THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MECHANICAL AND MANUFACTURING ENGINEERING THE STABILITY AND HANDLING CHARACTERISTICS OF BICYCLES David Stevens 3135047 BACHELOR OF MECHANICAL ENGINEERING June 2009 Supervisor: Dr. Nathan Kinkaid Abstract Developments have recently been made in analysis of bicycle self stability. Applicability of benchmarked linearized dynamics equations to a variation of modern bicycle designs is investigated. Results gained through experimentation on an instrumented bicycle with variable geometry are compared to predicted results. Precise three dimensional modeling is used to calculate bicycle mass properties, for use in dynamics equations. Strong correlations between experimental and predicted results are found over large variations in bicycle geometry. 2 Declaration I, David Stevens, declare that this thesis is my own work, except where acknowledge, and has not been previously submitted for academic credit. 3 Acknowledgements Over the course of writing this thesis, my supervisor Dr Nathan Kinkaid has offered support when it was needed, as well as technical advice. Your enthusiasm towards the project has been a great motivation throughout the year Russell Overhall has been generous in passing on his experience, and knowledge. Thank you for having trust in my decision to attach your computer to my uncontrolled bicycle. I would like to thank my girlfriend, friends and family, in particular my parents, for their support and encouragement throughout my education. 4 Table of Contents 1 Introduction......................................................................................................10 2 Bicycle Research History.................................................................................14 2.1 Linearized Dynamics Equation for Bicycles .............................................18 2.2 Solutions to the linearized equations .........................................................23 2.3 Data Analysis .............................................................................................25 3 Test Bicycle Design ..........................................................................................26 4 Modelling ..........................................................................................................29 4.1 Pro Engineer Mass Properties....................................................................33 5 Bicycle Instrumentation ..................................................................................35 Instrument Calibration ...........................................................................................40 6 Experimental Procedure..................................................................................42 6.1 Data Processing..........................................................................................43 7 Results ...............................................................................................................45 7.1 Experimental Observations........................................................................45 7.2 Calculated Stability Ranges .......................................................................46 7.3 Experimental Results .................................................................................59 8 Discussion of Results........................................................................................71 9 Conclusions and Recommendations for Future Work.................................73 10 Bibliography .................................................................................................74 11 Appendix.......................................................................................................77 5 11.1 Appendix A: Calculation of Coefficients of Linearized Dynamics Equation 77 11.2 Appendix B: Calculated coefficients for tested head angles .....................80 11.3 Appendix C Drawings................................................................................82 6 Table of Figures Figure 1: Frame Components Layout ........................................................................11 Figure 2 Head Angle and Trail ..................................................................................12 Figure 3 Commuter Bike ...........................................................................................13 Figure 4 Downhill Mountain Bike.............................................................................13 Figure 5 Bicycle Model .............................................................................................18 Figure 6 Configuration Space ....................................................................................21 Figure 7 Stability Diagram.........................................................................................23 Figure 8 Quadratic Recurrence Bifurcation Diagram................................................24 Figure 9 Residuals......................................................................................................25 Figure 10 Variable Head Angle .................................................................................27 Figure 11: Craftright Digital Angle Gauge................................................................29 Figure 12: Instrumented Bicycle................................................................................30 Figure 13: Variation of Trail and Wheelbase.............................................................30 Figure 14: Complete 3D Model .................................................................................31 Figure 15 Constraint System......................................................................................32 Figure 16: Head Angle Variation...............................................................................32 Figure 17: Sinus SoundBook .....................................................................................35 Figure 18: Voltage divider arrangement....................................................................36 Figure 19: Steering angle potentiometer, and restraint..............................................36 Figure 20: Rear Wheel Speedometer .........................................................................37 Figure 21: Speedometer Diagram ..............................................................................38 Figure 22: Cateye Wireless Speedometer..................................................................38 Figure 23: Accelerometer...........................................................................................39 7 Figure 24: Lean of Rear Frame..................................................................................39 Figure 25: Potentiometer Calibration.........................................................................40 Figure 26: Eigenvalue Plot, head angle 81.9°............................................................46 Figure 27: Eigenvalue Plot, head angle 80.8°............................................................47 Figure 28: Eigenvalue Plot, head angle 79.0°............................................................48 Figure 29: Eigenvalue Plot, head angle 78.0°............................................................49 Figure 30: Eigenvalue Plot, head angle 77.0°............................................................50 Figure 31: Eigenvalue Plot, head angle 75.2°............................................................51 Figure 32: Eigenvalue Plot, head angle 73.0°............................................................52 Figure 33: Eigenvalue Plot, head angle 70.3°............................................................53 Figure 34: Eigenvalue Plot, head angle 66.8°............................................................54 Figure 35: Eigenvalue Plot, head angle 62.6°............................................................55 Figure 36: Eigenvalue Plot, head angle 61.5°............................................................56 Figure 37: Eigenvalue Plot, head angle 58.6°............................................................56 Figure 38: Eigenvalue Plot, head angle 55.6°............................................................57 Figure 39: Stable Velocity vs Head Angle.................................................................58 Figure 40: Rear frame lean rate, for head angle 80.8°. ..............................................59 Figure 41: Steer angle output, for head angle 80.8°. .................................................60 Figure 42: Rear Wheel Speedometer Output, head angle 80.8°. ...............................61 Figure 43: Velocity vs time, head angle 80.8°...........................................................61 Figure 44: Lean rate curve fitting, for head angle 80.8°............................................62 Figure 45: Experimental eigenvalues, head angle 80.8°............................................63 Figure 46: Experimental eigenvalues, head angle 78.0°............................................64 Figure 47: Experimental eigenvalues, head angle 77.0°............................................64 Figure 48: Experimental eigenvalues, head angle 75.2°............................................65 8 Figure 49: Experimental eigenvalues, head angle 73.0°............................................66 Figure 50: Experimental eigenvalues, head angle 70.3°............................................66 Figure 51: Experimental eigenvalues, head angle 66.8°............................................67 Figure 52: Experimental eigenvalues, head