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Performance-Based Gear Metrology: Kinematic - Transmission - Error Computation and Diagnosis

Performance-Based Gear Metrology: Kinematic - Transmission - Error Computation and Diagnosis

William D. Mark

ISBN: 978-1-118-35789-7

Nov 2012

288 pages



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 examples

Preface xi

Acknowledgments xvii

1 Introduction 1

1.1 Transmission Error 2

1.2 Mathematical Model 4

1.3 Measurable Mathematical Representation of Working-Surface-Deviations 6

1.4 Final Form of Kinematic-Transmission-Error Predictions 10

1.5 Diagnosing Transmission-Error Contributions 12

1.6 Application to Gear-Health Monitoring 13

1.7 Verification of Kinematic Transmission Error as a Source of Vibration Excitation and Noise 14

1.8 Gear Measurement Capabilities 15

References 19

2 Parallel-Axis Involute Gears 21

2.1 The Involute Tooth Profile 21

2.2 Parametric Description of Involute Helical Gear Teeth 24

2.3 Multiple Tooth Contact of Involute Helical Gears 27

2.4 Contact Ratios 27

References 30

3 Mathematical Representation and Measurement of Working-Surface-Deviations 31

3.1 Transmission Error of Meshing-Gear-Pairs 32

3.2 Tooth-Working-Surface Coordinate System 34

3.3 Gear-Measurement Capabilities 36

3.4 Common Types of Working-Surface Errors 37

3.5 Mathematical Representation of Working-Surface-Deviations 38

3.6 Working-Surface Representation Obtained from Line-Scanning Tooth Measurements 45

3.7 Example of Working-Surface Generations Obtained from Line-Scanning Measurements 54

References 67

4 Rotational-Harmonic Analysis of Working-Surface Deviations 69

4.1 Periodic Sequence of Working-Surface Deviations at a Generic Tooth Location 69

4.2 Heuristic Derivation of Rotational-Harmonic Contributions 70

4.3 Rotational-Harmonic Contributions from Working-Surface Deviations 71

4.4 Rotational-Harmonic Spectrum of Mean-Square Working-Surface Deviations 75

4.5 Tooth-Working-Surface Deviations Causing Specific Rotational-Harmonic Contributions 79

4.6 Discussion of Working-Surface Deviation Rotational-Harmonic Contributions 83

5 Transmission-Error Spectrum from Working-Surface-Deviations 95

5.1 Transmission-Error Contributions from Working-Surface-Deviations 96

5.2 Fourier-Series Representation of Transmission-Error Contributions from Working-Surface-Deviations 99

5.3 Rotational-Harmonic Spectrum of Mean-Square Mesh-Attenuated Working-Surface-Deviations 101

5.4 Example of Rotational-Harmonic Spectrum of Mean-Square Mesh-Attenuated Working-Surface-Deviations 103

References 108

6 Diagnosing Manufacturing-Deviation Contributions to Transmission-Error Spectra 109

6.1 Main Features of Transmission-Error Spectra 109

6.2 Approximate Formulation for Generic Manufacturing Deviations 113

6.3 Reduction of Results for Spur Gears 119

6.4 Rotational-Harmonic Contributions from Accumulated Tooth-Spacing Errors 121

6.5 Rotational-Harmonic Contributions from Tooth-to-Tooth Variations Other Than Tooth-Spacing Errors 126

6.6 Rotational-Harmonic Contributions from Undulation Errors 131

6.7 Explanation of Factors Enabling Successful Predictions 158

References 162

7 Transmission-Error Decomposition and Fourier Series Representation 165

7.1 Decomposition of the Transmission Error into its Constituent Components 166

7.2 Transformation of Locations on Tooth Contact Lines to Working-Surface Coordinate System 171

7.3 Fourier-Series Representation of Working-Surface-Deviation Transmission-Error Contribution 175

7.4 Fourier-Series Using Legendre Representation of Working-Surface-Deviations 186

7.5 Fourier-Series Representation of Normalized Mesh Stiffness KM(s)/KM 191

7.6 Approximate Evaluation of Mesh-Attenuation Functions 195

7.7 Accurate Evaluation of Fourier-Series Coefficients of Normalized Reciprocal Mesh Stiffness KM/KM(s) 200

7.8 Fourier-Series Representation of Working-Surface-Deviation Transmission-Error Contributions Utilizing only Real (Not-Complex) Quantities 210

References 238

8 Discussion and Summary of Computational Algorithms 241

8.1 Tooth-Working-Surface Measurements 242

8.2 Computation of Two-Dimensional Legendre Expansion Coefficients 246

8.3 Regeneration of Working-Surface-Deviations 248

8.4 Rotational-Harmonic Decomposition of Working-Surface-Deviations 251

8.5 Explanation of Attenuation Caused by Gear Meshing Action 251

8.6 Diagnosing and Understanding Manufacturing-Deviation Contributions to Transmission-Error Spectra 252

8.7 Computation of Mesh-Attenuated Kinematic-Transmission-Error

Contributions 253

References 257

Subject Index 259

Figure Index 267

Table Index 269