Investigation on Structural Mapping Laws of Sensitive Geometric Errors Oriented to Remanufacturing of Three-Axis Milling Machine Tools
Abstract
:1. Introduction
2. Geometric Error Modeling of a Three-Axis Machine Tool
3. Analysis of Sensitive Geometric Errors Using Partial Differentiation Method
3.1. Partial Differentiation Method
3.2. An Example to Analyze the Sensitivity of a Machine Tool
4. Mapping Law between Sensitive Geometric Errors and the Structure of a Three-Axis Remanufactured Machine Tool
4.1. Mapping Law between Sensitive Geometric Errors of Volumetric Position Error and the Structure
4.1.1. Mapping Law of Sensitive Linear Error
4.1.2. Mapping Law of Sensitive Angular Error
- (1)
- Mapping law to identify sensitive angular PDGEs
- Sensitive angular PDGEs identification of motion axis on workpiece chain.Each moving axis on workpiece chain has two sensitive angular PDGEs in the corresponding identification direction. The two errors are the angular errors around another two directions in addition to the identification direction. The sensitive angular PDGEs of the X-axis, Y-axis and Z-axis can be identified as follows:The sensitive angular PDGEs of the X-axis in the x identification direction on workpiece chain are , . The sensitive angular PDGEs of the Y-axis in the x identification direction on workpiece chain are and . The sensitive angular PDGEs of the Z-axis in the x identification direction on workpiece chain are and .The sensitive angular PDGEs of the X-axis in the y identification direction on workpiece chain are and . The sensitive angular PDGEs of the Y-axis in the y identification direction on workpiece chain are and . The sensitive angular PDGEs of the Z-axis in the y identification direction on workpiece chain are and .The sensitive angular PDGEs of the X-axis in the z identification direction on workpiece chain are and . The sensitive angular PDGEs of the Y-axis in the z identification direction on workpiece chain are and . The sensitive angular PDGEs of the Z-axis in the z identification direction on workpiece chain are and
- Sensitive angular PDGEs identification of motion axis on tool chainThe relationship between the identification direction and the direction of cutter axis will affect the mapping law.If the identification direction is accordance with the direction of cutter axis. There are no sensitive angular PDGEs in the identification direction of all axes on the tool chain. As shown in Figure 1, the cutter axis is along with z direction, hence X-axis and Z-axis on the tool chain have no sensitive angular PDGEs in the identification z direction.If the identification direction is inconsistent with the cutter axis direction. Each motion axis on the tool chain has one sensitive angular PDGEs. The sensitive angular PDGEs rotated around the third axis except the identification direction and the cutter axis direction. See in Figure 1, the cutter axis is along with z direction, hence the sensitive angular PDGEs in x identification direction are and , the sensitive angular PDGEs in y identification direction are and .
- (2)
- Mapping law to identify sensitive angular PIGEs
- The identification direction consistent with the direction of motion axis next to the workpiece has two sensitive angular PIGEs. Squareness errors between the motion axis and the other two axes are the sensitive angular PIGEs. As shown in Figure 1, Y-axis is next to the workpiece, hence and are the sensitive angular PIGEs in the y direction.
- There are no sensitive angular PIGEs in the identification direction consistent with the motion axis next to the tool. As shown in Figure 1, Z-axis is next to the tool on the tool chain, hence there are no sensitive angular PIGEs in the z direction.
- The last one sensitive angular PIGE can be found in the remaining third identification direction. As shown in Figure 1, Y-axis is closest to the workpiece, and Z-axis is closest to the tool. Hence, the x identification direction corresponding to the remaining X-axis has a sensitive angular PIGE
4.2. Mapping Law between Sensitive Geometric Errors of Volumetric Posture Error and the Structure
- (1)
- Mapping law to identify sensitive angular PDGEs
- (2)
- Mapping law to identify sensitive angular PIGEs
5. Simulation and Experiments
5.1. Quick Identification of Geometric Errors Based on the Mapping Table
5.2. Simulation Verification
5.3. Experimental Validation
6. Conclusions
- (1)
- Based on the proposed method, sensitive geometric error terms of different three-axis machine structures can be obtained without geometric error modeling and sensitivity analysis.
- (2)
- The weak coupling relationship among the geometric errors of three-axis machine tools can be concluded, and the partial differential method was utilized to analyze the sensitivity of geometric errors.
- (3)
- Mapping tables between the sensitive geometric errors and the machine tool structure can be applied to the convenient identification of sensitive error terms without professional knowledge.
- (4)
- The identification results of sensitive geometric errors by using the proposed method are verified through simulation and experiment. The results showed that the straightness error of milling was improved up to 0.007 mm by compensating the identified sensitive geometric errors.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Movement Axis | Physical Meaning of Each Error Term | Symbols | Error Serial Number |
---|---|---|---|
X | Positioning error of the X-axis in the x direction | 1 | |
Straightness error of X-axis in y direction | 2 | ||
Straightness error of X-axis in z direction | 3 | ||
Rolling error of X-axis around x direction | 4 | ||
Angular error of X-axis around y direction | 5 | ||
Angular error of X-axis around z direction | 6 | ||
Y | Straightness error of Y-axis in x direction | 7 | |
Positioning error of Y-axis in y direction | 8 | ||
Straightness error of Y-axis in z direction | 9 | ||
Angular error of Y-axis around x direction | 10 | ||
Rolling error of Y-axis around y direction | 11 | ||
Angular error of Y-axis around z direction | 12 | ||
Z | Straightness error of Z-axis in x direction | 13 | |
Straightness error of Z-axis in y direction | 14 | ||
Positioning error of Z-axis in z direction | 15 | ||
Angular error of Z-axis around x direction | 16 | ||
Angular error of Z-axis around y direction | 17 | ||
Rolling error of Z-axis around z direction | 18 | ||
Assembly errors | Perpendicularity error between X-axis and Y-axis | 19 | |
Perpendicularity error between X-axis and Z-axis | 20 | ||
Perpendicularity error between Y-axis and Z-axis | 21 |
Identification Direction | Sensitive Geometric Errors of Volumetric Position Error | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Sensitive Linear Error | Sensitive Angular Errors | |||||||||||||
PDGEs | PIGEs | |||||||||||||
Workpiece Chain | Tool Chain | Motion Axis Next to the Workpiece | Motion Axis Next to the Tool | The Third Motion Axis | ||||||||||
If There Is No Motion Axis on the Corresponding Chain, the Corresponding PDGEs Are Removed | ||||||||||||||
X | Y | Z | Direction of Cutter Axis | X | Y | Z | X | Y | Z | |||||
x | y | z | ||||||||||||
x | -- | -- | The sensitive angular PIGEs in the third direction of motion axis is the squareness error that not appeared in the direction of motion axis next to the workpiece. For example: if X-axis is next to the workpiece, the sensitive angular PIGEs in the third direction is . | |||||||||||
y | -- | -- | ||||||||||||
z | -- | -- |
Identification Direction | Sensitive Geometric Errors of Volumetric Posture Error | |||||
---|---|---|---|---|---|---|
PDGEs | PIGEs | |||||
Direction of Cutter Axis | Direction of Cutter Axis | |||||
x | y | z | x | y | z | |
x | -- | -- | ||||
y | -- | -- | ||||
z | -- | -- |
Identification Direction | Sensitive Geometric Errors of Volumetric Position Error | ||||||
---|---|---|---|---|---|---|---|
Sensitive Linear Error | Sensitive Angular Errors | ||||||
PDGEs | PIGEs | ||||||
Workpiece Chain | Tool Chain | Motion Axis Next to the Workpiece | Motion Axis Next to the Tool | The Third Motion Axis | |||
If There Is No Motion Axis on the Corresponding Chain, the Corresponding PDGEs Are Removed | |||||||
X | Y | Direction of Cutter Axis | Y | Z | |||
z | |||||||
x | |||||||
y | |||||||
z | -- | -- |
Identification Direction | Sensitive Geometric Errors of Volumetric Posture Error | |
---|---|---|
PDGEs | PIGEs | |
Direction of Cutter Axis | Direction of Cutter Axis | |
z | z | |
x | ||
y | ||
z | -- | -- |
X/Y/Z Position (mm) | 0 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 |
---|---|---|---|---|---|---|---|---|---|---|---|
(µm) | −0.5 | 1 | 2.5 | 3.5 | 5 | 6 | 8 | 9 | 11 | 13 | 15 |
(µm) | 2 | 2.5 | 2.5 | 0.5 | −1.5 | −2 | −2 | −1 | 1 | 3 | 3 |
(µm) | 3 | 3 | 0 | −1 | −1 | −2 | −1 | −0.5 | 0 | 1 | 1 |
(s) | −0.5 | 1 | 2 | 2.5 | 3 | 4 | 4 | 5 | 6 | 6.5 | 7 |
(s) | 0.5 | 1.5 | 2.5 | 4 | 5 | 6.5 | 6 | 7 | 7.5 | 6 | 5.5 |
(s) | 0.5 | 1 | 2 | 3.5 | 4.5 | 6 | 6.5 | 6 | 6 | 6.5 | 6.5 |
(µm) | 0 | 0.5 | 1 | 0.5 | 0 | 0.5 | 0.5 | 0 | 1 | 0.5 | 0 |
(µm) | −3.5 | −7 | −7 | −7 | −9 | −7 | −5 | −1 | 3 | 5 | 5 |
(µm) | −0.5 | 0 | 1 | 5 | 1 | 0 | −0.5 | 0 | 0.5 | 1 | 3 |
(s) | −0.5 | 0 | 1 | 2 | 3 | 3.5 | 4.5 | 5 | 6 | 6 | 6 |
(s) | 1.5 | 3.5 | 5.5 | 6.5 | 8 | 9 | 9 | 9 | 7.5 | 7.5 | 7.5 |
(s) | 0 | 1 | 3 | 4 | 5 | 6 | 8 | 9 | 9 | 10 | 11 |
(µm) | −3 | −1 | 1 | 2 | 1 | 0 | 1 | 1 | 0 | −1 | −2 |
(µm) | 2 | −10 | 2 | 4 | 3 | 2 | 2 | 0 | 0 | −1 | −3 |
(µm) | −4 | −2.5 | 6 | −0.5 | 2 | −1.5 | −0.5 | −3.5 | 1.5 | 2.5 | −3.5 |
(s) | 0.5 | 1.5 | 1.5 | 3 | 4 | 3 | 4 | 3 | 4 | 4 | 5 |
(s) | −1.5 | −2 | 2.5 | −2.5 | −2.5 | −1.5 | −1.5 | −1.5 | −0.5 | 0.5 | 1 |
(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
(s) | 5 | ||||||||||
(s) | −22.5 | ||||||||||
(s) | 45.5 |
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Ding, W.; Song, Z.; Ding, S. Investigation on Structural Mapping Laws of Sensitive Geometric Errors Oriented to Remanufacturing of Three-Axis Milling Machine Tools. Machines 2022, 10, 341. https://doi.org/10.3390/machines10050341
Ding W, Song Z, Ding S. Investigation on Structural Mapping Laws of Sensitive Geometric Errors Oriented to Remanufacturing of Three-Axis Milling Machine Tools. Machines. 2022; 10(5):341. https://doi.org/10.3390/machines10050341
Chicago/Turabian StyleDing, Wenzheng, Zhanqun Song, and Shuang Ding. 2022. "Investigation on Structural Mapping Laws of Sensitive Geometric Errors Oriented to Remanufacturing of Three-Axis Milling Machine Tools" Machines 10, no. 5: 341. https://doi.org/10.3390/machines10050341
APA StyleDing, W., Song, Z., & Ding, S. (2022). Investigation on Structural Mapping Laws of Sensitive Geometric Errors Oriented to Remanufacturing of Three-Axis Milling Machine Tools. Machines, 10(5), 341. https://doi.org/10.3390/machines10050341