Analytical Investigations of Nonlinear Stiffness Characteristics of Halbach-Cylinder Magnetic Springs for Heavy-Load Capacity
Abstract
:1. Introduction
2. Force and Stiffness Characteristics of Circular Magnetic Springs
2.1. DifferentConfigurations of Circular Magnetic Springs
2.2. Magnetic Force and Stiffness of Each Basic CMS
2.2.1. Type I CMSs
2.2.2. Type II CMSs
2.2.3. Type III CMSs and Type IV CMSs
3. Analytical Stiffness of Halbach-Cylinder Magnetic Spring
3.1. Basic Configuration of an HCMS
3.2. Stiffness Formulationof the 3-3-Type HCMS
4. Parametric Analysis of Negative Stiffness of Basic CMSs
4.1. Stiffness Curves of the Four Basic CMSs
4.2. Effects of the Axial Thicknesson
4.3. Effects of the Radial Thickness Ratio on
4.4. Effects of the Gap () on
5. Parametric Analysis of Negative Stiffness of the 3-3-Type HCMS
5.1. Validation of the Analytical Model by Finite Element Modeling
5.2. Comparison of the 3-3-Type HCMS with a Traditional Single-Layer CMS
5.3. Effects of Geometric Parameters on
5.4. Nonlinear Stiffness Characteristics of All 3-3-Type HCMSs
5.5. A Case Study on Optimizing the Structure of the 3-3-Type HCMS
6. Potential Solution for HCMSs with Larger-Amplitude and Wider-Range Negative Stiffness
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
CMS | Circular magnetic spring |
FEM | Finite element model |
HCMS | Halbach-cylinder magnetic spring |
HSLD | High static and low dynamic |
QZS | Quasi-zero stiffness |
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Description | Type | Basic Structure | Denotation | Magnetic Force | Stiffness |
---|---|---|---|---|---|
Radial–radial magnetization | Type I | OUT ← IN ← | |||
Axial–axialmagnetization | Type II | OUT ↑ IN ↑ | |||
Perpendicular magnetization (radial–axial) | Type III | OUT → IN ↑ | |||
Perpendicular magnetization (axial–radial) | Type IV | OUT ↑ IN → |
Type | All Denotations | Magnetic Force | Stiffness |
---|---|---|---|
Type I | OUT ← IN ← OUT ← IN → OUT → IN ← OUT → IN → | ||
Type II | OUT ↑ IN ↑ OUT ↑ IN ↓ OUT ↓ IN ↑ OUT ↓ IN ↓ | ||
Type III | OUT → IN ↑ OUT → IN ↓ OUT ← IN ↑ OUT ← IN ↓ | ||
Type IV | OUT ↑ IN → OUT ↑ IN ← OUT ↓ IN → OUT ↓ IN ← |
Symbol | Physical Meaning | Unit |
---|---|---|
The magnetic-field vector | − | |
The residual flux density of the inner magnetic ring | T | |
The residual flux density of the outer magnetic ring | T | |
The axial force between the surface and the surface | N | |
The half-height of the inner magnetic ring | m | |
The half-height of the outer magnetic ring | m | |
The surface current density vector | − | |
The surface current density of surface 1 | A/m2 | |
The surface current density of surface 1′ | A/m2 | |
The surface current density of surface 3 | A/m2 | |
The surface current density of surface 3′ | A/m2 | |
The unit normal vector | − | |
The inside radius of the inner magnetic ring | m | |
Theoutside radius of the inner magnetic ring | m | |
The inside radius of the outer magnetic ring | m | |
Theoutside radius of the outer magnetic ring | m | |
The radius of the Q point | m | |
The radius of the P point | m | |
The vector product | − | |
The permeability of the vacuum | V⋅s/(A⋅m) | |
The included angles between Q and the axis | rad | |
The included angles between P and the axis | rad |
Parameter | ||||||
Value | 10 mm | 17.5 mm | 22.5 mm | 30 mm | 10 mm | 10 mm |
Parameter | |||||
Value | 10 mm | 17.5 mm | 22.5 mm | 30 mm | 5 mm |
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Chen, Z.; Zhang, Y.; Xiong, Y.; Wang, A. Analytical Investigations of Nonlinear Stiffness Characteristics of Halbach-Cylinder Magnetic Springs for Heavy-Load Capacity. Appl. Sci. 2025, 15, 5099. https://doi.org/10.3390/app15095099
Chen Z, Zhang Y, Xiong Y, Wang A. Analytical Investigations of Nonlinear Stiffness Characteristics of Halbach-Cylinder Magnetic Springs for Heavy-Load Capacity. Applied Sciences. 2025; 15(9):5099. https://doi.org/10.3390/app15095099
Chicago/Turabian StyleChen, Zhongsheng, Yangyi Zhang, Yeping Xiong, and Ankang Wang. 2025. "Analytical Investigations of Nonlinear Stiffness Characteristics of Halbach-Cylinder Magnetic Springs for Heavy-Load Capacity" Applied Sciences 15, no. 9: 5099. https://doi.org/10.3390/app15095099
APA StyleChen, Z., Zhang, Y., Xiong, Y., & Wang, A. (2025). Analytical Investigations of Nonlinear Stiffness Characteristics of Halbach-Cylinder Magnetic Springs for Heavy-Load Capacity. Applied Sciences, 15(9), 5099. https://doi.org/10.3390/app15095099