UUnniivveerrssiittyy ooff NNeebbrraasskkaa -- LLiinnccoollnn DDiiggiittaallCCoommmmoonnss@@UUnniivveerrssiittyy ooff NNeebbrraasskkaa -- LLiinnccoollnn Mechanical (and Materials) Engineering -- Mechanical & Materials Engineering, Dissertations, Theses, and Student Research Department of 4-2012 EErrrroorr RReedduuccttiioonn aanndd EEffffeecctt ooff SStteepp SSiizzee iinn AAddjjuussttmmeenntt CCaallccuulluuss ffoorr CCaamm AApppplliiccaattiioonnss Sai Siddhartha Nudurupati University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/mechengdiss Part of the Mechanical Engineering Commons Nudurupati, Sai Siddhartha, "Error Reduction and Effect of Step Size in Adjustment Calculus for Cam Applications" (2012). Mechanical (and Materials) Engineering -- Dissertations, Theses, and Student Research. 36. https://digitalcommons.unl.edu/mechengdiss/36 This Article is brought to you for free and open access by the Mechanical & Materials Engineering, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Mechanical (and Materials) Engineering -- Dissertations, Theses, and Student Research by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. ERROR REDUCTION AND EFFECT OF STEP SIZE IN ADJUSTMENT CALCULUS FOR CAM APPLICATIONS by Sai Siddhartha Nudurupati A THESIS Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Master of Science Major: Mechanical Engineering Under the Supervision of Professor Wieslaw M. Szydlowski Lincoln, Nebraska April, 2012 ERROR REDUCTION AND EFFECT OF STEP SIZE IN ADJUSTMENT CALCULUS FOR CAM APPLICATIONS Sai Siddhartha Nudurupati, M.S. University of Nebraska, 2012 Adviser: Wieslaw M. Szydlowski Any measurement, however carefully done, will never be free from errors. Similarly, machining of cams for automobiles is prone to contain errors. These errors are naturally a part and parcel of cam manufacturing. The nature of deviations of the manufactured cam profile from the theoretical cam determines its usability. Sometimes, allowable deviations in high speed cams may be in the order of 2540 µm. Larger deviations will disqualify the cams for applications. Velocity and acceleration of the cam are estimated from the measured displacement of the cam follower during quality control implementation. This data helps in eliminating the unfit cams. Existing methods deal with a notorious challenge from propagation of measurement errors in the displacement data to predicted velocity and acceleration values. J. Oderfeld developed a little known method called ‘Adjustment Calculus’ which is an alternative method for this purpose. This method combines the ‘marching point’ method that fits a polynomial to discrete data and a symmetric Stirling interpolation method. Until now, adjustment calculus has been applied to reduce errors in acceleration data. In this work, adjustment calculus is implemented to velocity predictions. ‘Weights’ for calculation of adjusted velocity are derived using a cubic polynomial fit and symmetric Stirling interpolation formula. The effect of step size on application of adjustment calculus to different cam profiles is probed using the Monte Carlo method. Effective step size for practical applications in automotive cam quality control is suggested for each cam profile. Practical pointers for application to cam inspection for velocity and acceleration analysis are formulated. Table of Contents 1. INTRODUCTION 1.1 Role of Errors in Engineering Process ................................................................. 1 1.2 Special Role of Cam Mechanisms ......................................................................... 2 1.2.1 Manufactured Cam-Follower System Errors .................................................... 3 1.2.1.1 Structural Errors ........................................................................................ 4 1.2.1.2 Fixed Backlash Errors ............................................................................... 4 1.2.1.3 Variable Backlash Errors .......................................................................... 4 1.2.1.4 Cyclic Errors ............................................................................................. 4 1.2.1.5 Cam Profile Errors .................................................................................... 4 1.2.1.6 Waviness ................................................................................................... 5 1.2.2 Generalized Classification of Errors ................................................................. 5 1.2.2.1 Systematic Errors: ..................................................................................... 5 1.2.2.2 Random Errors: ......................................................................................... 5 1.3 Need to determine “True Acceleration” from measured data........................... 6 1.3.1 Intelligent robots with vision ............................................................................ 6 1.3.2 Manufacture of missile shells ........................................................................... 6 1.3.3 Diagnostics of hand-arm vibration syndrome ............................................... 6 1.3.4 Position Tracking of Mechanical Linkages: ................................................. 7 1.3.5 Musculoskeletal analysis of race horse: ........................................................ 7 1.3.6 Cam’s velocity and acceleration estimation: ................................................ 7 1.4 Need for numerical differentiation & integration............................................... 8 1.5 Current Methods .................................................................................................... 8 1.5.1 Finite differences: ............................................................................................. 8 1.5.2 Johnson’s Method: .......................................................................................... 13 1.5.3 Polynomial Approximation: ........................................................................... 17 1.5.4 Chen’s Method: .............................................................................................. 19 1.5.5 Adjustment Calculus: ..................................................................................... 22 1.6 Objective of this work: ........................................................................................ 28 2. DERIVATION OF WEIGHTS FOR VELOCITY 2.1 Weights for velocity: ............................................................................................ 30 2.2 Weights for acceleration: .................................................................................... 37 3. APPLICATION OF ADJUSTMENT CALCULUS 3.1 Cams with Basic Curves and their Motion Characteristics ............................. 38 3.1.1 Cycloidal Cam: ............................................................................................... 38 3.1.2 Harmonic Cam:............................................................................................... 41 3.1.3 3-4-5 Polynomial Cam: .................................................................................. 44 3.1.4 Polynomial Cam P1P2: ................................................................................... 46 3.1.5 4-5-6-7 Polynomial Cam: ............................................................................... 49 3.1.6 Modified Trapezoidal Cam:............................................................................ 51 3.1.7 Sine Cam: ....................................................................................................... 60 3.2 Procedure to Analyze Effects of Errors ............................................................. 62 3.3 Determination of effect of errors using Monte Carlo method ......................... 63 4. MONTE CARLO METHOD AND STEP ANALYSIS 4.1 Effect of profile type on step size ........................................................................ 65 4.1.1 Cycloidal cam: ................................................................................................ 66 4.1.2 Harmonic cam: ............................................................................................... 71 4.1.3 3-4-5 Polynomial cam: ................................................................................... 74 4.1.4 Polynomial cam P1P2:.................................................................................... 77 4.1.5 4-5-6-7 Polynomial cam: ................................................................................ 80 4.1.6 Modified Trapezoidal cam: ............................................................................ 83 4.1.7 Sine cam: ........................................................................................................ 86 4.2 Effect of dwell location on step size .................................................................... 89 4.2.1 Cycloidal Cam: ............................................................................................... 89 4.2.2 3-4-5 Polynomial Cam: .................................................................................. 90 4.2.3 4-5-6-7 Polynomial Cam: ............................................................................... 90 5. CONCLUSIONS AND FUTURE WORK 5.1 Conclusions:.......................................................................................................... 92 5.1.1 Effect of profile type on optimum step selection: .......................................... 92 5.1.2 Effect of location of dwell on optimum step selection: .................................. 94 5.2 Future Work ......................................................................................................... 95 REFERENCES APPENDIX 1 ACKNOWLEDGMENT I wanted to express my sincere gratitude to Dr. Wieslaw M. Szydlowski for his encouragement, guidance and patience while supervising my thesis. I have been an admirer of his knowledge in engineering and teaching methods from his first course Intermediate Kinematics that I have attended. I am also grateful to Dr. Carl Nelson and Dr. Linxia Gu for their support and assistance. I wanted to express extreme gratitude to my parents Sri N. S. S. Sai and Smt. N. Sarada and my sister N. Sai Sravanthi for their love and support throughout my life. I want to thank Dr. Arun Natarajan for his invaluable advice and guidance. I would like to thank my friends Satish, Padma, Apoorva, Chaitanya, Dhairya, Hardik, Shailesh, Uday, Ke, Chris, Sri Harsha, Venkat, Ananth, Mahanth and Ved who have been very supportive and encouraging. I would like to prostrate to Shri Sai Baba for this life and beyond. LIST OF TABLES Table 1.5.1.1: Difference Table ....................................................................................... 12 Table 1.5.2.1: An Example for Application of Johnson’s method .................................. 16 Table 1.5.5.1: Application of Adjustment Calculus......................................................... 26 Table 1.5.5.2: Numerical Example for Application of Adjustment Calculus .................. 28 Table 2.1.1: Central differences for discrete values......................................................... 35 Table 4.1.1.1: Deviations at maximum velocity and acceleration of cycloidal cam at different step sizes............................................................................................................. 68 Table 4.1.2.1: Deviations at maximum velocity and acceleration of harmonic cam at different step sizes............................................................................................................. 72 Table 4.1.3.1: Deviations at maximum velocity and acceleration of 3-4-5 polynomial cam at different step sizes ................................................................................................. 75 Table 4.1.4.1: Deviations at maximum velocity and acceleration of polynomial cam P1P2 at different step sizes ............................................................................................... 78 Table 4.1.5.1: Deviations at maximum velocity and acceleration of 4-5-6-7 polynomial cam at different step sizes ................................................................................................. 81 Table 4.1.6.1: Deviations at maximum velocity and acceleration of modified trapezoidal cam at different step sizes ................................................................................................. 84 Table 4.1.7.1: Deviations at maximum velocity and acceleration of sine cam at different step sizes ........................................................................................................................... 87 Table 4.2.1.1: Deviations at maximum velocity and acceleration of cycloidal cam at different step sizes............................................................................................................. 89