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Performance-based gear metrology : kinematic-transmission-error computation and diagnosis
Personal Author:
Publication Information:
Chichester, West Sussex ; Hoboken : John Wiley & Sons, 2013
Physical Description:
xviii, 268, [269] p. : ill. ; 25 cm.


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30000010306287 TL262 M37 2013 Open Access Book Book

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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

Prefacep. xi
Acknowledgmentsp. xvii
1 Introductionp. 1
1.1 Transmission Errorp. 2
1.2 Mathematical Modelp. 4
1.3 Measurable Mathematical Representation of Working-Surface-Deviationsp. 6
1.4 Final Form of Kinematic-Transmission-Error Predictionsp. 10
1.5 Diagnosing Transmission-Error Contributionsp. 12
1.6 Application to Gear-Health Monitoringp. 13
1.7 Verification of Kinematic Transmission Error as a Source of Vibration Excitation and Noisep. 14
1.8 Gear Measurement Capabilitiesp. 15
Referencesp. 19
2 Parallel-Axis Involute Gearsp. 21
2.1 The Involute Tooth Profilep. 21
2.2 Parametric Description of Involute Helical Gear Teethp. 24
2.3 Multiple Tooth Contact of Involute Helical Gearsp. 27
2.4 Contact Ratiosp. 27
Referencesp. 30
3 Mathematical Representation and Measurement of Working-Surface-Deviationsp. 31
3.1 Transmission Error of Meshing-Gear-Pairsp. 32
3.2 Toom-Working-Surface Coordinate Systemp. 34
3.3 Gear-Measurement Capabilitiesp. 36
3.4 Common Types of Working-Surface Errorsp. 37
3.5 Mathematical Representation of Working-Surface-Deviationsp. 38
3.6 Working-Surface Representation Obtained from Line-Scanning Tooth Measurementsp. 45
3.7 Example of Working-Surface Generations Obtained from Line-Scanning Measurementsp. 54
Appendix 3.A Method for Estimating Required Number of Primary Line-Scanning Measurements Based on Surface-Roughness Criteriap. 58
Appendix 3.B Method for Estimating Required Number of Primary Line-Scanning Measurements for Case of Known Ghost-Tone Rotational-Harmonic Numberp. 61
Referencesp. 67
4 Rotational-Harmonic Analysis of Working-Surface Deviationsp. 69
4.1 Periodic Sequence of Working-Surface Deviations at a Generic Tooth Locationp. 69
4.2 Heuristic Derivation of Rotational-Harmonic Contributionsp. 70
4.3 Rotational-Harmonic Contributions from Working-Surface Deviationsp. 71
4.4 Rotational-Harmonic Spectrum of Mean-Square Working-Surface Deviationsp. 75
4.5 Tooth-Working-Surface Deviations Causing Specific Rotational-Harmonic Contributionsp. 79
4.6 Discussion of Working-Surface Deviation Rotational-Harmonic Contributionsp. 83
Appendix 4.A Formal Derivation of Equation (4.3)p. 88
Appendix 4.B Formulas for |B ke (n)| 2 and G ¬Ņ (n) Involving Only Real Quantitiesp. 90
Appendix 4.C Alternative Proofs of Equations (4.33a) and (4.33c)p. 91
Referencesp. 92
5 Transmission-Error Spectrum from Working-Surface-Deviationsp. 95
5.1 Transmission-Error Contributions from Working-Surface-Deviationsp. 96
5.2 Fourier-Series Representation of Transmission-Error Contributions from Working-Surface-Deviationsp. 99
5.3 Rotational-Harmonic Spectrum of Mean-Square Mesh-Attenuated Working-Surface-Deviationsp. 101
5.4 Example of Rotational-Harmonic Spectrum of Mean-Square Mesh-Attenuated Working-Surface-Deviationsp. 103
Referencesp. 108
6 Diagnosing Manufacturing-Deviation Contributions to Transmission-Error Spectrap. 109
6.1 Main Features of Transmission-Error Spectrap. 109
6.2 Approximate Formulation for Generic Manufacturing Deviationsp. 113
6.3 Reduction of Results for Spur Gearsp. 119
6.4 Rotational-Harmonic Contributions from Accumulated Tooth-Spacing Errorsp. 121
6.5 Rotational-Harmonic Contributions from Tooth-to-Tooth Variations Other Than Tooth-Spacing Errorsp. 126
6.6 Rotational-Harmonic Contributions from Undulation Errorsp. 131
6.7 Explanation of Factors Enabling Successful Predictionsp. 158
Appendix 6.A Validation of Equation (6.46)p. 161
Referencesp. 162
7 Transmission-Error Decomposition and Fourier Series Representationp. 165
7.1 Decomposition of the Transmission Error into its Constituent Componentsp. 166
7.2 Transformation of Locations on Tooth Contact Lines to Working-Surface Coordinate Systemp. 171
7.3 Fourier-Series Representation of Working-Surface-Deviation Transmission-Error Contributionp. 175
7.4 Fourier-Series Using Legendre Representation of Working-Surface-Deviationsp. 186
7.5 Fourier-Series Representation of Normalized Mesh Stiffness K M (s)/K Mp. 191
7.6 Approximate Evaluation of Mesh-Attenuation Functionsp. 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) Quantitiesp. 210
Appendix 7.A Integral Equation for and Interpretation of Local Tooth-Pair Stiffness K Tj (x,y) Per Unit Length of Line of Contactp. 217
Appendix 7.B Transformation of Tooth-Contact-Line Coordinates to Cartesian Working-Surface Coordinatesp. 220
Appendix 7.C Fourier Transform and Fourier Seriesp. 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 Seriesp. 236
Appendix 7.F Evaluation of Discrete Convolution of Complex Quantities Using Real Quantitiesp. 236
Referencesp. 238
8 Discussion and Summary of Computational Algorithmsp. 241
8.1 Tooth-Working-Surface Measurementsp. 242
8.2 Computation of Two-Dimensional Legendre Expansion Coefficientsp. 246
8.3 Regeneration of Working-Surface-Deviationsp. 248
8.4 Rotational-Harmonic Decomposition of Working-Surface-Deviationsp. 251
8.5 Explanation of Attenuation Caused by Gear Meshing Actionp. 251
8.6 Diagnosing and Understanding Manufacturing-Deviation Contributions to Transmission-Error Spectrap. 252
8.7 Computation of Mesh-Attenuated Kinematic-Transmission-Error Contributionsp. 253
Referencesp. 257
Subject Indexp. 259
Figure Indexp. 267
Table Indexp. 269