Performance-Based Gear Metrology

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

Autorentext

William D. Mark, The Pennsylvania State University, USA
Dr Mark is Senior Scientist in the Applied Research Laboratory and Professor Emeritus of Acoustics at The Pennsylvania State University. He has over 40 years experience working in the acoustics industry, including roles in Bolt, Beranek and Newman Inc., Sperry Rand Research Center, The US Air Force and the Cambridge Research Laboratories culminating in a Meritorious Civilian Service Award from U.S. Navy in 2001. He is widely thought of as the leading expert in the area of gear vibration excitation, and is Fellow of the Acoustical Society of America and Senior Member of the Institute of Electrical and Electronics Engineers. Dr Mark has published multiple journal papers as well as contributing to a number of books during his career.

Zusammenfassung
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

Inhalt

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…