1. State Key Laboratory of Intelligent Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China 2. School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
Passive vibration isolation systems have been widely applied due to their low power consumption and high reliability. Nevertheless, the design of vibration isolators is usually limited by the narrow space of installation, and the requirement of heavy loads needs the high supporting stiffness that leads to the narrow isolation frequency band. To improve the vibration isolation performance of passive isolation systems for dynamic loaded equipment, a novel modular quasi-zero stiffness vibration isolator (MQZS-VI) with high linearity and integrated fluid damping is proposed. The MQZS-VI can achieve high-performance vibration isolation under a constraint mounted space, which is realized by highly integrating a novel combined magnetic negative stiffness mechanism into a damping structure: The stator magnets are integrated into the cylinder block, and the moving magnets providing negative-stiffness force also function as the piston supplying damping force simultaneously. An analytical model of the novel MQZS-VI is established and verified first. The effects of geometric parameters on the characteristics of negative stiffness and damping are then elucidated in detail based on the analytical model, and the design procedure is proposed to provide guidelines for the performance optimization of the MQZS-VI. Finally, static and dynamic experiments are conducted on the prototype. The experimental results demonstrate the proposed analytical model can be effectively utilized in the optimal design of the MQZS-VI, and the optimized MQZS-VI broadened greatly the isolation frequency band and suppressed the resonance peak simultaneously, which presented a substantial potential for application in vibration isolation for dynamic loaded equipment.
Magnitudes of radial magnetic flux density generated by the equivalent current loop and magnetic charge loop, respectively
,
Magnitudes of axial magnetic flux density generated by the equivalent current loop and magnetic charge loop, respectively
,
Magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively
,
Radial magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively
,
Axial magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively
c
Damping coefficient
cideal
Ideal damping coefficient
d
Wire diameter of the metal spring
D
Middle diameter of the metal spring
E
Complete elliptic integral of the second kind
Fv
Damping force
Fz
Axial force
Fz* (* = u, m, l)
Axial force of moving magnets suffered from the top stator, the middle stator, and the lower stator respectively
,
Forces of the current loop acted by another current loop and another magnetic charge loop, respectively
,
Forces of the magnetic charge loop acted by another current loop and another magnetic charge loop, respectively
G
Complete elliptic integral of the first kind
H
Free height of the metal spring
H0, H3
Axial lengths of the moving magnets and the stator magnets on the top?bottom side, respectively
H1
Axial length of the assemble moving magnet
H2
Axial length of the middle stator magnet
Hc
Axial height of design space
Hp
Height of the piston head
Iib
Current of the inner current loop of the bottom moving magnet (i = 1) or the bottom stator magnet (i = 3)
Iiu
Current of the inner current loop of the upper moving magnet (i = 1) or the upper stator magnet (i = 3)
Ijb
Current of the outer current loop of the bottom moving magnet (j = 2) or the bottom stator magnet (j = 4)
Iju
Current of the outer current loop of the upper moving magnet (j = 2) or the bottom stator magnet (j = 4)
Im, Is
Surface currents of the equivalent current loop of the moving magnet and the stator magnet with radiative magnetization, respectively
Jl (l = 1,2,…,4)
Surface density of the equivalent ampere’s currents of the magnets with axial magnetization
ke
Stiffness of the vibration isolation system at equilibrium position
ki (i = 0,1,…,3)
Fitted coefficients of the nonlinear axail force
kn
Negative stiffness
kn0
Negative stiffness at the equilibrium position
kn(z)
Negative stiffness when the axial displacement is z
kp
Positive stiffness
kz
Total stiffness
L
Axial gap between the moving magnets and stator magnets on the top?bottom side
m
Mass of the payload
Mi (i = 1,2,…,4)
Magnitude of magnetization of the magnet
Mi (i = 1,2,…,4)
Magnetization of the magnet
n0
Effective turn number of the metal spring
Nb (b = c, v, s)
Number of segments for fictitious current loops, volume magnetic charge loops, and surface magnetic charge loops, respectively
nk (k = 1,2,…,8)
Unit vector normal to the surface of the ring magnets
P
Attenuation rate of vibration
Δp
Pressure difference
Qm, Qs
Magnetic charges of the equivalent magnetic charge loop of the moving magnet and the stator magnet with axial magnetization, respectively
Qsi
Magnetic charge of the micro unit of the equivalent inner surface magnetic charge loop of the middle moving magnet (i = 1) or the middle stator magnet (i = 3)
Qsj
Magnetic charge of the micro unit of the related outer surface magnetic charge loop of the middle moving magnet (j = 2) or the middle stator magnet (j = 4)
Qvk
Value of the magnetic charge of the micro unit of the equivalent volume magnetic charge loop of the middle moving magnet (k = 1) or the middle stator magnet (k = 3)
r1, r2
Inner and outer radii of the moving magnet, respectively
r3, r4
Inner and outer radii of the middle stator magnet, respectively
r5, r6
Inner and outer radii of the upper-bottom stator magnet, respectively
ri
Inner radius of all the magnets apart from the middle stator magnet
rm
Radial coordinate of micro units of equivalent loops of the moving magnet
rs
Radial coordinate of micro units of equivalent loops of the stator magnet
rvk, rvp
Radii of the equivalent volume magnetic charge loops of the middle stator magnet and the middle moving magnet, respectively
Rc
Radius of design space
Rp
Radius of the piston head
r
Radial unit vector
t
Time
T1, T3
Thicknesses of all the moving magnets and the stator magnets on the top?bottom side, respectively
T2
Thickness of the middle stator magnet
Td
Displacement transmissibility of the isolator
u
Flow index of the fluid
v
Moving velocity of the piston
W1, W2
Flow rates of the differential pressure flow and the shear flow, respectively
Wv
Total discharge of the fluid
x
Nondimensional displacement
X
Magnitude of the nondimensional displacement
z
Relative axial displacement
z1, z2
Axial coordinates of the lower plane of the upper stator magnet and the upper moving magnet, respectively
z1u, z2t
Axial coordinates of the current loop of the upper stator magnet and the upper moving magnet, respectively
z3, z4
Axial coordinates of the lower plane of the middle stator magnet and the middle moving magnet, respectively
z3n, z4i
Axial coordinates of the surface magnetic charge loop of the middle stator magnet and the middle moving magnet, respectively
z5, z6
Axial coordinates of the lower plane of the lower moving magnet and the lower stator magnet, respectively
z5q, z6w
Axial coordinates of the current loop of the lower moving magnet and the lower stator magnet, respectively
zb, zp
Displacements of the base excitation and the payload platform, respectively
zm
Axial coordinate of micro units of equivalent loops of the moving magnet
zp (p = 1,2,…,6)
Axial coordinate of the lower plane of each magnet
zs
Axial coordinate of micro units of equivalent loops of the stator magnet
zvj, zvm
Axial coordinates of the volume magnetic charge loop of the middle stator magnet and the middle moving magnet
Zb
Magnitude of the base displacement
z
Axial unit vector
αh
Ratio of H1 to H2
αv
Ratio of H0 to H3
βh
Ratio of T1 to T2
βv
Ratio of T1 to T3
η
Extent of stiffness nonlinearity
ηideal
Ideal stiffness nonlinearity
κ
Design width of the linear-stiffness interval of the CMNSM
ρ
Density of the damping fluid
γ
Kinematic viscosity of the damping fluid
δ
Damping gap
ι
Distance between the middle moving magnet and the top?bottom moving magnets
λ
Radial gap between the moving magnets and middle stator magnet
μ0
Permeability of the vacuum
εideal
Ideal stiffness counteraction ratio
θ
Circumferential unit vector
ω
Circular frequency
ω0
Natural frequency
τ
Nondimensional time
ξ
Relative damping ratio
ψ
Variable of integration
φ
Phase of nondimensional displacement
Ω
Nondimensional frequency
1
B WangZ JiangP D Hu. Study on 6-DOF active vibration-isolation system of the ultra-precision turning lathe based on GA-BP-PID control for dynamic loads. Advances in Manufacturing, 2023 (in press)
2
K Ikeda , K Kamimori , I Kobayashi , J Kuroda , D Uchino , K Ogawa , A Endo , T Kato , X J Liu , M H B Peeie , H Kato , T Narita . Basic study on mechanical vibration suppression system using 2-degree-of-freedom vibration analysis. Vibration, 2023, 6(2): 407–420 https://doi.org/10.3390/vibration6020025
3
M H Demir . Design and analysis of passive components-supported SLS-VI mechanism for the control of road-induced stretcher vibrations during ambulance movement. Journal of Vibration Engineering and Technologies, 2022, 11(8): 3959–3979 https://doi.org/10.1007/s42417-022-00795-3
4
F C Lin , H Zheng , B Xiang , R Xu , W Jiang , L Lang . Vibration-induced noise in extremely low frequency magnetic receiving antennas. IEEE Antennas and Wireless Propagation Letters, 2021, 20(6): 913–917 https://doi.org/10.1109/LAWP.2021.3066689
5
G X Dong , C C Ma , F Zhang , Y J Luo , C X Bi . Non-resonant response of the novel airborne photoelectric quasi-zero stiffness platform with friction damping. International Journal of Applied Electromagnetics and Mechanics, 2020, 64(1–4): 315–324 https://doi.org/10.3233/JAE-209336
6
S Jiang , J Z Wang , S K Wang , W Shen . Vibration isolation control performance for an innovative 3-DOF parallel stabilization platform. Journal of Mechanical Science and Technology, 2022, 36(7): 3677–3689 https://doi.org/10.1007/s12206-022-0642-4
7
H Z Liu , X Z Huang , M Yan , M X Chang . Dynamic response and time-variant reliability analysis of an eight-rod shock isolator. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2022, 236(13): 7041–7054 https://doi.org/10.1177/09544062211070464
8
Y Shin , S Moon , W Jung , S Bae . Experimental approach to active mounts using electromagnetic actuator and rubber with consideration of shock resistance for naval shipboard equipment. Shock and Vibration, 2019, 2019: 3958359 https://doi.org/10.1155/2019/3958359
9
D Qu , X D Liu , G T Liu , T He . Vibration isolation characteristics and control strategy of parallel air spring system for transportation under abnormal road and eccentric load conditions. Measurement and Control, 2021, 54(3–4): 252–268 https://doi.org/10.1177/0020294021996631
10
S H Li , G Z Feng , Q Zhao . Design and research of semiactive quasi-zero stiffness vibration isolation system for vehicles. Shock and Vibration, 2021, 2021: 5529509 https://doi.org/10.1155/2021/5529509
11
H Li , Y C Li , J C Li . Negative stiffness devices for vibration isolation applications: a review. Advances in Structural Engineering, 2020, 23(8): 1739–1755 https://doi.org/10.1177/1369433219900311
12
Y M Zhao , J N Cui , L M Zou . Genetic optimization of repulsive magnetic array negative stiffness structure for high-performance precision micro-vibration isolation. Journal of Vibration Engineering and Technologies, 2022, 10(4): 1325–1336 https://doi.org/10.1007/s42417-022-00449-4
13
Z Q Lu , W H Liu , H Ding , L Q Chen . Energy transfer of an axially loaded beam with a parallel-coupled nonlinear vibration isolator. Journal of Vibration and Acoustics, 2022, 144(5): 051009 https://doi.org/10.1115/1.4054324
14
J L Wu , L Z Zeng , B Han , Y F Zhou , X Luo , X Q Li , X D Chen , W Jiang . Analysis and design of a novel arrayed magnetic spring with high negative stiffness for low-frequency vibration isolation. International Journal of Mechanical Sciences, 2022, 216: 106980 https://doi.org/10.1016/j.ijmecsci.2021.106980
15
J L Zhao , G Zhou , D Z Zhang , I Kovacic , R Zhu , H Y Hu . Integrated design of a lightweight metastructure for broadband vibration isolation. International Journal of Mechanical Sciences, 2023, 244: 108069 https://doi.org/10.1016/j.ijmecsci.2022.108069
16
Q Wang , J X Zhou , K Wang , Q D Lin , D L Xu , G L Wen . A compact quasi-zero-stiffness device for vibration suppression and energy harvesting. International Journal of Mechanical Sciences, 2023, 250: 108284 https://doi.org/10.1016/j.ijmecsci.2023.108284
17
Q Wang , J X Zhou , D L Xu , H J Ouyang . Design and experimental investigation of ultra-low frequency vibration isolation during neonatal transport. Mechanical Systems and Signal Processing, 2020, 139: 106633 https://doi.org/10.1016/j.ymssp.2020.106633
18
Z Z Ma , R P Zhou , Q C Yang . Recent advances in quasi-zero stiffness vibration isolation systems: an overview and future possibilities. Machines, 2022, 10(9): 813 https://doi.org/10.3390/machines10090813
19
X J Jing , Y Y Chai , X Chao , J Bian . In-situ adjustable nonlinear passive stiffness using X-shaped mechanisms. Mechanical Systems and Signal Processing, 2022, 170: 108267 https://doi.org/10.1016/j.ymssp.2021.108267
20
M Abuabiah , Y Dabbas , L Herzallah , I H Alsurakji , M Assad , P Plapper . Analytical study on the low-frequency vibrations isolation system for vehicle’s seats using quasi-zero-stiffness isolator. Applied Sciences, 2022, 12(5): 2418 https://doi.org/10.3390/app12052418
21
F Zhao , J C Ji , K Ye , Q T Luo . Increase of quasi-zero stiffness region using two pairs of oblique springs. Mechanical Systems and Signal Processing, 2020, 144: 106975 https://doi.org/10.1016/j.ymssp.2020.106975
22
F Zhao , J C Ji , K Ye , Q T Luo . An innovative quasi-zero stiffness isolator with three pairs of oblique springs. International Journal of Mechanical Sciences, 2021, 192: 106093 https://doi.org/10.1016/j.ijmecsci.2020.106093
23
C Y Wei . Design and analysis of a novel vehicle-mounted active QZS vibration isolator. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 2023, 47(4): 2121–2131 https://doi.org/10.1007/s40997-023-00622-4
24
Z B Chen , S B Yu , B Wang , X Y Zhuang , F Lin . The research about application of quasi-zero stiffness vibration isolation technology in a large vehicle-mounted optic-electronic equipment. Ain Shams Engineering Journal, 2023, 14(2): 101841 https://doi.org/10.1016/j.asej.2022.101841
25
K Meng , Y Sun , H Y Pu , J Luo , S J Yuan , J L Zhao , S R Xie , Y Peng . Development of vibration isolator with controllable stiffness using permanent magnets and coils. Journal of Vibration and Acoustics, 2019, 141(4): 041014 https://doi.org/10.1115/1.4043413
26
J L Wu , J X Che , X D Chen , W Jiang . Design of a combined magnetic negative stiffness mechanism with high linearity in a wide working region. Science China Technological Sciences, 2022, 65(9): 2127–2142 https://doi.org/10.1007/s11431-022-2121-7
27
Y Zhang , Q H Liu , Y G Lei , J Y Cao , W H Liao . Halbach high negative stiffness isolator: modeling and experiments. Mechanical Systems and Signal Processing, 2023, 188: 110014 https://doi.org/10.1016/j.ymssp.2022.110014
28
S J Yuan , Y Sun , M Wang , J H Ding , J L Zhao , Y N Huang , Y Peng , S R Xie , J Luo , H Y Pu , F Q Liu , L Bai , X D Yang . Tunable negative stiffness spring using maxwell normal stress. International Journal of Mechanical Sciences, 2021, 193: 106127 https://doi.org/10.1016/j.ijmecsci.2020.106127
29
L X Tu , D H Ning , S S Sun , W X Li , H Huang , M M Dong , H P Du . A novel negative stiffness magnetic spring design for vehicle seat suspension system. Mechatronics, 2020, 68: 102370 https://doi.org/10.1016/j.mechatronics.2020.102370
30
F Zhang , M L Xu , S B Shao , S L Xie . A new high-static-low-dynamic stiffness vibration isolator based on magnetic negative stiffness mechanism employing variable reluctance stress. Journal of Sound and Vibration, 2020, 476: 115322 https://doi.org/10.1016/j.jsv.2020.115322
31
M T Wang , P Su , S Y Liu , K Chai , B X Wang , J F Lu . Design and analysis of electromagnetic quasi-zero stiffness vibration isolator. Journal of Vibration Engineering and Technologies, 2023, 11(1): 153–164 https://doi.org/10.1007/s42417-022-00569-x
32
T Wang , S Q Zhu . Resonant frequency reduction of vertical vibration energy harvester by using negative-stiffness magnetic spring. IEEE Transactions on Magnetics, 2021, 57(9): 1–7 https://doi.org/10.1109/TMAG.2021.3095589
33
M K Wu , J L Wu , J X Che , R Q Gao , X D Chen , X Q Li , L Z Zeng , W Jiang . Analysis and experiment of a novel compact magnetic spring with high linear negative stiffness. Mechanical Systems and Signal Processing, 2023, 198: 110387 https://doi.org/10.1016/j.ymssp.2023.110387
34
B Yan , N Yu , C Y Wu . A state-of-the-art review on low-frequency nonlinear vibration isolation with electromagnetic mechanisms. Applied Mathematics and Mechanics, 2022, 43(7): 1045–1062 https://doi.org/10.1007/s10483-022-2868-5
35
Y M Zhao , J N Cui , L M Zou , Z Y Cheng . Modeling and dynamics of magnetically repulsive negative stiffness permanent magnetic array for precision air/magnetic composite vibration isolation. International Journal of Structural Stability and Dynamics, 2022, 22(7): 2250031 https://doi.org/10.1142/S0219455422500316
36
Y S Zheng , Q P Li , B Yan , Y J Luo , X N Zhang . A Stewart isolator with high-static-low-dynamic stiffness struts based on negative stiffness magnetic springs. Journal of Sound and Vibration, 2018, 422: 390–408 https://doi.org/10.1016/j.jsv.2018.02.046
37
R B Hao , Z Q Lu , H Ding , L Q Chen . Orthogonal six-DOFs vibration isolation with tunable high-static-low-dynamic stiffness: experiment and analysis. International Journal of Mechanical Sciences, 2022, 222: 107237 https://doi.org/10.1016/j.ijmecsci.2022.107237
38
R B Hao , Z Q Lu , H Ding , L Q Chen . Shock isolation of an orthogonal six-DOFs platform with high-static-low-dynamic stiffness. Journal of Applied Mechanics, 2023, 90(11): 111004 https://doi.org/10.1115/1.4062886
39
E Diez-Jimenez , R Rizzo , M J Gómez-García , E Corral-Abad . Review of passive electromagnetic devices for vibration damping and isolation. Shock and Vibration, 2019, 2019: 1250707 https://doi.org/10.1155/2019/1250707
40
X F Yang , L Yan , Y J Shen , H C Li , Y L Liu . Dynamic performance analysis and parameters perturbation study of inerter–spring–damper suspension for heavy vehicle. Journal of Low Frequency Noise, Vibration and Active Control, 2021, 40(3): 1335–1350 https://doi.org/10.1177/1461348420962898
41
D Guan , X J Cong , J Li , P B Wang , Z W Yang , X J Jing . Theoretical modeling and optimal matching on the damping property of mechatronic shock absorber with low speed and heavy load capacity. Journal of Sound and Vibration, 2022, 535: 117113 https://doi.org/10.1016/j.jsv.2022.117113
42
X B Huang , B T Yang . Towards novel energy shunt inspired vibration suppression techniques: principles, designs and applications. Mechanical Systems and Signal Processing, 2023, 182: 109496 https://doi.org/10.1016/j.ymssp.2022.109496
43
Z Q Lu , M Brennan , H Ding , L Q Chen . High-static−low-dynamic-stiffness vibration isolation enhanced by damping nonlinearity. Science China Technological Sciences, 2019, 62(7): 1103–1110 https://doi.org/10.1007/s11431-017-9281-9
44
Z M Wu , H Li , X S Kong , Z H Deng . A novel design of vibration isolator with high and frequency dependent damping characteristics based on a large negative Poisson’s ratio (LNPR) structure. Mechanical Systems and Signal Processing, 2023, 186: 109818 https://doi.org/10.1016/j.ymssp.2022.109818
45
C R Liu , K P Yu , J Tang . New insights into the damping characteristics of a typical quasi-zero-stiffness vibration isolator. International Journal of Non-Linear Mechanics, 2020, 124: 103511 https://doi.org/10.1016/j.ijnonlinmec.2020.103511
46
H Y Ma , B Yan , L Zhang , W G Zheng , P F Wang , C Y Wu . On the design of nonlinear damping with electromagnetic shunt damping. International Journal of Mechanical Sciences, 2020, 175: 105513 https://doi.org/10.1016/j.ijmecsci.2020.105513
47
R S Ma , K M Bi , H Hao . A novel rotational inertia damper for amplifying fluid resistance: experiment and mechanical model. Mechanical Systems and Signal Processing, 2021, 149: 107313 https://doi.org/10.1016/j.ymssp.2020.107313
48
X Shi , S Y Zhu . Simulation and optimization of magnetic negative stiffness dampers. Sensors and Actuators A: Physical, 2017, 259: 14–33 https://doi.org/10.1016/j.sna.2017.03.026
49
W X Liu , E M Lui . Mathematical modeling and parametric study of magnetic negative stiffness dampers. Advances in Structural Engineering, 2020, 23(8): 1702–1714 https://doi.org/10.1177/1369433219900289
50
Z H Wang , Z P Cheng , G Z Yin , W A Shen . A magnetic negative stiffness eddy-current inertial mass damper for cable vibration mitigation. Mechanical Systems and Signal Processing, 2023, 188: 110013 https://doi.org/10.1016/j.ymssp.2022.110013
51
R Ravaud , G Lemarquand , V Lemarquand . Force and stiffness of passive magnetic bearings using permanent magnets. Part 1: axial magnetization. IEEE Transactions on Magnetics, 2009, 45(7): 2996–3002 https://doi.org/10.1109/TMAG.2009.2016088
52
R Ravaud , G Lemarquand , V Lemarquand . Force and stiffness of passive magnetic bearings using permanent magnets. Part 2: radial magnetization. IEEE Transactions on Magnetics, 2009, 45(9): 3334–3342 https://doi.org/10.1109/TMAG.2009.2025315
53
M F J Kremers , J J H Paulides , E Ilhan , J L G Janssen , E A Lomonova . Relative permeability in a 3D analytical surface charge model of permanent magnets. IEEE Transactions on Magnetics, 2013, 49(5): 2299–2302 https://doi.org/10.1109/TMAG.2013.2239976
54
A N Vučković , N B Raičević , S S Ilić , S R Aleksić . Interaction magnetic force calculation of radial passive magnetic bearing using magnetization charges and discretization technique. International Journal of Applied Electromagnetics and Mechanics, 2013, 43(4): 311–323 https://doi.org/10.3233/JAE-131703
55
A N Vučković , S S Ilić , S R Aleksić . Interaction magnetic force calculation of ring permanent magnets using Ampere’s microscopic surface currents and discretization technique. Electromagnetics, 2012, 32(2): 117–134 https://doi.org/10.1080/02726343.2012.645430
56
Q Yu , D F Xu , Y Zhu , G F Guan , Q Li . Damping characteristic modeling and numerical simulation analysis of the viscous damper of a clearance hydrocylinder. Journal of Vibration and Shock, 2020, 39(20): 161–167
57
X Shi , F L Zhao , Z D Yan , S Y Zhu , J Y Li . High-performance vibration isolation technique using passive negative stiffness and semiactive damping. Computer-Aided Civil and Infrastructure Engineering, 2021, 36(8): 1034–1055 https://doi.org/10.1111/mice.12681