reciprocating masses 1

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


2/11

"*.,(-.'.:-.,,)

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


3/11

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


4/11

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


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


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


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


8/11

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


9/11

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


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.


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

reciprocating masses 1

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