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Models of Compr
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Purdue University
Purdue e-Pubs
I3%!3!+ C% E'%%' C&%%#% S#+ & M%#!#!+ E'%%'
1992
Helical Coil Suspension Springs in Finite ElementModels of
Compressors
A. D. Kelly Virginia Polytechnic Institute and State
University
C. E. KnightVirginia Polytechnic Institute and State
University
F++6 3 !$ !$$3!+ 6* !3: ://$#.+.4$4%.%$4/#%#
$#4%3 ! %% !$% !5!+!+% 34' P4$4% %-P4, ! % 5#% & 3% P4$4%
U5%3 L!%. P+%!% #3!#3 %4@4$4%.%$4 &
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H%#*/E5%3/$%+3.3+
K%++, A. D. !$ K'3, C. E., "H%+#!+ C+ S4% S' F3% E+%%3 M$%+
& C%" (1992). InternationalCompressor Engineering
Conference. P!% 870.://$#.+.4$4%.%$4/#%#/870
http://docs.lib.purdue.edu/?utm_source=docs.lib.purdue.edu%2Ficec%2F870&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://docs.lib.purdue.edu/icec?utm_source=docs.lib.purdue.edu%2Ficec%2F870&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://docs.lib.purdue.edu/me?utm_source=docs.lib.purdue.edu%2Ficec%2F870&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://docs.lib.purdue.edu/icec?utm_source=docs.lib.purdue.edu%2Ficec%2F870&utm_medium=PDF&utm_campaign=PDFCoverPageshttps://engineering.purdue.edu/Herrick/Events/orderlit.htmlhttps://engineering.purdue.edu/Herrick/Events/orderlit.htmlhttps://engineering.purdue.edu/Herrick/Events/orderlit.htmlhttps://engineering.purdue.edu/Herrick/Events/orderlit.htmlhttp://docs.lib.purdue.edu/icec?utm_source=docs.lib.purdue.edu%2Ficec%2F870&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://docs.lib.purdue.edu/me?utm_source=docs.lib.purdue.edu%2Ficec%2F870&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://docs.lib.purdue.edu/icec?utm_source=docs.lib.purdue.edu%2Ficec%2F870&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://docs.lib.purdue.edu/?utm_source=docs.lib.purdue.edu%2Ficec%2F870&utm_medium=PDF&utm_campaign=PDFCoverPages
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F
ig
ur
e
1
.
R
e
fi
ne
d
S
u
s
p
en
s
io
n
S
p
rin
g
F
M
M
o
de
l
h
av
in
g
87
0
d
.
o
. f.
N
e
x
t.
s
p
ri
ng
ra
te
s
ar
e
c
a
lc
ul
at
ed
fr
o
m
s
t
at
ic
s
o
lu
ti
on
s
o
f
th
s
pr
in
g
m
o
d
el
.
T
h
e
p
ro
c
ed
u
re
i
s
t
o:
1
.
r
e
st
ra
in
a
n
od
e
(
se
t
a
ll
6
n
o
d
al
d
.o
.
f.
e
qu
a
l
to
z
e
ro
)
a
t
on
e
s
p
ri
ng
en
d
,
2
.
a
pp
l
y
a
u
n
it
d
e
fl
ec
ti
o
n
a
t
th
e
o t
h
er
s
p
ri
ng
e
nd
,
f
or
ex
a
m
p
le
,
s
et
e
1
0
t
o
c
a
lc
ul
at
e
t h
e
to
rs
io
n
a
l s
p
ri
ng
ra
te
, a
n
d
3
.
z
er
o
t
he
re
m
a
in
in
g
5
d .
o
.f.
a
t
t h
e
de
fl
e
ct
ed
en
d
.
T
h
e
re
a
c
tio
n
f
o
rc
es
ca
l
cu
la
te
d
at
t
he
de
f
le
ct
e
d e
n
d
co
r
re
sp
o
n
d
t o
th
e
s
p
ri
ng
ra
te
s
.
in
cl
ud
i
ng
th
e
c
ou
p
li
ng
te
rm
s
.
C
o
e
ff
ic
ie
n
ts
f
o
r t
h
e
fu
ll
s
pr
in
g
s
ti
ff
ne
s
s
m
a
tr
ix
a
re
fo
u
nd
w
ith
si
x
s
ta
t
ic
s
o
lu
ti
o
ns
,
o
n
e
f
o
r
e
ac
h
de
fl
e
ct
ed
d
.o
.
f.
C
o
nv
e
rg
e
n
ce
o
f
th
e
fin
i
te
el
em
en
t
pr
o
ce
d
u
re
s
h
ou
l
d,
o
f
co
u
rs
e
, b
e
a
s
s
ur
e
d
by
r
e
pe
a
ti
ng
t
h
e
a
na
ly
s
is
w
it
h
f
ur
ther dis cretization.
D
e
ri
vi
n
g
th
e
s
p
rin
g
r
at
e
s is
p
o
ss
ib
le
u
s
in
g
s
im
p
li
fie
d
v
e
rs
io
n
s
o
f b
e
a
m
t h
e
o
ry
o
r
s
tra
i
n
e
n
e
rg
y
e
x
pr
e
ss
io
n
s
in
co
n
ju
n
c
tio
n
w
ith
C
as
t
ig
lia
n
o
s
s
e
co
n
d
th
e
o
re
m
[2
).
T
hi
s
p
ro
c
e
du
re
w
as
n
o
t
fo
ll
ow
e
d
th
r
ou
g
h
fo
r
e
a
ch
co
e
ff
ic
ie
n
t o
f
th
e
st
if
fn
e
ss
m
a
tr
ix
,
h
o
w
e
ve
r
.
it
i
s
be
n
e
fic
ia
l
to
fin
d
a
fe
w
d
e
ri
ve
d
va
lu
e
s
f o
r
c
o
m
p
ar
is
o
n
to
t
h
e F
M
r
es
u
lts
d
ur
in
g
t h
e
sp
ri
ng
m
o
de
l
d
ev
e
lo
p
m
e
nt
.
B
a
sic
de
s
ig
n
f o
r
m
u
la
s f
o
r
th
e
p
ri
n
ci
pa
l-
d
ire
c
ti
on
sp
ri
n
g
ra
te
s
a
r
e
w
i
d
el
y
a
v
ai
la
b
le
[
2,
3.
4
1
a
nd
a
re
r
e
po
r
te
d
ly
q
u
ite
a
cc
u
ra
te
w
he
n
ce
rt
a
in
g
e
o
m
e
tr
ic
c
o
nf
ig
u
ra
ti
on
s
re
g
ar
d
in
g
th
e
s
p
ri
ng
in
d
ex
a
n
d
p
it
c
h
a
ng
le
ar
e
a
dh
e
re
d
t
o
(
2]
.
F
or
i n
s
ta
n
ce
,
in
a
he
li
ca
l
s
pr
in
g
h
a
vi
ng
en
d
c
o
nd
it
io
n
s
fi
xe
d
a
s
d
e
sc
ri
b
ed
a
b
o
ve
in
t h
e
F
M
p
ro
ce
d
u
re
w
e
h
av
e
t
he
fo
ll
o
w
in
g
fo
r
m
u
la
s.
A
x
ia
l
sp
r
in
g
ra
t
e:
2
)
L
a
te
ra
l
sp
r
in
g
ra
t
e:
(
3
)
Torsional spring rate:
4)
F
in
al
ly
, s
p
ri
ng
ra
te
s
m
a
y
be
m
e
a
su
r
ed
. I
t
is
d
if f
ic
u
lt ,
h
o
w
e
ve
r
,
to
m
e
a
su
r
e
sp
r
in
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ra
t
es
o
th
e
r t
h
an
th
o
se
i
n
t
he
si
m
p
le
a
x
ia
l
an
d
l
at
er
a
l
di
re
c
tio
n
s.
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hu
s
,
o
bt
a
in
in
g
t
he
sp
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d
at
a
n
e
c
es
s
ar
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to
co
m
p
l
et
e
th
e
f
u
ll
s
tif
fn
e
ss
m
a
tr i
x is
n
o
t
p
ra
c
tic
a
l.
As
w
i
th
th
e
d
e
ri
ve
d
s
p
ri
ng
7
8
-
8/16/2019 Helical Coil Suspension Springs in Finite Element
Models of Compr
5/10
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8/16/2019 Helical Coil Suspension Springs in Finite Element
Models of Compr
6/10
E
q
u
a
ti
on
(
S
a
l
w
i
ll
g
e
n
e
ra
l
ly
o
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t
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r
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fr
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a
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iv
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o
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er
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ra
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m
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ly
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o
ta
t
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al
in
e
rt
ia
o
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th
e
sp
r
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g
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is
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ed
.
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th
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le
s
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ro
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q
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r
a
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e.
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n
in
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al
t
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l
s
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rg
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c
a
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n
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lic
a
l
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y
a
d
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pt
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or
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o
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e
s
o
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ti
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n
o
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a
c
o
n
ti
nu
o
u
s
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o
d
.
S
u
b
s
ti
tu
ti
n
g
N
O
fo
r
the le ngth
o
f
the rod, one obta ins,
f
-
i
I
Q
_
r
;
N
O
~
y
(6
)
H
o
w
e
ve
r
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th
e
s
e
m
o
de
s
u
su
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ll
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t
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r
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ie
s
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q
u
a
ti
o
n
6)
y
ie
ld
s
t
h
e
f
ir
s
t
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te
r
n
a
l t
o
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ur
g
e
o
f
t
h
e
e
x
am
p
le
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p
ri
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t
o b
e
b
e
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n
3
7
7
0
a
n
d
4
3
5
0
H
z.
La
t
er
a
l
su
r
g
e
fr
e
q
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n
c
y
r
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la
tio
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a
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7
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re
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H
a
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o
r
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frequencies
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abou
t
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to
1
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e
s
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s
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e
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s
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ri
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n
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s
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ly
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e
l
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t
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lly
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F
o
r
t
h
is
r
e
as
o
n
, i
t
is
ad
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a
n
ta
g
e
o
u
s
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e
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e
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h
s
p
ri
ng
m
o
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ls
w
h
i
ch
m
a
in
t
a
in
d
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n
a
m
i
c
a
cc
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r
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cy
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r
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se
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e
d
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o
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ee
d
o
m
S
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g
M
o
d
e
l
D
e
ve
l
op
m
en
t
Figure 2 shows a c oarse-m esh
spring m odel whic
h
has
o
n
ly
fo
u
r
li
n
ea
r
b
e
a
m
e
l
e
m
e
nt
s
pe
r
c
o
il,
an
d
is
c
o
m
p
a
ra
b
l
e
to
th
e
r
e
fi
ne
d
m
o
d
el
o
f
f
ig
u
re
1
.
T
h
is
d
.o
.f
.
re
d
u
c
ti
o
n
r
es
u
lt
s
in
s
ig
n
if
ic
a
n
t
s
ta
t
ic
s
t
if f
e
n
in
g
.
T
hr
o
u
gh
a
c
o
m
b
i
n
at
io
n
o
f
m
a
te
r
ia
l
g
e
o
m
e
t
ri
c
vo
l
um
e
c
o
n
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e
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a
ti
o
n
a
n
d
m
o
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tu
n
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g
,
t
h
e
c
o
a
rs
e
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e
s
h
m
o
de
l
c
a
n
,
h
ow
e
ve
r
. b
e
c
o
a
xe
d
to
ag
r
e
e
w
it
h
th
e
f
in
e
-m
es
h
m
o
d
e
L
F
igu re 2 . Coarse-Mesh
FEM
Spring Model having 150 d.o.f
.
P
re
s
e
rv
in
g
v
o
l
~
m
e o
f
th
e
s
t
ru
c
tu
r
e
u
p
o
n
d
is
c
re
t
iz
a
ti
on
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g
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iz
e
d
a
s
g
o
o
d
m
od
e
li
n
g
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r
a
ct
•c
e
.
F
o
r
th
e
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n
n
g
m
od
e
l
a
p
pl
ic
a
ti
o
n,
t
h
e
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o
il
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ia
m
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te
r
o
f
th
e
co
a
r
se
m
o
d
e
l i
s
in
c
re
a
s
e
d
s
o
t
h
a t
th
e
~
o
m
b
i
n ~
o
f
fo
u
r
e
le
m
e
n
ts
is
e
q
u
iv
a
le
n
t
t
o
t
h
e
tr
u
e
c
o
il
c
i
rc
u
m
f
er
e
n
ce
.
H
o
w
e
v
e
r,
t
h
ts
m
o
d
tf•
c
a
t•
on
a
lo
n
e
d
o
e
s
n
o
t
re
s
o
lv
e
o
v
e
r-
s
ti
ff e
n
i
ng
b
y
t
he
c
o
ar
s
e
m
e
s
h
.
7
8
-
8/16/2019 Helical Coil Suspension Springs in Finite Element
Models of Compr
7/10
Co
n
se
q
ue
n
tly
,
na
tu
r
al
fr
eq
u
en
c
y p
r
ed
ic
ti
on
s
w
il l
b
e
el
ev
at
e
d,
fo
r
ex
a
m
pl
e,
fr
eq
u
en
c
ie
s
fo
un
d
w
ith
th
e
r
ed
u
ce
d
sp
r
ing
s
c
on
s
id
er
ed
h
e
re
ra
n
ge
d
f
ro
m
0
to
3
0
)
h
ig
h
.
S
pr
in
g
ra
te
s
a
re
c
o
m
pa
re
d
i
n
Ta
b
le
2
.
T
ab
le
2
R
ed
u
ce
d-
S
pr
in
g R
a
te
Co
m
p
ar
is
on
Direction
870
DOF
FEM
150
DOF
FEM
Difference
A
x i
a l
'
54
3
9
5
.1
)
7
0
1
12
.3
)
+
2
9
La
te
r
ia
l'
3
8
6
6
7.
6)
4
05
7
0.
9)
+4
.
9
T
or
si
on
11
8
1
3
.3
)
1
1
8
1
3
.3
) 0
B
e
nd
in
g
3
1
2
3
5.
2)
3
2
3
3
6.
5)
+
3
.
5
T
ra
n
sla
ti
on
a
l
un
it
s a
re
lb
/in
(
N
/m
m
)
R
o
ta
tlo
)
al
u
n
its
a
re
l
b •
in
/r
ad
(
N
•m
/
ra
dl
Next, model tuning
is
employed
to
im p
ro
v
e
d
yn
a
m
ic
c
or
re
la
tio
n
w
it
h
th
e
re
fi
ne
d
m
o
de
ls
.
T
he
b
e
st
tu
n
in
g
p
ar
am
e
te
r
w
as
f
ou
n
d
to
be
m
a
ss
d
e
ns
it
y
wh
ic
h
r
es
u
lts
in
a
ne
a
rly
u
n i
fo
rm
s
h
if
t o
f
na
tu
ra
l
fre
q
ue
n
ci
es
. I
n
a
d
di
tio
n
, t
he
m
o
d
ifi
ed
m
a
ss
d
e
ns
it
y
de
s
ire
d
, P
mo
di
tie
d•
ne
c
es
s
ar
y f
o
r c
o
rr
ela
t
ion
of
th
e
re
d
uc
ed
d
.
o .
f.
sp
ri
ng
m
o
d
el
s
m
ay
be
ca
lc
u
la
te
d
d i
re
ct
ly
fr
o
m
P
n I
OC
He
d
( f
wd
uc
ed )
P
8IIU
CO
d
f
oftn
ed
7
)
w
h
er
e P
red
uc
ed
a
nd
fre
du
ce
d
a
re
fo
u
nd
in
a
n
in
it
ial
e
ig
e
ns
o
lut
io
n
o
f
th
e
re
du
c
ed
m
od
e
l,
an
d
fre
fin
ed
i
s
th
e
ta
rg
e
t
fre
q
ue
n
cy
f
ou
n
d b
y
t
he
re
fin
e
d
m
od
e
l.
t
w
a
s
ne
c
es
s
ar
y
to
in
c
re
as
e
th
e
m
a
ss
d
e
ns
it y
of
th
e
s
p
rin
g
o
f t
h i
s
st
u
dy
b
y
a
f a
c
to
r
o
f a
b
ou
t 1
.3
to
re
ac
h
a
gr
ee
m
e
nt
w
it
h
the refined model.
T
ab
le
3
co
m
p
ar
e
s
th
e
f i
rs
t t
en
sp
ri
ng
m
o
d
es
fo
un
d
b
y
th
e
r
ef
in
ed
a
n
d
re
d
uc
e
d
si
de
s
pr
in
g
m
od
e
ls
.
Al
l m
o
d
e
sh
ap
e
s
ar
e
co
rr
el
at
ed
a
n
d
th
e
co
rr
e
sp
on
d
in
g
fr
eq
u
en
c
ies
a
re
w
i
th
in
5
.3
p
e
rc
en
t
d i
ff e
r
en
c
e.
S
p
ri
ng
r
at
es
of
th
e
re
d
uc
e
d
m
o
de
l,
h
ow
e
v
er
,
do
n
o
t i
m
pr
o
ve
w
ith
th
e
d
en
s
it y
c
o
rr
ec
ti
on
.
T
hi
s
ar
ti f
ic
ia
l s
ti
ffe
n
in
g
is
e
xp
e
cte
d
to
a f
fe
c
t o
n
ly
th
e
lo
w
-f
re
q
ue
n
cy
c
o
m
pr
es
so
r
-s
us
p
en
s
ion
m
od
es
of
th
e
a
ss
e
m
bl
y
m
od
e
l.
Ta
b
le
3
.
R
e
du
ce
d
S
p
rin
g
N
o
rm
a
l M
o
d
e
C
om
p
a
ris
o
n
M
od
e
S
h
ap
e
R
e
fin
e
d
M
o
de
l
R
e
d
uc
ed
M
od
e
l
D
if
fe
re
n
ce
Descrip tion
870
d.o.f. 150 d .o .f.
F
re
qu
e
nc
y,
H
z
F
re
q
ue
n
cy
, H
z
A
x
ia
l
6
53
6
5
3
0.
0
R
ad
ia
l
75
2
7
3
6
-
2
.1
X
La
te
ri
al
8
06
76
3
-5
.
3
Y
La
te
ra
l
82
2
77
8
-
5.
3
A
xi
al
12
7
6
12
7
8
+
0
.1
X Late ral
14
1
4
1
38
8
·1
.
8
Y
La
te
ra
l
14
3
8
1
41
3
-
1.
7
X
V L
a
te
ra
l
1
4
90
1
45
4
-
2.
4
Ax
ia
l
18
1
8
18
2
1
-0
.2
Y
X
L
at
er
al
1
99
5
19
7
8
-
0.
8
78
4
-
8/16/2019 Helical Coil Suspension Springs in Finite Element
Models of Compr
8/10
The
n atu
ral f
re qu
enci
es a
re se
nsiti
ve t
o oth
er pa
ram
ete r
s bu
t tun
in g t
he re
duc
ed m
ode
l
by
othe
r ave
nue
s we
re no
t su
cce
ssfu
l.
t f ir s
t the
to rs
iona
l con
sta n
t, J
was
c ha
nge
d in
an
e
ffo rt
t o c
orre
la te
natu
ra l fr
equ
encie
s. T
he
effec
t pr
oved
to b
e lo
caliz
ed , d
epe
ndin
g on
t he
mod
e sh
ape.
Fo
r ex
amp
le , r
educ
ing J
to
mat
ch t
he fi
rst a
xia l
m od
e has
n
o e
ffec
t on
a
ra
diall
y ex
pans
ive
mod
e an
d ha
s in
cons
is te n
t e f
fec ts
on
th e
la te
ra l m
ode
s. T
he l
ocal
ized
effe c
ts re
sult
in a
n on
-uni
fo rm
red
uctio
n of
natu
ra l f
re qu
enci
es a
nd c
onse
que
ntly
a ch
ange
in th
e nu
mer
ical
ord e
r of
mod
e e
xtrac
ti on
. Th
e ju
mble
d m
ode
s are
d if
f icu l
t to
corr
elate
.
Mod
if ica
tions
t o
area
m om
ent
s of
iner
tia, I
, ca
use
local
e ffe
cts
sim il
ar to
tho
se fo
und
wi
th
to rs
io na
l co
nsta
nt m
odi
fi cat
io ns
. In
ad
diti o
n, r
elatio
ns
betw
een
I
J an
d th
e n
atu r
al
frequ
enc
y are
n o
t rea
dily
avai
lable
, s
o itera
tion
s are
ne
cess
ary t
o ar
rive
at th
e ap
pro p
riate
mo
difie
d va
lues
.
Elas
tic m
odu
lus
is an
oth e
r po
te nt
ia l p
aram
ete r
f or
tu nin
g th
e co
arse
-m e
sh m
ode
l bu
t
w
as n
ot co
nsid
ered
her
e. T
unin
g w
ith th
e e l
astic
m od
ulus
m a
y pro
ve
to be
mor
e su
cces
sful
tha
n tu
nin g
w
ith m
ass
den
sity
beca
use
it w
il l i
mpro
ve
stiffn
ess
agre
eme
nt
as w
ell.
In
add
it ion
, nat
ural
freq
uenc
ies
are r
elate
d to
fE
s
o th
e m
odif i
ed v
alue
may
be
so lv
ed fo
r
d
irect
ly .
SP
RING
S IN
TH
E CO
MP
RES
SOR
SS
EM
BLY
DYN
MI
C FI
NITE
ELE
ME
NT M
OD
EL
In th
is s
tudy
, sp
ring
surg
e p
rove
d to
be
an
im p
orta
nt fa
ctor
in
the
com
pre s
sor
a
ssem
bly
s dy
nam
ic pr
oper
ties.
T ab
le 4
com
pare
s th e
n or
mal m
ode
s of
a co
mpr
esso
r ha
ving
th
e s
imple
m a
tr ix
spri
ngs
to a
c om
pre
ssor
m od
el h
avin
g th
e co
arse
-m e
sh s
pring
s.
The
si
gnif i
canc
e of
surg
e fr
eque
ncie
s on
the
c om
pre s
sor
is di
sc us
sed
more
t ho
roug
hly
by K
elly
an
d K
nig h
t [1
L bu
t th
e sp
ring
r ep
re se
ntati
on d
if fe r
ence
s a
re of
in t
eres
t he
re. F
or
th e
fr
eque
ncy
ra ng
e s h
own
, sp
ring
surg
e f re
quen
cies
a cc
ount
for n
earl
y on
e-ha
lf of
the a
ssem
bly
r
eson
ance
s. T
he m
esh
ed-s
prin
g rep
rese
nta t
ion
is ne
ce ss
ary t
o fin
d the
se m
ode
s. P
rese
ntly
,
the coarse-m esh spring
is
so
m ew
hat
s tiff
e r th
an
the
more
ref i
ned
mes
h m
ode
l. T
his
is
ev
iden
t in s
ligh
tly e
leva
ted c
omp
re ss
or s
uspe
nsio
n fre
que
ncies
.
785
-
8/16/2019 Helical Coil Suspension Springs in Finite Element
Models of Compr
9/10
T
ab
le
4.
C
o
m
p
re
ss
o
r
A
ss
e
m
b
ly
N
o
r
m
a
l M
o
d
e
s
(k
]
S
pr
in
g
s
M
es
h
ed
-
S
pr
in
g
s
M
M
F
r
e
g
z
Fr
e
a .
.
z
M
o
d
e
D
e
sc
ri
o
tio
n
1
-6
0
0
Ri
gi
d
B
o
d
y
7
8
.
6
1
8.
4
8
13.3
20.1
9
1
4
.8
2
2
.2
c
om
pr
e
ss
o
r
su
s
p
en
s
io
n
m
o
d
e
s
1
0
2
2
.2
3
1.
9
11
2
6.
2
33
.
6
1
2
3
2
.3
3
5
.
7
1
3
9
9.
4
9
9
.4
sh
o
c
kl
oo
p
1
4
16
3
1
6
3
sh
o
c
kl
o
op
1
5
20
8
20
8
sh
o
c
k l
o
op
1
6
3
4
7
to
p
s
p
rin
g
1
7
36
9
to
p
s
p
ri
ng
1
8
4
9
2
4
9
3
sh
o
c
k l
oo
p
19
497 top
spring
2
0
5
0
4
to
p
s
p
rin
g
2
1
5
3
5
5
3
6
sh
o
c
k l
oo
p
22
6
2
7
to
p
s
p
ri
ng
2
3
6
4
6
si
de
sp
r
in
g,
h
o
u
si
ng
M
8
(3
,
1 )
2
2
4
65
2
si
d
e
s
pr
in
g
25
6
6
2
66
2
m
o
d
ifi
e
d
h
ou
s
in
g
m
o
d
e
M
8
(
3
, 1
26
6
8
5
to
p
s
p
rin
g
2
7
6
9
0
to
p
s
pr
in
g
2
8
7
26
7
1
0
m
o
di
fie
d
h
o
u
s
in
g
m
o
d
e
M
9
(
3
, 1
2
9
7
1
9
to
p
s
pr
in
g
30
731
side springs, housing
M9(3,
1),
M1
0(2 ,
1
3
1
7
3
3
si
de
sp
r
in
g
3
2
7
7
2
7
4
7
m
o
di
fi
ed
ho
u
s
in
g
m
o
d
e
M
1
0
(2
,
1
3
3
7
6
0
si
de
sp
r
in
g
s
34
7
7
7
s
id
e
sp
ri
n
gs
,
h
o
us
in
g
M
9
(3
, 1
)
3
5
7
9
0
s
id
e
s
p
ri
ng
s
36
8
0
8
t
o
p
sp
r
in
g
3
7
8
1
0
a
ll
s
pr
in
g
s,
h
o
u
si
ng
M
1
0
(2
, 1
3
8
8
5
3
8
5
4
s
ho
c
k
lo
op
3
9
8
7
4
8
8
0
m
o
di
fi
ed
ho
u
s
in
g
m
o
d
e
M
1
1
(4
,
1
4
0
9
09
to
p
s
pr
in
g
41
937
to
p
s
p
rin
g
,
h
ou
s
in
g
M
1
2
(
to
p
l.
M
1
3
(
4,
1
l
4
2
9
6
8
to
p
s
pr
in
g
,
ho
u
s
in
g
M
1
2
(t
o
p)
4
3
9
6
8
9
7
7
m
o
d
if
ie
d
h
o
u
si
ng
m
o
de
M
1
2(
to
p
)
9
72
9
7
1
m
od
i
fie
d
h
o
u
si
ng
m
o
de
M
1
3
(
4,
1)
4
5
9
8
5
t
o
p
s
pr
in
g
. h
o
u
si
n
g
m
od
e
M
1
2(
to
p
),
M
1
3
14
, 1
46
1
0
1
6
1
0
2
0
s
ho
c
k
lo
op
4
7
1
10
2
1
1
0
0
m
od
if
ie
d
h
o
u
si
ng
m
o
de
M
1
4
(
2,
1
,
to
p
)
4
8
1
2
5
3
12
6
6
sh
o
c
kl
oo
p
4
9
1
2
8
6
t
op
sp
r
in
g
,
ho
u
si
n
g
m
o
d
e
M
1
5(
5
,
1
5
0
1
29
1
1
2
8
9
m
o
di
fi
ed
h
ou
s
in
g
m
o
d
e
M
1
5
(5
,
1
)
'Iden tif ied hou
s
in
g
d
e
fl
ec
ti
o
n
e
xp
e
ri
m
e
nt
a
lly
'H
o
u
s i
n
g
d
ef
le
c
ti
on
is
d
e
sc
r
ib
ed
by
an
(n
,m
)
t
y
pe
d
es
c
ri
p t
o
r,
w
h
e
re
n corresponds
to
the
c
ir
cu
m
f
e
re
n
tia
l
p
a
tt e
r
n
a
nd
m
c
o
rr
es
p
o
nd
s
to
th
e
ax
ia
l
p
at
te
r
n.
78
6
-
8/16/2019 Helical Coil Suspension Springs in Finite Element
Models of Compr
10/10
C
ONC
LUS
ION
S
One
app
roac
h to
repr
esen
t he
lical
susp
ensi
on s
pring
s in
a co
m pre
sso
r ass
em b
ly m
odel
i
s by
a si
m ple
spr
in g o
r sti f
fnes
s e
le m e
nt.
For t
his
m eth
od ,
an i
ndep
end
ent f
ini te
e lem
en t
sp
rin g
m od
el c
an effi
cien
tly
and
acc
ura t
e ly
ca lcu
late
all of
th
e ne
cess
ary
sprin
g st
iffne
ss
co
eff ic
ien t
s , inc
ludi
ng the
c
oup l
ing t
erm s
. In
addi
tio n,
mea
sure
m en
ts a
nd e
lem e
n ta r
y sp
ring
re la
tio n
s are
be
nefic
ial to
this
pro
ced
ure.
If
spri
ng s
urge
ex
ists
in t
he f
requ
enc
y ra
nge of
i
n te re
st, th
e
spri
ngs
sho
uld
be
rep
re se
nte d
with
th
eir o
wn
fi n i
te e
le m
ent
mesh
in
the
ass
em b
ly .
The
sp
ring
sarg
e
cha
ract
eris t
ic s m
ay
be c
alc u
lated
wit
h th
e Ind
epe
nden
t sp
ring
m od
el.
A lte
rna te
ly ,
s im p
le
axi
a l an
d to
rs io
na l
surg
e fre
quen
cies
ma
y be
qu
ick ly
app
rox i
m ate
d from
the
e lem
enta
ry
rel
ation
s pr
esen
te d.
CK
NOW
LE
DGM
ENT
S
T
he
au th
o rs w
ish t
o tha
nk
Bris
to l C
om
pre s
sors
, In c
. and
the
V irg
in ia
Cen
ter f
or In
nov
ative
Technology for
their
sponsorsh ip . We als o thank Dr.
L
D
M itche l l . Dr.
R
G. Mitch ine r. and
M r.
Davi
d G i
lliam
fo
r thei
r
e
ffort
s
an d
cont
ribut
ions
to t
h is
re se
arc h.
R
EFE
REN
CES
1.
K
elly
, A .
D .,
and
K n ig
ht.
C E
., D
yn am
1c F
in ite
Ele
m en
t M o
del i
ng a
nd A
naly
s is
of a
H
erm e
tic
R ec
ip ro
catin
g C
om p
ress
o r .
Pr
ocee
din g
s of
the
1992
In
tern
at ion
a l
C om
pres
sor
Engi
neer
in g Co
nfer
ence
-
at P
urd u
e, We
st
La fa
yette
, IN
, 199
2.
2
.
Coo
k, R
obe
rt D.
and
You
ng,
W arr
en C
• A
dvan
ced
M ec
han
ics of
M a
te ria
ls , M
acm
il lan
Publis hin g Co ..
New
York, NY,
1985.
3
W a
hl, A
. M .
. M e
cha
n ica
l Sor
in gs
, Pe
nto n
Pub
lishin
g C
o., C
le ve
la nd
, OH
, 19
44.
4
A
non
ymo
us
Spr
in g D
esig
n M a
nua
l, o
ciety
of
Aut
omo
tive E
ngine
ers,
Inc.
W a
rren
da le ,
PA,
19
Sg.
5
M itc
he ll
. L
D a
nd S
hig l
ey,
J. E
., M
ech
an ic
a l E
ngine
erin
g D
esign
, Fo
urth
Ed
ition
,
M c
G raw
-H i
ll, New
Y
ork ,
NY
, pp.
4
47-
463
1
983.
6
H am
ilt on
, Ja
m es
F.,
M ea
sure
m en
t an
d C o
ntro
l
of
C
om
ores
sor N
oise
. Ra
y w
H e
rrick
Labora torie s. Schoo l
of
M echanic a l Engin eerin g, Purdue U nivers ity , West La
fayette , IN,
1
988
. pp
.
1
2-13
.
7.
W a
hl, A.M
.
M e
cha
nica l
Spr
jn gs,
Sec
ond
Edit
io n,
McG
raw-
Hi ll .
N
ew
Yo
rk , N
Y, 19
63
