fatigue analysis of automative tensioner in front end
TRANSCRIPT
Fatigue Analysis of Automative Tensioner in Front End
Accessory Drive Systems
by
Maryam Talimi
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Mechanical and Industrial Engineering Department
University of Toronto
© Copyright by Maryam Talimi 2016
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Fatigue Life Assessment of Automative Tensioner in Front End
Accessory Drive Systems
Maryam Talimi
Doctor of Philosophy
Mechanical and Industrial Engineering Department
University of Toronto
2016
Abstract
Tensioners, as critical parts of automotive Front End Accessory Drive (FEAD) systems, are
subjected to a significant range of dynamic loads due to engine pulsations which can lead to
fatigue and operational failure. It is challenging to analytically investigate the fatigue life for
powertrain components given the parameters involved. In this paper, the fatigue life assessment
of a tensioner is studied through three main steps, namely stress analysis, fatigue properties
estimation, and fatigue life prediction. Series of finite element analyses are carried out to
investigate stress distribution in the tensioner spindle and pinpoint the critical areas. In addition,
the fatigue properties of the tensioner are estimated using the experimental data. The tensioner
fatigue behavior is analyzed through strain-life approach. The predicted fatigue parameters was
then used to generate the cyclic stress-strain curve for the material of the selected tensioner based
on Morrow theory. Finally, the fatigue estimation method was modified based on Neuber’s rule
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and using the generated cyclic stress-strain curve for the die cast A380 tensioner. This model
presents the Neuber parameter life data to account for plasticity correction.
The possibility of using the newly developed HPDC magnesium alloys, MRI 153M and MRI
230D as alternative materials for tensioner casting parts is investigated in this study. Wohler
curves for these different alloys AZ91D, MRI153M, and MRI230D are presented and then
compared to current material for casting parts of the tensioner, A380 aluminum alloy. Effects of
possible design variables were also investigated in the study using the response surface method.
A methodology to achieve an optimal design shape for the tensioner based on the application and
loading was proposed when considering life of the part. This research work presents an in-depth
quantitative modeling approach to estimate the fatigue life of the automotive tensioner under real
working condition by developing the stress histories. The developed modeling approach is
applicable for evaluating any cases involving power train mechanical components.
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Acknowledgments
I would like to express my appreciation to my supervisor, Professor Jean W. Zu, for her delicate
support, recommendations, inspiring guidance and encouragement throughout my PhD.
Professor Zu's insights, vision, and advice not only helped me in my research, but also has an
everlasting influence on my professional career and personal life. I feel very fortunate that I had
the opportunity to work with her. I would also like to acknowledge Litens Automotive Group for
support on this collaborative project.
I would like to extend my appreciation to my Ph.D. committee, Professor Kamran Behdinan and
Professor Tobin Filleter, for their insight, suggestions, and time in evaluating my research during
these four years.
I am very thankful for the financial support of the University of Toronto, Ontario Graduate
Scholarship (OGS) program, and the Graduate Student Endowment Fund (GSEF) which made
this research possible.
My gratitude goes to my family, specially my parents, for their continuous encouragement and
unconditional love during all these years. Their support was source of energy and they were
definitely my inspiration and motivation to reach higher.
In addition, I would like to thank my officemates who their friendship and companionship
enriched my graduate life.
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Table of Contents
Acknowledgments .......................................................................................................................... iv
Table of Contents ............................................................................................................................ v
List of Tables ............................................................................................................................... viii
List of Figures ................................................................................................................................. x
List of Appendices ....................................................................................................................... xiii
Abbreviations ............................................................................................................................... xiv
Nomenclature ................................................................................................................................ xv
1 Introduction ................................................................................................................................ 1
1.1 Belt Drive Systems ............................................................................................................. 4
1.2 Automative Tensioner ......................................................................................................... 4
1.3 Background and Motivation ............................................................................................... 5
1.4 Objectives ........................................................................................................................... 6
1.5 Contributions ....................................................................................................................... 7
1.6 Organization of the Thesis .................................................................................................. 7
2 Literature Review ....................................................................................................................... 9
2.1 Front End Accessory Drive Systems .................................................................................. 9
2.2 Tensioner Mechanics ........................................................................................................ 10
2.3 Dynamic Analysis of Front End Accessory Drive Systems ............................................. 11
2.4 Tensioner in Accessory Belt Drive Systems ..................................................................... 13
2.5 Fatigue Analysis of Automotive Components .................................................................. 15
2.5.1 Stress Life Approach ............................................................................................. 17
2.5.2 Strain Life Approach ............................................................................................. 19
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2.5.3 Effect of Mean Stress On Life Analysis ............................................................... 20
3 Dynamic Analysis of Front End Accessory Drive Systems ..................................................... 22
3.1 Mathematical Modelling of Belt Drive Systems .............................................................. 22
3.1.1 Systems Equation of Motion ................................................................................. 24
3.1.2 System Modal Analysis ........................................................................................ 27
3.1.3 Dynamic Tension .................................................................................................. 29
3.1.4 Dynamic Analysis of the Five-Pulley Belt Drive System .................................... 30
3.2 Summary ........................................................................................................................... 40
4 Fatigue Analysis ....................................................................................................................... 41
4.1 Stress Analysis of Tensioner Using Finite Element Method ............................................ 41
4.1.1 Tensioner Geometry .............................................................................................. 42
4.1.2 Mesh Generation ................................................................................................... 43
4.1.3 Boundary Conditions ............................................................................................ 44
4.1.4 Finite Element Analysis Results and Discussion .................................................. 46
4.1.5 FEA Validation ..................................................................................................... 49
4.2 Fatigue Behaviour and Life Predictions ........................................................................... 56
4.2.1 Strain Life Approach ............................................................................................. 57
4.2.2 Effect of Mean Stress ............................................................................................ 58
4.3 Tensioner Fatigue Parameters ........................................................................................... 60
4.4 Fatigue Behaviour at Critical Regions .............................................................................. 64
4.4.1 Stress Concentration Factor .................................................................................. 66
4.4.2 Plasticity Correction of Linear Elastic FEA Stress and Strain Data ..................... 73
4.5 Summary ........................................................................................................................... 76
5 Tensioner Design Optimization ............................................................................................... 77
5.1 Effect of Material on Tensioner Life ................................................................................ 78
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5.2 Effects of Possible Design Variables ................................................................................ 91
5.3 Summary ......................................................................................................................... 102
6 Random/Variable Amplitude Loading ................................................................................... 103
6.1 Realistic Load Histories .................................................................................................. 103
6.2 Rain Flow Counting Method .......................................................................................... 110
6.3 Palmgren-Miner Rule ...................................................................................................... 111
6.4 Summary ......................................................................................................................... 113
7 Conclusions ............................................................................................................................ 114
7.1 Conclusions and Contributions ....................................................................................... 114
7.2 Future Work .................................................................................................................... 117
References ................................................................................................................................... 119
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List of Tables
Table 1. Physical Properties of the Prototypical System .............................................................. 34
Table 2. Tensioner Arm Characteristics ....................................................................................... 35
Table 3. Natural Frequencies of the System ................................................................................. 37
Table 4. Dynamic Belt Span Tensions .......................................................................................... 38
Table 5. Mechanical Properties: Die Cast A380 Aluminum Alloy [45] ....................................... 47
Table 6. Equivalent Stress and Strain at the Fillet Area ............................................................... 48
Table 7. Comparison of Measured and Simulated Strains from the FE Model ............................ 55
Table 8. Results for Standard Die Cast Tensioner at Ambient Conditions .................................. 61
Table 9. Fatigue Parameters for Die-Casting Al Alloy Tensioner ................................................ 63
Table 10. Comparison of Cyclic Mechanical Properties of A380, A356 and A413 .................... 66
Table 11. Maximum Equivalent Von-Mises Stress Corresponding to Different Applied Forces at
the Critical Area of the Tensioner ................................................................................................. 67
Table 12. Chemical composition of AZ91D, AE44, and A380 alloys ......................................... 81
Table 13. Mechanical Properties of MRI Mg Alloys, AZ91D Mg, and A380 Al Alloy .............. 81
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Table 14. Comparison of Bolt Retention Percentage of MRI, AZ91D Mg Alloys with A380 Al
Alloy [56] ...................................................................................................................................... 83
Table 15. Comparison of Corrosion Resistance of MRI, AZ91D Mg Alloys with A380 Al Alloy
[56] ................................................................................................................................................ 83
Table 16. Fatigue Properties of the Component ........................................................................... 85
Table 17. Fatigue Properties of the Materials ............................................................................... 86
Table 18. Range of Design Variables ........................................................................................... 93
Table 19. Array Result of Central Composite Design (CCD) for Case Study 1 .......................... 95
Table 20. Array Result of Central Composite Design (CCD) for Case Study 2 .......................... 98
Table 21. Input Parameters for the Parametric Study of the Tensioner Spindle ......................... 100
Table 22. Output Parameters of the Parametric Design Study of Tensioner Spindle ................. 101
Table 23. 16-step Loading, Resultant Tensioner Hubload and Stress Generated at Critical Area
..................................................................................................................................................... 109
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List of Figures
Figure 1. Layout of Belt Drive System Including: Crankshaft, Air Conditioner, Steering Pump,
Idler, Water Pump, Alternator, Overrunning Alternator Pulley and Tensioner .............................. 2
Figure 2. A Typical Front End Accessory Drive System ............................................................. 23
Figure 3. A Five-Pulley Automotive Belt Drive System .............................................................. 32
Figure 4. Crankshaft Torque Signal .............................................................................................. 36
Figure 5. Frequency Response Function of the Angular Displacement of (a) Power Steering and
(b) Air Conditioner ....................................................................................................................... 38
Figure 6. Schematic of a Typical Tensioner ................................................................................ 42
Figure 7. Tensioner Assembly ...................................................................................................... 43
Figure 8. Meshed Geometry of the Die Cast Tensioner with 382021 Elements .......................... 45
Figure 9. Tensioner Base Schematic Showing the Boundary Conditions .................................... 46
Figure 10. Stress Distribution in Tensioner Spindle ..................................................................... 48
Figure 11. Tensioner Schematic (a) x Direction, and (b) z Direction ........................................... 50
Figure 12. Free-body-diagram of the studied tensioner ................................................................ 51
Figure 13. Strain Gauges Placed on the Tensioner Spindle .......................................................... 54
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Figure 14. Gauge Orientation in Rectangular Rosette .................................................................. 54
Figure 15. Strain Versus Force at Rosette Location (a) 1 and (b) 3 ............................................. 56
Figure 16. Tensioner Fatigue Parameters Derivation Flow Diagram ........................................... 63
Figure 17. Tensioner Spindle Fatigue Data (Analytical and Experimental SWT Parameter) ...... 64
Figure 18. Estimated Cyclic Stress-Strain Curve for A380 Compared with the Published Results
for A356 and A413 ....................................................................................................................... 66
Figure 19. Stress Contour Showing the Max Stress at the Critical Area When Applying 3570N
Load .............................................................................................................................................. 68
Figure 20. Path Created for the Stress Concentration Estimation ................................................ 69
Figure 21. Stress Gradient Generated on the Selected Path .......................................................... 70
Figure 22. Stress versus Distance to the Neutral Axis .................................................................. 71
Figure 23. Neuber Parameter-Reversals to Failure Curve for Die Cast A380 Tensioner ............. 75
Figure 24. Material S-N Curve and Component S-N Curve for A380 Al. Alloy ......................... 88
Figure 25. Material S-N Curve and Component S-N Curve for MRI 153M Mg. Alloy .............. 89
Figure 26. Material S-N Curve and Component S-N Curve for MRI 230D Mg. Alloy ............... 89
Figure 27. Material S-N Curve and Component S-N Curve for AZ91D Mg. Alloy .................... 90
Figure 28. Estimated Component S-N Curves for A380, AZ91D, MRI153M, and MRI230D .... 91
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Figure 29. Fillet Radius and Spindle Tower Height as Design Variables of the Parametric Study
....................................................................................................................................................... 92
Figure 30. Three Dimensional Response Surface and the Corresponding Contour Plot (where x1
is the fillet radius and x2 corresponds to the spindle tower height) .............................................. 96
Figure 31. General Optimization Algorithm ................................................................................. 97
Figure 32. Design Variables for the Parametric Study ................................................................. 99
Figure 33. Crankshaft Speed Versus Time ................................................................................. 106
Figure 34. Crankshaft Torque Signal .......................................................................................... 107
Figure 35. Real Time Sample of Crankshaft Speed and Torque Versus Time ........................... 108
Figure 36. Cycles Counting Method using Rain Flow Approach ............................................... 111
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List of Appendices
Appendix I ................................................................................................................................... 99
xiv
Abbreviations
FEAD Front End Accesory Dive System
ASTM American Society for Testing and Materials
DOF degrees of freedom
SWT Smith, Watson, and Topper
LCF Low Cycle Fatigue
FRF Frequency Response Function
HCF High Cycle Fatigue
FEA Finite Element Analysis
FEM Finite Element Model
RSM Response Surface Method
LSM Least Square Method
HPDC High Presuure Die Cast
OEM Original Equipment Manufacturers
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Nomenclature
b Fatigue Strength Exponent
c Fatigue Ductility Exponent
Nf Number of Cycles to Failure
R Stress Ratio
S Stress
SWT Smith, Watson, and Topper
εa Cyclic Strain Amplitude
εe Elastic Strain Component
εp Plastic Strain Component
εf Fatigue Ductility Coefficient
σa Cyclic Stress Amplitude
σar Transferred Stress Amplitude
σm Mean Stress
σmax Maximum Stress
σmin Minimum Stress
σf Fatigue Strength Coefficient
K Cyclic Strength Coefficient
n Cyclic Strain Hardening Exponent
N Number of Applied Cycles
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t Time
E Young’s Modulus
LCF Low Cycle Fatigue
FRF Frequency Response Function
Frequency
HCF High Cycle Fatigue
Greek Symbols
Poisson's Ratio
Density
Stress Tensor
y
Yield Stress
Rotational Response
Rotational Frequency
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Chapter 1
1 Introduction
Automotive Front End Accessory Drive (FEAD) systems are extensively employed in power
transmission systems, where the crankshaft supplies power to several critical accessories such as
alternator, power steering, pump, air conditioner, etc.
Figure 1 shows a typical FEAD system, which includes a number of accessories and an
automatic tensioning mechanism. An automatic tensioner is often used in vehicles with a single
serpentine belt. Generally, the purpose of using a belt tensioning system is to apply a constant
force to the belt under steady state condition, and to eliminate possible resultant belt slippage.
The tensioner itself consists of a spring-loaded mechanism housed behind a pulley. Excitations
from the vehicles power train can lead to fatigue failure of the FEAD components. Tensioner
failure due to cyclic dynamic loadings is one of the most critical conditions that can lead to
catastrophic failure of FEAD systems. Therefore, it is essential to obtain an accurate estimation
of fatigue life for the tensioner components.
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Figure 1. Layout of Belt Drive System Including: Crankshaft, Air Conditioner, Steering Pump,
Idler, Water Pump, Alternator, Overrunning Alternator Pulley and Tensioner
Fatigue in automotive components has been under investigation by several researchers. Rahman
et al. [1] studied the fatigue life prediction of the lower suspension arm using the strain-life
approach. In this study, Rahman et al. [1] presented a numerical method which predicted the life
of the suspension arm under variable amplitude loading conditions. Finally, the most suitable
material for the suspension arm was reported following an investigation on the life prediction of
the component using different materials. An accelerated fatigue testing procedure for automotive
chassis components was presented in a study by Beaumont et al. [2]. They presented a
comparison between the fatigue testing methods currently used in the automotive industry, and
evaluated the efficiency and performance of these testing methods.
In another study by Raju et al. [3], fatigue failure of aluminum alloy wheels were analyzed by
generating the S-N curve of the component. They considered the operational condition of the
aluminum wheels by applying a constant amplitude radial load to the component. Subsequently,
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the output life resulting from the developed finite element (FE) model was compared with the
experimental fatigue testing results. Fatigue life of the steel wheel was investigated by Topac et
al. [4] using the stress-life approach. In this study, the corresponding S-N curve was estimated
considering the wheel monotonic properties and the Marin modification factors such as surface
finish, size, and stress concentration were obtained. Steering knuckle fatigue life was estimated
in a study by Jhala et al. [5]. They compared the performance of steering knuckles using three
different materials, including forged steel, cast aluminum, and cast iron. The results of both
numerical and experimental analyses showed the superior performance of the forged steel
steering knuckle comparing to the other two knuckles. Jung et al. [6] evaluated the automatic belt
tensioner damage using the stress life approach and Palmgren-Miner rule. They proposed a
testing method to measure the equivalent damage in real driving condition.
Although the above-mentioned researchers have provided valuable insight into the numerical
analysis of fatigue life assessment of mechanical components, a comprehensive solution
concerning the fatigue life of power train components has not been addressed to this date. In this
work, a comprehensive solution to the fatigue life estimation of the tensioner spindle as a critical
part of FEAD systems is presented. Three major steps of stress analysis, fatigue properties
estimation, and fatigue life prediction are considered in this work. First, the stress distribution in
the tensioner spindle is generated using finite element analysis to identify the critical region
experiencing the highest stress. In addition, due to the lack of fatigue properties for the tensioner
spindle, fatigue parameters are optimized using the experimental data. Finally, an analytical
method is developed through strain-life approach to estimate tensioner lifetime based on the
optimized fatigue parameters.
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The objective of the present study is to predict the fatigue life of automotive tensioners by
developing a practical simulation methodology. The simulation work involves two parts. First, an
FE model is developed to obtain the stress developed in the tensioner when the part undergoes
static loading. Second, the FEA results are used as inputs for fatigue life calculation base on a
strain-life fatigue material model for life prediction of the part.
1.1 Belt Drive Systems
In belt drive systems, power and motion is transferred from the engine crankshaft to the pulleys
through belt. Belt drive systems include a number of fixed pulleys and an automatic tensioner
that is used to transmit power and motion. The power is transferred due to the torque generated
as a result of tangential force. This tangential force is the result of belt tension difference
between the tight and slack side of the pulley.
A belt tensioning system is often used in vehicles with a single serpentine belt. The automatic
tensioner itself consists of an arm and a pulley. The tensioner pulley rotates around a fixed pivot.
The tensioner comprises a component with a spring mechanism housed behind the pulley. The
main purpose of tensioner application in a belt drive system is to maintain the minimum constant
belt tension required for power transmission under steady state condition.
1.2 Automative Tensioner
Tensioners are used in vehicles with single serpentine belt. Serpentine belts require application
of a tensioner for proper power transmission. In front end accessory drive systems, the main
purpose of the tensioner is to provide the minimum tension on the belt that is required for proper
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transmission of power from the driver pulley to the driven pulleys. With introduction of
automatic tensioning mechanism, the need of frequent belt retensioning is eliminated.
If the proper tension is not achieved on the system, belt slip happens which consequently results
in belt wear. The belt will gradually wears smooth and hardens. The hardening happens due to
the heat aging. To avoid the effects of wear and any possible high preloading at the installation
and extreme fluctuation of the belt tension while in service, an automatic belt tensioning
mechanism is required. Another significant effect of application of automatic tensioners in
FEAD systems is to filter the belt tension fluctuations, which leads to improved dynamic
behavior of the transmission [7].
Tensioners are complex mechanical parts equipped with springs installed behind the pulley. The
spring mechanism applies proper torque to the system during different working modes. The
tensioner consists of different components: arm, bracket, damper or bushing, spring, pulley,
pulley bearing, dust shield and pulley center bolt. If one of the internal components of the
tensioner fails, it causes stress on the belt and other system accessories, and consequently allows
the belt to slip.
1.3 Background and Motivation
Since it is widely known that about 80% of component failures are related to structural fatigue,
fatigue life prediction has gained more attention in design and durability analysis [8]. As part of
design process engineers have to consider the design durability of the product over its life cycle.
A major cause of failure is the growth of cracks, which grow due to fatigue loadings until
fracture occurs [9]. Numerous studies on numerical analysis of fatigue life assessment have been
6
performed in the past. However an analytical model, which predicts the lifetime of components
with complex geometry under real dynamic loading and condition, has not been addressed. In
some cases, developing a fatigue model for such complex parts is a challenging task due to the
lack of access to test specimens. Moreover, fatigue tests performed on small specimens are not
sufficient for precisely establishing the fatigue life of a part. These tests are useful for rating the
relative resistance of a material and the baseline properties of the material to cyclic stressing. The
baseline properties must be combined with the load history of the part in a design analysis before
a component life prediction can be made [10]. The type of applied loading (uniaxial, bending, or
torsional), loading pattern (either periodic loading at a constant amplitude or random loading),
magnitude of peak stresses, overall size of the part, fabrication method, operating temperature
and environment are other important aspects which have not been well investigated analytically
in previous studies. Thus a comprehensive mathematical model for predicting fatigue damage of
automative tensioners will be developed in this study.
1.4 Objectives
This doctoral research is focused on the analytical study of the dynamic response of the front end
accessory drive systems, as well as fatigue investigation of the tensioner due to dynamic loading
applied on the tensioner and other accessories from the engine pulsations. The scope of the work
includes investigation of the accessory drive system and its dynamic behavior under specific
torsional vibrations of the engine. Automatic tensioner as one critical part of the system is
analyzed using finite element methods to evaluate the stress developed on the tensioner due to
the force applied on the part from the belt tension. Finally, an in depth study of the fatigue
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behavior of tensioners is performed to predict the component life under variable dynamic
loading.
1.5 Contributions
The work presented in this thesis resulted in one journal publication [40], and one manuscripts is
currently under review. The research contributions are summarized in the following list:
1. A dynamic model of a multi-degree-of-freedom system to investigate dynamic loads on
the tensioner was developed.
2. A finite element model to calculate the stress developed in the tensioner under static
conditions was developed.
3. The finite element belt model was correlated and validated with the experimental results
for the tensioner.
4. Fatigue properties for the tensioner life model were determined via experimental results.
5. A mathematical model to predict fatigue life based on strain life approach under constant
amplitude loading was provided.
6. A design enhancement solution for the tensioner was provided in order to obtain an
extended fatigue life.
7. Newly developed materials were investigated and proposed as alternatives for casting
components of the tensioner (arm and bracket).
8. Stress histories of tensioner under real working condition were determined for further
analysis of fatigue life of tensioners based on generated fatigue parameters and real
working condition.
1.6 Organization of the Thesis
This thesis is organized into seven chapters. Chapter 1 presents an introduction, background,
motivations and an overview of the contributions of the thesis. The literature review provided in
chapter 2 elaborates the history of this thesis contributions along with the aspects that were not
8
considered in the previous studies by other researchers. This chapter reviews the techniques
employed in fatigue analysis of different mechanical and structural components and provides an
introduction to the background of these methodologies. Chapter 3 discusses dynamic analysis of
front end accessory drive systems, and the system response to engine torsional vibration.
Thereafter, the details of the finite element model developed for evaluation of the stress
distribution on the tensioner under static loading are presented in chapter 4. Additionally, this
chapter provides a comprehensive methodology used for studying the fatigue behaviour of the
automative tensioners. Chapter 5 illustrates how the complex geometry of tensioner can be
optimized using parametric studies along with response surface method. Realistic load histories
applied to tensioners while in service are investigated in chapter 6. The real world usage profile
is then used to further evaluate the fatigue behaviour of the tensioners under real working
conditions. Chapter 6 also presents the details of the methodology employed for comprehensive
fatigue analysis of tensioners under variable amplitude loading. Finally, chapter 7 summaries the
conclusions and possible future plans of this research study.
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Chapter 2
2 Literature Review
This chapter presents a comprehensive literature review on the subject of fatigue life analysis of
automotive components, previously conducted by other researchers. First, the front end
accessory drive systems are introduced. The tensioner mechanics and the dynamic analysis of
belt drive systems are described. Thereafter, the fatigue life approaches used for complex
automotive components under static and dynamic loading are discussed. Lastly, the feasibility of
these approaches for automotive front end accessory drive systems is assessed.
2.1 Front End Accessory Drive Systems
Automotive front-end accessory drive systems use serpentine belts to transmit power from the
crankshaft simultaneously to the accessory components. Cassidy et al. [11] developed the idea of
tensioner device in belt and pulley system and performed a dynamic analysis on a belt drive
system. An automatic tensioning device is used to maintain the belt tension, which consequently
eliminates belt slippage. The dynamic analysis of the whole serpentine belt drive system is a
challenging subject and it has been investigated for over 15 years [12]. The rotational vibrations
10
of pulleys in belt drive systems is considered to be more dominant in belt drive systems. Hence,
different researchers including Abrate [13] investigated the rotational vibrations of the belt drive
systems.
2.2 Tensioner Mechanics
An automatic belt tensioning system is employed in vehicles using a single serpentine belt.
While the belt drives power from the crank to the accessories, an automatic tensioner is designed
to apply a constant force on the belt.
Due to the high axial stiffness, most of the serpentine belts require to be supported by an
automatic tensioner in the belt drive system. The tensioning mechanism is mainly used to
compensate for the effects of the wear, to avoid any possible high preloading at the installation
and excessive fluctuation of the belt tension during the operating conditions. Moreover, the
filtering of the belt tension fluctuations guaranteed by the tensioner allows an improved dynamic
behavior of the transmission [7].
Tensioners are equipped with spring-based mechanism housed behind the tensioner pulley. It
consists of basic parts such as base, damper, spring, arm, pulley bearing, dust, and pulley center
bolt. Tensioner internal components can fail, consequently resulting in stress on the belt and
other accessory components, and causing the belt slippage. Understanding the failures, and
having the ability to analyze what caused them should be considered in the design process.
In order to design accessory drive tensioners against fatigue, it is required to investigate the
dynamic loads applied to the system. These include the dynamic belt tension and dynamic hub-
load. Furthermore, the dynamic behavior of an axially moving belt introduces significant
11
challenges in the design process of such tensioning systems. In addition, a very small excitation
caused by the accessory system (pulley eccentricity or fluctuating input torque) induces
excessive vibrations to the accessory drive system. When the operating speed of the system is
close to the natural frequency, the vibrations become very significant. Achieving belt dynamic
stability is also a challenge in operations with high rotary speed. [14] The design of accessory
drive tensioners involves its own challenges in addition to the complex dynamic performance of
a typical belt drive system. To design an automatic tensioner that maintains the required tension
for power transmission on the belt during the service life of the belt drive system, a
comprehensive study of the system dynamic behavior is necessary.
2.3 Dynamic Analysis of Front End Accessory Drive Systems
Vibration in the front end accessory drive systems can be longitudinal and transversal. The
longitudinal type of vibration occurring in the belt drive systems is caused by the longitudinal
belt deformation due to oscillation of the accessory drive rotary inertias about their pin axes. In
this type of vibration, the belt can be modeled as linear springs. The belt vibration vertical to the
belt axial direction causes another type of vibration, transverse vibration. In this type of
vibration, the belt acts as a taut spring. Various sources cause these two major types of vibration:
torque fluctuation from the CS or accessory pulleys, pulley eccentricities, movement of pulley
supports, or abnormal belt properties [15]. This section explains how different approaches by
researchers are used to investigate the dynamic behavior of the accessory drive systems.
Hawker et al. [16] investigated natural frequencies of a damped drive system including a
dynamic tensioner. This study, however, does not consider the effect of the tensioner on either
the equilibrium state or the dynamic response. Design optimization of a four-pulley tensioner in a
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belt drive system with a conventional one-pulley tensioner is performed by Zu et al. [17]
assuming only longitudinal belt vibrations. Beikmann et al. [18] evaluated a prototypical drive
system consisting of a driven pulley, a driving pulley, and a dynamic tensioner. In this study, the
authors used a closed-form solution method to calculate the natural frequencies and mode
shapes. Zhang et al. [17] focused on the modal analysis of serpentine belt drive systems.
To model an axially moving belt with significant velocity, a string model is used. Different
researchers worked on the transversal and longitudinal vibrations of a moving belt since 1950s.
Sack et al. studied the behavior of waves developed in a belt under tension in 1950s [19]. The
bending effect of the belt was neglected in most of belt drive systems studies with acceptance of
small errors. This means the models developed for the axially moving belt ignored the belt
rigidity. Another assumption made in the models was that the wave motion of the belt was
superimposed to the longitudinal motion of the belt. These models neglected coupling the
transversal and longitudinal vibration of the belt. Later in 1978, Ulsoy et al. [7] investigated the
nonlinear transversal vibration of the belt in a system including a tensioning mechanism and a
number of belt spans. This study considered the torque variation effects on different spans
dynamic tension. The study concludes that parametrical excitations can be developed due to
dynamic tensions, which cause belt span lateral vibration.
Steady state response of an axially moving belt was also studied by Chanon et al. [20]. In this
study, the belt was subjected to constant lateral force and fixed in space. The main purpose of
this work was to investigate the lateral vibration of the belt under tension and in time domain.
Vibration of axially moving continua under tension was investigated in Wickert’s work in 1990
[21]. This developed model in this study considered coupling of transversal and longitudinal belt
13
vibrations. The presented model was used by different researchers to model an axially moving
belt in drive systems with tensioning mechanism with consideration of both lateral and
longitudinal vibrations.
Moon et al. [22] modeled the nonlinear vibrations of belt drive systems with the assumption of
belt excitation due to pulley eccentricity. This study showed dynamic instability can be caused
by pulley eccentricity when belt is moving with high linear velocity. Chaug et al. [23] used a
similar methodology to model an axially moving string vibration in presence of geometric
nonlinearity and translating acceleration.
A model for an Euiler-Bernouli beam under high tension was developed by Oz et al. [24]
considering the bending stiffness. Later on, Pellicano et al. [25] used same approach to
investigate the numerical and experimental analysis of axially moving belt with complex
dynamics. In another study by Ha et al. [26], the bifurcation and chaos developed by viscoelastic
moving string with three-dimensional vibration state. The model developed in this study
considered the non-linear geometry factor due to three-dimensional coupling of transversal and
longitudinal vibrations. FE model was developed by Kojvurov et al. [27] to analyze the response
of the system, which includes a moving belt under tension. The developed dynamic analysis
models have been applied in design investigation of complex belt drive systems.
2.4 Tensioner in Accessory Belt Drive Systems
Different researchers investigated dynamic behavior of belt drive systems with the focus of
coupled vibration between the belt and the tensioner. The idea of tensioning mechanism in a belt
drive system was first proposed by Cassidy et al. [22]. The pulley rotational vibration is
14
dominant comparing to the belt transversal vibration. Belt drive systems dynamics are often
considered with nonlinear behavior due to the coupling of belt transversal and longitudinal
vibration and tensioner vibrations.
Aberte et al. [23] studied the nonlinear vibrations of belt drive systems in presence of a
tensioning mechanism. The model developed by Aberte used Runge-Kutta method for solving
nonlinear equations of motion. This model was developed to investigate the behavior of the
system with rapid acceleration and deceleration when a friction type tensioner is present.
Beickman et al. [24] studied how the belt axial velocity affects the nonlinear behavior of
stretched belt, which results in coupling of the belt and tensioning mechanism vibrations. In
another study of belt drive system rotational vibrations; the belt slip onset was analyzed [25].
Zhang and Zu [26] investigated the nonlinear behavior of viscoelastic differential constitutive
law. In this work, natural frequencies and response of the system was predicted using
perturbation method. In another study, they studied forced vibration analysis of viscoelastic belts
when the excitation is caused due to pulley eccentricity [27]. Another work by same authors was
performed on modal analysis of the belt drive systems. Kwon et al. [28] studied the physical
characteristics of moving belts with the main focus on the lateral vibration power flow.
Furthermore, investigation of belt drive systems dynamics was presented by Parker et al. [29]. In
this study, a hybrid discrete-continuous model was used to analyze pulley rotations, tensioner
arm rotation and transversal vibration of adjacent belt spans to the tensioner. In another work, a
numerical model along with simulation software was developed and presented by Lavos et al.
[30]. This software simulates the timing drive transmission with the presence of a tensioning
15
mechanism. Dynamics of belt drive system with consideration of shear deformation on the
behavior was also analyzed by Tonoli et al. [31].
2.5 Fatigue Analysis of Automotive Components
Excitations from the vehicles power train can lead to fatigue failure of the FEAD components.
Tensioner failure due to cyclic dynamic loadings is one of the most critical conditions that can
lead to catastrophic failure of FEAD systems. Therefore, it is essential to obtain an accurate
estimation of fatigue life for the tensioner components.
Fatigue in automotive components has been under investigation by several researchers. Jhala et
al. [5] studied the fatigue behaviour of vehicle steering knuckle via finite element method.
Dynamic loadings applied on the steering knuckle during service life were considered in the
fatigue analysis of the component. The authors focused on different materials and manufacturing
processes used to make the component. The stress distribution of the steering knuckle was
predicted using finite element analysis. Fatigue properties were found using load-controlled
fatigue testing for case of different materials namely, forged steel, cast aluminium, and cast iron.
The results of comparison of fatigue properties and manufacturing processes of different cases
showed that forged steel has superior fatigue behaviour than the other two candidates.
In another study, Shim et al. [32] investigated the fatigue life of pulley in vehicle power steering
system. The stress state of the pulley was determined for the case of high tensile loading and
high torque using finite element method. The purpose of this study was to investigate the fatigue
life of the pulley with a focus on the durability analysis. An optimum design for the pulley was
proposed as the end result of the study. The authors used the response surface method to
16
optimize the pulley shape using two possible design variables. The results of this study showed
the critical zone of the pulley while under high tension and torque. The durability analysis was
performed to explain the probability of failure for the pulley design. A secure design was
concluded at fatigue failure probability of 1%.
Raju et al. [3] investigated the fatigue behaviour of the aluminium alloy wheels. This work was
focused on how the radial loads applied on the wheel during service lead to fatigue of the wheel.
In this study, the authors estimated the fatigue properties of the aluminium wheel using
experimental data taken from the fatigue testing at various stress ratios. A finite element model
was developed for the wheel to predict the stress distribution and the fatigue life of the wheel.
The authors showed the good agreement found between the FEA and the experimental results.
In another study by Zheng et al .[33], the fatigue life of steel wheel of vehicles was investigated
considering damage mechanics. In this work, the authors proposed an economical method to
stimulate the fatigue-bending test, which is usually used for fatigue testing of the wheel in
automotive industry. Finite element method was used to predict the stress state of the wheel
while under bending. The boundary conditions to stimulate the fatigue-bending test were
carefully studied in this work. Fatigue life of the steel wheel was predicted in the concept of
continuum damage mechanics. The comparison of the finite element results and experimental
results showed a good agreement and verified the accuracy of the proposed numerical method.
Fatigue behavior of automotive chassis with a focus on the reliability of the part in case of static
and dynamic loads was investigated by Beauumont et al. [2]. In General, the reliability analysis
of any part includes undergoing several time-consuming high-cycle fatigue testing. This study
proposed a method that considers the reliability framework of the part, while reducing the testing
17
number on specimens. Three different fatigue testing methods, StairCase, Locati, and StairCase-
Locati, are investigated and compared in this research. These methods were used at PSA Peugeot
Citroen at the time of the study. The results of the research showed that Locati, and the
StairCase-Locati approaches do not depend on the starting stress and step stress values. StairCase
and StairCase-Locati methods are reliable even with use of small sample size.
Augustins et al. [34] modified the Dang Van criterion for the case of biaxial tensile loading on
the automotive cylindrical head design. This study was developed at PSA Peugeot Citroen in
collaboration with the LAMPA (Laboratoire Arts et Metriers ParisTech d’Angers). The critical
region of the automotive engine cylinder head experiences high biaxial tensile loading with high
mean stress. An empirical method was developed based on the Dang Van criterion to compensate
the unreliability of one parameter of the criterion for equi-biaxial tensile loading. The authors
focused on the adjustment of the fatigue parameters of the criterion, currently used at PSA, to
predict the fatigue behaviour of the cylinder head design more accurately. The developed
approach was verified by test data available in the literature and also on automotive cylindrical
head design. However, this method as an empirical method is be used with extra caution in case
of complex loading cycle.
2.5.1 Stress Life Approach
To design a sound fatigue solution, different fatigue assessment methods are used based on the
application and in-service loading. In this content, the number of cycles of stress that causes
fatigue is of importance. Stress life methods are used for high cycle fatigue cases. The stress or
equivalent strain amplitude in which the part operates determines the principle of the approach. If
the applied stress is within elastic limit, and there is no plastic strain is present unless in the tip of
18
the crack, the stress life approach is used. In this approach, stress life curves are developed for
fatigue assessment of parts. The stress life curves (Wohler curves) are usually in log-log scale.
Knowing the linear elastic stress histories on the part and using the rainflow counting method,
fatigue cycles can be extracted. Stress life approach uses the linear elastic stresses, which are
estimated using a linear FE model. The stress range is then used to access the total damage on the
part using damage accumulation methods (e.g. Miner’s rule).
Different researchers studied the fatigue behaviour of automotive parts using stress life approach.
Topac et al. [4] studied the prediction of fatigue life of automobile wheel under dynamic radial
loading. The authors used finite element method for fatigue life prediction in this work. In heavy
vehicle, the fatigue failure of the steel wheel occurs on the air ventilation holes. The finite
element method was used to pinpoint the critical areas of the wheel, which happens to be the
same region for all the tested samples. The mechanical properties and the S-N curve of the steel
wheel were calculated using the tensile tests, and the estimated ultimate tensile strength along
with the Marin factors for the critical region. Finally the authors proposed methods of design
enhancement to increase the life of the component.
In another study by Raju et al. [3], fatigue failure of aluminum alloy wheels were analyzed by
generating the S-N curve of the component. They considered the operational condition of the
aluminum wheels by applying a constant amplitude radial load to the component. Subsequently,
the output life resulting from the developed finite element (FE) model was compared with the
experimental fatigue testing results. Fatigue life of the steel wheel was investigated by Topac et
al. [4] using the stress-life approach. In this study, the corresponding S-N curve was estimated
considering the wheel monotonic properties and the Marin modification factors such as surface
19
finish, size, and stress concentration were obtained. Steering knuckle fatigue life was estimated
in a study by Jhala et al. [5]. They compared the performance of steering knuckles using three
different materials, including forged steel, cast aluminum, and cast iron. The results of both
numerical and experimental analyses showed the superior performance of the forged steel
steering knuckle comparing to the other two knuckles.
In an study by Jung et al. [6], the automatic belt tensioner damage was evaluated using the stress
life approach and Palmgren-Miner rule. In this paper, the authors proposed a testing mode to
measure the equivalent damage of the automatic tensioners in real driving condition. The FTP-75
mode, as the fuel efficiency test mode, is used as the reliability assessment method for
automotive components when there is a timing issue for an actual testing. In this study, authors
proposed a test mode for real driving conditions, which was then compared and verified with the
FTP-75.
2.5.2 Strain Life Approach
Strain life approach is used for low cycle fatigue cases, where part is subjected to stresses higher
than yield. This introduces plastic deformation to the component and results in short lives. This
type of service loading is called low cycle fatigue, and is analyzed using strain life approach. For
ductile materials, the strain life approach is used for the cases of 100 and 100,000 cycles of
service operation. In this method strain-life curves are developed for analyzing the fatigue
behavior of the part. The strain life approach, a more recently developed method, can be applied
for both high and low cycle fatigue cases.
20
In this method, after the E-N curve is developed using the Coffin-Manson relation. The steps of
strain life approach are very similar to those in the stress life approach, with the difference of
using elastic plastic strains to estimate the damage. The elastic-plastic strains can be estimated
either using a non-linear FE model, or a linear FE model along with elastic-plastic correction
method (e.g. Neuber’s rule).
After the stress/strain histories are developed, the fatigue cycles for each stress/strain range can
be extracted. This is generally done using rainflow algorithm. Using the stress/strain range and
the developed damage curve, the damage can be assessed for each cycle. The accumulated
damage is then estimated using sum damage methods (e.g. Miner’s rule).
Researchers have studied the strain life approach for automotive components. Rahman et al. [1]
studied the fatigue life prediction of the lower suspension arm using the strain-life approach. In
this study, Rahman et al. [1] presented a numerical method which predicted the life of the
suspension arm under variable amplitude loading conditions. Finally, the most suitable material
for the suspension arm was reported following an investigation on the life prediction of the
component using different materials. An accelerated fatigue testing procedure for automotive
chassis components was presented in a study by Beaumont et al. [2]. They presented a
comparison between the fatigue testing methods currently used in the automotive industry, and
evaluated the efficiency and performance of these testing methods.
2.5.3 Effect of Mean Stress On Life Analysis
Different modification methods have been studied to account for the mean stress effects on the
life analysis of components. Ince et al. [35] proposed a modification to the Morrow and SWT
21
methods. In this study, the proposed method is compared to the original methods in terms of
accuracy and capability. They found that the Morrow method provides the least accurate results
comparing to the other methods. The proposed method showed superior accuracy with the
experimental results when compared to the original Morrow and SWT models for the case of the
Incoloy 901 and the ASTM A723. The level of accuracy for the case of 7075-T561 aluminium
alloy and 1045 HRD 55 steel was shown to be similar for both the proposed modified and the
original models [35].
Nieslony et al. [36] studied the effects of mean stress level on the fatigue life of constructional
components. The two proposed methods were validated using the experimental results from the
literature. The studied models showed a good correlation with the experimental results.
Although the above-mentioned researchers have provided valuable insight into the numerical
analysis of fatigue life assessment of mechanical components, a comprehensive solution
concerning the fatigue life of power train components has not been addressed to this date. In this
paper, a comprehensive solution to the fatigue life estimation of the tensioner spindle as a critical
part of FEAD systems is presented. Three major steps of stress analysis, fatigue properties
estimation, and fatigue life prediction are considered in this work. First, the stress distribution in
the tensioner spindle is generated using finite element analysis to identify the critical region
experiencing the highest stress. In addition, due to the lack of fatigue properties for the tensioner
spindle, fatigue parameters are optimized using the experimental data. Finally, an analytical
method is developed through strain-life approach to estimate tensioner lifetime based on the
optimized fatigue parameters.
22
Chapter 3
3 Dynamic Analysis of Front End Accessory
Drive Systems
Since the fatigue life of a part depends significantly on the dynamic load history, it is important
to investigate the dynamic loads on the part. In this section, a mathematical model which predicts
the dynamic loading on the components of front end accessory drive (FEAD) systems will be
explained. A multi-degree-of-freedom dynamic model will be developed assuming the input is
the fluctuating torque from the engine to the belt drive system.
3.1 Mathematical Modelling of Belt Drive Systems
Mathematical modelling to analyze the dynamic behaviour of belt drive systems is detailed in
this section.
23
Figure 2. A Typical Front End Accessory Drive System
Figure 2 shows a typical front end accessory drive system which consists of accessory pulleys,
and a dynamic tensioner driven by the engine crankshaft (driving pulley). Pulleys other than the
tensioner have fixed axes. We model the response of the system to arbitrary crankshaft
excitation. The crankshaft pulley rotates anti-clockwise and the belt drives other accessory
pulleys in the system. The belts strands are assumed to be coupled with accessory pulleys, and
have uniform properties ( m , EA ). The rotation of each accessory pulley and the tensioner arm
represent the dynamic responses (i
,i=1,...,n) caused by the crankshaft excitation [37].
Different assumptions are made in modeling the dynamic behavior of a typical system. These
assumptions are as follows:
The belt slippage is assumed to be negligible
The friction coefficient is assumed to be constant between the belt and the pulleys
The belt is assumed to be uniform, linearly elastic, and to act in a quasi-static manner
The lateral belt response is neglected when compared to the longitudinal response
24
3.1.1 Systems Equation of Motion
In this analysis, the Lagrange’s energy method is used to derive the equations of motion of the
above multi-degree-of-freedom system. The system modeling is developed by considering the
belt spans as linear springs with linear viscous dampers.
In the Lagrange`s method, the kinetic energy of the system can be written as the sum of the
kinetic energy of all the components in the system [42]. Hence, the system kinetic energy can be
described as follows:
2
1
2
2
1
tt
n
i
iiJJT
(3. 1)
where I denotes to each accessory pulley and t denotes to the tensioner arm.
The strain energy of the fixed pulley system is defined as:
n
i
si
s
bfs xKUi
1
2
2
1
(3. 2)
In this analysis, the belts are assumed to obey the Hooke’s law over the operating range.
Therefore, the stiffness value of each belt span is evaluated using ib LEAKi
/ (i=1,...,n), where
E is the belt modulus, A is the average cross sectional area, and Li is i
th
belt span length.
Knowing the system layout, each pulleys position, radius and tensioner arm pivot point, the belt
span length and contact arcs are obtained. Moreover, the belt span length change can be
described by iiiisi rrx 11 .
25
Considering the tensioner as the nth
pulley, the strain energy for the tensioner pulley is defined
as:
2
111
2
11 sin2
1sin
2
1
1tnarmnnnn
s
btnarmnn
s
bst lrrKlrrKUnn
(3. 3)
where tjj
, j
is the belt span direction angle, t
is the tensioner arm position angle,
ir is the radius of th
i accessory pulley, arm
l is the tensioner arm length, and t
is the tensioner
arm dynamic angle.
The total strain energy of the system which includes a tensioner is described as:
sfsts UUU (3. 4)
The pulleys, e.g. the tensioner pulley, with springs attached to their rotational axis have potential
energy associated with the spring rotation. This potential energy is defined as:
2
1
2
2
1
2
1
tt
n
i
irr kkUi
(3. 5)
Summation of the total strain energy and the potential energy of the spring rotation equals to the
total potential energy of the system as:
rs UUU (3. 6)
The system total potential energy is obtained using:
26
22
111
2
11
2
1
22
11
2
1sin
2
1
sin2
1
2
1
1tttnarmnnnn
s
b
tnarmnn
s
b
n
i
iriiii
s
b
KlrrK
lrrKKrrKU
n
ni
i
(3. 7)
The Lagrange equation is then written in terms of the kinetic and the potential energy of the
system:
i
iii
U
q
T
q
T
dt
d
ni ...,,2,1 (3. 8)
where tqq ii represents the generalized velocity and iQ represents the neoconservative
forces corresponding to iq .
Using the Lagrange method, the equations of motion is expressed as follows:
FXKXCXM ... (3. 9)
where M, C and K represent the mass, damping, and stiffness matrices, respectively. X stands for
physical variable vector and F is the applied forcing function to the system.
The stiffness matrix, K, is derived using the Lagrange equation. Assuming the Rayleigh
proportional damping, the damping matrix, C, is calculated. Generally, the damping effect on the
system natural frequencies and mode shapes is very small. Therefore, it can be neglected in the
obtaining the system natural frequencies and mode shapes. They can be obtained by solving the
following eigenvalue problem:
02
XKI (3. 10)
27
Thereafter, the modal analysis technique can be used to solve the above forced vibration
problem.
3.1.2 System Modal Analysis
In this part, we find the system response to the crankshaft excitation which is represented by
each accessory pulley and the tensioner arm rotations. In the modal analysis technique, the
dynamic variable vector is defined as:
t
t
t
X
t
n
1
(3. 11)
and the input forcing function is expressed as below:
0
0
sin
tQ i
F (3. 12)
In a forced dynamic analysis, the format of the response is the same as that of the forcing
function. The forced response is shown as following:
nt
nn
t
t
t
sin
sin
sin
11
11
X
(3. 13)
28
Modal analysis technique is used to solve this forced vibration problem. U is defined to be the
mass normalized eigen-vector matrix of M-1
.K. A modified matrix form of equations of motions
is decoupled in the following format:
FUqUKUqUCUqUMUTTTT
(3. 14)
where qi represents the modal coordinate. Hence, the equation of motions in the modal
coordinate is:
FqKqCqI dd (3. 15)
where Cd and Kd are diagonal and are in fact transformed form of stiffness and damping matrices
in modal coordinate. I represents the identity matrix. The equations of motion in modal
coordinate are as follows:
iiiniiniiFqqq
2
,,2
(3. 16)
The solution to the above equation is:
iii tqq sin (3. 17)
where
2
,
22
,
2
,
21iniin
ini
i
Fq
(3. 18)
and
29
2
,
,1
1
2tan
in
ini
i
(3. 19)
Thereafter, qi is transformed to the physical coordinate using the mass normalized Eigen-vector
matrixes:
qUX . (3. 20)
3.1.3 Dynamic Tension
The tension in each belt span can be calculated using the pulley angular displacements. This
tension is called the rotation tension as it is caused by the pulley rotations. When the tensioner is
placed as the nth
pulley on the FEAD system, the belt span tensions are determined by:
)..()..( 11110 iiiibiiiibi rrcrrkTTii
(3. 21. 1 )
ntotarmiinnb
ntototarmiinnbn
lrrc
lrrkTT
n
n
sin.
sin.
11
1101
1
1
(3. 21. 2 )
ntotarmnnb
ntototarmnnbn
lrrc
lrrkTT
n
n
sin.
sin.
11
1101
(3. 21. 3 )
where i
bk (i=1,..., n-1) and i
bc are the stiffness and damping coefficient of ith
belt span,
respectively. Here, i denotes to each pulley (i=1,..., n-1). The above equations explain the
relationship between the span tensions and each pulley rotational coordinates i and
t .
If we define:
30
0TTT ii (3. 22)
The tension in each belt span can be expressed in the following format:
tTtTT CK (3. 23)
where T is used to calculate the tension in each belt span only due to the dynamic response of
the system.
3.1.4 Dynamic Analysis of the Five-Pulley Belt Drive System
Various factors are involved in order to assess the fatigue life of the tensioner device: materials
properties, design geometry, and loading time history. Hence, it is important to investigate the
loading on the parts.
In this section, a mathematical model is developed which predicts the dynamic loading on the
components of front end accessory drive systems (FEADs). Figure 3 shows the schematic of the
belt and pulleys system which is examined in this part. An automatic tensioner is employed in
the system to maintain the required static tension over the operating range. The physical
properties of the system used in this part of the project are presented in Table 1.
In this study, the above analytical method is used to find the solution to the equations of motion
for the case of the sinusoidal excitation inputs.
31
3.1.4.1 System Properties
The properties of the five-pulley belt drive system, investigated in this study, are tabulated in
Table 1. This table presents the details of the system geometry, and other important physical
properties of the system such as the belt modulus, the effective length and width of the belt, and
etc.
Table 1The stiffness of each belt span,i
K )5...1( i , are respectively calculated as 0.73, 0.52,
0.65, 0.68, and 0.48 MN/m. The tensioner arm has a rotational stiffness,t
K of 69.04 N-m/rad
The belt spans and the tensioner mechanism are under static preload (340.66 N) causing initial
deflections. Knowing the amount of the crankshaft torsional activity, the following simulation
model can be used to determine static and dynamic responses of the system.
2
55
2
44
2
33
2
22
2
11
2
2
1
2
1
2
1
2
1
2
1
2
1 JJJJJJT tt (3. 24)
The potential energy due to the deflections of the belt and the tensioner spring is defined as:
2
555115
2
444554
2
33443
2
22332
2
11221
2
sin2
1
sin2
1
2
1
2
1
2
1
2
1
tarm
tarm
tt
lrrK
lrrKrrK
rrKrrKKU
(3. 25)
In this 5 pulley-belt drive system, the crankshaft is considered to be the first pulley. The static
pre-tension, T0, as a result of the tensioning mechanism, is measured to be 340.66 N, while the
static tension caused by the load, Load , is equal to 260 N. Tensioner alignment angles are
32
measured to be deg, and deg. A proportional viscous damping with a damping
ratio, of0.8 is assumed to be valid.
Figure 3. A Five-Pulley Automotive Belt Drive System
For the described five-pulley system, the equations of motion for each pulley derived using the
Lagrange method are as follows:
15415152211
2
1111155151
sin rarmbbbbb KlrKrrKrrKrKKIQ
25213322
2
222221212
rbbbb KrrKrrKrKKIQ
32324433
2
333332323
rbbbb KrrKrrKrKKIQ
44543435544
2
4444443434
sin rarmbbbbb KlrKrrKrrKrKKIQ
5555454541155
2
55555454545
sinsin rarmbarmbbbbb KlrKlrKrrKrrKrKKIQ
33
trtarmbarmb
armbarmbarmbarmb
t
KlKlK
lrKlrKlrKlrKIQ
2
5
22
4
2
55445454141566
sinsin
sinsinsinsin
45
4545
If the harmonic force applied to this five-belt pulley system is described by:
0
0
0
0
0
sin tQ
Q
i
(3. 26)
The angular displacement of each pulley, defined as the response of the system is as following:
t
t
t
t
t
t
X
t
5
4
3
2
1
(3. 27)
In more details, the frequency response of the system can be described by:
6
55
44
33
22
11
sin
sin
sin
sin
sin
sin
t
t
t
t
t
t
t
X
(3. 28)
In this study, the above analytical method is used to find the solution to the equations of motion
for the case of the sinusoidal excitation inputs.
34
3.1.4.2 System Properties
The properties of the five-pulley belt drive system, investigated in this study, are tabulated in
Table 1. This table presents the details of the system geometry, and other important physical
properties of the system such as the belt modulus, the effective length and width of the belt, and
etc.
Table 1. Physical Properties of the Prototypical System
Pulley 1
C/S
Pulley 2
A/C
Pulley 3
P/S
Pulley 4
Idler
Pulley 5
Tensioner Idler
Spin axis
coordinates (0,0) (173.25, 16.50) (240.75, 210) (123.00, 125.25) (-24.06,176.72)
Radii
(mm) 80.0 55.7 57.5 35.0 38.1
Wrap
Angle
(deg)
152.81 21.18 219.83 125.48 134.01
Ratio 1 1.4176 1.3863 2.2268 1.9617
Inertia
(kg.mm2)
1×106
5448 2040 200 500
Other
physical
properties
Belt modulus: EA=120000 N
Belt effective Length: lb=1323.52 mm
Belt Width: wb=20.00 mm
Span lengths: l1=106.42 mm, l2=169.49 mm, l3=110.07 mm, l4=135.21 mm,
l5=173.90 mm
The characteristics of the tensioner arm is presented in details in Table 2.
35
Table 2. Tensioner Arm Characteristics
Spin Axis Coordinates (xt, yt) (-114, 180)
Arm Inertia Iarm (kg.mm2) 300
Arm Length larm (mm) 90.0
Arm Angle at reference tr (deg) 357.911
Tensioner spring stiffness Kt (N.m/rad) 69.04
The belt and pulleys system is modeled as a six degree-of-freedom-system, assuming each belt
span as a linear spring with constant damper. From this dynamic model, the natural frequencies
and corresponding mode shapes, and angular displacement fluctuations of each pulley, and the
tensioner arm will be determined. The dynamic model is described in the next section.
3.1.4.3 Numerical solution for general excitation input from the crankshaft
To this end, a multi-degree-of-freedom model for a system (Figure 3) is developed assuming the
input is the fluctuating torque from the crankshaft to the belt drive system. In this model, the
rotational vibrations of the pulleys are assumed to be dominant compared to the transverse
vibrations of the belt. The belts are modeled as linear springs with proportional viscous damping.
As the motion of the crankshaft, , is typically given in practical applications, it is treated as a
specified excitation source in the following forced vibration analysis. The respective torque of
the crankshaft versus the frequency is shown in Figure 3.
36
Figure 4. Crankshaft Torque Signal
Using the Lagrange’s energy equation method, one can derive the mass, damping, and stiffness
matrices. In the next step, natural frequencies of the system are obtained based on analytical
modeling described in section 5.1.
3.1.4.4 System Natural Frequencies and Vibration Mode Shapes
As detailed in section 3.1.1, the natural frequency and the mode shapes are found by solving the
eigenvalue problem and the system inertia, damping and stiffness matrices. The dynamic
analysis shows that the dominant natural frequency of 78.4 Hz will result in dominant vibration
mode. Table 3 presents the comparison of the simulated natural frequencies and those predicted
by the FEAD software. The results show that both simulated natural frequencies and mode
shapes are in a good agreement with the predicted results by the FEAD software.
0 1000 2000 3000 4000 5000 60000
20
40
60
80
100Crank Shaft Torque Input
[RPM]
Torq
ue
[N
.m]
37
Table 3. Natural Frequencies of the System
Frequency Mode 1 2 3 4 5 6
Simulated (Hz) 78.40 165.34 321.83 573.47 781.26
1.59×106
FEAD Software
(Hz)
78.48 165.61 324.01 567.32 805.29 1.60×106
3.1.4.5 Dynamic Response of the System
The responses of pulley angular displacement are obtained in this step. The simulated results are
compared to the predicted results provided by the company using specific software (FEAD),
which was developed in collaboration with the University of Toronto [38]. Figure 5 shows the
angular displacement response of the power steering and air conditioner in the system. As it can
be seen from the graph, there is a good correlation between the frequency responses of the
components.
Therefore, the developed simulation scheme can be used to predict static and dynamic responses
of any typical accessory belt drive system with a good degree of accuracy. Using the applied
loading on the FEAD components, we will proceed to calculate the stress distribution followed
by fatigue analysis of the parts.
38
(a) (b)
Figure 5. Frequency Response Function of the Angular Displacement of (a) Power Steering and
(b) Air Conditioner
3.1.4.6 Dynamic Belt Span Tensions
In this section, the dynamic tensions for each belt span are calculated. The system dynamic
response (obtained in section 3.1.3) was used to estimate the dynamic tension of each belt span.
Table 4 presents the estimated belt span tension.
Table 4. Dynamic Belt Span Tensions
Span Dynamic Span Tension
Peak (N)
C/S to A/C 120 @ 2760 RPM
A/C to P/S 135 @ 2760 RPM
P/S to Idler 124 @ 2800 RPM
0 1000 2000 3000 4000 5000 60000
2
4
6
8
10Angular Displacement of Power Steering
[RPM]
Re
spo
nse [
rad
/ra
d]
FEAD Software
Simulated Response
0 1000 2000 3000 4000 5000 60000
1
2
3
4
5
6Angular Displacement of Air Conditioner
[RPM]
Re
spo
nse [
rad
/ra
d]
FEAD Software
Simulated Response
39
Idler to Ten. Idler 180 @ 2790 RPM
Ten. Idler to C/S 100 @ 2800 RPM
3.1.4.7 Dynamic Hubloads on FEAD Components
The dynamic hubloads are determined using the tensions calculated in section 3.1.4.6. The
following equation shows the relation between the dynamic hubload and dynamic tensions of the
belt spans wrapped around each pulley.
0TTT ii (3.29)
iiiiii TTTTH cos2 1
22
1 (3.30)
where i is the wrap angle of each pulley.
Table 6. Dynamic Hubload at Accessories
Span Dynamic Hubload Peak
A/C 50.4 N at 2760 RPM
P/S 243.6 N at 2800 RPM
Idler 271.4 N at 2790 RPM
CS 152.8 N at 2800 RPM
Tensioner Idler 217.8 N at 2760 RPM
40
3.2 Summary
In this chapter, a dynamic model was developed for the belt drive systems. The purpose of
developing this model was to evaluate dynamic loading that tensioners experience during
service. By taking this approach, the significance of real loading conditions in the fatigue
behavior of the part is considered. In this chapter, details of a multi-degree-of-freedom dynamic
model is given. The model was developed based on the assumption that the input is the
fluctuating torque from the engine to the belt drive system. Using a typical crankshaft torque
input over engine speed, the response of the system was evaluated and confirmed. Further
investigation was performed to explore the belt span tensions caused by the selected torque input
to the system. Finally, FEAD component hubloads, as the result of the tensions generated on the
wrapped belt along the accessory, are calculated. It can be concluded that this mathematical
model can be used for investigation of hubloads on each part of the belt drive system, including
accessories and automatic tensioner.
41
Chapter 4
4 Fatigue Analysis
Excitations from the vehicles power train can lead to fatigue failure of the FEAD components.
Tensioner failure due to cyclic dynamic loadings is one of the most critical conditions that can
lead to catastrophic failure of FEAD systems. Therefore, it is essential to obtain an accurate
estimation of fatigue life for the tensioner components. This chapter presents the fatigue analysis
of automatic tensioners by performing different steps. These results have been published in the
International Journal of Vehicle Performance [39].
4.1 Stress Analysis of Tensioner Using Finite Element Method
In this section the details of the tensioner finite element model for the purpose of an accurate
determination of the local stress distribution are presented.
42
4.1.1 Tensioner Geometry
Generally, a tensioner consists of a base, damper, spring, arm, pulley bearing, pulley, and pulley
center bolt. The schematic of a typical tensioner is shown in Figure 6. In this study, only the
tensioner pulley is considered for fatigue investigation.
Figure 6. Schematic of a Typical Tensioner
The resultant force on the tensioner base is calculated by force and moment analysis of the
assembly. Knowing the hubload and spring load and the geometry of the tensioner, one can find
the resultant force on the tensioner base. Another important reason in considering the tensioner
bracket for the fatigue study is due to the geometry of the tensioner assembly and the fact that the
cracks which cause the tensioner failure initiate in this part of the tensioner. In this work, the FE
model is developed which represents the assembly loading condition. Simplifying the assembly
FE model to the subcomponents FE method reduces the time and therefore cost of long duration
simulation runs. The tensioner assembly used in this work is shown in Figure 7.
43
Figure 7. Tensioner Assembly
In this work, in order to simulate the real fatigue testing conditions, tensioner was constrained
using three bolts. These bolts along with fixed elements which simulate the engine blocks were
generated and added to the solid model using SolidWorks software [40].
4.1.2 Mesh Generation
Finite element analysis is performed on a tensioner base for static and dynamic load analyses. In
this section, the geometry mesh is described in details for the die casting tensioners. Generally,
44
tetrahedral elements are used to mesh imported complex geometries in ANSYS [41]. In this
study, Quadratic Tetrahedral elements are used instead of linear tetrahedral elements to mesh the
tensioner spindle geometry. Quadratic Tetrahedral elements reduce the rigidity of the geometry
and increase the accuracy level of the model [42]. To achieve the quadratic tetrahedral elements
the patch conforming mesh method is used.
From a grid sensitivity study, a 382021 element mesh could provide a reasonable convergence
and was used for all simulations presented in this work. Figure 8 shows the meshed geometry of
the die cast tensioner with 382021 elements.
4.1.3 Boundary Conditions
A cyclic force with constant amplitude was applied on the tensioner spindle head (as shown in
Figure 8) to simulate the real working conditions of tensioners in FEAD systems. In FEAD
system, the tensioner is mounted on the engine using bolts. In his study, the tensioner is fixed
using three bolts to reflect the tensioner real operating conditions, as shown in Figure 8 . The bolt
heads were fixed using fixed support elements in the FE model. Since the applied boundary
condition at bolt locations (i.e. bolt pretension or clamp load) significantly affects the stress
distribution developed in the tensioner, they need to be considered in order to achieve accurate
results. Tightening torque applied to the bolt during installation causes clamp load. The amount
of clamp load generated in the bolt was calculated using:
KDPT (4.1)
45
where T is the tightening torque, K represents the coefficient of friction (0.20 for dry and 0.15
for lubricated joint), and D and P are bolt nominal diameter and clamp load, respectively.
Knowing the tightening torque, bolt size, and the friction coefficient, the clamp load was
estimated.
In this analysis, a clamp load of 21875 N was used which reflects the installation clamp load
produced due to the tightening torque applied on each M8 bolt. To account for these clamping
loads, pretension elements were used on the bolts in the simulation. Figure 9 shows the direction
of the load and other boundary conditions applied to the tensioner spindle in this study.
Figure 8. Meshed Geometry of the Die Cast Tensioner with 382021 Elements
Force and moment analysis of the tensioner assembly shows how the hubload and spring load are
transferred to the tensioner base. This investigation (details in section 4.1) shows that the
tensioner spindle operates under the cyclic loadings applied to the head of the spindle tower.
46
To find the critical region of the component, different range of loading reflecting the real
operation environment need to be investigated. In this work, various hub load values applied on
the tensioner are simulated using the finite element model developed, as shown in Figure 9. The
location on which the loading was applied in the present finite element model represents the
service loading condition of the tensioner spindle.
(a) (b)
Figure 9. Tensioner Base Schematic Showing the Boundary Conditions
4.1.4 Finite Element Analysis Results and Discussion
The considered tensioner in this work is made of die cast A380 aluminum alloy. The monotonic
material properties of die cast A380 aluminum alloy are presented in table 1. As nonferrous
F
Clamp load
Engine blocks
13mm
F
47
metals do not exhibit fatigue limit, cycles to fatigue limit for these materials are assumed to be at
108 cycles [43].
Table 5. Mechanical Properties: Die Cast A380 Aluminum Alloy [44]
Ultimate Tensile Strength 320 MPa
Yield Strength 165 MPa
Elongation to Failure 3.5%
Modulus of Elasticity 71 GPa
Density 2.71 g/cm3
Fatigue Strength 140 MPa
The equivalent von-Mises stresses of the tensioner were predicted through ANSYS Workbench
software [41] as shown in Figure 5. The results showed that the base of the spindle experiences
the highest stresses during the service life of the tensioner. The maximum equivalent stress
obtained from the tensioner FE model can be observed in Figure 10. As it is shown in Figure 10,
the maximum equivalent stress (114.9 MPa) occurs when the load of 3570 N was applied on the
spindle head.
48
Figure 10. Stress Distribution in Tensioner Spindle
Table 6 presents the generated equivalent stress and strain at the critical location of the spindle
when different loads were applied on the tensioner. As the force applied to the head of the
spindle increases higher stress is generated at the critical location. The stresses obtained from the
finite element model were used for further investigation of fatigue life of the tensioner after the
model is validated.
Table 6. Equivalent Stress and Strain at the Fillet Area
Force (N) Maximum equivalent stress (MPa) Maximum equivalent strain
5300 160 2.414e-3
4700 148 2.145e-3
4100 122 1.871e-3
49
3570 114 1.629e-3
4.1.5 FEA Validation
In fatigue analysis of components, inaccuracy of predicted stress and strains are considered to be
a potential cause of error. As a result, it is critical to validate the FE model in this work. In order
to validate the FE model, strains predicted by the simulation and strains measured from strain
gauges in the physical testing are compared.
In this study, the finite element model was validated through a series of experiments using strain
gauges. Experiment was conducted on the full tensioner assembly. However, the FE model was
only developed on the base part of the tensioner. Therefore, in order to precisely confirm the
model, the resultant force reflecting the experimental loading conditions was used in the finite
element model. This resultant force represents the actual loading on the spindle during the
experiment. This force is transferred to the tensioner base from the tensioner pulley is calculated.
The resultant force is calculated using the force and moment balance on the parts. The details of
the force calculation are presented in this section.
For this purpose, a static analysis is performed, deriving the hubload on the tensioner base.
Figure 11 shows a schematic of the studied tensioner.
50
(a) (b)
Figure 11. Tensioner Schematic (a) x Direction, and (b) z Direction
The system is analyzed part by part and step by step force analysis using the free body diagram
of each section. Writing force-balance and moment-balance equations based on the free-body
diagram of the section leads us to accurate determination of the force exerted on the tensioner
base.
The following shows the derivation of the forces transferred to part 2, the tensioner arm, from the
tensioner idler pulley (as shown in Figure 12). By summing the forces acting on the tensioner
pulley, force balance relationship:
0 xFxxxx
TTFF21211
0 yFyyxy
TTFF21211
51
2
1
2
11 yxFFF
where T1 and T2 are the tensions of to the belt spans wrapped along the tensioner pulley.
Figure 12. Free-body-diagram of the studied tensioner
By summing the forces acting on the tensioner arm, force balance relationship, and setting them
equal to zero, the resultant force on the tensioner arm is calculated. As mentioned earlier, this
resultant force is transferred to the tensioner base. The following force balance equations shows
how this resultant force is calculated:
52
0 xF cos.cos.1 SRx
FFF
0 yF sin.sin.1 SRy
FFF
The tensioner is designed in a way that the location on which the spring load is applied remains
in line with the hubload. In other words, when the hubload is changed, the arm rotates and the
spring load angle with respect to the arm remains constant. The design is made to meet
180 .
Figure 12 shows the free body diagram of the studied tensioner. In this figure, d1 is the distance
between the pulley centre and the reference face of the tensioner. ds is the distance between
where the spring load is applied and the reference face. Finally, d2 represents the distance on
which the resultant force of the hubload and spring load is applied with respect to the reference
face.
Writing the moment balance in x-z plane leads to:
0M 0..cos..cos.211
xRxss
dFdFdF
Moment balance in y-z plane leads to the following:
0M 0..sin..sin.211
yRyss
dFdFdF
Considering that 180 , 222
dddyx .
53
The strain gauge experiment is conducted on the full assembly. In this experiment, the load is
applied on the tensioner pulley. Therefore, the resultant force caused by the hubload and spring
load on the pulley was calculated. This resultant force was then applied to the head of the base at
distance of d2. The loading in the finite element model was stimulated to reflect the resultant
force on the tensioner base.
4.1.5.1 Experimental Set-up
The finite element model is validated through a series of experiments using strain gauges. The
experimental data was provided the industrial partner, Litens Automotive Group. Three rosettes
were positioned on the inside diameter of tensioner spindle. The strain gauge locations can be
observed in Figure 13. Figure 14 shows the orientation of three strain gauges in the rectangular
rosettes. It should be noted that in the validation process of the FE model, the strain gauge
locations and magnitudes of the applied loads on the tensioner were set to result in strains in
elastic range.
Three independent strains from gauges, (εa, εb, and εc) were measured at specific time, force and
displacement. The normal strains in x and y direction, and the shear strain between x and y axes
(εx, εy, and γxy) were calculated from the measured strains using the strain transformation
equations. The equivalent von-Mises strains were found at each rosette location.
54
Figure 13. Strain Gauges Placed on the Tensioner Spindle
The measured strains were then compared with the estimated strains from the finite element
model. Table 7 presents a comparison between these results for rosette locations 1 and 3. As it
can be observed from this table, an acceptable percentage error between the experimental and
simulated results was achieved at different applied force and locations. Similar trends were
observed for the rosette location 2.
Figure 14. Gauge Orientation in Rectangular Rosette
Rosette 1
Rosette 2
Rosette 3
116˚
127˚
117˚
Gauge b
Gauge aGauge c
y
x
45°
45°45°
55
4.1.5.2 Results
By measuring the three independent strains from gauges in a rectangular rosette, the equivalent
strains were calculated for each location. The measured strains were then compared with the
estimated strains from the finite element model. Table 7 presents a comparison between these
results for rosette locations 1 and 3. As it can be observed from this table, an acceptable
percentage error between the experimental and simulated results was achieved at different
applied force and locations. Similar trends were observed for the rosette location 2.
Table 7. Comparison of Measured and Simulated Strains from the FE Model
Force (N)
Equivalent strain at R1 (ε) Error Equivalent strain at R3 (ε) Error
Experimental FE % Experimental FE %
142.578 1.55e-05 1.53e-05 1 2.97e-05 2.59e-05 13
180.054 1.90e-05 1.93e-05 -1 3.57e-05 3.26e-05 9
203.735 2.19e-05 2.18e-05 1 3.56e-05 3.69e-05 -4
293.091 3.04e-05 3.14e-05 -3 5.22e-05 5.31e-05 -2
330.566 3.36e-05 3.54e-05 -5 6.02e-05 5.99e-05 0
373.901 3.96e-05 4.00e-05 -1 6.51e-05 6.78e-05 -4
416.016 4.22e-05 4.45e-05 -6 7.61e-05 7.54e-05 1
484.497 5.44e-05 5.18e-05 5 8.52e-05 8.78e-05 -3
534.546 6.15e-05 5.72e-05 7 9.73e-05 9.69e-05 0
56
533.813 6.09e-05 5.71e-05 6 8.99e-05 9.68e-05 -8
Figure 15 shows a comparison between the present finite element model and the experimental
data obtained from the strain gauges. In this figure, the strain at rosette locations 1 and 3 versus
the applied load are presented. As it can be seen, the comparison shows a good agreement
between the simulation results from the FE analysis and the measured data.
(a) (b)
Figure 15. Strain Versus Force at Rosette Location (a) 1 and (b) 3
4.2 Fatigue Behaviour and Life Predictions
In this part of the study, the life assessment of the tensioner is performed using strain-life criteria,
followed by Smith-Watson-Topper (SWT) mean stress correction theory [45]. Results obtained
57
from section II were used to estimate the tensioner life. In addition, an analytical method to
predict the fatigue life of the tensioner is described.
4.2.1 Strain Life Approach
Fatigue life methods attempt to assess the life of the material in terms of number of cycles to
failure, N. Generally, low-cycle fatigue condition is considered for the materials life when 1 ≤
N≤ 103, while high cycle fatigue is defined when N is higher than 10
3. In the stress-based method
[46], the material is assumed to have only elastic deformations, and local plasticity is neglected.
Thus, the stress-based method is applicable to the extent that the above assumptions are valid.
This method is accurate only for the case of high-cycle fatigue, where the applied load results in
only elastic deformation of the component. Although, most engineering components are
designed such that the material remains elastic under applied nominal loads, plastic strains are
often generated due to fatigue in the critical regions [1]. Additionally, the local strain-life
approach is preferred if the loading history is irregular and where the mean stress and the load
sequence effects are thought to be of importance. On the other hand, a strain-life method,
developed for analyzing low-cycle fatigue data, has proved to be useful for the analysis of the
high-cycle fatigue data as well [35][47].
In this study, the strain-life method based on Ince et al. study [35] was employed in order to
address the fatigue concerns due to elastic (εe) and plastic (εp) strains. The general fatigue-life
relation can be expressed in terms of the strain amplitude (εa) as:
58
c
ff
b
f
f
pa
pea NNEE
22
(4. 2)
where
εa cyclic strain amplitude
εe elastic strain component
εp plastic strain component
σa cyclic stress amplitude
σf regression intercepts named fatigue strength coefficient
εf regression intercepts named fatigue ductility coefficient
b fatigue strength exponent
c fatigue ductility exponent
Nf number of cycles to failure
Since Eq. 1 is a general form of fatigue-life equation, which considers both elastic and plastic
strains, it can be used in design and analysis under both stress-life and strain-life approaches.
Fatigue parameters σf, εf, b and c are the materials properties, and they were determined
empirically from experimental data of σa, εp and Nf in log-linear forms of Basquin equation (high
cycle fatigue), and Coffin-Manson equation (low cycle fatigue) [48].
4.2.2 Effect of Mean Stress
Since the materials fatigue strength is largely dependant of applied mean stress, mean stress
correction theories were considered in this study for the fatigue assessment. Therefore,
59
depending on the fatigue behaviour of the components, the Morrow mean stress correction [35]
or Smith, Watson, and Topper (SWT) stress correction method [36] need to be used to modify
the strain-life curve and to account for mean stress. These two theories of mean stress corrections
are as follows
Morrow [35]:
1
f
m
ar
a
(4. 3)
SWT [36]:
aar max
(4. 4)
In these mean stress correction theories, the transferred stress amplitude is calculated by [36]:
R
R
aar
1
2
2
1
max
(4. 5)
where R refers to the applied loading or stress ratio, maxmin R . The SWT model is extensively
used for the strain based fatigue analysis. Dowling et al. [45] found that the SWT stress
correction theory is more conservative and to correlate better results in case of nonferrous alloys
when compared to Morrow correction theory. Hence, this method was applied in this study for
further investigation of fatigue life of the die cast Aluminum alloy tensioner.
60
The tensioner spindle strength is determined in two steps of obtaining the nominal alternating
and mean stresses. These factors result in an equivalent alternating stress (uniaxial stress).
Equivalent alternating stress is then compared with the fatigue strength of the selected tensioner
material i.e. the selected E-N curve for selected material. This comparison predicts tensioner
spindle lifetime in terms of number of cyles to failulre. The alternating and mean stress can be
determined from the equations 21max Ra , and 21max Rm . Following calculation of the
dynamic and mean stress at the critical location in the component, one can use the following
extension of the SWT equation to account for the mean stress [35].
cb
fff
b
f
f
a NNE
22
2
2
max
(4. 6)
In this method, the SWT parameter, a max , is the product of total strain life relationship
(Equation 4.1) multiplied by Basquin equation, bffa N2 , [46]. The above-mentioned
extension of the SWT parameter accounts for both the mean stress effects and the strain
amplitude and can be applied in strain-life calculations [47].
4.3 Tensioner Fatigue Parameters
This section explains the testing apparatus employed in the fatigue testing of the tensioner.
Fatigue tests were conducted at room temperature with a frequency of 30 Hz, and various cyclic
loadings with nonzero stress ratios. During the experiment, the tensioner was fixed using three
bolts, and the load was applied on the head of the spindle to simulate the tensioner working
conditions, as shown in Figure 9. The following material properties were calculated using the
61
conducted tests as described later in this section: Fatigue strength coefficient, fatigue strenght
exponent, fatigue ductility coefficient and ductility exponent.
As detailed earlier, the finite element model was developed to calculate the stresses in the
tensioner. The FE model was carried out for various loading conditions applied to the tensioner
in the experimental test rig. The test loads and the resultant stress and strain amplitudes can be
combined to estimate the fatigue parameters of the tensioner. In this testing platform, fatigue
testing equipment was incapable of performing both tension and compression within the same
cycle. In other words, passing through zero value of load induces problem for the control system.
Therefore, the loading range of nonzero value to a maximum force was used for the experiment.
The results of the fatigue testing, which consists of constant mean, dynamic load applied, and
numbers of cycles to failure for the Ford Puma tensioner are presented in Table 8.
Table 8. Results for Standard Die Cast Tensioner at Ambient Conditions
Loading Case No.
Mean Load
(N)
Dynamic Load
(N)
Cycles to Failure
(Nf)
1 2700 2600 759,924
2 2400 2300 1,699,290
3 2100 2000 3,139,414
4 2100 1470 6,114,359
62
The Least Square (LS) optimization method [49] was used in this study to estimate the tensioner
fatigue parameters from the number of cycles to failure measured in testing. An optimization
with the concern of reducing the mean error between the predicted and mesaured life was
performed according to the method described in Figure 16. In optimizaing the fatigue
parameters, a range of initial estimated values were set for the fatigue coefficients. Subsituting
these set of parameters into equation 2, the new final fatigue parameters were identified based on
the minimum total error scheme. The fatigue parameters obtained are given in Table 9.
63
Durability Machine No. of Loading Conditions, N
Measured Cycles to Failure for N Loading
Conditions, Bf1, …, BfN
Create FEA Tensioner Model
Tensioner Equivalent Stress
Obtain Initial Estimated for Fatigue
Parameters
Fatigue CoefficientsRange
σf: σs to σe ; b: bs to be εf: εs to εe ; c: cs to ce
Step Sizesσ, sb, sε, sc
Calculate Cycles to Failure for
N Loading Conditions, Cf1, …, CfN
Calculate Mean % Error Between
Predicted, Cf’s and Measured Life, Bf’s
Are all Range Values
Done
Obtain Final Fatigue Parameters for Minimum Error
Figure 16. Tensioner Fatigue Parameters Derivation Flow Diagram
Table 9. Fatigue Parameters for Die-Casting Al Alloy Tensioner
f B f c
650 (MPa)
Following the determination of the fatigue parameters of the tensioner spinlde, the strain life data
can be predicted based on equation 5. Figure 17 presents the developed strain life curve for the
die cast aluminum tensioner. In this work, a set of specified stress ratios were used to examine
64
the tensioner life. In each case, cycles to failure of the tensioner was predicted based on the SWT
method described in section III. As it is shown in the figure, the lower stress ratio leads to a
longer fatigue life for the tensioner. The figure also shows how the results from the analytical
approach are in a very good agreement with the experimental data from the fatigue testing. The
relatively small percentage error confirms the reliability of the approach used for this analysis.
Figure 17. Tensioner Spindle Fatigue Data (Analytical and Experimental SWT Parameter)
4.4 Fatigue Behaviour at Critical Regions
The materials stress-strain behaviour during a fatigue test is predicted in form of a hysteresis
loop. The material stabilizes its deformation behaviour by passing the initial transient phase and
the similar hysteresis loop is obtained during each loading cycle. Each Strain range in an
experiment will have a corresponding stress range that can be measured. The cyclic-stress strain
curve can be obtained by plotting all of this data points [50].
65
The cyclic stress-strain curve explains the behaviour of the material after deformation occurred
in service for a few times, whereas the traditional stress-strain curve (monotonic stress-strain
curve) explains the material behaviour as it was first manufactured. A simple power function,
shown in Equation 6, explains the relationship using three material properties; cyclic strength
coefficient ( K ), cyclic strain-hardening exponent ( n ), and elastic modulus ( E ).
n
KE
1
222
(4. 7)
As the experimental data for measuring these parameters were not available, the cyclic strain-
hardening exponent was predicted using the fatigue strength exponent (b) and ductility exponent
(c) according to Morrow theory: cbn . Following the Morrow theory, the cyclic strength
coefficient was also determined by nffK
[51] for the A380 die cast tensioner.
To validate the estimated values for the cyclic strength coefficient and strain-hardening exponent
of die cast A380, the results are compared to the published data from the literature, that is A413
[50] and A356 [50]. As shown in Figure 18, the results from the current study show that A380
has higher cyclic strength coefficient when compared to A413, while the contrary behaviour can
be observed for the case of A356. Furthermore, A380 shows higher cyclic strain-hardening
exponent than both A356 and A413. These results are also tabulated in Table 10.
66
Figure 18. Estimated Cyclic Stress-Strain Curve for A380 Compared with the Published Results
for A356 and A413
Table 10. Comparison of Cyclic Mechanical Properties of A380, A356 and A413
A380 [Present Study] A356 [50] A413 [50]
K 655 (MPa) 676 (MPa) 128 (MPa)
n 0.156 0.137 0.028
E 71 (GPa) 72 (GPa) 66 (GPa)
4.4.1 Stress Concentration Factor
Generally, the maximum stress experienced on a localized region of a component, caused by a
large stress gradient is called stress concentration. Stress concentrations are often caused by
discontinuities in geometry or materials properties and/or forces on the contact areas. In this
67
work, the changes in geometry are the main cause of the stress concentration, particularly on the
notch area on the base of the spindle.
The stress concentration factor in the elastic region is defined by:
nom
tK
max
In this work, two methods are used to estimate the fatigue concentration factors at the critical
area of the tensioner pulley. First method uses the FEA model to investigate the stress gradients
and consequently the stress concentration factor. Second method uses analytical approach to
estimate the stress concentration factor. The factors obtained from both approaches are then
compared.
4.4.1.1 Stress concentration factor estimation using finite element analysis
Finite element analysis is used as the commonly used method for estimating the maximum stress
applied on the tensioner. Table 11 presents a list of maximum stress experienced at the base of
the spindle under different loading conditions.
Table 11. Maximum Equivalent Von-Mises Stress Corresponding to Different Applied Forces at
the Critical Area of the Tensioner
F (N) σmax (MPa)
3570 114
4100 122
4700 148
5300 160
68
Thereafter, the nominal stress is determined using the stress gradient. The distance from the
centre of the force applied and the area of maximum stress (start of the big fillet) is found to be
48.11mm. As for the case of pure bending, the nominal stress is calculated using the
corresponding maximum stress experienced by the bending moment:
nomI
MY max (if tK did not exist)
8
434
1073.94
1076.18
4
r
I
114 MPa
Figure 19. Stress Contour Showing the Max Stress at the Critical Area When Applying
3570N Load
69
MPaPanom 115.3310115.33
1073.9
1076.181011.483570 6
8
33
In order to calculate the stress concentration factor, a plot of the stress with respect to the
distance to the neutral axis of the spindle is needed. In other words, the stress gradient is to be
calculated for the purpose of estimating the nominal stress and hence the stress concentration
factor.
For this purpose, the stresses developed on the path of the neutral axis to the notch area are
obtained from the finite element analysis. Figure 21 shows the path on which the stress gradient
is calculated.
Figure 20. Path Created for the Stress Concentration Estimation
70
The stress generated in the path was then simulated using the validated finite element model. The
results are shown in Figure 21. The stress contour shows how the stress increases when
travelling from point 1 to 2 on the path. More details of the changes in the stress throughout the
path are given in Figure 22. Next section illustrates how these results can be used to estimate the
stress concentration factor.
Figure 21. Stress Gradient Generated on the Selected Path
To find the stress gradient on the critical part of the spindle, the area under the pure bending
stress gradient is calculated by:
eutralAxisDistnceToNAkt max21
eutralAxisDistnceToNAkt 2max
71
Using path operations and integration, the area under the stress-length curve of the shoulder with
circular shaft is:
51039.4 ktA m2
MPaPaeutralAxisDistnceToNA ktnom 9.5227.52891566)1066.1/(1039.42225
max
Figure 22. Stress versus Distance to the Neutral Axis
Therefore:
00.299.176151)()( max MPaMPaFEAFEAK nomt
The above calculation explained how the stress concentration factor is estimated through the
FEA analysis along analytical methods. This factor is found to be 2 at the hot spot of the
tensioner spindle.
σmax=151 MPa
σnom=76 MPa
0
20
40
60
80
100
120
140
160
0 0.005 0.01 0.015 0.02
Stre
ss (
MP
a)
Distance to Neutral Axis (m)
FEA
Analytical
72
4.4.1.2 Stress Concentration Factor Estimation Using Roark’s Analytical Model
One can find the stress concentration factor using analytical methods. Assuming the spindle as a
square shoulder with fillet in circular shaft under bending conditions and elastic stress, the
following equation is used to find the maximum stress experienced at the critical area [52].
3max
2
32
hD
MK t
, where3
4
2
321
222
D
hC
D
hC
D
hCCK t
where
There are two fillets at the notch area of the tensioner (Small fillet mmr 25.02 , big fillet
mmr 25.010 ). However, the important fillet is the small fillet with the radius of 25.02 mm, as
this is where the highest stress is found in the FE analysis. The following analytical approach in
calculating the stress concentration factor is based on this value for the notch radius.
Therefore, having the spindle height and internal diameter of 13.04 mm and 63.50mm,
respectively, the value of 52.6rh is calculated. The required coefficients in equation … are
calculated as:
0.225.0 rh 0.200.2 rh
C1 rhrh 086.0149.1927.0 rhrh 010.0831.0225.1
C2 rhrh 837.0281.3015.0 rhrh 257.0958.0790.3
C3 rhrh 506.0716.1847.0 rhrh 862.0834.4374.7
C4 rhrh 246.0417.0790.0 rhrh 595.0046.3809.3
73
Hence,
159.250.63
04.132089.0
50.63
04.132651.0
50.63
04.132019.3282.3
32
tK
The error between the calculated stress concentration factor using the FEA results and Roark’s
formula is as following.
100/% formulaKFEAKformulaKError ttt
%36.7100159.2/00.2159.2% Error
The result of the numerical approach in stress concentration factor calculation shows a good
agreement with the results of the analytical Roark’s approach. Hence, the FEA model can be
used for further calculation of the stress concentration factor for the critical area of the notch.
4.4.2 Plasticity Correction of Linear Elastic FEA Stress and Strain Data
When the applied load on a component critical area goes above the materials elastic limit, the
stress developed on the part changes from when the stress is within the elastic range. In this case,
the formula presented by Neuber [53] considering both stress and strain concentrations is more
accurate for the design of the component against fatigue. Neuber [53] defined an effective stress
concentration factor and an effective strain concentration factor as to account for the stress and
strain concentrations. These factors are defined as:
C1: 3.282
C2: -3.019
C3: 0.651
C4: 0.089
74
nom
K
max
nom
K
max
One significant factor, which needs to be considered when designing mechanical components
subjected to fatigue loading, is the durability design of the component so that the real local
stress-strain behavior of the component is predicted at the component critical area. The finite
element analysis, however, normally is performed under elastic conditions and does not account
for plasticity at the critical area of that specific part. Different methods can be used to transform
the elastic stress-strain data to elastic-plastic stress-strain data based on the conditions.
In the present study, the Neuber method modified by Topper et al. [54] for fatigue analysis was
used to account for localized plasticity at the hot spot of the component. This modified Neuber
correction theory can be expressed by:
nomnomtK 2
maxmax (4. 7)
where Kt refers to the fatigue concentration factor. In this analysis, the nom is found using the
materials elastic behavior and the nominal stress.
The following three-step approach is used in this work:
1. First, the stress-strain behaviour of the component is computed at the critical region,
using the elastic FEA.
75
2. Second, the energy, which equals to the product of the elastic stress and elastic strain, is
calculated.
3. Lastly, the cyclic stress-strain behaviour of the materials at the critical area is estimated
using the materials stabilized stress-strain.
Thereafter, the Neuber parameter, 21E , was estimated through combining the cyclic stress-
strain curve and strain-life curve. The Neuber parameter, derived from equation 4.7, equals to
nomtK . In other words, this method was used to scale the elastic stresses generated from FE
method and to provide more realistic results for the fatigue life prediction at the critical regions.
Figure 23 shows how the tensioner life is changed when the Neuber parameter increases. In other
words, this graph relates the fatigue life of the tensioner to stress concentration factor. Therefore,
it can be used in life prediction of tensioners with the same particular material, but different
geometry and hence different concentration factor.
Figure 23. Neuber Parameter-Reversals to Failure Curve for Die Cast A380 Tensioner
76
In the case of A380 Al alloy tensioner, the reversals to failure corresponding to each Neuber
parameter value is calculated based on equation 4.1. The plot shown in Figure 23 explains the
degree of life dependence on the stress concentration factor at the critical area.
4.5 Summary
A comprehensive fatigue model was developed in this chapter. The model was developed by
predicting the stress generated in the tensioner bracket as it undergoes loading. The linear elastic
FE model was validated by experimental data using strain gauges installed on an area close to the
critical area. After finite element model validation, this elastic model was used along with the
fatigue testing data to extract the fatigue properties of the die cast aluminium material. To
develop an accurate fatigue model, the strain life approach was used. The developed model was
followed and modified by Smith-Watson-Topper (SWT) mean stress correction theory. Finally,
the Neuber method modified by Topper et al. [54] for fatigue analysis was used to account for
localized plasticity at the critical area of the component. As a results, the reversals to failure
corresponding to each Neuber parameter value was calculated and presented in this chapter.
77
Chapter 5
5 Tensioner Design Optimization
The new regulations have made automotive industries to seek light weight components in order
to reduce fuel emissions. As a consequent Original Equipment Manufacturers (OEM) are keen to
use tensioners with optimized designs while not compromising the required characteristics. The
Response Surface Method (RSM) is a combination of statistical and mathematical techniques to
approximately model and analyze the response of the system when different design parameters
are used. This method focuses on the design enhancement in order to obtain an extended fatigue
life. The objective of the response surface methodology is to optimize the response (output)
reflecting changes in the independent design variables (inputs). This process is performed after
the range of the input variables are selected carefully. The series of experiments (simulation
runs) are designed to show the influence of the input variable change on the output response.
The response surface method application reduces the cost in case of expensive analysis methods
such as finite element method and the numerical noise associated with the expensive analysis. A
78
design enhancement solution using the possible design variables at the critical regions needs to
be investigated.
In this chapter, effects of different design variables including geometrical parameters and
materials are investigated.
5.1 Effect of Material on Tensioner Life
Reduction in the automotive weight has a significant impact on the fuel consumption and
consequently the CO2 emission. Researchers have been working to find a light weight material
suitable for the automotive industry. The automotive structures are often subjected to dynamic
loads and high temperature in service.
High-pressure die casting (HPDC) magnesium alloys are good candidates to replace the
aluminum alloys in automotive structures. Some of the properties that make HPDC magnesium
alloys attractive for the automotive applications include high strength or stiffness to weight ratio,
damping capacity, diecastability and the possibility of integrated designs [55].
The materials suitable for automotive applications need to meet significant requirements. In the
selection of material suitable for the powertrain components, different requirements need to be
considered including creep and fatigue resistance, tensile, compressive properties, corrosion
resistance, diecastability and cost of the production.
Until recently, the use of the magnesium alloys as the automotive components has been limited
due to two main drawbacks of these materials; the higher cost of magnesium compared to
79
aluminum alloys and lack of a competitive HPDC magnesium alloy with close properties to
HPDC aluminum alloys such as A380.
Throughout the past few years, major automotive manufacturers started searching for a new
creep-resistant HPDC magnesium alloys as substitutes of HPDC aluminum alloys for powertrain
components due to the concern of weight reduction and higher fuel efficiency. Replacing the
current aluminum alloy power train components with the high performance magnesium based
alloys has a significant impact on the automotive weight reduction.
DSM1 has introduced two new creep resistance HPDC magnesium alloys, MRI 153M and MRI
230D, which can be considered as candidates to replace the commonly used A380 aluminum
alloy in automotive applications [55].
i. MRI 153M
MRI 153M is a low cost candidate for power train components when the service temperature of
up to 150˚C and loading condition of up to 80 MPa are experienced. This Beryllium free alloy,
has superior castability capabilities due to the high content of Aluminum, 8 wt.%. The superior
tensile, compressive and fatigue resistant properties at room temperature are also due to the high
aluminum content of this alloy [55].
This inexpensive creep resistant alloy shows evidence of excellent properties, which makes it a
good alternative for the power train applications. To name a few, corrosion resistance, die
castability and same or better mechanical properties at ambient conditions when compared to the
commonly used traditional Mg alloys, AZ91D. The MRI 153M creep resistance property is
80
considerably higher than that of the commercial alloys at 130-150˚C under stresses of 50-80MPa
[55].
ii. MRI 230D
MRI 230D exhibits excellent properties when diecastability, high strength, corrosion and creep
resistance are considered in an application. As examples of the possible applications of this alloy
due to its superior properties, the engine blocks, transmission housings, bed plates and
converting stators can be mentioned.
The Ca content in MRI 230D is higher than that of MRI 153M. In MRI 230D a continuous
network is made between the lower secondary dendrite arm spacing (SDAS) and the
intermetallic compounds. This network is a result of higher Ca concentration compared to MRI
153M. Hence, the alloy exhibits more stability at higher temperatures (up to 190C) and stress
conditions (70-100 MPa) [55].
When compared to all new developed and commercially available magnesium alloys, this alloy
exhibits other superior properties such as adequate diecastability capability, higher tensile and
compressive yield strengths. Furthermore, the higher corrosion resistance level of this alloy
makes it a good candidate to replace the components which currently use A380 aluminum alloy
[55].
iii. Comparison of MRI alloys properties with A380 alloy
Table 13 shows a summary of mechanical properties of MRI 153M (Mg-Al-Ca-Sr Based alloy)
and MRI 230D (Mg-Al-Ca-Sr-Sn Based alloy) alloys compared to the conventional AZ91D
81
(Mg-Al-Zn), AE44 (Mg-Al-Mn-Si) Mg alloys and A380 Al alloy. The creep resistant AE44,
developed by Dow Magnesium in early 70’s, is currently used for Z06 engine cradle [55]. Other
examples of recent automotive applications of the creep-resistant Mg alloys are use of DSM
alloys in VW, AS series in transmission case, and AJ alloys in some BMW engine blocks.
Table 12. Chemical composition of AZ91D, AE44, and A380 alloys
U.S. ASTM %Al %Zn %Mn %Si
%Fe %Cu
Max
%Ni
Max
others
each
%Mg %Sn Rare
Earth
AZ91D 8.3-9.7 0.35-1.0 0.15-0.50 0.10 0.005 0.030 0.002 0.02 Balance -
A380 Balance 3.0 0.50 7.5-9.5 1.3 3.0-4.0 0.50 - 0.30 0.35
AE44 4.15 - 0.39 0.03 - - - - Balance - 4.01
Table 13. Mechanical Properties of MRI Mg Alloys, AZ91D Mg, and A380 Al Alloy
Properties MRI 153M [55] MRI 230D [55] AZ91D [56] AE44
Ultimate Tensile Strength (MPa)
20˚C
150˚C
175˚C
250
190
172
245
205
178
240
157
138
240
162
150
Yield Strength (MPa)
20˚C
150˚C
175˚C
170
135
125
180
150
145
160
108
89
136
115
110
Elongation to Failure (%)
20˚C
150˚C
175˚C
6
17
22
5
16
18
3
23
21
11
19
25
Modulus of Elasticity (GPa) 45 45 45 45
Density (g/cm3) 1.82 1.82 1.81 1.82
82
Fatigue strength (MPa) 120 110 100
Sources to elastic modulus: Hydro Magnesium September 2005, ASM Handbook; Magnesium & Magnesium
Alloys, Dead Sea Magnesium, Noranda Magnesium Data The mechanical properties of a die cast alloy depend
strongly on the fabrication variables involved, as well as on the alloy composition. *Hydro Magnesium nm: not
measured
Similar tensile and compressive yield strengths are shown for MRI and A380 alloys at
temperatures between 20˚C and 175˚C. The combination of tensile yield strength and ductility
properties of A380 aluminum alloy is inferior to that of MRI and AE series alloys. However, the
creep resistance of the MRI 230D is on the same level as the A380 at temperature range of 100-
180˚C and stresses of 70-110 MPa. It is also noticed that the AE44 shows lower creep resistance
than other alloys at 100˚C temperature under 110 MPa stress value [55].
The resistance of the material to release the stress under the bolts, called bolt load retention
capability, is another key property that needs to be investigated for automotive applications. Loss
of clamp load and/or oil leakage is caused by stress relaxation under the bolts. Hence, it is
important to study the bolt retention capability of materials used in power train components. The
test results from literature (Table 14) shows that MRI alloys have superior bolt retention property
compared to AZ91D at 150˚C and 175˚C for 200 h, while they show inferior characteristic
compared to A380 alloy at same conditions. However, the bolt retention capability of 50-75%,
exhibited by the MRI alloys at room temperature, meets the requirement for reliable use of the
alloy under 150-175˚C conditions [55].
83
Table 14. Comparison of Bolt Retention Percentage of MRI, AZ91D Mg Alloys with A380 Al
Alloy [55]
Property MRI 153M MRI 230D AZ91D A380
Retained stress at room temperature
after 200 h tetsing at (%)
150˚C/ 50 MPa
150˚C/ 70 MPa
175˚C/ 70 MPa
51
48
-
76
73
67
25
19
-
95
92
79
Corrosion resistance under service conditions, as another important factor in automotive
applications, needs to be investigated. The corrosion performance of the MRI alloys is compared
to that of AZ91D and A380 alloys. Table 15 summarizes the corrosion rate of these alloys
obtained from the two commonly used corrosion tests GM 9540P and SAEJ2334. As it can be
seen from the table, the corrosion resistance of the MRI alloys is lower than AZ91D, but higher
than A380 [55].
Table 15. Comparison of Corrosion Resistance of MRI, AZ91D Mg Alloys with A380 Al Alloy
[55]
Property MRI 153M MRI 230D AZ91D A380
Creep resistance
(Mils/year)
ASTM B117 (240 h)
GM 9540P (40 days)
SAEI2334 (8- days)*
7.2
1.01
0.72
7.9
1.25
0.72
7.2
1.1
0.56
15.6
4.59
-
84
* Equivalent to 5 years of real world test
Fuel consumption reduction resulted from weight reduction is a new topic that automotive
manufacturers are investigating. Considering the high tensile and compressive yield strengths at
room temperature and also at short term elevated temperatures, the use of HPDC MRI alloys as a
promising alternative to A380 alloy in tensioner is investigated in this work.
Alloys performance required for the automotive power train applications are defined as:
Creep resistance (tensile and compressive) up to 175˚C (min creep rate)
Bolt-load retention up to 175˚C (50% min)
Metallurgical/thermal stability
Tensile yield strength up to 175˚C (100 MPa)
Fatigue resistance (fatigue limit at 175˚C: 45 MPa min)
Ultimate tensile strength up to 175˚C (130 MPa)
Salt-spray corrosion resistance (0.1-0.25 mg/cm2/day)
Elongation (min 3% at room temperature)
Acceptable diecastability (comparable to AM or AE series)
Acceptable cost (5-10 cent over alloy prices)
Availability of raw materials
Alloy production (compatibility with plant processes)
Melt handling (oxidation, sludge formation)
Recyclability
Given the competitive properties of the newly developed HPDC magnesium alloys, MRI 153M
and MRI 230D, compared to the HPDC A380, the current alloy used in the tensioner, the
possibility of using these materials for the tensioner is investigated in this work.
85
The tensioner is optimized considering different materials and design parameters with the aim of
maximizing the fatigue life while minimizing the component weight. Four different alloys,
A380, MRI 153M, MRI 230D, and AZ91D are considered as the tensioner materials. Each
material has different fatigue properties, fatigue strength coefficient and fatigue strength
exponent, which need to be considered in the optimization process. Two different design
parameters, the fillet radius (R) and the height of the tapered tower (H), are also defined as the
inputs to the optimization procedure.
Table 16. Fatigue Properties of the Component
Component S-N curve f (MPa) b f c
A380 650
MRI 153M 435.84
MRI 230D 535.79
AZ91D 551.86
Due to the lack of information about the fatigue properties of the tensioner made of the proposed
materials, MRI 153M, MRI 230D, these properties are estimated using the specimen fatigue
properties. The fatigue properties of these materials are extracted from the literature [55]. These
properties are tabulated in table Table 17.
86
Table 17. Fatigue Properties of the Materials
literature estimated
Maetrials S-N curve f (MPa) b f (MPa) b
A380 -- -- 494.70
MRI 153M [57] 413.12
MRI 230D [57] 299.55
AZ91D [57] 329.26
The materials S-N curve is defined as:
(5.1)
where and are the strength limit and the symmetrical bending fatigue limit of the
material, respectively. Commonly, N0 is considered as 1×107.
3101 N bN
9.01
0NN
11
N
0
3101 NN
1
0
0
11log9.0log
3log
loglogloglog
bNN
NN
b
1
87
The component S-N curve is then determined using the above mentioned materials factors. The
relation between the materials and component fatigue parameters are as follows:
(5.2)
where
where represents the number of cycles corresponding to the fatigue limit. , , and
represent the fatigue notch factor, component size factor, and surface finish factor, respectively.
Using the monotonic mechanical properties of the A380 Al alloy (Table 17) and based on
equations 5.1, the materials S-N curve is estimated. Figure 24 shows a plot of the A380 Wohler
curve in log-log scale.
As detailed in section 4.3, the resultant fatigue coefficients are used to plot the component S-N
curve of the A380 Al alloy tensioner. The logarithmic scale of the component S-N curve is also
shown in Figure 24. Using these graphs, the is calculated to be 1.07.
3101 N DDN 11
0
3101 NN
DbDDNN
NN
1
0
0
11log9.0log
3log
loglogloglog
DDK
/loglog
11
11
NN
fN
D
KK
0N
fNK
N
N
DK
88
Figure 24. Material S-N Curve and Component S-N Curve for A380 Al. Alloy
To estimate the fatigue life of the tensioner made of MRI 153M and MRI 230D, the above
approach is used. In this approach, it is considered that the fatigue notch factor and the
component size factor are constant for the design of the tensioner using the new proposed
materials. This consideration is based on the same design dimensions and assuming the same
notch sensitivity factor for these aluminium and magnesium alloys. It is also assumed that the
change of the surface finish factor is negligible; hence the value of A380 alloy can be used to
estimate the component S-N curves for these proposed alloys as the tensioner material.
Using the described method, the component S-N curves of the MRI 153M, MRI 230D, and
AZ91D are estimated. The following figures show the material and the component Wohler
curves for the proposed alloys.
1.5
1.7
1.9
2.1
2.3
2.5
2.7
0 1 2 3 4 5 6 7 8
lg σ
a (M
Pa
)
lg N
Material S-N Curve
Component S-N curve
89
Figure 25. Material S-N Curve and Component S-N Curve for MRI 153M Mg. Alloy
Figure 26. Material S-N Curve and Component S-N Curve for MRI 230D Mg. Alloy
1.5
1.7
1.9
2.1
2.3
2.5
2.7
0 1 2 3 4 5 6 7 8
lg σ
a (M
Pa
)
lg N
Material S-N Curve
Component S-N curve
1.5
1.7
1.9
2.1
2.3
2.5
2.7
0 1 2 3 4 5 6 7 8
lg σ
a (M
Pa)
lg N
Material S-N Curve
Component S-N curve
90
Figure 27. Material S-N Curve and Component S-N Curve for AZ91D Mg. Alloy
Table 16 summarizes the resultant fatigue coefficients of the proposed alloys when used in
tensioner.
1.5
1.7
1.9
2.1
2.3
2.5
2.7
0 1 2 3 4 5 6 7 8
lg σ
a (M
Pa
)
lg N
Material S-N Curve
Component S-N curve
91
Figure 28. Estimated Component S-N Curves for A380, AZ91D, MRI153M, and MRI230D
The calculated Wohler curves for different alloys are shown in Figure 28. As described earlier,
these S-N curves are estimated according to Equation 5.2 and based on the calculated .
5.2 Effects of Possible Design Variables
The response of the model to different design variables can be analyzed using the response
surface method. In this method, the interesting responses corresponding to various design
variables are plotted as the response surfaces. This approach uses the least square methods as the
regression methods.
0
50
100
150
200
250
300
350
1.E+00 1.E+02 1.E+04 1.E+06 1.E+08
σa (M
Pa
)
Reversals to failure (2Nf)
A380
AZ91D
MRI 230D
MRI 153M
DK
92
To optimize the design of the tensioner with the aim of minimizing the weight within the specific
required life, the response surface method is used in this work. The central composition design
(CCD) is used to identify the limits of the design variables [32].
In order to obtain an extended fatigue life, a design enhancement solution, using the following
design variables at the critical regions is applied. Two design variables, spindle tower height (H)
Figure 29. Fillet Radius and Spindle Tower Height as Design Variables of the
Parametric Study
Tower Height
Fillet Radius
93
and fillet radius of the base of the tower (R) were established to a range that no interference with
other parts of the tensioner assembly is created.
Table 18 shows the range defined for these two design variables. This table summarizes the
variables used for design enhancement of the tensioner with the purpose of minimizing weight
while maintaining the required component life.
Table 18. Range of Design Variables
Factor Level
-1 0 1
x1 (Radius of fillet, R, mm) 1 3 5
x2 (Spindle tower height, H, mm) 38 43 48
Simulation was performed to study the effects of each of these design variables as well as
combination of the two variables. The results of the study are presented in the next sections.
In this research work, the response surface method is used to analyze the effect of different
design parameters on the response of the system. With investigation of various important process
variables using the RSM, the component can be modeled statistically. The response surface
approach is commonly formed based on the least square method as its regression model. In the
current study, the optimum design was achieved using the RSM to minimize the tensioner weight
within the specific life. Design of Experiments (DOE) was used to identify the specific design
variables [32].
94
The response of the system is usually presented graphically, i.e. three-dimensional space or
contour plots. These contours are combination of constant curves for each x1 and x2 planes where
no other parameter is changed. Each of these contours represents a specific height of response
surface.
The current study is focused on two possible design variables, the fillet radius (R) and the
spindle tower height (H). These design parameters were established up to ranges considering no
interference with other subcomponents of the assembly. Figure 29 shows the two selected design
variables for this part of the study. The established ranges of these design parameters are
presented in Table 18.
Simulation was performed for the whole range of each parameter according to Central
Composite Design (CCD) approach. The most common response surface approach for design of
experiments is the CCD approach. CCD uses factorial or fractional factorial design including
centre points and axial points.
In this study, effects of both fillet radius and spindle tower height are investigated using response
surface method. This method considers both variables with aiming minimizing the weight of the
component while meeting the life criteria.
For the case of constant amplitude loading with mean loading of 2100N and dynamic loading of
2000N, the following results are achieved using the response surface method. The mesh size of
0.003 is used in this response surface analysis as the converged solution is achieved by using this
mesh size (detailed in section 4.1.2). The current tensioner design develops the maximum
95
equivalent stress of 152 MPa on the critical area, while having the mass of 651.14 g. This design
has a fillet radius and cylinder height of 2 mm and 43 mm, respectively. The following response
surface analysis shows how changing these two design variables affects the maximum equivalent
stress developed on the base of the spindle and the mass of the tensioner base.
Table 20 presents the results of stress analysis of the CCD points. The maximum stress and
geometry mass are presented as the direct output parameters of the parametric study.
Table 19. Array Result of Central Composite Design (CCD) for Case Study 1
Design Point Fillet Radius
(mm)
Cylinder height
(mm)
Max. Equivalent Stress
(MPa)
Geometry Mass
(g)
1 3 43 165.0 651.2
2 1 43 198.3 651.1
3 5 43 149.2 651.5
4 3 38 139.3 640.2
6 1 38 164.5 640.1
7 5 38 131.3 640.5
8 1 48 211.8 662.1
9 5 48 162.6 662.5
10 3 48 172.8 662.2
Some of the design points result in stresses higher than 90% of the yielding point at the critical
area of the part. Meaning that the corresponding design variables cannot be used for such
operating service conditions. Therefore, these design points are removed from the list of
acceptable variables. Finally, the parameters, which lead to minimum weight while meeting the
fatigue criteria, are determined as the optimal design shape for the tensioner.
96
Figure 30. Three Dimensional Response Surface and the Corresponding Contour Plot (where x1
is the fillet radius and x2 corresponds to the spindle tower height)
The general optimization algorithm used in this study is shown in Figure 31. After estimation of
life using the predicted load/stress histories, the optimization process continues searching for the
optimal design shape. If the optimal design meets the termination condition for the optimization
process, the process is stopped and the optimal design is found.
97
Figure 31. General Optimization Algorithm
To optimize the design with specific input parameters against fatigue, the method developed in
this work is to be used. Considering dynamic force of 2000 N and mean load of 2100 N, the
stress developed in the part under maximum loading condition is estimated using the model. This
linear elastic model developed for the tensioner spindle can be used to estimate the stress at
different design points. After achieving the stress for maximum hubload experienced in service,
the stress corresponding to minimum hubload can also be estimated. Thereafter, the fatigue
graphs developed in section … can be used to insure the design of the part against fatigue.
98
Figure 30 shows the three-dimensional response surface of the tensioner bracket where two
design variable, as described earlier, are used. The response surface contour corresponds to the
response of the system where x1 represents the fillet radius range of input and x2 represents the
spindle tower height range of input.
Table 20. Array Result of Central Composite Design (CCD) for Case Study 2
Model x1 x2 Max Stress
(MPa)
Min Stress
(MPa)
Fatigue Life
(×106)
Mass
(g)
1 1 1 162.6 3.9 3.2 662.5
2 1 -1 131.3 3.2 8.5 640.5
3 -1 1 211.8 5.2 0.3 662.1
4 -1 -1 164.5 4.0 3.2 640.1
5 0 0 165.0 4.0 3.1 651.2
6 1 0 149.2 3.6 6.2 651.5
7 -1 0 198.3 4.8 0.5 651.1
8 0 1 172.8 4.2 1.7 662.2
9 0 -1 139.3 3.4 6.8 640.2
Two other possible design variables are also investigated, namely the tower hole radius (x3) and
depth (x4). Figure 32 shows the design variables selected for the parametric study. Same
approach as described above is taken to find the optimal design aiming the minimum weight
while meeting the minimum required life.
99
Considering the part is under mean load of 2400 N and dynamic load of 2300N, the stress
developed in the part is estimated for various design parameters and combination of these input
parameters. The developed finite element model is used to estimate the stress distribution and the
maximum stress generated at the critical area of the tensioner spindle.
The design of experiments is set while considering no interference between the spindle and the
surrounding environment. The initial fillet radius, tower hole radius and depth are 2mm, 10mm
and 20mm, respectively. The fillet radius is set to be changed from 2mm to 5mm. The tower hole
Hole Depth
Hole Radius
Figure 32. Design Variables for the Parametric Study
100
radius range is defined as 7-10mm. The range of 10mm to 22mm is used for the hole depth input.
Table 21 presents the input parameters for the parametric study.
Table 21. Input Parameters for the Parametric Study of the Tensioner Spindle
Design Point Fillet Radius
(mm)
Hole Radius
(mm)
Hole Depth
(mm)
1 2 8.5 16
2 2.28 7.28 11.12
3 2.28 9.72 11.12
4 3.5 8.5 16
5 3.5 7 16
6 3.5 10 16
7 3.5 8.5 10
8 3.5 8.5 22
9 4.72 7.28 11.12
10 4.72 7.28 20.88
11 4.72 9.72 20.88
12 5 8.5 16
The direct outputs of this parametrical study are maximum stress generated at the critical area
and the mass corresponding to the design inputs. Table 22 summarizes how changing the inputs,
based on the design of experiments, affects the life and geometry mass of the tensioner spindle.
As descried earlier, the developed linear elastic FE model is used to estimate the maximum stress
at the critical area. If the strain amplitude is then estimated using the estimated cyclic stress-
101
strain curve, described in section Fatigue Behaviour and Life Predictions4.2. The next step is to
estimate the life of the part using the SWT parameter and the Neuber parameter.
The results of the series of experiments show that all these design points meet the fatigue design
criteria. Therefore, optimal shape can be concluded to be corresponding to the minimum weight.
However, in optimizing the shape of a component, one should note the increase in the stress due
to weaker design at critical regions. In other words, the design with less weight might lead to the
stress generated to be higher than 90% of the yield, i.e. 148.5 MPa for the A380 die casting
aluminum. Such design points are not acceptable and should be deleted from the optimization
process.
Table 22. Output Parameters of the Parametric Design Study of Tensioner Spindle
Design Point Max. Equiv. Stress
(MPa)
Min Equiv. Stress
(MPa)
Fatigue Life
(106)
Geometry Mass
(g)
1 152.2 3.3 6.1 652.4
2 150.9 3.2 6.2 657.3
3 150.3 3.2 6.2 653.3
4 147.3 3.1 6.3 652.5
5 146.1 3.1 6.4 655.7
6 149.8 3.2 6.2 648.7
7 146.9 3.1 6.3 656.2
8 150.3 3.2 6.2 648.8
9 137.9 2.9 7.0 657.5
10 140.4 3.0 6.9 653.1
11 137.3 2.9 7.1 645.8
102
12 142.4 3.0 6.7 652.8
5.3 Summary
Given the competitive properties of the newly developed HPDC magnesium alloys, MRI 153M
and MRI 230D, compared to the A380, the current alloy used in the tensioner, the possibility of
using these materials for the tensioner is investigated in this research work. Due to the lack of
information about the fatigue properties of the tensioner made of the proposed materials, MRI
153M, MRI 230D, these properties are estimated using the specimen fatigue properties from
literature.
Assuming the fatigue notch factor and the component size factor remains the same, life curves of
the tensioner made of MRI 153M and MRI 230D are developed according to Equation 5.2. This
consideration is based on the same design dimensions and assuming the same notch sensitivity
factor for these aluminium and magnesium alloys. Using this approach, Wohler curves for
different alloys A380, AZ91D, MRI153M, and MRI230D were estimated.
An applicable shape optimization methodology for the tensioner was also proposed in this work.
In this approach, the optimal design shape was achieved through response surface method and
considering the durability analysis of the tensioner in FEAD systems. In this study, the critical
part of tensioner was investigated using the finite element analysis when maximum hubload in
service is applied to tensioner. Effects of possible design variables were investigated in the
optimization study using the response surface method. The optimal design shape of the tensioner
was then achieved when considering life of the part.
103
Chapter 6
6 Random/Variable Amplitude Loading
In this chapter, the stress histories of the tensioner under real working conditions are
investigated. The tensioner service environment is replicated using the realistic load histories
prior to the application of the finite element model. Thereafter, the stress histories of the
tensioner are developed under the real working conditions using the method described in this
chapter.
6.1 Realistic Load Histories
In fatigue analysis of the components, stress/strain estimation can be approached in the time or
frequency domains. The stress/strain histories developed in a component due to dynamic loading
can be estimated using different approaches. One of the following approaches that can be taken
in analyzing the stress/strain histories of dynamically loaded part is the standard time domain
approach. This method applies the quasi-static stress analysis approach. Transient dynamic
analysis is applied in a more sophisticated time domain approach. Other methods to approach the
104
stress/strain histories of such parts are the frequency domain approach which applies the
harmonic stress analysis method, and frequency domain approach with a focus on the random
vibration analysis approach. In this work, the quasi-static stress analysis method is used to
calculate the stress/strain histories of the tensioner in the FEAD system.
The quasi-static analysis method considers the variable external loading in a linear elastic
analysis approach. In this method, a static load replaces each external load history acting at the
same direction and location as the history. For each load unit, the static stress analysis is used to
estimate the stress developed due to that individual load. Using the static stress influence
coefficients, the dynamic stress developed from each load history is calculated. In this method,
the dynamic stress developed from each applied load history can be estimated using that history
along with the static stress influence coefficients from that specific unit load. thereafter, the total
dynamic stress histories are calculated by superimposing the stresses within the component.
Following equations are used to calculate the stress histories at each FE node in this method with
plane stress conditions assumption [2]:
n
i
ixixtFt
1
(6.1)
n
i
iyiytFt
1
n
i
ixyixytFt
1
105
where , and represent the stress influence coefficients and n corresponds to the
number of load histories applied. To define the stress coefficients, the stress developed as a result
of unit load applied at the same direction and location as the load history is considered [58].
Customer usage profile data is usually shown in frequency or probability (pdf) with respect to
load or stress. In this research study, a real world usage profile (RWUP) is used to evaluate the
fatigue life of tensioners in belt drive systems. RWUP considers ranges of loads/stress,
environmental conditions such as temperature, humidity, vibration, operating frequency or
amplitudes such as tensioner arm movement. The engine usage profile represents the time spent
(in hrs, %) at each engine speed (RPM) or load (%). The usage pattern data corresponds to the
specific vehicle drive train (tire size, gear ratios) and drive cycles (road and traffic conditions).
The vehicle speed of a drive cycles specified for each vehicle provides the engine speed.
A periodically varying engine speed (crankshaft pulley RPM) with respect to time is shown in
Figure 33. The RPM profile can be described by Foruier series as shown in Figure 35. This
Figure shows how the crankshaft torque changes with the RPM. Combining the two profiles, one
can find the crankshaft torque profile with respect to time. Hence, the span tension variation at
every RPM level can be investigated using the analytical method described in Chapter 3.
Using the span tensions of the wrapped belt around the tensioner, the tensioner hubload can be
calculated as described earlier in chapter 3. Thereafter, this accelerated scheme is used to
estimate the stresses developed on the tensioner over the varying crankshaft RPM.
xi yi xyi
)( tF i
106
Figure 33. Crankshaft Speed Versus Time
0 1000 2000 3000 4000 5000 60000
20
40
60
80
100Crank Shaft Torque Input
[RPM]
Torq
ue
[N
.m]
107
Figure 34. Crankshaft Torque Signal
Knowing how the crankshaft speed changes over time, and also how the crankshaft torque
changes with respect to crankshaft speed, one can find the crankshaft torque versus time. As
detailed earlier in chapter 3, the dynamic model developed for the specific belt drive system can
be used to find the response of the system to each crankshaft torque input. In other words, the
belt span tensions with for each torque input can be estimated. These belt span tensions can be
further analyzed for estimation of hubloads on each accessory drive at each period of time. These
hubloads over range of time are called the loading history applied to the part in service, and is
used in estimating the damage experienced during service.
Figure 34 shows a typical crankshaft RPM change over time and the corresponding crankshaft
torque input. This graph is used to simplify the variable amplitude loading histories to small
blocks of constant amplitude loading. Using this simplified model, one can estimate the hubload
for each block of time. Dynamic analysis of these blocks of loading input leads to estimation of
the hubload (resultant force) on the part over time. Thereafter, the finite element model
developed in chapter 4 is used to predict the stress caused by each loading block at the critical
area.
108
Figure 35. Real Time Sample of Crankshaft Speed and Torque Versus Time
Table 23 presents a 16-step real time span loading and the crankshaft torque corresponding to
each time block. As explained earlier, each span tension of the belt drive system is estimated
using the dynamic model developed for that specific belt drive system. After running the
dynamic model, the results are recorded. Table 23 presents these recorded span tensions for the
16-step crankshaft torque input. The hubload on tensioner is then calculated considering the wrap
angle at the nominal position of the tensioner. Last step was to estimate the maximum stress at
each time span at the critical area.
0
10
20
30
40
50
60
70
80
90
100
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500 600 700 800
CS To
rqu
e (N.m
)
CS
Spee
d (
RP
M)
Time (sec)
109
Table 23. 16-step Loading, Resultant Tensioner Hubload and Stress Generated at Critical Area
Step
No.
Time Span
(s)
Crankshaft Speed.
(RPM)
CS Torque
(N.m)
Resultant Force
(N)
Stress at Critical Area
(MPa)
1 0-50 2600 87 2000 64.8
2 50-100 2250 78 1780 57.7
3 100-150 2500 83 1820 58.9
4 150-200 2350 80 1800 58.3
5 200-250 1850 68 1400 45.4
6 250-300 1800 66 1260 40.8
7 300-350 1250 46 1100 35.6
8 350-400 2350 80 1800 58.3
9 400-450 2500 83 1820 58.9
10 450-500 2650 90 2200 71.3
11 500-550 2600 88 2120 68.7
12 550-600 2550 85 1850 59.9
13 600-650 2300 79 1790 57.9
14 650-700 1750 66 1380 44.7
15 700-750 1600 62 1300 42.1
16 750-800 1100 33 890 28.8
By applying this crankshaft torque profile, the belt span tension corresponding to each CS speed
level can be calculated. Thereafter, the resultant force applied on the tensioner due to the belt
span variation is evaluated. To calculate the corresponding force to each CS torque input, the
mathematical model developed in chapter 3 is used.
110
This accelerated method is used along with the span tensions to evaluate the stress history of the
tensioner. The life model for the bet drive system is then generated using these results to estimate
the life of the tensioner.
6.2 Rain Flow Counting Method
There are two main steps in life assessment of components in time domain. First step is to count
the cycles and the second task is to predict damage using cumulative damage theories. In cases
of irregular/random stress/ strain histories applied to the part, the cycle counting helps to
simplify the load history to blocks with constant amplitude. There exists various counting
approaches in the literature. One of the most common approaches with accurate prediction is the
rain flow counting method [9].
The cycles are counted as to meet the method shown in Figure 36. In this approach, the cycles
are counted as peak-valley-peak or valley-peak-valley if the second stress/load range is greater or
equal to the first range. Figure 36 shows how details of the cycles counting using the rain flow
method. As 2.1 is less than 3.2 , the peak-valley-peak of 1-2-3 is considered as one cycle. In
this case, the cycle is assumed to have a range equal to the first range. This process can be
performed at the start of any peak or valley. Each peak to valley cycle, which has been recorded,
is removed for further cycle counting analysis. After the history is fully investigated and
accounted for, the counting procedure is stopped.
111
Figure 36. Cycles Counting Method using Rain Flow Approach
Generally, the cycles of the load/stress history by rain flow counting method are shown in a
matrix, which includes the number of cycles corresponding to each mean and range of stress. In
order to convert the matrix to a manageable size, these mean and dynamic stress are usually
rounded off.
6.3 Palmgren-Miner Rule
Second step in fatigue life assessment of components in time domain is to account for damage
accumulated in service time. One of the most common methods used in automotive industries is
the use of Palmgren-Miner's rule for damage summation. Palmgren-Miner’s rule is used as the
failure criteria to induce the effect of multiple blocks of variable amplitude cyclic loading. The
second step in fatigue analysis of parts under variable amplitude loading is to account for the
112
damage accumulated in the material by different blocks of loading. The most common rule used
for damage summation is Miner’s rule. This method is confirmed to provide accurate results. In
1945, Miner [10] proposed an estimation model for the damage caused by a specific stress
range.
Di =ni
N f( )i
(6.2)
where ni implies to the number of cycles corresponding to the stress range, and ifN represents
the number of cycles to failure at that particular stress range. The numbers of cycles, ni, are
estimated from the stress histories along with the cycles counting method.
Materials S-N curve is usually used to evaluate the numbers of cycles to failure at each specific
stress range. However, ifN are generally determined from the material strain–life curve rather
than the stress– life curve [11]. In this work, the SWT parameter to cycles to failure, developed
for the particular die cast aluminum, is used to evaluate the number of cycles to failure for each
stress range. As mentioned earlier, the SWT parameter accounts for mean stress effects.
Therefore, it results in more accurate results, which considers effects of both stress range and
mean stress.
The damage accumulated within the part, produced by the full stress history, can be evaluated by
adding the damage caused by each stress range. The mathematical formula for accumulative
damage is calculated by:
113
sn
i if
i
N
nD
1
(6.3)
In Miner’s rule, the life is assumed to be equal to the reciprocal of the damage [10]. The failure
of the material occurs when damage equals to 1 [58].
6.4 Summary
Stress histories of the tensioner under real working conditions were investigated in this
chapter. Prior to the application of the finite element model, the realistic load histories were used
to simulate the tensioner service environment. Subsequently, the stress histories the tensioner
experiences while in service were developed under the real working conditions.
To assess the tensioner life under variable amplitude loading and in time domain, two main steps
were then taken. After using a common method like rain flow counting method, the variable
amplitude loading was simplified to multiple blocks of constant amplitude loading. Thereafter,
the damage accumulated in the part was evaluated upon calculation of the realistic load history.
The failure criteria was defined to be when the damage accumulated is equal or greater than 1.
114
Chapter 7
7 Conclusions
7.1 Conclusions and Contributions
In this work, a dynamic model was developed for the belt drive systems. The purpose of
developing this model was to evaluate dynamic loading that tensioners experience during
service. By taking this approach, the significance of real loading conditions in the fatigue
behavior of the part is considered. In this chapter, details of a multi-degree-of-freedom dynamic
model is given. The model was developed based on the assumption that the input is the
fluctuating torque from the engine to the belt drive system. Using a typical crankshaft torque
input over engine speed, the response of the system was evaluated and confirmed. Further
investigation was performed to explore the belt span tensions caused by the selected torque input
to the system. Finally, FEAD component hubloads, as the result of the tensions generated on the
wrapped belt along the accessory, are calculated. It can be concluded that this mathematical
115
model can be used for investigation of hubloads on each part of the belt drive system, including
accessories and automatic tensioner.
A comprehensive fatigue model was developed in this chapter. The model was developed by
predicting the stress generated in the tensioner bracket as it undergoes loading. The linear elastic
FE model was validated by experimental data using strain gauges installed on an area close to the
critical area. The results of the finite element study showed that the base of the spindle
experiences the highest stress and therefore is the critical location of the part. Since the fatigue
properties of the material is one of the significant factors in fatigue assessment of components,
the Least Square method along with the experimental data were used to obtain these parameters
for the die cast aluminum tensioner. The predicted fatigue parameters was then used to generate
the cyclic stress-strain curve for the material of the selected tensioner based on Morrow theory.
Finally, the fatigue estimation method was modified based on Neuber’s rule and using the
generated cyclic stress-strain curve for the die cast A380 tensioner. This model presents the
Neuber parameter life data to account for plasticity correction.
The possibility of using the newly developed HPDC magnesium alloys, MRI 153M and MRI
230D as alternative materials for tensioner casting parts was also investigated. Fatigue properties
of the tensioner made of the proposed materials were estimated using the specimen fatigue
properties from literature. In this approach, assumptions were made in consideration of constant
fatigue notch factor and component size factor . This assumption was made based on the same
design dimensions and assuming the same notch sensitivity factor for these aluminium and
116
magnesium alloys. Using this approach, Wohler curves for different alloys A380, AZ91D,
MRI153M, and MRI230D were estimated.
To study the shape optimization of the tensioner, the response surface method was used. In this
approach, the optimal design shape was achieved considering the durability analysis of the
tensioner in FEAD systems. In this optimization study, the critical part of tensioner was
investigated using the finite element analysis when maximum hubload in service is applied to
tensioner. Effects of possible design variables were investigated in the optimization study using
the response surface method. The optimal design shape of the tensioner was then achieved when
considering life of the part.
Stress histories of the tensioner under real working conditions were investigated in this research
work. Prior to the application of the finite element model, the realistic load histories were used to
simulate the tensioner service environment. Subsequently, the stress histories the tensioner
experiences while in service were developed under the real working conditions.
To assess the tensioner life under variable amplitude loading and in time domain, two main steps
were taken. After using a common method like rain flow counting method, the variable
amplitude loading was simplified to multiple blocks of constant amplitude loading. Thereafter,
the damage accumulated in the part was evaluated upon calculation of the realistic load history.
The failure criteria based on Miner rule was used and defined to where the damage accumulated
in the part is equal or greater than 1.
In conclusion, a mathematical model was fully developed to predict the tensioner spindle fatigue
117
life with the consideration of the mean stress correction theories and plasticity correction. The
results showed an excellent correlation between the analytically calculated life to failure of a
typical tensioner and the experimental results which confirms the reliability of the proposed
method. Therefore, this approach can be considered for examination of both the fatigue
properties and the fatigue life of any tensioner with the same material with the consideration of
the specific part constraints such as stress concentration factor.
7.2 Future Work
Hybrid drive systems are recently being developed by alternator suppliers and car manufacturers
to reduce fuel consumption. Hybrid cars work with an internal combustion engine (with a fuel
tank) along with a type of electric motor (with a battery). Hybrid drive systems have a Motor
Generator Unit (MGU) or a Belt Integrator Starter Unit (B-ISGU). In the MGU, an electric
motor contributes to power by transferring the energy to the crankshaft through serpentine belt.
The MGU is usually installed in the alternator location and transfers the system to a mild hybrid
drive system. The use of MGU in FEAD systems leads the manufacturers to investigate
tensioners with dual arms to maintain the tension at the slack side of the belt during all engine
modes. During the belt start and boost modes, where the MGU drives the crankshaft and other
accessories, the belt slack side is different from the regeneration or steady state engine modes,
where the crankshaft is the driving pulley and the MGU acts as an alternator. Effects of
temperature on fatigue behaviour was not studied in this research work and is recommended as
118
future work. The temperature effects need to be investigated not only considering the dual arm
tensioners, but also the conventional tensioners as part of the belt drive systems.
Car manufacturers are interested in developing and investigating hybrid tensioning devices with
dual arms to maintain the required tension in the slack side of the system based on the driving
modes.
Dynamic simulation of hybrid drive systems is more complex than a regular FEAD system with
no hybrid mechanism. Investigation of the dynamic loading on the tensioners for all engine
modes is a challenging task. Hybrid systems usually need application of an isolating device to
engage and disengage the high inertia of MGU from the system during acceleration and
deceleration. Different isolation devices have been proposed by automotive part suppliers and
are still under development. Use of such devices induces non-linearity to the system and adds to
its complexity. Additionally, development of a fatigue model due to the very complex geometry
as well as applied loading is of significant interest of car manufacturers and automotive
suppliers. Therefore, multi-axial fatigue damage models need to be developed for analyzing the
life of hybrid tensioning devices to investigate the effects of multi-axial loadings applied to these
tensioners during their service life.
119
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124
Appendix I
In this section, the details of the six-degree-of-freedom FEAD system (studied at section 3.1.4)
are presented. The system mass, stiffness and damping matrices are presented here.
armJ
J
J
J
J
J
M
00000
00000
00000
00000
00000
00000
5
4
3
2
1
tarmarm
armarm
arm
arm
KlKlK
rlKrlKrKrK
rlKrrKrKrK
rrKrKrK
rrKrKrKrrK
rlKrrKrrKrKrK
K
)(sin..)(sin..
)sin(...)sin(.....
)sin(.......
00....
000......
)sin(.....00.....
5
22
54
22
4
555454
2
55
2
54
444544
2
44
2
43
433
2
33
2
32
322
2
22
2
21211
515515211
2
15
2
11
tarmarm
armarm
arm
arm
ClClC
rlCrlCrCrC
rlCrrCrCrC
rrCrKrC
rrCrCrCrrC
rlCrrCrrCrCrC
C
)(sin..)(sin..
)sin(...)sin(.....
)sin(.......
00....
000......
)sin(.....00.....
5
22
54
22
4
555454
2
55
2
54
444544
2
44
2
43
433
2
33
2
32
322
2
22
2
21211
515515211
2
15
2
11
The dynamic variable vector is:
SYM
SYM
125
t
t
t
t
t
t
X
t
5
4
3
2
1
t1 : known
t2 , t3 , t4 , t5 , tt : unknown
and the input forcing function is expressed as below:
0
0
0
0
0
0
0
0
0
0
sin 0TtQ i
F
0515551522111
2
15
2
1111 .sin............. TrlKrrKrrKrKrKJ tarm
0.......... 33222
2
22
2
21121122 rrKrKrKrrKJ
0.......... 44333
2
33
2
32232233 rrKrKrKrrKJ
0.sin............. 44455444
2
44
2
43343344 tarm rlKrrKrKrKrrKJ
0.sin...sin............. 5554544
2
55
2
544544151555 tarmarm rlKrlKrKrKrrKrrKJ
0.sin..sin..sin...sin.... 5
22
54
22
444441515 ttarmarmarmarmtarm KlKlKrlKrlKJ
Natural frequencies and mode shapes of the system are calculated to be as follows:
Natural Frequencies and Mode Shapes
126
Mode 1 2 3 4 5 6
Frequency (Hz) 78.40 165.34 321.83 573.47 781.26 1.59e6
Mode Shapes
Crank Shaft 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000
Air Conditioner 0.3663 -0.3794 0.0306 -0.0010 0.0002 0.0000
Power Steering 0.5768 0.4042 -0.3231 0.0382 -0.0164 0.0000
Idelr 1.0000 1.0000 1.0000 -0.6296 0.5418 0.0000
Tensioner Idler 0.5373 0.5898 0.9087 1.0000 -0.1627 0.0000
Tensioner Arm -0.2402 -0.2507 -0.2986 0.3402 1.0000 0.0000
0000.00000.13402.02986.02507.02402.0
0000.01627.00000.19087.05898.05373.0
0000.05418.06296.00000.10000.10000.1
0000.00164.00382.03231.04042.05768.0
0000.00002.00010.00306.03794.03663.0
0000.10000.00000.00000.00000.00000.0
U
Results from FEAD software (developed by Litens automotive group) are presented as follows:
127
Natural Frequencies and Mode Shapes
Mode 1 2 3 4 5 6
Frequency (Hz) 78.48 165.61 324.01 567.32 805.29 1.60e6
Mode Shapes
Air Conditioner 0.3670 -0.3781 0.0293 -0.0010 0.0002
Power Steering 0.5773 0.4071 -0.3145 0.0384 -0.0150
Idelr 1.0000 1.0000 1.0000 -0.6170 0.5318
Tensioner Idler 0.5216 0.5721 0.8851 1.0000 -0.1836
Tensioner Arm -0.2329 -0.2416 -0.2784 0.3451 1.0000
0000.00000.13451.02784.02416.02329.0
0000.01836.00000.18851.05721.05216.0
0000.05318.06170.00000.10000.10000.1
0000.00150.00384.03145.04071.05773.0
0000.00002.00010.00293.03781.03670.0
0000.10000.00000.00000.00000.00000.0
U
Comparison of simulated results and FEAD software by Litens shows a good agreement
between the results.
128
Natural Frequencies (Hz)
1 2 3 4 5 6
Simulated 78.40 165.34 321.83 573.47 781.26 1.59e+006
FEAD Software 78.48 165.61 324.01 567.32 805.29 1.60e+006