# Stability Evaluation for a Damped, Constrained-Motion Cutting Force Dynamometer

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Time-Domain Simulation

- enable the force/deflection amplitudes to be predicted and validated;
- allows for a variety of tool geometries including an arbitrary number of cutting teeth, variable teeth spacing, variable helix angles, and cutter teeth runout;
- discern signal quality between alternative dynamometer systems using the force/deflection frequency content.

- The instantaneous chip thickness is calculated based on the nominal, tooth angle-dependent chip thickness, the current normal vibration, and the vibration of the previous tooth at the same angle.
- The tangential (t) and rotating-normal (n) components of the cutting force is determined by (1) and (2), where b is the axial depth of cut, and h(t), is the instantaneous chip thickness. The cutting (shearing) force coefficients are denoted by k
_{tc}and k_{nc}while the edge (rubbing) coefficients are denoted by k_{te}and k_{ne}:

_{t}= k

_{tc}bh(t) + k

_{te}b

_{n}= k

_{nc}bh(t) + k

_{ne}b

- The force components are used to find the new displacements by numerical solution of the differential equations of motion in the x (feed) and y directions, shown by (3) and (4):

- The tooth angle, φ, is incremented, and the process is repeated. Modal parameters are used to describe the system dynamics in the x (feed) and y directions, where multiple degrees of freedom in each direction can be incorporated; see Figure 2.

#### 2.2. Milling Stability

_{s}is the vector of OPT sampled x displacements and N is the length of the x

_{s}vector. For a stable cut, the absolute value of the difference between subsequent points is zero; as a result, their normalized sum remains zero. For an unstable cut, the difference between subsequent points is non-zero and their normalized sum is greater than zero [45,46]. For this research, the metric, M, was selected to be 1 µm.

#### 2.3. Experimental Setup

## 3. Results

_{c}, and the tooth passing frequency, f

_{t}, and its integer harmonics, N. The cutting test cases for the CMDs are presented in Table 4 and Table 5.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Direction | m (kg) | k (N/m) | c (N-s/m) |
---|---|---|---|

x | 190.873 | 3.38 × 10^{7} | 120,529 |

1457.395 | 2.21 × 10^{9} | 20,952 | |

1170.143 | 2.20 × 10^{9} | 97,233 | |

376.291 | 9.51 × 10^{8} | 46,771 | |

111.083 | 3.36 × 10^{8} | 12,033 | |

99.394 | 4.12 × 10^{8} | 17,079 | |

27.683 | 1.81 × 10^{8} | 12,348 | |

12.988 | 3.38 × 10^{8} | 10,606 | |

40.247 | 1.38 × 10^{9} | 13,900 | |

2.455 | 1.35 × 10^{8} | 907 | |

109.275 | 7.70 × 10^{9} | 10,239 | |

2.061 | 1.54 × 10^{8} | 1975 | |

11.237 | 1.05 × 10^{9} | 6515 | |

6.408 | 6.78 × 10^{8} | 3155 | |

3.028 | 4.32 × 10^{8} | 2320 | |

15.432 | 3.70 × 10^{9} | 8989 | |

1.356 | 3.67 × 10^{8} | 1327 | |

10.041 | 5.72 × 10^{9} | 4165 | |

0.251 | 1.55 × 10^{8} | 403 | |

4.122 | 4.25 × 10^{9} | 2143 | |

0.061 | 6.57 × 10^{7} | 158 | |

0.174 | 2.22 × 10^{8} | 683 | |

1.145 | 1.80 × 10^{9} | 1610 | |

0.026 | 4.62 × 10^{7} | 171 | |

1.532 | 3.70 × 10^{9} | 1801 |

**Figure A1.**Tool tip FRF for the x (feed) direction. The real (

**top**) and imaginary (

**bottom**) parts of the complex valued FRF are presented.

Direction | m (kg) | k (N/m) | c (N-s/m) |
---|---|---|---|

y | 109.319 | 2.18 × 10^{7} | 67,573 |

105.099 | 7.15 × 10^{8} | 32,947 | |

14.532 | 6.25 × 10^{8} | 4626 | |

1.222 | 6.66 × 10^{7} | 1248 | |

4.407 | 2.91 × 10^{8} | 35,341 | |

4.074 | 3.75 × 10^{8} | 15,300 | |

1.204 | 1.91 × 10^{8} | 2594 | |

4.211 | 7.78 × 10^{8} | 3286 | |

1.938 | 5.36 × 10^{8} | 1840 | |

0.284 | 1.75 × 10^{8} | 449 | |

0.065 | 7.04 × 10^{7} | 157 | |

0.461 | 6.17 × 10^{8} | 1331 | |

0.030 | 5.20 × 10^{7} | 193 | |

0.153 | 3.75 × 10^{8} | 10,985 | |

0.783 | 2.48 × 10^{9} | 42,620 | |

2.520 | 8.83 × 10^{9} | 51,484 |

**Figure A2.**Tool tip FRF for the y direction. The real (

**top**) and imaginary (

**bottom**) parts of the complex valued FRF are presented.

**Table A3.**CMD (no damping) modal parameters for the stability measurement setup in the x (feed) direction.

Direction | m (kg) | k (N/m) | c (N-s/m) |
---|---|---|---|

x | 0.689 | 2.08 × 10^{7} | 43 |

**Figure A3.**CMD (no damping) FRF for the x (feed) direction. The real (

**top**) and imaginary (

**bottom**) parts of the complex valued FRF are presented.

Direction | m (kg) | k (N/m) | c (N-s/m) |
---|---|---|---|

y | 328.120 | 1.03 × 10^{8} | 36,110 |

170.733 | 4.99 × 10^{8} | 30,917 | |

90.087 | 1.78 × 10^{9} | 16,172 | |

126.798 | 5.91 × 10^{9} | 83,599 | |

76.151 | 4.45 × 10^{9} | 44,629 | |

38.573 | 4.01 × 10^{9} | 48,575 | |

11.282 | 2.18 × 10^{9} | 9849 | |

4.596 | 1.11 × 10^{9} | 6002 | |

4.868 | 1.27 × 10^{9} | 3633 | |

64.251 | 1.83 × 10^{10} | 12,751 | |

45.307 | 1.44 × 10^{10} | 9664 | |

66.769 | 2.17 × 10^{10} | 12,612 | |

1.078 | 3.77 × 10^{8} | 1372 | |

27.137 | 9.96 × 10^{9} | 11,231 | |

3.288 | 1.40 × 10^{9} | 3628 | |

9.618 | 4.40 × 10^{9} | 5130 | |

2.596 | 1.26 × 10^{9} | 2302 | |

7.566 | 4.14 × 10^{9} | 5516 | |

1.228 | 7.35 × 10^{8} | 1519 | |

2.538 | 1.58 × 10^{9} | 1206 | |

17.166 | 1.22 × 10^{10} | 4618 | |

9.875 | 7.96 × 10^{9} | 10,479 |

**Figure A4.**CMD (no damping) FRF for the y direction. The real (

**top**) and imaginary (

**bottom**) parts of the complex valued FRF are presented.

**Table A5.**CMD (with damping) modal parameters for the stability measurement setup in the x (feed) direction.

Direction | m (kg) | k (N/m) | c (N-s/m) |
---|---|---|---|

x | 0.701 | 2.07 × 10^{7} | 99 |

**Figure A5.**CMD (with damping) FRF for the x (feed) direction. The real (

**top**) and imaginary (

**bottom**) parts of the complex valued FRF are presented.

**Table A6.**CMD (with damping) modal parameters for the stability measurement setup in the y direction.

Direction | m (kg) | k (N/m) | c (N-s/m) |
---|---|---|---|

y | 4334.195 | 1.23 × 10^{10} | 204,353 |

182.268 | 5.93 × 10^{8} | 16,828 | |

106.509 | 2.14 × 10^{9} | 24,751 | |

42.190 | 1.95 × 10^{9} | 60,633 | |

152.618 | 8.50 × 10^{9} | 40,556 | |

182.646 | 1.44 × 10^{10} | 53,112 | |

14.408 | 1.35 × 10^{9} | 15,197 | |

36.573 | 4.42 × 10^{9} | 16,729 | |

12.639 | 1.97 × 10^{9} | 21,354 | |

1.555 | 3.17 × 10^{8} | 2508 | |

43.098 | 9.35 × 10^{9} | 7871 | |

453.258 | 1.00 × 10^{11} | 1,077,191 | |

93.694 | 2.23 × 10^{10} | 30,614 | |

25.323 | 6.35 × 10^{9} | 15,557 | |

2.805 | 9.32 × 10^{8} | 2065 | |

27.777 | 1.14 × 10^{10} | 22,029 | |

2.990 | 1.44 × 10^{9} | 4892 | |

1.072 | 5.83 × 10^{8} | 1030 | |

9.755 | 5.40 × 10^{9} | 3260 | |

177.937 | 1.00 × 10^{11} | 144,264 | |

1.078 | 6.18 × 10^{8} | 449 | |

9.715 | 8.46 × 10^{9} | 2695 | |

4.884 | 4.71 × 10^{9} | 6702 |

**Figure A6.**CMD (with damping) FRF for the y direction. The real (

**top**) and imaginary (

**bottom**) parts of the complex valued FRF are presented.

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**Figure 1.**Constrained-motion dynamometer design and optical interrupter placement for displacement measurement.

**Figure 2.**Chip regeneration in milling. The feed direction is identified, and the tool rotation is assumed to be clockwise with respect to the fixed x-y coordinate system.

**Figure 6.**Time-domain stability limit (TDS), solid black line with teal infill, expressed as a function of spindle speed, Ω, for the CMD (no damping).

**Figure 7.**Time-domain stability limit (TDS), solid black line with teal infill, expressed as a function of spindle speed, Ω, for the CMD (with damping).

**Figure 8.**Al-6061-T6 CMD (no damping) vibration behavior for stable cutting at {4900 rpm, 1 mm}. Predicted displacement (

**a**), velocity (

**b**), and once-per-tooth (OPT) Poincaré map (

**c**) and measured displacement (

**d**), velocity (

**e**), and OPT Poincaré map (

**f**).

**Figure 9.**Al 6061-T6 CMD (no damping) Poincaré maps for stable cutting at {4900 rpm, 2 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 10.**Al 6061-T6 CMD (no damping) Poincaré maps for stable cutting at {4900 rpm, 3 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 11.**Al 6061-T6 CMD (no damping) Poincaré maps for stable cutting at {4900 rpm, 4 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 12.**Al 6061-T6 CMD (no damping) Poincaré maps for regenerative chatter (secondary Hopf bifurcation) at {4900 rpm, 5 mm}. Predicted (

**a**) and measured (

**b**). Note the difference in scale from Figure 11.

**Figure 13.**Al 6061-T6 CMD (no damping) Poincaré maps for regenerative chatter (secondary Hopf bifurcation) at {4900 rpm, 6 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 14.**Al-6061-T6 CMD (with damping) vibration behavior for stable cutting at {4900 rpm, 1 mm}. Predicted displacement (

**a**), velocity (

**b**), and OPT Poincaré map (

**c**) and measured displacement (

**d**), velocity (

**e**), and OPT Poincaré map (

**f**).

**Figure 15.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 2 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 16.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 3 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 17.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 4 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 18.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 5 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 19.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 6 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 20.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 7 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 21.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 8 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 22.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 9 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 23.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 10 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 24.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 11 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 25.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 12 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 26.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 13 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 27.**Al 6061-T6 CMD (with damping) Poincaré maps for stable cutting at {4900 rpm, 14 mm}. Predicted (

**a**) and measured (

**b**).

**Figure 28.**Milling stability validation for the CMD (no damping) (

**a**) and the CMD (with damping) (

**b**).

**Figure 29.**Milling stability lobe comparison for the CMD (no damping) (

**a**) and the CMD (with damping) (

**b**).

Principle | Output Response | Output Quantity |
---|---|---|

Piezoelectric | Change piezoelectric material deformation generating charge | Electric charge, C |

Capacitive | Change in capacitance | Electric capacitance, F |

Inductive | Change in electromagnetic induction | Electrical potential, V |

Piezoresistive | Change in resistance (semiconductor strain gauge) | Electrical resistance, Ω |

Resistive | Change in resistance (wire/metal film strain gauge) | Electrical resistance, Ω |

Drive current | Change in current consumption by the driving motors of the machine tool | Electrical current, A |

Compliance | Change in mechanical elastic deformation | Mechanical displacement, m |

Diameter (mm) | Teeth | Insert Material | |
---|---|---|---|

15.88 | 1 | Coated carbide (Kennametal EC1402FLDJ) | |

Cutting parameters | |||

Spindle speed (rpm) | 4900 | ||

Feed per tooth(mm) | 0.1 | ||

Axial depth (mm) | Various | ||

Radial depth (mm) | 3 (19% radial immersion) | ||

Cutting direction | Conventional (up) milling | ||

Cutting force coefficients | |||

k_{tc} (N/mm^{2}) | 1250 | ||

k_{rc} (N/mm^{2}) | 400 | ||

k_{te} (N/mm) | 5 | ||

k_{re} (N/mm) | 7 |

**Table 3.**x-direction modal parameters for the single degree of freedom (SDOF) constrained-motion dynamometers (CMDs).

Modal Parameters | Al 6061-T6 CMD (No Damping) | Al 6061-T6 CMD (with Damping) |
---|---|---|

Direction | x | x |

m (kg) | 0.689 | 0.701 |

k (N/m) | 2.08 × 10^{7} | 2.07 × 10^{7} |

c (N-s/m) | 43 | 99 |

Spindle Speed (rpm) | Radial Depth (mm) | Axial Depth (mm) | Metric, M (μm) |
---|---|---|---|

4900 | 3 | 1 | 0.32 |

2 | 0.57 | ||

3 | 0.51 | ||

4 | 1.16 | ||

5 | 58.80 | ||

6 | 49.35 |

Spindle Speed (rpm) | Radial Depth (mm) | Axial Depth (mm) | Metric, M (μm) |
---|---|---|---|

4900 | 3 | 1 | 0.10 |

2 | 0.20 | ||

3 | 0.18 | ||

4 | 0.24 | ||

5 | 0.24 | ||

6 | 0.32 | ||

7 | 0.33 | ||

8 | 0.30 | ||

9 | 0.88 | ||

10 | 0.31 | ||

11 | 0.73 | ||

12 | 0.36 | ||

13 | 0.34 | ||

14 | 0.83 |

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## Share and Cite

**MDPI and ACS Style**

Gomez, M.; Schmitz, T.
Stability Evaluation for a Damped, Constrained-Motion Cutting Force Dynamometer. *J. Manuf. Mater. Process.* **2022**, *6*, 23.
https://doi.org/10.3390/jmmp6010023

**AMA Style**

Gomez M, Schmitz T.
Stability Evaluation for a Damped, Constrained-Motion Cutting Force Dynamometer. *Journal of Manufacturing and Materials Processing*. 2022; 6(1):23.
https://doi.org/10.3390/jmmp6010023

**Chicago/Turabian Style**

Gomez, Michael, and Tony Schmitz.
2022. "Stability Evaluation for a Damped, Constrained-Motion Cutting Force Dynamometer" *Journal of Manufacturing and Materials Processing* 6, no. 1: 23.
https://doi.org/10.3390/jmmp6010023