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### Summary

### Summary

A mathematically rigorous explanation of how manufacturingdeviations and damage on the working surfaces of gear teeth causetransmission-error contributions to vibration excitations

Some gear-tooth working-surface manufacturing deviations ofsignificant amplitude cause negligible vibration excitation andnoise, yet others of minuscule amplitude are a source ofsignificant vibration excitation and noise. Presentlyavailable computer-numerically-controlled dedicated gear metrologyequipment can measure such error patterns on a gear in a few hoursin sufficient detail to enable accurate computation and diagnosisof the resultant transmission-error vibration excitation. How to efficiently measure such working-surfacedeviations, compute from these measurements the resultanttransmission-error vibration excitation, and diagnose themanufacturing source of the deviations, is the subject of thisbook.

Use of the technology in this book will allow quality spotchecks to be made on gears being manufactured in a production run,to avoid undesirable vibration or noise excitation by themanufactured gears. Furthermore, those working in academiaand industry needing a full mathematical understanding of therelationships between tooth working-surface deviations and thevibration excitations caused by these deviations will find the bookindispensable for applications pertaining to both gear-quality andgear-health monitoring.

Key features:

Provides a very efficient method for measuring parallel-axishelical or spur gears in sufficient detail to enable accuratecomputation of transmission-error contributions fromworking-surface deviations, and algorithms required to carry outthese computations, including examples Provides algorithms for computing the working-surfacedeviations causing any user-identified tone, such as ?ghosttones,? or ?sidebands? of the tooth-meshingharmonics, enabling diagnosis of their manufacturing causes,including examples Provides explanations of all harmonics observed in gear-causedvibration and noise spectra. Enables generation of three-dimensional displays and detailednumerical descriptions of all measured and computed working-surfacedeviations, including examples### Author Notes

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.

### Table of Contents

Preface | p. xi |

Acknowledgments | p. xvii |

1 Introduction | p. 1 |

1.1 Transmission Error | p. 2 |

1.2 Mathematical Model | p. 4 |

1.3 Measurable Mathematical Representation of Working-Surface-Deviations | p. 6 |

1.4 Final Form of Kinematic-Transmission-Error Predictions | p. 10 |

1.5 Diagnosing Transmission-Error Contributions | p. 12 |

1.6 Application to Gear-Health Monitoring | p. 13 |

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

1.8 Gear Measurement Capabilities | p. 15 |

References | p. 19 |

2 Parallel-Axis Involute Gears | p. 21 |

2.1 The Involute Tooth Profile | p. 21 |

2.2 Parametric Description of Involute Helical Gear Teeth | p. 24 |

2.3 Multiple Tooth Contact of Involute Helical Gears | p. 27 |

2.4 Contact Ratios | p. 27 |

References | p. 30 |

3 Mathematical Representation and Measurement of Working-Surface-Deviations | p. 31 |

3.1 Transmission Error of Meshing-Gear-Pairs | p. 32 |

3.2 Toom-Working-Surface Coordinate System | p. 34 |

3.3 Gear-Measurement Capabilities | p. 36 |

3.4 Common Types of Working-Surface Errors | p. 37 |

3.5 Mathematical Representation of Working-Surface-Deviations | p. 38 |

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

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

Appendix 3.A Method for Estimating Required Number of Primary Line-Scanning Measurements Based on Surface-Roughness Criteria | p. 58 |

Appendix 3.B Method for Estimating Required Number of Primary Line-Scanning Measurements for Case of Known Ghost-Tone Rotational-Harmonic Number | p. 61 |

References | p. 67 |

4 Rotational-Harmonic Analysis of Working-Surface Deviations | p. 69 |

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

4.2 Heuristic Derivation of Rotational-Harmonic Contributions | p. 70 |

4.3 Rotational-Harmonic Contributions from Working-Surface Deviations | p. 71 |

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

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

4.6 Discussion of Working-Surface Deviation Rotational-Harmonic Contributions | p. 83 |

Appendix 4.A Formal Derivation of Equation (4.3) | p. 88 |

Appendix 4.B Formulas for |B ke (n)| 2 and G ¿ (n) Involving Only Real Quantities | p. 90 |

Appendix 4.C Alternative Proofs of Equations (4.33a) and (4.33c) | p. 91 |

References | p. 92 |

5 Transmission-Error Spectrum from Working-Surface-Deviations | p. 95 |

5.1 Transmission-Error Contributions from Working-Surface-Deviations | p. 96 |

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

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

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

References | p. 108 |

6 Diagnosing Manufacturing-Deviation Contributions to Transmission-Error Spectra | p. 109 |

6.1 Main Features of Transmission-Error Spectra | p. 109 |

6.2 Approximate Formulation for Generic Manufacturing Deviations | p. 113 |

6.3 Reduction of Results for Spur Gears | p. 119 |

6.4 Rotational-Harmonic Contributions from Accumulated Tooth-Spacing Errors | p. 121 |

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

6.6 Rotational-Harmonic Contributions from Undulation Errors | p. 131 |

6.7 Explanation of Factors Enabling Successful Predictions | p. 158 |

Appendix 6.A Validation of Equation (6.46) | p. 161 |

References | p. 162 |

7 Transmission-Error Decomposition and Fourier Series Representation | p. 165 |

7.1 Decomposition of the Transmission Error into its Constituent Components | p. 166 |

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

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

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

7.5 Fourier-Series Representation of Normalized Mesh Stiffness K M (s)/K M | p. 191 |

7.6 Approximate Evaluation of Mesh-Attenuation Functions | p. 195 |

7.7 Accurate Evaluation of Fourier-Series Coefficients of Normalized Reciprocal Mesh Stiffness K M /K M (s) | p. 200 |

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

Appendix 7.A Integral Equation for and Interpretation of Local Tooth-Pair Stiffness K Tj (x,y) Per Unit Length of Line of Contact | p. 217 |

Appendix 7.B Transformation of Tooth-Contact-Line Coordinates to Cartesian Working-Surface Coordinates | p. 220 |

Appendix 7.C Fourier Transform and Fourier Series | p. 225 |

Appendix 7.D Fourier Transform of Scanning Content of the Line Integral (Equation (7.24b,c))$ | p. 229 |

Appendix 7.E Fractional Error in Truncated Infinite Geometric Series | p. 236 |

Appendix 7.F Evaluation of Discrete Convolution of Complex Quantities Using Real Quantities | p. 236 |

References | p. 238 |

8 Discussion and Summary of Computational Algorithms | p. 241 |

8.1 Tooth-Working-Surface Measurements | p. 242 |

8.2 Computation of Two-Dimensional Legendre Expansion Coefficients | p. 246 |

8.3 Regeneration of Working-Surface-Deviations | p. 248 |

8.4 Rotational-Harmonic Decomposition of Working-Surface-Deviations | p. 251 |

8.5 Explanation of Attenuation Caused by Gear Meshing Action | p. 251 |

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

8.7 Computation of Mesh-Attenuated Kinematic-Transmission-Error Contributions | p. 253 |

References | p. 257 |

Subject Index | p. 259 |

Figure Index | p. 267 |

Table Index | p. 269 |