A novel high-static-low-dynamic stiffness isolator with enhanced negative stiffness magnetic spring using variable reluctance stress

Abstract

To isolate low frequency vibration, an enhanced negative stiffness magnetic spring using variable reluctance stress (ENS-MS-VRS) is proposed, which utilizes the designed double mover configuration and relocated nonworking air-gap. The ENS-MS-VRS is parallel connected with spiral flexure spring to construct a high-static-low-dynamic stiffness vibration isolator. The analytical model of the ENS-MS-VRS is initially established and subsequently validated by simulation analysis, then the negative stiffness enhancement mechanism of ENS-MS-VRS is theoretically elucidated by comparing it with the negative stiffness magnetic spring using variable reluctance stress (NS-MS-VRS). Furthermore, the results of experiment demonstrate that the isolator with ENS-MS-VRS exhibits a significantly wider isolation band than that with NS-MS-VRS.

Keywords

vibration isolation enhanced negative stiffness magnetic spring variable reluctance stress

Introduction

Vibrations at low frequencies have a significant impact on the performance of precision equipment in submarine and aerospace applications, thus, it is necessary to control the low frequency vibrations. However, the traditional linear passive isolator fails to simultaneously satisfy the requirement of low frequency vibration isolation and high load-capacity.¹ To address this issue, negative stiffness spring (NSS) such as buckled beam² and bio-inspired structure³ were proposed for low frequency vibration isolation. Magnetic negative stiffness mechanisms^4,5 also garnered much attention of scholars due to their advantages of frictionless and small size. Furthermore, researchers have been pursuing the NSSs with high negative stiffness due to their remarkable potential in enhancing both low frequency vibration isolation performance and load capacity of the isolator. Zhang⁶ proposed a negative stiffness magnetic spring using variable reluctance stress (NS-MS-VRS), and it exhibits significantly higher amplitude of negative stiffness compared to other magnetic NSSs utilizing magnetic attractive or repulsive force.

In this study, an enhanced negative stiffness magnetic spring based on variable reluctance stress (ENS-MS-VRS) is proposed, in which the double mover configuration (DMC) and relocated nonworking air-gap are employed to increase negative stiffness, and the high-static-low-dynamic stiffness isolator with ENS-MS-VRS is developed. A theoretical investigation is conducted on the ENS-MS-VRS, followed by experimental validation of the improvement of negative stiffness through a comparison between the isolator equipped with ENS-MS-VRS and that with NS-MS-VRS.

Theoretical investigation of the ENS-MS-VRS

Configuration of the isolator equipped with ENS-MS-VRS

The proposed isolator is mainly composed of a rod, three shells, two spiral flexure springs and the ENS-MS-VRS, as depicted in Figure 1(a). And the ENS-MS-VRS comprises three iron stators, two identical iron movers, two sets of permanent magnetic tiles and two aluminum grippers. In the ENS-MS-VRS, all magnetic tiles are magnetized radially and centripetally, hence a negative stiffness mechanism is formed based on variable reluctance stress. In contrast, the spiral flexure spring acts as a positive stiffness spring and support the isolated objective, which is fixed on the top of the rod. The ENS-MS-VRS can counteract the positive stiffness to reduce the dynamic stiffness of the isolator, thus the isolation performance for low frequency vibration is effectively enhanced.

Figure 1.

Scheme of the proposed isolator with ENS-MS-VRS: (a) half-section diagram of the isolator; (b) magnetic circuit of the ENS-MS-VRS.

Magnetic stiffness analysis of the ENS-MS-VRS

In the developed ENS-MS-VRS, the magnetic flux generated by magnetic tiles would initiate from the N pole of magnetic tiles, subsequently flow through the movers and working air-gap (WA), and then passes through stators and nonworking air-gap (NA), finally returns to the S pole to form close circuits, as shown in Figure 1(b). The movers are magnetized by the permanent magnetic tiles, resulting in interaction with the surrounding magnetic field. This interaction is referred to as variable reluctance stress, where the stress direction aligns with the flux direction. According to the formula for variable reluctance stress,⁶ the total axial magnetic force $F_{m}$ exerted on the upper mover and lower mover can be expressed as follows

F_{m} = (B_{u 1}^{2} - B_{d 1}^{2} + B_{u 2}^{2} - B_{d 2}^{2}) \cdot A_{0} / (2 μ_{0})

(1)

where

B_{u 1}

and

B_{d 1}

denote the magnetic flux density on the upper and lower surface of the upper mover, respectively,

B_{u 2}

and

B_{d 2}

separately represent the magnetic flux density on the upper and lower surface of the lower mover,

μ_{0}

is the permeability in the vacuum,

A_{0} = π (R_{a}^{2} - r_{a}^{2})

is the pole face area of the mover at each WA, where

R_{a}

and

r_{a}

refer to the outer radii and inner radii of the movers.

The diagram of the equivalent magnetic circuit of ENS-MS-VRS is illustrated in Figure 2(a). According to magnetic circuit analysis,⁶ the magnetic flux density $B_{u 1}$ , $B_{d 1}$ , $B_{u 2}$ and $B_{d 2}$ can be obtained as

B_{u i} = R_{d i} / (R_{u i} + R_{d i}) \cdot A_{m} / A_{0} \cdot B_{m i} (i = 1, 2)

(2)

B_{d i} = R_{u i} / (R_{u i} + R_{d i}) \cdot A_{m} / A_{0} \cdot B_{m i} (i = 1, 2)

(3)

where

B_{m 1}

and

B_{m 2}

refer to the magnetic flux density of the upper and lower magnetic tiles, respectively, the equivalent pole area of magnetic tiles is represented by

A_{m}

, which can be calculated as

π (R_{pm} + r_{pm}) t \cdot N θ / (360 \circ)

, where

R_{pm}

and

r_{pm}

separately represent the outer and inner radii of magnetic tiles, t and

θ

denote the thickness and angle of magnetic tiles, respectively, N corresponds to the number of magnetic tiles on each gripper;

R_{u 1}

and

R_{d 1}

respectively represent the magnetic reluctance of the upper and lower WA of upper mover; similarly,

R_{u 2}

and

R_{d 2}

denote the magnetic reluctance of the upper and lower WA of lower mover, respectively. It is worth noting that

R_{u 1} = R_{u 2} = (x_{0} - x) / (μ_{0} A_{0})

and

R_{d 1} = R_{d 2} = (x_{0} + x) / (μ_{0} A_{0})

, where

x_{0}

refers to the thickness of WA and x indicates the axial displacement of the movers. By substituting equation (2) and (3) into equation (1), the magnetic force

F_{m}

can be derived as follows

F_{m} = \frac{A_{m}^{2} B_{m}^{2}}{μ_{0} A_{0}} \frac{x}{x_{0}}

(4)

Figure 2.

Magnetic circuit and magnetic force of the ENS-MS-VRS: (a) scheme of the equivalent magnetic circuit; (b) magnetic force calculated by analytical model and FEM.

Then the magnetic stiffness, denoted as $K_{m}$ , is defined as the negative derivative of magnetic force $F_{m}$ with respect to displacement x, and it can be derived as $K_{m} = - (A_{m}^{2} B_{m}^{2}) / (μ_{0} A_{0} x_{0})$ . It is apparent that the magnetic stiffness $K_{m}$ exhibit a negative value, thereby indicating the ENS-MS-VRS as a negative stiffness spring.

Furthermore, in order to validate the accuracy of the theoretical model, the magnetic force of ENS-MS-VRS is simulated with COMSOL software and the obtained simulation results are compared with analytical results, as shown in Figure 2(b). The parameters of ENS-MS-VRS are listed in Table 1. Figure 2(b) depicts that the simulated magnetic force increases with displacement x, and when the movers are far from the equilibrium position, a strong nonlinear relationship between simulated magnetic force and displacement x occurs. Conversely, the analytical magnetic force shows a linear correlation with displacement x. However, the theoretical results exhibit good agreement with simulation results in close proximity to the equilibrium position, thus the theoretical model is applicable for micro-amplitude vibration. Additionally, under identical displacement x, reducing WA thickness $x_{0}$ can effectively increase magnetic force $F_{m}$ . As a result, the ENS-MS-VRS can generate high negative stiffness with small $x_{0}$ .

Table 1.

Parameters of the ENS-MS-VRS.

$r_{pm} (mm)$	$R_{pm} (mm)$	$r_{a} (mm)$	$R_{a} (mm)$	$t (mm)$	$B_{r} (T)$	$N$	$θ (\circ)$
6.5	10.0	2.0	6.5	3.5	1.35	6	42

The mechanism of negative stiffness enhancement

To demonstrate the enhanced negative stiffness of the proposed ENS-MS-VRS, it is compared with the NS-MS-VRS. The NS-MS-VRS is equipped with only a mover, and this configuration is called single mover configuration (SMC). The distinguishing feature of the ENS-MS-VRS is its utilization of double movers with a corresponding configuration referred to as DMC, as depicted in Figure 3(a) and (b). And this configuration refinement leads to a twofold increase in negative stiffness compared to SMC, as shown in Figure 3(c). What's more, the volume of DMC is merely 1.5 times that of SMC, thus DMC can generate higher negative stiffness than SMC when considering equal volumes.

Figure 3.

Comparison between DMC and SMC ( $x_{0} = 1.3 mm$ ): (a) half section representation of DMC; (b) half section depiction of SMC; (c) magnetic force of DMC and SMC.

On the other hand, the proposed ENS-MS-VRS positions the NA outside of the magnetic tiles, whereas in NS-MS-VRS, the NA is located inside of the magnetic tiles, as shown in Figure 4(a) and (b). This relocation of NA leads to an increase in the equivalent pole area of NA, thereby reducing the magnetic reluctance of NA. Consequently, the negative stiffness $K_{m}$ can be enhanced,⁷ as shown in Figure 4(c). Besides, regardless of whether it is before relocation or after relocation, the volume of magnetic spring remains constant, indicating that the relocation of NA can improve negative stiffness without change on the size of the structure.

Figure 4.

Relocation of nonworking air-gap ( $x_{0} = 1.3 mm$ , utilizing SMC configuration): (a) location of NA before relocation; (b) location of NA after relocation; (c) magnetic force before and after relocation.

Experimental validation of the ENS-MS-VRS

To verify the effectiveness of the ENS-MS-VRS and evaluate the isolation performance of the proposed isolator, the acceleration transmissibility of the isolators with ENS-MS-VRS, with NS-MS-VRS and without ENS-MS-VRS were examined. The set-up of experiment is presented in Figure 5(a). During the experimental test, a swept sine signal generated by the B&K Pulse analyzer and subsequently amplified by the power amplifier (B&K 2712) is transmitted to the vibration exciter (B&K 4808), for the purpose of inducing a swept sine base excitation on the isolator. The isolator is affixed to the surface of exciter, with a mass being attached to the upper end of the isolator rod. Two accelerometers (B&K 4507) are fixed on the surface of exciter and mass, respectively, for measuring their respective vibration acceleration. Afterwards, the two acceleration signals are recorded by the LAN-XI data acquisition hardware (B&K 3160). The acceleration transmissibility of the isolator can be determined by calculating the ratio between the fast Fourier transformation (FFT) of the mass's acceleration and that of the base.

Figure 5.

The experiment for acceleration transmissibility test: (a) the photograph of testing site; (b) the comparison of transmissibility between the isolators with ENS-MS-VRS, with NS-MS-VRS and without ENS-MS-VRS.

In all experiments, the mass of isolated objective m was determined to be $0.985 kg$ , and the parameters of ENS-MS-VRS can be found in Table 1, with $x_{0}$ being equal to 1.3 mm. The restoring force of the spiral flexure spring was simulated by FEM to determine the simulated positives stiffness $K_{S}$ of the isolator, yielding a value of 69,804 N/m. The measured transmissibility curves of various isolators are shown in Figure 5(b). Based on the data presented in Figure 5(b), it was observed that the isolator without ENS-MS-VRS had a measured natural frequency of 42.0 Hz, while the one with NS-MS-VRS exhibited a slightly lower value of 38.7 Hz. In contrast, the natural frequency of the isolator incorporating ENS-MS-VRS was significantly reduced to 28.7 Hz. Subsequently, the experimental negative stiffness of the NS-MS-VRS and the ENS-MS-VRS can be determined as −10,205 N/m and −36,453 N/m, respectively. The remarkable enhanced negative stiffness of the ENS-MS-VRS validates the effectiveness of the design of DMC and relocation of NA. Consequently, the isolator with ENS-MS-VRS exhibited a much wider isolation band and superior isolation performance compared to the isolator with NS-MS-VRS. Besides, as illustrated in Figure 5(b), the robust agreement between the experimental and theoretical results validates the correctness of theoretical model.

Conclusions

In this study, a novel high-static-low-dynamic stiffness vibration isolator with ENS-MS-VRS is proposed. Theoretical investigation and experimental result demonstrate that the designed double mover configuration, along with the relocation of nonworking air-gap, remarkably enhances the negative stiffness of the ENS-MS-VRS in comparison to its previous NS-MS-VRS counterparts. As a result, the dynamic stiffness of the isolator utilizing ENS-MS-VRS is effectively reduced, resulting in a significant expansion of the isolation band towards low frequency. Consequently, the designed isolator with ENS-MS-VRS can be used to isolate low frequency vibration.

Footnotes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11872290).

ORCID iDs

Wuhui Pan

Hongyu Xie

Feng Zhang

Juntao Ye

Ziqiang Geng

Chong Liu

Shilin Xie

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Yan

Zhao

, et al. A vari-stiffness nonlinear isolator with magnetic effects: theoretical modeling and experimental verification. Int J Mech Sci 2018; 148: 745–755.

Liu

Huang

Hua

. On the characteristics of a quasi-zero stiffness isolator using Euler buckled beam as negative stiffness corrector. J Sound Vib 2013; 332: 3359–3376.

Yan

Shi

, et al. Bionic paw-inspired structure for vibration isolation with novel nonlinear compensation mechanism. J Sound Vib 2022; 525: 116799.

Yan

Wang

, et al. Hybrid time-delayed feedforward and feedback control of lever-type quasi-zero-stiffness vibration isolators. IEEE Trans Ind Electron 2024; 71: 2810–2819.

Dong

Zhang

Xie

, et al. Simulated and experimental studies on a high-static-low-dynamic stiffness isolator using magnetic negative stiffness spring. Mech Syst Signal Process 2017; 86: 188–203.

Zhang

Shao

, et al. A new high-static-low-dynamic stiffness vibration isolator based on magnetic negative stiffness mechanism employing variable reluctance stress. J Sound Vib 2020; 476: 115322. doi: https://doi.org/10.1016/j.jsv.2020.115322

Yuan

Sun

Wang

, et al. Tunable negative stiffness spring using Maxwell normal stress. Int J Mech Sci 2021; 193: 106127. doi: https://doi.org/10.1016/j.ijmecsci.2020.106127