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Epicyclic Gear Train Solution Techniques
with Application to Tandem Bicycling
Christopher A. Corey
Thesis submitted to the faculty of Virginia Polytechnic
Institute and State University in partial fulfillment of the
requirements for the degree
Master of Science
in
Mechanical Engineering
Dr. Charles F. Reinholtz, Chair
Dr. Alfred L. Wicks
Dr. Robert L. West, Jr.
8 December 2003
Blacksburg, VA
Keywords: epicyclic, planetary, gear train, kinematics,
design
Epicyclic Gear Train Solution Techniques with
Application to Tandem Bicycling
Christopher A. Corey
(Abstract)
This thesis presents a unification of kinematic and force-based methods for the design
and analysis of planetary gear trains along with a discussion of potential applications in tandem
biking. Specifically, this thesis will provide a simple solution technique for the general case of a
two-degree of freedom (2DOF) planetary gear train along with new graphical design aids. It will
also address the use of epicyclic gear trains as a power coupling in a tandem bike.
In the current literature, planetary gear trains are given a clear treatment with regard to
the pure kinematics of the system, but little or no literature exists that includes the torques
present in the system. By treating both the kinematics and torque balance of the most general
case, this thesis attempts to fill a void in the current literature. After developing the solution to
the general two-degree of freedom case using the Willis formula, a force analysis will be
performed using the conservation of energy principle assuming zero losses. Once the total
solution is known, nomographs will be presented as a simple design tool. These graphical aids
enable the designer to simultaneously approximate both speeds and torques for the mechanism.
After fully developing a satisfactory solution technique and design tools, these will be applied to
the problem of coupling the power provided by the riders of a tandem bicycle.
Acknowledgments
Six years ago, if someone had told me I would eventually receive a Master of Science
degree in mechanical engineering, I would have laughed. When I entered school here at Virginia
Tech, I was treating engineering as a secondary degree to my “real” major, music performance.
Many thanks are due to my parents, who were there for me in my transition from a music student
to an engineer, even as they watched tuition money fly out the door in a seemingly endless
stream. Thanks to my mother, for her constant questions. As frustrating as it was to answer the
same question repeatedly, it was a constant reminder of how much she cared about me. I want to
thank my father for spending countless hours with me in a workshop as a child, for this is where
my appetite for engineering was born. Special thanks to my sister, Tricia, for constantly
reminding me that I was smart enough and stubborn enough to accomplish any goal I set for
myself.
I want to thank all of my friends here at Tech that helped me through my journey. Each
one of you has made it different and enjoyable in your own way. Thank you all for listening to
me complain and offering me perspective that you can’t find in a library. Moreover, thanks to
my friends that helped me put academics in perspective and helped me to “take the edge off”
when plans went haywire and things didn’t work the way I intended. I would like to thank Dave
McKee in the music department. Without the time I spent with the Marching Virginians every
day, I surely would have gone insane long before completing my degree.
All the thanks in the world to my advisor, Dr Reinholtz, for seeing in me the potential to
study at the graduate level and encouraging me every step of the way. Without his honest advice
and calming influence, this would not have been possible.
Thank you all for everything.
iii
Contents
Acknowledgements iii
Table of Contents iv
List of Figures and Tables v
Chapter 1: Motivation and Background 1
1.1 Introduction 1
1.2 Motivation 1
1.3 Background 3
1.4 Literature Review 6
Chapter 2: Development of Solution Technique and Design Aids 9
2.1 Motion Analysis 9
2.2 Torque Analysis 10
2.3 Nomography 12
Chapter 3: Case Study: Multi-Rider Human Powered Vehicle 17
Chapter 4: Conclusions and Recommendations 23
4.1 Conclusions 23
4.2 Recommendations 23
References 24
Appendix A: MATLAB Transmission Ratio Selection Code 25
Appendix B: MATLAB Power Contribution Code 26
Vita 28
iv
Figures and Tables
Figure 1: Gear train to be used in the Human Powered Vehicle Team’s design effort 2
Figure 2: (a) The elementary epicyclic gear train and (b) its kinematical representation 3
Figure 3: The simple and complex epicyclic gear trains 4
Figure 4: Epicyclic gear train of the lower arrangement of quadrant I in figure 3 5
Figure 5: Epicyclic gear train with a basic transmission ratio of 1.0 10
Figure 6: General layout of nomograph for solution of epicyclic gear trains 12
Figure 7: Completed nomograph, with ranges of basic transmission ratio, R, labeled 14
Figure 8: Nomograph with torques represented as vectors 15
Table 1: Design Constraints For Tandem Bike Design 17
Figure 9: Nomograph With Target Regions for ωA highlighted 18
Figure 10: Nomographs for (a) 1


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