Dynamic balancing of irregular shafts
by Ralph C Challender
A THESIS Submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree
of Master of Science in Mechanical Engineering at Montana State College
Montana State University
© Copyright by Ralph C Challender (1951)
Abstract:
A production-line balancing machine, obtained through the War Assets Administration and received at
Montana State College, is discussed in this paper. A discussion of the general problem of balancing
along with the application of the principles of mechanics applying to balancing is given.
Some of the various types of balancing machines are described and the features of each are discussed.
Apparatus, which can be used to determine unbalance and indicate the corrections to be made without
removing the part from its location in a machine, is described.
Results obtained from the production-line balancing machine are included and discussed. The results
obtained seem to indicate certain undesirable characteristics of the machine. An attempt is made to
evaluate possible reasons for these characteristics. DYNAMIC BALANCING OF IRREGULAR SHAFTS
by
RALPH C. CHALLENDER
A THESIS
Submitted, t o th e Graduate F a c u l t y
in
p a r t i a l f u l f i l l m e n t o f the requirements
f o r the d e g r e e o f
M aster o f S c i e n c e i n M e c h a n ic a l E n g i n e e r i n g
at
Montana S t a t e C o l l e g e
Approved:
II%a , Major Department
Chairman, Examining Committee
T 'd u ate /D iv isio n
Bozeman, Montana
May, 1951
. A' 3 7f
Of
^
- 2‘
TABLE OF CONTENTS
Acknowledgment ................................................................................................
3
A b s t r a c t .................................................................................
4
I n t r o d u c t i o n .....................................................................................................
5
B a l a n c i n g ..........................................................................................
7
B a l a n c i n g Machines ....................................................................................... 17
The s u r p l u s M a c h i n e ..................................................................................... 24
A n a l y s i s o f R e s u l t s ..................................................................................... 27
D i s c u s s i o n ............................................................................................................. 38
Appendix ..........................................................
. ........................
. . . . 4 0
L i t e r a t u r e C i t e d and C o n s u l t e d ................................................ ....
. 47
100306
-3ACKNOWLEDOMENT
T h i s t h e s i s was unde rtak en a t t h e s u g g e s t i o n o f Dr. H.
F , M u l l l k i n 1 Head o f t h e M e c h a n ic a l E n g i n e e r i n g Department
a t Montana S t i t e C o l l e g e .
f u l l y a c k n o w le d g e d ,
H is h e l p and guid ance a r e g r a t e ­
Thanks a r e a l s o due t o P r o f e s s o r F . C,
Komann o f t h e M e c h a n ic a l E n g i n e e r i n g Departm ent, Dr. Henry
A n t o s i e w i c z o f t h e M athem atics D e p a r tm en t, and P r o f e s s o r J .
B . T. Downs, f o r m e r l y o f t h e M e c h a n ic a l E n g i n e e r i n g Depart­
m en t.
The a u t h o r w i s h e s t o e x p r e s s h i s a p p r e c i a t i o n t o
o t h e r s t a f f members f o r v a l u a b l e h e l p and s u g g e s t i o n s .
- UADSTRiiCT
A p r o d u c t i o n - l i n e b a l a n c i n g m a ch in e , o b t a i n e d through
t h e War A s s e t s A d m i n i s t r a t i o n and r e c e i v e d a t Montana S t a t e
C o lle g e , i s d is cu sse d in t h i s paper.
A d i s c u s s i o n o f the
g e n e r a l problem o f b a l a n c i n g a l o n g w i t h the a p p l i c a t i o n o f
t h e p r i n c i p l e s o f m ec h a n ic s a p p l y i n g t o b a l a n c i n g i s g i v e n .
Some o f t h e v a r i o u s t y p e s o f b a l a n c i n g m a ch ine s are
d e s c r i b e d and t h e f e a t u r e s o f e a c h are d i s c u s s e d .
Apparatus,
which can be used t o d e t e r m in e un balance and i n d i c a t e the
c o r r e c t i o n s t o be made w i t h o u t removing t h e p a r t from i t s
l o c a t i o n i n a m a ch in e , i s d e s c r i b e d .
R e s u l t s o b t a i n e d from the p r o d u c t i o n - l i n e b a l a n c i n g
machine are i n c l u d e d and d i s c u s s e d .
The r e s u l t s o b t a i n e d
seem t o i n d i c a t e c e r t a i n u n d e s i r a b l e c h a r a c t e r i s t i c s o f the
m a ch in e .
An a tt e m p t i s made t o e v a l u a t e p o s s i b l e r e a s o n s
for these c h a r a c t e r is t ic s .
-5 INTRODUCTION
The purp ose o f the r e s e a r c h and work l e a d i n g t o t h i s
t h e s i s was t o i n s t a l l and p l a c e i n o p e r a t i o n a b a l a n c i n g
machine r e c e i v e d from war s u r p l u s .
The machine was d e s i g n e d
and b u i l t by a company engaged i n t h e manu facture o f a i r ­
cra ft engines.
I t was used i n t h e a c t u a l p r o d u c t i o n l i n e t o
b a l a n c e t h e c r a n k s h a f t o f a 12 c y l i n d e r , V - t y p e e n g i n e .
I n o r d e r t o adapt t h e machine f o r l a b o r a t o r y e x p e r i m e n t s ,
a s t r a i g h t s h a f t was u s e d .
J i x d i s k s were made and p l a c e d
a l o n g the s h a f t l e n g t h and h o l e s were d r i l l e d i n t h e s e d i s k s ,
p a r a l l e l t o t h e s h a f t a x i s a t a u n ifo rm d i s t a n c e from the
c e n t e r s , t o be used f o r t h e a d d i t i o n o f w e i g h t s .
A f t e r t h e p r e l i m i n a r y work was f i n i s h e d and t h e s h a f t
and d i s k s were i n s t a l l e d , t h e a c t u a l o p e r a t i o n o f t h e
b a l a n c i n g machine was o b s e r v e d and s t u d i e d t o d e t e r m i n e t h e
c h a r a c t e r i s t i c s o f th e m ach ine.
At t h i s p o i n t c e r t a i n r e s u l t s
were q u e s t i o n e d and an a t t e m p t t o e v a l u a t e t h e r e s u l t s was
made.
One o f t h e primary p u rp o ses o f a b a l a n c i n g machine i s
t o d e t e r m in e t h e c o r r e c t i o n s needed t o b a l a n c e t h e u n i t in
q u e s t i o n i n the most r a p i d and e c o n o m i c a l f a s h i o n p o s s i b l e .
The b a l a n c i n g machine w i l l a c c o m p l i s h the purpose o f d e t e r ­
m ining t h e r e q u i r e d c o r r e c t i o n s , but t h e proced ure i n v o l v e d
i s n o t o f t h e e a s i e s t and most d i r e c t n a t u r e .
The c h a r a c ­
t e r i s t i c s p r e v i o u s l y r e f e r r e d t o are such t h a t t h e magnitude
-6-
o f t h e un b a la n ce a f f e c t s t h e a n g u l a r p o s i t i o n o f the r e a d i n g
of the unbalance.
In t h e a c t u a l b a l a n c i n g o f r o t a t i n g u n i t s i t i s o f t e n
n o t p r a c t i c a b l e or p o s s i b l e t o c o m p l e t e l y remove o r make
a l l o w a n c e f o r the e n t i r e un balan ce p r e s e n t .
I n t h i s paper
t h e proced ure i n v o l v e d w i l l be s a t i s f a c t o r y f o r th e p a r t i c u l a r
c a s e when t h e r o t a t i n g p a r t can be assumed c o m p l e t e l y r i g i d .
7/hen c a s e s a r e i n v o l v e d where t h e e l a s t i c d e f l e c t i o n o f the
p a r t s sh o u ld be c o n s i d e r e d t h e problem becomes more complex
i n n a t u r e and u s u a l l y b a l a n c i n g can o n l y be done f o r a
p a r t i c u l a r sp eed or range o f s p e e d s .
Standard a b b r e v i a t i o n s a r e used f o r terms i n v o l v e d i n
the d is c u s s io n .
When a new a b b r e v i a t i o n i s i n t r o d u c e d , a
b r i e f d e s c r i p t i o n f o l l o w s t h e formula or e q u a t i o n where t h e
symbol i s u s e d .
R e f e r e n c e i s made t o the so u r c e o f some o f
t h e s t a t e m e n t s and e q u a t i o n s u s e d . I n t h e B a l a n c i n g s e c t i o n ,
ho w ev er, no s p e c i f i c r e f e r e n c e s a r e made s i n c e th e m a t e r i a l
i s p r e s e n t e d i n a number o f books d e a l i n g w i t h t h e s u b j e c t o f
v ib ration s.
-7BALiiNCINGr
Moving p a r t s o f a machine have i n common, a s a r u l e ,
e i t h e r r e c i p r o c a t i n g m o tio n or a m o t i o n o f r o t a t i o n such
a s t h a t o f t h e r o t o r o f an e l e c t r i c m otor.
I f t h e moving
p a r t s are no t b a l a n c e d , f o r c e s must be a p p l i e d t o produce
th e a c c e l e r a t i o n o f t h e moving m a s s e s from t h e s t a t i o n a r y
p a r t s o f t h e m a ch in e , such as t h e b e a r i n g s and t h e machine
frame.
It is
in su pp lyin g th e se a c c e le r a t i n g f o r c e s that
s e r i o u s t r o u b l e may a r i s e , such a s v i b r a t i o n s i n t h e machine
he a v y b e a r i n g p r e s s u r e , which cause undue wear; d e f e c t i v e
work w i t h g r i n d i n g d i s k s ; and d e f e c t i v e l u b r i c a t i o n .
It is ,
t h e r e f o r e , o f g r e a t importance t o b a l a n c e or n e u t r a l i z e
t h e s e f o r c e s i n v a r i o u s t y p e s o f m a c h in e s .
I n t h i s p a p e r , m o t i o n s o f r o t a t i o n o n l y w i l l be con­
sid ered .
The r o t a t i n g p a r t s o f a machine may be i n s t a t i c
or dynamic b a l a n c e .
S t a t i c balance e x i s t s in a r o t a t i n g
p a r t when t h e f o r c e s which a c t on t h e p a r t , i f t h e p a r t i s
not running, are in e q u ilib r iu m r e g a r d le s s o f the p o s i t i o n
i n which t h e p a r t i s p l a c e d .
Dynamic ba la n ce e x i s t s i n th e
s y s t e m when the f o r c e s e x e r t e d by t h e m asses o f t h e moving
s y s t e m on t h e s t a t i o n a r y p a r t s o f t h e machine are i n e q u i l ­
i b r i u m , and h e n c e e x e r t no v a r y i n g r e s u l t a n t f o r c e s on the
s t a t i o n a r y p a r t s o f t h e machine.
From an a n a l y s i s o f a s i m p l e r o t o r c o n t a i n i n g o n l y one
unbalanced w e i g h t , the e q u a t i o n o f t h e r e s u l t i n g f o r c e may
be d e t e r m i n e d .
Assume t h a t a s i n g l e w e i g h t o f m a g n it u d e , W,
i s l o c a t e d a t a d i s t a n c e , r , from t h e c e n t e r o f r o t a t i o n o f
a -'Jlind ^r or d i s k .
v<hcn the d i s k or c y l i n d e r i s r o t a t e d ,
the unbalanced w e i g h t w i l l have an a c c e l e r a t i o n toward the
c e n t e r o f th e body.
In orde r t o s u p p l y t h i s c e n t r i p e t a l
a c c e l e r a t i o n the b e a r i n g s s u p p o r t i n g the r o t a t i n g d i s k or
cv l i n g e r and the a s s o c i a t e d s h a f t and j o u r n a l s w i l l have t o
*
e ^ e r t f o r c e s t h a t w i l l overcome or b a la n ce t h e r e s u l t i n g
i n e r t i a or c e n t r i f u g a l f o r c e c a u se d by the un balan ced w e i g h t .
The c e n t r i f u g a l f o r c e may be r e p r e s e n t e d by t h e e q u a t i o n
F ""-g*r 00 where W i s t h e un balan ced w e i g h t i n po unds, g i s
t h e a c c e l e r a t i o n due t o g r a v i t y e x p r e s s e d i n f e e t p er secon d
p er s e c o n d , r i s t h e d i s t a n c e from t h e c e n t e r o f r o t a t i o n t o
the w eight expressed in f e e t , and^ i s the angular v e l o c i t y
i n r a d i a n s pe r s e c o n d .
I f a c o n a i t i o n o f t h i s kind e x i s t s i n a machine the
f o r c e s r e q u i r e d t o r e s i s t the c e n t r i f u g a l f o r c e c a u se d by
t h e un balanced w e i g h t w i l l bo t r a n s m i t t e d from t h e r o t a t i n g
body t o t h e frame o f th e m a ch in e .
The f o r c e e q u a t i o n shows
t h a t t h e magnitude o f t h e r e s u l t i n g f o r c e s w i l l v a r y d i r e c t l y
w i t h the square o f t h e sp eed o f the r o t a t i n g p a r t .
F ig u r e I
shows th e v a r i a t i o n o f t h e un balanced f o r c e w i t h th e speed o f
r o t a t i o n f o r a u n i t unbalance o f I o u n c e - i n c h .
An o u n c e - i n c h
i s t h e s t a n d a r d u n i t used t o measure unbalance and i s d e f i n e d
a s a w e i g h t o f one ounce p l a c e d a t a d i s t a n c e o f one in c h
~9—
o.
/
CD
/
/
CENTRIFUGAL
FORCE - POUNDS
O
IO
~~r
/
O
/
/
/
/
/
O
rO
/
/
/
O
CSJ
/
/
/
/
z
IOOO
2000
3000
4000
5000
REVOLUTIONS PER MINUTE
VARIATION
FIG. I
:
OF FORCE
WITH
SPEED
6000
-10
from, t h e c e n t e r o f r o t a t i o n .
Unbalance o f t h i s n a tu re can be d e t e c t e d and c o r r e c t e d
by s i m p l e means.
By t a k i n g t h e r o t a t i n g body and a l l o w i n g i t
t o r o t a t e f r e e l y w i t h i t s a x i s i n a h o r i z o n t a l p l a n e the he av y
s p o t w i l l come t o r e s t a t th e low s i d e i n d i c a t i n g t h a t w e ig h t
s h o u l d be removed a t t h i s l o c a t i o n or t h a t w e i g h t sh o u ld be
added d i a m e t r i c a l l y o p p o s i t e .
The magnitude o f t h e weight
removed or t h e w e i g h t added s h o u l d be o f such a q u a n t i t y t h a t
t h e e q u a t i o n W1 r,
= W2 r^ i s f u l f i l l e d .
In t h i s e q u a t i o n W1
r e p r e s e n t s th e unbalanced w e i g h t ; r, , t h e r a d i a l d i s t a n c e o f
t h e unbalanced w e i g h t ; and 'W
2 and r 2 t h e w e i g h t and r a d i a l
d is ta n c e r e s p e c t i v e l y o f the balancing w eig h t.
T h i s sim ple
c a s e i s an example o f s t a t i c b a l a n c i n g s i n c e the unbalance
can be d e t e c t e d by s t a t i c means.
A l s o th e dynamic f o r c e s
c a used by the un balanced w e i g h t i n a r o t a t i n g body w i l l be
b a l a n c e d by the a d d i t i o n or removal o f a s i n g l e w e i g h t .
T h is
can be shown by a comparison o f t h e e q u a t i o n s i n v o l v e d i n th e
two c a s e s .
In t h e c a s e o f r o t a t i n g unbalance t h e c o r r e c t i o n
i s made by a dding w e i g h t d i a m e t r i c a l l y o p p o s i t e t h e i n i t i a l
un bal anced w e i g h t or by removing w e i g h t a t t h e same angular
location .
The v a l u e o f the w e ig h t t o be added or removed
can be d e t e r m in e d by e q u a t i n g th e c e n t r i f u g a l f o r c e caused
by t h e un balan ced w e i g h t t o th e c e n t r i f u g a l f o r c e caused by
balancing w eight.
S i n c e th e b a l a n c i n g w e i g h t i s added
d i a m e t r i c a l l y o p p o s i t e , t h e f o r c e s a r e c o l l i n e a r and the
-11-
resu ltant i s zero.
relation sh ip
E q u a t in g t h e s e f o r c e s , we o b t a i n the
r,
.
S i n c e the a n g u l a r s p e e d , ,
i s t h e same f o r both w e i g h t s i t can be e l i m i n a t e d from the
e q u a t i o n , l e a v i n g .7, r , » Wx r x , which i s t h e same e q u a t i o n
as f o r t h e s t a t i c b a l a n c e c a s e .
In t h e p r e v i o u s d i s c u s s i o n , i t was assumed t h a t the r e ­
moval or a d d i t i o n o f t h e w e i g h t n e c e s s a r y t o a c h i e v e the
s t a t i c b a l a n c e was a p p l i e d i n t h e same p l a n e , p e r p e n d i c u l a r
t o t h e s h a f t a x i s , a s t h e un balanced w e i g h t .
I f t h e method
o f s t a t i c b a l a n c i n g i s a p p l i e d t o r o t a t i n g b o d i e s t h a t are
o f a c o n s i d e r a b l e l e n g t h , the p o s i t i o n o f t h e proper p l a n e ,
a l o n g t h e s h a f t a x i s , f o r t h e a d d i t i o n o f th e c o r r e c t i n g
w e i g h t , w i l l no t be i n d i c a t e d .
Assuming t h a t th e method i s
used f o r b o d i e s o f c o n s i d e r a b l e l e n g t h , the r e s u l t may be
more u n d e s i r a b l e than t h e i n i t i a l u n b a l a n c e ,
I f a w eig ht
o f t h e pr o pe r magnitude and r a d i a l d i s t a n c e i s added , u n l e s s
t h e p l a n e o f t h e w e i g h t i s t h e same p la n e a s t h a t o f the un­
b a l a n c e , a dynamic c o u p l e w i l l be i n t r o d u c e d .
Since i t i s
n o t p o s s i b l e t o d e t e r m i n e t h e l o c a t i o n o f t h e pr o pe r p la n e t o
make c o r r e c t i o n s by s t a t i c means, a d i f f e r e n t procedure i s
r e q u i r e d f o r b a l a n c i n g l o n g r o t a t i n g b o d i e s , e v e n though t h e
u n ba la n ce may be c a u se d by a s i n g l e w e i g h t .
I n f i g u r e 2 t h e s i m p l e c a s e o f s t a t i c un b a la n ce i s
illu stra ted .
F i g u r e 3 shows t h e c o r r e c t i o n o f t h i s unbalance
by t h e a d d i t i o n o f a s i n g l e w e i g h t p l a c e d d i a m e t r i c a l l y
-12-
*
FIG. 2
STATIC
UNBALANCE
FIG. 3
STATICALLY
FIG 4
STATIC BALANCE
WITH
DYNAMIC UNBALANCE
BALANCED
-13-
o p p o s i t e t h e unbalanced w e i g h t .
F ig u r e 4 shows t h e r e s u l t
o f i n t r o d u c i n g a w e i g h t o f c o r r e c t magnitude and r a d i a l
d i s t a n c e , but n o t i n t h e same p l a n e a s th e o r i g i n a l un­
b a la n c e d w e i g h t .
The g e n e r a l c a s e , i n c l u d i n g both t y p e s o f u n b a l a n c e ,
w i l l now be c o n s i d e r e d .
Assume t h a t a r o t o r c o n t a i n s v a r i o u s
un balanced w e i g h t s a s shown i n f i g u r e 5 .
It i s p o ssib le to
d e t e r m in e i f t h e r o t o r i s i n s t a t i c b a la n ce by t h e method
described p r e v io u sly .
However, t h i s approach w i l l be by­
p a s s e d and a method o f c o r r e c t i n g t h e unbalance w i l l be d e s ­
c r i b e d t h a t d e t e r m i n e s t h e m a g n it u d e , a n g u la r p o s i t i o n , and
r a d i a l d i s t a n c e o f s i n g l e w e i g h t s t o be added i n two d i f f e r e n t
planes.
F i r s t , c h o o se p la n e s I and I I as t h e two c o r r e c t i o n
p l a n e s f o r r e a s o n s o f c o n v e n i e n c e i n a dding w e i g h t s .
c o n s i d e r p l a n e I as a r e f e r e n c e p l a n e .
Now,
Taking moments w i t h
r e s p e c t t o p l a n e I , f i n d the w e i g h t t o be added i n plane I I
t o make t h e r e s u l t a n t moment o f th e f o r c e s o f t h e unbalanced
w e i g h t s and t h e w e i g h t i n p l a n e I I z e r o .
This s t e p determines
t h a t t h e o n l y u n ba la n ce r e m a i n i n g w i l l have t o be l o c a t e d In
plane I ,
By c o n s i d e r i n g p la n e I I a s t h e r e f e r e n c e p l a n e , t h e
same p r o ced ure can be f o l l o w e d t o f i n d t h e b a l a n c i n g w e ig h t
required in plane I .
Thus, t h e s y s t e m can be brought t o a
c o n d i t i o n o f dynamic b a l a n c e w i t h a w e i g h t p l a c e d i n each o f
t h e c o r r e c t i o n or r e f e r e n c e p l a n e s .
In f i g u r e 6 , t h i s method
i s i l l u s t r a t e d , u s i n g v e c t o r s t o r e p r e s e n t the moments o f the
-14-
FIG. 5
FIG. 6
GENERAL CASE
MOMENTS
WITH
OF
UNBALANCE
RESPECT
TO
PLANE
I
-15-
f o r c e s with r e s p e c t to th e r e fe r e n c e p la n e s ,
A d i f f e r e n t meth od, which w i l l ^ i v e i d e n t i c a l r e s u l t s ,
i s now d e s c r i b e d .
C o n s i d e r , as b e f o r e , a r o t a t i n g body con­
t a i n i n g v a r i o u s unbalanced w e i g h t s and p i c k two r e f e r e n c e
p l a n e s , p l a n e s I and I I ,
Add e q u a l and o p p o s i t e f o r c e s i n
p l a n e I f o r each o f the f o r c e s due t o t h e unbalanced w e i g h t s .
T h is r e s u l t s i n a f o r c e and a c o u p l e r e p l a c i n g e a ch o f the
f o r c e s o f unbalance.
In o t h e r w o r d s , the o r i g i n a l unbalanced
f o r c e s have been r e p l a c e d by a c o u p l e and a f o r c e i n p la n e I .
N e x t , f i n d t h e r e s u l t a n t o f t h e c o u p l e s and add two f o r c e s ,
one i n p l a n e I and one i n p l a n e I I ,
t o form a c o u p l e t h a t
w i l l be o p p o s i t e t o the r e s u l t a n t o f t h e c o u p l e s .
system i s
Now, th e
i n b a l a n c e w i t h the e x c e p t i o n o f t h e f o r c e s i n t r o ­
duced i n p l a n e I ,
Find t h e r e s u l t a n t o f th e f o r c e s in t r o d u c e d
i n plane I and add a f o r c e i n p l a n e I t o b a l a n c e t h i s r e ­
su ltan t.
A c t u a l l y th e o n l y e x t e r n a l changes made t o the s y s ­
tem have been the a d d i t i o n o f a f o r c e i n p l a n e I I t o form p a r t
o f t h e c o u p l e added t o b a l a n c e t h e o r i g i n a l c o u p l e s and the
a d d i t i o n o f a f o r c e i n p l a n e I t o b a l a n c e t h e f o r c e s added
in plane I .
These two f o r c e s a r e added by i n c l u d i n g w e i g h t s
o f t h e c o r r e c t m a g n i t u d e , a n g u l a r p o s i t i o n , and r a d i a l
d is ta n c e in the r e fe r e n c e p la n e s .
Thus, the g e n e r a l c a s e o f u n ba la n ce i n a r o t a t i n g body
can be b a la n ce d by t h e a d d i t i o n o f a w e ig h t in e a ch o f two
d if f e r e n t reference planes.
As s t a t e d b e f o r e , i t i s assumed
-16-
t h a t t h e r o t a t i n g b o d i e s are r i g i d and t h a t e l a s t i c d e f l e c ­
t i o n s can be n e g l e c t e d .
r o t a t in g bodies i s
The problem o f b a l a n c i n g r i g i d
then one o f f i n d i n g t h e v a l u e s o f t h e
w e i g h t s , a n g l e s , and r a d i a l d i s t a n c e s o f w e i g h t s t o be
added i n two d i f f e r e n t p l a n e s p e r p e n d i c u l a r t o t h e a x i s o f
rotation .
-17-
BALANCING MACHINES
A d i s k mounted on a r o t a t i n g s h a f t which c o n t a i n s an
unbalanced w e i g h t w i l l v i b r a t e and a l s o cause r o t a t i n g f o r c e s
t o be t r a n s m i t t e d t o t h e b e a r i n g s .
The v i b r a t i o n and the
b e a r i n g f o r c e s can be made t o d i s a p p e a r by a t t a c h i n g a sm a ll
w e i g h t on t h e " l i g h t s i d e " o f t h e d i s k .
The d e t e r m i n a t i o n o f
t h e l o c a t i o n o f the c o r r e c t i n g w e i g h t i s a s i m p l e problem i n
sta tics.
The s h a f t can be p l a c e d on p a r a l l e l h o r i z o n t a l r a i l s
and t h e h e a v y s i d e w i l l t h e n r o l l down.
A t r i a l correction
w e i g h t can be added t o the t o p a i d e o f the d i s k and the pr o ­
cedure c o n t i n u e d by c h a n g in g th e w e i g h t u n t i l t h e s h a f t and
d i s k a re i n a p p a re n t e q u i l i b r i u m .
Then th e s h a f t w i l l remain
i n any p o s i t i o n and the o r i g i n a l unbalanced w e i g h t i s
a p p r o x i m a t e l y b a la n ce d by t h e c o r r e c t i o n w e i g h t .
The s h a f t
i s p l a c e d on the p a r a l l e l h o r i z o n t a l r a i l s t o e l i m i n a t e a s
much a s p o s s i b l e the f r i c t i o n a l r e s i s t a n c e t o r o t a t i o n .
The s e t o f h o r i z o n t a l r a i l s i s t h e s i m p l e s t s t a t i c
b a l a n c i n g machine i n e x i s t e n c e .
For r o t a t i n g b o d i e s i n which
th e main p a r t o f t h e mass i s i n d i s k form, s t a t i c b a la n ce i s
the only balance th a t i s n e c e s s a r y to insure v i b r a t io n f r e e
operation at a l l speeds.
I n c a s e t h e r o t a t i n g body i s e l o n g a t e d and t h e mass or
w e i g h t cannot be c o n s i d e r e d a s c o n c e n t r a t e d i n a t h i n d i s k ,
t h e c o n d i t i o n o f s t a t i c b a la n c e i s no t s u f f i c i e n t t o e l i m i n a t e
vib ration .
As shown i n f i g u r e 4 , a r o t o r may be i n s t a t i c
—1 8 -
b a l a n c e and y e t e x e r t r o t a t i n g f o r c e s on th e b e a r i n g s when
the sh a ft i s r e v o lv in g ,
A dynamic b a l a n c i n g machine or
arrangement i s r e q u i r e d t o d e t e r m in e t h e s i z e and l o c a t i o n
o f the unbalance,
A s i m p l e t y p e o f a dynamic b a l a n c i n g machine i s now
described.
The r o t o r i s p la c e d i n two b e a r i n g s which are
attached to a l ig h t t a b l e .
The t a b l e i s su p p o r t e d on s p r i n g s
a t t h e f o u r c o r n e r s and can be made r o t a t a b l e about e i t h e r
one o f two a x i s , l o c a t e d in t h e p l a n e s t h a t w i l l be used as
correction planes.
The r o t o r i s d r i v e n by a f l e x i b l e s h a f t
or b e l t , or can be d r i v e n by a motor r i g i d l y mounted on th e
tab le.
The l a t t e r scheme i s no t a s d e s i r a b l e a s t h e o t h e r s
s i n c e I t i n c r e a s e s t h e w e i g h t o f t h e t a b l e which d e c r e a s e s
t h e s e n s i t i v i t y o f t h e m a ch in e .
as fo llo w s.
The b a l a n c i n g proced ure i s
The t a b l e i s h e l d a t one o f t h e a x i s and r e l e a s e d
a t t h e o t h e r and t h e r o t o r s o t i n m o t i o n .
A fter the unit
comes t o r e s o n a n c e , t h e a m p lit u d e o f t h e v i b r a t i o n a t th e f r e e
end o f t h e t a b l e can be measured and t h e p o s i t i o n o f the
maximum a m p litu d e d e t e r m i n e d .
The p o s i t i o n o f t h e maximum
a m p l i t u d e w i l l depend upon t h e v i b r a t i o n c h a r a c t e r i s t i c s o f
t h e t a b l e and r o t o r ,
As we w i l l show i n t h e s e c t i o n d e v o t e d
t o an a n a l y s i s o f t h e s u r p l u s machine i n s t a l l e d , t h e s h a f t
w i l l d e s c r i b e t h e he a v y s p o t when i t runs below t h e c r i t i c a l
sp e e d ; t h e l i g h t s p o t when i t runs above th e c r i t i c a l sp eed;
and a p o i n t which i s 90 d e g r e e s b e hin d the he a v y s p o t when i t
-19-
i s run a t e x a c t l y th e c r i t i c a l s p e e d .
In a c t u a l p r a c t i c e
o h i s /aethod i s v e ry i n a c c u r a t e s i n c e the a n g l e i s dependent
upon the sp eed o f th e r o t o r and v a r i e s through a maximum o f
130 d e g r e e s ,
' m° r e s a t i s f a c t o r y m e t h o d u s e s the b a s i c arrangement
d e s c r i b e d a b o v e , but do es no t depend upon t h e a n g l e o f the
maximum a m p l i t u d e .
The method c o n s i s t s o f o b s e r v i n g the
a m p litu d e o f v i b r a t i o n o n l y and t h e n making two a d d i t i o n a l
runs w i t h a s m a l l unbalance w e i g h t added a t a c o n v e n i e n t
l o c a t i o n f o r t h e f i r s t run and f o r t h e second run t h e un­
b a l a n c e w e i g h t i s added i n a d i a m e t r i c a l l y o p p o s i t e l o c a t i o n ,
it
can be shown t h a t t h e t h r e e t r i a l runs and t h e r e s p e c t i v e
a m p l i t u d e s w i l l d e t e r m in e t h e unknown unbalance w i t h a
p o s s i b i l i t y o f one a m b i g u i t y .
T h i s a m b i g u it y can be removed
by a i o u r t h run which w i l l a l s o c h e ck the
-LIr su ohree r u n s .
a c c u r a c y o f the
I f a f t e r t h e m o t i o n s d e s c r i b e d above have
been made, and t h e c o r r e c t i o n w e i g h t added, v i b r a t i o n i s s t i l l
p r e s e n t t h e p r o c e d u r e may be r e p e a t e d once m ore,
The v i b r a ­
t i o n , i f p r e s e n t , s h o u ld be much l e s s than t h e o r i g i n a l
vib ration .
The p r o ced ure d e s c r i b e d f o r b o th o f the above methods
must be a p p l i e d t w i c e , once w i t h t h e a x i s i n p l a n e I a s the
r o t a t i n g a x i s and a g a i n w i t h t h e a x i s i n pla n e I I a s the
TT
ben Hartog, J. P . , !947 I.^wil«UICAL VlBkhTlONb,
PP• 2 9 6 - 2 9 7 , McGraw-Hill Book C o . , I n c . , New York, N. Y.
—S c ­
rota t i n g a x i s .
7/he re l a r g e numbers o f u n i t s have t o be
b a l a n c e d a s a r e q u i r e m e n t o f p r o d u c t i o n , t h e two methods
d e s c r i b e d t a k e t o o much t i m e .
A s i m i l a r arrangement i s used i n a n o t h e r t y p e o f
b a l a n c i n g machine^-.
The p a r t t o be b a la n c e d i s su pp orted
and r o t a t e d on a p i v o t e d c a r r i a g e .
One o f t h e c o r r e c t i o n
p l a n e s i s l o c a t e d o ver t h e p i v o t a x i s and th e unbalance i s
de ter m in e d f o r t h i s p o s i t i o n o f t h e u n i t and t h e n w i t h the
sp ecim en t urned end f o r end, t h e unbalance i n t h e o t h e r
c o r r e c t i o n p l a n e can be d e t e r m i n e d .
The machine c o n t a i n s a
c o m p en sa tin g w h e e l r o t a t i n g w i t h t h e p a r t i n which both the
magnitude and a n g u l a r p o s i t i o n o f a b a l a n c i n g w e i g h t may be
changed w h i l e t h e r o t o r and com p en sa ting w h e e l a r e r u n n i n g .
By v a r y i n g t h e a n g u l a r p o s i t i o n o f t h e b a l a n c i n g w e i g h t or
w e i g h t s t h e o p e r a t o r can s e t t h e w h ee l t o g i v e t h e s m a l l e s t
a m plitud e o f v i b r a t i o n .
Then by v a r y i n g t h e r a d i a l d i s t a n c e
o f t h e b a l a n c i n g w e i g h t s the s y s t e m can be brought i n t o
balance.
The method d o e s n o t r e q u i r e t h a t t h e c o r r e c t i v e
w e i g h t s be p l a c e d in t h e c o r r e c t i o n p l a n e s .
S i n c e t h e two
unbalanced f o r c e s a r e determ ined and assum ing t h e r o t a t i n g
u n i t i s r i g i d , any two w e i g h t s t h a t w i l l g i v e t h e same e x ­
t e r n a l r e s u l t may be added.
A method^, which i s s i m p l e r than t h e methods p r e v i o u s l y
TI
2.
Thompson,
T . , 194^, i.x.C!. ::i /,L VISR-.TICNS
p . 9 2 , P r e n t i c e - H a l l , I n c . , New York, New York
D e n H a r t o g , J . P . , 19 4 7 , pp. 3 0 0 - 3 0 2
9
-21-
d e s c r l b e d , can be used i f means are a v a i l a b l e t o measure
t h e a n g l e between t h e l o c a t i o n o f t h e unbalance i n a r o t o r
and the p o s i t i o n o f th e maximum a m p l i t u d e .
Let the rotor
be su p p o r t e d i n two b e a r i n g s , a and b , which a r e s t i f f l y su p p o rte d i n the v e r t i c a l d i r e c t i o n and f l e x i b l e
horizontal d ir e c tio n .
i n the
Two b a l a n c i n g p l a n e s w i l l a g a i n be
used and t h e b a l a n c i n g or c o r r e c t i o n p l a n e s t a k e n a t con­
v e n i e n t l o c a t i o n s f o r th e a d d i t i o n or removal o f w e i g h t s ,
Imagine t h e r o t o r t o be i d e a l l y b a la n ced so t h a t no v i b r a t i o n
a pp ea r s w h i l e i t i s r o t a t i n g .
Now a u n i t w e i g h t i s added a t
c o r r e c t io n plane I in a d e f i n i t e angular l o c a t i o n .
T h is w i l l
c a u se a v i b r a t i o n i n both b e a r i n g s and t h e s e v i b r a t i o n s are
d e n o te d
as Odltrand Sobr .
The f i r s t s u b s c r i p t
bearing
where t h e v i b r a t i o n
r e f e r s t o the
i s measured andt h e second sub­
s c r i p t r e f e r s to the l o c a t i o n o f the unbalance.
For the
general
c a s e , where damping i s p r e s e n t , t h e
numbers Z are
complex
b e c a u s e t h e damping and t h e speed o f t h e r o t o r d e t e r ­
mine a phase a n g l e between t h e u n ba la n ce f o r c e and t h e maximum
am plitude.
I n a s i m i l a r manner th e i d e a l r o t o r may be unbalanced
w i t h a u n i t w e i g h t i n p la n e I I 1 which w i l l ca use b e a r i n g
vib ration s
and
,
The f o u r numbers
t h e complex dynamic i n f l u e n c e numbers.
are known as
These f o u r i n f l u e n c e
numbers c o m p l e t e l y d e ter m in e t h e i n e r t i a and e l a s t i c pro­
p e r t i e s o f the s y s t e m f o r the sp eed a t which t h e y a r e d e t e r ­
—2 2 —
m ined.
They e r e i n d e p e n d e n t , h o w e v e r , o f t h e amount o f
unbalance p r e a e n t,
Now s u p p o s e t h a t an u n b a l a n c e , w h ic h n u m e r i c a l l y
a s w e l l a s a n g u l a r l y d i f f e r s fr o m t h e u n i t u n b a l a n c e , e x i s t s
in plane I ,
R e p r e s e n t i n g t h e u n b a l a n c e by a co m p lex num ber,
Uj , we s e e t h a t t h e r e s u l t i n g v i b r a t i o n a t b e a r i n g <su w i l l be
r e p r e s e n t e d by t h e p r o d u c t o f two complex n u m b e r s , SZ1jiul U1 .
The r e s u l t i n g v i b r a t i o n a t b e a r i n g Jr w i l l be r e p r e s e n t e d by
r o d UCt o f two complex n um bers OZTfarI j ^ .
notations i t
J i t h these
i s p o s s ib le to w r ite the e q u a tio n s f o r the
b e a r i n g v i b r a t i o n s i n t e r m s o f t h e f o u r complex num bers ^
t h e co m p lex n um bers U1
i n p l a n e s I and I I .
and U3t
and
r e p r e s e n t i n g t h e unbalance
L e t t i n g t h e co mplex B tia b era Vjiu and Vb
r e p r e s e n t t h e v i b r a t i o n s a t b e a r i n g s a and b r e s p e c t i v e l y , we
can w r i t e th e f o l l o w i n g e q u a t i o n s :
flu = c c CLX
-X+ 0 c O-D: l IL
The b e a r i n g v i b r a t i o n s ,
and V^
, can be
u r ed and
t h e p r o b l e m t h e n r e q u i r e s t h e s o l u t i o n o f t h e two e q u a t i o n s
c o n t a i n i n g t h e s i x known co m p lex n u m b ers f o r t h e two complex
n um bers
1Ur and ITlt w h ic h w i l l r e p r e s e n t t h e e q u i v a l e n t un­
b a l a n c e i n p l a n e s I and I I ,
B a l a n c i n g a p p a r a t u s I s c o m m e r c i a l l y a v a i l a b l e which
u t i l i z e s t h e t h e o r y m e n t i o n e d ab o v e f o r t h e p u r p o s e o f
b a l a n c i n g r o t o r s w i t h o u t r e q u i r i n g them t o be removed from
-23-
th e b e a r i n g s in which th e y n o rm a lly o p e r a t e .
The v i b r a t i o n s
a t t h e two b e a r i n g s o f a m ac h in e c a n be m e a s u r e d w i t h t h e use
o f an e l e c t r i c a l c i r c u i t w h ich e n a b l e s t h e v e r y s m a l l r e ­
s u l t i n g v i b r a t i o n s o f t h e m achin e f r a m e t o be a m p l i f i e d and
th e m agnitudes m easured.
The a n g l e o f t h e maximum a m p l i t u d e
can be m e a s u r e d e a s i l y by e m p l o y i n g a s t r o b o s c o p i c lamp which
i s made t o f l a s h a t t h e h i g h s p o t e v e r y r e v o l u t i o n o f t h e
ro to r.
V a r i o u s o t h e r m eth o d s c a n be employed t o d e t e r m i n e t h e
h ig h spot of the v i b r a t i o n .
-24
THE oURPLUo MACHINE
The s h a f t t o be b a la n c e d I s s e t i n b e a r i n g s r i g i d l y
fastened to a ta b le .
The t a b l e i s supp orted by f o u r h e l i c a l
s p r i n g s l o c a t e d a t the c o r n e r s .
The s h a f t i s d r i v e n e x t e r ­
n a l l y through u n i v e r s a l j o i n t s .
The u n i v e r s a l j o i n t s p r o v i d e
th e f l e x i b i l i t y n e c e s s a r y t o a l l o w t h e s h a f t and t a b l e t o
v i b r a t e w i t h o u t r e s t r i c t i o n from t h e d r i v i n g s o u r c e .
A
m e c h a n i c a l l i n k a g e i s c o n n e c t e d t o th e t a b l e a t e a c h o f t h e
r e a r c o r n e r s t o measure th e h o r i z o n t a l v i b r a t i o n s o f the
tab le.
Through t h e l i n k a g e from each c o rn er o f t h e t a b l e
m i r r o r s are g i v e n s m a l l r o t a t i o n s .
L i g h t h i t t i n g t h e m ir r o r s
i s r e f l e c t e d and t h e h o r i z o n t a l v i b r a t i o n s o f t h e t a b l e cause
t h e a n g l e s o f t h e m i r r o r s t o c h a n g e , thu s c h a n g i n g th e path
o f th e r e f l e c t e d l i g h t .
An a u t o s y n motor c o n n e c t e d t o the
d r i v i n g s h a f t d r i v e s i n turn an a u t o s y n motor wh ic h i s
g ea red t o a p o l i s h e d h e x a g o n a l s h a f t .
The l i g h t r e f l e c t e d
from t h e m i r r o r s I s i n tur n r e f l e c t e d from t h e r e v o l v i n g
h e x a g o n a l s h a f t and s t r i k e s a t a r g e t s c r e e n .
The two l i g h t
p a t t e r n s on t h e t a r g e t s c r e e n each have m o tio n s i n two
d irection s.
The h o r i z o n t a l m o tio n i s caused by t h e h o r i z o n t a l
v i b r a t i o n or movement o f t h e t a b l e and the v e r t i c a l m otion i s
due t o t h e r e v o l v i n g h e x a g o n a l m irr o r s h a f t and r e p r e s e n t s
the r o t a t i o n of the s h a f t .
The t a r g e t s c r e e n i s d i v i d e d i n
t h e v e r t i c a l d i r e c t i o n t o r e p r e s e n t one r e v o l u t i o n o f the
s h a f t and i n t h e h o r i z o n t a l d i r e c t i o n t o r e p r e s e n t t h e
—2 $ —
mag nitude o f t h e v i b r a t i o n s o f t h e t a b l e .
H e l i c a l s p r i n g s are a t t a c h e d t o the t a b l e a t both o f t h e
f r o n t c o r n e r s and s p r i n g s are e f f e c t i v e l y a t t a c h e d t o t h e r e a r
c o r n e r s t hr o ug h th e e x t e r n a l l i n k a g e s .
A c o n v e n i e n t means
i s p ro v ided by which t h e t e n s i o n i n th e r e a r s p r i n g s may be
v a r i e d i n o rder t o p o s i t i o n t h e l i g h t f o r each end o f the
t a b l e on t h e t a r g e t s c r e e n .
For c o n v e n i e n c e th e l i g h t f o r t h e
l e f t end o f t h e t a b l e i s w h i t e and f o r the r i g h t end o f t h e
table i s red.
A means o f l o c k i n g t h e t a b l e i s p r o v id e d t o a l l o w the
s h a f t t o be p l a c e d i n the b e a r i n g s and removed w i t h the t a b l e
r ig id ly h eld .
A c i r c u i t i s p r o v i d e d t o r e l e a s e t h e t a b l e when
t h e s h a f t i s up t o speed a f t e r s t a r t i n g and l o c k t h e t a b l e
a g a i n when t h e b a l a n c i n g machine i s turned o f f .
A reversing
arrangement i s p r o v id e d f o r t h e d r i v i n g motor and i s used as
a brake t o s h o r t e n the time r e q u i r e d f o r t h e s h a f t and d r i v i n g
mechanism t o come t o r e s t .
A method i s p r o v id e d f o r e f f e c t i v e l y s h i f t i n g the
l o c a t i o n o f t h e o b s e r v e d v i b r a t i o n s w i t h r e s p e c t t o the
le n g th o f the t a b l e .
The e l e c t r i c a l c o n t r o l s a r e m ainly
c o n t a i n e d i n a c o n t r o l box a t a r e a r c o rn er o f t h e machine
and th e s t a r t - s t o p s w i t c h i s l o c a t e d a t the f r o n t .
The b a l a n c i n g machine i s made t o pe rm it s i m u l t a n e o u s
b a l a n c i n g i n two p l a n e s s i n c e the t a b l e i s f r e e t o v i b r a t e
a t any or a l l l e n g t h w i s e p o s i t i o n s .
However, t h e auth o r
—2 6 —
was u n a b l e , by c o n s i d e r a b l e e x p e r i m e n t a t i o n , t o a c c o m p l i s h
the sim u ltan eou s b a la n cin g proced u re.
B u t , by r e s t r i c t i n g
t h e movement o f one end o f t h e t a b l e a t a t i m e , t h e s h a f t
was a p p r o x i m a t e l y b a l a n c e d .
From a comparison o f p h o t o g r a p h s ,
r e c e i v e d from t h e m a n u f a c t u r e r , o f a s i m i l a r machine and t h e
a c t u a l m a c h in e , i t
i s a pp a re nt t h a t t h e makers have made
c e r t a i n changes i n t h e d e s i g n o f t h e machine s i n c e t h e s u r p l u s
machine was made.
A few o f t h e c h a n g e s w i l l be d i s c u s s e d i n
a l a t e r s e c t i o n o f t h i s p a p er .
-27-
ANALYSIS OF RESULTS
C o n sid e r a v e r t i c a l s h a f t w i t h a d i s k a t t a c h e d a s shown
in f ig u r e 7 .
The d i s k c o n t a i n s an unbalanced w e i g h t which
moves t h e c e n t e r o f g r a v i t y , G, o f t h e combined d i s k and
w e i g h t away from th e g e o m e t r i c c e n t e r o f the d i s k , 3 , toward
t h e un balance w e i g h t .
C o n s id e r t h e d i s k t o be l o c a t e d midway
be tween two b e a r i n g s and r o t a t i n g w i t h a c o n s t a n t an g u la r
speed.
The s h a f t w i l l have a d e f l e c t i o n , when r o t a t i n g ,
caused by t h e un balance i n t h e d i s k .
Let t h e d e f l e c t i o n , r ,
be t h e d i s t a n c e from t h e g e o m e t r i c c e n t e r o f the d i s k t o the
i n t e r s e c t i o n o f t h e c e n t e r l i n e o f t h e b e a r i n g s and th e p l a n e
o f the d is k at 0 ,
The d i s k i s under t h e a c t i o n o f two f o r c e s ,
t h e s t o r i n g f o r c e o f t h e s h a f t , k r , and t h e c e n t r i f u g a l
force, m
neglected .
(r+e).
The e f f e c t o f g r a v i t y and f r i c t i o n i s
I n o rder f o r t h e s e two f o r c e s t o be i n e q u i l i b r i u m
t h e y must be c o l l i n e a r , e q u a l , and o p p o s i t e .
By e q u a t i n g t h e
two f o r c e s 1 and s o l v i n g f o r r , we o b t a i n
wherece*= ^k/m, t h e n a t u r a l f r e q u e n c y o f l a t e r a l v i b r a t i o n
o f th e d i s k and s h a f t a t z e r o s p e e d .
The e q u a t i o n i n d i c a t e s t h a t t h e c r i t i c a l s p e e d o f the
s h a ft i s equal to the natural frequency of l a t e r a l v ib r a tio n
TI
Thompson,
J. T . , 1 9 4 3 , p. 5 2 -8 3
—2 8 —
O SG
FIG. 7
ROTATING SHAFT
-29-
o f t h e s h a f t and d i s k .
For s p e e d s below t h e c r i t i c a l speed
t h e s h a f t and d i s k w i l l v i b r a t e w i t h the heavy s i d e out s i n c e
r is p ositive.
For s p e e d s above t h e c r i t i c a l r i s n e g a t i v e
and t h e s h a f t w i l l v i b r a t e w i t h th e l i g h t s i d e o u t .
For a s y s t e m , such a s u t i l i z e d i n the b a l a n c i n g m a c h in e ,
where the u n i t i s s u p p o rte d on s p r i n g s , i t can be shown! t h a t
t h e a n g l e o f t h e maximum a m p litu d e w i l l l a g b e hin d t h e un­
balance f o r c e .
The a n g l e w i l l v a r y be tween O and 180 d e g r e e s ,
d e p e n d in g on the damping p r e s e n t and th e s u p p o r t i n g s p r i n g
constants.
In t h e s u r p l u s b a l a n c i n g machine the t a b l e i s
f r e e t o move i n both t h e h o r i z o n t a l and v e r t i c a l d i r e c t i o n s .
Thus, s i n c e t h e s p r i n g c o n s t a n t s and t h e damping are d i f f e r e n t
f o r t h e two d i r e c t i o n s , th e a m p l i t u d e s o f t h e v i b r a t i o n s and
th e phase a n g l e s w i l l be d i f f e r e n t f o r th e two c a s e s .
The p r e v i o u s d i s c u s s i o n i s i n c l u d e d t o p r e s e n t t h e b a s i c
d i f f e r e n c e between t h i s b a l a n c i n g machine and t h e v a r i o u s
t y p e s d e s c r i b e d i n th e s e c t i o n on B a l a n c i n g M a c h in e s .
In
t h a t s e c t i o n , t h e d e v i c e s were assumed t o have f l e x i b i l i t y
i n o n l y one d i r e c t i o n , whereas t h e dynamic b a l a n c i n g machine
i n v e s t i g a t e d d o e s n o t l i m i t th e v i b r a t i o n s t o one d i r e c t i o n .
The r e s u l t s o f th e i n v e s t i g a t i o n are r e p r e s e n t e d by s i x
sample graphs rep ro d uced from t h e p a t t e r n s on t h e t a r g e t
s c r e e n f o r s i x d i f f e r e n t p o s i t i o n s f o r the a d d i t i o n o f known
unbalance.
Tl
The t r a c e s on th e l e f t s i d e o f th e s c r e e n
Thompson,
7. T . , 1 9 4 8 , pp. 7 2 - 7 5
-30-
20
40
60
80
IO O
120
140
160
UNBALANCE
6 OZ-IN----- 4 OZ-IN-------2 OZ-IN--------LOCATION
LEFT END
180
200
220
240
ANGLE - 30
260
280
300
320
SAMPLE CHART.
FIG. 8
340
- 3 1-
20(6
20
22a
40
UNBALANCE
240
260
280
6 OZ-IN---4 OZ-IN ——
2 OZ-IN---LOCATION
LEFT END
300
60
80
IO O
120
ANGLE-150
320
140
340
160
0
180
20
200
40
220
60
240
80
260
100
280
120
300
140
320
160
SAMPLE CHART.
FIG. 9
340
200
220
240
260
28c/
300
3 io
3410
UNBALANCE
6 OZ-IN---4 OZ-IN —
2 OZ-IN —
LOCATION
20
40
LEFT END
A N G L E -2 7 0 6
60
80
100
120
140
160
SAMPLE
CHART
FIG. 10
\
•**
200
-33-
20
220
40
240
60
260
80
280
IO O
300
120
320
140
340
160
0
20
40
60
80
UNBALANCE
6 OZ-IN---4 OZ-IN —
2 OZ-IN---LOCATION
RIGHT END
A N G L E -3 0
180
200
220
240
260
100
280
120
300
140
320
160
SAMPLE CHART
FIG. Il
340
I
-34~
20
40
60
80
IO O
120
140
160
UNBALANCE
180
6 OZ-IN
4 OZ-IN
2 OZ-IN
200
220
LOCATION
RIGHT END
240
A N G LE -150
260
280
300
320
SAMPLE CHART
340
FIG. 12
#
-35-
200
20
220
40
240
60
260
80
280
IO O
300
120
320
140
340
160
20
UNBALANCE
6 O Z - IN 4 OZ-IN—
2 OZ-IN—
40
LOCATION
0
RIGHT END
60
t
180
200
220
240
A N G L E -2 7 0
80
260
100
280
120
300
140
320
160
SAMPLE CHART
FIG. 13
340
-36r e p r e s e n t t h e v i b r a t i o n a t t h e l e f t end o f t h e t a b l e and th e
t r a c e s on t h e r i g h t s i d e i n d i c a t e t h e v i b r a t i o n s a t t h e r i g h t
end o f t h e t a b l e .
F ig u r e 8 shows th e r e s u l t s o f a dding t h r e e
d i f f e r e n t w e i g h t s , one a t a tim e t o t h e l e f t d i s k a t the
app roxim ate l o c a t i o n o f t h e l e f t l i n k a g e and a t an an g u la r
lo ca tio n of 30°.
F i g u r e 9 shows t h e r e s u l t s f o r t h e same
t h r e e w e i g h t s added t o th e same d i s k a t an a n g u l a r l o c a t i o n
of 150°.
F i g u r e 10 r e p r e s e n t s t h e r e s u l t s o b t a i n e d by a d d in g
t h e w e i g h t s a t 270° i n t h e same d i s k .
F ig u r e 11 shows the
r e a d i n g s o b t a i n e d by t h e w e i g h t s added a t t h e r i g h t end o f
t h e s h a f t a t a 30° a n g u l a r l o c a t i o n .
F ig u r e 12 shows the
r e s u l t s w i t h t h e w e i g h t s added a t t h e 150° a n g l e on t h e d i s k
a t t h e r i g h t end.
F i g u r e 13 r e p r e s e n t s the r e a d i n g s w it h t h e
w e i g h t s added a t t h e 2 7 0 ° l o c a t i o n on the r i g h t d i s k .
R e f e r r i n g t o f i g u r e s 8 - 1 0 , i t w i l l be s e e n t h a t the
magnitude o f t h e w e i g h t added a t any a n g u l a r l o c a t i o n on th e
l e f t d i s k has a d e f i n i t e e f f e c t upon t h e a n g l e o f t h e maximum
a m p litu d e as w e l l a s upon t h e a m p l i t u d e .
The r e a d i n g s o f th e
v i b r a t i o n s a t t h e r i g h t end o f t h e t a b l e cause d by the
a d d i t i o n o f u n ba la n ce w e i g h t s on t h e l e f t d i s k are o bserved
t o be a p p r o x i m a t e l y 180° out o f phase w it h t h e r e a d i n g s at
t h e l e f t end.
The l a g g i n g phase a n g l e o f t h e v i b r a t i o n s a t
t h e l e f t e nd, cause d by unbalance i n the l e f t d i s k , i s
depend en t upon t h e s i z e o f th e u n b a l a n c e .
This discrep ancy
cannot be d e f i n i t e l y e x p l a i n e d by t h e author and p o s s i b l e
-
37-
r e a s o n s f o r t h i s f a l l a c y a re g i v e n i n the D i s c u s s i o n .
F i g u r e s 11 - 1 3 , r e p r e s e n t i n g t h e v i b r a t i o n s caused by
un balance i n t h e d i s k l o c a t e d a t t h e r i g h t and, show t h a t the
v i b r a t i o n s o f t h e r i g h t and o f t h e t a b l e vary o n l y i n a m pli­
tud e f o r v a r i o u s v a l u e s o f unbalance a t a d e f i n i t e ang u la r
lo ca tio n .
T h i s phenomena i s i n agreement w i t h t h e m a t e r i a l
i n c l u d e d f o r t h e c a se o f s i m u l t a n e o u s b a l a n c i n g i n two p l a n e s
and d i s c u s s e d i n t h e s e c t i o n on B a l a n c i n g M a c h in e s .
The a u th o r found t h a t a p p a r e n t l y o n l y one o f t h e f o u r
dynamic i n f l u e n c e numbers u. remains c o n s t a n t f o r t h i s
b a l a n c i n g m ach ine.
Numerous t r i a l s were made i n a t t e m p t i n g
t o v a r y t h e a d j u s t m e n t s o f t h e b a l a n c i n g machine i n some
manner t h a t would g i v e c o n s t a n t v a l u e s f o r the dynamic
i n f l u e n c e numbers, but t o no a v a i l .
DISCUSSION
Two p o s s i b l e e x p l a n a t i o n s f o r t h e n o n - c o n s t a n t dynamic
i n f l u e n c e numbers od arc made by t h e a u t h o r .
The f i r s t r e a s o n i s s u g g e s t e d by the method o f supp ort
'iv e n the t a b l e .
As was p r e v i o u s l y m e n t i o n e d , t h e v i b r a t i o n s
are n o t r e s t r i c t e d t o one d i r e c t i o n and the r e s u l t i n g con­
f i g u r a t i o n o f th e v i b r a t i o n w i l l depend upon t h e ang u la r
speed and th e c o n s t a n t s o f t h e s y s t e m i n the two d i r e c t i o n s .
As an e x a m p le , c o n s i d e r the v i b r a t i o n i n th e v e r t i c a l d i r e c ­
t i o n t o have a ph ase a n g l e be tween 9 0 ° and 1 8 0 ° .
’,'/hen the
un balance i s i n a v e r t i c a l d i r e c t i o n the r e s u l t i n g a m plitude
w i l l be l a g g i n g and c o u l d p o s s i b l y have a n o t i c e a b l e e f f e c t
upon t h e a m p litu d e i n t h e h o r i z o n t a l d i r e c t i o n a t t h e p a r t ­
icu lar in sta n t.
In t h e p h o to g ra p h s men tion ed p r e v i o u s l y , t h e
method o f s u p p o r t i n g t h e t a b l e h a s been changed.
i s su p p o r t e d by r o ds about a f o o t l o n g .
The t a b l e
The r o d s are so
l o c a t e d t h a t th e t a b l e i s i n a s t a b l e p o s i t i o n and th e
f l e x i b i l i t y o f the t a b l e i n t h e v e r t i c a l d i r e c t i o n i s g r e a t l y
reduc ed i n comparison w i t h s p r i n g m o un tin g .
The f l e x i b i l i t y
i n t h e h o r i z o n t a l d i r e c t i o n i s p r o b a b ly no t d e c r e a s e d v e ry
much i f a t a l l .
The se cond e x p l a n a t i o n i s s u g g e s t e d by t h e method used
t o c o n n e c t t h e s h a f t t o th e d r i v i n g head.
A double u n iv er sa l
j o i n t was used t o p r o v i d e f l e x i b i l i t y end i t i s p o s s i b l e t h a t
i n some way t h i s arrangement may i n t r o d u c e v i b r a t i o n s t o the
-39-
s h c f t and t a b l e .
I f s o , th e v i b r a t i o n t r a n s m i s s i o n could be
b a la n c e d w i t h the o r i g i n a l b a l a n c e and when v a r i o u s un ba la n ces
a re I n t r o d u c e d , th e r e s u l t i n g m o tio n o f t h e t a b l e could ca use
f o r c e s t o be t r a n s f e r r e d through t h e u n i v e r s a l j o i n t s to the
s h a f t end t a b l e .
—Z^Ow
APPENDIX
The r e s u l t s o b t a i n e d by p u r p o s e l y u n b a l a n c i n g t h e s h a f t
in t h e b a l a n c i n g machine seem t o i n d i c a t e t h a t t h e u n i t i s
n o t u s a b l e f o r a l a b o r a t o r y e x p e r i m e n t in a c t u a l b a l a n c i n g .
The v a r i a t i o n i n t h e phase a n g l e betw een t h e un balance and
t h e p o s i t i o n o f maximum a m p litu d e d e s t r o y s t h e main purpose
of a bala n cin g machine.
However, t h e b a l a n c i n g machine can
be used t o i l l u s t r a t e t h e p r i n c i p l e s i n v o l v e d i n dynamic
balancing.
A pr oposed l a b o r a t o r y e x p e r i m e n t , u s i n g t h i s
m a c h i n e , w i l l be g i v e n .
P ho to g r a phs o f th e b a l a n c i n g machine
a r e i n c l u d e d a s f i g u r e s 1 4 , 1 5 , 1 6 , and 1 7 .
The a u t h o r f e e l s t h a t t h e machine can be u sed f o r two
d if f e r e n t purposes.
The f i r s t p u r p o s e , t o i l l u s t r a t e p r i n ­
c i p l e s o f b a l a n c i n g t o underg ra du a te s t u d e n t s i n a l a b o r a t o r y
course, i s given in t h i s s e c t i o n .
The second p u r p o s e , t o
supplement an advanced c o u r s e i n I f e c h a n i c a l V i b r a t i o n s , i s
n o t d e s c r i b e d i n t h i s paper.
The l a b o r a t o r y e x p e r i m e n t , proposed by t h e a u t h o r ,
c o n s i s t s o f two p a r t s .
The f i r s t p a r t I l l u s t r a t e s th e i n t r o ­
d u c t i o n o f dynamic unbalance by t h e improper a d d i t i o n o f a
w e i g h t i n a p l a n e n o t c o i n c e d e n t w i t h the p l a n e o f a s i n g l e
un balanced w e i g h t .
The second p a r t shows the b a l a n c i n g o f
l o n g r i g i d r o t o r s by t h e a d d i t i o n o f two w e i g h t s i n two
d iffe re n t a x ia l p lan es.
I t i s assumed t h a t t h e s t u d e n t w i l l
have completed t h e c o u r s e s i n S t a t i c s and Dynamics b e f o r e t h i s
«
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FIG. 14
RIGHT
FIG. 15
TOP
41
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FRONT
VIEW
VIEW
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FIG. 17
RIGHT
42
END
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VIEW
*•43l a b o r a t o r y e x p e r im e n t i s g i v e n .
P rop osed L a b o r a t o r y Experiment - B a l a n c i n g
I . Preface
Unbalance i n r o t a t i n g b o d i e s c a u s e s u n d e s i r a b l e e f f e c t s
i n m a ch in e s such a s he avy b e a r i n g p r e s s u r e , undue w e a r ,
v i b r a t i o n s , d e f e c t i v e work w i t h g r i n d i n g d i s k s , and d e f e c t i v e
lub rication.
B a l a n c i n g o f r o t a t i n g b o d i e s can be a c c o m p l i s h e d
by s e v e r a l m eth o d s .
For t h i n d i s k s a c o n d i t i o n o f s t a t i c
b a l a n c e i s g e n e r a l l y s u f f i c i e n t , w h er ea s l o n g r o t o r s r e q u i r e
an a d d i t i o n a l t y p e o f b a l a n c i n g , known a s dynamic b a l a n c e .
S t a t i c b a la n ce r e f e r s t o a c o n d i t i o n o f s t a t i c e q u i l i b r i u m .
R e f e r t o t h e c a s e o f a t h i n d i s k mounted on a s h a f t .
I f the
d i s k c o n t a i n s an unbalanced w e i g h t , and b e a r i n g f r i c t i o n i s
e l i m i n a t e d a s much a s p o s s i b l e , t h e he a v y s i d e o f t h e d i s k
w i l l s e e k th e l o w e s t p o s i t i o n when t h e a x i s o f t h e s h a f t i s
h o rizon tal.
By r e p e a t e d t r i a l s , th e pr oper magnitude and
r a d i a l d i s t a n c e o f a b a l a n c i n g w e i g h t , p la c e d d i a m e t r i c a l l y
o p p o s i t e t h e he avy s i d e , can be d e t e r m i n e d .
in a s t a t e of s t a t i c balance.
The d i s k i s now
I f the d i s k i s r o t a t e d , t h e
c e n t r i f u g a l f o r c e caused by t h e he av y s i d e w i l l be b a l a n c e d
by t h e c e n t r i f u g a l f o r c e caused by t h e b a l a n c i n g w e i g h t .
Thus t h e d i s k i s both s t a t i c a l l y and d y n a m i c a l l y b a l a n c e d .
I n t h e p r e v i o u s d i s c u s s i o n , we have assumed t h a t t h e
b a l a n c i n g w e i g h t was added i n th e same a x i a l plane a s t h e
o r i g i n a l un balance w e i g h t .
However, i f t h i s method i s used
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44-
t o ba lu n c e a l o n g r o t o r w i t h a s i n g l e unbalanced w e i g h t , the
a x i a l p o s i t i o n o f t h e un balance w i l l n o t be i n d i c a t e d ,
A
w e i g h t o f t h e proper v a l u e and r a d i a l d i s t a n c e and d i a m e t r i ­
c a l l y o p p o s i t e t h e u n b a l a n c e , added a t any p la n e o t h e r than
t h e p l a n e o f o r i g i n a l u n b a l a n c e , can l e a d t o a c o n d i t i o n f a r
worse than the o r i g i n a l u n b a l a n c e .
',Vhen r o t a t i n g ,
i f th e
c e n t r i f u g a l fo r c e o f the balan cing weight i s not c o l l i n e a r
w i t h t h e c e n t r i f u g a l f o r c e o f t h e unbalance w e i g h t , a dynamic
couple i s in tro d u ced .
T h i s dynamic c o u p l e , s i n c e i t
cannot
be d e t e c t e d by s t a t i c means, c o n s t i t u t e s a dynamic u n b a la n c e .
For r i g i d b o d i e s , a c o n d i t i o n o f t h i s kind can be c o r r e c t e d
by a d d in g w e i g h t s t o g i v e an e q u a l and o p p o s i t e c o u p l e which
w i l l b a la n ce t h e f i r s t c o u p l e .
II.
P ro c e d u r e
The i n s t r u c t o r w i l l e x p l a i n t h e b a l a n c i n g machine t o be
used i n t h i s e x p e r i m e n t .
The s t u d e n t sh ould become f a m i l i a r
w i t h th e machine b e f o r e p r e c e d i n g .
Caution must be e x e r c i s e d
s i n c e th e r o t a t i n g s h a f t i s n o t c o v e r e d and t h e w e i g h t s
p r o j e c t from t h e d i s k s .
S e l e c t a nut and a b o l t and a
s u f f i c i e n t number o f washers t o g i v e a weight o f I 1 / 2 t o
2 ounces.
S e l e c t a seco n d group t o match t h e w e i g h t o f t h e
first.
P l a c e one w e i g h t a t a c o n v e n i e n t l o c a t i o n on t h e r i g h t
disk .
S t a r t t h e machine and o b s e r v e t h e unbalance r e a d i n g s .
K e x t , add t h e second w e i g h t d i a m e t r i c a l l y o p p o s i t e t h e f i r s t
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45-
w e i g h t on th e same d i s k and o b s e r v e t h e r e a d i n g s when the
machine i s r u n n i n g .
The se cond w e i g h t sh ould n u l l i f y the
un balance caused by the f i r s t w e i g h t .
Remove t h e second w e i g h t from t h e r i g h t d i s k and add i t
t o t h e l e f t d i s k a t t h e same a n g u l a r l o c a t i o n .
'Vhen ru nning
t h e e f f e c t o f the dynamic c o u p le which has been i n t r o d u c e d
sh o u ld be a p p a r e n t .
The p r e c e d i n g t r i a l s w i l l i l l u s t r a t e
t h e need f o r dynamic b a l a n c i n g .
The l a b o r a t o r y i n s t r u c t o r w i l l g i v e the v a l u e and
l o c a t i o n o f s e v e r a l w e i g h t s t o be added t o th e s h a f t d i s k s .
Remove the two w e i g h t s used i n the f i r s t p a r t and p l a c e t h e
s t a t e d w eights at the in d ic a te d l o c a t i o n s .
Measure the
d i s t a n c e s between t h e d i s k s and t h e r a d i u s o f t h e c e n t e r s o f
th e h o l e s d r i l l e d
in the d is k s .
Observe t h e u n b a l a n c e .
o f t h e d i s k s w i l l be used a s c o r r e c t i o n d i s k s .
Two
Considering
one o f t h e d i s k s a s a r e f e r e n c e p l a n e , t h e moments o f the
auded u n ba la n ce w e i g h t s w i t h r e s p e c t t o th e r e f e r e n c e pla n e
can be t a k e n .
C a l c u l a t e th e v a l u e o f a s i n g l e w e i g h t t o
be added a t t h e seco n d c o r r e c t i o n d i s k t o make t h e r e s u l t a n t
o f th e moments o f t h e w e i g h t s , i n c l u d i n g t h e c o r r e c t i o n
w eig h t, equal to zero .
Repeat t h e proced ure t o f i n d a s i n g l e
w e i g h t t o be added t o t h e f i r s t c o r r e c t i o n d i s k .
Add the two
w e i g h t s t o th e c o r r e c t i o n d i s k s a t t h e c a l c u l a t e d l o c a t i o n s .
The s h a f t s h o u ld now be b a l a n c e d .
The g e n e r a l c a s e o f un­
balance in r i g i d r o t a t i n g bodies i s I l l u s t r a t e d
i n t h e above
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46-
exam ple,
The main purp ose o f a dynamic b a l a n c i n g machine i s t o
d e t e r m i n e th e two c o r r e c t i o n s t h a t must be made t o b alan ce
rotatin g u n its ,
The methods used v a r y c o n s i d e r a b l y and t h i s
machine was b u i l t t o pe rm it s i m u l t a n e o u s b a l a n c i n g i n two
plan es.
Other t y p e s r e q u i r e two runs f o r b a l a n c i n g .
The
r o t o r i s f i r s t b a la n c e d by t h e a d d i t i o n o f a w e i g h t i n one
o f t h e p l a n e s and t h e n turned end f o r end t o d e t e r m i n e the
w e i g h t needed f o r t h e second p l a n e .
I I I . R esu lts
The r e p o r t s h o u ld i n c l u d e the c a l c u l a t i o n s made t o
d e t e r m i n e the v a l u e s o f the two w e i g h t s added t o t h e s h a f t .
D i s c u s s t h e r e s u l t i n g b a l a n c e o b t a i n e d by the a d d i t i o n o f
t h e two w e i g h t s .
-47LITERATTIRE CITED AND CONSULTED
Den H a r t o g , J . P . , 1 9 4 7 , MECHANICAL VIBRATIONS,
McCraw-Hlll Book Company, I n c . , New York
and London,
Myklea t a d ,
Lieely, F . B
[. 0 . ,
1 9 4 4 , VIBRATION ANALYSIS
McGraw-Hill Book Company, I n o . , New York
and London.
and E n s i g n , N . E . , 1940
ANALYTICAL MECHANICS FOR ENGINEERS
John YIlley & S o n s , I n c , , New York.
Thomson, W. T . , 1 9 4 8 , MECHANICAL VIBRATIONS
P r e n t i c e - H a l l , I n c . , New York,
100906
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