Machining Dynamics - Frequency Response to Improved Productivity

Machining Dynamics

Machining Dynamics - Frequency Response to Improved Productivity, Tony L. Schmitz, Kevin S. Smith

Preface

Through this book we demonstrate the importance of considering the role of process dynamics in machining performance. We based the book on graduate courses in mechanical vibrations and manufacturing that we have previously offered, but also included aspects of our research programs in machining dynamics and precision engineering. 

We developed the text to be applied in a traditional 15 week course format with an intended audience of upper division and graduate level engineering students, as well as the practicing engineer. We organized the book into seven chapters. The chapter topics are summar- ized here.

• Chapter 1 – We provide a of list potential obstacles to machining productiv-ity and highlight the focus areas for this text. We direct the reader to Fig. 1.1.1 for a graphical identification of these areas.
• Chapter 2 - We first review the fundamentals of single and two degree of freedom free and forced vibrations. We then continue with a description of the frequency response function, including experimental techniques.
• Chapter 3 – The purpose of this chapter is to describe regenerative chatter in turning and introduce the stability lobe diagram. We detail both analytical and time-domain simulations to determine stable and unstable cutting conditions.
• Chapter 4 – In this chapter we focus on milling and describe the correspond- ing analytical and time-domain simulations for stability prediction.
• Chapter 5 – Our goal for this chapter is to investigate the influence of forced vibrations during stable milling on part geometric accuracy. Both analytical and time-domain approaches are provided.
• Chapter 6 - We analyze the frequency content of stable and unstable milling signals; update the time-domain simulations developed in Chapters 4 and 5 to include runout of the cutter teeth and variable teeth spacing; discuss stability of low radial immersion milling; and describe the uncertainty evaluation for stability boundaries.
• Chapter 7 - In this chapter, we apply receptance coupling to prediction of the tool point frequency response function. We also review Euler-Bernoulli beam theory and provide expressions for beam frequency response functions under various boundary conditions.

Books Contents:

1 Introduction ............................................ 1

1.1 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 A Brief Reviewe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Modal Analysis ......................................... 7

2.1 Single Degree of Freedom Free Vibration . . . . . . . . . . . . . . . . 7

• 2.1.1 Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
• 2.1.2 Forced Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
• 2.1.3 Self-Excited Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 9
• 2.1.4 Viscous Damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
• 2.1.5 Coulomb Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
• 2.1.6 Solid Damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Single Degree of Freedom Forced Vibration . . . . . . . . . . . . . . 16
2.3 Two Degree of Freedom Free Vibration . . . . . . . . . . . . . . . . . 23
2.4 Two Degree of Freedom Forced Vibration . . . . . . . . . . . . . . . 34

• 2.4.1 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
• 2.4.2 Complex Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . 39

2.5 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

• 2.5.1 Modal Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
• 2.5.2 Model Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
• 2.5.3 Modal Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.6 Modal Testing Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

• 2.6.1 Force Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
• 2.6.2 Vibration Measurement. . . . . . . . . . . . . . . . . . . . . . . . . 53

2.7 Measurement Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Turning Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1 Turning Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Regenerative Chatter in Turning . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Stability Lobe Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 The Oriented FRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Turning Time-Domain Simulation. . . . . . . . . . . . . . . . . . . . . . 85

• 3.5.1 Chip Thickness Calculation. . . . . . . . . . . . . . . . . . . . . . 85
• 3.5.2 Force Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
• 3.5.3 Displacement Calculation . . . . . . . . . . . . . . . . . . . . . . . 88
• 3.5.4 Multiple Degree of Freedom Modeling. . . . . . . . . . . . . 94

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4 Milling Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.1 Milling Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

• 4.1.1 Tooth Passing Frequency . . . . . . . . . . . . . . . . . . . . . . . 108
• 4.1.2 Multiple Teeth in the Cut . . . . . . . . . . . . . . . . . . . . . . . 110

4.2 Regenerative Chatter in Milling. . . . . . . . . . . . . . . . . . . . . . . . 113
4.3 Stability Lobe Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

• 4.3.1 Average Tooth Angle Approach . . . . . . . . . . . . . . . . . . 117
• 4.3.2 Oriented FRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
• 4.3.3 Fourier Series Approach . . . . . . . . . . . . . . . . . . . . . . . . 126

4.4 Milling Time-Domain Simulation with Straight Teeth . . . . . . 136

• 4.4.1 Chip Thickness Calculation. . . . . . . . . . . . . . . . . . . . . . 136
• 4.4.2 Force Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
• 4.4.3 Displacement Calculation . . . . . . . . . . . . . . . . . . . . . . . 140
• 4.4.4 Simulation Summary and Implementation . . . . . . . . . . 140

4.5 Milling Time-Domain Simulation with Helical Teeth . . . . . . . 147
4.6 Ball Milling Time-Domain Simulation with Helical Teeth . . . 157
4.7 Experimental Cutting Force Coefficients. . . . . . . . . . . . . . . . . 161

• 4.7.1 Updated Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . 161
• 4.7.2 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
• 4.7.3 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . 167

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5 Surface Location Error in Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.1 Surface Location Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.2 Frequency-Domain Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 176

• 5.2.1 Fourier Force Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 176
• 5.2.2 Variation in Surface Location Error with Axial Location. . . . . . . . . . . . . . . . . . . . . . . .185
• 5.2.3 Combining Stability and Surface Location Error in a Single Diagram . . . . . . . . . .  . . 188

5.3 Cycloidal Tool Path Time-Domain Simulation . . . . . . . . . . . . 189
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6 Special Topics in Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.1 Frequency Content of Milling Signals . . . . . . . . . . . . . . . . . . . 199
6.2 Runout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

• 6.2.1 Simulation Modification . . . . . . . . . . . . . . . . . . . . . . . . 217

6.3 Variable Teeth Spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

• 6.3.1 Simulation Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.4 Low Radial Immersion Milling . . . . . . . . . . . . . . . . . . . . . . . . 224
6.5 Uncertainty Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

7 Tool Point Dynamics Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.2 Basic Receptance Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

• 7.2.1 Two Component Rigid Coupling . . . . . . . . . . . . . . . . . 237
• 7.2.2 Two Component Flexible Coupling . . . . . . . . . . . . . . . 241
• 7.2.3 Two Component Flexible, Damped Coupling . . . . . . . 248
• 7.2.4 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
• 7.2.5 Complex Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . 252
• 7.2.6 Receptance Coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7.3 Advanced Receptance Coupling . . . . . . . . . . . . . . . . . . . . . . . 255
7.4 Beam Receptances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
7.5 Assembly Receptance Predictions . . . . . . . . . . . . . . . . . . . . . . 268
7.6 Tool-Holder-Spindle-Machine Receptance Predictions. . . . . . 275

• 7.6.1 Spindle-Machine Receptances. . . . . . . . . . . . . . . . . . . . 279
• 7.6.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

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