# Development and Testing of a High-Frequency Dynamometer for High-Speed Milling Process

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Preliminary Description of the SGBD

#### 2.2. Setup of the Data Acquisition System

_{x}, V

_{y}, V

_{z}) into force signals (F

_{x}, F

_{y}, F

_{z}).

#### 2.3. Basic Mythology for the Test

_{x}, F

_{y,}and F

_{z}under static loads conditions to establish the connection between the input and output data. All the loads were applied at the same point (center of each plane of the workpiece, e.g., plane XY in top view, plane YZ in the side view, and plane XZ in front view). Then, the load with a 40 N interval was applied (up to 160 N) in this calibration, and the corresponding strain values were recorded by the measurement system for each load interval.

## 3. Design and Construction of the SGBD

#### 3.1. Specifications and Material of the Main Structure

- Natural frequency (without workpiece) higher than 9 kHz;
- Natural frequency (with the reference workpiece, oversize less than 48 mm × 20 mm × 10 mm) higher than 8 kHz;
- Suitable for measuring milling forces, especially at high or super high rotational speed, the highest rotational speed allowed is about 40,000 r/min or more;
- Three cutting force components F
_{x}, F_{y}, F_{z,}should be available, and the measuring range of transversal components F_{x}, F_{y}, F_{z}≈±1 kN; - The sensitivity of the thin elastic part should be more than 1.5 µε/N;
- Cross-sensitivity should be lower than about 10% for all sensing routes;
- Relatively simple to do and of relatively low cost.

_{i}b

_{i,}and l

_{i}(i = 1, 2, 3, 4, 5) are used to represent the width, length, and height of each part (as shown in Figure 4).

#### 3.2. Determination of Dimensions of the Main Structure

_{ii}|

_{i = 1, 2, …, n}is the natural frequency of the system in which the quality of the ith part m

_{i}is only considered.

_{k}, ω

_{1}, and ω

_{4}, where ω

_{k}is the angular frequency of the system in which the m

_{1}and m

_{4}are not considered, ω

_{1}is the angular frequency of the system in which the m

_{1}is only considered, and ω

_{4}is the angular frequency of the system in which the m

_{4}is only considered. Thus, the angular frequency of the simplified model of the SGBD can be written as Equation (2):

_{1}, K

_{1}can be calculated as follows:

_{1}was calculated as Equation (4):

_{1}and m

_{4}. As for a vibration system, if its movement equation is expressed as Equation (5):

_{max}is:

_{max}is:

_{max}= V

_{max}, and then:

_{eq}and k

_{eq}are the equivalent mass and equivalent stiffness of the vibrator, respectively, ω

_{o}is the angular frequency of the vibrator, and A is the amplitude.

_{1}and m

_{4}, if we define the displacement of point P

_{1}: x

_{1}= l, the displacements of points P

_{2}, then P

_{2}and P

_{4}are l + (k

_{5}/k

_{3})/l and 2l + (k

_{5}/k

_{3})/l respectively. Assume the displacement of each cross-section of spring is proportional to the distance between the fixed end and the cross-section, which is the same as that in the condition of static deformation. When the instantaneous velocity of a mass m at any time is defined to $\dot{x}$, the corresponding velocity of the micro section dy on the spring located at the position y will be $y\dot{x}/l$. Furthermore, when ρ is defined as the mass of per unit of the spring (linear density), and ρdy is the quality of the micro section dy of the spring, then the kinetic energy of the dy can be written as $dT=\rho dy{\left(y\dot{x}/l\right)}^{2}/2$. Therefore, the kinetic energy of the spring is $T={{\displaystyle \int}}_{0}^{1}\frac{1}{2}\rho dy{\left(y\dot{x}/l\right)}^{2}=\rho l\dot{x}/6$. Assume the quality of the thick elastic part which is close to the fixed end m

_{k51}= ρl, then the maximum kinetic energy of the element will be ${T}_{k51max}=\rho {\dot{x}}_{max}^{2}/6$. Similarly, the maximum kinetic energy of the thin elastic part, the thick elastic part which is far to the fixed end, and the quasi-rigid part can be calculated as T

_{k3max}, T

_{k52max,}and T

_{k2max}.

- The oversize of the main structure is limited to 150 mm × 60 mm × 60 mm to gain a high natural frequency (without workpiece), which is higher than 9 kHz.
- The thickness of the thin elastic part (b
_{3}) should not less than 0.7 mm to measure the range of transversal components F_{x}, F_{y}, F_{z}≈±1 kN. - The distance between the quasi-rigid part and the force-sensing elastic elements should be within 0.3 mm ~0.8 mm, and the cross-section area of the thick elastic part should be more than 10 times than that of the thin elastic part, which can maintain the sensitivity of the thin elastic part more than 1.5 µε/N.

## 4. Dynamometer Calibration

#### 4.1. Static Calibration

_{x}, F

_{y,}and F

_{z}. The output voltages of microvolt were averaged for each direction. To decrease the dependency on the location of the acting point, all the loads were applied at the same point in the static calibration. Additionally, because of the small values of the cutting parameters used in the following milling experiments, the maximum cutting force is lower than 100 N. When the load was up to 160 N, a 40 N interval was applied in this calibration, and the corresponding strain values were recorded for each load interval. Thus, the final calibration curves were obtained by plotting the curve of load values and output readings. Figure 5 shows the calibration curves for F

_{x}, F

_{y,}and F

_{z,}respectively. The effect of loading towards one direction on the other force elements was also analyzed, and minor fluctuations were detected. It can be found from Figure 5 that the cross-sensitivity is smaller than about 10% for all sensing directions.

_{x}, V

_{y}, V

_{z}) and the corresponding forces (F

_{x}, F

_{y}, F

_{z}) for the dynamometer can be calculated, and a calibration matrix [18] can be derived. Using the calibration matrix, the three-dimensional milling forces can be computed automatically by the developed data processing software according to the output voltages. However, it was found in the process of static calibration that the location of the acting point has a direct influence on the output voltages, which results in a large deviation of measuring results for a constant input loading. Therefore, the conversion matrix for the input and output data should consider the influence of dependency on the location of an acting point. For this purpose, a new conversion matrix model combined with moment effect was developed, which will be in the following Section 4.2.

#### 4.2. Conversion Matrix Model

_{x}, V

_{y}, and V

_{z}are the out voltages in X, Y, and Z directions, respectively, F

_{x}, F

_{y}, and F

_{z}are the input loads (i.e., cutting forces) in X, Y, and Z directions, respectively, a

_{ij}|

_{i, j = x, y, z}is the calibration coefficient between F

_{i}and V

_{j}, b

_{ij}|

_{i, j = x, y, z}is the calibration coefficient between M

_{i}and V

_{j}.

#### 4.3. Dynamic Identification

## 5. Milling Test of the SGBD

#### 5.1. Experimental Setting

#### 5.2. Results and Discussion

_{x}and F

_{y}) obtained by SGBD and Kistler except for the cutting force signals in the Z direction (F

_{z}), although the cutting force of one flute is smaller than that of another due to the imbalance of the end mill. It seems that the SGBD is better than Kistler in the measurement of cutting force F

_{z}. The maximum cutting forces (F

_{xmax}, F

_{ymax}, F

_{zmax}) obtained by SGBD and Kistler are as shown in Figure 10. Where the value of each column shown in the figure indicates the mean value of the maximum cutting force in stable cutting conditions, it was found that there is also a fair agreement for the maximum cutting forces in X and Y directions (F

_{xmax}, F

_{ymax}). The difference values between SGBD and Kistler are 2.3%~5.8% in the X direction and 3.5%~8.8% in the Y direction. However, for the maximum cutting forces in the Z direction (F

_{zmax}), the difference values are 17.2%~30.2%. It is the big fluctuation of cutting force in the Z-direction measured by Kistler (as shown in Figure 9c), which results in the bigger difference of F

_{zmax}between SGBD and Kistler. It is mainly because that the natural frequency of the Kistler dynamometer in the Z direction is about 1.6 kHz, which is far less than that of the SGBD (about 9.15 kHz). As to the measurement of high-frequency dynamic force, the frequency content (the higher harmonics) of the milling forces that exceed the bandwidth of the dynamometer may become distorted by its dynamics. When the cutting force (in the Z direction) is small, the Kistler dynamometer is very difficult to acquire the actual force signals because of the signal distortion combined with noise interference. That is why it is difficult to observe clear periodic cutting force signals in the Z direction by the Kistler dynamometer in this test.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The strain gauges in dynamometer design involving (

**a**). Distribution of strain gauges (

**b**). Bridge connections of strain gauges for Fx, Fy, and Fz.

**Figure 4.**The strain gauge-based dynamometer design involves (

**a**). Decomposition of the main structure (

**b**). Physical model of the main structure.

**Figure 5.**The Three-dimensional forces measurement through dynamometer in (

**a**). F

_{x}, (

**b**). F

_{y,}and (

**c**). F

_{z}directions.

**Figure 7.**Frequency response function (FRF) and phase response function of the main structure in (

**a**) X- direction (

**b**). Y-direction (

**c**). Z-direction.

**Figure 9.**Cutting force signals obtained by strain-gauge-based dynamometer (SGBD) and Kistler (10,000 r/min), (

**a**) cutting force signals in X direction Fx (

**b**) cutting force signals in Y direction Fy (

**c**) cutting force signals in Z direction Fz.

Properties | Values |
---|---|

Density | 2.78 kg/m^{3} |

Poisson ratio | 0.33 |

Modulus of elasticity | 73 GPA |

Tensile strength | 410 MPa |

Yield strength | 265 MPa |

Hardness | 115 HB |

**Table 2.**The dimension of each part and its relationship with the natural frequency of the main structure.

Dimension of Each Part of the Main Structure | Dimension (mm) | Relation to the Natural Frequency |
---|---|---|

Length of the quasi-rigid part l_{2} | 30 | Inversely proportional |

Length of the thin elastic part l_{3} | 8 | Inversely proportional |

Length of the thick elastic part l_{5} | 11 | Inversely proportional |

Cross-section area of the quasi-rigid part A_{2}(a_{2} × b_{2}) | 30 × 30 | Proportional |

Cross-section area of the thin elastic part A_{3}(a_{3} × b_{3}) | 8 × 1 | Proportional |

Cross-section area of the thick elastic part A_{5}(a_{5} × b_{5}) | 30 × 5 | Inversely proportional |

Volume of the upper support plate V_{1}(a_{1} × b_{1} × l_{1}) | 48 × 48 × 4 | Inversely proportional |

Volume of the lower support plate V_{4}(a_{4} × b_{4} × l_{4}) | 48 × 100 × 8 | Inversely proportional |

Cutting Parameters | Machine Tool | Cutting Tool | Workpiece | Dynamometers | |||
---|---|---|---|---|---|---|---|

n (r/min) | f_{z}(mm/z) | a_{e}(mm) | a_{p}(mm) | ||||

10,000 | 0.1 | 0.5 | 0.5 | Mikron UPC 710 machining center, maximum rotational speed of 18,000 rpm, maximum feed of 20 m/min [27]. | Solid carbide end mills, diameter: 10 mm, 2 flutes, nose radius: 0.5 mm, helix angle: 30°. | Material: 2A12 aluminum alloy, overall size: 20 × 20 × 4 mm | SGBD / Kistler 9265B |

12,000 | 0.1 | 0.5 | 0.5 | ||||

15,000 | 0.1 | 0.5 | 0.5 | ||||

18,000 | 0.1 | 0.5 | 0.5 |

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**MDPI and ACS Style**

Lyu, Y.; Jamil, M.; He, N.; Gupta, M.K.; Pimenov, D.Y.
Development and Testing of a High-Frequency Dynamometer for High-Speed Milling Process. *Machines* **2021**, *9*, 11.
https://doi.org/10.3390/machines9010011

**AMA Style**

Lyu Y, Jamil M, He N, Gupta MK, Pimenov DY.
Development and Testing of a High-Frequency Dynamometer for High-Speed Milling Process. *Machines*. 2021; 9(1):11.
https://doi.org/10.3390/machines9010011

**Chicago/Turabian Style**

Lyu, Yanlin, Muhammad Jamil, Ning He, Munish Kumar Gupta, and Danil Yurievich Pimenov.
2021. "Development and Testing of a High-Frequency Dynamometer for High-Speed Milling Process" *Machines* 9, no. 1: 11.
https://doi.org/10.3390/machines9010011