# Tool Orientation Optimization for Disk Milling Process Based on Torque Balance Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Torque Balance Method

#### 2.1. Introduction of High-Efficiency and Powerful Compound Milling Process

#### 2.2. Torque Balance Algrithm Derivation Based on Minimum Residual Amount

_{S}) formatted as a rectangular coordinate.

_{S}) formatted as polar coordinates; r is the distance from the pole to the edge of the integral region (D). At the same time, the integral region D also changes correspondingly.

#### 2.3. Concentric Circle Ray Point (CCRP) Method

_{1}′ on the plane has been generated, the torque M

_{1}can be easily calculated according to Equation (7), ${M}_{1}=k\left|{P}_{1}{{P}^{\prime}}_{1}\right|\left|{P}_{1}{{P}^{\prime}}_{0}\right|$.

_{1}.

**Way 1:**points on the line of P

_{0}P

_{2}′.

_{1}on line P

_{0}P

_{2}′.

**Way 2:**points on the plane.

_{0}P

_{2}′ to match M

_{1}, if n points can be found on the plane to satisfy the formula $\sum _{i=2}^{n}k\left|{P}_{n1}{}^{\prime}{P}_{n1}\right|\left|{P}_{0}{}^{\prime}\right|{P}_{n1}{}^{\prime}\mathrm{cos}{\alpha}_{i}}=0$. Since the number of n is not unique, it indicates that there may be more than one such newly synthesized torque matching M

_{1}. Obviously, the newly generated torque may not be the same as the torque generated by the point on the line.

**Step****1:**- Extend the free surface. The method to extend the surface is shown in Figure 5a.
**Step****2:**- Create a plane that does not intersect the existing free surface. As shown in Figure 5b, find a point on the free surface closest to the plane and defined as P
_{0}. **Step****3:**- Project the point P
_{0}on to the plane, defined as P_{0}′. As shown on the plane in Figure 5c, R, 2R, …, mR are the radii of the concentric circles centered on the P_{0}′, and n rays are created with P_{0}′ as the start point and 2π/n radian adjacent to the two rays. After that, m concentric circles generate m × n intersection points with n rays, defined as ${{P}^{\prime}}_{m,n}$. **Step****4:**- Project the points ${{P}^{\prime}}_{m,n}$ on the extended free surface, which are defined as ${P}_{m,n}$. The projection method is shown in Figure 5d.
**Step****5:**- Get the distance ${D}_{m,n}$ of ${P}_{m,n}$ and ${{P}^{\prime}}_{m,n}$, then multiply the stiffness factor k before ${D}_{m,n}$, and change the distance variable ${D}_{m,n}$ into the mechanical variable ${F}_{m,n}$.
**Step****6:**- Get the distance ${L}_{mn,0}$ of ${{P}^{\prime}}_{m,n}$ and ${{P}^{\prime}}_{0}$, then multiply the mechanical variable ${F}_{m,n}$ and the distance variable ${L}_{mn,0}$, calculate the torque value of M
_{m}_{,n}. **Step****7:**- Find the angle value α
_{n}between the nth ray and the x axis, then get the torque value of each point which was generated by the concentric circles and rays, defined as ${\stackrel{\rightharpoonup}{M}}_{r,\alpha}$.

#### 2.4. The Calculation of the Optimal Tool Orientation of the Blisk Milling Cutter Based on the Torque Balance Method

**Step 1:**Decompose the torque vector ${\stackrel{\rightharpoonup}{M}}_{r,\alpha}$. According to Equation (9), there are innumerable points on the plane and free surface. Since a point corresponds to a torque vector, that means there are an infinite number of torque vectors. Based on the uncertainty of the torque vector value and direction, it is difficult to achieve balance through the calculation method. Therefore, it is necessary to decompose an infinite number of torque vectors in rectangular coordinates for unity. So, all the vectors ${\stackrel{\rightharpoonup}{M}}_{r,\alpha}$ included can be decomposed into a vector along the x direction and y direction by using the formula as follows:

**Step 2:**Accumulate torque values in the x and y directions. Through the calculation method of Equation (9), all the vectors after decomposed are concentrated in the x and y directions. Meanwhile, the vectors, after being decomposed, can be added and subtracted in either x direction or y direction. The torque values in x direction and y direction should be accumulated respectively to facilitate the torque balancing algorithm by using the following formula

**Step 3:**Calculate the total torque and the radian. According to Equations (9) and (10), the un-neutralized part of the torque of the free surface and plane should be calculated after the decomposition and accumulation effect, so as to calculate the rotation radian of the plane and realize the torque balance method between the plane and the free surface. Since all the initial torque vectors are decomposed into the x direction and y direction, so that all the accumulated torque vectors are located in the x direction and y direction, the total torque and angle value are calculated as shown in Equation (11).

**Step 4:**Calculate the rotation axis of the final plane. According to Equation (11), the total torque value and the direction in which the total torque is located have been calculated. It follows that the sum of the components of all torque values in the perpendicular direction of the total torque vector is zero. As shown in Figure 6a, we define the direction perpendicular to the total torque vector as ${\stackrel{\rightharpoonup}{M}}_{\perp}$. Therefore, the line that vector ${\stackrel{\rightharpoonup}{M}}_{\perp}$ sits on is the axis of rotation of the final plane.

**Step 5:**Calculate the rotation direction. According to the definition of symmetry, if the plane is rotated along the vector ${\stackrel{\rightharpoonup}{M}}_{\perp}$ by a certain radian γ, the components of the torque value on both sides of $\stackrel{\rightharpoonup}{M}$ in the direction of ${\stackrel{\rightharpoonup}{M}}_{\perp}$ increase or decrease at the same time, without affecting its symmetry. Since the total torque value is obtained after generating countless points between the plane and the free surface, in order to achieve the state of balance and symmetry, the total torque increase and decrease of all points must be equal to the total torque value after the plane rotates a certain radian γ around axis ${\stackrel{\rightharpoonup}{M}}_{\perp}$. To facilitate calculation, points on the plane are divided into two parts along the rotation axis ${\stackrel{\rightharpoonup}{M}}_{\perp}$- the right part (RP) and the left part (LP). If the torque values on the right part (RP) are decreasing, the plane is rotated clockwise along axis ${\stackrel{\rightharpoonup}{M}}_{\perp}$, and if the torque values on the left part (LP) are increasing, on the contrary, the plane changes accordingly.

**Step 6:**Calculate the rotation radian of the final plane. As shown in Figure 6b, the new plane is formed in the normal of the plane ($\stackrel{\rightharpoonup}{n}$) and total torque directions ($\stackrel{\rightharpoonup}{M}$), and the intersecting lines and intersecting curves are generated respectively with the plane and the free plane. As shown in Figure 6b, a certain number of points are generated on the intersecting line and intersecting curve, so as to illustrate the relationship between the change of torque and the position of points when the plane rotates radian γ along axis ${\stackrel{\rightharpoonup}{M}}_{\perp}$, and the formula is calculated as follows:

_{i}is the distance from point P

_{i}to axis ${\stackrel{\rightharpoonup}{M}}_{\perp}$. When the points belong to the LP, the torque values are negative; otherwise, they are positive.

_{0}′), an axis of rotation (${\stackrel{\rightharpoonup}{M}}_{\perp}$), and a rotation radian (γ) are calculated. If a plane is rotated according to these parameters, the optimal cutting plane of disk milling cutter will be obtained. In the other words, the normal direction of the rotated optimal plane is the optimal tool orientation of the disk milling cutter.

## 3. Simulation and Calculation

#### 3.1. Secondary Development for Torque Balance Method on UG NX

- (1)
- Creating menu tools and dialogue box. The files “torquebalancemethod.men” and “torquebalancemethod.dlg” were created by using the models of MenuScript and UIStyler supported by UG NX software. After that, saving the two files into a specific folder.
- (2)
- Loading the generated C++ file. The app and CPP (C++ language prepared by the source code file suffix) files corresponding to the UG NX version number are created by using Visual Studio 2008 and named “torqueblancemethod.app” and “torqueblancemethod.cpp”.
- (3)
- Writing the torque balance method code based on the toque balance algorithm and the CCRP method. The file named “torquebalancealgorithm.dll” is created after compiling the program, and then putting the dll in the startup folder in a special position.

#### 3.2. Simulation and Measurement Results

_{x}), angle value between the tool orientation of the disk milling cutter and the y axis (α

_{y}), and angle value between the tool orientation of the disk milling cutter and the z axis (α

_{z}) are gathered together as shown in Table 1. In the table, point number denotes the number of points where concentric circles intersect with rays. In this simulation experiment, the same number of concentric circles and number of rays are used, and the range is 9 to 31. For the angle value between the tool orientation of the disk milling cutter and the x axis (α

_{x}), the maximum and minimum values are 91.1938° and 88.3465°, respectively. For the angle value between the tool orientation of the disk milling cutter and the y axis (α

_{y}), the maximum and minimum values are 42.1519° and 36.4559°, respectively. Lastly, for the angle value between the tool orientation of the disk milling cutter and the z axis (α

_{z}), the maximum and minimum values are 53.5567° and 47.8486°, respectively.

#### 3.3. Calculation of Tool Orientation of the Disk Milling Cutter

## 4. Algorithm Verification and Experimental Verification

#### 4.1. Comparison Algorithm and Model

#### 4.2. Algorithm Contrast

#### 4.2.1. Calculation Accuracy

^{−4}. As shown in Table 2, after comparing the simulation results, it is found that the four methods—the steepest descent method, Newton method, conjugate gradient method, and torque balance algorithm can reach the level of 10

^{−5}without the simulation time. At the same time, the simulation results can meet the precision requirements of the CNC system of the machine tool. Therefore, the conclusion can be inferred that these four methods meet the requirements in terms of calculation accuracy.

#### 4.2.2. Operation Speed

#### 4.2.3. Convergence Speed

#### 4.3. Experimental Verification

## 5. Conclusions

- (1)
- By analyzing the main principle of disc milling and the position of the cutting edge of the disc milling cutter, the maximum cutting amount is converted to the minimum residual amount. Through the calculus formula, the torque balance algorithm is deduced.
- (2)
- Because of the asymmetry of random points in the torque balance algorithm, the generation of random points is controlled by the CCRP method (concentric circle ray point method). On the basis of the CCRP method, the suitable formula for the torque balance algorithm is deduced. At the same time, a torque balance method including the torque balance algorithm and CCRP method is formed.
- (3)
- After comparison with the other three algorithms (steepest descent method, Newton method, and conjugate gradient method) the operation speed and the convergence of the torque balance method are better than the others. In particular, the operation speed of torque balance method was reduced by 0.35 times, 1.5 times, and 2.25 times compared to the other three methods.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of efficient and powerful CNC (Computer Numerical Control) milling machine.

**Figure 3.**Schematic diagram of integral. (

**a**) Diagram of surface integral; (

**b**) Schematic diagram of polar integral region.

**Figure 5.**Concentric circle ray point (CCRP) method. (

**a**) Extend the free surface; (

**b**) Create the plane and find the rotate center; (

**c**) Create a point with concentric circles and rays on the plane; (

**d**) Project the points on the plane to the free surface.

**Figure 6.**The schematic diagram for searching the optimal machining plane. (

**a**) The schematic diagram for calculating the final torque and search the rotation axis of the plane; (

**b**) The schematic diagram for calculating the radians of a plane rotation.

**Figure 8.**Realizing the torque balance method by using UG NX secondary development. (

**a**) The interface created by UG NX; (

**b**) The effect diagram of the torque balance method is realized.

**Figure 10.**Simulation results of torque balancing algorithm (

**a**) The angle between tool orientation and the x, y, and z axis. (

**b**) The angle between the tool orientation and x axis; (

**c**) The angle between the tool orientation and y axis; (

**d**) The angle between the tool orientation and z axis.

**Figure 11.**Comparison algorithm and model. (

**a**) The convergence path of four methods; (

**b**) The blank model of the blade.

**Figure 12.**The operation speed of the torque balance method, Newton method, steepest descent method, and conjugate gradient method.

**Figure 13.**Convergence speed comparison. (

**a**) Convergence speed comparison of the torque balance method and Newton method; (

**b**) Convergence speed comparison of the torque balance method and steepest descent method; (

**c**) Convergence speed comparison of the torque balance method and conjugate gradient method.

**Figure 14.**The process of blisk milling. (

**a**) Contact segment of the blisk milling process; (

**b**) Milling section of the blisk milling process; (

**c**) The effect drawing after the blisk milling process.

No. | Point Number | α_{x} | α_{y} | α_{z} |
---|---|---|---|---|

(Degree) | (Degree) | (Degree) | ||

1 | 81 | 89.8347 | 42.1519 | 47.8486 |

2 | 100 | 89.9485 | 36.9819 | 53.0181 |

3 | 121 | 89.1692 | 36.4559 | 53.5567 |

4 | 144 | 88.9838 | 37.3696 | 52.6491 |

5 | 169 | 88.7202 | 38.831 | 51.1982 |

6 | 196 | 88.3465 | 39.8124 | 50.2361 |

7 | 225 | 89.237 | 39.7183 | 50.292 |

8 | 256 | 88.8145 | 40.311 | 49.7139 |

9 | 289 | 89.6196 | 40.4243 | 49.5783 |

10 | 324 | 89.6333 | 40.6267 | 49.3757 |

11 | 361 | 89.4733 | 41.4041 | 48.6007 |

12 | 400 | 91.1938 | 39.8228 | 50.2025 |

13 | 441 | 90.0157 | 41.1163 | 48.8837 |

14 | 484 | 89.4752 | 41.0811 | 48.9238 |

15 | 529 | 90.0607 | 41.0045 | 48.9956 |

16 | 576 | 90.0241 | 40.9742 | 49.0258 |

17 | 625 | 90.2489 | 40.811 | 49.1901 |

18 | 676 | 90.0824 | 40.9737 | 49.0264 |

19 | 729 | 90.0839 | 41.0566 | 48.9435 |

20 | 784 | 90.0253 | 41.1894 | 48.8107 |

21 | 841 | 90.036 | 40.9206 | 49.0794 |

22 | 900 | 90.0325 | 40.8421 | 49.1579 |

23 | 961 | 90.0324 | 40.8420 | 49.1581 |

No. | The Algorithm Name | The Residual Amount | α_{x} | α_{y} | α_{z} |
---|---|---|---|---|---|

(mm^{3}) | (Degree) | (Degree) | (Degree) | ||

1 | Steepest descent method | 18,040.75523 | 90.03241 | 40.84201 | 49.15810 |

2 | Newton method | 18,040.75522 | 90.03240 | 40.84202 | 49.15813 |

3 | Conjugate gradient method | 18,040.75528 | 90.03242 | 40.84204 | 49.15812 |

4 | Torque balance method | 18,040.75524 | 90.03243 | 40.84201 | 49.15814 |

No. | Section Number | Channel 1 | Channel 2 | Channel 3 |
---|---|---|---|---|

(mm) | (mm) | (mm) | ||

1 | Section 1 | +0.040 ~ +0.054 | +0.054 ~ +0.091 | +0.077 ~ +0.132 |

2 | Section 2 | +0.089 ~ +0.192 | +0.072 ~ +0.238 | +0.057 ~ +0.173 |

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## Share and Cite

**MDPI and ACS Style**

Li, Z.; Shi, Y.
Tool Orientation Optimization for Disk Milling Process Based on Torque Balance Method. *Symmetry* **2020**, *12*, 60.
https://doi.org/10.3390/sym12010060

**AMA Style**

Li Z, Shi Y.
Tool Orientation Optimization for Disk Milling Process Based on Torque Balance Method. *Symmetry*. 2020; 12(1):60.
https://doi.org/10.3390/sym12010060

**Chicago/Turabian Style**

Li, Zhishan, and Yaoyao Shi.
2020. "Tool Orientation Optimization for Disk Milling Process Based on Torque Balance Method" *Symmetry* 12, no. 1: 60.
https://doi.org/10.3390/sym12010060