Computational Surface and Roundness Metrology (eBook)

Bala Muralikrishnan has a PhD in Mechanical Engineering from the University of North Carolina at Charlotte, USA. He works for the National Institute of Standards and Technology, where he is a guest researcher in the Engineering Metrology Group of the Precision Engineering Division - one of the divisions of the Manufacturing Engineering Laboratory. Jayaraman Raja has a PhD in Mechanical Engineering from the Indian Institute of Technology, Madras, India. He is a professor and chairman in the Department of Mechanical Engineering & Engineering Science at the University of North Carolina at Charlotte, USA. His research interests include Surface & Form Metrology; Computational Metrology; and Precision Engineering.

Preface 6
Acknowledgment 7
Contents 8
Notation 15
Introduction 17
1.1 Surface and Roundness Metrology 17
1.2 Scope and Objectives 17
1.3 Organization 19
References 20
Part I Filtering 21
A Brief History of Filtering 22
2.1 Introduction 22
2.2 Electrical Filters 22
2.3 Digital Filters 23
2.4 The Envelope Method 23
2.5 The Gaussian Filter 23
2.6 Overlap of Measurement Techniques 24
2.7 Recent Advances in Filtering 24
2.8 Summary 25
References 25
Filtering in the Frequency Domain 27
3.1 Surface Wavelengths 27
3.2 Fourier Transform, Discrete Fourier Transform, and Fast Fourier Transform 28
3.3 Filtering in the Frequency Domain 30
3.4 Wrap-Around Effect 33
3.5 Amplitude Transmission and Phase Characteristics 33
3.6 Summary 34
Exercises 34
References 35
Filtering in the Time Domain 36
4.1 Filtering as an Averaging Process 36
4.2 Relationship Between Frequency-Domain and Time- Domain Filtering 39
4.3 Filtering Profiles: Putting It All Together 40
4.4 Summary 42
Exercises 43
References 44
Gaussian Filter 45
5.1 Introduction 45
5.2 High-Pass and Low-Pass Filters 47
5.3 Roughness, Waviness, and Form Using the Gaussian Filter 47
5.4 Effect of Cutoff 48
5.5 Phase Characteristics 48
5.6 Summary 49
Exercises 49
References 50
The 2RC Filter 51
6.1 Introduction 51
6.2 The 2RC High-Pass Filter 51
6.3 More on the 2RC Filter 54
6.4 Comparison of the 2RC and Gaussian Filters 56
6.5 Summary 56
Exercises 57
References 58
Filtering Roundness Profiles 59
7.1 Introduction 59
7.2 Gaussian Filter for Roundness 59
7.3 Amplitude Transmission Characteristics 61
7.4 Filtering Roundness Profiles in the Time Domain 62
7.5 Circular Convolution in the Frequency Domain 62
7.6 Summary 64
Exercises 64
References 65
Filtering 3D Surfaces 66
8.1 Areal Surface Texture Analysis 66
8.2 2D Convolution 66
8.3 Gaussian Filter in 3D 67
8.4 A Note on Indices 67
8.5 Frequency-Domain Filtering in 3D 70
8.6 Summary 73
Exercises 73
References 74
Part II Advanced Filtering 75
Gaussian Regression Filters 76
9.1 Introduction 76
9.2 Zero-Order Gaussian Regression Filter 77
9.3 Second-Order Gaussian Regression Filter 80
9.4 3D Zero-Order Gaussian Regression Filter 82
9.5 3D Second-Order Gaussian Regression Filter 83
9.6 Summary 84
Exercises 84
References 85
Spline Filter 86
10.1 Introduction 86
10.2 Amplitude Transmission Characteristics 87
10.3 Implementation of the Non-periodic Spline Filter 88
10.4 Implementation of the Periodic Spline Filter 91
10.5 Summary 95
Exercises 95
References 95
Robust Filters 96
11.1 Rk Filter 96
11.2 Robust Gaussian Regression Filter 99
11.3 Summary 100
Exercises 101
References 101
Envelope and Morphological Filters 102
12.1 Envelope Filters 102
12.2 Stylus Tip Convolution 103
12.3 Morphological Filters 105
12.4 A Word on the Fundamentals 107
12.5 3D Morphological Filtering 107
12.6 Summary 109
Exercises 109
References 110
Multi-scale Filtering 111
13.1 Introduction 111
13.2 Alternate Sequence Filters or Scale Space Analysis 111
13.3 Wavelet-Based Filters 115
13.4 Summary 117
Exercises 118
References 119
Part III Fitting 120
Introduction to Fitting Substitute Geometry 121
14.1 Introduction 121
14.2 Fitting Criteria 122
14.3 Solution Methodologies 123
References 124
Least-Squares Best-Fit Line and Plane 126
15.1 Introduction 126
15.2 Closed-Form Solution for LS Best-Fit Line 126
15.3 Matrix Formulation 127
15.4 Centroid as a Point on the LS Line 128
15.5 Normal LS in Parametric Form 129
15.6 Lagrange Multiplier Method 131
15.7 Back to the LS Line 132
15.8 LS Best-Fit Plane 133
15.9 Summary 134
Exercises 134
References 135
Non-linear Least-Squares I: Introduction 136
16.1 Introduction 136
16.2 Formulating the Circle in a Plane Problem 136
16.3 The Steepest Descent Algorithm 137
16.4 The Gauss-Newton Algorithm 139
16.5 The Levenberg-Marquardt Algorithm 141
16.6 Summary 142
Exercises 143
References 143
Non-linear Least-Squares II: Circle, Sphere, and Cylinder 144
17.1 Introduction 144
17.2 Initial Estimates for Center and Radius of Circle 144
17.3 Best-Fit Sphere 147
17.4 Best-Fit Cylinder 148
17.5 Summary 149
Exercises 149
References 149
Fitting Radius-Suppressed Circle Data 150
18.1 Introduction 150
18.2 The Lima°on Approximation 150
18.3 LS Best-Fit Circle 151
18.4 LS Best-Fit Cylinder 153
18.5 Errors in the Lima°on Approximation 154
18.6 Summary 155
Exercises 156
References 156
Exchange Algorithms for Minimum Zone 157
19.1 Introduction 157
19.2 Exchange Algorithms 157
19.3 Exchange Algorithm for Line 158
19.4 Exchange Algorithm for Plane 161
19.5 Exchange Algorithm for Circle 163
19.6 Summary 166
Exercises 167
References 167
Reference Circle-Fitting Using Linear Programming Simplex 168
20.1 Introduction 168
20.2 LP Simplex 168
20.3 Formulating the MI Problem 170
20.4 Formulating the MC Problem 174
20.5 Duality 175
20.6 LP and Exchange Algorithms 177
20.7 Exchange Algorithms for MI and MC Circles 179
20.8 Summary 179
Exercises 179
References 180
Part IV Parameterization 181
Surface Finish Parameters I: Amplitude, Spacing, Hybrid, and Shape 182
21.1 Introduction 182
21.2 Amplitude Parameters 182
21.3 Spacing Parameters 184
21.4 Hybrid and Shape Parameters 187
21.5 Summary 189
Exercises 190
References 191
Surface Finish Parameters II: Autocorrelation, Power Spectral Density, Bearing Area 192
22.1 Autocovariance and Autocorrelation Function 192
22.2 Power Spectral Density 194
22.3 Amplitude Density Function and Bearing Area Curve 196
22.4 Summary 199
Exercises 200
References 200
3D Surface Texture Parameters 201
23.1 Introduction 201
23.2 Amplitude and Shape Parameters 201
23.3 AACV, APSD, and BAC for 3D Surfaces 202
23.4 Spacing and Hybrid Parameters 205
23.5 Summary 206
Exercises 206
References 206
Part V Errors and Uncertainty 207
Uncertainty Considerations 208
24.1 Introduction 208
24.2 Random and Systematic Components 208
24.3 Uncertainty Modeling 209
24.4 Uncertainty Propagation 209
24.5 Systematic Errors: an Example 209
References 210
Uncertainty Propagation in Computations 211
25.1 Introduction 211
25.2 Intervals at Some Level of Confidence 212
25.3 The Central Limit Theorem 213
25.4 Limitations of GUM 215
25.5 The Monte Carlo Method 215
25.6 Filtering Surface Profiles 216
25.7 Fitting Substitute Geometry 219
25.8 Summary 221
Exercises 222
References 222
Error Separation Techniques in Roundness Metrology 223
26.1 Introduction 223
26.2 Full Reversal 223
26.3 Two-Position Method 224
26.4 Two-Probe Method 227
26.5 Three-Probe FFT Method 227
26.6 Three-Probe Sequential Method 229
26.7 Three-Position FFT Method 231
26.8 Error Separation by Solving Linear Equations 232
26.9 Uncertainty Propagation 234
26.10 Summary 234
Exercises 234
References 235
Other Relevant Topics 237
27.1 Introduction 237
27.2 Parameter Extraction 237
27.3 Tools for Correlation and Diagnostics 238
27.4 Challenges 239
References 239
Answers to Selected Exercises 241
Index 258