Author Contributions
Conceptualization, X.L.; methodology, X.L. and D.L.; investigation, X.L., D.L., and C.D.; validation, D.L. and X.L.; formal analysis, Y.L. and Z.F.; project administration, X.L. and C.W.; software, C.W.; supervision, X.L. and Z.F.; visualization, D.L. and Y.L.; writing—original draft, C.D. and Y.L.; writing—review and editing, X.L. and D.L.; funding acquisition, X.L. and C.W. All authors have read and agreed to the published version of the manuscript.
Figure 1.
A spindle–holder–tool system.
Figure 1.
A spindle–holder–tool system.
Figure 2.
Timoshenko element.
Figure 2.
Timoshenko element.
Figure 3.
The original shape and deformation of a beam in the X-Z plane.
Figure 3.
The original shape and deformation of a beam in the X-Z plane.
Figure 4.
Rotating beam with centrifugal force.
Figure 4.
Rotating beam with centrifugal force.
Figure 5.
Effect of gyroscopic moment.
Figure 5.
Effect of gyroscopic moment.
Figure 6.
Diagrams of tool runout and cutting forces.
Figure 6.
Diagrams of tool runout and cutting forces.
Figure 7.
Experimental setup for testing tool-tip FRFs.
Figure 7.
Experimental setup for testing tool-tip FRFs.
Figure 8.
Original experimental FRFs in the X-direction.
Figure 8.
Original experimental FRFs in the X-direction.
Figure 9.
Original experimental FRFs in the Y-direction.
Figure 9.
Original experimental FRFs in the Y-direction.
Figure 10.
Comparisons of theoretical and experimental FRFs in the X-direction.
Figure 10.
Comparisons of theoretical and experimental FRFs in the X-direction.
Figure 11.
Comparisons of theoretical and experimental FRFs in the Y-direction.
Figure 11.
Comparisons of theoretical and experimental FRFs in the Y-direction.
Figure 12.
Flowchart of the simulation experiment.
Figure 12.
Flowchart of the simulation experiment.
Figure 13.
Frequency responses in X-direction for condition 1.
Figure 13.
Frequency responses in X-direction for condition 1.
Figure 14.
Frequency responses in Y-direction for condition 1.
Figure 14.
Frequency responses in Y-direction for condition 1.
Figure 15.
Frequency responses in X-direction for condition 2.
Figure 15.
Frequency responses in X-direction for condition 2.
Figure 16.
Frequency responses in Y-direction for condition 2.
Figure 16.
Frequency responses in Y-direction for condition 2.
Figure 17.
Stability lobe diagrams for condition 1.
Figure 17.
Stability lobe diagrams for condition 1.
Figure 18.
Stable and unstable time responses for condition 1.
Figure 18.
Stable and unstable time responses for condition 1.
Figure 19.
Stability lobe diagrams for condition 2.
Figure 19.
Stability lobe diagrams for condition 2.
Figure 20.
Stable and unstable time responses for condition 2.
Figure 20.
Stable and unstable time responses for condition 2.
Figure 21.
Time responses for condition 1.
Figure 21.
Time responses for condition 1.
Figure 22.
Time responses with tool runout for condition 1.
Figure 22.
Time responses with tool runout for condition 1.
Figure 23.
Time responses without tool runout for condition 1.
Figure 23.
Time responses without tool runout for condition 1.
Figure 24.
Time responses in X-direction for condition 2.
Figure 24.
Time responses in X-direction for condition 2.
Figure 25.
Time responses in Y-direction for condition 2.
Figure 25.
Time responses in Y-direction for condition 2.
Figure 26.
Time responses in X-direction with tool runout for condition 2.
Figure 26.
Time responses in X-direction with tool runout for condition 2.
Figure 27.
Time responses in Y-direction with tool runout for condition 2.
Figure 27.
Time responses in Y-direction with tool runout for condition 2.
Figure 28.
Time responses in X-direction without tool runout for condition 2.
Figure 28.
Time responses in X-direction without tool runout for condition 2.
Figure 29.
Time responses in Y-direction without tool runout for condition 2.
Figure 29.
Time responses in Y-direction without tool runout for condition 2.
Figure 30.
Time responses for cases A and B with tool runout.
Figure 30.
Time responses for cases A and B with tool runout.
Figure 31.
Time responses for cases A and C with tool runout.
Figure 31.
Time responses for cases A and C with tool runout.
Figure 32.
Time responses for cases A and B with time-varying cutting force coefficients.
Figure 32.
Time responses for cases A and B with time-varying cutting force coefficients.
Figure 33.
Time responses for cases C and D with time-varying cutting force coefficients.
Figure 33.
Time responses for cases C and D with time-varying cutting force coefficients.
Figure 34.
The stable and unstable regions predicted by SLD.
Figure 34.
The stable and unstable regions predicted by SLD.
Figure 35.
Experimental setup for micro-milling.
Figure 35.
Experimental setup for micro-milling.
Figure 36.
Machined surfaces of workpiece. (a) Machined surface for cutting condition A ( rpm, mm). (b) Machined surface for cutting condition B ( rpm, mm).
Figure 36.
Machined surfaces of workpiece. (a) Machined surface for cutting condition A ( rpm, mm). (b) Machined surface for cutting condition B ( rpm, mm).
Table 1.
Material properties and geometric parameters of spindle–holder–tool system for condition 1.
Table 1.
Material properties and geometric parameters of spindle–holder–tool system for condition 1.
Parameters | Spindle | Holder | Tool | Cutter |
---|
Elastic modulus E (Pa) | | | | |
Poisson ratio | 0.3 | 0.3 | 0.23 | 0.23 |
Density | 7900 | 7850 | 14,000 | 14,000 |
Length | 0.198 | 0.031 | 0.0136 | 0.003 |
Radius | 0.0135 | 0.0105 | 0.002 | 0.0002 |
Table 2.
Material properties and geometric parameters of spindle–holder–tool system for condition 2.
Table 2.
Material properties and geometric parameters of spindle–holder–tool system for condition 2.
Parameters | Spindle | Holder | Tool | Cutter |
---|
Elastic modulus E (Pa) | | | | |
Poisson ratio | 0.3 | 0.3 | 0.3 | 0.25 |
Density | 7850 | 7850 | 7850 | 8100 |
Length | 0.219 | 0.031 | 0.0014 | 0.002 |
Radius | 0.0125 | 0.0105 | 0.0015 | 0.00015 |
Table 3.
Cutting force coefficients for condition 1.
Table 3.
Cutting force coefficients for condition 1.
Number of Teeth | Ktc | Krc | Kte | Kre |
---|
2 | | | | |
Table 4.
Cutting force coefficients for condition 2.
Table 4.
Cutting force coefficients for condition 2.
Number of Teeth | Ktc | Krc | Kte | Kre |
---|
2 | | | | |
Table 5.
Comparisons of peak values in X-direction for condition 1.
Table 5.
Comparisons of peak values in X-direction for condition 1.
Rotation Speed (rpm) | Amplitude () | Frequency (Hz) |
---|
Theoretical | Simulation | Theoretical | Simulation |
---|
0 | 1.530 | 1.654 | 9971 | 9880 |
10,000 | 0.997 | 1.042 | 9821 | 9520 |
20,000 | 1.375 | 1.488 | 9571 | 9400 |
30,000 | 0.575 | 0.654 | 9221 | 9120 |
Table 6.
Comparisons of peak values in Y-direction for condition 1.
Table 6.
Comparisons of peak values in Y-direction for condition 1.
Rotation Speed (rpm) | Amplitude () | Frequency (Hz) |
---|
Theoretical | Simulation | Theoretical | Simulation |
---|
0 | 1.530 | 1.342 | 9971 | 9960 |
10,000 | 0.997 | 1.020 | 9821 | 9560 |
20,000 | 1.375 | 1.371 | 9571 | 9480 |
30,000 | 0.575 | 0.572 | 9221 | 9120 |
Table 7.
Comparisons of peak values in X-direction for condition 2.
Table 7.
Comparisons of peak values in X-direction for condition 2.
Rotation Speed (rpm) | Amplitude () | Frequency (Hz) |
---|
Theoretical | Simulation | Theoretical | Simulation |
---|
0 | 6.362 | 6.640 | 12,301 | 12,200 |
10,000 | 4.805 | 4.628 | 12,300 | 11,840 |
20,000 | 3.864 | 3.900 | 12,211 | 11,760 |
30,000 | 1.150 | 1.732 | 11,780 | 11,745 |
40,000 | 3.925 | 4.184 | 11,760 | 11,680 |
50,000 | 1.011 | 1.209 | 11,390 | 11,520 |
Table 8.
Comparisons of peak values in Y-direction for condition 2.
Table 8.
Comparisons of peak values in Y-direction for condition 2.
Rotation Speed (rpm) | Amplitude () | Frequency (Hz) |
---|
Theoretical | Simulation | Theoretical | Simulation |
---|
0 | 6.362 | 6.459 | 12,301 | 12,200 |
10,000 | 4.805 | 4.896 | 12,300 | 11,840 |
20,000 | 3.864 | 3.900 | 12,211 | 11,800 |
30,000 | 1.150 | 1.254 | 11,791 | 11,760 |
40,000 | 3.925 | 3.639 | 11,760 | 11,680 |
50,000 | 1.011 | 1.387 | 11,390 | 11,520 |