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Hybrid Machine Modelling and Control HYBRID MACHINE MODELLING AND CONTROL by Lale Canan Tokuz This thesis is submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy of the Council for Xational Academic A wards. Mechanisms and Machines Group Liverpool Polytechnic February 1992 CONTENTS Acknowledgements ............................................................................................................. (5i) Abstract ............................................................................................................................. {6i) In trod uction Non- Uniform mechanism motion .................................................................................. 1 Two degrees of freedom mechanisms ............................................................................. 7 Thesis Structure .......................................................................................................... 10 Chapter 1 The Hybrid Arrangement 1.1. Introduction ................................................................................................................. 13 1.~. The General Description of the Experimental Set-Up .................................................... 13 1.2.1. The Drive Motors .............................................................................................. 15 1.2.2. The Differential Gear-Unit ................................................................................ 15 1.2.3. The Design of Slider-Crank Mechanism ............................................................. 16 1.3. System Control and Measurement ................................................................................ 17 1.4. Torque and Power Measurement ................................................................................... 18 1.5. (~onclusion .................................................................................................................... 18 Chapter 2 Motion Design ~.l. Int.roduction ................................................................................................................. 19 :!.:!. Molioll Design .............................................................................................................. 19 2.2.1. Polynonlial Laws ............................................................................................... 20 2.2.2. Solution of Polynomial Coefficients ................................................................... :?O 2.3. l'he Exanlple Motions .................................................................................................. 2:~ 2.3.1. Rise- Return l\1otion ........................................................................................... 24 2.3.2. Risc-Dwell-Rt'turn \Iotion · .......... · ........ · ................ · .... ·· ..................................... 27 2.3.3. Risl'- Rt·t url1- Dwell ~fotion ................................................................................. 30 1i 2.4. Conclusion .................................................................................................................... 33 Chapter 3 Kinematic and Dynamic Issues 3.1. Introduction ................................................................................................................. 34 :3.2. Kinematic Analysis of Slider-Crank .............................................................................. J:I 3.2.1. Inverse Kinematics ............................................................................................ Jli 3.3. Dynamic Analysis of Slider-Crank ................................................................................ :1~ 3.3.1. Generalized Coordinates .................................................................................... :~~ 3.3.2. Lagrange's Equations ........................................................................................ 42 3.3.3. Inverse Dynamics for slider-crank ...................................................................... 43 3.4. Determination of the Separate Inputs for The Hybrid Arrangement ............................... ,~ 3.5. Conclusion .................................................................................................................... 48 Chapter 4 Mathematical Modelling of the Hybrid Arrangement 4.1. Introduction ................................................................................................................. 53 4.2. Modelling of a System .................................................................................................. 53 4.2.1. Mathemat.ical Model .......................................................................................... 54 4.2.2. Classification of Models ..................................................................................... 54 4.2.3. Development of a mathematical model .............................................................. 55 4.:3. The Derivat.ion of The Equat.ions of Motion ................................................................. :)5 4.3.1. The Differential Equations of Motion for The Hybrid Arrangement ................... 55 4.3.2. Matrix Form Representation of Equations of Motion ......................................... 62 4.4. Example Tests for The System Response ...................................................................... 64 4.4.1. DC-Motor Characteristics .................................................................................. 64 4.4.2. The Servo-Motor Response for Standard Inputs ................................................. 67 4.5. The System Responses for The Hybrid Arrangement ..................................................... 71 4.5.1. The System Response for R-R Motion ............................................................... i'1. 4.5.2. The System Response for R-D-R Motion ........................................................... 76 4.;).3. The System Response for R-R-D Motion ........................................................... 1'\0 -1.6. (~onclusion .................................................................................................................... 84 Chapter 5 Computer Control Isslies 5.1. Introduction ................................................................................................................. 1'i:1 5.'1.. Control Schelilc ............................................................................................................ 85 ;).2.1. Sanlplin~ Rate .................................................................................................. 86 :1.2.'1.. Coordination of Constant Spt·t.'<i \tolor and Servo- \totor ................................... 87 'r_I 5.2.3. Other Issues ...................................................................................................... ,,~ 5.2.4. Controller Hardware Requirements .................................................................... "9 5.3. Hardware Architecture and Interface for the Controller ................................................. 90 5.4. Comrnand and R.esponse Curves ................................................................................... 93 5.5. Command l\10tion Tuning ............................................................................................ ~16 5.5.1. Tuning Algorithm ............................................................................................ 100 5.6. Conclusion ................................................................................................................... 10:! Chapter 6 Power Transmission and Flow in the Hybrid Arrangement 6.1. Introduction ................................................................................................................ 104 6.2. Differential Transmissions ........................................................................................... 10-1 6.3. General Analysis of a Differential Mechanism .............................................................. 106 6.3.1. Torque Distribution .......................................................................................... 107 6.3.2. Effect of Losses ................................................................................................. 109 6.3.3. Torque and Power Calculation in the Program ................................................. III 6.4. The Experimental Set-Up ............................................................................................ 118 6.4.1. Torque Measurement ........................................................................................ 11 t' 6.4.2. Angular Velocity Measurement ......................................................................... 118 6.4.3. Torque, Angular Velocity and Power Curves .................................................... 120 6.5. Conclusion ................................................................................................................... 1~5 Chapt.er 7 Comparison of The Model and The Experimental Results 7.1. Introduction ................................................................................................................ 126 7.2. Comparison of Model and Experimental Responses ...................................................... 126 7.2.1. Standard Inputs ............................................................................................... 1~6 7.2.2. Different Motion Considerations ....................................................................... 129 7.3. Comparison of Torque, Angular Velocity and Power Outputs .....................................
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