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Article

Analytical Investigations of Nonlinear Stiffness Characteristics of Halbach-Cylinder Magnetic Springs for Heavy-Load Capacity

by
Zhongsheng Chen
1,2,*,
Yangyi Zhang
2,
Yeping Xiong
3 and
Ankang Wang
2
1
School of Engineering Science, Shandong Xiehe University, Jinan 250107, China
2
College of Railway Transportation, Hunan University of Technology, Zhuzhou 412007, China
3
Faculty of Engineering and Physical Sciences, University of Southampton, Southampton SO16 7QF, UK
*
Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(9), 5099; https://doi.org/10.3390/app15095099
Submission received: 16 February 2025 / Revised: 1 May 2025 / Accepted: 2 May 2025 / Published: 3 May 2025

Abstract

Quasi-zero stiffness (QZS) has become a promising way of realizing low-frequency vibration isolation, where magnetic springs have been widely adopted for constructing negative stiffness. However, existing single-layer magnetic springs often have a small-amplitude negative stiffness, so the loading capacity is low. In order to address this issue, this paper presents novel Halbach-cylinder magnetic springs (HCMSs) by using the Halbach array. Firstly, stiffness formulas of basic single-layer magnetic springs are analytically built based on the Amperian current model. The stiffness of the HCMS is derived from combining multiple single-layer magnetic springs. Then, nonlinear stiffness characteristics of both single-layer magnetic springs and HCMSs are investigated in terms of the amplitude, the uniformity, and the displacement range of negative stiffness. Analytical results show that HCMSs can generate negative stiffness with different equilibrium positions, and the amplitude of negative stiffness of HCMSs is much larger than that of single-layer magnetic springs. The amplitude of negative stiffness is in conflict with the uniformity, so a trade-off design is needed. In addition, increasing the number of layers of Halbach cylinders can be adopted to realize larger-amplitude and wider-range negative stiffness. This study will provide new insights into designing QZS with heavy-load capacity.
Keywords: quasi-zero stiffness; heavy load; large-amplitude negative stiffness; magnetic spring; Halbach arrays; Halbach-cylinder magnetic spring quasi-zero stiffness; heavy load; large-amplitude negative stiffness; magnetic spring; Halbach arrays; Halbach-cylinder magnetic spring

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MDPI and ACS Style

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

AMA Style

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 Style

Chen, 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 Style

Chen, 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

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