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

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.

Key words： vibration isolation; quasi-zero stiffness; damping; magnetic spring; integrated design

Nomenclature

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
Br	Residual flux density
B_{\text{r}}^{\text{c}}, B_{\text{r}}^{\text{q}}	Magnitudes of radial magnetic flux density generated by the equivalent current loop and magnetic charge loop, respectively
B_{\mathit{\text{z}}}^{\text{c}}, B_{\mathit{\text{z}}}^{\text{q}}	Magnitudes of axial magnetic flux density generated by the equivalent current loop and magnetic charge loop, respectively
{\mathit{B}}^{\text{c}}, {\mathit{B}}^{\text{q}}	Magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively
{\mathit{B}}_{\text{r}}^{\text{c}}, {\mathit{B}}_{\text{r}}^{\text{q}}	Radial magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively
{\mathit{B}}_{\mathit{\text{z}}}^{\text{c}}, {\mathit{B}}_{\mathit{\text{z}}}^{\text{q}}	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
F_{\mathit{\text{z}}\text{c}}^{\text{c}}, F_{\mathit{\text{z}}\text{c}}^{\text{q}}	Forces of the current loop acted by another current loop and another magnetic charge loop, respectively
F_{\mathit{\text{z}}\text{q}}^{\text{c}}, F_{\mathit{\text{z}}\text{q}}^{\text{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
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

Acknowledgements

This work was supported by the National Key R&D Program of China (Grant Nos. 2020YFB2007300 and 2020YFB2007601), the National Natural Science Foundation of China (Grant Nos. 52075193, 52305107, and 52275112), the National Science and Technology Major Project of China (Grant No. 2017ZX02101007-002), and the Postdoctoral Science Foundation of China (Grant No. 2022M711250).

Electronic Supplementary Material

The supplementary material can be found in the online version of this article at https://doi.org/10.1007/s11465-023-0778-7 and is accessible to authorized users.

Conflict of Interest

The authors declare that they have no conflict of interest.

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