Fatigue Analysis of Rear Axle Shaft for Ultra Light Car PDF

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Fatigue Analysis of Rear Axle Shaft for Ultra Light Car by Using Modified Goodman Method Cho Cho Myint1, Htay Htay Win2, Aung Ko Latt3 Mechanical Master Candidate, Department of Mechanical Engineering, Mandalay Technological University 1 Professor, Department of Mechanical Engineering, Mandalay Technological University 2 Associate Professor, Department of Mechanical Engineering, Mandalay Technological University 3 Email: [email protected] : [email protected] : [email protected] Abstract – This research presents the failure analysis of rear axle shafts for ultra light car. Considering the system, forces and torque acting on a shaft are used to calculate the stresses induced. In this study, the fatigue analysis of rear axle shafts are conducted by the theoretical and numerical approaches using the Distortion energy theory, and Modified Goodman Method based on three different materials. Three types of materials (structural steel, stainless steel and alloy steel) are selected. Firstly, the designs of rear axle shafts are calculated for three materials. The lengths of shafts are 610mm, 420mm and the calculated diameters of rear axle shafts are 30mm. The analysis of fatigue behaviours for rear axle shafts are also conducted using ANSYS 14.5 finite element analysis (FEA) by numerical approach. According to these methods, equivalent or von-Mises stresses are almost nearly the same, fatigue factors of safety are greater than one and the life of shaft is infinite for three materials. Keywords –Failure analysis, Fatigue, Rear Axle Shaft, Shaft, Stress concentration, Ultra Light Car.

I. INTRODUCTION An axle shaft is a rotating member usually of circular cross-section (solid or hollow), which is used to transmit power and rotational motion in machinery and mechanical equipment in various applications. An axle is a central shaft for a rotating wheel. The wheel may be fixed to the axle, with bearings or bushings provided at the mounting points where the axle is supported. The axles maintain the position of the wheels relative to each other and to the vehicle body. Dead axle does not transmit power like the front axle, in a rear wheel drives are dead axles. On the dead axle suspension system is mounted, so it’s also called suspension axle. Generally axle shafts are generally subjected to torsional stress and bending stress due to self-weight or weights of components or possible misalignment between journal bearings. Most shafts are subjected to fluctuating loads of combined bending and torsion with various degrees of stress concentration. For such shafts the problem is fundamentally fatigue loading. Eccentric Shaft is widely appreciated for its features like corrosion resistant, long service, effective performance and reliability. The main components of rear axle shaft are shown in Figure 1.

Figure 1. Components of Rear Axle Shaft [8]

Power is transmitted from the drive gear to the axle shafts through differential pinions and side gear. When the vehicle is moving in a straight line, the ring gear is spinning the case. The differential pinions and axle side gears are moving around with the case, with no movement between the teeth of the pinions and axle side gears [4].

II. DESIGN AND FATIGUE ANALYSIS OF REAR AXLE SHAFT

There are three steps in the design and fatigue analysis of rear axle shaft. They are (i) design calculation of axle shaft, (ii) stress analysis of rear axle shaft, and (iii) fatigue analysis of axle shaft. In this paper, firstly these analyses were computed by using theoretical and numerical analysis. (i) Design calculation of axle shaft The axle shaft is a rotating member, in general, has a circular cross-section and is used to transmit power. The shaft is generally acted upon by bending moment, torsion and axial force. The specification data of rear axle shaft is shown in Table I. TABLE I SPECIFICATION OF REAR AXLE SHAFTS

Physical Properties

Symbol

Value

Unit

Gross weight

WG

4806.9

N

Motor power

P

3

kW

Motor speed

Nm

1500~2500

rpm

Existing shaft diameter

D

26

mm

Length of long axle

L1

610

mm

Length of short axle

L2

420

mm

Weight of ring gear

Wr

20.4686

N

Weight of bevel gear

Wb

2.0022

N

Weight of pinion

Wp

1.2459

N

Weight of motor

Wm

235.44

N

Weight of solar

Ws

451.26

N

Diameter of ring gear

dR

182

mm

Diameter of bevel gear

dB

58.6

mm

dp

49.7

mm

Ng

15

teeth

Np

12

teeth

Diameter of pinion Number of teeth of gear Number of teeth of pinion

961 Three types of materials are used in this paper to analysis the fatigue endurance limit or fatigue life and the fatigue strength of the materials. The mechanical properties for three types of materials are shown in Table II.

373.2617 N

RF

G E

0.06

F

0.305

0.055

H

RG

TABLE II MATERIAL PROPERTIES OF STEEL

Materials

1314.54 N

Density (ρ), kg/m3

Yield Strength (Sy), MPa

Tensile Strength (Sut ), MPa

Young's Modulus GPa,

7870

250

460

200

8000

205

515

193

7850

417

655

205

373.2617 N SFD 325.9075 N

Structural Steel Stainless Steel Alloy Steel

78.8724 Nm

BMD

20.5294Nm

The torque can be computed from the known power transmitted and the rotational speed. 60 P Maximum Torque, Tmax  2π N (1) max

60 P Minimum Torque, Tmin  2π N min

(2)

T  Tmin Mean Torque, (3) Tm  max 2 T  Tmin Alternative Torque,Ta  max (4) 2 The transmitted load acts tangential to the pitch cone and is the force on the pinion and the gear. Bending moment diagrams show the variation of bending moment along the length of the shaft in Figure 2 and 3 for long axle and short axle. Ft 

2M t

(5)

dP

Fa  Ft tanφ sinγ

(6)

Fr  Ft tanφ cos γ

(7)

Figure 3. Shear Force and Bending Moment Diagram for Rear Short Axle Shaft

To calculate the diameter of shaft, the ASME Code equation for a solid shaft having torsion, bending and axial loads is as follow:

d3 

16 πSs

(K bM b 

αFad 2 )  (Kt Mt )2 8

(8)

where, For rotating shaft, K b is 1.5 and Kt is 1. [10] The column factor  is unity for a tensile load. For a compressive load  can be computed by Equation (9) and (10).

α

1 , For L/k115

(9)

(10)

Allowable shear stress (S s ) with keyway is

Ss  0.75 x 0.18 Su (or) 0.75 x 0.3 Sy

(11)

By using the ASME Code equation, the standard diameters of shafts for three types of materials are 30mm.

RC

373.2617 N

C

B A 0.045

0.485

D 0.08 1314.54 N

RB 251.4639 N

SFD 373.2617 N 1314.54 N

105.1632 Nm

BMD 16.7968 Nm

Figure 2. Shear Force and Bending Moment Diagram for Rear Long Axle Shaft

(ii) Stress analysis of rear axle shaft There are several methods to perform fatigue analysis and predicting the lifetime of shaft including stress-life, stain-life, crank propagation, and spot weld methods. To analyze fatigue behaviour of a shaft, firstly stress values of that shaft must be computed. Distortion energy theory is used for fatigue failure analysis to find the maximum stress values due to combined loading of bending and torsion. It is also known as von Mises- Hencky Theory.

σm  σ 2m  3τ 2m

(12)

σa  σ2a  3τ2a

(13)

Where, σ m and σa are mean and alternating bending stresses, τm and τa are mean and alternating shear stresses.

σm 

32Mm πd 3

(14)

962 32Ma πd 3

(15)

16Tm 3 πd 16T τa  3a πd

(16)

σa 

τm 

Fatigue strength at 103 cycles is, Sm= 0.9 Su. [1] The equation can be written to give corrected endurance limit (fatigue strength), Se as follows: Se  k a  k  k  k  k  k  S b c d e f e

(19)

The coefficients are detailed below, (17)

(iii) Fatigue Analysis of Axle Shaft by Modified Goodman Method There are three stages of fatigue failure, fatigue-crackinitiation, fatigue crack-propagation and sudden fracture due to unstable crack growth. The first stage can be of short duration, the second stage involves most of the life of shaft and the third stage is instantaneous. The fatiguecrack initiation life (Ni) is defined as the number of cycles required to initiate a fatigue crack. The number of cycles required to propagate a fatigue crack to a critical size is called the fatigue-crack-propagation life (Np). The total fatigue life (Nt) is the sum of the initiation and propagation lives, that is shown in Figure 4.

Figure 4. The Stages of Total Fatigue Life [1]

Fatigue limit, endurance limit, and fatigue strength are all expressions used to describe a property of materials. Endurance limit of the material is very important because it indicates the boundary of the infinite life. If the alternating stresses stay below this limit then the material can be used very long time with no fatigue failure. The nonferrous metals and alloy do not have an endurance limit. Fatigue life is the number of cycles of fluctuating stress and strain of a specified nature that a material before failure occurs. These are shown in stress-cycle (SN) diagram. S-N diagram is a plot of fatigue life at various levels of stress. Figure 5 shows the S-N curve for steel.

Where, a and b are constants, they are to be found in Table A-1.

 d  kb     7.62 

If the number of cycles is infinite, the stress level is called the endurance limit (S'e). For structural steel, relationship is given as (18) Se  0.5Su

0.1133

2.79mm  d  51mm

bending 1  k c  0.85 axial  0.59 torsion  S kd  T SRT

(21)

(22)

(23)

Assume reliability factor is 99.99%, ke =1 is in Table A- 2. Miscellaneous-effects factor includes all the effects not covered in such factors; residual stresses, corrosion, and metal spraying and so on. So, this factor kf is 1 [1]. There are four methods for the fatigue analysis of a general fluctuating stress, namely: (1)Soderberg, (2)Modified Goodman, (3)Gerber and (4)ASME-Elliptic. These methods are defined as lines or curves in a diagram shown below, where the midrange stress/strength components are shown on the horizontal axis and the alternating stress/strength components on the vertical axis.

Figure 6. Goodman, Soderberg, and Gerber Lines [3]

Fatigue factor of safety (nf) can be calculated by using Modified Goodman Method,

nf 

Figure 5. S-N Curve for Steel [3]

(20)

k a  aSbu

SeS u σ'a Su  σ' m Se

(24)

If the factor of safety is greater than one, the design is safe and the factor of safety is less than one, the design is fail [6]. The fatigue strength of engineering materials is in general lower than their tensile strength. A ratio of the fatigue strength to the tensile strength is called the fatigue ratio. It is normally observed in the case of steels. The fatigue strength (Sf) can be expressed as,

Sf 

σ a σ 1 m Su

(25)

963 Number of cycles to failure (N) can be expressed by using the fatigue strength of material; f is the fatigue strength fraction as shown in Table A-3 [2]. 1 (26) S N ( f )b a 2 (27) (f S )

a

u

Se

1 fS b   log( u ) 3 Se

(28)

The equivalent von-Mises stresses, Alternating Shear Stress, Actual Endurance Strength, fatigue factor of safety, and number of cycles to failure of axle shaft are calculated by using Distortion energy theory and Modified Goodman Method. These results are shown in Table III and IV. TABLE III THEORETICAL RESULTS OF REAR LONG AXLE SHAFT

Value

Symbol

Unit

AISI1020

AISI304

AISI4140

σ'm

36.2461

32.3661

59.829

MPa

σ'a

25.5993

22.8589

42.255

MPa

Se

72.99

78.22

93.48

MPa

nf

2.33

2.8

1.84

-

N

5.48×107

16.07×107

1.84×107

cycles

Figure 8. S-N Diagram of Rear Short Axle Shaft

According to the Figure 7 and 8, the fatigue strength at 103 cycles are 414MPa, 463.5MPa, and 589.5MPa for AISI1020, AISI304, and AISI4140. These diagrams are usually located at the 1×106 cycles of infinite life. According to the results from table III and IV, the life cycles of shafts are infinite life. So, the axle shaft designs are safe.

TABLE IV THEORETICAL RESULTS OF REAR SHORT AXLE SHAFT

Value

Symbol

Unit

AISI1020

AISI304

AISI4140

σ'm

38.0256

33.6426

65.7078

MPa

σ'a

21.5116

19.0322

37.1718

MPa

Se

72.82

78.22

93.48

MPa

nf

2.31

N

3.20 7

10.15×10

2.00 7

33.09×10

Figure 9. Graphical Approach Using the Goodman Theory for Rear Long Axle Shaft

7

2.2×10

cycles

After calculating the values of endurance strength and number of cycles of the shaft, the S-N diagram for the structural steel shaft on log-log coordinates is plotted by using MATLAB code as shown in Figure 7 and 8.

Figure 10. Graphical Approach Using the Goodman Theory for Rear Short Axle Shaft

Figure 7. S-N Diagram of Rear Long Axle Shaft

Figure 9 and 10 show the Goodman diagram including the stresses applied in the shaft. In these figures, the mean stress is plotted at the X-axis and alternating stress is plotted at the Y-axis. The green, blue, and pink lines are the Goodman line for structural steel, stainless steel, and alloy steel. These are connected to the actual endurance strength and ultimate strength of material. Calculated stress lines, which are connected von-Mises stress and alternating combine stress. The calculated stress lines are inside the Goodman line, so

964 the shaft design is safe. However, they had been outside the Goodman line, and then the design is not safe.

III. FATIGUE ANALYSIS BY FEA APPROACH

After finishing set up the boundary conditions on rear long axle shaft, run the solution and get the results of equivalent (von-Mises) stress, factor of safety and life cycles as shown in figures below.

In this paper, stress and fatigue behaviours of rear axle shafts are also computed by Finite Element Analysis using ANSYS software package to verify with theoretical results. Rear axle shaft designs are drawn by SolidWorks software and saved in type of IGES file to understand ANSYS software.

Figure 14. Von-Mises Stress in Rear Long Axle Shaft Using Structural Steel

Figure 11. Geometry of Rear Long Axle Shaft

The static structural shaft SolidWorks model was added to the geometry in ANSYS Workbench. This geometry model was meshed with high smoothing as shown in Figure 12.

According to the Figure14, the maximum von-Mises stress is 35.521MPa which is occurred at the side of wheel. And, the minimum von-Mises stress is 3.049×10-4 MPa at the another side of the shaft.

Figure 15. Factor of Safety on Rear Long Axle Shaft Using Structural Steel Figure 12. Meshing of Rear Long Axle Shaft

Firstly, give the input conditions to the model, which applied reaction force from wheel at point A of rear long axle shaft. Then applying force 373.2617N at Y- axis of point B and bending moment 15.279Nm is applied to the shaft as shown in Figure 13. Structural steel, stainless steel and alloy steel are used for engineering data.

The value of safety factor of the shaft is shown in Figure 15. According to this figure, the safety factor range is 0 to 15. The minimum value is 2.427and the maximum value is 15.

Figure 16. Life Cycle of Rear Long Axle Shaft Using Structural Steel

Figure 13. Loading Condition of Rear Long Axle Shaft

Figure 16 shows fatigue life cycle of shaft and the number of cycles is 1×106 cycles from the numerical

965 analysis. According to the fatigue theory, number of 106 cycles is defined as infinite life.

TABLE V COMPARISON OF THEORETICAL AND FEA RESULTS FOR REAR LONG AXLE SHAFT

VonMises stress, σ'm (MPa)

Figure 17. Von-Mises Stress in Rear Short Axle Shaft Using Structural Steel

According to the Figure17, the maximum von-Mises stress is 38.405MPa which is occurred at the side of wheel. And, the minimum von-Mises stress is 3.766×10-4 MPa at the another side of the shaft.

Fatigue factor of safety (nf )

Deviation Theoretical Approach FEA Approach Deviation

Life (cycles)

Alloy Steel

36.246

32.366

59.829

35.521

34.280

62.948

2.0%

5.6%

4.9%

2.33

2.80

1.84

2.42

2.51

1.37

4.1%

10.2%

25.5%

Infinite Life

Infinite Life

Infinite Life

TABLE VI

Fatigue factor of safety (nf )

Theoretical Approach FEA Approach Deviation Theoretical Approach FEA Approach Deviation

Life (cycles)

The value of safety factor of the shaft is shown in Figure 18. According to this figure, the safety factor range is 0 to 15. The minimum value is 2.245 and the maximum value is 15.

Stainless Steel

COMPARISON OF THEORETICAL AND FEA RESULTS FOR REAR SHORT AXLE SHAFT

VonMises stress, σ'm (MPa)

Figure 18. Factor of Safety on Rear Short Axle Shaft Using Structural Steel

Theoretical Approach FEA Approach

Structural Steel

Structural Steel

Stainless Steel

Alloy Steel

38.025

33.642

65.708

38.405

32.875

62.444

0.99%

2.3%

4.9%

2.31

3.20

2.00...