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

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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.

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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

A :

Effective surface area of the piston motion

B r :

Residual flux density

\(B_{\rm{r}}^{\rm{c}},\,B_{\rm{r}}^{\rm{q}}\) :

Magnitudes of radial magnetic flux density generated by the equivalent current loop and magnetic charge loop, respectively

\(B_z^{\rm{c}},\,B_z^{\rm{q}}\) :

Magnitudes of axial magnetic flux density generated by the equivalent current loop and magnetic charge loop, respectively

B c, Bq :

Magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively

\({\boldsymbol{B}}_{\rm{r}}^{\rm{c}},\,{\boldsymbol{B}}_{\rm{r}}^{\rm{q}}\) :

Radial magnetic flux densities generated by the equivalent current loop and the magnetic charge loop, respectively

\({\boldsymbol{B}}_z^{\rm{c}},\,{\boldsymbol{B}}_z^{\rm{q}}\) :

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

\(F_{z{\rm{c}}}^{\rm{c}},\,\,F_{z{\rm{c}}}^{\rm{q}}\) :

Forces of the current loop acted by another current loop and another magnetic charge loop, respectively

\(F_{z{\rm{q}}}^{\rm{c}},\,\,F_{z{\rm{q}}}^{\rm{q}}\) :

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

H 0, H 3 :

Axial lengths of the moving magnets and the stator magnets on the top–bottom side, respectively

H 1 :

Axial length of the assemble moving magnet

H 2 :

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

n 0 :

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)

r 1, r 2 :

Inner and outer radii of the moving magnet, respectively

r 3, r 4 :

Inner and outer radii of the middle stator magnet, respectively

r 5, r 6 :

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

T 1, T 3 :

Thicknesses of all the moving magnets and the stator magnets on the top–bottom side, respectively

T 2 :

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

W 1, W 2 :

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

z 1, z 2 :

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

z 3, z 4 :

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

z 5, z 6 :

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

βh :

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

μ 0 :

Permeability of the vacuum

ε ideal :

Ideal stiffness counteraction ratio

θ :

Circumferential unit vector

ω :

Circular frequency

ω 0 :

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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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).

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Authors and Affiliations

  1. State Key Laboratory of Intelligent Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, China

    Wei Zhang, Jixing Che, Zhiwei Huang, Ruiqi Gao, Wei Jiang, Xuedong Chen & Jiulin Wu

  2. School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Wei Zhang, Jixing Che, Zhiwei Huang, Ruiqi Gao, Wei Jiang, Xuedong Chen & Jiulin Wu

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  1. Wei Zhang
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Zhang, W., Che, J., Huang, Z. et al. Novel modular quasi-zero stiffness vibration isolator with high linearity and integrated fluid damping. Front. Mech. Eng. 19, 5 (2024). https://doi.org/10.1007/s11465-023-0778-7

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