Skip to main content

Advertisement

SpringerLink
Log in

An investigation on the improved magnetic stiffness model and characteristic analysis for two cylindrical permanent magnets

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

As a functional material, permanent magnets (PMs) are widely used in magnetic actuation, magnetic bearings, energy harvesters, and lithography semiconductor industries. The magnetic stiffness determines the stability and performance of the magnetic systems. Therefore, an accurate and efficient magnetic stiffness model tool is needed. This paper newly establishes an improved high precision magnetic stiffness model between two cylindrical PMs for parallel and perpendicular magnetization directions, which takes the PMs' relative permeability, residual magnetic flux density, magnetization direction, dimensions, and relative positions into account. The magnetic stiffness model is solved by a numerical algorithm. Moreover, the magnetic stiffness calculation results are indirectly validated by the finite-element method (FEM) and experimental measurement, and the accuracy and efficacy of the established model are demonstrated. Furthermore, the sensitivity of magnetic stiffness to magnets’ size and position parameters is explored, and the influence of these parameters on magnetic stiffness characteristics is studied and discussed. The results indicate that the relative positions between PMs have a significant influence on magnetic stiffness. Thus the accurate control of the relative position is important to design PM devices. In addition, the developed magnetic stiffness model has lower computational efforts than the FEM and lower costs than experimental measurement. The model includes all magnetic properties and relative positions parameters, which can calculate the principal magnetic stiffness and cross-coupling magnetic stiffness and doesn't produce any principle error. The established model makes it easy to design and optimize PM devices that depend on magnetic stiffness, such as magnetic bearings, magnetic vibration isolator, magnetic piezoelectric cantilever beam energy harvester, etc.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Abbreviations

B :

Magnetic flux density

B r :

Residual magnetic flux density

df :

Differential magnetic force

e :

Relative position between the two parallel PMs’ central axes along z-axis

F :

Magnetic force

F x, F y, F z :

Magnetic force components along the x/y/z-direction

h :

Distance between two perpendicular PMs along the x-axis

K :

Magnetic stiffness

L :

Length

M :

Magnetization vector

n :

Outward unit normal vector

Q m :

Magnetic charges

r 1 :

Distance vector from the center of surface 2 to point P

r 2 :

Distance vector from the center of surface 3 to point Q

R :

Radius

S :

Surface

V :

Volume

x 0 :

Axial displacement between two parallel PMs along x-axis

α :

Angle between the vector r1 and the z-axis

β :

Angle between the vector r2 and the z-axis

μ 0 :

Permeability of vacuum

μ r :

Relative permeability

\(\sigma_{m}\) :

Surface magnetic charge density

\(\rho_{m}\) :

Volume magnetic charge density

\(\Delta A\) :

Area elements

References

  1. Zhou W, Wang B, Lim CW, Yang Z (2020) A distributed-parameter electromechanical coupling model for a segmented arc-shaped piezoelectric energy harvester. Mech Syst Signal Process 146:107005. https://doi.org/10.1016/j.ymssp.2020.107005

    Article  Google Scholar 

  2. Seong S, Hu C, Lee S (2017) Design under uncertainty for reliable power generation of piezoelectric energy harvester. J Intell Mater Syst Struct 28:2437–2449. https://doi.org/10.1177/1045389X17689945

    Article  Google Scholar 

  3. Yang Z, Wang YQ, Zuo L, Zu J (2017) Introducing arc-shaped piezoelectric elements into energy harvesters. Energy Convers Manage 148:260–266. https://doi.org/10.1016/j.enconman.2017.05.073

    Article  Google Scholar 

  4. Jung J, Kim P, Lee JI, Seok J (2015) Nonlinear dynamic and energetic characteristics of piezoelectric energy harvester with two rotatable external magnets. Int J Mech Sci 92:206–222. https://doi.org/10.1016/j.ijmecsci.2014.12.015

    Article  Google Scholar 

  5. Cao Y, Huang H, Zhu ZH, Su S (2019) Optimized energy harvesting through piezoelectric functionally graded cantilever beams. Smart Mater Struct. https://doi.org/10.1088/1361-665X/aaf761

    Article  Google Scholar 

  6. Song HC, Kim SW, Kim HS et al (2020) Piezoelectric energy harvesting design principles for materials and structures: material figure-of-merit and self-resonance tuning. Adv Mater 32:1–34. https://doi.org/10.1002/adma.202002208

    Article  Google Scholar 

  7. Hu G, Liang J, Lan C, Tang L (2020) A twist piezoelectric beam for multi-directional energy harvesting. Smart Mater Struct. https://doi.org/10.1088/1361-665X/abb648

    Article  Google Scholar 

  8. Li H, Liu D, Wang J et al (2020) Broadband bimorph piezoelectric energy harvesting by exploiting bending-torsion of L-shaped structure. Energy Convers Manage 206:112503. https://doi.org/10.1016/j.enconman.2020.112503

    Article  Google Scholar 

  9. Nie X, Tan T, Yan Z et al (2019) Broadband and high-efficient L-shaped piezoelectric energy harvester based on internal resonance. Int J Mech Sci 159:287–305. https://doi.org/10.1016/j.ijmecsci.2019.06.009

    Article  Google Scholar 

  10. Wu Y, Qiu J, Zhou S et al (2018) A piezoelectric spring pendulum oscillator used for multi-directional and ultra-low frequency vibration energy harvesting. Appl Energy 231:600–614. https://doi.org/10.1016/j.apenergy.2018.09.082

    Article  Google Scholar 

  11. Jeong S, Cho JY, Sung TH, Yoo HH (2017) Electromechanical modeling and power performance analysis of a piezoelectric energy harvester having an attached mass and a segmented piezoelectric layer. Smart Mater Struct. https://doi.org/10.1088/1361-665X/aa550b

    Article  Google Scholar 

  12. Castagnetti D (2019) A simply tunable electromagnetic pendulum energy harvester. Meccanica 54:749–760. https://doi.org/10.1007/s11012-019-00976-7

    Article  Google Scholar 

  13. Ravaud R, Lemarquand G, Lemarquand V (2009) Force and stiffness of passive magnetic bearings using permanent magnets. Part 2: radial magnetization. IEEE Trans Magn 45:3334–3342. https://doi.org/10.1109/TMAG.2009.2025315

    Article  Google Scholar 

  14. Ravaud R, Lemarquand G, Lemarquand V (2009) Force and stiffness of passive magnetic bearings using permanent magnets. Part 1: axial magnetization. IEEE Trans Magn 45:2996–3002. https://doi.org/10.1109/TMAG.2009.2025315

    Article  Google Scholar 

  15. Jungmayr G, Marth E, Amrhein W et al (2014) Analytical stiffness calculation for permanent magnetic bearings with soft magnetic materials. IEEE Trans Magn 50:8300108

    Article  Google Scholar 

  16. Van Casteren DTEH, Paulides JJH, Janssen JLG, Lomonova EA (2015) Analytical force, stiffness, and resonance frequency calculations of a magnetic vibration isolator for a microbalance. IEEE Trans Ind Appl 51:204–210. https://doi.org/10.1109/TIA.2014.2328780

    Article  Google Scholar 

  17. Santra T, Roy D, Choudhury AB, Yamada S (2019) Experimental verification of force and stiffness between two ring magnets calculated by Monte Carlo integration technique. J Instit Eng India Ser B 100:123–129. https://doi.org/10.1007/s40031-019-00373-4

    Article  Google Scholar 

  18. Zhu P, Ren X, Qin W, Zhou Z (2017) Improving energy harvesting in a tri-stable piezomagnetoelastic beam with two attractive external magnets subjected to random excitation. Arch Appl Mech 87:45–57. https://doi.org/10.1007/s00419-016-1175-z

    Article  Google Scholar 

  19. Stanton SC, McGehee CC, Mann BP (2010) Nonlinear dynamics for broadband energy harvesting: INVESTIGATION of a bistable piezoelectric inertial generator. Physica D 239:640–653. https://doi.org/10.1016/j.physd.2010.01.019

    Article  MATH  Google Scholar 

  20. Zhou Z, Qin W, Zhu P (2016) Improve efficiency of harvesting random energy by snap-through in a quad-stable harvester. Sens Actuators A 243:151–158. https://doi.org/10.1016/j.sna.2016.03.024

    Article  Google Scholar 

  21. Zhou Z, Qin W, Zhu P (2018) Harvesting performance of quad-stable piezoelectric energy harvester: modeling and experiment. Mech Syst Signal Process 110:260–272. https://doi.org/10.1016/j.ymssp.2018.03.023

    Article  Google Scholar 

  22. Leng Y, Tan D, Liu J et al (2017) Magnetic force analysis and performance of a tri-stable piezoelectric energy harvester under random excitation. J Sound Vib 406:146–160. https://doi.org/10.1016/j.jsv.2017.06.020

    Article  Google Scholar 

  23. Yan B, Ma H, Zhao C et al (2018) A vari-stiffness nonlinear isolator with magnetic effects: theoretical modeling and experimental verification. Int J Mech Sci 148:745–755. https://doi.org/10.1016/j.ijmecsci.2018.09.031

    Article  Google Scholar 

  24. Zheng Y, Zhang X, Luo Y et al (2016) Design and experiment of a high-static-low-dynamic stiffness isolator using a negative stiffness magnetic spring. J Sound Vib 360:31–52. https://doi.org/10.1016/j.jsv.2015.09.019

    Article  Google Scholar 

  25. Zhou Z, Chen S, Xia D et al (2019) The design of negative stiffness spring for precision vibration isolation using axially magnetized permanent magnet rings. JVC/J Vib Control 25:2667–2677. https://doi.org/10.1177/1077546319866035

    Article  MathSciNet  Google Scholar 

  26. Zhang F, Xu M, Shao S, Xie S (2020) A new high-static-low-dynamic stiffness vibration isolator based on magnetic negative stiffness mechanism employing variable reluctance stress. J Sound Vib 476:115322. https://doi.org/10.1016/j.jsv.2020.115322

    Article  Google Scholar 

  27. Li Q, Li S, Li F et al (2020) Analysis and experiment of vibration isolation performance of a magnetic levitation vibration isolator with rectangular permanent magnets. J Vib Eng Technol 8:751–760. https://doi.org/10.1007/s42417-019-00188-z

    Article  Google Scholar 

  28. Wang G, Liao WH, Zhao Z et al (2019) Nonlinear magnetic force and dynamic characteristics of a tri-stable piezoelectric energy harvester. Nonlinear Dyn 97:2371–2397. https://doi.org/10.1007/s11071-019-05133-z

    Article  Google Scholar 

  29. Wang G, Wu H, Liao WH et al (2020) A modified magnetic force model and experimental validation of a tri-stable piezoelectric energy harvester. J Intell Mater Syst Struct 31:967–979. https://doi.org/10.1177/1045389X20905975

    Article  Google Scholar 

  30. Jansen JW, Van Lierop CMM, Lomonova EA, Vandenput AJ (2007) Modeling of magnetically levitated planar actuators with moving magnets. IEEE Trans Magn 43:15–25. https://doi.org/10.1109/TMAG.2006.886051

    Article  Google Scholar 

  31. Qu C, Pei YC, Li ZX (2021) a study on magnetic force characteristics between two cuboidal permanent magnets. J Supercond Novel Magn 34:2441–2454. https://doi.org/10.1007/s10948-021-05927-6

    Article  Google Scholar 

  32. Furlani EP (1993) Formulas for the force and torque of axial couplings. IEEE Trans Magn 29:2295–2301

    Article  Google Scholar 

  33. Janssen JLG, Paulides JJH, Compter JC, Lomonova EA (2010) Three-dimensional analytical calculation of the torque between permanent magnets in magnetic bearings. IEEE Trans Magn 46:1748–1751. https://doi.org/10.1109/TMAG.2010.2043224

    Article  Google Scholar 

  34. Kremers MFJ, Paulides JJH, Ilhan E et al (2013) Relative permeability in a 3D analytical surface charge model of permanent magnets. IEEE Trans Magn 49:2299–2302. https://doi.org/10.1109/TMAG.2013.2239976

    Article  Google Scholar 

  35. Furlani EP (1993) A formula for the levitation force between magnetic disks. IEEE Trans Magn 29:4165–4169. https://doi.org/10.1109/20.280867

    Article  Google Scholar 

  36. Qu C, Pei YC, Xin QY et al (2021) A reciprocating permanent magnetic actuator for driving magnetic micro robots in fluids. Proc Inst Mech Eng C J Mech Eng Sci. https://doi.org/10.1177/09544062211014547

    Article  Google Scholar 

  37. Challa VR, Prasad MG, Shi Y, Fisher FT (2008) A vibration energy harvesting device with bidirectional resonance frequency tunability. Smart Mater Struct. https://doi.org/10.1088/0964-1726/17/01/015035

    Article  Google Scholar 

  38. Qu C, Pei YC, Xu L et al (2020) A study on electromagnetic field and force for magnetic micro-robots applications. Progr Electromag Res M 97:201–213. https://doi.org/10.2528/PIERM20073005

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of Jilin Province [Grant Number 20170101060JC]; and the Graduate Innovation Fund of Jilin University.

Author information

Authors and Affiliations

  1. Key Laboratory of CNC Equipment Reliability, Ministry of Education, School of Mechanical and Aerospace Engineering, Jilin University, Nanling Campus, Changchun, 130025, People’s Republic of China

    Chuan Qu, Yong-Chen Pei, Fan Yang, Zhen-Xing Li & Qing-Yuan Xin

Authors
  1. Chuan Qu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong-Chen Pei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fan Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhen-Xing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Qing-Yuan Xin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong-Chen Pei.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qu, C., Pei, YC., Yang, F. et al. An investigation on the improved magnetic stiffness model and characteristic analysis for two cylindrical permanent magnets. Meccanica 57, 677–696 (2022). https://doi.org/10.1007/s11012-021-01461-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-021-01461-w

Keywords

Search

Navigation