Increase of Fatigue Strength of Coiled Compression and Tension Springs

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29 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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Increase of Fatigue Strength of Coiled Compression and


Tension Springs



Gennady Aryassov (Ph.D.) and Viktor Strizhak (Ph.D.)





Tallinn Technical University, Institute of Mechatronics






19086 Tallin
n, Ehitajate tee 5, Estonia




tel.: +3726203300, fax:+3726203196,




E
-
mail:
garjasov@staff.ttu.ee
, strizhak@staff.ttu.ee



1. Abstract


Coiled compression and tension springs are widely used in
different machines. Capacity
for work of the springs often influences on capacity for work and reliability of machines
and devices as a whole. Consequently, to increase the reliability of machines and devices
it is necessary more precise determination of t
he values of the endurance limits of the
springs. Usually, in many strength problems, the major components of stress are static,
with less accurately alternating stresses superposed. Most failures originate with stress of
this type. The problem presents gr
eat difficulties because of the two sources of stress.

When the springs are loaded by non
-
steady loading then to take care of the unlimited
number of combinations of range and average stress, the line of failure of the material
must be used. The line of fa
ilure is curve line on a diagram where the average stresses for
the test are plotted as the abscissa, and the range stresses as the ordinate. Since
experimental data for the line of failure are usually not at hand, it is customary to make
the conservative
approximation that it is a straight line. However, use this line may give
considerable mistakes on determination of the limit range stresses. To avoid this fact the
method of approximation of the failure curve line by the line with the properly chosen
para
meters for the coiled compression and tension springs in the article is proposed and
calculation results are given as well.

Key worlds:
Coiled compression and tension springs, Fatigue failure, Alternate

stress,
Limit stress.


2. Introduction


In different
machines and devices when an impact takes place elastic elements are widely
used. One of the most used elastic elements is coiled compression and tension springs.
These springs usually are accumulator of energy and subjected to vibration or repeated
loadin
g. On such loading in the springs fatigue fractures begin with a minute (usually
microscopic) crack at a critical area of high local stress.

Extensive research over the last century has given a partial understanding of the basic
mechanism associated with f
atigue failures. Many references

for example

1

,

2


contain
a summary of much of the current knowledge as it applies to engineering practices. In
many strength problems, the major components of stress are static, with less accurately
known alternating st
resses superposed. However, the most failures originate with stresses
of this type. The problem presents great difficulties because of the fundamentally
different mechanism of failure in the two courses of stress.

To increase the reliability of the coiled
(or helical) springs it is necessary to know how
these stresses influence on fatigue life. The test data for springs are very limited and often
it is impossible to find the values of the limited stresses for different springs. Many
researches the diagram o
f limit stresses for non
-
steady loading use. The line of the
limited stresses is represented by curved line in
τ
m
-

τ
a

coordinates. For simplification
some portions of this curve are substituted by straight lines. More widely used the
method of approximation known as Serensen
-
Kinasoshvili’s

2


method. (Fig. 1).














Fig.1. Approximation of diagram of

limit amplitudes of stresses

-

-

-

-

real curve

AB and KD are approximating straight lines.


The stresses on the diagram shown in Fig.1 are marked as:

τ
в

–

ultimate stress, shear

τ
T

–

yield point stress, shear

τ
a

–

alternating stress (or range stress, or

stress amplitude), shear

τ
-
1

–

endurance limit stress for reversed shearing (reversed stress)

τ
0


–

endurance limit stress for pulsating shearing (pulsating stress)

τ
a

–

mean stress (average stress), shear

The approximation gives the possibility to find t
he limit stress for different combinations
of
τ
a

and

τ
m


with use of short
–

time and limited tests. The diagram shown in Fig.1
gives the values of the
τ
a

close to real one for reversed cycles, however when pulsating
stresses take place the values of

τ
a

will be higher as compared with the real stresses, i.e.
a safety factor will be decreased. To avoid this fact the approximation can be made by
curve line with taking into account specially selected factors.



45
o

τ
-
1

τ
o
/
2


τ
в


τ
т


τ
m


τ
a

A

B

O

K

45
o

C

D

3
. Approximation of diagram of limited stre
sses


Let the diagram of limited stresses in co
-
ordinates
τ
m
–

τ
a


is approximated by equation

2
1
m
B
m
a











,



(1)

where


2
min
max





a
,


2
min
max





m
,

τ
max

–

maximal stress,
τ
min

–

minimal stress,


and



are
the factors which must be
determined.

The curve (1) naturally is at point
A

(Fig.1) when the pulsation cycle of loading takes
place. Let us consider that on determination of factors


and



the curve (1) passes the
point B (
τ
0
/2,
τ
0
/2) and
C

(
τ
B
, 0) for

pulsating cycle of loading and constant load,
accordingly. Substituting these coordinates in Eq.1 gives

2
0
0
1
0
2
2
2
















В
,

В
В









1
0
.

Having solved these equation one can obtain the factors


and


as follows
























В
В
В
В
В
В













0
0
1
2
0
0
1
2
2
4
,





(2)
























В
В
В
В
В
В













0
0
1
0
1
0
2
2
2

.

The equation of yield line
DK

(Fig.2) is
Т
m
a





,

(3)

From Esq. (1) and (3) after exception of
τ
B

one can obtain the following equation





0
1
1
2















В
Т
m
в
m

,

and from this one the abscissa of point
K

(Fig.2) can be found as

В
В
В
Т
OK











































1
2
1
2
1
2
1
.

Now the line
OK
is determined by angle

*

1
1
2
1
2
1
1
tan
1
1
2
*


























В
В
В
Т















(4)


















Fig.2. The diagram of approximating curve (1) crossing the points
A
,
B

and
C


The line
OK

separates the area of the cycles of loading limited by yielding
(
*



) from
area of cycles of loading limited by fatigue limit (
*



).


4
. Determination of safety factor on fatigue


When
*




the safety factor usually concerning yield limit is determined. This
calculation is not difficult. However, if
*




then the safety factor can be calcu
lated by
next way. Let us assume that we have the cycle of loading when the point
M

(Fig.3) with
co
-
ordinates (
a
m


,
) correspond to the working stresses. The limit cycle like the cycle
of point M will be marked by point
L
.














F
ig.3. The diagram to determine the safety factor.


From the diagram shown in Fig.3 one can determine the safety factor
FS

as

45
o

45
o

τ
-
1

τ
в


τ
T

τ
m

τ
a

A

B

C

D

K

O

τ
o
/
2

k

φ
*

45
o

τ
-
1

τ
в


τ
т

τ
m

τ
a

A

L

C

D

K

O

k

φ

M

l

m


Om
Ol
OM
OL
FS



.

The equation of the straight line
OL

is




tg
m
a


.



(5)

Having
excepted
τ
a

from Eqs.(1) and (5) one can obtain the following equation




0
tan
1
2














В
m
В
m
.

Then the abscissa of the point
L

has the value


В
В
Ol



































1
2
1
2
tan
2
tan

and the equation for safety factor gives

m
В
В
FS





































1
2
1
2
tan
2
tan

.

(6)


5
. Sample of problem


Use of the obtained equations is illustrated in the following sample problem.

The equations for stress and deformation of a closely coiled helical spring are derived
from the corresponding equations

for the torsion of a round bar. However it is assumed
when the helix angle
0
0
20
...
10


. When the



12
0

it is necessary to take into account
bending stresses.

Let the coiled spring is made of chromium
-
vanadium steel having the
following proper
ties:

shear strength,
τ
в

= 1200 MPa

yield strength in shear,
τ
T

= 950 MPa

endur
a
nce limit stress for pulsating shearing,
τ
0

= 550 MPa

endur
a
nce limit stress for reversed shearing,
τ
-
1

=0.6
τ
0

= 330 MPa

modulus of elasticity in shear,
G

= 0.8∙10
5
MPa.

The spring is made of the wire having round cross
-
section. The wire diameter d

= 7

mm,
the external diameter

D
e
= 60 mm, the number of spring coils
i
= 6, the initial spring
deflection

f
1

= 3,8

mm, the working spring deflection
h

= 28

mm. It is necessar
y to
determine the safety factor
FS

.



Problem solution.

The mean spring diameter

D

=

D
e

–

d

=
60
–

7

=
53 mm.


Coefficient taking into consideration the curvature of the spring coil

18
.
1
3
7
/
53
4
2
7
/
53
4
3
/
4
2
/
4









d
D
d
D
k
.

Rigidity of the spring (it is equal to the force

for spring deformation on one unit).

88
.
26
6
53
8
7
10
8
.
0
8
3
4
5
3
4







i
D
d
G
c
H/mm

The force for initial deflection of the spring

F
min

= c∙ f
1


=
26.88 · 3.8 = 102.1 Н.

The maximal deflection of the spring

f
2

=
f
1

+
h

= 3.8 + 28 = 31.8 mm.

The maximal force on the sprin
g

F
max

=
c
.

f
2
= 26.88 · 31.8 = 854.7 Н.

The minimal and maximal stresses

4
.
47
7
14
.
3
53
1
.
102
8
18
.
1
8
3
3
min
min






d
D
F
k


MPa,

0
.
397
7
14
.
3
53
7
.
854
8
18
.
1
8
3
3
max
max






d
D
F
k



MPa,

Then

2
.
222
2
4
.
47
0
.
397
2
min
max








m

MPa,

8
.
174
2
4
.
47
0
.
397
2
min
max








a

MPa.

The coiled springs usually have no stress
concentrations, and then characteristic of the
loading cycle is given as

787
.
0
2
.
222
8
.
174
tan



В
a



.

The value of

tan
must be compared with the value of
*
tan


by Eq. (4). For
comparison it is necessary to determine t
he factors


and



by Eqs. (2)

,
176
.
0
1200
550
2
1200
550
1200
330
1200
550
1200
550
2
1200
330
4
2
2
4
2
0
0
1
2
0
0
1













































В
В
В
В
В
В














.
097
.
0
1200
550
2
1200
550
1200
330
2
1200
550
1200
330
1200
550
2
2
2
2
0
0
1
0
1
0













































В
В
В
В
В
В















When the values


and



are substituted into Eq. (4) the
*
tan

is equal to

.
16
.
0
1
1200
330
1200
330
1200
950
097
.
0
1
097
.
0
2
176
.
0
1
097
.
0
2
176
.
0
1
1
1
1
2
1
2
1
1
tan
2
1
1
2
*




















































В
В
В
Т













So the
*




the safety factor is determined by Eq. (6)

.
46
.
1
2
.
222
1200
1200
330
097
.
0
1
097
.
0
2
79
.
0
176
.
0
097
.
0
2
79
.
0
176
.
0
1
2
tan
2
tan
2
1
2





















































m
В
В
FS













Such value of safety factor is satisfactory, however is close to limit value.



6
. Dependence of safety factor from mechanical properties and cycle
characteristics


Let us consider the inf
luence of mechanical properties of
material and

coefficient of cycle
asymmetry on variable loading on factor of safety of fatigue. When the different method
of approximation of real diagram of limit stresses. In the first case the calculation is
doing with

help of the equations of present work and in the two others by methods given
in [2].

In the case of approximation by Serensen it is necessary to use three points
A
,
B

and
D


(Fig.1)

,
1
m
a
FS












(7)


where



0
0
1
5
.
0
5
.
0








.



(8)

When the more simple diagram by Rabinovich [3] is considered only the two point A and
C are used

then




b



1


.




(9)

The calculation results is given in Fig.4. The error of calculation by Serensen l
e
ads to
increased

FS
as compared with present method (Fig.4 a, b). However the Rabinovich
method gives decreased values
of

FS
.


a)




b)





Fig.4. Dependence of safety
factor on endurance

limit of shearing stress.



O
n the increased values of
τ
в

the different in calculation results is decreased up to zero
(when
τ
в

is equal to 1200 MPa and more).

More effectively on the calculation results the asymmetry factor of
max
min



r

influences.

On the increase
of r

when sign of stresses is

constant (Fig.5) the different in calculation
results is risen (Fig.6 a
, b
) considerably.














Fig.5. Asymmetry
cycle of

constant sign.


a)




b)



a

t


m


min


max






Fig.6. Dependence of safety
factor on asymmetry

factor o
f loading cycle.



7
. Conclusion


It will be noted that it the helix angle of the spring


is more 10
0
…12
0

then the bending
stresses must be taken into account. To determine the safety factor on bending

FS

one
can use the equations fo
r determination of the

FS

by replacing


by

. The total safety
factor
FS

will be taken from following equation

2
2
2
1
1
1


FS
FS
FS


.

This method of calculation of the coiled springs working under the axial forc
es on
tension and compression can be spread for calculation of the safety factors for coiled
springs subjected to torsion when the bending stresses in the spring wire take place.




References


1.

Fuchs, H.O., and R.I.Stephens. Metal Fatigue in Engineering, Willey, New York, 1980.

2.

Serensen, S.V., Kogaev, V.P., Schneiderovitch, R.M. The Load Capacity and Strength
Calculation of Machine Parts. Machenebuilding. Moscow. 1975.