Analysis of Axial Load Distribution in a Jet Engine Disk-Shaft Spline
Coupling
by
Justin Paul McGrath
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
Master of Engineering
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2009
(For Graduation December, 2009)
i
© Copyright 2009
by
Justin P. McGrath
All Rights Reserved
ii
CONTENTS
Analysis of Axial Load Distribution in a Jet Engine Disk-Shaft Spline Coupling ............ i
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS ........................................................................................................ vi
ACKNOWLEDGMENT ................................................................................................. vii
ABSTRACT ................................................................................................................... viii
1. Introduction.................................................................................................................. 1
2. Methodology ................................................................................................................ 5
2.1
Analytical Solution ............................................................................................ 5
2.2
Finite Element Solution ..................................................................................... 8
3. Results........................................................................................................................ 13
4. Conclusion ................................................................................................................. 23
5. References.................................................................................................................. 24
6. Appendix 1................................................................................................................. 25
7. Appendix 2................................................................................................................. 27
8. Appendix 3................................................................................................................. 30
9. Appendix 4................................................................................................................. 31
iii
LIST OF TABLES
Table 1 – Material Properties of 3D Spline Coupling Model .......................................... 13
Table 2 – Geometric Properties of 3D Spline Coupling Model ...................................... 14
Table 3 – Analytical results for axial load distribution at root fillet radius ..................... 16
Table 4 – Finite element results for axial load distribution at root fillet radius .............. 17
iv
LIST OF FIGURES
Figure 1-1: Finite element illustration of a spline coupling .............................................. 1
Figure 1-2: Pratt & Whitney’s F119 turbojet engine cutaway .......................................... 2
Figure 2-1: 3D model of the spline coupling assembly ..................................................... 9
Figure 2-2: A representative 3D symmetrical slice of the spline coupling assembly ....... 9
Figure 2-3: Finite element model of the representative section of the spline coupling ... 10
Figure 2-4: Applied boundary conditions of the finite element model ........................... 11
Figure 2-5: Pair of contact elements representing the mating of the spline teeth ............ 12
Figure 3-1: 3D vector sum of deflection in the spline coupling from applied torque ..... 15
Figure 3-2: Schematic showing the path used to extract the load distribution ................ 16
Figure 3-3: Finite element model axial load distribution in the shaft spline teeth .......... 18
Figure 3-4: Plot of contact pressure at root fillet radius verses contact length................ 19
Figure 3-5: Plot of normalized contact pressure, p(x)/p(avg).......................................... 19
Figure 3-6: 10X distortion of the 3D deflection of the sleeve spline teeth...................... 21
Figure 3-7: 10X distortion of the 3D deflection of the shaft spline teeth ........................ 21
v
LIST OF SYMBOLS
Symbol
m(x)
Description
Unit
Axial torque distribution along the root fillet radius
lb-in
φ1
Angle of twist of the sleeve
rad
φ2
Angle of twist of the shaft
rad
Cθ
Torsional stiffness constant of the spline assembly
lb/in-rad
M1(x)
Axial torque distribution in the sleeve
lb-in
M2(x)
Axial torque distribution in the shaft
lb-in
x
Axial distance along the root fillet radius of the teeth
in
L
Contact length of the coupling system
in
G1
Shear modulus of the sleeve material
Ksi
G2
Shear modulus of the shaft material
Ksi
J1
Polar moment of inertia of the sleeve geometry
in4
J2
Polar moment of inertia of the shaft geometry
in4
α
Constant of integration
in-3/2
A
Constant of integration
lb-in
B
Constant of integration
lb-in
c
Effective tooth height of spline
in
R
Pitch radius of the spline coupling system
in
N
Number of spline teeth in the coupling system
#
p(x)
Axial load distribution along the root fillet radius
psi
pavg
Average load seen along root fillet radius of the teeth
psi
p(x)max
Maximum load seen along root fillet radius of the teeth
psi
PR
The pressure ratio of maximum load over average load
-
τ
Applied torque of the coupling system
n
Number of nodes in the finite element model
p
Density
lb/in3
w
Weight
lb
E
Modulus of elasticity of the material
r
Root fillet radius of spline teeth
vi
lb-in
#
Ksi
in
ACKNOWLEDGMENT
I take this opportunity first of all thank my Lord and Savior, Jesus Christ. He has
given me the strength, knowledge, and perseverance to conduct and complete this
project. Without him, none of this would be possible.
Secondly, I thank my thesis advisor Dr. Ernesto Gutierrez-Miravete for his
mentorship, advice, and patience with me during the entire process. He has provided me
with valuable knowledge to make my project formulation process flow smoothly. Along
that same vein, I thank all of my graduate professors at Rensselaer Polytechnic Institute
at Hartford. Through their courses, they have provided me with significant knowledge
and theoretical background necessary to perform the project work.
Finally, I thank my employer United Technologies: Pratt & Whitney, for their
financial support and investment in my graduate education. They have given me the
opportunity to achieve an advanced degree in engineering. The employee scholars
program is one of the finest in the world.
vii
ABSTRACT
A spline coupling system is the method of choice in transferring torque between
rotating parts in a gas turbo fan engine. The elongated gear teeth distribute the torque
load axially along the centerline of the shaft. The challenge in spline design lies in
distributing that load evenly across the pressure face of the teeth.
Tatur [7], employs a methodology to analyze the axial load distribution in involute
splines. This methodology is modified and used for the general spline case. Assuming
one hundred percent torque transfer between the parts, and assuming the torque is
applied with axial symmetry, an analytical equation of load distribution is derived.
For comparison, a 3D model of the analytical spline geometry is produced, and a
representative 3D finite element model is generated. Identical boundary conditions and
torque are applied to the finite element model to match the theoretical case. Solving the
model reveals that the axial load distribution curve matches the curve of the analytical
equation. Both cases show that the load due to torque peaks at the axial ends of the root
fillet radius of the spline teeth. But there is a discrepancy in the magnitude of the load
predicted.
The analytical solution predicts a max load of 97.6 ksi and an average load of 57.6
ksi, while the finite element model shows a max load of 68.2 ksi and an average load of
45.3 ksi. This discrepancy is explained by the 3D model which demonstrates that
approximately 25% of the torque load provides the force that bends the spline teeth and
twist the coupling assembly. The theoretical solution does not account for these 3D
effects as it assumes one hundred percent torque transfer between parts.
Based on the findings, the analytical solution is the more conservative approach to
design a spline coupling system as it predicts higher loads. The finite element method,
although it is a significantly lengthier analysis, more accurately simulates the response
the spline coupling will see in an engine. Because it predicts 12.5% lower loads, it
correlates that the finite element model uncovers greater life capability of the coupling
system when compared to the analytical equation.
viii
ix
1. Introduction
A spine coupling is an effective mode of torque transfer between two rotating parts.
It employs the use of equally spaced teeth in the angular direction of a cylindrical
coordinate system. The teeth on the male spline interlock with the female spline along
the longitudinal axis of the cylindrical shaped coupling system (see Figure 1-1). To
learn more about spline couplings and the finite element model displayed in Figure 1-1,
review the findings Barrot, Paredes, and Sartor [2]. They have conducted several studies
to understand the loading and constraints necessary to accurately model spline torque
transfer. Their work investigates geometric design choices and their effects on load
distribution.
With a substantially larger contact area between teeth when compared to a
traditional gear system, spline couplings are used in high torque applications. The load
can be distributed in the longitudinal direction of the pressure faces of the teeth.
Figure 1-1: Finite element illustration of a spline coupling with the yellow graphic
representing the male spline and the blue representing the female spline [2]
Because of their robustness in handling torque, spline couplings are very common to gas
turbo fan and turbo jet engines in the aerospace industry. In one example of a gas
turbofan engine, a spline coupling is used to transfer torque from the hub of a turbine
disk to the low-pressure shaft that drives the front fan. This system can be seen in
engines such as Pratt & Whitney’s PW4000 series, and the military applications of F119
1
(Figure 1-2) and F135. The other aerospace gas turbine engine manufacturers, namely
Rolls Royce and General Electric also use splines to transfer shaft torque.
Figure 1-2: Pratt & Whitney’s F119 turbojet engine cutaway, where splines are
used to transfer torque to/from concentric shafts which run on the engine
centerline. [9]
The challenge in spline design, particularly in aerospace applications is the effect
that the load distribution across the pressure faces of the teeth (see Figure 1-3). Although
the pressure faces of the spline teeth provide a large contact area to distribute the torque
load; the challenge lies in distributing that load evenly. Uneven load distribution comes
from two main sources, geometric inconsistencies and deflection during torque transfer.
2
Figure 1-3: Schematic diagram of a spline coupling system of a gas turbofan engine
which employs the use of a spline to transfer torque from the turbine disk to the
low pressure shaft [9]
The manufacturing process of a spline carries with it inherent tolerances. These
tolerances manifest themselves in the root fillet radii, the flatness of the pressure faces,
and the circular run out of the pitch diameter of the spline coupling. The slight variance
of the geometry due to tolerances causes the spline to mate unevenly during torque
transfer. The result is that each set of spline teeth sits differently and a flush contact is
3
not achieved. This lack of symmetry causes each tooth to see a slightly different load
circumferentially. The tolerances and uneven contact allows the coupling system to
vibrate during torque transfer. The vibratory stresses cause the spline teeth the wear
prematurely from fretting at the pressure faces. The fretting wear ultimately decreases
the life of the part.
Of even more concern is the axial load distribution seen on the pressure faces of the
spline teeth. This has a greater effect on the wear and the life of the spline than the
tolerance induced vibratory stresses. Because of deflection in the coupling system during
torque loading, only certain parts of the pressure face of the teeth mesh. As a result there
is an uneven load distribution along the face of the teeth. This load distribution is most
clearly seen at the root fillet radius along the axial edge of the pressure face of the spline
teeth. This is the area where stresses are the highest due to 3D stress concentration
factors. Understanding the load distribution in this area during torque transfer gives a
designer the ability to more accurately estimate the life of the spline coupling system.
The load ratio is directly correlated with the life of the coupling; the higher the pressure
ratio the less life can be obtained with all else being equal.
In the analysis to follow a spline coupling is modeled analytically to determine the
equation of the axial load distribution when a specified torque load is applied to the
system. This equation is then compared to a finite element model of the same coupling
system using the equivalent boundary conditions. The peak stress on the pressure face is
determined and normalized by the average stress across the contact area of the spline
teeth. The calculated ratio is used as a comparison point between the analytical and finite
element models to see which is the more conservative approach.
The analysis offers designers insight on the stress profile they can expect during
operation of the coupling system. They can use the load distribution as feedback to
pinpoint parameters and dimensions to be changed to arrive at a more favorable load
distribution.
4
2. Methodology
2.1 Analytical Solution
The goal of the mathematical analysis is to develop an equation outlining the load
distribution on the contact face of the spline teeth. The first step is to identify the
governing equation that represents the torque transfer of the coupling system. In Tatur’s
analysis of involute splines [7], he determined that the axial distribution of torque in the
coupling system, m(x), is related to the torsional stiffness of the spline geometry and the
difference of the angle of twist between the shaft and the sleeve.
m( x) 
d
dx
M 2 ( x)  c 1 ( x)  2 ( x)
(1)
Although Tatur used this equation for involute splines it can be applied to splines of
all types. The differential relationship is the governing equation for the subsequent
analysis. When analyzing the equation 1 certain assumptions are made. The first is that
there is one hundred percent perfect transfer of torque from the sleeve to the shaft. The
second is that the torque is applied evenly across the contact length of the spline teeth.
These assumptions and the fact that this analysis is done in a steady state condition allow
the use of the global torque equilibrium equation which states:
M1 (0)  M 2 (0)  M1 ( L)  M 2 ( L)
(2)
Equation 2 articulates that the torque applied at axial position x = 0 of the contact face is
the same as the torque applied at the end of the contact face, axial position x = L, where
L is the total contact length of the system.
Proceeding one step further with the assumption, the torque at every axial distance
along the contact length is the same with in the spline coupling system. This is the Local
torque equilibrium equation:
M 1 ( x)  M 2 ( x)  M1 ( x)  M 2 ( x)
(3)
To obtain a general equation m(x) of axial torque distribution along the pressure face the
first step is to differentiate the relationship established in the first equation.
d
dx
m( x ) 
d2
dx 2
M 2 ( x)  c
d
dx
1 ( x)   2 ( x)
(4)
The derivative of the angle of twist φ, is related to the shear modulus and second
moment of inertia of the spline geometry. This can be obtained two ways; a tedious hand
5
calculation or by querying a three dimensional computer aided solid model of the
coupling system.
d
dx
i ( x) 
 M i ( x)
Gi J i
(5)
The subscript i, represents either the shaft material and geometry or the sleeve
material and geometry. Applying the relationship in the above equation to the derivative
of the axial torque distribution yields:
d
dx
m( x) 
d2
dx2
 M ( x) M 1 ( x) 
M 2 ( x)  c  2


G
J
G1 J 1 
 2 2
(6)
In equation 6, the subscript 1 is the sleeve, and the subscript 2 is the shaft. Algebraic
factoring of the axial torque distribution leads to the following.
d
dx
 1
 M 1 ( x)  M 2 ( x) 
1 
m( x)  c M 2 ( x) 

  c 

G1 J 1
 G2 J 2 G1 J 1 


(7)
Substituting the local torque equilibrium relationship from equation 2 into equation 7
results in further simplification of the derivative of torque distribution.
d
dx
 1
 M 1 (0)  M 2 (0) 
1 
m( x)  c M 2 ( x) 

  c 

G1 J 1
 G2 J 2 G1 J 1 


(8)
The next step is to calculate the second derivative of m(x) taking into account that M1(0)
and M2(0) are known constants which represent the initial applied torque to the system
and therefore the second term of equation 8 drops out of the equation.
d2
dx2
m( x)  c
d
dx
 1
1 
M 2 ( x) 


 G2 J 2 G1 J 1 
(9)
The derivative of M2(x) is equaled to m(x) by the relationship established in equation 1.
Substituting m(x) into the equation above and rearranging yields a second order
differential equation that describes the evolution of torque transfer along the axial
direction of the spline teeth.
d2
dx2
 1
1 
m( x)  c 

 m( x)  0
 G2 J 2 G1 J 1 
(10)
The analytical solution for the second order differential equation has the form:
m( x)  Aex  Be x
(11)
6
Where the constant α equals:
 1
1 
  c 


 G2 J 2 G1 J 1 
(12)
In order to solve for the coefficients A and B, boundary conditions are applied.
Based off the assumption set in place at the beginning of the analysis, torque at the
contact interface x = 0, equals τ, the known applied torque of the system. Furthermore,
torque at the contact interface x = L also equals the applied torque of the system. These
conditions are represented mathematically:
m ( 0)  
m( L)  
Applying the boundary conditions to the analytical solution yields values for A and B.
 eL  1 

A    2L
e

1


(13)
 e 2L  eL 

B    2L
 e 1 
(14)
The overall axial torque distribution equation of the spline coupling system is
represented by the analytical solution:
 eL  1  x
 e 2L  eL
e    2L
m( x)    2L
 e 1
 e 1
 x
e

(15)
In equation 15, τ, α, and L are all known constants; while x represents the axial
distance along the pressure face of the spline coupling. With axial distribution of torque
fully defined, the load distribution is determined. Many scientists such at Barrot,
Paredes, and Sartor [2], have employed the following relationship in their analysis to
relate torque transfer in splines to axial load distribution.
cRNp ( x)  m( x)
(16)
Load distribution, p(x), is related to torque transfer by three geometric constants; c is the
effective tooth height, R is the pith radius and N is the number of teeth in the coupling
system. Solving for p(x) from equation 16 and substituting the expression from equation
15 yields:
7
  e L  1  x
 e 2L  e L
e    2L
p( x)    2L
 e 1
  e 1
 x  1 
e 

cRN




(17)
The analytical equation of load distribution is based on the known values of applied
torque, as well as material and geometric properties of the spline coupling. The curve of
equation 17 will show pressure peaks due to torque transfer of the designed spline
system along the axial direction of the contact face. Pressure peaks provide an important
indicator to designers of the effectiveness of the coupling system. When comparing the
pressure peak to the overall average pressure along the axial direction of the contact face
a ratio is formed:
PR 
p( x) max
p avg
(18)
The comparison of the max pressure or max load seen on the contact face to the
average pressure in the axial direction is a direct correlation to the overall life of the
spline coupling system. The higher the pressure ratio, PR, the less life cycles the system
can obtain. Therefore PR is an important parameter in the design of splines. Designers
can use the metric to change geometry and loads to minimize the ratio and even the load
distribution.
The pressure ratio of the analytical solution is compared to the ratio produced by the
finite element model of the same coupling system to see which method of analysis is
more conservative from a design standpoint.
2.2 Finite Element Solution
The finite element solution is a lengthier method of analyzing the load distribution
due to torque transfer in a coupling system. In many cases it is more accurate, especially
when analyzing three dimensional geometry. The analytical solution does not account
for three dimensional effects such as stress risers and 3D deflections.
To make an accurate comparison, a 3D CAD model of the spline coupling system is
rendered using Unigraphics NX-4 software (Figure 2-1). The model contains the exact
geometric parameters found in the analytical methodology. Two separate part files are
created; one to model the spline sleeve and the other to model the shaft. The two
components are mated along the pressure teeth in an assembly file (Figure 2-1).
8
Figure 2-1: 3D model of the spline coupling assembly, the red is the shaft and the
silver is the sleeve.
Following mating of the spline teeth and a check for interference and misalignment
of the contact faces, the 3D geometry is prepared for structural analysis. To reduce the
size of the model and curtail computing and post-processing time, a representative
section of the 3D model (Figure 2-2) is used for structural analysis.
Figure 2-2: A representative 3D symmetrical slice of the spline coupling assembly
used for structural analysis.
9
The 3D section is imported into ANSYS and meshed (Figure 2-3). Material
properties are assigned to the volumes representing the sleeve and the shafts. For both
the analytical and finite element solutions, the sleeve material is chosen as IN-100 nickel
super-alloy which is a typical application for turbine disks in gas turbo fan engines. The
shaft material is chosen as INCO-718 nickel super-alloy, also a common shaft material
in the aerospace industry.
Both the sleeve and the shaft are meshed using Solid 45 elements. Brick mesh is
used for the full hoop part of the shaft and the sleeve. However due to the irregular shape
of the volumes representing the spline teeth, tetrahedral mesh is used there. The size of
the mesh at the spine teeth interface is imperative to ensure the nodes capture the details
of the load distribution. In this finite element model the tetrahedral mesh is 0.02 in/in,
which is composed of solid95 and solid92 tetrahedral elements.
Figure 2-3: Finite element model of the representative section of the spline coupling
10
After meshing the boundary conditions are applied. First, cyclic symmetry is
applied to the left and right faces of the sectioned geometry. Second, a surface to surface
contact is created between the areas representing the pressure faces of the two pairs of
spline teeth. The contact pair consists of Conta174 and Targe170 surface elements to
simulate the mating condition of the teeth. In the third boundary condition considered,
the full hoop ends of the shaft and the sleeve are constrained axially. In the fourth
condition, the full hoop portion of the sleeve is constrained so it cannot rotate about the
axial centerline of the model. Finally, a uniform force is applied to each node at the full
hoop ends of the shaft to simulate the applied torque received from the sleeve (see
Figure 2-4). The force is determined using the following relationship:
F
 1
 
Rn
(19)
In equation 19, the coupling system torque τ, is divided by the pitch radius and the
number of nodes, n, on one face of the full hoop end of the shaft. The expression
produces an accurate nodal force to match the analytical methodology.
Figure 2-4: Applied boundary conditions of the finite element model
11
Figure 2-5: Pair of contact elements representing the mating of the spline teeth
The finite element model contains with in it 29,258 nodes. The model is evaluated
as a steady state system at room temperature which is consistent with the analytical
methodology.
Careful post-processing of the model extracts the resultant load distribution from the
nodes at the root fillet radius of both spline teeth. The extracted data is plotted and
compared with the corresponding analytical curve.
12
3. Results
Both the analytical model and the finite element solution use IN-100 powder nickel
super alloy for the sleeve and INCO-718 for the shaft when analyzing the coupling
system. To learn more about the composition and application of these two alloys see
Appendix A1. This is a representative combination for the turbine section of a gas turbo
fan engine. The sleeve represents the hub portion of the disk which is closer to the gas
path than the shaft which is why IN-100 powder is used. It has more high temperature
capability than INCO-718 while still maintaining high strength. A summary of the
relevant material properties used in the coupling calculations are summarized in Table 1.
Table 1 – Material Properties of 3D Spline Coupling Model
Specification
Symbol
Sleeve
Shaft
Unit
Material
-
IN-100
INCO718
-
Density
p
0.284
0.297
lb/in3
Weight
w
0.118
0.173
lb
Modulus of Elasticity
E
30.1
31.0
Ksi
Shear Modulus
G
11.94
11.10
Ksi
Polar Moment of Inertia
J
0.085
0.037
in4
Along with material properties, there are two coupling metrics displayed in Table 1.
The weight of each component, w, and the polar moment of inertia, J, of each
component were calculated by querying the 3D Unigraphics model. The weight and
polar moment of inertia can be verified by hand calculations by resolving both
geometries into simple shapes and adding the value of each shape in an iterative process.
To compute the axial load distribution on the spline teeth of the shaft, all geometric
parameters of the coupling system are defined. Along with geometry the user applied
torque is selected as well. Again to match the conditions of a typical gas turbo fan
engine, a torque value of 350in-lb is used in calculation of the results. This value is
13
comparable to that transferred between a turbine disk and low pressure shaft. A
summary of all geometric parameters and applied torque are displayed in Table 2.
Table 2 – Geometric Properties of 3D Spline Coupling Model
Specification
Symbol
Value
Unit
Applied Torque
τ
350
in-lb
Contact Length
L
0.30
in
Pitch Radius
R
0.70
in
Number of Teeth
N
56
#
Tooth Height
c
0.032
in
Root Fillet Radius
r
0.010
in
Pressure Angle
θ
30
deg
Torsional Stiffness
Cθ
3332488
lb/in-rad
The last parameter listed in table 2 is the torsional stiffness of the coupling system.
This value is difficult to determine without the use of the finite element model. Equation
1 defines that the applied torque is equaled to the difference of the angle of twist of the
shaft and sleeve multiplied by the torsional stiffness constant, Cθ. Querying the results of
the finite element model the difference in the angle of twist between the two components
is determined. Plotting the vector sum of deflection for the coupling system, (see Figure
3-1) the maximum and minimum points of deflection of each component are located.
The maximum deflection on the shaft is seen at the extreme axial ends of the full hoop
section. The minimum deflection of the shaft is seen at the middle of the shaft where the
spline teeth are located. This result makes sense intuitively because the outer ends of the
shaft should be more flexible than the middle section where the teeth add hoop stiffness.
The sleeve has the opposite response due to the fact the full hoop section is much
thicker than the sleeve spline teeth. Therefore, Figure 3-1 maximum defection is seen in
the spline teeth and the minimum is seen at the outer edges of the full hoop section.
For each component the difference between the maximum and minimum deflections
are determined and resolved into radians based on the pitch radius of the coupling
14
system. Using the relationship that the circumference, s = rφ, assuming that deflections
represent a small arc. Knowing the calculated angle of twist and the applied torque the
torsional stiffness constant is estimated as shown in table 2.
Figure 3-1: 3D vector sum of deflection in the spline coupling from applied torque
With all of the coupling specifications defined, the values can be substituted into
equations: 12, 13, 14, 17 and 18. These equations represent the analytical solution of
load distribution at the root fillet radius of the spline teeth of the shaft. The calculated
values of the equations are shown in table 3.
15
Table 3 – Analytical results for axial load distribution at root fillet radius
Parameter
Value
Unit
α
10.67
(lb/in-rad)1/2
A
13.67
-
B
336.3
-
p(x)max
97.66
ksi
pavg
57.61
ksi
PR
1.70
-
The analytical response is now compared to the finite element solution. The axial
load distribution at the root fillet radius of the spline teeth of the shaft is extracted from
the finite element model. See Figure 3-2, which shows the path used to extract the load
seen at the root fillet radius of both teeth. The path distance is equaled to the contact
length of the spline teeth, L = x = 0.3 inches.
Figure 3-2: Schematic showing the path used to extract the load distribution in the
finite element model
16
The extracted load distribution along the contact length of the spline teeth is
summarized in table 4 for comparison to the analytical model summarized in table 3.
Table 4 – Finite element results for axial load distribution at root fillet radius
Parameter
Left Tooth
Right Tooth
Unit
p(x)max
71.13
68.51
ksi
pavg
47.06
45.34
ksi
PR
1.51
1.51
-
d(x)max
0.00015
0.00015
in
davg
0.0014
0.00014
in
DR
1.1
1.1
-
The parameters d(x), davg, and DR represent the deflection of the spline teeth as a
function of x, the average deflection over the contact length, and the deflection ratio
respectively. The deflection ratio, DR, is the ratio of the maximum deflection, d(x) over
the average deflection seen across the root fillet radius of the shaft spline teeth.
Deflection is book kept in the finite element model to determine whether axial deflection
in the spline teeth influences the load distribution. However, table 4 shows that the
deflection across the root fillet radii of the spline teeth is uniform, with a DR of 1.1.
The finite element solution of load distribution along the plotted path in figure 3-2 is
not uniform. The load peaks at either end of the contact length as displayed by the finite
element response of torque transfer presented in figure 3-3.
17
Figure 3-3: Finite element model axial load distribution in the shaft spline teeth due
to the applied torque
When observing the entire pressure face of the spline teeth, the red color shows that the
highest stress is found in the root fillet radius along the contact length. It also shows that
the load in the fillet is not distributed evenly. This result is consistent with the plot of the
analytical solution (Figure 3-4).
18
Axial Load Distribution in Spline Coupling
120000
Contact Load (psi)
100000
80000
Analytical Sol.
60000
Finite Element Sol. Left
Finite Element Sol. Right
40000
20000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Spline Tooth Contact Length (in.)
Figure 3-4: Plot of contact pressure at root fillet radius verses contact length
Normalized Axial Load Distribution
1.8
1.6
1.4
p(x)/p(avg)
1.2
1
Analytical Sol.
Finite Element Sol. Left
0.8
Finite Element Sol. Right
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Spline Tooth Contact Length (in.)
Figure 3-5: Plot of normalized contact pressure, p(x)/p(avg), at the root fillet radius
of the spline teeth verses contact length
19
The finite element and analytical models produce the same shape of axial load
distribution due to torque transfer. The load peaks at both ends of the contact length of
the spline teeth in both cases. This response is undesirable because it causes more
fretting between the pressure faces and as a result reduces the life and effectiveness in
transferring torque.
It is important to note that although both the analytical and finite element solution
produce the same shape of load distribution, the analytical model predicts higher peaks.
The maximum analytical load is 97.6 ksi, while the finite element equivalent is 68.2,
which leaves a 30 ksi disparity between the two. Furthermore, referring to tables 3 and 4,
the pressure ratio of the analytical model is 1.70 while the finite element pressure ratio is
1.51.
The disparity between the analytical and finite element solutions can be explained
by the two facts. First, is that the analytical solution has with it inherent assumptions
discussed in the methodology. It assumes one hundred percent torque transfer from the
sleeve to the shaft, which means the analytical load distribution is a result of zero loss of
energy to friction or deformation. In actuality, as the finite element solution shows, there
is not one hundred percent torque transfer. A portion of the torque goes into deforming
the spline teeth. Looking at the pressure faces of the teeth in Figure 3-3, the blue and
green represents the portion of the tooth under compression, while the red and yellow
represent the portion under tension. Each shaft tooth is acting as a cantilever beam where
the torque transfer from the sleeve supplies the load at the end of each shaft tooth.
Because some of the torque causes the teeth to bend, the root fillet radius does not see
the entire load. That is why, in Figure 3-4, and Figure 3-5, the analytical solution has
higher peaks.
The second reason for the disparity in load distribution and pressure ratio between
the analytical and finite element solution is the fact that the analytical does not take into
account the distorted shape during torque transfer. The 3D finite element model has two
pairs of deformed teeth that mesh with each other (see Figure 3-6, and Figure 3-7).
Because the pressure faces of the teeth do not meet flush during torque transfer, the root
fillet radius does not see the entire load. This explains why the analytical solution has a
higher pavg and pmax which results in a 12.5% higher pressure ratio.
20
Figure 3-6: 10X distortion of the 3D deflection of the sleeve spline teeth under
torque load
Figure 3-7: 10X distortion of the 3D deflection of the shaft spline teeth under toque
load
21
Figure 3-4 and figure 3-5 show a disparity between the analytical solution and the
finite element solution although both cases exhibit the same shaped curve of load
distribution. The normalized load distribution plot causes both cases to more closely
align, which is a sign that the analytical and finite element solutions agree when the
normalized plot is used. Also, it is important to note that the finite element model shows
consistent behavior when comparing the right tooth to the left tooth. This symmetry
points to the fact that the boundary conditions are properly applied and match those used
in the analytical methodology.
Figure 3-3, figure 3-6 and figure 3-7 help to explain why the finite element model
predicts a lower load distribution and lower pressure ratio. The 3D effects of the tooth
bending and the fact that the pressure faces of the teeth do not exhibit a flush mating
surface because of torsional distortion, cause an imperfect torque transfer system. From
the current analysis this loss of torque can be quantified to approximately 25%. The 3D
effects reduce the load seen at the root fillet radius. However, these effects simulate the
loading condition a designer can expect during operation in an engine.
When comparing the finite element to the analytical both have advantages and
disadvantages. The finite element solution gives an accurate analysis of the coupling
design. The finite element model predicts a lower PR than the analytical equation. With
all else being equal, by correlation, the finite element model proves that there is
approximately 12.5% more life in the coupling system. Although the finite element
model is more realistic it is a much lengthier method of determining the effectiveness of
the spline design. The analytical equation is conservative but it gives an instant answer
when the geometric and material properties are plugged into equations 12-18. The
designer will have to weigh his or her options on whether or not 12.5% more predicted
life is value added when compared to the time savings of a shorter analysis.
22
4. Conclusion
Tatur’s method has been used for involute spline analysis to the general spline case
and has yielded comprehensive results. The analytical equation for load distribution at
the root fillet radius of the spline teeth is dependent upon the material properties of the
shaft and sleeve, the spline coupling geometry, and the torque applied to the system.
The finite element model which incorporates the spline coupling material properties,
geometry, and boundary conditions into a 3D analysis validates the analytical solution.
Both cases predict the same axial load distribution curve. Symmetry and accuracy of the
boundary conditions in the finite element model are justified by the fact that both the left
and right tooth exhibit the same response to the applied torque load.
The disparity between the analytical and finite element solutions in predicting
maximum load and pressure ratio can be explained by 3D effects. The analytical solution
assumes one hundred percent torque transfer from the sleeve to the shaft. The 3D finite
element solution shows that this assumption is flawed because approximately 25% of the
torque is dissipated during transfer. A portion of that coupled force goes into bending the
spline teeth and twisting the overall spline geometry.
Both methods provide the designer a valuable picture of the load distribution across
the pressure face of the spline teeth. This load, and the ratio of max load over average
load is directly correlated to the amount of life expected in the coupling system. Since
the analytical solution predicts the higher load ratio (PR), it is the more conservative
approach. The 3D finite element model is more accurate, and based on the results
predicts 12.5% lower pressure ratio which indicates there is more life in the system. The
disadvantage of the finite element solution is the time it takes to create the model and
verify the correctness of the boundary conditions.
Ultimately the designer must decide if the 12.5% more predicted life capability is
worth the extra time spent on the finite element analysis, or if the conservative analytical
equation will suffice. There is value in a quick conservative answer. Both approaches
can be used to help the designer look at geometry changes to the coupling system and
observe the effect they have on the axial load distribution. Of course the goal is to make
the axial load on the teeth uniform. That is the next step in optimizing the spline design.
23
5. References
[1] Adey, R. A., Baynham, J., Taylor, J. W., 1999. “Development of analysis tools
for spline couplings,” Proceedings of the Institution of Mechanical Engineers,
Part G: Journal of Aerospace Engineering, v 214, n 6, 2000, pp. 347-357.
[2] Barrot, A., Paredes, M., and Sartor, M., 2008, “Extended equations of load
distribution in the axial direction in a spline coupling,” Engineering Failure
Analysis, 16(2009), pp. 200-211.
[3] Ding, J., McColl, I. R., Leen, S. B., 2006. “The application of fretting wear
modeling to a spline coupling,” Wear, 262(2007), pp. 1205-1216.
[4] Orberg, E., Jones, F. D., Horton, H. L., and Ryffel, H. H., 2003. Machinery’s
Handbooks Guide. 27th Edition, Industrial Press Inc, New York, 2004.
[5] Sum, W. S., 2005. “Efficient finite element modeling for complex shaft
couplings under non-symmetric loading,” Journal of Strain Analysis for
Engineering Design, v 40, n 7, October, 2005, pp. 655-673.
[6] Sum, W. S., 2002. “Parametric Study on the Frictional Contact Behaviour
between Spline Teeth,” Materials Science Forum, v 440-441, 2003, pp. 69-76.
[7] Tatur, G. K., Vygonnyi, A. G. “Irregularity of load distribution along a splined
coupling,” Russian Engineering Journal, 1969; XLIX: 23-7.
[8] Tjernberg, A., 2000. “Load distribution and pitch errors in a spline coupling,”
Materials and Design 22(2001), pp. 259-266.
[9] Pratt & Whitney Design Manual, 2002. Section C1.4, Military applications, pp
102-104.
24
6. Appendix 1
[A1.1]: Spline Coupling Material General Information
INCONEL 718
PRIMARY APPLICATION: Disk-turbine Disk-compressor Ring
MAXIMUM USE TEMPERATURE: 1200F
DENSITY: 0.297 lb per cu in.
GRAIN SIZE: Equiaxed grains ASTM 6 - 4
GENERAL DESCRIPTION: Heat treatable nickel-base alloy which has good
strength at temperatures up to 1200F coupled with good oxidation and corrosion
resistance in that temperature range. Yield strengths Inconel 718 are superior to
those of Waspaloy and Incoloy 901.
APPLICATION DETAILS: Primarily for rotor parts including compressor and
turbine disks operating at temperatures up to 1200F which require: good strength
good oxidation and corrosion resistance
[A1.2]: Spline Coupling Material General Information
IN-100 (GATORIZED)
PRIMARY APPLICATION: Disk-turbine Disk-compressor Seal Spacer
MAXIMUM USE TEMPERATURE: 1250F (1300-1350F should be restricted to
limited exposure time applications only)
25
DENSITY: 0.284 lb per cu in.
GRAIN SIZE: ASTM 10 - 12.5
GENERAL DESCRIPTION: IN-100 is a heat treated nickel-base alloy gatorized
after consolidation into billet from pre-alloyed powder by extrusion. It has good
strength at temperatures up to 1300F with good oxidation and fair to good
corrosion resistance. Extended use above 1250F (greater than 1000 hours at 1300F,
greater than 500 hrs at 1350F) will result in the formation of grain boundary
carbides, which can reduce material properties (stress-rupture). Ultimate tensile
strength is superior to Waspaloy, Astroloy and Inconel 718. Yield strength is
inferior to Inconel 718, but superior to Waspaloy and Astroloy. Creep strength is
superior to Inconel 718 and initially superior to Waspaloy but drops off rapidly
above 1300F becoming inferior to that alloy.
APPLICATION DETAILS: Primarily applicable to rotating parts such as disks,
seals and spacers in compressor and turbine sections operating at temperatures up
to 1250-1350F which require: High temperature tensile strength superior to
Waspaloy and Inconel 718. Maximum cyclic life creep requirements limited to
below 1250F
MELTING PRACTICE: Powder produced by gas atomization from alloy made by
vacuum induction melting process.
26
7. Appendix 2
[A2]: ANSYS 3D Spline Coupling Model Log File
/BATCH
/COM,ANSYS RELEASE 11.0SP1 UP20070830
15:45:14
10/21/2009
pwautom
*SET,symmeshFailed_,0 ! Initialize counter for area symmetry mesh failure to 0
*SET,meshKey_,arg1
! 0=free, 1=mapped, 2=try mapped then free
*SET,shapeCode_,arg2 ! 0=use whatever, 1=use tria
*SET,midKey_,arg3
! mshmid key
*SET,volindx_,arg4
!Debug /NERR,0,9999999,-1,0,0 !! Allow for mesh failures....
! Mesh cyclic symmetry faces
! ... For structural and thermal cyclic symmetry faces, ...
MSHKEY, meshKey_
MSHMID, midKey_
MSHAP, shapeCode_, 2D
TYPE, etsurf_
CSYS, 5 ! 04-05-2007
*DO, ar43, 1, ar31
*IF, symmarry(volindx_,ar43) ,LT, 1 ,EXIT
ASEL,S, , ,%symmarry(volindx_,ar43)%
ALLS, BELO, AREA
ASLV, R
*IF, ARINQR(0,13) ,LT, 1 ,CYCLE ! Area was on a meshed volume
! Attempt to area mesh symmetry face
ALLS, BELO, AREA ! 04-05-2007
ESIZE, globvlsz(volindx_), 0
*IF, ARINQR(symmarry(volindx_,ar43),-6) ,EQ, 0 ,THEN
AMESH, ALL
*IF, ARINQR(symmarry(volindx_,ar43),-6) ,EQ, 0 ,THEN
27
am_amesh, globvlsz(volindx_) ! Attempt to iterate to an area
! mesh size that works
*ENDIF
*ENDIF
! If source area was meshed, attempt to copy mesh to target area
*IF, ARINQR(symmarry(volindx_,ar43),-6) ,GT, 0 ,THEN
! Copy a KP from the master side to the presumed copy side location
! RNG 04-05-2007
*SET,sctr_angl_,-360/%matsf(maxc+volindx_)%
*SET,keypt1_,KPNEXT(0)
ASEL,A, , ,%symcarry(volindx_,ar43)%
*IF, ARINQR(symcarry(volindx_,ar43),-6) ,GT, 0 ,CYCLE ! See if area already
meshed
ALLS, BELO, AREA
! If the new KP isn't with 0.001-in of the "copied" kp, reverse the sector angle
! RNG 04-05-2007
!RNG 10-9-2007 Modify logic so new KP isn't literally created.
*SET,keypt2_,kp(kx(keypt1_),ky(keypt1_)+SCTR_ANGL_,kz(keypt1_))
*AFUN, DEG
*SET,ar51,(KZ(keypt1_)-KZ(keypt2_))**2
*SET,ar52,KX(keypt1_)*COS(KY(keypt1_)+SCTR_ANGL_)
-
KX(keypt2_)*COS(KY(keypt2_))
*SET,ar51,ar51+ar52*ar52
*SET,ar52,KX(keypt1_)*SIN(KY(keypt1_)+SCTR_ANGL_)
KX(keypt2_)*SIN(KY(keypt2_))
*SET,ar51,SQRT(ar51+ar52*ar52)
*AFUN, RAD
*if, ar51, gt, 0.001, then
*SET,sctr_angl_,-sctr_angl_
*ENDIF
! Success for AGEN,2,ALL; NUMM,NODE; NUMM,KP guarantees
28
-
! face symmetry. It may also be more robust than MSHCOPY.
MSHCOPY, AREA, %symmarry(volindx_,ar43)%, %symcarry(volindx_,ar43)%,
5, 0, sctr_angl_, 0, 0.0001, ,
*SET,ar22,_STATUS
! Failed to copy mesh to the symmetry target face
*IF , ar22 ,GT, 2 , THEN
*SET,symmeshFailed_,symmeshFailed_+1
ACLEAR, ALL
! Success. Save the areas as meshed.
*ELSE
CMSEL, A, cycsym_
CM, cycsym_, AREA
*ENDIF
! Failed to mesh the symmetry master face
*ELSE
*SET,symmeshFailed_,symmeshFailed_+1
*ENDIF
*ENDDO ! ar43
*SET,eshap_,
*SET,shapeCode_,
*SET,keypt1_,
*SET,keypt2_,
*SET,meshKey_,
*SET,midKey_,
*SET,sctr_angl_,
*SET,volindx_,
C*** AutoModeler SYMmetry MESHing MACro END
SAVE
FINISH
! /EXIT,MODEL
29
8. Appendix 3
[A3.1]: Contour Plot of Shaft Spline Teeth 3D deflection
[A3.2]: Excel Plot of Spline Tooth Deflection vs. Contact Length
3D Deflection of Spline Tooth
1.6E-04
Deflection (in.)
1.5E-04
1.5E-04
Finite Element Sol. Left
Finite Element Sol. Right
1.4E-04
1.4E-04
1.3E-04
0.000
0.050
0.100
0.150
0.200
0.250
Spline Tooth Contact Length (in.)
30
0.300
0.350
9. Appendix 4
[A4.1]: Load Distribution Data of the Analytical Solution
Torque
Dist.
350
317.4838
288.5896
262.9879
240.3866
220.5277
203.1848
188.16
175.2818
164.4033
155.4005
148.1706
142.6312
138.7189
136.3893
135.6157
136.3893
138.7189
142.6312
148.1706
155.4005
164.4033
175.2818
188.16
203.1848
220.5277
240.3866
262.9879
288.5896
317.4838
350
x
in.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
0.3
m(x)
lb-in.
122500
111119.3
101006.4
92045.78
84135.3
77184.71
71114.68
65855.99
61348.62
57541.17
54390.18
51859.72
49920.9
48551.62
47736.26
47465.5
47736.26
48551.62
49920.9
51859.72
54390.18
57541.17
61348.62
65855.99
71114.68
77184.71
84135.3
92045.78
101006.4
111119.3
122500
31
p(x)th
psi
97656.25
88583.64
80521.66
73378.33
67072.15
61531.17
56692.19
52499.99
48906.75
45871.47
43359.52
41342.25
39796.64
38705.06
38055.05
37839.2
38055.05
38705.06
39796.64
41342.25
43359.52
45871.47
48906.75
52499.99
56692.19
61531.17
67072.15
73378.33
80521.66
88583.64
97656.25
p(x)/pavg
psi/psi
1.695247
1.537753
1.397802
1.273798
1.164327
1.06814
0.984138
0.911365
0.848988
0.796298
0.752692
0.717674
0.690843
0.671894
0.66061
0.656863
0.66061
0.671894
0.690843
0.717674
0.752692
0.796298
0.848988
0.911365
0.984138
1.06814
1.164327
1.273798
1.397802
1.537753
1.695247
[A4.2]: Load Distribution Data of the Left Tooth of the Finite Element Model
x_fea_lft
in.
0.000
0.010
0.020
0.029
0.039
0.049
0.059
0.069
0.079
0.088
0.098
0.108
0.118
0.128
0.138
0.147
0.157
0.167
0.177
0.187
0.196
0.206
0.216
0.226
0.236
0.246
0.255
0.265
0.275
0.285
0.295
p(x)fea_lft
psi
7112.9
6910.1
6678.9
6388.9
5936.1
5529.4
5167.8
4838.3
4542
4276.8
4077.9
3908.1
3767
3687.3
3621.4
3569.9
3631.7
3682.9
3722.4
3816.9
3967.2
4171.2
4426
4735.2
5099.4
5447.9
5816.5
6208.7
6555.9
6757.2
6872.1
p(x)fea_lft
psi
71129
69101
66789
63889
59361
55294
51678
48383
45420
42768
40779
39081
37670
36873
36214
35699
36317
36829
37224
38169
39672
41712
44260
47352
50994
54479
58165
62087
65559
67572
68721
32
p(x)/pavg
psi/psi
1.511333
1.468242
1.419117
1.357499
1.261289
1.174874
1.098042
1.028031
0.965074
0.908725
0.866463
0.830384
0.800404
0.783469
0.769467
0.758524
0.771655
0.782534
0.790927
0.811006
0.842942
0.886287
0.940426
1.006124
1.083509
1.157557
1.235877
1.31921
1.392983
1.435754
1.460168
3D defl.
in.
1.52E-04
1.48E-04
1.44E-04
1.41E-04
1.39E-04
1.38E-04
1.36E-04
1.36E-04
1.35E-04
1.35E-04
1.34E-04
1.34E-04
1.34E-04
1.34E-04
1.33E-04
1.33E-04
1.34E-04
1.34E-04
1.34E-04
1.34E-04
1.34E-04
1.35E-04
1.35E-04
1.36E-04
1.36E-04
1.38E-04
1.39E-04
1.41E-04
1.44E-04
1.48E-04
1.52E-04
[A4.3]: Load Distribution Data of the Right Tooth of the Finite Element Model
x_fea_rt
in.
0.000
0.010
0.020
0.029
0.039
0.049
0.059
0.069
0.079
0.088
0.098
0.108
0.118
0.128
0.138
0.147
0.157
0.167
0.177
0.187
0.196
0.206
0.216
0.226
0.236
0.246
0.255
0.265
0.275
0.285
0.295
p(x)fea_rt
psi
6850.5
6414.8
6017.7
5559.1
4992.2
4528.4
4157.1
4059
3890.7
3674.1
3440.7
3254.7
3116.1
3111.3
3196.2
3411.8
3329.5
3330
3368.9
3474.4
3601.5
3744.1
4048.4
4371
4709.7
5048.1
5546.3
6249.8
6535.4
6707.7
6820.5
p(x)fea_rt
psi
68505
64148
60177
55591
49922
45284
41571
40590
38907
36741
34407
32547
31161
31113
31962
34118
33295
33300
33689
34744
36015
37441
40484
43710
47097
50481
55463
62498
65354
67077
68205
33
p(x)/pavg
psi/psi
1.510856
1.414764
1.327185
1.226042
1.101014
0.998724
0.916835
0.8952
0.858082
0.810311
0.758836
0.717814
0.687246
0.686187
0.704912
0.752462
0.734311
0.734421
0.743
0.766268
0.7943
0.825749
0.892862
0.96401
1.03871
1.113343
1.223219
1.378374
1.441362
1.479362
1.50424
3D defl.
in.
1.52E-04
1.48E-04
1.44E-04
1.41E-04
1.40E-04
1.38E-04
1.37E-04
1.36E-04
1.36E-04
1.35E-04
1.35E-04
1.34E-04
1.34E-04
1.34E-04
1.34E-04
1.34E-04
1.34E-04
1.34E-04
1.34E-04
1.34E-04
1.35E-04
1.35E-04
1.36E-04
1.36E-04
1.37E-04
1.38E-04
1.40E-04
1.42E-04
1.45E-04
1.48E-04
1.52E-04