron fie
ftont end
The followiag
analyli.al
derivatioD
applies
only
lo intine iyp€s
or engines
in
ehich
dre cllinders
are
in
line oD
the
same
ide
ofihe
cradkshafi.
The reciF
r@ating
nas
M and
the
R/, ratio
is the
sane
for each
cylinder
As
shoqal.in
Fig
10
13,8, ofthe
first crank
locares
rhe clockwis€
posirion
of
the
dalcshaJt in Lhe elginE cycle,
dr
and
0r
de the
fixed ansles of crdks
2
and
3,
resp€ctively, measured
clockwise fron crark
Alrhougl
three
cyliDders
are
shown;n Fig 10.13,
any number of
cylinders
may
be @nsidered-
The
inenia
force Fof ady
siven
cyhnder at
0
is
=
M.R.,
cos
o + M
f
.,
cos
20
The two rrght-hand
tefms
in Eq l0
6 ale
the
li6r
tko
rems
of a series,
th.
renaiiilrB terds
ol
which
are
uslally cons;dered
negligible The firsr rerril
(first
ha.monic) is known
as
the primary
force
4
and the
second
te.m (second
harnonic)
as
the seconda.v fo.ce
F..
Thus.
.
F=F,+F,
wle.e
Fp
=
MRd'?
cos
0 and
4 =
(.R:/L)o,
cos 2e
/D
M,ad:icdo+;cosro]
The sumnrarion
of the
inertia
forces ofa mullicylinde.
engi.e
is
rhe
resuiranr
force
or
lhaki.g lorce
S,
\chich
represenrs
rhe unbalance:
S=:F=:,ff+:4
In some
engines,
rhe
pnmaryforces
may
be balanced
although
rhe
secondary
fo.ces mal, nor
The
reverse
may also
be rhe
case
The equarion
of
unbalance
ot
p.imary
torces is developed
as follows,
in
whrch
0
=
0r
+
O.
I
Fo
=
; ,Y71":
-.
U
=
MRor
:
cos
0
=
MRo: :
cos
(0r
+
d)
=
MRo':
:
IGs
o,Xcos
d)
-
Gin
o,)Gin
O)l
ANALYTICAL
DETERMINATION
OF
UNBAT"ANCE
517
cos e,
and sin
e'
are constant
ior
all
terms of rhe
summatiori.
:,
4
=
MR@:lGc
e)
t
(ms
,D
- Gin
0,)
t
GiD
O)l
The
equahon
of unbaldce
of
secondarv
for€s
is
simila.
in form.
C,-tF'3=MRdiIi,cos€)
=
MRol[(cos
oJ
:
(,
cos
d) -
Gin
0
):
(d
sin
d)]
a
=
MtL'
11"*
ze,; :
td
cos
2+)
- G;n
20,)
r
(a
sin
2+)l
q+c,
,r
r,
=
M
f
"'tGos
20,)
:
(cos
20)
- Gin
2e,)
:
Gin
2o)l
(10.10)
(10.9)
(10.11)
Il
may
b. seeo
6om
E4s
10.
of
cmnks
h a muldcylinder
€nFne
t
si
+,:
cDs2d,
ard:
sin2O
maY
b€.one
tuncrions
onlY
of
er.
lt maY
for@,
fie
fouowing
summations
must
all
be
zero
/
:@s0=0
(10.6)
tcos2$=0
:stn26
=
0
/
Anorher
mode
olshaktng
musl
be
mnsidered
for
multicyl'nde.
engines Bv
\ie$nq
rhe
ena
ne
o'
f,e
l0 ljfiontheside
ir
m"\
be
seen
lhdL
lhe
line
of
action;i rhe
r;ul,anr
shdl,rne
1o(e
rn
rhe
a$o ol4ne
mal nor liF
on a I'ne or
symneLrybetweentbemainbearings
Moreover,the
S nav he rhifline
dxiallv
tn
rhe
arial
plane
as a lunc
eneine
o'cillaesln
an
end
ovet
"nd
'orarronalnod'
be deternined
fion
the
PrinciPle
of moments
tn
ter
and
a
secondarv
momenl
Ci.
iD
which
moments
ar€
taken
*ith
respect
to
a
refercn.e
Dlane
ar
rhe fi'sr
cylinder'
In
Ir8 l0 lJ .
L
rhe
dr'ran(e from
lhe
'eferencr
;Lne
ro
rhe line
of
acrion
or
rhe
inenra
lor(e
or
"nl
given
cvlinder'
(10,?
(r0.r)
(10.12)
(10.13)
The distanceax
oftbe
line
ofaqion
ofthe
slrakingiolce
S may
be
deler ined
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518
BA -ANCE
OF
MAC[m ER1
from the rsultant
moment
C
about
rhe
reference plaDe
as
fouowsl
2 F,
=
4ry
@s 20,
S=>F,+>4
r=a.@s20,
=ffe.r-"ru,
ANALyTTCAL
DETERMINATToN
oF
UNBALANCE
lll
C
'.s
(10.14)
(10.rJ)
FIGUR
IO.I.I
IO
ll and
l0
12:
t
(,
@s
d) =
0(1)
+
a(-1)
+
8(
1)+12(1)=0
>
(d
sin
{)
=
0(o)
+
a(0)
+
8(0)
+
12(0)
=
0
>
(, cos 2S)
=
0(l)
+
a(r)
+
80)
+ 120)
:
24
:
(, sm 2S)
=
0(0)
+
4(0)
+
8(0)
+ 12(0)
=
0
B
elel,na
to Eq
. .O
'I
tld
L0
2
ir
nay be
an
thar
"
s"'onda1
m
r_r'
{]
utour
rtr"
'eren"..
p,.e
,na
lhat
rhe
primarv momenr'
irP
/ero
MR''')
__
C=q+C,=2t
L
@s
tr\
rhe
nbatee.t offte r.ciprocarin8
mas
otrhe
@nv.q-
h r Fit.
10
14, in
wh,cb
the
dants
aE
at
l8{r. Dercmind
I
stcting @uple.
.Sofufion. The
hled
eeies
6
de shown m
Fig.
tO.]4
Ir
shoutd
b.
notcd
rhat.
ahholstr
6,
and
d,
are /ero.
rheu
coqne funcrions are
uniry
and
mur be
taren
inlo
a@unr
in
ri.
eguations thar
deleimine trnbalance
The foltowing
summarjons
are
nade
ro
deremin€
lle osb6
$hich
appar
in the
egualions
of unbalance:
:@sO
=
I I
-
I
+
I
=
0
>sin6=0+0+0+0:0
tos26=t+1+l+1=4
:sin24=0+0+0+0=0
By reie rng
ro
Eqs 10 9 and
10 10,
ir
nay be seen
that
rhe
primary
forc*
are
balaned
and that lhe
sKndary
locs are
not:
r
4
-
Mx,{(@s
0,x0)
-
Gin
0,x0)l
=
0
)
t
=
l-
t(cos
2BL)ra)
-
(sin
20J(0)l
Equadoi
10.15, *hich
giles
rhe
shaking force
enginc
dafunctionof0,,
is
shoM
ploued
itr
Fig-
10.1
for@ atre
Gasiaplc
hatuonicdrue sho*
orcular
The
follo*ir8
sunmarions give
the.onstanrs which
appty
in
theeooent
equalios
-2
FIGUR
10.15
-
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5/11
520
BALANCE
OF
MACHINERY
The
lhe
of
a;i.i of
the shaldng
fore s is
deremined
as
fo oNs:
C
Z4(MR\11Lt 6\
2a,
r
4(MA,U,/L)
@s 2€,
_
flrd
lin.
ol
actio
of rhe
shaking fore is
@nstat
si@
ar
is nor
s
tuncrion
of 0,.
Als,
by
iisshing
thsr
rhc
lne or acrion
of
S is rhrouBh
rbe
enter
of
Faviry
6t
ihc
engii;
at
d, =
6
rtr.. no
dal
shEkinC
couple
en\ts.
A five{linder
engiie
ls sho\T
in Frg.
reset
engrner
whrle
rhcling
foLe\
are bdtanced,
ling
couples
are
not
balanced
as
vill be
shown
in
E:amplc
0.).
AndtJdhe
batau
of
the riveaylinder.nSrne
shorn in
Frs. ro.t7
re
arrve
ro shakinq
torc6
and shakrns
couDtcs.
FIGURE
1ll.I?
Solution,
Frcnttl
10 l7
the
fixed
ansles
d
are
O,
=
28r
From
Eqs-
10.9
aid 10.10,
:
4
=
MRo)t(6s
0,) >
(6s
d) -
Gin
0L) >
Gin
4)l
,{f,t:u.
: f,
t
i(@r
20,)
:
ro)
7@)
-
hir
20,)
:
(sir
2d)l
>s
+ =
c6ff
+
@s 216' + coj 1.r4"
+
cos 72'
+
cos
28tr
-
=
I
-
0-8090
-
0.8090
+
0
309{
+ 01090
=0
:
sin
O
=
sin
Cr +
si
216'
+
sin
144"
+
sin
?2'
+
sin
28tr
=
0
-
05878 +
058'A
+ 0951r
-
0.9511
=0
ANALYTICAL DETERMINATToN
oF
I,NBAI-ATJCE
5 1
4s
I
I
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8/9/2019 Reciprocating Masses 1
6/11
522
IALiNCE
OF
MACTTTNERY
I cos
26
=
@s
Cf +
6s 432'
+
@s
288. +
cos 14-j.
+
cos 576"
=
1+
03090
+ 03090
-
08090
-
o,8o9o
=0
:
si
2{
=
sin
0'
+
sin 432" +
sin 2a8. +
sin 144. +
sin 5?6'
:0+0951r_09511
+05876
-
05878
r
=0
Tncre,op,
s
qn
b€
qen
abov.,
thc
Firur}
dd
F@nddry
tores aF
bat ed
Frod
Egs.
10 ll
ad
10,12,
c,
=
MlRo{(6
0J
I
(d
6
6)
- Gii
o,) r
k
sin
d)l
MR'.'
.
c,
=
-
t-
[i6
]0r):
(d
c rdr
(srn
:0
r:
t,
rn )6)l
IGc6d)=0(t)+ (
0.6oc0)
+%(-o6txr)
+
lz(o
3oeo)
+
4d(0.3090)
=
_0
264,
>
(d
sin
6)
=
00) + d(
o
s8?8)
+ 2"(+0
5878)
I
3d(0
9511)
+ 4,(
-0
951r)
=
0 3633d
;
(,
cos 2+)
=
00) + d(0
3090)
+
2i(0
3090)
+
34(
_0
E090)
+
4a(
0.8090)
-
4 73gJo
I(dsin24)
=
0(0)
+ d(0
95t1)
+
2r(
ogslD
+
t,(05876)
+.{d(
0.5878)
=
I 51892
q
.=
MRa,l(cos
0)(
0.26a,)
-
(srn
6,X-o
363ja)l
=
Mno,c[-o
2616
B] +
0,3633
sin
0,1
.
,
,
-
j-
l(6
2e.)( a.7lo0,t
-
r$n
ro.).
l
5j89d)l
-r
l-a
?lao coi
?0,
+
t.5i69.in
20l
ns
10 Eqs
10
9, 10
10, 1011, and
1012 tusunerhat
cylindcrs
trrc b
iq
Ir.3. ShoR
tbat
the
conventionalsix{ylinderengine
ofFiI]
l0,12isiF
ANALYNC,AL
DETERMINATICN
OI
UNIIA -ANCF 5"
,I
itidn.1o
show rhdr
,he.,\
cYrnderrnsine
ii in
brlance,,,
i\
n.i
(.
Jry
ro
sho
io
lowing
summdrion(
are .e-o.
F om
I rg.
10.12,
rle
fi/cd
ane s
r'
1re
6'=4,=r
O:=+'=2a0'
+,=d,:120.
:s4=1
i
t-l-l+1=0
\6
\,5
\6 \,5
:sin+=0
+-+--l-+0=0
:6s2d:1- I
l-l-i+1=o
\/t
'"5
'"5
\/1
__+_r0=c
:GcosO)
=
00)
+
b(
i)
+
21,(
l)
+rr(
)+4b(-4)
+
5b(1)
=
0
/ \6\ /\'/.r-
:(dnns)
=
uro)
+
bl
2
/+rb\t
+4bl-
rJ+{b(o)
=o
)."(
r("cos26)=0(1)
+D(
t+2b(-i)
+3r(
,")+4b(
i)
+5b(1)
=o
\n
'f
,rl.
tll
) ,ar.a
2dr
0,0) ,1,
,
I
'
-rl
-,
)* \
.
t
/\A\
+
4b\
2.1
+5r(or
=o
\rF
-
8/9/2019 Reciprocating Masses 1
7/11
524
BALANCE
OF
MAClJINERY
The shakins force
of
rhe
niddle
set
of four cylinden
in
terms
of 03 of
the
nlsi
cylinde.
in rle
m'ddle
set
i
^
'
MR"]
s,
_,__
("s
2er
IjoweYer,
sin@
0
=
0r +
27{P,
^
.
MR\}
s,=a
r
cosz(q t
27v)
HR+}
=
_1
r
co6
Ze,
Sire S,
=
-S:,
tbe
resdteit
shaking
force is zero.
Also,
sine
the
lines
of
action
of S, add
S:
ee
cDiEidcnr at
rbe
ceote.
of
the
engine, dere
is rc
a '.at
shakiEgouple.
Thur, [te r €
six
-
8/9/2019 Reciprocating Masses 1
8/11
l
I
l
l
I
i
I
I
I
I
I
i
526
BAI_^NCE
OF
MACHTNERY
The
following
summations apply
to eithe.
bank
sin@ rhe
fxed
ansles
S
are
the
same
for both
banks:
4,:4e
00
,=0
d,=e0"
:@s+
=
0
isin4
Sr
=
18f
=0
: cos 20
:
(a
cos
C)
t
(a
cos
24)
=0
=
-3a
=0
-
rsin26
=
0
:Gsi.4)
=
-d
:(dsin&)
=
0
10.20.
rr0.
l
CR
C,
tan (8,
+
.y)
=Co
CL
_cR
CL
CR
C.
tan
0r
CL
+
CR ran
€,
Subsrrruiron
of the values
of
C.
and
CR from
Eqs l0
16 and
10 17
eives
1
1 =
198
43'
(Lhird
quadran )
resulranl
a{ial
sha
kine
coufh
e$ ,an,
unba
ded d
a
co",'".,",,.*,,".,
,iii;::#i,;i*Tiii,J
I and
4-
As
shown, rhe €ngine
is complerely
balanced
by
rhe
introducrion
ol0
balancingcouple
C.,
rhe
equilibranrofC,
in rhe
torm
ofrwo
counteNeighk
sucll
$n.22)
C.
=
MRo:[-3a
@s
0, +
a
sin
0r] (10.16)
For
fte
risht
bank,
rhe
crank
angle
of the
fiBt
cylinder
is
-(p
€,)
=
Or
_
p
=
o+:jgrThe
axial
shakins
couple
CR for
lhe righr
bank
is
Ck
:
MRoz[-3d
cos
(0,
,
90)
+
a sin
(0,
-
eo)]
=
MRu:[-Ja
sin
0, d
cos
e,]
)
(10.17)
.
V
ENGINES
.
Since
tle couples
C.
and
Cr
are
in axtalplanes
at 90"
as
shown
in
Fis
1
nagnitude
of the
resultanr
couple C
6
C=\/(-,+c"^
ubstitutiotr
of the values
of
C.
and
Cx
from
Eqs. t0 16
and
10
t?
gives
c
=
..ii
ua.,"
h
may
be
s9€o
that
the resuhant unbalanced
coupte
of the
V,8
engine
is in(
pendent
of €,
4nd
Ls
derefore
consran(
in nagnitude
for aI
anguta?-posirion;
the
darlshalt-
The
ardat
plane
in
which
the
resqlrant
oupte
L
Ues^;
gi,en
d'e
angle
d
measued
clockwi5e
from
rhe ptane
of
the
lefr bank
as
shoq;
in
I
It
may
be
seen
ihat a is a
funclion of
O,
and
rhat
rhe vecror
C
rotates
vid
th
engine
crankshafr. The
angle
that
C
makes
with
the
first
clank is.y,
Fhich
nrll
be
dete.mined
as
follows,
since a
=
€,
+
.y:
FIGURE
10.?O
c.
=
F,
= -\nO
MR-'o
-
8/9/2019 Reciprocating Masses 1
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52t
BALANCE
O' MACHINERY
RIGHT
BANK
LEFT
BANK
Cylinders
1,
3,5
Cylinders
2. 4,
6
(No
I
reference
c_1,lrnder)
(No.
2
.eference
cyti.der)
4,,=
0'
o2=
0.
=
210"
di =
240.
6j
=
120'
d6
:
120"
FTGURE
10.22
V
ENGINES
529I
From Eqs
10.9
and 1010,
:
Fp
=
MlRoz(cos
0,)
r
(cos
S)
_
Gin
€,)
:
Grn
+)l
MR'''1
2
f,
-
-l-
l(cos
20,)
: r€s
26)
_
(\in
2€,)
:
{srn
20,
Right Bank
:
cos
6
=
cr"
[P
+
cos
Z4{F
+
cos
12ff
=
I
-
0J
-
05
=0
I sin
d
=
siD 0'
+ sir
2ff +
sin 120,
=0-0866+0866
=0
>
cos 2+
=
cos
0.
+ cos
480"
+
cos
2:10.
:1_05
05
=0
t
tin 2+
=
sin 0' +
sin
480'
+
sin
240.
=0+0866
0866
=0
Lei
Bank
Icos4=s
Isind=0
:cos2S =
0
:sii26=0
Tlererore.
rhe
pnmat
and
scondaryshakingforces
are batanced
for borh
banks
Shaking
Couples
From
Eqs.
l0 I1 and
10
12.
cp
=
MRo:{Gos
0,): (o
cos
+)
-
Gin
0,)
:
(d
sin
O)l
c =
ftt-'
:e,l
:
f.,:os
20)
Grn
20,)
: (d
sin
2d)l
-
8/9/2019 Reciprocating Masses 1
10/11
530
EAI-ANCE
OF MACIIINERY
RErl
Br.k
(0r
reference
angle):
:
(d cc
0)
-
0(1)
-
a(0 5)
-
2a(0.s)
:
(a
sin
d)
=
0(0)
-
"(0
866)
+
2a(0 866)
=
>
(a
cor 2d)
-
0(l
)
4(0 5)
-
2,(0 5)
.
:
(,
sin
2dr
*
0(0)
-
a(0.166)
-
2a(O
S) -
Left
Bank
(+z
rcfercDcE
ande\
L
(a
cos
4)
=
1.5d
:
(a
si
0) =
+0.8{,64
:
(d
@s
20)
=
-1.5d
:,
(a
sin 20)
=
-0.866d
Right
Bank
-
l.5o
+
0.866a
q
=
MRo:tGd
€)(
15,)
- Gin
€,)(0.8664)l
.
=
MRolal(
r.5
cos
e,)
-
(0
866
en 0,)l
1-
-
M8'-111so,.2e
11-t
5d)
(srn
le
)t-0
8oo4rl
-'t''
-
R'e 1,
1 s .or:0,;
,
(o.8oo
sn )grj
t'
Lefr Bank
(wherc
0,
:
0, + 90)
C,
=
MR61a[-r.5
@s0,
-
0.866 sin
0,]
=
MRod[
r.s ms
(0, +
90")
-
0.866
sin
(0,
+ 90")l
=
MRola[-l.s
(-sin
E)
-
0
866
(cos
0,)]
C,
=
MR@?o[1.5
sin
0r
-
0
866 cos 0'l
c,
-
MR:d'aI
Ls
cos 2€:
r
o.8bo
sin
2eJ
'r'
,?,,Ii.*iL1i|,Li #"*
"Buick\
Ncw
Even
Firins
ec v'6 E
ene
" sAE
r{rnr
V
ENGINI4
r,
=
-L-11.5
cos
26r
-
0
866
sb
20J
,
A5
c"ri
be
srer-froD
rh.
alalFis
above,
tb.
pm:rt
d
s€.od.t
rry
tor6
ar€
balanred
for
each
t-t .
Ho-**,
,r,"
i.i-,,y
ard
s€.Dndar,
dr
b3dly
our
or
balaoa
for
u",r
r""r..
"nJ,r.,.ri
"
oo
",y
or.r"ry
lnis
a
was
doDe
tor
rhe
V,8
encine
-;"ddr,i;;
;;'#qu" ,;'"lli',r,"
u",-*
"r,r,.
i.,€nia rorce
and
oi rne
v-o
engne,
rs
irrerenirg
ro
onsider
rhe deyetopEeDt of rhat
dd_rbe p,obreh
encoudre*d
;;
;';;;;;;;.q"e rariar,on du€
ro
config rarion.
A
VSenBine
wa\drletopcd
in
t9o2
br
Buict
wirh
a
lhreerlhrodc,n
l
spaced llu"
apdn
as shown
in
rig.
t0.22
Tiis
design
"irt
a
firiig
oilcr 0l
5-4-3 2
sives
uneven
inrerva,
"i
*""t.l"ri
i.tJ.,'" .f 150._90._-150._,
90'tetweei
cylrnder
firings
which
.esutrs
in a
high fluctuaring
output
,
As.a
relulr
or
lhe
fuet
.risrs
in
l97l_1974.
l]le
Buick
v.6
engrnc
w
t
'rodLced;n
lo/5 ds
a
means
of
improvinB
fuet
e@nomy I*esmoorh
o
t
trerc \a,
inp,o,ed
by
v/i*,g.i.r
.,.i,i
pi
ii.,"
,;.ruded
rnsrc
or
lul
rrrrcw
sa.
ddraned
15"
tor
rhe
cvl,ndp,.
"i"".i..r ".a
*,".a-.a
rS,fi
qlindeD
of
rhe
orher
bank.
Thir
p;od
rltra
c\liode's
.The
resulr
$"s
"
smo;rher
l;
u/l
Tlis engine
was
disconr;nued
in
196?
earlier c.ankshair
desrSn.
-
8/9/2019 Reciprocating Masses 1
11/11
532
BAI,ANCE OF MACHINERY
BALANCE
OF
FOUR.BAR
TINKAOFJ
533
es
offour
and
six
cylindem
arecomjrotrty
applicarrons.
An
eighrcylinder
opposed
Figures
10.24
and
10 25
show
rtetrinder
,lTangemar
for
ihe
four-cyrrnder
and six-cylinde.
opposed
ensines
FIGUR
IOI4
10.9 OPPOSED
ENGINES
As
shown
in Fig- 10 23, the
opposed cngrne
consisLs of lwo
banks
of cylinders,
or
two
inli.e
engines
oo opposite
sides of
the
crankshaft
in a
common
horizonral
plane
The
opposed engine is
a special
case
of
the
V
ensine
rD
which
P
=
180'
and
rhe
dete.m;nation of balane or
unbalance
may be
dade
as h V
engines.
The
resultant
sbaking
fofce
S
and the resultant unbalanced
axial
momenr
C
lie
10.10
BALANCE
OF
FOUR.BAR
LINKAGES
l.
Equivalent
toral
mass
Ma+Ms=M1
2.
€quivalenr
mass
cenler
MAIA
MBLS
=
O
3.
Equivalenr
moment
of rne
a
M^l1a
+
Mstls
=
I
FIGURE
10.25
L
FIGURE
10.26