Opis: Performance-Based Gear Metrology - William D. Mark

A mathematically rigorous explanation of how manufacturing deviations and damage on the working surfaces of gear teeth cause transmission-error contributions to vibration excitations Some gear-tooth working-surface manufacturing deviations of significant amplitude cause negligible vibration excitation and noise, yet others of minuscule amplitude are a source of significant vibration excitation and noise. Presently available computer-numerically-controlled dedicated gear metrology equipment can measure such error patterns on a gear in a few hours in sufficient detail to enable accurate computation and diagnosis of the resultant transmission-error vibration excitation. How to efficiently measure such working-surface deviations, compute from these measurements the resultant transmission-error vibration excitation, and diagnose the manufacturing source of the deviations, is the subject of this book. Use of the technology in this book will allow quality spot checks to be made on gears being manufactured in a production run, to avoid undesirable vibration or noise excitation by the manufactured gears. Furthermore, those working in academia and industry needing a full mathematical understanding of the relationships between tooth working-surface deviations and the vibration excitations caused by these deviations will find the book indispensable for applications pertaining to both gear-quality and gear-health monitoring. Key features: * Provides a very efficient method for measuring parallel-axis helical or spur gears in sufficient detail to enable accurate computation of transmission-error contributions from working-surface deviations, and algorithms required to carry out these computations, including examples * Provides algorithms for computing the working-surface deviations causing any user-identified tone, such as 'ghost tones,' or 'sidebands' of the tooth-meshing harmonics, enabling diagnosis of their manufacturing causes, including examples * Provides explanations of all harmonics observed in gear-caused vibration and noise spectra. * Enables generation of three-dimensional displays and detailed numerical descriptions of all measured and computed working-surface deviations, including examplesPREFACE i General Background References x CHAPTER 1. INTRODUCTION 1 1.1 Transmission Error 1 1.2 Mathematical Model 3 1.3 Measurable Mathematical Representation of Working-Surface Deviations 4 1.4 Final Form of Kinematic Transmission Error Predictions 7 1.5 Diagnosing Transmission-Error Contributions 9 1.6 Application to Gear-Health Monitoring 10 1.7 Verification of Kinematic Transmission Error as a Source of Vibration Excitation and Noise 11 1.8 Gear Measurement Capabilities 12 CHAPTER 2. PARALLEL-AXIS INVOLUTE GEARS 15 2.1 The Involute Tooth Profile 15 2.2 Parametric Description of Involute Helical Gear Teeth 16 2.3 Multiple Tooth Contact of Involute Helical Gears 18 2.4 Contact Ratios 19 CHAPTER 3. MATHEMATICAL REPRESENTATION AND MEASUREMENT OF WORKING-SURFACE DEVIATIONS 21 3.1 Transmission Error of Meshing Gear Pairs 21 3.2 Tooth-Working-Surface Coordinate System 22 3.3 Gear Measurement Capabilities 24 3.4 Common Types of Working-Surface Errors 25 3.5 Mathematical Representation of Working-Surface Deviations 25 3.6 Working-Surface Representation Obtained From Line-Scanning Tooth Measurements 32 3.7 Example Working-Surface Generations Obtained from Line-Scanning Measurements 38 Appendix 3A. Method for Estimating Required Number of Primary Line-Scanning Measurements Based on Surface-Roughness Criteria 40 Appendix 3B. Method for Estimating Required Number of Primary Line-Scanning Measurements for Case of Known Ghost-Tone Rotational-Harmonic Number 43 CHAPTER 4. ROTATIONAL-HARMONIC ANALYSIS OF WORKING-SURFACE DEVIATIONS 49 4.1 Periodic Sequence of Working-Surface Deviations at a Generic Tooth Location 49 4.2 Heuristic Derivation of Rotational-Harmonic Contributions 49 4.3 Rotational-Harmonic Contributions from Working-Surface Deviations 50 4.4 Rotational-Harmonic Spectrum of Mean-Square Working-Surface Deviations 55 4.5 Tooth-Working-Surface Deviations Causing Specific Rotational-Harmonic Contributions 59 Appendix 4A. Formal Derivation of Eq. (4.3) 65 Appendix 4B. Formulas for |B-kl (n)|^2 and G (n) Involving Only Real Quantities 66 Appendix 4C. Alternative Proofs of Eqs. (4.33a) and (4.33c) 67 CHAPTER 5. TRANSMISSION-ERROR SPECTRUM FROM WORKING-SURFACE DEVIATIONS 70 5.1 Transmission-Error Contributions from Working-Surface Deviations 70 5.2 Fourier-Series Representation of Transmission-Error Contributions from Working-Surface Deviations 73 5.3 Rotational-Harmonic Spectrum of Mean-Square Mesh-Attenuated Working-Surface Deviations 76 5.4 Example Rotational-Harmonic Spectrum of Mean-Square Mesh-Attenuated Working-Surface Deviations 78 CHAPTER 6. DIAGNOSING MANUFACTURING-DEVIATION CONTRIBUTIONS TO TRANSMISSION-ERROR SPECTRA 82 6.1 Main Features of Transmission-Error Spectra 82 6.2 Approximate Formulation for Generic Manufacturing Deviations 85 6.3 Reduction of Results for Spur Gears 90 6.4 Rotational-Harmonic Contributions from Accumulated Tooth-Spacing Errors 91 6.5 Rotational-Harmonic Contributions from Tooth-to-Tooth Variations Other Than Tooth-Spacing Errors 96 6.6 Rotational-Harmonic Contributions from Undulation Errors 101 6.7 Explanation of Factors Enabling Successful Predictions 116 Appendix 6A. Validation of Equation (6.46) 118 CHAPTER 7. TRANSMISSION-ERROR DECOMPOSITION AND FOURIER SERIES REPRESENTATION 120 7.1 Decomposition of the Transmission Error into its Constituent Components [Mark (1978)] 121 7.2 Transformation of Locations on Tooth Contact Lines to Working-Surface Coordinate System [Mark (1978)] 125 7.3 Fourier-Series Representation of Working-Surface-Deviation Transmission-Error Contribution [Mark (1978), (1979)] 129 7.4 Fourier-Series Using Legendre Representation of Working-Surface Deviations [Mark (1979)] 141 7.5 Fourier-Series Representation of Normalized Mesh Stiffness K-M (s)/K -M 146 7.6 Approximate Evaluation of Mesh-Attenuation Functions [Mark (1978), (1979)] 149 7.7 Accurate Evaluation of Fourier-Series Coefficients of Normalized Reciprocal Mesh Stiffness K -M/K-M (s) 154 7.8 Fourier-Series Representation of Working-Surface-Deviation Transmission-Error Contributions Utilizing Only Real (Not-Complex) Quantities 165 Appendix 7A. Integral Equation for and Interpretation of Local Tooth-Pair Stiffness KTj(x,y) Per Unit Length of Line of Contact [Mark (1978)] 173 Appendix 7B. Transformation of Tooth-Contact-Line Coordinates to Cartesian Working-Surface Coordinates 176 Appendix 7C. Fourier Transform and Fourier Series 180 Appendix 7D. Fourier Transform of Scanning Content of the Line Integral f (~j) (s) secpsi-b ?-(y-A (s))^(y-B (s)) f-Cj (y,betas+gammay)dy, Eq. (7.24b,c) 185 Appendix 7E. Fractional Error in Truncated Infinite Geometric Series 192 Appendix 7F. Evaluation of Discrete Convolution of Complex Quantities Using Real Quantities 193 CHAPTER 8. DISCUSSION AND SUMMARY OF COMPUTATIONAL ALGORITHMS 196 8.1 Tooth Working-Surface Measurements 196 8.2 Computation of Two-Dimensional Legendre Expansion Coefficients 200 8.3 Regeneration of Working-Surface Deviations 203 8.4 Rotational-Harmonic Decomposition of Working-Surface Deviations 205 8.5 Explanation of Attenuation Caused by Gear Meshing Action 206 8.6 Diagnosing and Understanding Manufacturing-Deviation Contributions to Transmission-Error Spectra 206 8.7 Computation of Mesh-Attenuated Kinematic-Transmission-Error Contributions 207 REFERENCES 212 Figure Locations Table Locations Index

Szczegóły: Performance-Based Gear Metrology - William D. Mark

Tytuł: Performance-Based Gear Metrology
Autor: William D. Mark
Producent: John Wiley
ISBN: 9781119961697
Rok produkcji: 2012
Ilość stron: 288
Oprawa: Twarda
Waga: 0.6 kg

Recenzje: Performance-Based Gear Metrology - William D. Mark