NEIL P. DOLL. A thesis submitted in ...... riser. Stress risers might exist from the onset of pitting. Also, tooth breakage on only one part of the tooth face might be a ...
MODELING THERMOMECHANICAL BEHAVIOR OF POLYMER GEARS BY
NEIL P. DOLL
A thesis submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE (MECHANICAL ENGINEERING)
AT THE
UNIVERSITY OF WISCONSIN-MADISON 2015
APPROVED BY:
_____________________________________ DATE: ___________________ PROFESSOR TIM A. OSSWALD DEPARTMENT OF MECHANICAL ENGINEERING UNIVERSITY OF WISCONSIN-MADISON
i
ABSTRACT This thesis presents a methodology for material characterization, thermomechanical modeling, and experimentation with polymer gearing. The material used for all experiments in this study was Victrex PEEK 450G. Material characterization began with dynamic mechanical analysis of PEEK 450G at loading frequencies of 0.1 Hz to 100 Hz and temperatures from 25 to 180 °C. This characterization is performed on a forced resonance machine, and time-temperature superposition principles were used to quantify mechanical properties at frequencies as high as 10 kHz. A tribology study of PEEK was performed using a pin-on-disk tribometer. A custom heated stage was developed for this study. The friction coefficient was measured as a function of temperature (25 to 175 °C), velocity (0.1 to 100 mm/s), and pressure (20 to 70 MPa). During these measurements, the friction coefficient varied from 0.4 to 0.6. The measured material data was then utilized in a coupled thermomechanical finite element model using ANSYS® Workbench and Mechanical APDL. An approach that used the concept of linear elastic strain energy density, hysteresis, and a fractional gear model was developed. The model predicted the transient temperature of a polymer gear drive using heat generated from hysteresis and friction. The simulation data was compared to experiments conducted on a gear test bench at Kleiss Gears, Inc. After adjusting the heat transfer coefficients, the simulation data was found to predict the transient thermal behavior of the polymer gears with reasonable accuracy.
ii
TABLE OF CONTENTS Abstract ................................................................................................................................................... i Acknowledgements ................................................................................................................................ v Terminology .......................................................................................................................................... vi Index of Figures................................................................................................................................... xiii Index of Tables ..................................................................................................................................... xv CHAPTER 1 – Introduction ................................................................................................................... 1 CHAPTER 2 – History and Background................................................................................................ 5 2.1 2.1.1
History of Polymer Gears ....................................................................................................... 5 Limitations of Metal Gears and Polymer Gears ............................................................. 6
2.2
Motivation .............................................................................................................................. 7
2.3
Material Selection................................................................................................................... 8
2.4
Design and Processing of Polymer Gears............................................................................... 9
2.4.1
Composite Gears .......................................................................................................... 15
2.5
Failure Mechanisms of Polymer Gears ................................................................................ 16
2.6
Literature review .................................................................................................................. 18
CHAPTER 3 – Dynamic Mechanical Analysis.................................................................................... 27 3.1
Theory .................................................................................................................................. 27
3.1.1
Viscoelastic Behavior ................................................................................................... 27
3.1.2
Time-Temperature Superposition ................................................................................. 30
3.2
Application to Gears ............................................................................................................. 32
3.3
Measurement Techniques ..................................................................................................... 34
3.3.1
Forced and Free Resonance Analyzers ......................................................................... 34
3.3.2
Atomic Force Microscopy ............................................................................................ 37
3.3.3
Dielectric Thermal Analysis ......................................................................................... 38
3.3.4
Custom Apparatus ........................................................................................................ 39
3.4
Experiments .......................................................................................................................... 39
3.4.1
Equipment and Methods ............................................................................................... 39
3.4.2
Results .......................................................................................................................... 41
CHAPTER 4 – Tribological Analysis .................................................................................................. 48 4.1
Gear Tribology ..................................................................................................................... 48
4.1.1
Sliding Friction and Rolling Resistance ....................................................................... 48
4.1.2
Wear ............................................................................................................................. 51
4.1.3
Lubrication and Additives ............................................................................................ 52
iii Acoustics ...................................................................................................................... 52
4.1.4 4.2
Measurement Techniques ..................................................................................................... 53
4.2.1
Pin-on-Disk................................................................................................................... 53
4.2.2
Twin-Disk ..................................................................................................................... 54
4.2.3
Block-on-Ring .............................................................................................................. 56
4.3
Experiments .......................................................................................................................... 57
4.3.1
Instrumentation ............................................................................................................. 57
4.3.2
Heater Design ............................................................................................................... 57
4.3.3
Probe Design ................................................................................................................ 60
4.3.4
Methods ........................................................................................................................ 61
4.3.5
Results .......................................................................................................................... 63
CHAPTER 5 – Finite Element Analysis .............................................................................................. 67 5.1
Description of Model ............................................................................................................ 67
5.1.1
Geometry ...................................................................................................................... 67
5.1.2
Material Properties ....................................................................................................... 69
5.1.3
Mesh ............................................................................................................................. 69
5.1.4
Contact.......................................................................................................................... 72
5.1.5
Boundary Conditions .................................................................................................... 74
5.2
Thermomechanical Coupling ............................................................................................... 78
5.2.1
Coupled Simulations .................................................................................................... 78
5.2.2
Hysteresis Thermal Generation .................................................................................... 78
5.2.3
Frictional Thermal Generation ..................................................................................... 81
5.3
Simulations ........................................................................................................................... 82
5.3.1
Hardware and Software ................................................................................................ 82
5.3.2
Methods ........................................................................................................................ 82
5.3.3
Results .......................................................................................................................... 84
CHAPTER 6 – Experimental Validation ............................................................................................. 90 6.1
Gear Test Bench ................................................................................................................... 90
6.1.1
Overview ...................................................................................................................... 90
6.1.2
Instrumentation ............................................................................................................. 91
6.2
Results .................................................................................................................................. 94
CHAPTER 7 – Conclusion and Recommendations ............................................................................. 99 7.1
Conclusion ............................................................................................................................ 99
7.2
Recommendations .............................................................................................................. 100
iv References .......................................................................................................................................... 102 Appendix ............................................................................................................................................ 106 ANSYS® Command Examples....................................................................................................... 106 Original Plastic Gears Interest Group at UW-Madison .................................................................. 107
v
ACKNOWLEDGEMENTS First and foremost, I would like to thank Mom and Dad for their unconditional love and support. It is from both of you that I learned how to pursue my dreams and never look back. Thank you for a great education. My unforgettable experience at the Polymer Engineering Center would not have been possible without the support of my advisor, professor, mentor, and friend, Tim A. Osswald. Thank you for bringing out the potential within me, and the occasional laugh or two. I will miss this place. I would like to thank my companion and best friend, Amanda Gonzalez, for pushing me through the hard times and helping me see the good. With you, life is always an adventure. Anthony Verdesca, you are one of the hardest workers I have ever met. Your future is bright. Thank you for your help and friendship. It was a pleasure working with you on this project. Many thanks to the team at Kleiss Gears: Rod Kleiss, Michael Weiss, Dusty Brunner, and Joseph Elmquist. This project was possible because of your efforts. Thanks for teaching me a few lessons from your gearing expertise. Thank you Frank Ferfecki for the PEEK 450G samples and helping me learn about a thermoplastic with considerable potential. You have got me convinced. Thanks to the local tribology experts, Professor Melih Eriten and Ahmet Usta, for letting me use the tribometer and having patience. Thank you Professor Rudolph and Professor Turng for joining my thesis committee and for teaching students about the possibilities of engineering with polymers and composites. Lukas Duddleston, many thanks for the peer edits. I owe you one.
vi
TERMINOLOGY AGMA
American Gear Manufacturers Association
ANSYS® Workbench
A multi-physics engineering simulation software
CMM
Coordinate measuring machine
CNC
Computer numerical control
Contact ratio
Average number of gear tooth pairs in contact on a pair of meshing gears
DETA
Dielectric thermal analysis
EDM
Electrical discharge machining
Flank
Gear tooth contacting surface
Flash temperature
The instantaneous temperature rise on a gear tooth’s surface
Gear test bench
Machine used to test gears at various frequencies and loads
Loss modulus
Viscous or lost response of a material
LPV
Limiting pressure velocity; maximum pressure at a given velocity that can be applied to a surface without failure
MATLAB®
High-level language and interactive environment used by engineers and scientists
Mechanical APDL
The original ANSYS® that does not require any user interface because it can be programmed with scripts
Module
Pitch circle diameter divided by the number of teeth
NVH
Noise, vibration, and harshness
PA
Polyamide
PEEK
Polyetheretherketone
PID
Proportional integral derivative
Pitch point
Point of contact between the pitch line of two gears
POM
Polyoxymethylene
vii
PPS
Polyphenylene sulfide
PTFE
Polytetrafluoroethylene
Roll tester
Method of checking the functional accuracy of the gear by measuring displacements while rolling the gear in mesh with a calibration gear
Storage modulus
The elastic response of a material
Tribometer
An instrument that measures tribological quantities, such as coefficient of friction and wear volume, between two surfaces in contact
TTS
Time-temperature superposition
UHMWPE
Ultra high molecular weight polyethylene
WLF
Williams-Landel-Ferry
𝑇𝑚𝑎𝑥,1
Flash temperature
𝐹𝐹
Form factor
𝑞𝑎𝑣
Average heat flux
𝑤
Width of heat source
𝑣
Velocity
𝑏
Coefficient for flash temperature
𝑘
Thermal conductivity
𝜌
Density
𝑐𝑝
Specific heat capacity
𝜇
Kinetic friction coefficient
𝑃𝑎𝑣
Average contact pressure
𝑣𝑠
Relative sliding velocity
𝑣1
Velocity of surface 1
viii
𝑣2 𝑇𝑚𝑎𝑥,2
Velocity of surface 2 Special case of flash temperature
𝐸ℎ
Volumetric hysteresis
𝛿
Out of phase component between stress and strain responses
𝑓𝑙𝑜𝑎𝑑 𝜎0
Loading frequency Stress amplitude
𝑇𝑚𝑎𝑥
Maximum gear tooth temperature
𝑇𝑎𝑚𝑏
Ambient temperature
𝑊𝑡
Transmitted load per unit width of the tooth
𝑣𝑝
Pitch line velocity
𝑚
Module
𝑏0 , 𝑏1 , 𝑏2 , 𝑏3
Regression fitting coefficients
𝐸𝑓,𝑤
Frictional heating for a unit face width
𝑊𝑖 𝑊𝑛
Load sharing factor
𝜃𝑝
Pressure angle
Δ𝑠
Displacement of the contact point along the line of action
𝐾
Geometrical constant
𝑖1
Driver tooth number
𝑖2
Driven tooth number
𝑆
Distance between the pitch point and contact point on the line of action
𝑝𝑛
Normal base pitch
𝐸ℎ,𝑤 𝑡𝑝
Hysteresis energy losses for a unit face width Tooth thickness at pitch circle
ix
𝐻
Hysteresis
𝐸′
Storage modulus
𝐸′′
Loss modulus
𝐸∗
Complex modulus
𝑈𝑚𝑎𝑥
Maximum strain energy density
𝑈𝑙𝑜𝑠𝑠
Dissipated strain energy density
𝑓
Frequency
𝜔
Angular velocity
𝑟𝑏
Base circle radius
𝑍
Tooth number
𝑃𝑇
Transmitted power
𝐸𝑓
Frictional power loss for one tooth
𝐸𝑓,1.5
Frictional power loss for a contact ratio of 1.5
𝑉𝑓𝑙𝑢𝑖𝑑
Volume of fluid transported between gear teeth
𝑟𝑜𝑢𝑡
Outside radius
𝑟𝑎
Reference radius
ℎ
Tooth width
Δ𝑇𝐵
Temperature rise of the gear
𝑇𝑇
Transmitted torque
𝜎
Normal stress
𝜀
Normal strain
𝐸
Young’s modulus
𝜏
Shear stress
𝐺
Shear modulus
x
𝛾
Shear strain
𝜂
Viscosity
𝜀̇
Strain rate
𝑡
Time
𝜀0
Strain amplitude
𝜏𝑅
Relaxation time
𝑡𝑟𝑒𝑓
Reference time
𝑎𝑇
Shift factor
𝐶1
Material constant
𝐶2
Material constant
𝑇
temperature
𝑇𝑟𝑒𝑓
Reference temperature
𝑏𝑇
Vertical shift factor
𝑇1
Temperature 1
𝑇2
Temperature 2
𝐸𝑎
Activation energy
𝑅
Universal gas constant
𝑇𝑔
Glass transition temperature
𝑁𝑡
Number of teeth on gear
𝐺′
Storage shear modulus
𝐺′′
Loss shear modulus
𝐺∗
Complex shear modulus
𝜐
Poisson’s ratio
𝜎𝐻
Maximum Hertzian contact stress or pressure
xi
𝐷𝑝
Pitch diameter
𝐸𝑝
Pinion modulus
𝐸𝑔
Gear modulus
𝑚𝑔
Speed ratio
𝑁𝑡,𝑔
Number of gear teeth
𝑁𝑡,𝑝
Number of pinion teeth
𝑇𝑚
Melting temperature
𝐹𝑛
Normal force
𝑅1
Sphere radius
𝑅2
Plate radius
𝜐1
Sphere Poisson’s ratio
𝜐2
Plate Poisson’s ratio
𝐸1
Sphere Modulus
𝐸2
Plate Modulus
𝑑1
Roller diameter 1
𝑑2
Roller diameter 2
ℎ𝑡
Teeth heat transfer coefficient
̅̅̅̅𝑡 𝑁𝑢
Teeth Nusselt number
𝑅𝑒𝑡
Teeth Reynolds number
𝑣𝑘
Kinematic viscosity
ℎ𝑠
Side heat transfer coefficient
̅̅̅̅𝑠 𝑁𝑢
Side Nusselt number
𝑟𝑖
Inner gear diameter
𝑅𝑒𝑠
Side Reynolds number
xii
𝑃𝑟𝑎𝑑
Radiation power
𝜖
Emissivity
𝐴𝑠
Surface area
𝐸ℎ,𝑉 𝐸𝑓,𝑎𝑣𝑒
Volumetric hysteresis power Average frictional heat flux rate
xiii
INDEX OF FIGURES Figure 1 - Depiction of project work flow.............................................................................................. 4 Figure 2 - Polymer repeat unit of PEEK ................................................................................................ 9 Figure 3 - Configurations of spur gears (left) and helical gears (right) [10] ........................................ 10 Figure 4 - Hybrid gear tooth geometry used in polymer test gears from Kleiss Gears ........................ 11 Figure 5 - Example of tooth shrinkage after cooling within mold cavity [7] ....................................... 12 Figure 6 - Injection molding gate designs for polymer gears [11] ....................................................... 13 Figure 7 - Coordinate measuring machine used for dimensional quality control [7] ........................... 14 Figure 8 - PEEK test gear and powdered metal hub (left) with gate and runner design (right) ........... 15 Figure 9 - Failure modes of gears [14] ................................................................................................. 16 Figure 10 - Diagram showing the cross section of a line heat source between two contacting surfaces [15] ....................................................................................................................................................... 18 Figure 11 - Energy losses incurred during loading of a polymer gear tooth [16]................................. 21 Figure 12 - Polymer gear heat transfer related to the concept of a gear pump [21] ............................. 25 Figure 13 - Comparison of simple elastic and viscoelastic models ...................................................... 28 Figure 14 - Plot of strain energy density versus time for one revolution of a polymer gear at 1,000 RPM and 10 Nm torque for an arbitrary element located on the surface of a loaded gear tooth ......... 33 Figure 15 - Depiction of a forced strain input and the time-lagging stress response ........................... 35 Figure 16 - 3 point bending fixture commonly used for stiff materials in a forced resonance DMA test [31] ....................................................................................................................................................... 36 Figure 17 - Normalized DMA data for PEEK 450G as a function of temperature (6.2 Hz, 0.2 % strain, 𝐺𝑚𝑎𝑥′=1.26 GPa, 𝐺𝑚𝑎𝑥′′=0.11 GPa) ................................................................................................. 37 Figure 18 - Schematic of atomic force microscope [33] ...................................................................... 38 Figure 19 - PEEK 450G flexural bar placed symmetrically on a three point bending fixture for DMA tests ....................................................................................................................................................... 40 Figure 20 - PEEK 450G’s storage and loss moduli as a function of temperature at select frequencies .............................................................................................................................................................. 41 Figure 21 - PEEK 450G temperature and frequency data prior to a time-temperature superposition shift ....................................................................................................................................................... 42 Figure 22 - PEEK 450G temperature and frequency data after time-temperature superposition at 110 °C .......................................................................................................................................................... 43 Figure 23 - PEEK 450G shift factors for storage and loss moduli at a reference temperature of 110°C .............................................................................................................................................................. 44 Figure 24 - PEEK 450G Arrhenius fit for a range of 80-130°C at a reference temperature of 117°C . 45 Figure 25 - PEEK 450G WLF fit for a range of 140-180°C at a reference temperature of 150°C ...... 46 Figure 26 - PEEK 450G master curve enforcing shift factors from WLF fit at a reference temperature of 150 °C .............................................................................................................................................. 47 Figure 27 - Directions of rolling and sliding friction at different points of contact in a gear mesh ..... 49 Figure 28 - Common energy loss mechanisms for rolling contact ....................................................... 50 Figure 29 - Diagram of a pin-on-disk tribometer ................................................................................. 54 Figure 30 - Diagram of a twin-disk tribometer..................................................................................... 56 Figure 31 - A schematic of a block-on-ring tribometer ........................................................................ 57 Figure 32 - Isometric view of custom heater design for friction measurements .................................. 58 Figure 33 - Cross-sectional view of custom heater for friction measurements .................................... 59 Figure 34 - The custom heater mounted to screw-driven linear stage on CSM Nano Tribometer ....... 60 Figure 35 - PEEK 450G spherical contact probe with 1.5mm radius .................................................. 61
xiv Figure 36 - Friction measurement of PEEK on PEEK contact............................................................. 61 Figure 37 - Coefficient of friction versus temperature for PEEK on PEEK and steel on PEEK contact (4 mm/s sliding velocity and 30 MPa Hertzian contact pressure) ........................................................ 64 Figure 38 - Coefficient of friction versus sliding velocity for PEEK on PEEK and steel on PEEK contact (25 °C sample temperature and 30 MPa Hertzian contact pressure) ....................................... 65 Figure 39 - Coefficient of friction versus maximum Hertzian pressure for PEEK on PEEK and steel on PEEK contact (25 °C sample temperature and 4 mm/s sliding velocity) ........................................ 66 Figure 40 - Simplification of the research gear's geometry for simulation .......................................... 68 Figure 41 - Depiction of fractional model used in final simulations .................................................... 68 Figure 42 - Full model and fractional model meshes ........................................................................... 70 Figure 43 - Mesh skewness statistics for full model ............................................................................ 71 Figure 44 - Mesh skewness statistics for fractional model ................................................................... 72 Figure 45 - Diagram of normal contact stiffness oscillation as a function of time steps [49] .............. 73 Figure 46 - Illustration of contact (red) and target (blue) surfaces ....................................................... 74 Figure 47 - Boundary conditions for coupled structural-thermal model .............................................. 75 Figure 48 - Heat transfer coefficients specific to the gears in this study .............................................. 76 Figure 49 - PEEK 450G hysteresis as a function of temperature (𝜀 = 0.1 %) ...................................... 80 Figure 50 - Image of a high contact ratio predicted by the simulation ................................................. 84 Figure 51 - Depiction of thermal gradients on polymer gear teeth....................................................... 85 Figure 52 - Comparison of hysteresis and frictional heating (1,000 RPM, 20 N-m) ........................... 86 Figure 53 - Comparison of hysteresis and frictional heating (1,000 RPM, 15 N-m) ........................... 87 Figure 54 - Comparison of results obtained with full model and fractional model (2,000 RPM, 10 Nm) ......................................................................................................................................................... 88 Figure 55 - Simulation of gear test bench conditions prior to experiments.......................................... 89 Figure 56 - Image of 38 tooth gears in the test bench (gears not used in simulation) [54] .................. 90 Figure 57 - one of two identical pairs of torque transducers and couplings on the test bench [54] ..... 91 Figure 58 - Overhead layout of the test bench and instrumentation ..................................................... 92 Figure 59 - Photo of gear test bench and instrumentation .................................................................... 93 Figure 60 - A look at a test gear after it was removed from the test bench .......................................... 93 Figure 61 - Thermal image of 30 tooth test gears at 1,000 RPM and 8 N-m load ............................... 94 Figure 62 - High speed photo of load sharing occurring during 1000 RPM at 8 N-m ......................... 94 Figure 63 - Adjusted simulation heat transfer coefficients................................................................... 95 Figure 64 - Experimental data compared to simulation data for 1,000 RPM at 8 N-m........................ 96 Figure 65 - Experimental data compared to simulation data for 2,000 RPM at 4 N-m........................ 97
xv
INDEX OF TABLES Table 1 - Examples of engineering thermoplastics used in gears ........................................................... 2 Table 2 - Material properties for PEEK 450G [8] .................................................................................. 9 Table 3 - Geometric parameters of test gears ....................................................................................... 14 Table 4 - LPV values of select resins, compounds, and metals ............................................................ 51 Table 5 - Normal load values used to calculate contact pressure for each probe ................................. 63
1
CHAPTER 1 – INTRODUCTION The invention of the gear is one of mankind’s earliest and most notable engineering feats. Early uses of the gear date back to 2600 B.C. when the Chinese used a set of wooden toggle gears in a chariot [1]. Today, gears are still a prominent choice for mechanical power transmission due to their high efficiency. Common examples include automotive transmissions, clocks, power drills, and aircraft landing gear. When considering material selection of gears, the first thing that generally comes to mind are metals, but development of stronger, more thermally stable grades of thermoplastic polymers have been replacing metal gears in many industrial applications. Polymers offer several competitive advantages as an alternative for gear manufacturing. To begin, polymers have a significantly lower density and therefore weigh less than steel gears of equal size. Thus, polymer gears improve efficiency because rotational inertia is decreased and less drivetrain torque is required. Time-dependent material properties of polymers (e.g. viscoelasticity) provide increased acoustic damping and quieter operation. Furthermore, traditional gear hobbing and machining practices limit metal gear geometries due to tooling restrictions. In most cases, polymer gears are manufactured via injection molding, a process by which molten thermoplastic resin is injected under pressure into a mold cavity, where it is then cooled and ejected as a solid part. The injection molding process allows for unique modifications to be made to the involute profiles of gear teeth since the final shape of the gear is governed by the geometry of the mold walls, which is normally limited by the accessibility of tooling during the machining process of metal gears. Lastly, in mass production, polymer gears are cheaper to manufacture since the majority of the investment emanates from the
2
amortizable mold fabrication and not from raw material costs. Further cost-savings arise from significantly shorter cycle times than machined metal gears. Polymer gearing has been in use since the 1950s for low power applications because stiffer, tougher grades of polymers did not yet exist [2]. Today, the market has seen tremendous growth due to increasing demands for gear efficiency. Much of the success can be attributed to the development of stronger, higher-temperature, and more chemically resistive thermoplastics. A list of common engineering thermoplastics used for gear applications is presented in Table 1. Trade Names
Chemical Name
Abbreviation
Nylon, Stanyl®
Polyamide 46
PA46
Nylon, Zytel®
Polyamide 66
PA66
Nylon
Polyamide 12
PA12
Delrin®, Hostaform®
Polyoxymethylene
POM
Fortron®
Polyphenylsulfide
PPS
KetaSpire®
Polyetheretherketone
PEEK
Table 1 - Examples of engineering thermoplastics used in gears
Among these engineered thermoplastics, PEEK is the most recent entrant to the market. PEEK was first synthesized by Imperial Chemical Industries in 1982 and is now manufactured by Victrex, among others, after a management buyout of Imperial Chemical Industries in 1993. PEEK is known for its high temperature performance, mechanical strength and stability, wear resistance, chemical resistance, hydrolysis resistance, electrical performance, and low smoke and toxic gas emission [3]. For gearing applications, PEEK is often compounded with lubricant additives, but could also be compounded with glass or carbon fibers for increased mechanical
3
performance. Due to its industry reputation as a superior engineering thermoplastic and its relatively modern debut, PEEK will be the resin of focus for the research outlined in this thesis. To advance the field of polymer gear design, it is necessary to develop a methodology that outlines the required steps for understanding polymer specific behavior in gear mechanics. This begins with a fundamental understanding of the time, temperature, and strain dependent material properties of the polymer. The reason being that polymer gears have a substantially greater viscoelastic response than metal, where viscoelasticity refers to the part viscous (i.e. liquid) and part elastic (i.e. solid) behavior in a material’s response to deformation. Simply said, a deeper understanding of material properties leads to more accurate predictions of polymer gears’ mechanical and thermal responses. In many cases, gears operate at higher temperatures, relative to ambient conditions, either from self-induced heating due to friction and hysteretic losses or from their environment, such as internal combustion engines. In either case, the polymer gear is more susceptible to higher operating temperatures than a metal gear because of its lower melting temperature. Furthermore, polymers experience a transformation from hard and brittle to soft and leathery at the glass transition temperature (𝑇𝑔 ), which is below the melting temperature. Polymer gears are most susceptible to rapid wear and tooth breakage when operating above the 𝑇𝑔 . Therefore, the ability to predict the operating temperature of a polymer gear based on its load and speed is critical. A methodology has been developed to predict the thermomechanical behavior of polymer gears. It begins in Chapter 2 with a review of the history and background of polymer gearing. This is followed by material characterization through dynamic mechanical analysis (DMA) in Chapter 3. Next, a tribological material characterization is presented in Chapter 4. The
4
measured material properties are implemented in a coupled thermomechanical analysis using ANSYS® in Chapter 5, which has the ability to predict the transient thermal behavior of a polymer gear based on prescribed loading conditions. Finally, the thermal simulation results are verified with experimental results on a gear test bench in Chapter 6. The project work flow is illustrated in Figure 1.
Figure 1 - Depiction of project work flow
5
CHAPTER 2 – HISTORY AND BACKGROUND 2.1 HISTORY OF POLYMER GEARS When polymer gears were first broadly distributed in the 1960s, they were ordered out of catalogues and were designed to the same specifications as metal gears. At this time, the American Gear Manufacturer’s Association (AGMA) had already been in existence since 1916 and had well-established design guidelines and manufacturing standards for metal gearing. Replicating design practices from metal gears to polymer gears was sufficient for low demand applications like consumer electronics, but problems arose for high load bearing applications. The greatest problem was excessive heating on the tooth flanks, since polymers have relatively poor thermal conductivities compared to metals. The first attempt at standardizing design and manufacturing processes of polymer gears was developed in Germany in 1981 with the release of VDI 2545. The primary problem with this standard was the limited material data available for engineering polymers being used for gear design [4]. This problem became a reoccurring theme over the next few decades since copious alterations were being made to thermoplastics by resin manufacturers through the addition of fillers and additives. Moreover, most classical gearing equations were being upheld as the only viable option by which polymer gears could be analyzed. Even with its limitations, VDI 2545 initiated early research efforts to describe the increased contact ratios (i.e. load sharing) that resulted from the less stiff moduli of polymeric materials. The work of Takanashi, Hachmann, and Strickle provided the earliest models for predicting steady-state operating temperatures based on the work of Blok [4]. They suggested the most important prediction factor is temperature because many failure criteria are dependent upon it.
6
Fifteen years later, in 1996, the standard VDI 2545 was withdrawn due to its growing deficiencies. Today, limited material data can still be an issue, but resin suppliers have also become more efficient at providing this data. Advancements in polymer analysis instrumentation have made the once tedious, difficult, and erroneous task of material characterization more user friendly, which has resulted in a more efficient and accurate design process. Many people still feel that polymer gearing should be designed on the premise of strength, stiffness, wear, operating temperature, chemical compatibility, and noise reduction [4]. Though all these criteria are valid, one critical component of material characterization that is still often overlooked is the time-dependency that affects these material properties. As of 2014, parts of a new German polymer gearing standard were released, VDI 2736, which is intended to replace its predecessor VDI 2545. When it is completed, it will contain a more updated material library and cover more elaborate involute designs (e.g. cross helical geometry). In conclusion, polymer gearing applications and materials are perpetually dynamic, and the gear engineer should be prepared to approach problems that leave him or her with a certain level of ambiguity. Also, experiments are expensive and time consuming, and testing production gears on a custom test bench is not always a viable option. Therefore, a methodology that educates the gear engineer on material characterization, simulation, and experimental verification is useful in many circumstances. 2.1.1
Limitations of Metal Gears and Polymer Gears
Though metal gears have their niche for extreme temperature and loading applications, improvements in polymer gear technology have provided opportunities for advancements into applications once thought only suitable for metal gears. To begin, an example of a success story was a case study that replaced a ground iron AGMA 10 gear used in an engine mass
7
counterbalance system with polymer gears made from PEEK, which provided not only a more efficient gear for the application, but also provided a gear that was superior in dimensional accuracy [5]. In this study, the gears made from PEEK provided a 69% reduction in weight, 78% reduction in inertia, which led to 30% reduction in system inertia and 9% reduction in torque in comparison to the ground iron gear. Overall, plastic has a lower investment in high volume applications due to the injection molding process. Depending upon the application, it has been proven that a high performance polymer gear cost 10-50 % less than a metal gear [6]. Part of this cost saving is because no secondary machining, hardening or finishing processes are required [3]. When comparing injection molding to the subtractive nature of machining, greater design freedom is achieved. Furthermore, polymer gears are superior in applications that require a decrease in noise, vibration, and harshness (NVH) due to their damping characteristics. Polymer gears do have disadvantages. First, polymers have significantly less strength than performance metal gears, which makes them only suitable for lower load applications. Also, polymers have large thermal expansion coefficients, which can cause increased wear from meshing offsets. Another common issue is the concentricity of polymer gears due to the tolerance stack-up of mold cores, which are not made with a turning process [7]. Lastly, polymer gears wear rapidly and are more prone to catastrophic failure when exposed to higher temperatures, especially temperatures above 𝑇𝑔 .
2.2 MOTIVATION The highest priority for studying polymer gears is to find current applications exclusive to metal gears that can now benefit from the efficiencies, cost savings, and design freedoms made
8
possible by engineered thermoplastics, such as PEEK. Today, the largest market for strategic polymer gear initiatives is the automotive industry. Legislation and demand for increased fuel economy and lighter vehicles has compelled the automotive industry to seek alternative materials, such as plastics and composites, to replace metal components. The Polymer Engineering Center at the University of Wisconsin-Madison collaborated with Kleiss Gears, Inc. in an effort to provide a fundamental understanding of the thermomechanical behavior of high-speed polymer gearing for automotive and other heavy-duty applications. The project had the following ongoing objectives: material characterization, injection molding simulation, thermomechanical simulation, and experimental validation. The goal of the research in this thesis was to present the first look at a methodology for creating a robust material model that can be utilized by ANSYS® engineering simulation software to model thermomechanical behavior in polymer gears manufactured from PEEK. Based on previous work by Kleiss Gears, it is known that polymer gears in automotive applications can experience temperatures up to 150 °C, torques of 25 N-m, and speeds up 14,500 RPM [7].
2.3 MATERIAL SELECTION The resin used in all measurements and experiments was PEEK 450G from Victrex. POM and PPS were not selected because they do not meet the operating temperature requirement of 150 °C, which is the outer temperature bound a gear might experience when being lubricated with hot engine oil in automotive applications [7]. PA resins were not selected based upon their sensitivity to moisture during operation. Some basic mechanical properties for PEEK 450G are shown in Table 2 and the polymer repeat unit is shown in Figure 2.
9
Table 2 - Material properties for PEEK 450G [8]
Figure 2 - Polymer repeat unit of PEEK
PEEK is a good choice for internal and external engine applications because it will not experience oil particle contamination to all known automotive fluids, it has great corrosion resistance, and does not experience swelling due to moisture [9]. The high average molecular weight of PEEK 450G decreases the crystallinity of the polymer, which reduces shrinkage effects during processing, increases the mechanical strength, and increases ductility.
2.4 DESIGN AND PROCESSING OF POLYMER GEARS The design and processing of polymer gears are heavily reliant upon each other, and need to be considered cohesively in order to produce a dimensionally stable product. There are many types of gear geometries and configurations (e.g. spur, helical, worm, bevel, hypoid, etc.), but
10
the most common gear profiles used are spur and helical, as shown in Figure 3. Spur gears have teeth that are projected radially and parallel to the axis of rotation, are the easiest gears to manufacture, and have the best power transmission efficiency. Helical gears have teeth that are projected at an angle with respect to the axis of rotation, which allows the contact of meshing gear teeth to occur gradually, minimizing noise and vibration. Also, helical gear teeth are longer with respect to spur gear teeth, which allows for better face contact ratios and higher strength.
Figure 3 - Configurations of spur gears (left) and helical gears (right) [10]
The problem with polymer helical gears is that they become increasing difficult to eject at larger helix angles. This is made possible by the use of a rotating mold core, but it also requires a more expensive mold [7]. There is no specific cutting tool required to machine the polymer gear tooth shape, as is necessary with metal gears. Moreover, most polymer gears only mate with one other gear, so the root trochoid can be modified for acceptance of only one mating gear, which offers more
11
load sharing potential and tolerance relief [7]. Kleiss Gears calls this process, “shape-forming the trochoid,” as depicted in Figure 4.
Figure 4 - Hybrid gear tooth geometry used in polymer test gears from Kleiss Gears
Optimized polymer tooth forms are generally referred to as hybrid tooth forms. Polymer tooth forms are easier to modify than metal tooth forms because instead of buying a new hobbing tool, which only makes one tooth form, electron discharge machining (EDM) can be used to modify the mold insert for the specific gear. Wire EDM is capable of cutting gear mold inserts with up to a 15° helix angle. Gear inserts with helix angles greater than 15° must be made by first machining an electrode with the gear’s geometry, and then the electrode is sunk into the mold material to obtain the final profile [7]. Polymer gears used in more demanding applications are often produced by over-molding the polymer around a metal gear hub. Gears are usually located on a shaft by means of a keyway or press fitting. If press fitted, a residual stress is developed in the hub from the compression forces. The purpose of the metal hub is to provide a stiffer internal gear structure that mitigates the potential effects of creep and stress relaxation within the surrounding polymer from the residual stresses.
12
When designing a polymer gear for the injection molding process, a constant wall thickness and symmetric geometry is ideal because it allows for more uniform cooling, shrinking, and residual stress development [7]. Thus, the gear’s hub, web, and rim thickness should equal. Ribs can also be used to provide lateral stiffness to the gear, but the ribs should be offset on opposite sides of the web to prevent thick geometries [10]. Since shrinkage is inevitable, the mold cavity must be properly oversized so that the gear shrinks to its desired dimensions, as shown in Figure 5.
Figure 5 - Example of tooth shrinkage after cooling within mold cavity [7]
The gate design and location is critical when processing polymer gears. A few common gate designs are displayed in Figure 6. The side gate is not desirable in many situations because it creates a weld line on the side opposite of the gate location as well as an anisotropic flow field that causes non-uniform shrinkage. The diaphragm gate is a common preference among many gear manufacturers because the problem of anisotropic shrinkage is eliminated by the development of a radial flow field, but a burr will exist where the runner is detached and the gear core can’t be supported during mold filling. The best option, but the most expensive
13
option, is the multi-pin gate. The multi-pin gate requires a 2 or 3 plate mold to remove the runner system, but it creates the most dimensionally accurate gear and provides a more uniform viscosity during the transient mold filling process [11].
Figure 6 - Injection molding gate designs for polymer gears [11]
The injection molding process requires different quality control metrics than traditional metal gears because the gear is not being turned on a table during forming. The tendencies of semicrystalline thermoplastics to shrink and warp during the molding process provides another challenge for controlling dimensional stability. Therefore, polymer gears should not be tested with traditional gear roll testers because errors can be mismatched or go undetected [7]. In situations that require extreme dimensional stability, the best way to measure the geometry of the teeth and the concentricity of the gear is with a coordinate measuring machine (CMM) [7]. All things considered, production set-up and verification for a polymer gear is on the scale of 3 to 4 months, while a comparable metal gear takes 12-18 months [7].
14
Figure 7 - Coordinate measuring machine used for dimensional quality control [7]
Considering all aforementioned design guidelines, the polymer gear selected for use in the research presented throughout this thesis was a 30 tooth spur gear provided by Kleiss Gears, as represented in Figure 8. A handful of geometric parameters for this gear is presented in Table 3. The gear was manufactured in an over-molding process using neat PEEK 450G resin from Victrex. The hub was manufactured by powdered metal pressing and has a serrated pattern on its outer diameter that promotes adhesion between itself and the PEEK. The tooth is a hybrid design from the shape-forming process performed at Kleiss Gears. A five pin gate design was chosen for a balance of filling pressure, weld line viscosity, and production capability [7]. Operating Pitch Diameter, 𝐷𝑝
Pressure Angle, 𝜃𝑝
Tooth Face Width, ℎ
Operating Tooth Thickness, 𝑡𝑝
Web Thickness
2.3622 in
30°
0.5092 in
0.1179 in
0.1738 in
Table 3 - Geometric parameters of test gears
15
Figure 8 - PEEK test gear and powdered metal hub (left) with gate and runner design (right)
2.4.1
Composite Gears
Composite gears, commonly referred to as reinforced polymer gears, with a thermoplastic matrix and various types of reinforcement have become an increasingly popular option to increase the operating strength. Short and long glass fibers, carbon fibers, milled glass particles, glass beads, and graphene have all been used in composite gear manufacturing [10]. Long fiber reinforced gears have been able to increase the impact and flexural strength of gear teeth without having a significant impact on the wear rate, whereas short fibers tend to cause increased wear rates because the fiber ends abrade contacting surfaces [10]. Predicting processing and mechanical behavior of composite gears can be complex. Composite gears tend to experience greater anisotropic shrinkage than neat polymer gears. Fibers orient themselves parallel to the weld lines and outer edges of tooth flanks which causes less shrinkage to occur in those areas respectively [10][12]. The mechanisms that cause wear among gear engagements of dissimilar materials is also not well understood for composites [13].
16
2.5 FAILURE MECHANISMS OF POLYMER GEARS The ability to efficiently identify gear failure mechanisms and relate that information into design adjustments and operating conditions is an important part of gear design. Luckily, gears have been failing for well over a century, so there are plenty of lessons to learn from. As seen in Figure 9, many gear failure modes can be related to both metal and polymer gears, but the lower operating temperature and viscoelastic nature of polymer gears also give them some unique failure modes aside from metal gears.
Figure 9 - Failure modes of gears [14]
Gear pitting occurs from a combination of surface and sub-surface fatigue stresses which cause micro cracks to form along planes of constant shear stress. This phenomena is a well understood element of Hertzian contact mechanics. Pitting tends to occur less in helical gears with large helix angles, but lower helix angles are better for small loads because of lower power transmission loss [5]. Adhesive normal wear occurs from microscopic welding between opposing wear faces of meshing gear teeth, and the effects are significant at the pitch line of unlubricated polymer gear
17
teeth [10]. Abrasive wear occurs when loose materials or contaminants are caught between wear surfaces. This is a common form of wear for gears that are lubricated with engine oil, especially when the oil degrades and/or becomes dirty. Tooth flank breakage and root breakage are both the product of fatigue, shock loading, stall, or high loads. Crack propagation at the root tends to occur after prolonged bending stress fatigue, while crack propagation at the tooth flank is usually the result of fatigue at a stress riser. Stress risers might exist from the onset of pitting. Also, tooth breakage on only one part of the tooth face might be a sign of misalignment or poor dimensional tolerances during gear manufacture. The two failure modes that are specific to polymer gears are melting and deformation. Since polymers have a much lower critical operating temperature than metals, polymer gears often find themselves near the threshold of melting in heavy duty applications. Thermally induced melting failures are result from heat generated through surface friction and hysteretic losses. For many semi-crystalline polymers, friction coefficients and hysteretic losses increase near and above the 𝑇𝑔 . Thus, as the gear begins to heat up, the rate at which the heat is generated also increases, even if the gear is operating at a constant load and frequency. As a result, localized temperatures at the gear mesh can then exceed the melting temperature of the polymeric material. Deformation in polymer gears is the result of shock loading with low material yield stresses as well as viscoelastic mechanisms, such as creep. Since polymers are a viscoelastic solid, their mechanical response is partially time-dependent. Under continuous operation, the teeth are exposed to a cyclic stress which causes creep and permanent deformation.
18
2.6 LITERATURE REVIEW Thermomechanical analysis between two rubbing surfaces began with the work of Shore in his B.S. thesis at the Massachusetts Institute of Technology in 1925 [15]. Shore studied the temperature effects of machine tools and chips on the wear of the tool. Most importantly, he was the first to apply a creative technique that treated the tool and work piece as the hot junction of a thermocouple [15]. This technique was later applied to a pin-on-disk tribometer by Bowden in 1935 [15]. In 1937 Blok was the first to apply this technique to metal spur gears [15]. In 1963, Blok’s research led to the development of the flash temperature concept, which describes the heat generated between cylindrical contact (e.g. gears, cams, tappets) as “line heat sources.” Blok suggested that temperatures tend to consolidate in a surface layer that is very thin in comparison to the width of the contact line and that the highest temperature tends to occur at the trailing edge of the line’s path of movement, as illustrated in Figure 10.
Figure 10 - Diagram showing the cross section of a line heat source between two contacting surfaces [15]
When the speed is high enough to make the affected temperature surface layer thin enough, the flash temperature, or maximum temperature, can be approximated by:
19
𝑇𝑚𝑎𝑥,1 = 𝐹𝐹 ∙
𝑞𝑎𝑣 𝑤 ∙√ 𝑏 𝑣
(2.1)
where 𝐹𝐹 is form factor that depends on the distribution of the incoming heat flux, 𝑞𝑎𝑣 is the average heat flux applied to the surface, 𝑤 is the width of the heat source, and 𝑣 is its velocity. The variable, 𝑏, is described by:
𝑏 = √𝑘𝜌𝑐𝑝
(2.2)
where 𝑘 is the thermal conductivity, 𝜌 is the density, and 𝑐𝑝 is the specific heat capacity. Furthermore, the average heat flux generated due to sliding friction can be approximated by: 𝑞𝑎𝑣 = 𝜇𝑃𝑎𝑣 𝑣𝑠
(2.3)
where 𝜇 is the kinetic friction coefficient, 𝑃𝑎𝑣 is the contact pressure between the two surfaces. The relative sliding velocity is denoted by: 𝑣𝑠 = |𝑣1 − 𝑣2 |
(2.4)
where 𝑣1 and 𝑣2 represent the velocity by which the conjunction area, or heat source, moves along the two contacting surfaces. In the case of two meshing gears that are made of the same material (i.e. same value of 𝑏), if the relative velocities are fast enough to assume a sufficiently thin thermal layer at the point of contact and the portions of heat provided to each surface are of the ratio 𝑏1 /𝑣1 to 𝑏2 /𝑣2 , then the maximum flash temperature is defined by:
𝑇𝑚𝑎𝑥,2 =
𝜇𝑤𝑃𝑎𝑣 |√𝑣1 − √𝑣2 | 𝑏 √𝑤
(2.5)
20
Blok’s flash temperature equation was developed to predict the maximum temperature at any angular position of a gear mesh. The equation is limited to one dimensional heat transfer and all empirical observations were made using metal spur gears, as the equation was developed before the broad use of polymer gears. In 1983, Gauvin supplemented the work of Blok and others by initializing the discussion and theory of polymer gear thermomechanical analysis. Gauvin’s model considers hysteretic energy losses from the viscoelastic material response of the polymer as well as frictional energy losses, which can be seen in Figure 11 [16]. The volumetric hysteretic energy losses are modeled by: tan(𝛿) 𝑓𝑙𝑜𝑎𝑑 𝜎02 𝐸ℎ = 1 + 𝑡𝑎𝑛(𝛿) 2 𝐸′
(2.6)
where 𝛿 is the out of phase component between the stress and strain responses, 𝑓𝑙𝑜𝑎𝑑 is the loading frequency, 𝐸′ is the storage modulus, and 𝜎0 is the stress amplitude.
21
Figure 11 - Energy losses incurred during loading of a polymer gear tooth [16]
Gauvin concluded that hysteresis at the surface is the result of compression effects, and bending hysteresis effects are likely negligible because the heat generation occurs into the bulk of the tooth. In his experiments, a nitrogen cooled indium antimonide detector (radiometer) was used for surface temperature measurements with a measuring spot diameter of 1.5 mm. Gears made from PA, POM, and ultra-high molecular weight polyethylene (UHMWPE) were tested. Gauvin’s results indicated that temperature increases with increased velocity, load, and module. A regression equation was developed based on experimental data as follows: 𝑏
𝑏
(𝑇𝑚𝑎𝑥 − 𝑇𝑎𝑚𝑏 ) = 𝑏0 𝑊𝑡 1 𝑣𝑝 2 𝑚𝑏3
(2.7)
where 𝑇𝑚𝑎𝑥 is the maximum gear tooth surface temperature, 𝑇𝑎𝑚𝑏 , is the ambient temperature, 𝑊𝑡 is the transmitted load per unit width of the tooth, 𝑣𝑝 is the pitch line velocity, 𝑚 is the module, and 𝑏0 , 𝑏1 , 𝑏2 , 𝑏3 are the regression fitting coefficients.
22
Koffi was the first to propose an iterative technique to be performed by a computer as well as a simplified hand calculation approach, which was presented in his thesis in 1983 [17]. The computational approach considered both forms of heat generation, frictional and hysteresis, but he found that hysteresis energy loss is very minimal and can be neglected for simplified hand calculations [17]. Koffi was able to describe a theoretical approach to calculating heat generation from higher contact ratios that arise from less stiff materials such as plastics. His model for frictional heating energy for a unit face width is:
𝐸𝑓,𝑤 = 𝜇𝑊𝑡
𝑊𝑖 1 𝑣∆ 𝑊𝑛 𝑐𝑜𝑠𝜃𝑝 𝑠 𝑠
(2.8)
where 𝑊𝑖 /𝑊𝑛 is the load sharing factor, 𝜃𝑝 is the pressure angle, and Δ𝑠 is the displacement of the contact point along the line of action. To determine the sliding velocity, Koffi proposed a geometrical condition that relates the pitch line velocity to a constant 𝐾 for contact occurring along the line of action that is calculated by: 𝑣𝑠 = 𝐾𝑣𝑝
𝐾 = [𝑐𝑜𝑠𝜃𝑝
𝑖1 + 𝑖2 𝑆 2𝜋 | |] 𝑖1 𝑖2 𝑝𝑛
(2.9)
(2.10)
where 𝑖1 and 𝑖2 represent the driver and driven gear tooth number, 𝑆 is the distance between the pitch point and the contact point along the line of action, and 𝑝𝑛 is the normal base pitch. Koffi also developed his own approach to calculating the energy losses due to hysteresis for a unit face width [17]:
23
𝐸ℎ,𝑤
tan(𝛿) 𝜎02 𝑡𝑝 𝜋 = 𝐾𝑣 ∆ 1 + 𝑡𝑎𝑛2 (𝛿) 4 𝐸′𝑐𝑜𝑠𝜃𝑝 𝑠 𝑠
(2.11)
where 𝑡𝑝 is the tooth thickness at the pitch circle. Koffi’s equations for heat generation due to friction and hysteresis were then solved iteratively on a computer by incrementing Δ𝑠 in very small steps. Koffi demonstrated reasonably good correlation between his iterative computer model and his simplified approach, which could be solved without the assistance of a computer [17]. Another approach to computationally solving the hysteresis, which is clever because it is not specific to the application of gears, is the work of Lin in 2004 [18]. Lin studied the temperature prediction of rolling tires by applying material measurements to a finite element analysis (FEA) software. Using a dynamic mechanical analysis (DMA), Lin measured the storage modulus 𝐸′ and loss modulus 𝐸′′ at different strain rate frequencies and temperatures, from which the hysteresis of the rubber tire was calculated [18]:
𝐻=
𝐸′′ 𝐸∗
𝐸 ∗ = √(𝐸′)2 + (𝐸 ′′ )2
(2.12)
(2.13)
where 𝐸 ∗ is the complex modulus. Lin related the applied hysteresis value to the strain energy density of each finite element within the model, and assumed that all lost energy was dissipated as heat. He then applied the volumetric heat dissipation per radial distance of the tire to a twodimensional tire cross section. The hysteresis energy losses were calculated as follows:
24
𝐻=
𝑈𝑙𝑜𝑠𝑠 𝑈𝑚𝑎𝑥
𝐸ℎ = 𝑈𝑙𝑜𝑠𝑠 𝑓
(2.14)
(2.15)
where 𝑈𝑚𝑎𝑥 is the total strain energy density provided by the FEA, 𝑈𝑙𝑜𝑠𝑠 is dissipated strain energy density, and 𝑓 is the frequency of the tire. The following year in 2005, Senthilvelan studied the effect of rotational speed and surface stress on unreinforced and reinforced PA66 gears and noted that both speed and stress (i.e. transmitted load) positively correlated to an increased operating temperature of the gears. He also noted that the torque had the most significant effect on the temperature increase [19]. Between 1993 and 2010, Hooke and Mao collectively conducted several studies on unreinforced and reinforced polymer gears and examined wear as well as thermal properties [20][21][22][23]. Their studies were primarily focused on POM and PA composite gears (55 % PA, 30 % glass fiber, 15 % polytetrafluoroethylene (PTFE)). During his studies, Mao found that the operating temperature rise of gears is nearly independent of the angular velocity and completely dependent upon transmitted load [23]. The gears were tested at a maximum speed of 2,500 RPM. Mao provided empirical and numerical explanations of the heat transfer occurring with the gears. He concluded that the heat transfer is extremely complex, but by using a thermocouple he was able to make a prediction about the polymer gear acting in a similar manner to a gear fluid pump, where the air becomes trapped between adjacent teeth on each surface, as depicted in Figure 12. Mao assumes that all air between adjacent gear teeth remains trapped during rotation and that the low thermal conductivity of polymers means that all generated heat from friction is convected to the surroundings during steady-state operation.
25
Figure 12 - Polymer gear heat transfer related to the concept of a gear pump [21]
From Mao’s theory, the frictional power loss for one tooth is approximated by:
𝐸𝑓 = 𝜇𝑃𝑎𝑣 𝑣𝑠 =
𝜇𝑃𝑎𝑣 𝜋𝜔𝑟𝑏 𝜋𝜇𝑃𝑇 = 𝑍 𝑍
(2.16)
where 𝜔 is the angular velocity, 𝑟𝑏 is the base circle radius, 𝑍 is the tooth number, and 𝑃𝑇 is the transmitted power. Equation 2.16 can be modified to accommodate load sharing between teeth, which is a common phenomenon for polymer gear drives. A contact ratio of 1.5 is represented by:
𝐸𝑓,1.5 = 𝐸𝑓 +
𝐸𝑓 0.5 1.25𝜋𝜇𝐸𝑓 = 2 𝑍
(2.17)
The volume of fluid transported between the teeth of a gear pump can be approximated by: 𝑉𝑓𝑙𝑢𝑖𝑑 = 2𝜋ℎ(𝑟2𝑜𝑢𝑡 − 𝑟2 )
(2.18)
where 𝑟𝑜𝑢𝑡 is the outside radius, 𝑟𝑎 is the reference radius, and ℎ is the tooth face width. Assuming that all frictional heat is transmitted to the pumped fluid, Equations 2.17 and 2.18 are combined to yield:
26
𝐸𝑓,1.5 = 𝜔𝑉𝑓𝑙𝑢𝑖𝑑 Δ𝑇𝐵 𝑐𝑝 𝜌
(2.19)
where Δ𝑇𝐵 is the temperature rise of the gear, which when combing the above equations can now be approximated by:
Δ𝑇𝐵 =
0.625𝜇𝑇𝑇 𝑐𝑝 𝑍ℎ(𝑟2𝑎 − 𝑟2 )
(2.20)
where 𝑇𝑇 is the transmitted torque. Mao was able to prove sufficient correlation of this formula with his measurements up to 1,500 RPM. He concluded that that surface temperature rise of the gear is from three components: the ambient temperature of surrounding air, bulk temperature rise of the gears above ambient temperature, and the short duration flash temperature rise of the contact zone [21]. Following the work of his gear pump approximation, Mao described a procedure for a two-dimensional numerical heat transfer prediction of the flash temperature, but he neglected the effects of hysteresis [22]. Most recently, in 2010 Letzelter developed a custom experimental apparatus for focusing an infrared camera on a refined part of the gear mesh. He used this apparatus to compare the effect of humidity on PA66 gears, and he correlated the effect of humidity on PA66’s mechanical properties with measurements from a DMA [24]. Letzelter concluded from his thermomechanical measurements that it is essential to have robust friction measurements to predict temperature rise with any level of accuracy.
27
CHAPTER 3 – DYNAMIC MECHANICAL ANALYSIS 3.1 THEORY 3.1.1
Viscoelastic Behavior
Polymeric materials have a mechanical response that is part liquid and part solid. Whether the response is dominated by the liquid attributes or solid attributes is a function of the time scale (i.e. strain rate), temperature, strain magnitude, and average molecular weight of the specific polymer. Hence, materials that are classified as having both a viscous and an elastic response are referred to as being “viscoelastic.’ When a viscoelastic material experiences a stress or strain input, part of the energy is stored and part of the energy is dissipated. For polymers, the dissipated energy can take the form of thermal, acoustic, or chemical energy. For ease of understanding and mathematical convenience, viscoelastic models are often modeled using spring and dashpot elements. This allows for constitutive relationships to be developed between the stress and strain response. From Hooke’s law, for a spring the stress is linearly related to the strain through the stiffness of the material: 𝜎 = 𝐸𝜀
(3.1)
𝜏 = 𝐺𝛾
(3.2)
where 𝜎 is the normal stress, 𝐸 is the Young’s modulus, 𝜀 is the normal strain, 𝜏 is the shear stress, 𝐺 is the shear modulus, and 𝛾 is the shear strain. For a dashpot (i.e. viscous damper) there is a Newtonian stress-strain relationship, which follows: 𝜎 = 𝜂𝜀̇ where 𝜂 is the viscosity and 𝜀̇ is the strain rate.
(3.3)
28
Figure 13 - Comparison of simple elastic and viscoelastic models
Figure 13 illustrates a Hookean solid and the simplest forms of viscoelastic models: the Maxwell model and the Kelvin-Voight model. The Hookean solid is governed by Hooke’s law from Equation 3.1. In the Maxwell Model, a single spring and dashpot are placed in series. The inelastic strains occurring in the viscous component of the Maxwell Model are not recoverable, which makes it useful for the Boltzmann superposition principle [25]. The Boltzmann superposition principle states that the final deformation of a polymer is equal to the summation of all load induced strains that occurred at different times in that polymer’s life cycle. Thus, the Maxwell Model represents solid behavior on short time scales and liquid behavior on long time scales [25]. The governing equation for the Maxwell Model is:
𝜎+
𝜂 𝑑𝜎 𝑑𝜀 =𝜂 𝐸 𝑑𝑡 𝑑𝑡
(3.4)
where 𝑡 is time. The Kelvin-Voight Model, often simply called the Kelvin Model, is also a single spring and dash pot, but aligned in a parallel. As a result of this arrangement, the stress in a Kelvin Model remains constant, and does not relax. In other words, the Kelvin Model
29
allows all deformation to be recovered over time [25]. Therefore, the Kelvin Model is best used to model the behavior of viscoelastic solids. The governing equation for the Kelvin Model is: 𝜎 = 𝐸𝜀 + 𝜂𝜀̇
(3.5)
When the Kelvin Model is subjected to a sinusoidal strain input, such as the input of a forced resonance DMA test, the strain as a function of time is [25]: 𝜀(𝑡) = 𝜀0 sin(𝜔𝑡)
(3.6)
where 𝜀0 is the strain amplitude. By combining Equations 3.5 and 3.6 and differentiating, the time dependent stress becomes [25]: 𝜎(𝑡) = 𝐸𝜀0 sin(𝜔𝑡) + 𝜂𝜔𝜀0 cos(𝜔𝑡)
(3.7)
From Equation 3.7, the components of the storage and loss modulus can be isolated: 𝐸′ = 𝐸
(3.8)
𝐸′′ = 𝜂𝜔
(3.9)
When a viscoelastic model is held under constant stress, the resulting strain is called creep. When the model receives a constant input strain, the resulting stress change is called stress relaxation. Both stress relaxation and creep are related to the same molecular mechanisms, and the time scales at which they occur is related to:
𝜏𝑅 =
𝜂 𝐸
(3.10)
30
where 𝜏 is relaxation time in the Maxwell Model and the retardation time in the Kelvin Model [26]. Generally, experimental data can be more accurately modeled by adding more springs and dashpots to a given model in different configurations. Examples of these elaborated models are the Jeffrey Model, Standard Model, and Generalized Maxwell Model, which have a more robust ability to model experimental data. The challenge with fitting these models, though, is that the models require higher order coefficients to be predicted. In reality, molecular structures and mechanisms do not correspond precisely to springs and dashpots, but the concept provides accurate results when executed properly. 3.1.2
Time-Temperature Superposition
As discussed, polymers have a time dependent response to stress or strain inputs. The time scales are a function of the supramolecular rearrangements of the polymer chains. The premise of time-temperature superposition (TTS) principles is that the rate of molecular motion is directly related to the temperature. Therefore, the time scale of the response will decrease at higher temperatures, and increase at lower temperatures. As a result, relatively short time scale behavior that is observed at higher temperatures can be shifted to longer time scales by decreasing the given reference temperature. In order to carry out the procedure of TTS, experimental stress-strain data needs to be generated under a range of isothermal conditions over similar time scales. One of the isothermal data sets is then chosen as a reference temperature, and all other data is shifted horizontally along the time or frequency axis until the ends of all curves become superimposed [25]. Once all data has been shifted, the resulting combination of superimposed curves is referred to as a “master
31
curve.” The amount each data set is shifted relative to the reference temperature data is called a shift factor and is calculated by:
log 𝑡 − log 𝑡𝑟𝑒𝑓 = log (
𝑡 𝑡𝑟𝑒𝑓
) = log 𝑎 𝑇
(3.11)
where 𝑡𝑟𝑒𝑓 is the reference time and 𝑎 𝑇 is the shift factor as a function of the reference temperature. After the shift factors are obtained, they can be plotted against temperature for fitting to TTS equations. Two well-known and widely used TTS equations are the WilliamsLandel-Ferry (WLF) equation and the Arrhenius equation [27]. The WLF equation assumes that the polymer is isotropic, homogeneous, and amorphous and is given by:
log(𝑎 𝑇 ) = −
𝐶1 (𝑇 − 𝑇𝑟𝑒𝑓 ) 𝐶2 + (𝑇 − 𝑇𝑟𝑒𝑓 )
(3.12)
where 𝐶1 and 𝐶2 are material contants, 𝑇 is temperature corresponding to the shift factor, and 𝑇𝑟𝑒𝑓 is the reference temperature that all data sets were shifted with respect to. The WLF equation works well for a range from 𝑇𝑔 , to temperatures as high as 𝑇𝑔 + 100°C [27]. For more precise curve fitting, a vertical shift factor is sometimes used to account for density changes as a function of temperature, which is given by:
𝑏𝑇 =
𝜌(𝑇1 )𝑇1 𝜌(𝑇2 )𝑇2
(3.13)
where 𝑇1 and 𝑇2 represent the respective temperatures of the densities. The Arrhenius equation is useful for curve fitting shift factors outside the range of the WLF equation. Also, it has been universally used to model many other temperature dependent mechanisms, such as chemical reaction kinetics. The Arrhenius equation for shift factors is:
32
log(𝑎 𝑇 ) =
𝐸𝑎 𝑅(𝑇 − 𝑇𝑟𝑒𝑓 )
(3.14)
where 𝐸𝑎 is the activation energy and 𝑅 is the universal gas constant. A combination of Arrhenius and WLF equations can be used to predict the shift factors 𝑎 𝑇 at a wide range of temperatures both near the 𝑇𝑔 and well outside. It should be noted, though, that such techniques are best fit for amorphous polymers and limited in application to semi-crystalline polymers [26].
3.2 APPLICATION TO GEARS A firm understanding of the polymer gear’s operating conditions is necessary for an efficient DMA. Since polymer gears are prone to failure when operating above their 𝑇𝑔 , the temperature range in this vicinity is perhaps the most critical focal point of the analysis. Furthermore, the time scale of the DMA measurements can be related to the loading cycle for a single tooth. For example, consider the strain energy density as a function of time for a polymer gear tooth obtained from a finite element analysis, as displayed in Figure 14.
33
Figure 14 - Plot of strain energy density versus time for one revolution of a polymer gear at 1,000 RPM and 10 Nm torque for an arbitrary element located on the surface of a loaded gear tooth
The time scale along the x-axis is representative of a single rotation of the gear, and the strain energy density for an arbitrary finite element located on the loaded surface of the polymer gear tooth is shown by the thick continuous line. By observation, it can be seen that the energy storage behavior can be approximated by a sinusoidal function, as indicated by the dotted line. This concept is important since equipment for the majority of DMA apparatuses apply a sinusoidal strain or stress to the test sample and measure the resulting material response. Therefore, the sinusoidal energy storage behavior and strain rate behavior of polymer gears is very fitting for a DMA analysis. As a result, the loading frequency of a polymer gear can be related to the rotational frequency 𝑓 and number of teeth 𝑁𝑡 by: 𝑓𝑙𝑜𝑎𝑑 = 𝑓𝑁𝑡
(3.15)
34
Others have successfully applied dynamic mechanical analysis to polymer gears. Letzelter used dynamic mechanical analysis and an Arrhenius shift function to describe the retardation times and moduli of PA66 gears [24]. Many DMA tests have been performed on PEEK 450G as a function of temperature, strain, and frequency, but limited data is available for higher frequency ranges characteristic of gears [28]. Proof exists that the time-dependency and hysteresis of PEEK 450G can be mitigated by increasing the fiber content in composite gears [29][30]. Furthermore, the operating temperature of PEEK can be lowered by addition of carbon fibers, which allows surface temperatures to thermally diffuse more rapidly.
3.3 MEASUREMENT TECHNIQUES 3.3.1
Forced and Free Resonance Analyzers
DMA is achievable by both forced and free resonance methods. Forced resonance methods are more common due to their commercialization for industry and versatility. A forced resonance DMA uses mechanical actuation to oscillate the test specimen at a desired frequency. Strain controlled inputs are the most common choice for mechanical actuation, but some machines do allow the user to also perform stress controlled inputs. For clarity, a sinusoidal strain input and its respective stress response are shown in Figure 15, where the time lag in response Δ𝑡 is evidence of viscoelastic behavior.
35
Figure 15 - Depiction of a forced strain input and the time-lagging stress response
Many fixtures exist that allow the sample to be tested in either compression, shear, extension, or flexural positions. The flexural test fixture was used for all measurements of PEEK 450G throughout this thesis, as portrayed in Figure 16. The test fixtures and sample were located in a precision convection oven for temperature control during experiments. The DMA machine is normally accompanied by analysis software to efficiently calculate the storage and loss moduli of a material. Some common relationships among these values are [25]:
tan(𝛿) =
𝐸′′ 𝐺′′ = 𝐸′ 𝐺′
(3.16)
𝛿 = 𝜔Δ𝑡
(3.17)
𝐸∗ 𝐺 = 2(1 + 𝜐)
(3.18)
𝐺 ∗ = √(𝐺′)2 + (𝐺 ′′ )2
(3.19)
∗
36
𝜂=
3𝐺 ∗ 𝜔
(3.20)
where 𝐺′ is the storage shear modulus, 𝐺′′ is the loss shear modulus, 𝐺 ∗ is the complex shear modulus, and 𝜐 is the Poisson’s ratio. The relationship among some of these variables for PEEK 450G as a function of temperature is depicted in Figure 17. The drawback with forced resonance machines is that they are typically limited to frequencies of 100 Hz on the high end. In comparison, high speed gearing will see loading frequencies in excess of 10 kHz. Thus, forced resonance DMA was coupled with TTS principles to predict behavior at higher loading frequencies.
Figure 16 - 3 point bending fixture commonly used for stiff materials in a forced resonance DMA test [31]
37
′ Figure 17 - Normalized DMA data for PEEK 450G as a function of temperature (6.2 Hz, 0.2 % strain, 𝐺𝑚𝑎𝑥 =1.26 GPa, ′′ =0.11 GPa) 𝐺𝑚𝑎𝑥
Free resonance DMA, or resonant ultrasound spectroscopy, requires that the specimen be cubic, cylindrical, spherical, or annulus in shape [32]. The specimen is placed between two ultrasonic transducers in which one transmits the resonant frequency and the other receives the signal. The specimen can only be analyzed at its resonant modes, which are a function of the specimen geometry. Therefore, free resonance DMA is limited to frequency ranges of 50 kHz to 20 MHz, which is generally above the loading frequencies of polymer gears [32]. 3.3.2
Atomic Force Microscopy
Atomic force microscopy (AFM) was developed as a more resolute alternative to optical diffraction methods for topographical analysis of microstructures. The device adjusts the position of the sample using a piezo driven stage while the deflection of a probe in constant
38
contact with the sample surface is measured using a laser and photodiode, as depicted in Figure 18. DMA measurements are made possible by applying a forced resonance to the sample through the piezo driven stage and measuring the response of the cantilever deflection. The resulting amount of indentation into the sample and the phase lag between the stage and cantilever displacement signals can be used to determine the storage modulus, loss modulus, and tan 𝛿. Current AFM technology can be used to measure material properties as frequencies as high as 1 kHz.
Figure 18 - Schematic of atomic force microscope [33]
3.3.3
Dielectric Thermal Analysis
Dielectric thermal analysis (DETA) operates on the same principles as electrical capacitance. The insulating polymeric sample is place between two electrodes, where one electrode is transmitting an electrical signal at a fixed frequency, and the other electrode is receiving the response signal. DETA is not suitable for directly measuring storage and loss moduli, but it is useful for measuring the tan 𝛿 at frequencies greater than 100 MHz.
39 3.3.4
Custom Apparatus
There are many other forms of DMA machines other than those aforementioned, but the techniques presented in this thesis have been found to be the most suitable for loading frequencies of polymer gear drives. Custom DMA devices have been constructed for research that are capable of measuring data frequencies from 100 kHz to the sub-resonant range (mHz) (over 11 decades on the log scale) [32]. This was achieved by fixing one end of a cylindrical specimen and cementing a magnetic to the free end. A torque was then applied to the magnetic on the free end by use of a Helmholtz coil. The resulting displacement was measured using a laser and split-silicon light detector [32]. Such a device could be highly useful to understanding polymer gear dynamics because it eliminates the limitations of TTS.
3.4 EXPERIMENTS 3.4.1
Equipment and Methods
A forced resonance DMA machine, TA RSA III Rheometrics System Analyzer, was used to characterize PEEK 450G’s dynamic response using the TA Orchestrator software. Due to the relatively high stiffness of PEEK in comparison to other thermoplastics and machine transducer limitations, a three point flexural bending fixture with a 40 mm span was used for analysis rather than a tensile fixture, as might be used for thin films. Test specimens were cut in 50 x 12 x 2 mm3 rectangular bars from 150 x 150 x 2 mm3 injection molded plaques of PEEK 450G using a CNC micro end mill. The rectangular bars were then deburred and measured before testing as shown in Figure 19.
40
Figure 19 - PEEK 450G flexural bar placed symmetrically on a three point bending fixture for DMA tests
Testing was performed with the intent of determining PEEK 450G’s compatibility with TTS by carrying out a dynamic frequency-temperature sweep. A dynamic frequency-temperature sweep measures the mechanical properties (e.g. 𝐸′and 𝐸′′) through a range of temperatures and discrete frequencies. The input sinusoidal strain amplitude was set to a value of 0.1 % and a maximum of ten values of frequency were chosen for testing: 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100 Hz. It is important to note that the maximum test frequency for the machine was 100 Hz, as is the case for many forced resonance DMAs. The specified temperature range was from 80 °C to 180 °C, which includes two important operating conditions for this analysis: the temperature of automotive engine oil used to lubricate the gears (~110 °C) and the 𝑇𝑔 of PEEK 450G (~143 °C) [8]. A temperature increment of 2 °C was chosen, which implies that the ten discrete frequencies were tested every 2 °C. After the temperature was ramped 2 °C, the specimen was allowed to equilibrate for two minutes to establish isothermal conditions within
41
the sample. Lastly, a static load of 50 g was maintained during testing because it was observed prior to testing that this load ensured full contact with the three point bending fixture. 3.4.2
Results
As seen in Figure 20, both the storage and loss modulus were a function of temperature and frequency. The rapid decrease in 𝐸′ or increase in 𝐸′′ indicated the onset of the glass transition for PEEK 450G. By increasing the loading frequency from 1 Hz to 100 Hz, the 𝑇𝑔 increased nearly 10 °C. Hence, a polymer gear operating at higher frequencies should have an increased thermal operating limit when compared to a polymer gear at lower frequencies.
Figure 20 - PEEK 450G’s storage and loss moduli as a function of temperature at select frequencies
To generate a more versatile and collective understanding of the data, a TTS was performed. The TTS began with grouping each of the ten frequencies with their respective temperature, and then the storage or loss modulus were plotted versus temperature for all the groups on the
42
same chart as displayed in Figure 21. In this case, storage modulus was chosen for the dependent variable. A TTS with loss modulus as the dependent variable was performed to verify the result, because if TTS is applicable to the material, the choice of storage or loss modulus as the dependent variable will yield the same results.
Figure 21 - PEEK 450G temperature and frequency data prior to a time-temperature superposition shift
Next, the 110 °C data set was chosen as the reference temperature and all data sets were shifted either to the right or left with respect to that data set so that the ends of each set were superimposed. Though this process can be done visually, it is more accurate to fit each data set with a cubic spline function and allow the alignment of adjacent sets to be optimized by minimizing the regression residual error, as was done in Figure 22. As a result, based on the
43
chosen reference temperature, frequency behavior well outside the measured frequency range of 10-1 to 102 Hz was predicted.
Figure 22 - PEEK 450G temperature and frequency data after time-temperature superposition at 110 °C
Using Equation 3.11, where time is the inverse of frequency, the duration that each data set is shifted with respective to the reference temperature data set was calculated as a dimensionless shift factor 𝑎 𝑇 . The shift factors corresponding to a TTS done separately with the storage modulus and loss modulus as the dependent variable are overlaid in Figure 23. At temperatures below 160 °C, shift factors between 𝐸′ and 𝐸′′ were nearly identical, but some deviation did occur at temperatures above 160 °C. This could be attributed to cold crystallization, which is characteristic of semi-crystalline polymers above 𝑇𝑔 .
44
Figure 23 - PEEK 450G shift factors for storage and loss moduli at a reference temperature of 110°C
There are two distinct trends among the shift factors in Figure 23. As discussed previously, the Arrhenius equation was adequate for modeling shift factors below the 𝑇𝑔 , where the logarithm of the shift factor is linear with respect to temperature. For PEEK 450G, there was only a subtle change in shift factors below the 𝑇𝑔 . The degree of change in shift factors with respect to temperature for this range was modeled with the Arrhenius equation as exemplified in Figure 24. For this model, the calculated activation energy was 31.984kJ/mol and the reference temperature was 117 °C. If the goal of TTS is to predict mechanical behavior at frequencies well outside the range of test frequencies, then the small change in relative shift factors predicted by the Arrhenius equation was limited in its applicability to TTS.
45
Figure 24 - PEEK 450G Arrhenius fit for a range of 80-130°C at a reference temperature of 117°C
In contrast, the WLF equation was best for modeling the behavior of shift factors from the 𝑇𝑔 to 𝑇𝑔 + 100°C. The WLF fit was designed for amorphous thermoplastics, but as demonstrated in Figure 25, the WLF fit also worked well with semi-crystalline PEEK 450G. The calculated fitting parameters were 𝐶1 = 24.476 and 𝐶2 = 78.284 at a reference temperature of 150 °C. A reference temperature of 150 °C was the minimum reference temperature that allowed the TTS to predict mechanical behavior above loading frequencies of 1 kHz, which is common with polymer gears.
46
Figure 25 - PEEK 450G WLF fit for a range of 140-180°C at a reference temperature of 150°C
Finally, once a successful fit to a TTS model was achieved, the data was shifted with respect to the shift factors predicted by the model. For example, the WLF fit was used to shift the data sets ranging from 140 °C to 180 °C, as displayed in Figure 26. At this point, the ultimate goal of the TTS was partially achieved, which was predicting the mechanical response of PEEK 450G at loading frequencies ranging from 100 Hz to 10 kHz. Based on the shift factors obtained with the WLF fit, a minimum reference temperature of 150 °C was required to predict behavior in this frequency range. Behavior at this temperature is important because it is a critical temperature for rapid wear of PEEK gears, but this temperature would likely be avoided in design considerations. Thus, a better understanding of the mechanical behavior of PEEK below 150 °C is necessary.
47
Figure 26 - PEEK 450G master curve enforcing shift factors from WLF fit at a reference temperature of 150 °C
Ideally, the Arrhenius fit would have provided the TTS needed to determine the mechanical behavior at lower operating temperatures, but the shift factors were not significant enough to produce this data. That provided two valuable conclusions from the TTS procedure for PEEK 450G. First, there was minimal shift in material response below 130 °C, which allowed the assumption to be made that PEEK 450G mechanical properties were not heavily dependent on frequency in that range. Second, TTS was not entirely sufficient for analysis of PEEK 450G and that the data should be verified with AFM, resonant ultrasound spectroscopy, or a custom technique that can explicitly measure at the operating frequencies of polymer gears.
48
CHAPTER 4 – TRIBOLOGICAL ANALYSIS 4.1 GEAR TRIBOLOGY 4.1.1
Sliding Friction and Rolling Resistance
Dry sliding friction is governed by three laws based on the work of Amontons and Coulomb: 1. Friction force is proportional to the normal load. 2. Friction force is independent of the apparent area of contact. 3. Friction force is independent of the sliding velocity. The frictional behavior of polymers, including gears, deviates from these laws in varying degrees [34]. Consequently, failure to inquire upon these relationships during the design of polymer gears could generate haphazard results. Modern gearing has superb power transmission efficiency, however, it is the energy losses due to friction that tend to have the largest implications on metal and polymer gear failure mechanisms. The involute profile of a gear is designed to operate in pure rolling contact at the pitch point, but as the line of contact deviates from the pitch point near the root and tip of the tooth, there is an abrupt change from rolling to sliding friction, as shown in Figure 27. The change from rolling to sliding friction is caused by different relative velocities between gear teeth since the point of contact no longer lies on the pitch circle [35]. The majority of sliding energy lost during sliding friction is dissipated as heat and can be approximated using Equation 2.3.
49
Figure 27 - Directions of rolling and sliding friction at different points of contact in a gear mesh
Furthermore, the radii at the region of contact vary during the translation of the gear due to the involute geometry, thus, the sliding and rolling velocities are not constant. Rolling and sliding phenomena eventually lead to surface cracking, surface fatigue, and heat build-up, which all result in increased wear [10]. The surface contact stress at the point of contact, which can be approximated with Hertzian theory, also varies along the line of action, and can be approximated from Hertz’s theory for two contacting cylinders:
𝜎𝐻 =
𝑊𝑡 √ 𝐷𝑝
1 𝜐𝑝2
1− 𝜋( 𝐸 𝑝
+
𝜐𝑔2
1− 𝐸𝑔
𝑚𝑔 =
1 𝑐𝑜𝑠𝜃𝑝 𝑠𝑖𝑛𝜃𝑝 𝑚𝑔 ) 2 𝑚𝑔 + 1
𝑁𝑡,𝑔 𝑁𝑡,𝑝
(4.1)
(4.2)
where 𝐷𝑝 is the pitch diameter, 𝐸𝑝 is the pinion modulus, 𝐸𝑔 is the gear modulus, 𝑚𝑔 is the speed ratio, 𝑁𝑡,𝑔 is the number of gear teeth, and 𝑁𝑡,𝑝 is the number of pinion teeth. In gear
50
contact, the highest Hertzian stresses occur when the driving tooth tip initially contacts the lower flank of the driven gear near the root [36]. Since friction implies that two surfaces are moving at relative velocities, the term “rolling friction” causes some ambiguity. Therefore, the energy losses associated with the rolling action of gear teeth are better referred to as rolling resistance [37]. In general, for rigid materials, energy losses for rolling resistance are much less than those for sliding friction. The energy lost during rolling friction can be attributed to, but is not limited to, several factors: elastic hysteresis, plastic deformation, adhesion hysteresis, electrostatic effects, and interfacial slip [37]. Elastic hysteresis occurs from the damping properties of the material, as depicted in Figure 28. Plastic deformation occurs when surface imperfections are deformed, which is not a significant issue for gearing since wear tracks and work-hardening occur early in the life cycle of a gear. Adhesion hysteresis is the pulling of surface atoms, and energy losses ensue when the atoms return to equilibrium. In dielectric and semi-conductor applications, an asymmetric electric double layer is formed between contacting surfaces, which creates an opposing moment on the direction of roll. Certainly, any form of rolling resistance that causes deformation can potentially cause interfacial slip on a micro-scale.
Figure 28 - Common energy loss mechanisms for rolling contact
51 4.1.2
Wear
In metal and polymer gears, wear is the product of cyclic fatigue and friction mechanisms, but melting is specific to polymer gears. In polymer to polymer contact, there is a limited capacity to conduct heat away from the surface, which results in a higher flash temperature, possibly greater than 𝑇𝑚 , at the mating surface. The limiting pressure and velocity (LPV) is a common metric used to quantify the tribological performance of materials. The LPV is calculated as the product of the sliding velocity and contact pressure at which excessive thermal wear from friction occurs, or crack growth and material ploughing occur from pressure [38]. Table 4 compares the LPV value of several thermoplastic resins, compounds and bearing metals, including PEEK 450G [38]. Material
LPV [MPa-m/min]
PEEK 450G
145
PEEK 450FC30 (30% Carbon + PTFE)
794
PA66 + Graphite, Glass Fiber
71
Polyimide + Graphite
895
POM
71
PTFE + Carbon
447
White Metal
265
Oil Impregnated Bronze
804
Table 4 - LPV values of select resins, compounds, and metals
For polymer gears, Mao found that a critical operating torque existed, similar to the LPV [39]. He categorized the wear into three stages: an initial running-in, a nearly linear stage, and a final fracture stage. The final fracture stage only occurred at loads above the critical torque. The
52
critical torque is directly correlated to a rapid increase in heat, which in effect increased the wear rates. 4.1.3
Lubrication and Additives
Lubrication is used in the majority of polymer gear applications, and is either blended into the polymer during processing, applied once during initial assembly of the gear drive, or applied continuously through lubrication system during the polymer gear’s life. Widely used lubricants in non-continuous applications include PTFE powder and silicone fluid. The PTFE powder works well because it smears into a thin film during long-term cyclic use [10]. Silicone works well during start-up, and is often mixed with PTFE to provide optimal lubrication during the early and late stages of a polymer gear’s operation. It is also common for various amounts of graphite and PTFE to be blended into the polymer during processing. For automotive applications, polymer gears are lubricated with the oils already in the vehicle components. For example, polymer gears being tested for use in engine transmissions should be lubricated with transmission fluid (e.g. Dexron VI) [9]. 4.1.4
Acoustics
The majority of energy loss in polymer gears is dissipated as heat, but some there is also energy dissipation due to sound. Acoustical noise in polymer gears can be generated by tooth-slap and friction, which is dependent upon tooth profiles, contact ratios, lubrication, and manufacturing tolerances. The magnitude of sound for PEEK gears is known to increase with speed and load, but for POM gears the sound is inversely proportional to the speed [40]. Furthermore, fiber reinforced gears increase in sound after the initial break-in period because the fibers on the flank surface become more exposed after initial wear [40]. Load sharing (i.e. increased contact
53
ratios) has also been successfully correlated to acoustical signals from the mesh of polymer gears [40].
4.2 MEASUREMENT TECHNIQUES 4.2.1
Pin-on-Disk
Both commercial and custom tribology machines are available to measure the friction and wear behavior of gears. The three types of tribometers commonly used for analysis of polymer gears are pin-on-disk, twin-disk, and block-on-ring apparatuses. Choice of instrumentation should be made based on ability to replicate test conditions (e.g. temperature, pressure, and velocity). The pin-on-disk tribometer consists of a pin that is pressed onto a rotating sample specimen, as shown in Figure 29. The normal force, transverse force, and penetration distance are precisely controlled and measured while the material specimen rotates or oscillates at an input frequency to determine the friction coefficient and wear depth. Pin-on-disk tribometers have great precision and are capable of making nano-scale measurements. However, Wright has suggested pin-on-disk machines have a limited ability to predict the amount of wear seen in actual gear test bench experiments [41]. Pins come in many different shapes and materials, but the most common pin geometry is a spherical tip. For a spherical tip on a flat plate, the maximum Hertzian contact pressure can be calculated by sphere on sphere contact with the plate radius being set to infinity [1]:
1 1 2 𝐹𝑛 (𝑅 + 𝑅 ) 1 2 = 0.578 √ 1 − 𝜐12 1 − 𝜐22 ( 𝐸 + 𝐸 ) 1 2 3
𝑃𝑚𝑎𝑥
(4.3)
54
where 𝐹𝑛 is the normal force, 𝑅1 is the sphere radius, 𝑅2 is the plate radius, 𝜐1 and 𝜐2 are the sphere are plate Poisson’s ratios, and 𝐸1 and 𝐸2 are the sphere and plate moduli.
Figure 29 - Diagram of a pin-on-disk tribometer
Hanchi conducted tribology studies on samples of neat PEEK 380G and 30% composite short carbon fiber by weight using with an environmental temperature controller using a 3.175 mm steel ball in a pin on disk configuration. It was found that that steady-state coefficient of friction was temperature dependent for the neat PEEK. With a 15 N load and sliding speed of 0.15 m/s, the friction coefficient increased from approximately 0.2 before the 𝑇𝑔 to 0.5 after the 𝑇𝑔 [42][43]. 4.2.2
Twin-Disk
Twin-disk tribometers are widely used to study rolling contact in bearing and gearing applications. As the operating principles are simple, custom built and application specific instrumentation is common. Two interchangeable cylindrical rollers, one fixed and one translatable, are brought into contact by an applied normal force, as shown in Figure 30. The goal is to measure the slip ratio, which is a measure of the amount of rolling to sliding contact along the line of action at the pitch point. Some machines operate both rollers at the same
55
angular velocity, so the slip ratio can be modified by changing the diameters of the rollers [35]. The slip ratio for constant operating angular velocity between two rollers is given by:
𝑠=(
𝑑2 − 𝑑1 ) ∗ 100 𝑑1
(4.4)
where 𝑑1 and 𝑑2 are the diameters of the individual rollers. The friction force can be measured if the fixed roller is mounted to a load cell so that the resulting tangential force can be resolved. Other twin-disk machines allow for the angular velocity of each roller to be controlled, so that the slip ratio is given by [44]:
𝑠 =2∗(
𝜔2 − 𝜔1 ) 𝜔1 + 𝜔2
(4.5)
A handful of studies have been conducted using a twin-disk tribometer and PEEK 450G. First, Hoskins found an increase in the friction coefficient from 0.1 to 0.5 from below to above the 𝑇𝑔 when characterizing PEEK at a 14% slip ratio and 400 N load [44]. Berer used a twin-disk to compare rolling friction of PEEK cylinders that were manufactured using pin or sprue gates [45]. The morphology of the specimens was compared after testing using a microtome. The results suggested that neat PEEK failed from micro pitting while carbon fiber reinforced PEEK failed from delamination and macro pitting.
56
Figure 30 - Diagram of a twin-disk tribometer
4.2.3
Block-on-Ring
The last type of tribometer is the block-on-ring machine, as displayed in Figure 31. The blockon-ring machine presses a specimen, usually a rectangular bar, onto a rotating metal ring with a constant load. As the ring rotates, a load cell measures the frictional force, typically within a test chamber that is temperature controlled and has an oil sump for lubrication studies. Victrex reports its friction coefficients from measurements made on an Amsler block-on-ring test machine using a sliding velocity of 183 m/min [38]. For PEEK 450G, the friction coefficient was 0.58 at 20 °C and 0.51 at 200 °C, which does not show the drastic change across the 𝑇𝑔 that Hoskins had reported. This suggests further experimentation is required to explain discrepancies in the reported friction coefficient of PEEK 450G.
57
Figure 31 - A schematic of a block-on-ring tribometer
4.3 EXPERIMENTS 4.3.1
Instrumentation
Of the three mentioned, the pin-on-disk CSM NTR2 Nano Tribometer was selected for this research and was equipped with a PI VT-75 screw-driven linear reciprocating stage. The maximum achievable sliding velocity was 100 mm/s and the largest applicable normal force was 1 N. The machine was not equipped with environmental temperature control. For spherical probes, a holder with a 100 Cr6 steel ball was chosen with a radius of 1 mm. The machine was controlled and monitored with InstrumX software from CSM. 4.3.2
Heater Design
A custom-heating device was fabricated for experiments, as shown in Figure 32. The custom heater was designed so that it could be mounted to the carriage of the screw-driven stage, fit within a maximum build envelope of 45 x 45 x 45 mm3, and heat the test sample to 250 °C. To protect the bearings of the screw-driven carriage from heat, Corning MACOR® machinable
58
glass ceramic was chosen as a base for the heater for its manufacturability, resistance to high temperature, and low thermal conductivity.
Figure 32 - Isometric view of custom heater design for friction measurements
An aluminum block was chosen as the housing for the heater elements for its high thermal conductivity to ensure an even distribution of heat. Two holes were reamed into the aluminum block to allow for a low thermal contact resistance with the cartridge heaters (40 W each, 120 Volts/AC). The top of the aluminum block was polished to a mirrored finish to provide low thermal contact resistance to the test sample. The PEEK 450G test samples were machined to a 40 x 40 x 2 mm3 from 150 x 150 x 2 mm3 injection molded plaques. A galvanized sheet metal clamping plate was fabricated with a window to allow access for the tribometer probe. A thin layer (1/32 in) of ceramic insulation was placed between the clamping plate and test sample to ensure an isothermal test surface. Through holes in all parts of the assembly enabled the heater to be screwed to the carriage.
59
Figure 33 - Cross-sectional view of custom heater for friction measurements
A type J micro-thermocouple from Omega was used to monitor surface temperature of the PEEK 450G test sample and was placed within 5 mm of the probe, as depicted in Figure 33. The thermocouple provided feedback to an Omega CN132 PID temperature controller, which maintained the set point temperature within ±1 °C. Polyimide tape was used to adhere the thermocouple hot junction point to the surface of the PEEK 450G. A handheld non-contact Omega infrared thermometer was used to verify the PEEK’s surface temperature. The heaters were powered at an 80 % duty cycle to avoid overshooting the temperature set-point during the controller’s auto-tune functionality. The exposed surfaces of the aluminum heater block were wrapped with ceramic insulation and glass fiber tape. The final assembly is shown in Figure 34.
60
Figure 34 - The custom heater mounted to screw-driven linear stage on CSM Nano Tribometer
4.3.3
Probe Design
As the standard probes for the tribometer were hardened steel, a custom probe was manufactured from PEEK 450G, since the subsequent gear test bench experiments would be between two PEEK gears. In past studies, PEEK gears have been observed to wear better when in contact with another PEEK gear [5]. A literature review suggested no pin-on-disk tribology with PEEK on PEEK contact had been conducted. An injection molded tensile bar was first machined into a small piece of square stock, which was turned in a precision lathe to obtain the final dimensions, as shown in Figure 35. The post and shoulder on top of the probe were designed to provide an adequate fit into the tribometer’s load cell. A sharp carbide 1.5 mm radius tool was used to turn the spherical end of the probe. The tribometer, PEEK 450G probe, and custom heater are shown in unison in Figure 36.
61
Figure 35 - PEEK 450G spherical contact probe with 1.5mm radius
Figure 36 - Friction measurement of PEEK on PEEK contact
4.3.4
Methods
Friction values were obtained for PEEK on PEEK and steel on PEEK sliding contact as a function of temperature, sliding velocity, and maximum Hertzian pressure. The samples and
62
the probe were cleaned with isopropanol before testing. Six test samples of PEEK 450G were used, with one sample being used per probe material for each of the three variables. The sliding direction was always in the direction of polymer melt flow from the injection molding process and each test began in a new wear track. Since the stage motion was reciprocating, all tests were run for 100 cycles, which ensured that steady state was reached. Data was acquired at a rate of 3 kHz, and the friction coefficient was determined by averaging the last 20 cycles of friction data for each test. The friction coefficient versus temperature tests were performed at a sliding velocity of 4 mm/s and maximum Hertzian contact pressure of 30 MPa for both steel and PEEK probes. Temperatures were tested in 25 °C increments from 25 °C to 175 °C. This range was chosen because it covered operating conditions of ambient start-up, engine oil temperature (110 °C), and PEEK 450G’s 𝑇𝑔 (~143 °C). The temperature controller PID settings were auto-tuned for each set point and the heater was allowed to equilibrate at its set point for 10 minutes prior to the start of each test. During equilibration, the probe was placed on the threshold of contacting the sample surface so that its contact point would also be at the temperature set point. The friction coefficient versus velocity tests were performed at a temperature of 25 °C and a maximum Hertzian contact pressure of 30 MPa for both steel and PEEK probes. The sliding velocity was tested at speeds of 1, 10, and 100 mm/s for both PEEK on PEEK and steel on PEEK contact. The friction coefficient versus maximum Hertzian contact pressure tests were performed at a temperature of 25 °C and a sliding velocity of 4 mm/s for both steel and PEEK probes. The maximum Hertzian contact pressure was calculated using Equation 4.3 from the applied normal load, probe contact radii, and probe material properties. The tested pressure range was
63
chosen with prior knowledge from structural simulation studies, knowing that maximum Hertzian contact pressures of 20-30 MPa were common for the experimental range of interest. The maximum tested pressure was dictated by the upper bounds of the tribometer load cell (1 N). The calculated contact pressure is listed as a function of normal load in Table 5. 𝑷𝒎𝒂𝒙 [MPa]
𝑭𝒏 PEEK 450G Probe [mN]
𝑭𝒏 100 Cr6 Probe [mN]
20
19.2
2.22
30
64.8
7.49
40
153.6
17.8
50
300
34.7
60
518.3
59.9
70
823
95.2
Table 5 - Normal load values used to calculate contact pressure for each probe
4.3.5
Results
For friction coefficient versus temperature tests, the friction coefficient for both PEEK on PEEK and steel on PEEK were negatively correlated with temperature for the tested range, as displayed in Figure 37. The PEEK on PEEK contact showed the largest drop in friction coefficient from approximately 0.55 to 0.4 between 50 °C and 75 °C, while the steel on PEEK showed the largest decrease from 0.47 to 0.35 between 50 °C and 125 °C. For both PEEK on PEEK and steel on PEEK contact, a change in friction coefficient may have been explained by glass transition temperature, which correlated well with the largest drop in friction coefficient for the steel on PEEK contact. It is possible that a localized flash temperature between the PEEK probe and PEEK surface was present at temperatures below 75 °C, but this would require further micro-scale investigations. The data agrees well with the measurements made
64
on the block-on-ring machine by Victrex, since the friction coefficient decreased slightly with increased temperature. This disagrees with the results of Hoskins, where a significant increase of friction coefficient from 0.2 to 0.5 was recorded around the 𝑇𝑔 on a twin-disk machine [44]. Unlike a block-on-ring or pin-on-disk, a twin-disk machine simulates rolling contact, which suggests that the change in friction coefficient at 𝑇𝑔 was related to rolling resistance mechanisms.
Figure 37 - Coefficient of friction versus temperature for PEEK on PEEK and steel on PEEK contact (4 mm/s sliding velocity and 30 MPa Hertzian contact pressure)
For the friction coefficient versus velocity tests, there was a slight increase in friction coefficient from 0.48 to 0.56 from 1 to 100 mm/s for steel on PEEK, while the PEEK on PEEK contact was nearly constant as a function of sliding velocity, as shown in Figure 38. Both probe
65
materials agreed well with Coulomb’s law of friction, which states that the kinetic friction coefficient is independent of velocity.
Figure 38 - Coefficient of friction versus sliding velocity for PEEK on PEEK and steel on PEEK contact (25 °C sample temperature and 30 MPa Hertzian contact pressure)
For the friction coefficient versus maximum Hertzian contact pressure tests, both PEEK on PEEK and steel on PEEK friction coefficients were negatively correlated with contact pressure for the tested range, as shown in Figure 39. The friction values between the two types of contact were nearly the same, with steel on PEEK being slightly higher for the majority of pressures by about 0.01 to 0.05. Both types of friction coefficients decreased from about 0.55 to 0.42 over the pressure range. A decrease in friction coefficient for increased pressure was likely related to the decrease in friction coefficient with temperature. As contact pressure increased, the amount of heat dissipation increased, which in turn resulted in a higher localized flash
66
temperature. Whether the higher temperatures were a result of environment or induced frictional heating, if the local surface temperatures exceeded the 𝑇𝑔 enough to reach the coldcrystallization temperature, then increased surface crystallinity may ensue. Increased crystallinity on the surface could provide a more rigid sliding plane with less hysteresis for the probe.
Figure 39 - Coefficient of friction versus maximum Hertzian pressure for PEEK on PEEK and steel on PEEK contact (25 °C sample temperature and 4 mm/s sliding velocity)
67
CHAPTER 5 – FINITE ELEMENT ANALYSIS 5.1 DESCRIPTION OF MODEL 5.1.1
Geometry
As stated previously, the test gear used for simulation and experimental studies was a 30 tooth spur gear manufactured by over-molding PEEK 450G around a pressed powdered metal hub. To improve the mesh quality and limit the number of finite elements needed for solution convergence, the gear geometry was simplified per Figure 40. The powdered metal hub was removed because it is over 50 times more stiff than the PEEK 450G, which was simplified by modeling the inner surface diameter as a rigid. For manufacturing design by injection molding, the web of the gear had many fillets and tapers to allow for quick removal from the mold. These were removed by filling in the web flush with the sides of the gear rim, though this added stiffness and thermal mass to the gear body. However, it was observed in the initial studies of this project that the strain energy was almost entirely contained within the teeth and rim of the gear, and that filling in the web had a negligible effect on maximum model stresses occurring at the teeth roots. Furthermore, the additional thermal mass had negligible effects on the maximum surface temperature due to high frictional heat fluxes at the flanks (i.e. flash temperature) and PEEK’s low thermal conductivity. Lastly, the notches formed on the inner surface of the PEEK hub from the outer profile of the metal hub were removed to mitigate convergence issues from stress anomalies.
68
Figure 40 - Simplification of the research gear's geometry for simulation
In later simulation studies, a fractional model was used to further improve the computational efficiency and accuracy, as shown in Figure 41. Whereas the full model was used to study behavior during a full rotation (360°), the fractional model was used to study a rotation of only 36° for which a maximum of four teeth were loaded in a single gear. The fractional gear considered a total of 8 teeth.
Figure 41 - Depiction of fractional model used in final simulations
Three-dimensional models were used instead of two-dimensional models for several reasons. Pressure gradients across the width of the flanks were parabolic in shape, which made this a
69
three-dimensional stress problem. This led to a three-dimensional thermal problem through hysteresis, frictional heat dissipation, and heat transfer. In conclusion, a three-dimensional model was necessary to model the thermomechanical behavior of polymer gears. 5.1.2
Material Properties
All simulations were modeled using elastic mechanical properties, most of which are listed in Table 2, and non-linear deformation effects were included. From the elastic results, linear viscoelastic approximations were made using data from DMA measurements, and thermal behavior was then predicted, as will be further discussed. From the friction experiments and material measurements by Victrex, the coefficient of friction for PEEK 450G was found to vary between approximately between 0.4 and 0.6 as a function of temperature, velocity, and contact pressure. Thus, a constant friction coefficient of 0.5 was used for all simulations.
Thermal radiation effects were included during heat transfer
simulations, and an emissivity of 0.90, based on PEEK 450G measurements done by NASA, was used [46]. 5.1.3
Mesh
Both the full model and fractional model were built entirely with ANSYS’s SOLID226 elements. SOLID226 elements were chosen because they are a coupled-field element that can provide both structural and thermal results. SOLID226 elements are offered in either hexahedral, tetrahedral, pyramid, or wedge options. Both the full model mesh and fractional model mesh consisted of primarily 20-node hexahedral elements and minimal 15-node wedge elements. These elements are quadratic elements and are a superior choice for linear elastic simulations due to their location of integration points and ability to capture stress concentrations on the surface of structures (i.e. contact problems) [47].
70
Both automatic and user controlled mesh options were used to create the full model and fractional methods. A mesh sweeping method was used to generate a symmetrical mesh across the width of the gear. This was done by mapping a two-dimensional mesh on the side of the gear with quadrilaterals and triangles, and the mesh was then swept axially across the width of the gear to extrude the three-dimensional element shape. Three equal elemental divisions were made along the width of the gear during the sweep. Edge sizing was also enforced on the gear tips, flanks, and roots to provide a higher element density in areas where high stresses were expected and on contacting surfaces. The meshes for both the full and fractional models are shown in Figure 42. The full model mesh consisted of 16,560 finite elements and the fractional mesh consisted of 16,269 elements. A finer mesh was used in the fractional model to capture more accurate contact pressures, which were used in the calculation of frictional heat dissipation.
Figure 42 - Full model and fractional model meshes
71
Mesh quality is essential to producing accurate results, thus, many metrics exist (e.g. orthogonal quality, aspect ratio, skewness) to quantify the quality of a particular mesh. One of the most powerful meshing metrics is skewness [48]. Skewness ranges from 0 to 1 and is defined as the difference in shape of an element compared to an equilateral element of equivalent volume; 0 having no skewness and 1 being highly skewed. A common rule in practice is to have a mesh skewness that does not exceed a maximum value of 0.95 [48]. Values near or above this threshold will likely lead to convergence issues. The skewness metric for both the full model and fractional model are shown in Figure 43 and Figure 44.
Figure 43 - Mesh skewness statistics for full model
72
Figure 44 - Mesh skewness statistics for fractional model
5.1.4
Contact
Understanding contact formulations is an essential part of accurately simulating gears. Contact occurs when two adjacent surfaces have become mutually tangent. The surfaces then transmit compressive normal forces or tangential friction forces, though “adhesion” (i.e. tensile forces) is not simulated during contact. Contact is a non-linearity that changes status so the stiffness matrix must be updated for every time step, which is computationally time consuming. A contact stiffness must be specified in order to limit the amount of penetration. An infinite stiffness would be desirable because it would limit the amount of penetration, but it is not numerically possible. Thus, a “negligible” amount of penetration must be accepted. A stiffness factor of 1 is used for bulk deformation problems, but a value of 0.1 may be useful for bending problems [49]. Contact stiffness is the most important parameter affecting simulation accuracy.
73
Ideally, the stiffness should be set as large as possible to prevent penetration, but if contact is too stiff, then the simulation will oscillate (i.e. diverge), as depicted in Figure 45. The acceptable stiffness factor found for simulations in this thesis was 0.5.
Figure 45 - Diagram of normal contact stiffness oscillation as a function of time steps [49]
Several normal contact formulations exist, but Augmented Lagrange was chosen for this study because it is less sensitive to contact stiffness and can be used with non-linear structural analyses, such as friction. Augmented Lagrange uses the integration points (between nodes) to detect contact. In all cases, tangential contact formulation is handled by the Pure Penalty method. For frictional contact, the contact and target surfaces must be specified prior to beginning the simulation. For the case of the gears, the contact surfaces were on the pinion and the target surfaces were on the driven gear. In Figure 46, the contact surfaces are shown in red and the target surfaces are shown in blue. Though the flanks and tips were expected to be the contact surfaces, it was good practice to consider all root surfaces as well.
74
Figure 46 - Illustration of contact (red) and target (blue) surfaces
Each time step, the solver checked which surfaces were close enough to be considered for contact. The distance that told the solver whether or not a surface should be considered for contact is called the pin ball region, which can be imagined as a spherical distance extending from the boundaries of the surface. The pinball region can be defined by the user or controlled by the solver, where the latter was chosen for this simulation. Other useful controls implanted in the simulation included automatic bisection and prediction for impact, which evaluated the amount of penetration at the end of each time step. If too much penetration existed, the time step was cut in half (bisected) and the simulation was re-run to minimize the penetration. 5.1.5
Boundary Conditions
The boundary conditions presented in Figure 47 were representative of both the full model and fractional model. The exception is that the “cut” ends of the fractional model were also treated as adiabatic. The structural and thermal boundary conditions were specific to two different simulations that were coupled. Each gear was translationally fixed and only had rotational freedom about its respective axis of rotation. The gears were separated along the y-axis by their pitch diameter. The pinion had a constant angular velocity, while the driven gear had a constant
75
moment opposing the pinion’s motion. Inertial effects were not included in the simulation. The gears had a surface friction coefficient of 0.5 and no axial friction.
Figure 47 - Boundary conditions for coupled structural-thermal model
Ambient temperature in the model was set to 23 °C. The inner surface of the gear hub (blue) was adiabatic. The gears had an external heat flux on the flanks of the teeth and an internal volumetric heat generation from hysteresis. The heat was transferred internally by conduction, and externally by radiation and convection (red + yellow). Different heat transfer coefficients were used for the sides of the gear (yellow) and the gear teeth (red). The heat transfer coefficients specific to the gears in this study are shown in Figure 48.
76
Figure 48 - Heat transfer coefficients specific to the gears in this study
Simplified bulk temperature prediction models have been performed previously using ANSYS® for a worm wheel operating in conjunction with a worm gear. The heat transfer coefficient on the flanks was predicted as that of a rotating cylinder [50]:
̅̅̅̅𝑡 ∗ ℎ𝑡 = 𝑁𝑢
𝑘 𝐷𝑝
(5.1)
The average Nusselt number for a rotating cylinder’s outer surface in quiescent air predicted by Ozerdem’s experiments was [51]: ̅̅̅̅ 𝑁𝑢𝑡 = 0.318𝑅𝑒𝑡 0.571
(5.2)
77
The Reynolds number, for a range of 2,000 to 40,000, for a rotating cylinder is given by [51]:
𝑅𝑒𝑡 =
ω𝐷𝑝 2 2𝑣𝑘
(5.3)
where 𝑣 is the kinematic viscosity. Similarly, the heat transfer for the sides of the gear was predicted using experimental correlations for a rotating disk in quiescent air [52]:
̅̅̅̅𝑠 ∗ ℎ𝑠 = 𝑁𝑢
𝑘 2𝑟𝑖
(5.4)
where 𝑟𝑖 is the inner radius of the gear hub. Assuming a constant Prandtl number for dry air, the average Nusselt number for a rotating disk’s side surface in quiescent air predicted by Latour’s experiments was [52]: ̅̅̅̅𝑠 = 0.556𝑅𝑒𝑠0.5 𝑁𝑢
(5.5)
The Reynolds number, for a range of 2,150 to 17,200, for a rotating disk is given by [52]:
𝑅𝑒𝑠 =
ω2𝑟𝑏 𝑟𝑖 𝑣𝑘
(5.6)
Equations 5.1 through 5.6 and the corresponding gear geometry of the test gears were used to produce the data in Figure 48. Dry air at 25 °C was used to determine the value of kinematic viscosity (𝑣𝑘 = 1.557 x 10-5 m2/s). For radiation heat transfer, ANSYS® assumed constant emissivity (𝜖) and the radiation power was calculated by [48]: 𝑃𝑟𝑎𝑑 = 𝜖𝜎𝑏 𝐴𝑠 𝑇 4
(5.7)
78
where 𝜎𝑏 is the Stefan-Boltzman constant and 𝐴𝑠 is the surface area. Radiation between different objects was calculated using the Monte Carlo method.
5.2 THERMOMECHANICAL COUPLING 5.2.1
Coupled Simulations
Coupled analyses are more computationally expensive and difficult to implement, but they provide more realistic results then analyses made with uncoupled assumptions. Examples of situations where coupled analyses are useful are brake rotors, friction stir welding, and gear drives. In ANSYS®, it is easy to implement a sequential thermal to structural coupling, but implementing direct coupling of a structural to thermal model is more difficult because it requires the use of scripts written in the ANSYS® parametric design language [47]. 5.2.2
Hysteresis Thermal Generation
There were two sources of heat generation within the finite element model. The first was internal volumetric heat generation due to hysteresis (i.e. viscoelastic heat dissipation). Each element within the model was strained and unstrained once per revolution of the gear, though the magnitude of that strain varied based on the element’s location within the mesh. For example, an element located on a driven flank or tooth root had a larger strain than an element located in the hub. The stress in an element was integrated with respect to the strain to obtain the strain energy density [53]: 𝜀
𝑈0 = ∫ 𝜎𝑑𝜀 0
(5.8)
79
An example of an arbitrary element’s strain energy density behavior as a function of time for one rotation is shown in Figure 14. The maximum value of the strain energy density was obtained for each element per cycle and the energy loss was calculated using Equation 2.14: 𝑈𝑙𝑜𝑠𝑠 (𝑇) = 𝑈𝑚𝑎𝑥 𝐻 Hysteresis varied only as a function of a finite element’s temperature since the gear was assumed to be operating at a constant frequency within the linear elastic range (i.e. independent of strain). The hysteresis of PEEK 450G as a function of temperature at a frequency of 100 Hz is shown in Figure 49. The DMA and TTS of PEEK 450G showed that there was not a significant change in hysteresis below 130 °C as a function of frequency. Thus, the hysteresis values at 100 Hz as a function of temperature from Figure 49 were used to simulate loading frequencies as high as 1 kHz.
80
Figure 49 - PEEK 450G hysteresis as a function of temperature (𝜀 = 0.1 %)
The volumetric heat generation from hysteresis was calculated as an average volumetric power over the duration of one revolution of the gear described by Equation 2.15: 𝐸ℎ = 𝑈𝑙𝑜𝑠𝑠 (𝑇)𝑓 The volumetric power as a function of temperature was calculated for each element within the model based on a single isothermal rotation, assuming that the strain energy density per element did not change significantly as the temperature of the gear increased. The coupling occurred when the volumetric power per element as a function of temperature was applied to the same mesh in a transient thermal model. The power each element received was calculated by integrating the volumetric heat generation with respect to the element volume:
81 𝑉
𝐸ℎ,𝑉 (𝑇) = ∫ 𝐸ℎ (𝑇)𝑑𝑉
(5.9)
0
Now, each element had its own temperature dependent hysteresis function in the transient thermal model, given the specific loading condition. 5.2.3
Frictional Thermal Generation
The second source of heat generation within the model came from the sliding friction at the flanks of the gear teeth. The heat generation rate due to sliding friction was calculated with Equation 2.3: 𝑞𝑎𝑣 = 𝜇𝑃𝑎𝑣 𝑣𝑠 The heat flux rate resulting from the sliding friction was calculated as a function of time during the structural simulation. The average heat flux rate for one rotation was calculated by integrating the heat flux with respect to time of the rotation, and then dividing the integral by the time of the rotation: 𝑡
𝐸𝑓,𝑎𝑣𝑒
∫ 𝑞𝑎𝑣 𝑑𝑡 = 0 𝑡
(5.10)
The average heat flux value was then applied to the operating flanks of the gear teeth in the transient thermal model. The heat flux was partitioned equally among the contacting flanks because both gears were assumed to have the same thermal conductivity (i.e. same material). Given all boundary conditions, the transient thermal model could now be used to predict operating temperatures at any duration.
82
5.3 SIMULATIONS 5.3.1
Hardware and Software
ANSYS® Workbench 14.5 and Mechanical APDL 14.5 are multi-physics software products that were chosen to perform the polymer gear simulation for their superior contact solvers and the ability to be programmed using the ANSYS® parametric design language. All simulations were performed on a Dell Precision T5500 work station with an Intel® Xeon® E5630 2.53 GHz CPU (2 processors) and 30 GB of RAM. The operating system was Windows 7 64-bit. 5.3.2
Methods
The simulation methods for the full model and fractional model were nearly identical, where the only difference was defining the cut ends of the fractional model as adiabatic in the thermal simulation. The process began with uploading the gear geometry to an ANSYS® Workbench transient-structural module and entering the material properties into the engineering data. All previously mentioned structural boundary conditions and meshing controls were applied and the gears should were set in a position with the meshed teeth in tangency at the pitch point. Mechanical APDL commands were used in the Workbench interface to convert all elements in the mesh to SOLID226 coupled elements, apply thermal degrees existing structural boundary conditions, and turn off inertial effects. The analysis settings were set so that one load step was used for every 12 degrees of rotation. The simulation enforced 10 sub-steps per load step at the start, and then allowed a minimum of 5 sub-steps or a maximum of 20 substeps for each subsequent load step. Hence, a full rotation of the gear required 30 load steps. The simulation was then solved using the ANSYS®’ internal Newton-Raphson implicit numerical method.
83
When the solution was finished, the result file (.rst) was imported into the time-history post processor in Mechanical APDL. Using Mechanical APDL commands, the maximum strain energy density values were extracted per element and saved to a text file (.txt) with their corresponding element number. The strain energy density values were then imported as an array into a MATLAB® macro that uses Equations 2.14, 2.15, 5.8, and 5.9 to solve the temperature dependent hysteresis function for each element using hysteresis values from Figure 49. The function was saved as a set of data pairs (temperature, hysteresis) in an array and the MATLAB® macro placed the necessary Mechanical APDL commands into the array so that the data could be read back into the ANSYS® Workbench solver. The frictional heat flux was extracted from the transient-structural solution as a function of time, then integrated and averaged using Equation 5.10 to the average heat flux. A transientthermal module was then opened and the geometry and mesh from the transient-structural simulation was imported. All previously mentioned thermal boundary conditions were applied and the gears were set in a position with the meshed teeth in tangency at the pitch point. The average heat flux was applied to the driving and driven flanks on both sets of gears and the element specific hysteresis functions were read into the solver using Mechanical APDL commands. ANSYS® Workbench linearly interpolated the (temperature, hysteresis) data pairs during the transient thermal simulation. The analysis settings were set so that five thermal time steps were used for a duration of 8,000 seconds. The first four thermal steps each had a duration of 900 seconds, and the fifth thermal step had a duration of 4,400 seconds. The simulation enforced 10 sub-steps per thermal time step. Results were analyzed after the solution was reached.
84 5.3.3
Results
All initial results were obtained only using the full model. Before proceeding to the analysis of operating temperature prediction, there are a couple important phenomena to observe. First, Figure 50 shows load sharing of polymer teeth, which is ubiquitous at high torques relative to the polymer gear size. It was important that the model was able to simulate this because load sharing leads to increased sliding friction contact at the tips and roots, which generates more heat. Also, frictional heating associated with load sharing is not easily predicted by theory. Second, the thermal gradients predicted by the model in Figure 51 correlated well with the location of wear patterns, as shown in Figure 9. The aforementioned mechanisms assured that the model was functioning before proceeding with the thermal analysis.
Figure 50 - Image of a high contact ratio predicted by the simulation
85
Figure 51 - Depiction of thermal gradients on polymer gear teeth
Figure 52 and Figure 53 compare the effects of hysteresis and frictional heating at two different loading conditions. At 1,000 RPM and 20 N-m, the gear was near its maximum operating torque, which resulted in significant hysteresis based on strain energy density. Over a duration of 6,000 seconds, hysteresis provided an additional temperature rise of 6 °C over just frictional heating alone. This was not a tremendous temperature rise in comparison to the nearly 100 °C temperature rise frictional heating produced from ambient conditions. Also, when comparing the predicted temperature of approximately 137 °C to Figure 49, the model was near 𝑇𝑔 where a large increase in hysteresis occurs, which was contributing to the temperature increase.
86
Figure 52 - Comparison of hysteresis and frictional heating (1,000 RPM, 20 N-m)
For operating conditions of 1,000 RPM and 15 N-m, hysteresis provided an additional 2 °C of temperature increase over frictional heating effects for the same duration. Both operating conditions assumed continuous operation, but when analyzing the transient temperature rise, there was almost no difference in temperature from the addition of hysteresis. These results agreed with work of Blok, Gauvin, Koffi, and Mao, that in most cases hysteresis was small enough that it could be neglected [15][16][17][21]. Regardless, hysteresis effects were considered in all subsequent simulation results.
87
Figure 53 - Comparison of hysteresis and frictional heating (1,000 RPM, 15 N-m)
The full model had limited efficiency because of the time it took to perform the transientstructural segment of the simulation. Using the aforementioned computer hardware, this model took nearly 30 hours to solve. At this point, the fractional model was utilized to improve computation time and mesh quality at the teeth. A comparison of the fractional model to the full model for operating conditions of 2,000 RPM at 10 N-m are compared in Figure 54. Over a test duration of 6,000 seconds, only a 1.5 °C difference in predicted temperature existed, which provided the confidence needed to proceed with the fractional model. Only four teeth are loaded in the fractional model, but a sequence of extra teeth in the loaded direction were included to capture the region of strain energy density that occurred in the gear rim and web, as shown in Figure 42.
88
Figure 54 - Comparison of results obtained with full model and fractional model (2,000 RPM, 10 N-m)
For the remainder of conducted simulations, the fractional model was now used. In anticipation for gear test bench experiments, two operating conditions were chosen for testing: 1,000 RPM at 8 N-m, and 2,000 RPM at 4 N-m. The test conditions were chosen based on the stable operating conditions of the newly built gear test bench at Kleiss Gears and because the operating conditions were equal in power, but different in frequency and load. Prior to experiments, the chosen operating conditions were simulated using the numerical model, and the results are shown in Figure 55. The large temperature increase in the 1,000 RPM at 8 N-m simulation is attributed to a much larger frictional heat generation. The increase in heat generation was due to increased contact ratios at higher loads, which caused more instances of contact.
89
Figure 55 - Simulation of gear test bench conditions prior to experiments
90
CHAPTER 6 – EXPERIMENTAL VALIDATION 6.1 GEAR TEST BENCH 6.1.1
Overview
The simulation results were evaluated at Kleiss Gears, Inc. in December 2014 using a recently built gear test bench capable of simulating the loading conditions and frequencies that the test gears may be subjected to in their end environment. Since the model did not yet consider the effects of lubrication, the gears were tested in a state of dry surface contact. Two identical 30 tooth production gears were used in each experiment with one gear being driven at a constant frequency, while the adjacent gear applied a constant load. Each test began by installing a new pair of gears within the test bench. The gears were powered for approximately 15 minutes, which was determined a sufficient time for thermal equilibrium based on several trial runs. A 38 tooth version of the test gears are shown in Figure 56 (not used for simulation).
Figure 56 - Image of 38 tooth gears in the test bench (gears not used in simulation) [54]
91
The gear test bench had a drive shaft and output shaft, each equipped with its own servo motor for speed and load control as well as a torque transducer. The end of each shaft had a collet where the test gears could be quickly swapped. Precision axles were press fit into the metal gear hub and then connected to the collets on the test bench. The entire test bench was controlled by custom software developed by Kleiss Gears. A picture of a torque transducer and coupling used on the test bench is shown in Figure 57.
Figure 57 - one of two identical pairs of torque transducers and couplings on the test bench [54]
6.1.2
Instrumentation
During each experiment a FLIR E40 infrared (IR) camera was used for gear surface temperature measurements and an IDT high-speed camera provided visualization of contact ratios. The atmospheric temperature on the IR camera was set to 21 °C, the emissivity of the PEEK 450G surface was set constant at 0.90, and the reference distance to the gears was set to 1 m. The high speed camera was controlled using Motion Studios software. External lighting was used to illuminate the gears during testing for the high speed photography, but more lighting would have allowed for higher resolution photos and slower frame speeds. Figure 58
92
and Figure 59 show the overall layout of the gear test bench and its instrumentation. Figure 60 provides a look at the screen of the IR camera as it examines a test gear that was removed from the test bench and still warm.
Figure 58 - Overhead layout of the test bench and instrumentation
93
Figure 59 - Photo of gear test bench and instrumentation
Figure 60 - A look at a test gear after it was removed from the test bench
94
6.2 RESULTS Videos taken with the IR camera were post-processed to obtain the maximum surface temperature of the gears as a function of time for both tests, as shown in Figure 61. The 1,000 RPM at 8 N-m test produced a maximum surface temperature of 110 °C and the 2,000 RPM at 4 N-m test produced a maximum surface temperature of 49 °C. Load sharing was also confirmed among the gear teeth for the 1,000 RPM at 8 N-m test as shown in Figure 62.
Figure 61 - Thermal image of 30 tooth test gears at 1,000 RPM and 8 N-m load
Figure 62 - High speed photo of load sharing occurring during 1000 RPM at 8 N-m
95
After comparing the simulation data to the experimental results, it was observed that the simulation under-predicted the transient temperature rise of both test conditions. Upon further investigation, the work of Terauchi showed that theoretical heat transfer coefficients for gears can significantly over-predict actual heat transfer coefficients [55]. Furthermore, based on previously mentioned theory by Mao, a gear operating in quiescent air acts as a fluid pump, where air is actually trapped in vortices between adjacent gear teeth [21]. Based on this theory, as ambient air becomes trapped and heated between gear teeth, the heat transfer slowly decreases during the rotation until the teeth re-mesh and new ambient air is once again trapped.
Figure 63 - Adjusted simulation heat transfer coefficients
96
Both of these claims support the fact that a polymer gear’s heat transfer coefficient in quiescent air would be less than a rotating disc or cylinder, as used for the correlation. The heat transfer coefficient was lowered on both simulations in a proportional manner to the original heat transfer coefficients in order to predict the operating temperature of the gears more accurately. The new heat transfer coefficients are shown in Figure 63 and the experimental data compared to simulation data is shown in Figure 64 and Figure 65.
Figure 64 - Experimental data compared to simulation data for 1,000 RPM at 8 N-m
With the new heat transfer coefficients, the 1,000 RPM at 8 N-m test was under predicted by approximately 5 °C, while the 2,000 RPM at 4 N-m test was over predicted by approximately 8 °C. The reason for the deviation in experimental and simulation values was related to the simplification of simulated gear geometry, the heat transfer coefficients, and the friction
97
coefficients. The simulated gear geometry was simplified to improve efficiency and help solver convergence. Filling in the gear web added additional thermal mass and reduced the surface area, while removal of the metal hub eliminated any heat transfer at the inner hub surface. PEEK 450G is an excellent thermal insulator, but both these changes had implications on steady state operating temperatures.
Figure 65 - Experimental data compared to simulation data for 2,000 RPM at 4 N-m
The heat transfer coefficients could have been improved through a convection correlation study of PEEK 450G gears in dry air, although it would have been hard to justify since PEEK is normally operated under lubricated conditions. It was mentioned in the literature review that a discrepancy exists in measured friction coefficients among different tribology machines for
98
PEEK 450G. A study that compares PEEK friction measurements on a variety of tribometers would have been useful to in providing reasoning to this moot point.
99
CHAPTER 7 – CONCLUSION AND RECOMMENDATIONS 7.1 CONCLUSION Modeling thermomechanical behavior of polymer gears begins with a thorough material characterization process. The loading frequencies of polymer gears tend to be outside the limits of many commercially available DMA techniques, and a custom measurement apparatus is the preferred method for this task. TTS principles are also useful, but vary in their applicability from polymer to polymer and temperature range of interest. Tribological measurements made with a pin-on-disk machine in this thesis found friction data for PEEK 450G to vary from 0.4 to 0.6 as a function of temperature, velocity, and pressure. Other experiments have found PEEK 450G’s friction coefficients to be highly affected by the 𝑇𝑔 . For tribology studies related to gears, a twin-disk machine is the preferred choice for its ability to measure a combination of rolling and sliding friction. However, a cross-comparative study of different tribology machines would be useful for research involved with PEEK 450G. Developing a simulation model for a polymer gear is time-consuming and the results are only as accurate as the input data. Always start with a simple model, and elaborate as time permits. Even if the data seems to correlate well with experiments, question everything. A gear test bench is the most superior of all gearing analysis tools, but also the most economically exhausting. Conducting research on polymer gears is difficult without continued access to a gear test bench or processing equipment to make experimental polymer gears. Before deciding to embark on a polymer gear research endeavor, make note of the tools within immediate reach.
100
7.2 RECOMMENDATIONS Although this thesis presented the groundwork for a methodology on material characterization, thermomechanical simulation, and experimental design of polymer gearing, there is an abundance of related thesis and research journal opportunities that still remain. For DMA techniques, dielectric thermal analysis, atomic force microscopy, and resonant ultrasound should be explored as methods for analysis of polymer gearing frequencies. Also, the thermal conductivity of PEEK 450G was assumed constant in this thesis, a transient line source or laser flash method should be used to look at the conductivity as a function of temperature. The research presented considered linear viscoelastic theory of PEEK, but non-linear DMA should be explored because others have found PEEK to strain soften in the non-linear viscoelastic range [28]. Since PEEK 450G is a semi-crystalline polymer, the effect of varying degrees of crystallinity on the mechanical properties should be studied as well. For further tribology studies, it would be beneficial if the friction measurements could be carried out in an environmental chamber because it would guarantee that both the probe and the substrate were at the test temperature. Also, a tribology study of PEEK 450G should be conducted that compares the friction coefficients obtained on different types of tribometers. For the numerical model, the simulation likely does not predict stick slip behavior as well as it should, which could be improved by very small time steps and a refined mesh. The augmented Lagrange normal contact method should be compared to the normal Lagrange method for frictional heat generation. A viscoelastic model should be incorporated into the study, such as a Prony series, to see if the time-dependent stress response has a significant effect on the friction coefficient at high frequencies. In new releases of ANSYS®, a Prony series can be used to output dissipated energy as a result. Also, a micro-scale simulation study on gear tooth
101
surfaces that simulates material response above the glass transition temperature should be implemented [56]. Custom Mechanical APDL programming could be implemented in ANSYS® to accept temperature dependent friction coefficients. ANSYS®, and other multiphysics software packages now allow processing simulation software, such as Moldex3D®, to be coupled. Therefore, the effects of molded residual stress and crystallinity could be incorporated into the thermomechanical simulations. Likewise, PEEK 450G has anisotropic thermal expansions coefficients that should be correlated to mold flow direction to see how that affects the meshing tolerances at higher temperatures. More test bench experiments at different operating conditions should be explored to develop a thermal correlation model as a function of temperature and time-dependent material properties as well as time, temperature, and pressure dependent tribological measurements. Lastly, a high speed IR camera should be used to capture flash temperature effects on polymer gear flanks.
102
REFERENCES [1]
Juvinall, Robert C., and Kurt M. Marshek. Fundamentals of machine component design. Vol. 83. New York: John Wiley & Sons, 2006.
[2]
Celanese "Optimize Your Gear Material and Gear Design for Performance and Profit." Brochure, September 2013.
[3]
It’s Time to Scrap Metal Gears. Victrex, 2014.
[4]
Beermann, Stefan. "Estimation of lifetime for plastic gears." In AGMA Conference, Fall Meeting, pp. 1-17. 2007.
[5]
Ferfecki, Frank J., and Allen Hale. Polymer Gear Development to Improve Efficiency and NVH Performance of an Engine Mass Balance System. No. 2011-01-0405. SAE Technical Paper, 2011.
[6]
Elmquist, Joseph. "Deciding When to Go Plastic." In Gear Technology, 46-47. July ed. Rancho Cucamonga, CA, 2014.
[7]
Kleiss, Rod, and Frank J. Ferfecki. "Designing a Polymer Gear for Use in the Environment of an Internal Combustion Engine." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 515-521. American Society of Mechanical Engineers, 2011.
[8]
Victrex. "Victrex PEEK 450G." Material Data Sheet, June 2015.
[9]
Ferfecki, Frank. The Performance Envelope of PEEK Polymer Gears. Victrex, 2013.
[10] A Guide to Plastic Gearing. GE Plastics, 2006. [11] Design of Plastic Gears and General Considerations of Plastic Gearing. QTC, 2015. [12] Senthilvelan, S., and R. Gnanamoorthy. "Influence of reinforcement on composite gear metrology." Mechanism and machine theory 43, no. 9 (2008): 1198-1209. [13] Kurokawa, Masaya, Yoshitaka Uchiyama, and Susumu Nagai. "Performance of plastic gear made of carbon fiber reinforced poly-ether-ether-ketone." Tribology International 32, no. 9 (1999): 491-497. [14] VDI 2736 Blatt 2, Thermoplastische Zahnräder, Stirnradgetriebe Tragfähigkeitsberechnung, 2013. [15] Blok, H. "The flash temperature concept." Wear 6, no. 6 (1963): 483-494. [16] Gauvin, Raymond, Patrick Girard, and Henri Yelle. Prediction of the Peak Temperature on the Surface of Thermoplastic Gear Teeth: Pres. at the Fall Technical Meeting; Montreal, Canada, October 17-19, 1983. AGMA, 1983.
103
[17] Koffi, D., R. Gauvin, and H. Yelle. "Heat generation in thermoplastic spur gears." Journal of Mechanical Design 107, no. 1 (1985): 31-36. [18] Lin, Yeong-Jyh, and Sheng-Jye Hwang. "Temperature prediction of rolling tires by computer simulation." Mathematics and Computers in Simulation 67, no. 3 (2004): 235-249. [19] Senthilvelan, S., and R. Gnanamoorthy. "Effect of rotational speed on the performance of unreinforced and glass fiber reinforced Nylon 6 spur gears." Materials & design 28, no. 3 (2007): 765-772. [20] Hooke, C. J., K. Mao, D. Walton, A. R. Breeds, and S. N. Kukureka. "Measurement and prediction of the surface temperature in polymer gears and its relationship to gear wear." Journal of Tribology 115, no. 1 (1993): 119-124. [21] Mao, K. "A new approach for polymer composite gear design." Wear 262, no. 3 (2007): 432-441. [22] Mao, K. "A numerical method for polymer composite gear flash temperature prediction." Wear 262, no. 11 (2007): 1321-1329. [23] Mao, K., W. Li, C. J. Hooke, and D. Walton. "Polymer gear surface thermal wear and its performance prediction." Tribology International 43, no. 1 (2010): 433-439. [24] Letzelter, Eric, Jean-Pierre de Vaujany, Laurent Chazeau, and Michele Guingand. "Quasi-static load sharing model in the case of Nylon 6/6 cylindrical gears." Materials & Design 30, no. 10 (2009): 4360-4368. [25] Osswald, Tim A., and Georg Menges. Materials science of polymers for engineers. Carl Hanser Verlag GmbH Co KG, 2012. [26] Herdy, Michael. "Introductory Theory Manual ViscoData and ViscoShift." IBHIngenierbüro, Grafschaft (2003). [27] T.A Instruments. "Thermal analysis application brief." TA-125. [28] Rae, P. J., E. N. Brown, and E. B. Orler. "The mechanical properties of poly (etherether-ketone)(PEEK) with emphasis on the large compressive strain response." Polymer 48, no. 2 (2007): 598-615. [29] Altstaedt, Volker, Philipp Werner, and Jan Sandler. "Rheological, mechanical and tribological properties of carbon-nanofibre reinforced poly (ether ether ketone) composites." Polímeros 13, no. 4 (2003): 218-222. [30] Senthilvelan, S., and R. Gnanamoorthy. "Damping characteristics of unreinforced, glass and carbon fiber reinforced nylon 6/6 spur gears." Polymer testing 25, no. 1 (2006): 5662. [31] TA Instruments. “RSA III Rheometric Series User Manual.” 2005.
104
[32] Lakes, Roderic S. "Viscoelastic measurement techniques." Review of scientific instruments 75, no. 4 (2004): 797-810. [33] The Opensource Handbook of Nanoscience and Nanotechnology. July 2015. [34] Walton, D., A. B. Cropper, D. J. Weale, and P. Klein Meuleman. "The efficiency and friction of plastic cylindrical gears Part 1: Influence of materials." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 216, no. 2 (2002): 75-78. [35] Sukumaran, Jacob, Matyas Ando, Patrick De Baets, V. Rodriguez, Laslo Szabadi, Gabor Kalacska, and V. Paepegem. "Modelling gear contact with twin-disc setup." Tribology international 49 (2012): 1-7. [36] Van Melick, Ir HGH. "Tooth-Bending Effects in Plastic Spur Gears." Gear Technology (2007): 58-66. [37] Jayadeep, U.B. "Rolling Friction." Lecture, October 26, 2010. [38] Victrex. "Material Properties Guide." June 2015. [39] Mao, K., W. Li, C. J. Hooke, and D. Walton. "Friction and wear behaviour of acetal and nylon gears." Wear 267, no. 1 (2009): 639-645. [40] Hoskins, T. J., K. D. Dearn, S. N. Kukureka, and D. Walton. "Acoustic noise from polymer gears–A tribological investigation." Materials & Design 32, no. 6 (2011): 3509-3515. [41] Wright, N. A., and S. N. Kukureka. "Wear testing and measurement techniques for polymer composite gears." Wear 251, no. 1 (2001): 1567-1578. [42] Hanchi, J., and N. S. Eiss Jr. "Tribological behavior of polyetheretherketone, a thermotropic liquid crystalline polymer and in situ composites based on their blends under dry sliding conditions at elevated temperatures." Wear 200, no. 1 (1996): 105121. [43] Hanchi, J., and N. S. Eiss. "Dry sliding friction and wear of short carbon-fiberreinforced polyetheretherketone (PEEK) at elevated temperatures." Wear 203 (1997): 380-386. [44] Hoskins, T. J., K. D. Dearn, Y. K. Chen, and S. N. Kukureka. "The wear of PEEK in rolling–sliding contact–Simulation of polymer gear applications." Wear 309, no. 1 (2014): 35-42. [45] Berer, M., Z. Major, and G. Pinter. "Elevated pitting wear of injection molded polyetheretherketone (PEEK) rolls." Wear 297, no. 1 (2013): 1052-1063.
105
[46] Clatterbuck, Carroll H., Mary Ayres-Treusdell, Gloria Park, and Diane Kolos. Atomic Oxygen and Space Environment Effects on Aerospace Materials Flown with EOIM-III Experiment. National Aeronautics and Space Administration, Goddard Space Flight Center, 1996. [47] Silva, Roberto. "Direct Coupled Thermal-Structural Analysis in ANSYS WorkBench." Lecture, April 2013. [48] Guide, ANSYS Meshing User’s. "Rel. 14.5." ANSYS Inc (2012). [49] ANSYS, Inc. "Lecture 3: Introduction to Contact." Lecture, December 2010. [50] Thyla, P. R., and R. Rudramoorthy. "Investigations On Gear Tooth Surface And Bulk Temperatures Using ANSYS." PSG College of Technology, Coimbatore (2004). [51] Özerdem, Bariş. "Measurement of convective heat transfer coefficient for a horizontal cylinder rotating in quiescent air." International communications in heat and mass transfer 27, no. 3 (2000): 389-395. [52] Latour, Benjamin, Pascale Bouvier, and Souad Harmand. "Convective heat transfer on a rotating disk with transverse air crossflow." Journal of Heat Transfer 133, no. 2 (2011): 021702. [53] Vable, Madhukar. Intermediate mechanics of materials. New York, NY: Oxford University Press, 2008. [54] "Testing." Kleiss Gears. http://www.kleissgears.com/engineering/testing/ (retrieved July 2015). [55] Terauchi, Yoshio, Kazuteru Nagamura, and Chang-Lin Wu. "On heat-balance of gearmeshing apparatus (Experimental and analytical heat transfer coefficient on tooth faces)." JSME international journal. Ser. 3, Vibration, control engineering, engineering for industry 32, no. 3 (1989): 467-474. [56] Lin, Ah-Der, and Jao-Hwa Kuang. "Dynamic interaction between contact loads and tooth wear of engaged plastic gear pairs." International Journal of Mechanical Sciences 50, no. 2 (2008): 205-213.
106
APPENDIX ANSYS® COMMAND EXAMPLES Change element type to SOLID226: et,matid,226,11
Set contact thermal degrees of freedom: keyopt,cid,1,1 keyopt,cid,10,2 keyopt,cid,9,1 RMODIF,cid,14,1E4
!
UX, UY, UZ, TEMP
CNCHECK,ADJUST
Turn off inertial effects and apply thermal boundary condition to surface: TIMINT,OFF,STRU cmsel,s,surface d,all,surface,23 allsel
Apply temperature dependent hysteresis to a specific element: *DIM,HYSTERESIS,TABLE,18,1,1,TEMP HYSTERESIS(1,0,1)= 20,79,109,119,124,129,134,139,144,149,154,159,164,168,174,178,193,203 HYSTERESIS(1,1,1)= 0.001831436,0.002140856,0.002858833,0.004442581,0.006869233,0.013960722,0.026932675,0.04 0308644,0.047098940,0.045368492,0.039936816,0.033775469,0.028604869,0.026202614,0.023843 822,0.023536876,0.024974886,0.026216763 BFE,1,hgen,,%HYSTERESIS%
107
ORIGINAL PLASTIC GEARS INTEREST GROUP AT UW-MADISON
