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INFLUENCE OF MOVING TOOTH LOAD ON GEAR CRACK PATH SHAPE AND FATIGUE LIFE
Damir T. Jelaska, Srdjan Podrug
The complete process of fatigue failure of mechanical elements may be divided into the following stages: 1) microcrack nucleation; 2) short crack growth; 3) long crack growth; and 4) occurrence of final failure The first two stages are usually termed as “crack initiation period” , while long crack growth is termed as “crack propagation period”.
The total number of stress cycles
N
can than be determined from the number of stress cycles
N
i fatigue crack initiation and the number of stress cycles
N
p required for the required for a crack to propagate from the initial to the critical crack length, when the final failure can be expected to occur:
N
N
i
N
p log
U
Wöhler curve
N i N p
crack initiation Podrug, Jelaska 1/4
FL N q N FL
log
N
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Critical plane methods
Critical plane approaches are based upon the physical observation that fatigue cracks initiate and grow on certain material planes, called critical planes, the orientation of which is determined by both stresses and strains at the critical location. Depending upon strain amplitude, material type and state of stress, materials generally form one of two types of cracks – shear cracks or tensile cracks.
1) Tensile based damage model has been found to be superior in correlating fatigue lives for materials whose damage development was tensile dominated:
d
t max a
E
f 2
N
2
b
f ' f
2
N
i
2) Shear based damage model has been found to be superior for materials whose damage development was shear dominated:
d
s a 1
k
max
R
t ' f
G
Podrug, Jelaska 2
N
i
b
0 f
N
i
c
0 The critical plane for tensile model is identified as the plane for which the tensile damage parameter
d
t has maximal value, and similarly, in the shear model, the critical plane is the plane for which the shear damage parameter
d
s is maximal. INFLUENCE OF MOVING TOOTH LOAD... 3
Gear models
Model where load is approximated with force acting at the HPSTC
A loading cycle of gear meshing is presumed as pulsating acting at the HPSTC
Moving force model
The magnitude as well as the position of the force, changes as the gear rotates through the mesh Podrug, Jelaska INFLUENCE OF MOVING TOOTH LOAD... 4
Practical example
Material data Gear data
Title Number of teeth 1 Number of teeth 2 Module Addendum modification coefficient 1 Addendum modification coefficient 2 Gear width 1 Gear width 2 Flank angle of tool Radial clearance factor Relative radius of curvature of tool tooth Addendum of tool Dedendum of tool Tip diameter Symbol
z
1
z
2
m
, mm
x
1
x
2
b
1 , mm
b
2 , mm n
c
* * f
h
a *
h
f *
d
a
E
, MPa
G
, MPa
R
m ,MPa
R
t , MPa D , MPa
K
th , MPa mm 206000 80000 0,3 1000 800 550
b
i
c
i
42 Cr Mo 4 – AISI 4142
n
'
K
' f ', MPa 0,14 2259 f ' 1820 -0,08 0,65 -0,76
C
, 269 Value 11 39 4,5 0,526 0,0593 32,5 28 24 o 0,35 0,25 1 1,35 Standard clearance mm
k
m
f ', MPa
b
0i f '
c
0i e,D
K
Ic , MPa mm 2620
m
1 1051 -0,08 1,13 -0,76 700 3,31 10 17 4,16 Podrug, Jelaska INFLUENCE OF MOVING TOOTH LOAD... 5
Gear tooth root stress
Podrug, Jelaska The stress has always the maximum value when load acts in the highest point of the single tooth contact (HPSTC). It follows that the maximum value of the tensile damage parameter
d
t will be at the point of the maximum main stress of the plane perpendicular to the root curve surface. Also, the maximum value of the shear damage parameter
d
s plane inclined at 45 o regarding to the plane of the maximum main stress. will be in the INFLUENCE OF MOVING TOOTH LOAD... 6
Lives to crack initiation
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Threshold crack length
Kitagawa-Takahashi type diagram
a
i
a
a
2 2 e 2 4 2 t,D D 1
K
th,eff D 2 The threshold crack length below which LEFM is not valid, i.e. transition point between initiation and propagation period Podrug, Jelaska INFLUENCE OF MOVING TOOTH LOAD... 8
Fatigue crack propagation
The application of the linear elastic fracture mechanics (LEFM) to fatigue is based upon the assumption that the fatigue crack growth rate, d
a
/d
N
, is a function of the stress intensity range
K
=
K
max -
K
min.
In this study the simple Paris equation is used to describe the crack growth rate: d
a
d
N
C
m
Fatigue crack closure
Fatigue crack closure occurs when crack faces contact during cyclic loading.
The three closure mechanisms considered most important are: Podrug, Jelaska INFLUENCE OF MOVING TOOTH LOAD... 9
Effective stress intensity range
Crack closure is reducing the stress intensity range. Reduced stress intensity range is effective stress intensity range
K
eff .
d
a
d
N
C
K
eff
m
K
eff
K
max 1 1 2 1
g
0,8561 0,0205
R
0,1438
R
2 0,2802
R
3 0,3007
R
4 Podrug, Jelaska INFLUENCE OF MOVING TOOTH LOAD...
g
e
K
max
K
th 1 10
Model where load is approximated with force acting at the HPSTC
The initial crack is placed perpendicularly to the surface at the point where maximum principal stress occurs in a gear tooth root for load acting in the HPSTC. The determination of the stress intensity factor (SIF) mode I and mode II is based on a
J
integral technique. In order to predict the crack extension angle the maximum tensile stress criterion (MTS) is used. 2 tan 1 1 4
K K
II I
K K
II I 2 8 The equivalent SIF is then:
K
eq cos 2 0 2
K
I cos 0 2 3
K
II sin 0 2 The effective stress intensity range:
K
eff
K
eff
K
eq,max
K
eq,max
K
cl for K eq,min
K
eq,min for K eq,min
K
cl ,
K
cl , Podrug, Jelaska INFLUENCE OF MOVING TOOTH LOAD... 11
Moving force model
Crack’s trajectory is computed at the end of the load cycle
j
1,
j
2 arctan 1 4
K
I
j
max
K
II
j
max 1 4
K
I
j
max
K
II
j
max 2 8 For load step from extension angle is calculated according to MTS criterion
K
eq cos 2
j
1,
j
2
K
I cos
j
1,
j
2 3
K
II sin
j
1,
j
2 For
j j
-1 to j crack th load case combined stress intensity factor is calculated
K
cl
K
eq, max d
a
c 1 2 1
g
eq, max
K
cl
m
The SIF when closure occurs The crack extension after one load cycle d
a
j
1,
j
2
K
eq
K
eq, max
K
eq
K
cl
da
c The amount of extension between load steps is proportional to the ratio of the change in equivalent SIF to the effective SIF arctan d
a
d
a
j
1,
j
j
1,
j
sin
j
1,
j
cos
j
1,
j
The final crack trajectory is approximated by a straight line Podrug, Jelaska INFLUENCE OF MOVING TOOTH LOAD... 12
Crack propagation results
Differences in crack paths for two gear models, and differences in number of loading cycles for the crack propagation to the critical length.
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Comparison with the experimental results
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CRACK PROPAGATION SIMULATION
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Conclusion
The numerical model used to predict the crack initiation life in tooth root is based on the critical plane methods as the most recent and numerically most demanded method for this purpose.
The fact that in actual gear operation the magnitude as well as the position of the force changes as the gear rotates through the mesh, is taken into account. In such a way, a more realistic stress cycle in gear tooth root is obtained. It resulted in significantly more exact assessments of the gear crack initiation life, and consequently in the entire fatigue life. It is reasonable, because the tensile damage parameter comprehends the amplitude of normal deformation, and the shear damage parameter comprehends the amplitude of shear deformation. Consequently, by neglecting the part of compression region stress cycle, which behave if dealing with load in the highest point of the single tooth contact, the significant error takes rise, especially in thin-rim spur gears.
By so completed numerical procedure, the predictions of crack propagation lives and crack paths in regard to the gear tooth root stresses are obtained, which are significantly closer to experimental results then existing methods.
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