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Kurzbeschreibung This book is a mathematically rigorous explanation of how manufacturing deviations and damage on the working surfaces of gear teeth cause transmission-error contributions to vibration excitations. It provides an efficient method for measuring parallel-axis helical or spur gears in sufficient detail and explanations of all harmonics observed in gear-caused vibration and noise spectra.
Aus dem Inhalt PREFACE 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
7.6 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 integral _(y_A (s))^(y_B (s))f_Cj (y,ßs+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
