Novel modular quasi-zero stiffness vibration isolator with high linearity and integrated fluid damping

doi:10.1007/s11465-023-0778-7

Front. Mech. Eng.

2024, Vol. 19

Issue (1) : 5 https://doi.org/10.1007/s11465-023-0778-7

Novel modular quasi-zero stiffness vibration isolator with high linearity and integrated fluid damping

Wei ZHANG^1,², Jixing CHE^1,², Zhiwei HUANG^1,², Ruiqi GAO^1,², Wei JIANG^1,², Xuedong CHEN^1,², Jiulin WU^1,²(

)

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

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Abstract

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.

Keywords vibration isolation quasi-zero stiffness damping magnetic spring integrated design

Corresponding Author(s): Jiulin WU

Just Accepted Date: 30 November 2023 Issue Date: 05 March 2024

Cite this article:

Wei ZHANG,Jixing CHE,Zhiwei HUANG, et al. Novel modular quasi-zero stiffness vibration isolator with high linearity and integrated fluid damping[J]. Front. Mech. Eng., 2024, 19(1): 5.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-023-0778-7
https://academic.hep.com.cn/fme/EN/Y2024/V19/I1/5

Fig.1 Schematic of the proposed modular quasi-zero stiffness vibration isolator (MQZS-VI). 1?Stator magnet, 2?Moving magnet, 3?Cylinder block, 4?Positioning bushing, 5?Adjustable base, 6?Annular sleeve, 7?End cover, 8?Metal spring, 9?Piston rod.

Fig.2 Configuration of the modular quasi-zero stiffness vibration isolator: distribution of geometric parameters (a) that affect negative stiffness and (b) that affect damping coefficient.

Fig.3 Magnetization diagram of the novel combined magnetic negative stiffness mechanism.

Fig.4 Discretizing model of the novel combined magnetic negative stiffness mechanism.

Fig.5 Analytical and finite element analysis (FEA) solution of the novel combined magnetic negative stiffness mechanism: (a) FEA model and (b) comparison of magnetic force and stiffness. Residual magnetic flux density B_r = 1.4 T, T₁ = T₃ = 7 mm, λ = 2 mm, T₂ = 2 mm, L = 7 mm, H₁ = 26 mm, H₂ = 20 mm, H₀ = H₃ = 5 mm, ι = 2 mm, and r_i = 5 mm.

Fig.6 Subsystems of the novel combined magnetic negative stiffness mechanism: configurations of (a) Group-1 and (b) Group-2; (c) stiffness characteristics of the subsystems. B_r = 1.4 T, T₁ = T₃ = 7 mm, λ = 2 mm, T₂ = 2 mm, L = 7 mm, H₁ = 26 mm, H₂ = 20 mm, H₀ = H₃ = 5 mm, ι = 2 mm, and r_i = 5 mm.

Tab.1 Initial value and varying range of geometric parameters

Fig.7 Effect of α_h on the stiffness of the combined magnetic negative stiffness mechanism: (a) 3D view and (b) 2D view.

Fig.8 Effect of β_h on the stiffness of the combined magnetic negative stiffness mechanism: (a) 3D view and (b) 2D view.

Fig.9 Effect of λ on the stiffness of the combined magnetic negative stiffness mechanism: (a) 3D view and (b) 2D view.

Fig.10 Effect of α_v on the stiffness of the combined magnetic negative stiffness mechanism: (a) 3D view and (b) 2D view.

Fig.11 Effect of β_v on the stiffness of the combined magnetic negative stiffness mechanism: (a) 3D view and (b) 2D view.

Fig.12 Effect of L on the stiffness of the combined magnetic negative stiffness mechanism: (a) 3D view and (b) 2D view.

Fig.13 Effect of ι on the stiffness of the combined magnetic negative stiffness mechanism: (a) 3D view and (b) 2D view.

Fig.14 Effect of geometric parameters and viscosity on damping coefficient: damping coefficients versus (a) δ and γ and (b) R_p and H_p.

Fig.15 Design flowchart for the modular quasi-zero stiffness vibration isolator.

Tab.2 Optimized parameters of CMNSMs

Fig.16 Comparison of the extent of nonlinearity under the same value of negative stiffness at the equilibrium position: (a) stiffness characteristic and (b) nonlinearity. CMNSM: combined magnetic negative stiffness mechanism.

Fig.17 Dynamic model of quasi-zero stiffness vibration isolation system.

Tab.3 Coefficients of the seventh-order polynomial

Fig.18 Characteristics of quasi-zero stiffness vibration isolation system with the proposed modular quasi-zero stiffness vibration isolator: (a) approximate force?displacement and stiffness–displacement curves and (b) transmissibility with different disturbance displacement.

Tab.4 Design parameters of the prototype

Fig.19 Experimental system of static characteristics. MQZS-VI: modular quasi-zero stiffness vibration isolator.

Fig.20 Results of the static experiments with different preload: (a) force–displacement curve and (b) fitted stiffness–displacement curves by 5th-order polynomial. MQZS-VI: modular quasi-zero stiffness vibration isolator.

Fig.21 Validation of the analytical model: (a) axial magnetic force and stiffness of the combined magnetic negative stiffness mechanism versus the displacement and (b) experimental error.

Fig.22 Experimental system of dynamic characteristics: (a) schematic diagram and (b) photo. 1?Velocity sensor, 2?Modular quasi-zero stiffness vibration isolator prototype, 3?Exciter, 4?DC power, 5?Dynamic signal analyzer, 6?Controller, 7?Power amplifier.

Fig.23 Results of the dynamic experiments: (a) response of velocity and (b) transmissibility. CMNSM: combined magnetic negative stiffness mechanism.

Tab.5 Vibration response in the time domain

Tab.6 Transmissibility in the frequency domain

Abbreviations
CMNSM	Combined magnetic negative stiffness mechanism
DOF	Degree-of-freedom
FEA	Finite element analysis
MNSD	Magnetic negative stiffness damper
MNSM	Magnetic negative stiffness mechanism
MQZS-VI	Modular quasi-zero stiffness vibration isolator
QZS	Quasi-zero stiffness
RMS	Root mean square
Variables
A	Effective surface area of the piston motion
B_r	Residual flux density
$B r c$ , $B r q$	Magnitudes of radial magnetic flux density generated by the equivalent current loop and magnetic charge loop, respectively
$B z c$ , $B z q$	Magnitudes of axial magnetic flux density generated by the equivalent current loop and magnetic charge loop, respectively
$B c$ , $B q$	Magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively
$B r c$ , $B r q$	Radial magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively
$B z c$ , $B z q$	Axial magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively
c	Damping coefficient
c_ideal	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
F_v	Damping force
F_z	Axial force
F_z^* (* = u, m, l)	Axial force of moving magnets suffered from the top stator, the middle stator, and the lower stator respectively
$F z c c$ , $F z c q$	Forces of the current loop acted by another current loop and another magnetic charge loop, respectively
$F z q c$ , $F z q q$	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
H_c	Axial height of design space
H_p	Height of the piston head
I_ib	Current of the inner current loop of the bottom moving magnet (i = 1) or the bottom stator magnet (i = 3)
I_iu	Current of the inner current loop of the upper moving magnet (i = 1) or the upper stator magnet (i = 3)
I_jb	Current of the outer current loop of the bottom moving magnet (j = 2) or the bottom stator magnet (j = 4)
I_ju	Current of the outer current loop of the upper moving magnet (j = 2) or the bottom stator magnet (j = 4)
I_m, I_s	Surface currents of the equivalent current loop of the moving magnet and the stator magnet with radiative magnetization, respectively
J_l (l = 1,2,…,4)	Surface density of the equivalent ampere’s currents of the magnets with axial magnetization
k_e	Stiffness of the vibration isolation system at equilibrium position
k_i (i = 0,1,…,3)	Fitted coefficients of the nonlinear axail force
k_n	Negative stiffness
k_n0	Negative stiffness at the equilibrium position
k_n(z)	Negative stiffness when the axial displacement is z
k_p	Positive stiffness
k_z	Total stiffness
L	Axial gap between the moving magnets and stator magnets on the top?bottom side
m	Mass of the payload
M_i (i = 1,2,…,4)	Magnitude of magnetization of the magnet
M_i (i = 1,2,…,4)	Magnetization of the magnet
n₀	Effective turn number of the metal spring
N_b (b = c, v, s)	Number of segments for fictitious current loops, volume magnetic charge loops, and surface magnetic charge loops, respectively
n_k (k = 1,2,…,8)	Unit vector normal to the surface of the ring magnets
P	Attenuation rate of vibration
Δp	Pressure difference
Q_m, Q_s	Magnetic charges of the equivalent magnetic charge loop of the moving magnet and the stator magnet with axial magnetization, respectively
Q_si	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)
Q_sj	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)
Q_vk	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
r_i	Inner radius of all the magnets apart from the middle stator magnet
r_m	Radial coordinate of micro units of equivalent loops of the moving magnet
r_s	Radial coordinate of micro units of equivalent loops of the stator magnet
r_vk, r_vp	Radii of the equivalent volume magnetic charge loops of the middle stator magnet and the middle moving magnet, respectively
R_c	Radius of design space
R_p	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
T_d	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
W_v	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
z_1u, z_2t	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
z_3n, z_4i	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
z_5q, z_6w	Axial coordinates of the current loop of the lower moving magnet and the lower stator magnet, respectively
z_b, z_p	Displacements of the base excitation and the payload platform, respectively
z_m	Axial coordinate of micro units of equivalent loops of the moving magnet
z_p (p = 1,2,…,6)	Axial coordinate of the lower plane of each magnet
z_s	Axial coordinate of micro units of equivalent loops of the stator magnet
z_vj, z_vm	Axial coordinates of the volume magnetic charge loop of the middle stator magnet and the middle moving magnet
Z_b	Magnitude of the base displacement
z	Axial unit vector
α_h	Ratio of H₁ to H₂
α_v	Ratio of H₀ to H₃
β_h	Ratio of T₁ to T₂
β_v	Ratio of T₁ to T₃
η	Extent of stiffness nonlinearity
η_ideal	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
ε_ideal	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

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