Stress Analysis of a Chicago Electric 4½ Angle Grinder
By
Khalid Zouhri
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
In Partial Fulfillment of the
Requirements for the degree of
MASTER OF MECAHNICAL ENGINEERING
Approved:
_________________________________________
Dr.Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
October, 2009
(For Graduation May 2010)
1

© Copyright 2009 by
Khalid Zouhri
All Rights Reserved
2

Table of Contents
1. Problem Statement................................................................................................................................ 3
1.1. Photos of the Device, Subassembly, and Components...................................................................... 3
1.2. Given and Measured Data ................................................................................................................. 9
1.1.1. Given Data ..................................................................................................................................... 9
1.1.2. Measured Data .............................................................................................................................. 10
1.1.3. Schematic Drawing ...................................................................................................................... 15
2. Force Analysis .................................................................................................................................... 15
2.1.1. Free Body Diagram of Shaft ........................................................................................................ 15
2.1.2. Gear Force Calculations: .............................................................................................................. 16
2.1.3. Shaft Force Calculations: .............................................................................................................. 17
2.1.4. Shear Force and Bending Moment Diagrams .......................................................................... ….19
3. Stress Analysis ............................................................................................................................... ….20
3.1.1. Bending Stress and Bending Factor of Safety .............................................................................. 20
3.1.2. Wear Stress and Bending Factor of Safety ................................................................................... 22
4. Stress Analysis of Output Shaft.................................................................................................................. 24
4.1.1. Shaft Force Calculations Using Vector Cross Product ................................................................... ….24
4.1.2. Static Factor of Safety Calculations for Shaft ...................................................................................... 25
4.1.3. Fatigue Factor of Safety Calculations .................................................................................................. 26
5. Finite Element of Output Shaft .................................................................................................................. 27
5.1.1. Loads and Restraints Setup .................................................................................................................. 27
5.1.1. COSMOS Stress Analysis .................................................................................................................. 28
6. Discussion ................................................................................................................................................. 30
7. References ................................................................................................................................................. 31
3

List of Figures
Figure 1: Chicago Electric Angle Grinder ..................................................................................... ...... 3
Figure 2: Complete Component Assembly ..................................................................................... ..... 4
Figure 3: Bevel Pinion in Gear Casting Subassembly .................................................................... ..... 4
Figure 4: Bevel Gear in Gear Casing Subassembly ........................................................................ ...... 5
Figure 5: Component Subassembly without Gear Casing ............................................................ ....... 5
Figure 6: Shaft with Bevel Pinion Gear Removed ......................................................................... ..... 6
Figure 7: Shaft with Bevel Pinion Gear......................................................................................... .. ..... 6
Figure 8: Bevel Pinion .................................................................................................................. …....7
Figure 9: Components in Position ................................................................................................. ...... 7
Figure 10: SolidWorks Model Assembly ..................................................................................... ...... 8
Figure 11: SolidWorks Assembly Exploded................................................................................ ........ 8
Figure 12: Components Analyzed Exploded......................................................................................... 9
Figure 13: ME-2 Rockwell Hardness Testing System .................................................................... ... 10
Figure 14: Hardness Testing for Shaft ............................................................................................ ... 10
Figure 15: Hardness Testing for Bevel Pinion .............................................................................. .... 11
Figure 16: Pinion Close-up with Reference Lines ......................................................................... ... 14
Figure 17: Schematic Drawing ....................................................................................................... ... 15
Figure 18: Free Body Diagram of Shaft ............................................................................................. 15
Figure 19: Pinion- Bevel Schematic .................................................................................................... 16
Figure 20: Force Analysis 2-D Planes ................................................................................................ 17
Figure 21: Load, Shear Force, and Bending Moment Diagrams ....................................................... 19
Figure 22: Restraints at Bearing D ............................................................................................................ 27
Figure 23: Restraints at Bearing C ............................................................................................................ 27
Figure 24: Loads at Pitch Point Applied Using Remote Load .................................................................. 28
Figure 25: Shaft with Constraints and Loads and Study Run ................................................................... 28
Figure 26: Probe Value of Maximum Stress ............................................................................................. 29
4

Table of Tables
Table 1: Manufacturing Specifications ................................................................................................. 9
Table 3: Hardness Testing Results ....................................................................................................... 11
Table 4: Hardness Conversion Chart ................................................................................................... 12
Table 5: AGMA Strength Graph for Gears .......................................................................................... 13
Table 6: Gear and Shaft Information ................................................................................................... 14
5

Abstract:
The mechanical device Chicago electric 4.5 inch angle grinder use on many of application in our world. This device have two gear bevel pinion gear and spiral pinion gear, the spiral bevel gear attached into the shaft and two bearing in the same line, all of them turn with electrical motor. The study on this project is to investigate the stress analysis on bevel pinion gear and shaft using the theoretical calculations and finite element analysis (FEA) and also stress analysis using the COSMOS software.
The purpose of this project is to give global idea about all the stress analysis that apply on the shaft and gear during the mechanical work of the device.
6

1.
Objective
The purpose of this Engineering Project for this semester is to do stress analysis of mechanical device
Chicago Electric 4.5 inch angle grinder ITEM 91223-1VGA. The grinder utilizes spiral bevel gears whose output angular velocity (ω) is 11,000 rpm. The first component I'm going to analyzed it’s the spiral bevel pinion gear .This pinion gear is directly coupled to the rotor shaft of the electric motor and supplies torque to the accompanying spiral bevel gear which is closed coupled to the output shaft. The pinion gear is held onto the end of the rotor shaft by a nut which forces the pinion against one of the two shaft bearings (Figure 7).
The second component I decided to analyze is the rotor shaft of the electric motor (Figure 6).
1.1
Photos of the Device, Sub assembly, and Components
Figure 1 show the Mechanical device tool Chicago Electric angle grinder
Figure 1: Chicago Electric Angle Grinder
7

Figure 2 show the complete component assembly (motor, shaft, bearing, bevel gear casting)
Figure 2: Complete Component Assembly
Figure 3 show the bevel pinion gear inside the gear casting.
Figure 3: Bevel Pinion in Gear Casting Sub assembly
8

Figure 4 show the bevel gear casting in subassembly.
Figure 4: Bevel Gear in Gear Casing Subassembly
Figure 5 show the two gears on the Grinder device, bevel pinion gear and bevel gear .
Figure 5: Component Subassembly without Gear Casing
9

Figure 6 it show part of the assembly, bevel pinion gear removed for hardness test
Figure 6: Shaft with Bevel Pinion Gear Removed
The picture bellow shows the shaft with bevel pinion gear.
Figure 7: Shaft with Bevel Pinion Gear
10

The picture bellow show the bevel pinion gear separated from the assembly device.
Figure 8: Bevel Pinion
The picture below show the shaft with bevel pinion gear and bevel gear in position.
Figure 9: Components in Position
11

1.2 Given Data
In Table 1 below [2], manufacturing specifications are given.
Manufacturing Specifications
Angular Velocity
Horse Power
Voltage
Frequency (AC)
Power
Amperage
11,000 rev/min
½ hp
110 volts
60 Hz single Phase
570 watts
4.5 amps
Table 1: Manufacturing Specifications
The Table below shows the Nomenclature with givens and unknowns symbols using in this project.
Symbol Definition b d avg
P
T
V av
K m
K o
K v
Φ
Φ n
ψ
γ d out
F
S
F a
N g
N p
J
F r
F t
K v n p
Tooth width (mm) or (in)
Avg. diameter of pinion (mm) or (in)
Outside diameter of pinion (mm) or (in)
Factor of Safety
Axial (thrust-bearing) load (N) or lbf
Radial (separating) load (N) or (lbf)
Tangential (torque-producing) load (N) or (lbf)
Velocity Factor
Rotating speed of pinion (rpm)
Number of teeth on ring (driven) gear
Number of teeth on pinion (driver) gear
Geometry factor for bending strength
Diametral pitch: large end of tooth (mm)
Torque
Avg. velocity (m/s.)
Load distribution factor
Overload factor
Velocity Factor
Pressure angle (°)
Pressure angle in the plane normal to the tooth (°)
Helix angle (°)
Pitch cone angle (°)
Table 3 – Nomenclature with givens and unknowns
12

Figure 10 show the SolidWork of the model assembly (bevel pinion gear, pinion shaft, bevel shaft, bearing, bevel gear, bevel casting, chuck washer)
Figure 10: SolidWorks Model Assembly
13

Figure 11 show the solid works assembly exploded into pieces (shaft, bevel pinion gear, bevel gear, bevel casting, bearings)
Figure 11: SolidWorks Assembly Exploded
Figure 12 show the two components that I’m going to analysis, the bevel pinion gear and shaft.
Figure 12: Components Analyzed Exploded
14

1.2
Given and Measured Data
1.2.1
Given Data
In Table 1 bellow [2], manufacturing are given.
In Table 1, all specifications are given by the manufacturer except horsepower. Knowing that for single phase armature motors, one hp is generated by 720 watts. Assuming that the most efficiency expected out of a single phase motor is approximately 70%, 60% is used in this analysis. hp

570 w
720 w

3
4 hp
1.2.2
Measured Data
For proper analysis, certain measurable data needed to be collected. The material used in the angle grinder was unknown; therefore it was necessary to perform hardness testing to gather useful material data. The shaft and bevel pinion gear were easily accessible by the Rockwell Hardness Testing
Machine. This made the process more reliable in that the integrity of the material was not compromised by forcefully removing the components from the remaining subassemblies. The bevel gear was completely isolated from its main subassembly. This ensured no cutting of material or extended surface hardening from clamping and pulling. The shaft was permanently fixed to its immediate subassembly.
Luckily the shaft was just long enough to get true readings without similar structural compromises as the bevel gear.
15

The picture bellow show the hardness test machine that I use to get result for hardness test.
Figure 13: ME-2 Rockwell Hardness Testing System
Before the hardness test could begin, proper surface preparation needed to be performed to ensure proper results. Three readings were done on the shaft as well as the bevel in different locations to and an average hardness number. The location of testing on the material was also changed to get a variety of readings and to also not get tainted results from nearby surface hardening created by testing. The hardness testing operation can be seen below.
It is imperative to note that the material will be much harder on the tooth of the gear rather than in the makeup material. There was no way to test the hardness of the material directly on the tooth without risking safety for the holder. The reading, however, was done as close as possible to the side of the tooth without risking the ball sliding down the face of the tooth.
16

Table 2: Hardness Testing Result
Using the hardness conversation chart [2] and the information from Shigley’s Mechanical
Engineering Design in Table 5 [3], material grade and tensile strength can be determined .grade1 used for limiting case material, steel.
17

The chart below shows the hardness conversion chart that I use to get Tensile strength.
Table 3: Hardness Conversion Chart
18

The figure below show the AGMA strength graph for gear it use to calculate the tensile strength.
Tensile Strength Calculation
Figure 4: AGMA Strength Graph for Gears
S t

77 .
3 H b

12 , 800 psi
S tpiniom

77 .
3 * 294

12 , 800 psi

35 .
526 kpsi
S tpiniom

77 .
3 * 294

12 , 800 psi

35 .
526 kpsi
Calculations of Diametric Pitch
P

N p d
Np =13 Teeth d=20.0mm (outside diameter) as shown in figure 17.
Equation below was used to calculate pitch.
19

P t

N p d

13
20 teeth / mm

0 .
65 teeth / mm
Before the normal diametral pitch can be determined, the helical angle “ψ” needed to be determined. In the illustration below, lines were added and a protractor was used to measure ψ.
Figure 16: Pinion Close-up with Reference Lines
By using the reference lines the angle ψ was measured to be 40 degree.
To get normal diametral pitch we use the formula bellow;
P n

P t cos

P n

0 .
65 cos 40
 teeth / mm

0 .
8485 teeth / mm
So we use the formula bellow to calculate the Pitch diameter; d p

N
P n

13 teeth
0 .
8485 teeth / mm

15 .
32 mm
20

1.2.3
Schematic Drawing
Table 5: Gear and shaft information
The figures 17 show the schematic drawing for the shaft and both bearing and bevel pinion gear.
Figure 17: Schematic Drawing
1.1. Free Body Diagram of Shaft
This figure 18 show the Free body diagram for all the force that apply to the shaft and bevel pinion gear and bevel gear, for each force have tangential and axial and radial force
21

Figure 18: Free Body Diagram of Shaft
22

1.2. Gear Force Calculations:
The figure 19 show the pinion bevel schematic
Figure 19: Pinion- Bevel Schematic
To get the angle T we have to use formula bellow to get y angle and from there we get T.
  tan

1
(
PinionTeet h
)
Bevelteeth
  tan

1
(
PinionTeet
Bevelteeth h
)

22 .
1

 
22 .
1

 
90
 
22 .
1
 
67 .
9

We use formula bellow to get the distance t
2
from the end of the bevel pinion gear to the pitch point.
Same way to get the y axis distance from the end of the bevel pinion gear to the pitch point. t t
2

3 .
25 mm
 cos( 22 .
1

)

3 .
011 mm
3

3 .
25 mm
 sin( 22 .
1

)

1 .
222 mm
23

These thicknesses represent the location of the pitch point and are illustrated above in Figure 19.
Distance from bearing D to horizontal pitch point location P is:

9.5 mm is from outside of bevel pinion to far side of bearing

Bearing is 7 mm wide so half of bearing is 3.50
DP ( x )

19 .
5

( 3 .
011 mm

3 .
50 mm )

13 mm
Distance from bearing D to horizontal pitch point location P is:
DP ( y )

20 mm

1 .
222 mm

8 .
78 mm
2 d p

8 .
78 mm

2

17 .
56 mm ( pitch
 diameter )
Bellow shows the calculation to find the axial and Tangential and radial force that apply from the bevel gear into the bevel pinion gear.
W t
BP

60 , 000

* d
 hp
* n
W t
BP


60 , 000

3
4 hp
* 17 .
56 mm * 11 , 000 rpm

74 .
2 N
W r
BP

W t
BP
 tan(

)
 cos(

)
W a
BP

W t
BP
 tan(

)
 sin(

)
W r
BP

74 .
2 N
 tan 20
  cos 22 .
1
 
25 .
02 N
W a
BP

74 .
2 N
 tan 20
  sin 22 .
1
 
10 .
16 N

W


10 .
16 i

25 .
02 j

74 .
2 k N
W

ABS ( W )

78 .
96 N
24

1.3. Shaft Force Calculations:
The figure 20 show the free body diagram for the shaft , it showing all the force that apply from the both bearing and bevel gear.
Taking the moment around D:
Figure 20: Force Analysis 2-D Planes
(x-y) plane :
So we take the moment at the point D to get the one equation with one unknown and than from there we solve for force F c y

M
D
 
F c y 
136 .
5 mm

10 .
16 N

8 .
78 mm

25 .
02 N .
13 mm

0
F c y  
1 .
73 N ( change
 direction

)

1 .
73 N

The total force in X direction equal to zero (static condition), to calculate the force F x
D


F x

W a 
F
X
D

0
25

F x
D

10 .
16 N

The total force in y direction equal to zero (static condition), to calculate the force F
D y . .


F y

W r 
F
D y 
F c y 
0

F
F y
D

26 .
75 N

(x-z ) Plane:
So we take the moment at the point D to get the one equation with one unknown and than from there we solve for force F c z

M
D

W t 
13 mm

F c z
.
136 .
5 mm

0
F c z  
7 .
07 N ( change
 direction

)

7 .
07 N

The total force in z direction equal to zero (static condition), to calculate the force F z
D


F z
 
W t 
F
D
Z 
F
C
Z 
0
F
Z
D

81 .
27 N
 c


1 .
73 j

7 .
07 K N

F
C

ABS ( F
C
)

7 .
28 N
F
D

{

10 .
16 i

26 .
75 j

81 .
3 K } N
F
D

ABS ( F
D
)

86 .
2 N
26

1.4. Shear Force and Bending Moment Diagrams
The figure 21 shows the load that we calculated and shear force and bending moment diagram.
Figure 21: Load, Shear Force, and Bending Moment Diagrams
27

2.
Stress Analysis
1.
Bending Stress and Bending Factor of Safety
We use the formula bellow to calculate the Dynamic Factor (K v
) .
Assume quality number is 6. Transmission accuracy level number (Q v
) could be taken as the quality number. Q v
= 6.
K v

(
A

200 .
V
)
B
A
A

50

56 ( 1

B )
 ifB

0 .
25 ( 12

Q v
)
2
3
B

0 .
25 * ( 12

6 )
2
3

0 .
8254
A

50

56 ( 1

0 .
8254 )

59 .
77
So we use the formula bellow to calculate the velocity in m sec
V


* d p
* n p
60
V


* 17 .
56 mm * 11 , 000 rev / min
60

10 , 113 .
8 mm sec

10 .
114 m sec
K v

(
59 .
77

200

10 .
114 m / sec
)
.
8254
59 .
77

1 .
59
Overload Factor (K o
)
Assuming uniform loading, so K o
= 1.
Size Factor K s
.
28

From Table 14-2 [5] in page 653 Shigley’s Mechanical Engineering Design. 8 th
Ed. (s, Y p
(13T) is 0.261
Face width 6.5 mm →6.5 mm/25.4 mm/in= 0.2559 in.
P = 0.8485 Teeth/mm = 21.552 Teeth/in
K s

1 k b

1 .
192

(
F
P
Y
)
0 .
0535
This is for standard units
K s

1 k b

1 .
192

(
F
P
Y
)
0 .
0535 
1 .
192 * (
0 .
2559 in *
21 .
552 T
0 .
261
)
0 .
0535
/ in

0 .
907
K
H

C mf

1

C mc
* ( C pf *
C pm

C ma *
C e
)
Assume uncrowned so (load correction factor) Cmc=1
F

10 .
00
In (Pinion Proportion factor) Cpf;
C pf

F
10 * d

0 .
025

6 .
5 mm
10 * 17 .
56 mm

0 .
025

0 .
0120
Cpm ( pinion proportion modifier)=1.0
Cma(Mesh alignment factor)
C ma

A

BF

CF
2
Using table 14.9 [6] , Value for A,B,C are as follows ( Commercial,Enclosed units);
A=0.127
B=0.0158
C
 
.
930

10

4
C ma

0 .
127

0 .
0158

0 .
2559
 
.
930

10

4
* .
2559
2 
.
1310
Ce(Mesh alignment correction factor )1
K
H

C mf

1

1 * ( 0 .
0120 * 1

0 .
1310 * 1 )

1 .

Rim Thickness factor Kb=1 due to consistent thickness of gear.
Speed Ration mG; m
G

N
G
N
P
 32 T
13 T

2 .
46
Load Cycle factor (Yn) using 13T for pinion and life cycle 10
Y
N ( P )

1 .
3558 * N p

0 .
0178 
1 .
3558 * ( 10
8
)

0 .
0178 
0 .
977
From Table 14-10 [7] and a reliability of 0.90, K
R
(Y
Z
) = 0.85
;
From Figure 14-6 [8], The Y
J(P)
= 0.21
K
T
(Temperature Factor) Temperature is less than 250 °F so K
T
= 1
Brinell hardness number 294
S t
(Allowable Bending Stress for Hardened Steels) for 294 Brinell.
Grade 1:
S t

( 0 .
533 H
B

88 .
3 ) MPa

( 0 .
533 * 294 * 88 .
3 )

245 MPa
Sc ( Contact fatigue strength ) for brinell and grade 1.
S c

( 2 .
22 H
B

200 ) Mpa

( 2 .
22 * 2 .
94

200 )

852 .
7 Mpa
Pinion Tooth bending stress
 p

W t
K
0
K v k s
P
*
F
*
K
H
K
B
Yj
 p

16 .
7 ibf * 1 * 1 .
59 * .
907 *
21 .
552 T
0 .
2559
/ in in
*
1 .
143
.
21
* 1

11040 psi

76 .
1 Mpa
We use the formula bellow to calculate the bending fatigue factor of safety for pinion;
30

S
F ( P )

 p
S
 t
*
K
T
Y
N

K
R
S
F ( P )



245 MPA * .
977
76 .
1 Mpa * 1 * .
85



3 .
7
 p

76 .
1 MPa S
F ( P )

3 .
7
1.2. Wear Stress and Bending Factor of Safety
Surface Condition Fatctor Cf=Zr=1( No surface defects specified)
We use the formula bellow to show the stress and bending factor of safety Z
Z

[( r p
 a )
2  r
2 bp
]
1
2

[( r
G
 a )
2  r
2 bg
]
1
2

( r p
 r g
) sin
 t
Where; r p

8 .
78 mm r
G

18 .
74 mm r bG , bp
 r cos
 t
, r bg

16 .
93 mm r bP

7 .
93 mm a

1 p n

1
0 .
8485

1 .
17855
 t
 tan

1 tan
 cos
 n  tan 20 cos 40



25 .
41

Z

[( 8 .
78 mm

1 .
17855 )
2 
( 7 .
93 mm )
2
]
1
2

[( 18 .
74 mm

1 .
17855 )
2 
( 16 .
93 mm )
2
]
1
2

[( 8 .
78 mm

18 .
74 mm ) sin 25 .
41
 
4 .
71 mm
 n

20

31

p n
 p n cos
 n

 p n cos
 n


0 .
8485 mm cos 20
 
3 .
479 mm m
N
 p
N
0 .
95 Z

3 .
479 mm
.
95 * 4 .
71 mm

.
778
I

Z
1
 cos 20
 sin 20

2 * m
N
* m m
G
G

1
I

Z
1
 cos

* sin

2 * 0 .
778
2 .
46
*
2 .
46

1

0 .
147
From Table 14-8 [9] from shigly book page 543 we get the formula to calculate the Ze ( Elastic coefficients factor);
C p

Z
E

191 MPa

2300 psi
We use the formula Stress cycle factors for pinion;
Z
N ( P )

1 .
4488 *
  p

0 .
023 
1 .
4488 * ( 10
8
)

0 .
023 
.
948
So we use the formula below to calculate the contact stress:
 c

C p
W t
K
0
K v
K s
K
H d
( p )
* F
*
C f
I
 
2300 psi 16 .
7 ibf * 1 * 1 .
59 * .
907
1 .
1385
*
.
69134 in * .
2559
1
*
.
147

74 .
7 ksi

515 MPa
We sue the formula below to calculate the wear Factor of safety:
S
H ( P )

[ s
C
* z
N
K
T

*
C
K
R
]

[
852 .
7 MAP * .
948
1 * .
85
]

1 .
85
515 MPa
32

 
515 MPa S
H ( P )

1 .
85
4.
Stress Analysis of Output Shaft
4.1.1. Shat Force calculations Using Vector Cross Product
So we use the formula below to get the absolute value for W
W

{ 10 .
16 i

25 .
02 j

74 .
2 k } N
W

ABS ( W )

78 .
96 N
R
DP

{

13 i

8 .
78 j } mm
R
DC

{ 136 .
5 i } mm
From both equation above we use the values to calculate the torque T
T

8 .
78 mm * W t
Bp

8 .
78 mm * 74 .
2 N

651 N .
mm
T

651 .
5 iN .
mm
So Taking the moment around D to calculate the Force F
Z
C :

M
D

R
DP
* W

R
DC
* F
C

T

0

M
D

{

13 i

8 .
78 j } mm * { 10 .
16 i

25 .
02 j

74 .
2 k } N

{ 136 .
4 i } mm * { F y c j

F z c k } N

{ Ti }

0
F
Z
C 

964 .
6 N .
mm
136 .
5 mm
 
7 .
07 KN
F
Y
C 

236 .
1 N .
mm

136 .
5 mm

1 .
73 j ..
N
33

F
C

{ 1 .
73 j

7 .
07 k } N
F c

ABS ( F
C
)

7 .
28 N
To get Fd, sum the forces equal to zero.
F d

F
C

W

0
So the sum of the force equal to zero (static condition)

F

{ F x d i

F y
D 
F
Z
D
} lbf

{ 0 i

1 .
73 j

7 .
07 } N

{ 10 .
16 i

25 .
02 j

74 .
2 } N

0
F x d  
10 .
16 i .
N F y d  
26 .
75 .
j .
N
F z d 
81 .
3 K .
N
F

{

10 .
16 i

26 .
75 j

81 .
3 k } N
So the absolute value for Fd is calculated below.
F
D

ABS ( F
D
)

86 .
2 N
4.1.2.
Safety Factor ,Safety Calculations for Shaft
We use the formula below to calculate the moment on the shaft at point D of the bearing.
M max

(

965 N .
mm )
2 
( 236 .
2 N .
mm )
2 
993 .
5 N .
mm
M max

993 .
5 N .
mm
T max

651 .
5 N .
mm
D / d

10 .
10 mm / 8 mm

1 .
26
34

Assume r=2.54 mm (0.1in) r / d

2 .
54 mm / 8 mm

.
3175
K t

1 .
4 [ 12 ]
K ts

1 .
2 [ 13 ]
Notch sensitivity (q)=0.95[14]
Shear notch sensitivity (qshear)=1 [15]
K f

1
 q ( K t

1 )

1

0 .
95 ( 1 .
4

1 )

1 .
38
K f

1
 q s
( K ts

1 )

1

1 ( 1 .
2

1 )

1 .
2
 " m
 " a

K f
32 .
M

.
d max
3

1 .
38
32 * 993 .
5 N .
mm

.( 8 mm )
3

27 .
28 MPa

3 .[ K fs
16 .
T max

.
d
3
]
2
 " m

3 .[ 1 .
2
16 .
*

651 .
5 N .
mm
.( 8 mm )
3
]
2 
13 .
47 MPa
Assume Sy=0.775Sut
Sut=106Ksi = 730.8MPa
So the safety factor of the shaft calculated using the formula bellow. n yield

S y
 ' m
  ' a n yield

0 .
775 * 730 .
8 MPa
13 .
47 MPa

27 .
28 MPa

13 .
9
35

4.1.3.
Fatigue Factor of Safety Calculations
Analysis based on Modified Goodman Design Theory
S
' e

0 .
5 * S ut
S e
'
0 .
5 * 730 .
8 MPa

365 .
4 MPa

Using Table6-2[5] from Shigly book page and assuming a surface finishes of mechanized or cold drawn: a=4.51 b=-0.265
Surface Factor Ka;
K a
 a * S ut b 
4 .
51 * 730 .
8 MPa

0 .
265 
0 .
786
Combined Loading so Kc=1
Room Temperature operation so kd=1
From table6-5[4] page 785 from Shigley’s Mechanical Engineering Design. 8 th
Ed. Book and assuming
90% reliability ke=0.897
S e

K a
K b
K c
K d
K e
S e
' 
.
786 * .
993 * 1 * 1 * .
897 * 365 .
4 MPa

255 .
82 MPa n
1 fatigue

 ' a
S e

 ' m
S ut
 n fatigue

S ut
S e
*
 ' a
*

S ut
S e
.
 ' m n fatigue

730 .
8 Mpa *
255 .
82 MPa * 730 .
8 MPa
27 .
28 MPa

255 .
82 MPa

13 .
47 MPa

8
Comparing N yield and N fatigue n yield

13 .
9
 n fatigue

8
This shows that the shaft will fail by fatigue.
36

5.
Finite Element of Output Shaft
5.1.1.
Loads and Restraints Setup
The figure 22 show the load and restraint setup on Cosmos software.
Figure 22: Restraints at Bearing D
37

At bearing D, radial and axial restraints were set. Restraints were applied to a 2 mm cylindrical face placed at the midpoint of the bearing location.
Figure 23: Restraints at Bearing C
38

The bearing restraint at C was set for only a rotational restraint. This was to ensure that over-constraining the shaft was avoided. This restraint was also applied to a 2 mm cylindrical face placed at the midpoint of the bearing location.
Figure 24: Loads at Pitch Point Applied Using Remote Load
39

5.1.1.
COSMOS Stress Analysis
The figure 25 shows the shaft with constraints and loads and study run .using the Cosmos software .as you see the maximum stress appear on the bearing area.
Figure 25: Shaft with Constraints and Loads and Study Run
The figure 26 show the prob value of the maximum stress
 ' max
Figure 26: Probe Value of Maximum Stress
40

As seen in the figure above, the maximum stress calculated by the study is 28.12 MPa
Using Von Misses to get an analytical value for  ' max
 ' max
  2 m
  m
*
 a
  2 a

3 .
 2 m
 ' max

( 1 .
38 *
32

* 965 N .
mm
* ( 8 mm )
)
2 
3 .( 1 .
2
16 * 236 .
2 N .
mm

.( 8 mm )
3
)
2 
26 .
9 MPa
% Difference :
% Difference

% Difference

26 .
9 MPa
26 .
9 MPa


28 .
12 MPa
28 .
12 MPa x
1 x
1

 x
2 x
2
2
* 100
* 100

4 .
43 % diffrence
2
Using COSMOS, the maximum value found was 28.12 MPa. There is a difference of 4.43 % from a calculated value of 26.9MPa. This could be attributed to model building techniques in SolidWorks. There could also be variations in part measurements which could skew calculations slightly. Changes in restraint locations and “bearing” restraints types may bring different results. Better understanding and practice of
COSMOS could also produce improved results. Overall the stress analysis attributed by COSMOS is an excellent representation of the type of stress the part may succumb to.
6.
Discussion
The project exemplified real world tasks that definitely sharpened my skills as young engineers. The handson experience gained by using the equipment and software tools was invaluable. The experience of using specific tools, acquiring data, and using this data in calculations was a particularly useful experience. It provided some relevance of the knowledge learned from work experience and my other engineering courses.
Also COSMOS in particular was beneficial not only because I was able to use it as a comparison tool but also because of the practical experience gained which will be useful in the a ‘real-world’ application. This
41

tool gives a visual representation of the types of stresses and possible deformations that may occur in design.
This quick reference will give the design engineer an approximation of what to expect under certain loads or applications. This brief exposure to COSMOS will certainly be invaluable.
7.
Reference:
[1] Manufacturing Specifications link, http://www.harborfreight.com/cpi/ctaf/displayitem.taf?Itemnumber=43471 p/english/gear/myset/spiral.html
[2] Hardness conversion chart, http://www.carbidedepot.com/formulas-hardness.htm
[3] AGMA Strength Graph for Hardened Steels,
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. pp 727
[4] Reliability Assumption,
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Table 6-5, pp 285
[5] Parameters for Marin Surface Modification Factors,
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Table 6-2, pp 280
[6] Lewis Form Factor,
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Table 14-2, pp 718
[7] Empirical Constants for Load Distribution Factor
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Table 14-9, pp 740
[8] Reliability Factor K
R
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Table 14-10, pp 744
[9] Geometry Factor,
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Figure 14-6, pp 727
[10] For Elastic Coefficient,
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
42

York NY, 2008. Table 14-8, pp 737
[12] For Stress Concentration Value K t,
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Appendix Figure 15-6, pp 1007
[13] For Stress Concentration Value K ts.
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Appendix Figure 15-8, pp 1008 32
[14] Notch Sensitivity q,
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Figure 6-20, pp 287
[15] Shear Notch Sensitivity q,
Budynas, R., Nisbett, K., Shigley’s Mechanical Engineering Design. 8 th
Ed. McGraw Hill, New
York NY, 2008. Figure 6-21, pp 288
43