Low-Carbon Optimization Design of Grinding Machine Spindle Based on Improved Whale Algorithm
Abstract
:1. Introduction
2. Spindle Carbon Emissions and Performance Analysis
2.1. Carbon Emissions Function
2.2. Static and Dynamic Performance Analysis
2.2.1. Static Performance
2.2.2. Dynamic Performance
3. Low-Carbon Design Approach for Spindles
3.1. Structural and Objective Selection
3.1.1. Structural Parameters
3.1.2. Design Variables
3.1.3. Determination of Design Variables
3.2. Integrated Optimization Objectives
3.2.1. Establishment of Test Methods
3.2.2. Response Surface Modeling
3.2.3. Establishment of Response Surface Model
- Carbon emissions regression equation.
- 2.
- Solving the maximum deformation regression equation.
- 3.
- Solving the first-order natural frequency regression equation.
3.2.4. Comprehensive Optimization Modeling
4. Algorithms and Results of Optimization
4.1. Algorithm for Optimization
- In comparison to similar classical algorithms, this algorithm has the more important ability to jump out of local optimization.
- The algorithm offers three search methods: surrounding, random, and bubble net, demonstrating robust local search capabilities. Finding discrete and discontinuous optimal solutions is the key to spindle optimization.
4.2. Improvement of the Whale Algorithm
4.2.1. Initialization of the Cat Chaotic Map
4.2.2. Golden Sine Algorithm
4.3. Analyses and Results of Optimization
4.4. Design Variables and Their Impact on Goals
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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References | Energy Consumption | Static Characteristic | Dynamic Characteristic | Optimization Method |
---|---|---|---|---|
Reference [9] | 10% reduction | Not considered | Not considered | Optimize motor parameters |
Reference [10] | 7% reduction | Not considered | Not considered | Optimized spindle drive mode |
Reference [11] | 10.6% reduction | Not considered | Not considered | Optimize processing time |
Reference [21] | 17.6% reduction | The maximum deformation is reduced by 0.8% | Not considered | Optimize topology |
Reference [23] | Not considered | Not considered | The first four natural frequencies increased by 17% | Optimize structure |
District | Chinese North | Chinese Northeast | Chinese East | Chinese Central | Chinese Northwest | Chinese South |
---|---|---|---|---|---|---|
Carbon emissions factor () | 0.9419 | 1.0826 | 0.792 | 0.8587 | 0.8922 | 0.804 |
Objectives | D1 | l1 | D2 | l2 | D3 | l3 | D4 | l4 | D5 | l5 | D6 | l6 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Carbon emissions | 0.146 | 0.289 | 0.017 | 0.287 | 0.9 | 0.048 | 0.321 | 0.103 | 0.224 | 0.092 | 0.209 | 0.357 |
Maximum deformation | 0.52 | 0.65 | 0.66 | 0.297 | 0.278 | 0.257 | 0.227 | 0.245 | 0.199 | 0.185 | 0.129 | 0.104 |
First-order frequency | −0.327 | −0.965 | −0.512 | −0.167 | −0.451 | −0.283 | −0.069 | −0.504 | −0.169 | −0.598 | −0.421 | −0.037 |
Serial Number | /mm | /mm | /mm | /Kg | /mm | /Hz |
---|---|---|---|---|---|---|
1 | 50.00 | 65.00 | 70.00 | 0.129 | 6.489 × 10−3 | 1120.09 |
2 | 50.00 | 65.00 | 63.00 | 0.127 | 6.493 × 10−3 | 1119.82 |
3 | 50.00 | 65.00 | 77.00 | 0.131 | 6.489 × 10−3 | 1120.55 |
4 | 50.00 | 58.50 | 70.00 | 0.117 | 6.072 × 10−3 | 1120.38 |
5 | 50.00 | 71.50 | 70.00 | 0.141 | 6.778 × 10−3 | 1119.97 |
6 | 45.00 | 65.00 | 70.00 | 0.129 | 6.327 × 10−3 | 1120.55 |
7 | 55.00 | 65.00 | 70.00 | 0.129 | 6.659 × 10−3 | 1120.69 |
8 | 45.93 | 59.72 | 64.31 | 0.118 | 6.032 × 10−3 | 1120.47 |
9 | 45.93 | 59.72 | 75.69 | 0.121 | 6.029 × 10−3 | 1120.82 |
10 | 45.93 | 70.28 | 64.31 | 0.137 | 6.579 × 10−3 | 1120.50 |
11 | 45.93 | 70.28 | 75.69 | 0.141 | 6.575 × 10−3 | 1120.53 |
12 | 54.07 | 59.72 | 64.31 | 0.118 | 6.279 × 10−3 | 1120.50 |
13 | 54.07 | 59.72 | 75.69 | 0.121 | 6.277 × 10−3 | 1121.33 |
14 | 54.07 | 70.28 | 64.31 | 0.137 | 6.859 × 10−3 | 1120.64 |
15 | 54.07 | 70.28 | 75.69 | 0.141 | 6.848 × 10−3 | 1120.24 |
Carbon Emissions | ||||
---|---|---|---|---|
1.097 × 10−3 | 9 | 4.74 × 10−9 | 5 |
Maximum Deformation | ||||
---|---|---|---|---|
1.084 × 10−6 | 9 | 2.672 × 10−10 | 5 |
First-Order Natural Frequency | ||||
---|---|---|---|---|
1.84 | 9 | 0.19 | 5 |
Design Variables/ Objectives | Before Optimization | Lightweight Optimization | Traditional Whale Algorithm | Improved Whale Algorithm |
---|---|---|---|---|
/mm | 50 | 51.4 | 46.2 | 44.5 |
/mm | 65 | 61.5 | 62.5 | 61 |
/mm | 70 | 67 | 66.4 | 65.5 |
/Kg | 0.129 | 0.1143 | 0.1232 | 0.1184 |
U/mm | 0.0065 | 0.00706 | 0.00614 | 0.00601 |
/Hz | 1120.09 | 1119.64 | 1120.52 | 1120.41 |
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Lu, Q.; Gao, X.; Chan, F.T.S. Low-Carbon Optimization Design of Grinding Machine Spindle Based on Improved Whale Algorithm. Mathematics 2024, 12, 69. https://doi.org/10.3390/math12010069
Lu Q, Gao X, Chan FTS. Low-Carbon Optimization Design of Grinding Machine Spindle Based on Improved Whale Algorithm. Mathematics. 2024; 12(1):69. https://doi.org/10.3390/math12010069
Chicago/Turabian StyleLu, Qi, Xubo Gao, and Felix T. S. Chan. 2024. "Low-Carbon Optimization Design of Grinding Machine Spindle Based on Improved Whale Algorithm" Mathematics 12, no. 1: 69. https://doi.org/10.3390/math12010069
APA StyleLu, Q., Gao, X., & Chan, F. T. S. (2024). Low-Carbon Optimization Design of Grinding Machine Spindle Based on Improved Whale Algorithm. Mathematics, 12(1), 69. https://doi.org/10.3390/math12010069