Design and experimental investigation of ultra-low frequency vibration isolation during neonatal transport

Mechanical Systems and Signal Processing 139 (2020) 106633

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Design and experimental investigation of ultra-low frequency vibration isolation during neonatal transport Qiang Wang a,b, Jiaxi Zhou a,b,⇑, Daolin Xu a,b, Huajiang Ouyang c a

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Changsha 410082, PR China College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, PR China c School of Engineering, University of Liverpool, Liverpool L69 3GH, UK b

a r t i c l e

i n f o

Article history: Received 21 August 2019 Received in revised form 28 November 2019 Accepted 8 January 2020

Keywords: Vibration isolation Quasi zero stiffness Incubator Neonatal transport

a b s t r a c t To resolve the tough issue of ultra-low frequency vibration isolation during neonatal transport, a new incubator with quasi-zero-stiffness (QZS) isolators is proposed. In the new incubator, the infant housing unit is supported by the QZS isolators to mitigate mechanical vibration being transmitted from the ambulance floor to the infant. Each of the QZS isolators is devised by combining a pair of permanent magnet rings and a coil spring, and the design parameters are realized by the equivalent magnetic charge method. Both the prototypes of the QZS isolator and incubator are manufactured, and their vibration isolation performances are tested. The results show that the QZS isolator can fulfil vibration isolation from 2 Hz, and the incubator with four such QZS isolators is effective from 3.2 Hz under the designated payload. Most importantly, the acceleration response of the infant model in the housing unit is much lower than the base excitation, when the incubator is driven by a random base excitation with frequency bandwidth from 2 Hz to 15 Hz. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction Neonatal transport is used to move premature or sick infants to medical facility for intensive care [1]. During the journey by road ambulance, the infant usually is subjected to mechanical vibration transmitted from the vehicle body of the ambulance, which mainly ranges from 3 Hz to 18 Hz [2]. It was reported that a human body is sensitive to vibrations from 1 Hz to 80 Hz [3], and vibration in low-frequency range from 1 Hz to 20 Hz has the greatest impact on a human body [4]. The lowfrequency mechanical vibration during neonatal transport can induce irreversible injury for the newborn baby whose physiological structure is not yet fully developed [5]. However, low-frequency vibration isolation is still a tough issue to be resolved, and there are rare studies on vibration isolation for neonatal transport. This situation necessitates the design and development of vibration-isolation devices to reduce mechanical vibrations and protect infants during neonatal transport. In the past few years, a few studies on vibration reduction during neonatal transport have been reported. Bailey-Van Kuren et al. [6] proposed semi-active air springs to achieve vibration isolation for a neonatal transport unit, showed a strategy that could provide better care quality. However, their active vibration isolation system was somewhat complicated and energy-consuming. Shukla et al. [7] devised a neonatal transport cart with a magnetorheological active suspension system, and numerical results showed that the cart could reduce shock and vibration transmitted to a neonate. Beale et al.[8] ⇑ Corresponding author at: College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, PR China. E-mail address: [email protected] (J. Zhou). https://doi.org/10.1016/j.ymssp.2020.106633 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.

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Q. Wang et al. / Mechanical Systems and Signal Processing 139 (2020) 106633

designed an active vibration control system to reduce acceleration received by infants in incubators. Sabota and Aghili [9] installed a passive vibration isolator consisting of a linear spring and a damper between the tray and the rigid frame of an incubator. Unfortunately, the vibration isolation performances of these two kinds of vibration isolators were not given in the patent applications [8,9]. Sherwood et al. [10] tried to improve comfort of an infant by using different types of mattresses. Gajendragadkar et al. [11] also conducted a study on finding out which kind of mattress would be the best for vibration reduction through experiments, which showed that gel-filled mattress was the most effective one. However, the acceleration of the infant was always higher than the disturbance from the floor of the ambulance, no matter which kind of mattress was placed under the infant. This implied that the mattress was unable to isolate the low-frequency vibrations. As well known, a linear vibration isolator can perform its function only when the external excitation frequency is greater pffiffiffi than 2xn , where xn is the natural frequency of the vibration isolation system. To isolate low-frequency vibration, the stiffness of the isolator should be selected as low as possible. However, this would induce large static deformation and even failure of the vibration isolation system. Fortunately, a kind of nonlinear isolator with high static stiffness and low dynamic stiffness is an ideal way for low-frequency vibration isolation [12]. The dynamic stiffness at the static equilibrium even can be designated to be zero by connecting a negative-stiffness (NS) mechanism in parallel with a positive-stiffness element, and then a quasi-zero-stiffness (QZS) isolator is fulfilled [13]. There are many ways to achieve a QZS isolator, and the main difference among them is how to design the NS mechanism, such as inclined springs[14–17], a horizontal spring connecting with inclined link [18,19], bistable structures resulted from buckling [20–22], X-shaped structures [23–25], cam-rollerspring mechanisms[26,27], bionic structures[28–30] and magnetic mechanisms [31–34]. Along these NS mechanisms, the magnetic one is the most compact for fulfilling negative stiffness. Carrella et al. [35] proposed a QZS isolator consisting of mutually attracting magnets and coil springs to verify the function of low-frequency vibration isolation through experiments. Xu et al. [18] used magnetic springs and inclined links to realize negative stiffness. Wu et al. [34] designed an NS mechanism by using three mutually repelling cuboidal magnets. Dong et al. [32] constructed a QZS isolator by combining a magnetic negative stiffness spring with a spiral flexural spring in parallel. Zheng et al. [31] utilized a pair of mutually repelling permanent magnets rings to achieve negative stiffness. Shan et al. [36] proposed a compact QZS isolator by connecting two magnetic rings in parallel with a pneumatic spring. Yan et al [37] proposed a bistable nonlinear vibration isolator consisting of a linear spring and permanent magnets, and they also utilized nonlinear electromagnetic shunt damping to enhance the vibration isolation performance of the nonlinear isolator [38]. To deal with payload changes, Zhou et al. [39] designed a QZS isolator by combining a mechanical spring in parallel with an electromagnet, and the negative stiffness could be tuned by adjusting the current of the electromagnet. All the above theoretical and experimental investigations indicated that the magnetic mechanism is a compact and effective way to realize negative stiffness. Consequently, a conceptual design of a compact QZS isolator with magnetic NS mechanisms was proposed in the authors’ previous work [40] to isolate mechanical vibrations for neonatal transport. Although the authors have given a conceptual design [40] of the QZS isolators for neonatal transport, an experimental investigation on the prototype incubator with QZS isolators for ultra-low frequency vibration isolation during neonatal transport was never reported. In this paper, a new incubator with QZS isolators is designed and fabricated to isolate the ultra-low frequency mechanical vibration. This paper presents the authors’ further research work on top of their previous work [40], and mainly focuses on design, manufacture and experimental test of a real prototype incubator with QZS isolators. The contributions of this paper can be summarized in three aspects. First, the static characteristics of the pair of mutually repelling magnet rings (NS mechanism) is revealed by the equivalent magnetic charge method and verified by a finite element analysis, which is much more accurate than the approximate method in the authors’ previous work [40]. Second, the prototype of the QZS isolator is fabricated, which is compact and tunable for payload changes. Finally, and most importantly, the prototype incubator with QZS isolators is tested under both sinusoidal and random excitations to validate its function of ultra-low frequency vibration isolation. This paper is organized as follows. In Section 2, the design scheme of the incubator with QZS isolators is demonstrated. In Section 3, theoretical analysis and finite element simulation of the QZS isolator are conducted to reveal its stiffness characteristics. In Section 4, both the prototypes of the QZS isolator and the incubator are fabricated, and experimental tests are carried out to evaluate the vibration isolation performances. Some conclusions are summarized in Section 5.

2. Design of the incubator with QZS isolators The QZS incubator consists of a box for auxiliary equipment (a1), an infant house unit (a2), and four QZS isolators (b1), as shown in Fig. 1. The QZS isolators separate the house unit from the auxiliary equipment, to prevent mechanical vibrations from both the ambulance vehicle body and the auxiliary equipment. The QZS isolator is constructed by combining a coil spring (c2) with a pair of mutually repelling permanent magnet rings (c4 and c7) in parallel, as shown in Fig. 1c. The inner magnet ring (c7) is fixed onto the smooth shaft (c9) by Nylon nuts (c6), and the outer magnet ring (c4) is fixed by the upper and lower casings (c8 and c5). The lower end of the smooth shaft is supported by the coil spring, while the upper end is connected to the housing unit. Additionally, the shaft is also guided by two linear bearings (c3) to ensure that it moves only along the axial direction. At the static equilibrium position, the isolator is compressed by the payload, the centre of the inner magnet ring aligns with that of the outer one, and the payload is totally supported by the coil spring. However, when the inner magnet ring

Q. Wang et al. / Mechanical Systems and Signal Processing 139 (2020) 106633

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Fig. 1. (a) Conceptual design of the incubator with QZS isolators for neonatal transport: a1- box for auxiliary equipment, a2- infant housing unit; (b) Connecting between the housing unit and isolators: b1- QZS isolator, b2- connecting screw; (c) QZS isolator: c1- Adjusting screw nut, c2– coil spring, c3– liner bearings, c4- outer magnet ring, c5- lower casing, c6- nylon nut, c7- inner magnet ring, c8- upper casing, c9- smooth shaft.

moves away from the static equilibrium, the pair of mutually repelling magnet rings acts as a negative-stiffness (NS) mechanism along the axial direction, and thus the stiffness of the coil spring can be substantially counteracted by the NS mechanism. To adapt to any payload change, a screw nut (c1) is installed at the end of the QZS isolator. It is used to adjust the position of the inner magnet ring, and thus to ensure the inner magnet ring aligns with the outer one at the static equilibrium position.

3. Design of the QZS isolator Firstly, the parameter design of the pair of magnet rings (NS mechanism) is conducted. The magnetic force of the NS mechanism is calculated by using the equivalent magnetic charge method and verified by finite element analysis, which is much more accurate than the approximate method in the authors’ previous work using empirical formula [40]. And then, the quasi-zero stiffness is fulfilled by combining the NS mechanism and the coil spring.

Fig. 2. Mutually repelling magnet rings for negative stiffness: (a) Cutaway view, (b) Front view.

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3.1. Theoretical analysis of the NS mechanism The schematic diagram of the pair of magnet rings (NS mechanism) is shown in Fig. 2. The inner ring is allowed to move away from the static equilibrium position but the outer ring remains fixed. Note that the centres of both magnet rings lie on the axis z of the smooth shaft, and both magnet rings have the same magnetization directions along the axial direction. Therefore, the mutually repelling magnet rings act as axial bearings with negative stiffness. The geometrical parameters of the pair of magnet rings are listed as follows. The inner ring: inner radius R1, outer radius R2, height l1, thickness t1. The outer ring: inner radius R3, outer radius R4, height l2, thickness t2. The air gap between the inner and outer ring is R3- R2. According to the equivalent magnetic charge method [41], the magnet ring can be represented by two charged planes, namely the upper and lower faces of the ring. Taking face 1 and face 3 as example, the magnetic charge qP at point P on face 1 of the inner ring and the magnetic charge qQ at point Q on face 3 of the outer ring can be given by

qP ¼ r1 r 1 dadr 1

ð1Þ

qQ ¼ r3 r3 dbdr3

ð2Þ

where r 1 and a are polar coordinates of the point on face 1, r 3 and b are polar coordinates of the point on face 3, as shown in Fig. 2b, and r1 and r3 are surface densities of magnetic charges on face 1 and face 3, respectively. According to electromagnetic theory [42], the magnetic field strength HP at point Q generated by the magnetic charge qP at point P is given by

HP ¼

1 qP r13 4pl0 r 313

ð3Þ

where l0 is the permeability of vacuum, r13 is the distance between point P and point Q, and r13 is the vector between point P and point Q. Therefore, the magnetic force between point Q and point P can be given by

dF13 ¼ qQ HP ¼

r1 r3 r1 r3 dr1 dr3 dadb r13 4pl0 r 313

ð4Þ

For the permanent magnetic materials, the surface density of magnetic charge is equal to the intensity of residual magnetic induction [43]

r1 ¼ Br1 ; r3 ¼ Br3

ð5Þ

where Br1 and Br3 are the residual magnetic induction intensities of the outer and inner magnet rings, respectively. Then, Eq. (4) can be rewritten as

dF13 ¼

Br 1 Br3 r 1 r 3 dr 1 dr 3 dadb r13 4pl0 r313

ð6Þ

The component in the z-axis direction dF 13z of the magnetic force dF13 is yielded

dF 13z ¼ dF13  i ¼

Br 1 Br3 r 1 r 3 dr 1 dr 3 dadb z0 4pl0 r313

ð7Þ

where i is the unit vector in the z axis. According to geometric relationships in Fig. 2, the distance from point P to point Q is

r13 ¼ jr13 j ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z20 þ ðr 1 sina  r 3 sinbÞ þ ðr 1 cosa  r 3 cosbÞ2

ð8Þ

Due to the geometric symmetry, the magnetic force dF 24z produced by the magnetic charges at any two points on face 2 and face 4, respectively, is equal to dF 13z

dF 24z ¼ dF 13z ¼

Br 1 Br3 r 1 r 3 dr 1 dr 3 dadb z0 4pl0 r313

ð9Þ

Similarly, the z-axial magnetic forces generated by the magnetic charges on face 2 and face 3, dF 23z , and by the magnetic charges on face 1 and face 4, dF 14z , can be given by

dF 23z ¼ dF23  i ¼

Br 2 Br3 r 2 r 4 dr 2 dr 3 dadb ðl2 þ z0 Þ 4pl0 r323

ð10Þ

dF 14z ¼ dF14  i ¼

Br 1 Br4 r 1 r 4 dr 1 dr 4 dadb ðl1  z0 Þ 4pl0 r314

ð11Þ

where

r23 ¼ jr23 j ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðl2 þ z0 Þ þ ðr 1 sina  r 3 sinbÞ þ ðr 1 cosa  r 3 cosbÞ2

ð12Þ

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r14 ¼ jr14 j ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðl1  z0 Þ þ ðr 1 sina  r 3 sinbÞ þ ðr 1 cosa  r 3 cosbÞ2

ð13Þ

As mentioned before, the outer ring is fixed, while the inner ring moves along the z-axis direction. When the inner ring deviates from the static equilibrium position O by a distance z, the inner ring is subjected by three forces dF 13z , dF 24z , dF 14z along positive z-axis direction and one attracting force dF 23z along the minus z-axis direction. Therefore, the differential form of the resultant magnetic force in z-axis direction dF z should be the sum of these four forces,

dF z ¼ dF 13z þ dF 24z þ dF 14z  dF 23z ¼

! Br 1 Br3 z0 z0 l1  z0 l2 þ z0 þ þ  r 1 r3 dr 1 dr3 dadb 4pl0 jr13 j3 jr24 j3 jr14 j3 jr23 j3

ð14Þ

Substituting z0 ¼ z þ ðl1  l2 Þ=2 into Eq. (14), the above expression can be rewritten as

dF z ¼

  Br 1 Br3 2z þ l1  l2 2z þ l1  l2 l1 þ l2  2z l1 þ l2 þ 2z r 1 r 3 dr1 dr 3 dadb þ þ  4pl0 2r 314 2r 323 2r 313 2r324

ð15Þ

And then, the axial resultant magnetic force F z can be obtained by integrating dF z over the whole ring face

Fz ¼

Br1 Br3 4pl0

Z 2p Z 2p Z 0

0

R2 R1

Z

R4

R3

  2z þ l1  l2 2z þ l1  l2 l1 þ l2  2z l1 þ l2 þ 2z r 1 r 3 dr 1 dr 3 dadb þ þ  2r 314 2r 323 2r313 2r 324

ð16Þ

It should be noted that, when an external force fz is applied on the inner ring along z-axial direction, one can derive that fz is equal to -Fz. Therefore, the restoring force of this magnetic bearing is

f z ¼ F z

ð17Þ

By differentiating the force fz with respect to the displacement z, the stiffness of the pair of magnet rings (NS mechanism) can be obtained

kN ¼

df z dF z ¼ dz dz

ð18Þ

3.2. Finite element analysis of the NS mechanism In order to validate the analytical expressions of z-axial magnetic force and stiffness, a finite element analysis of the magnetic field produced by the magnet rings is carried out by using ANSYS Maxwell. The parameters of the permanent magnets are listed in Table 1, where Br is the residual flux density and lr is the relative permeability. A 2D axisymmetric model of the magnet rings is built, and the distribution of magnetic flux lines can be calculated, as shown in Fig. 3. The restoring force fz and stiffness kN of the pair of mutually repelling magnet rings is depicted with respect to the displacement z, as shown in Fig. 4. The analytical restoring force and stiffness are calculated by using Eqs. (17) and (18), respectively, which are denoted by solid lines. The results obtained by the finite element method (FEM) are represented by square dots. It can be seen that the analytical predictions match well with the finite element analysis, especially in the displacement range close to the static equilibrium position. Therefore, the analytical expressions Eqs. (17) and (18) can be used to calculate the restoring force and stiffness of the magnet rings accurately. Moreover, as the inner ring moves from 0.04 m to 0.04 m, the restoring force firstly increases towards a peak value, and then suddenly falls to a valley value, which is a typical snap through phenomenon. Correspondingly, negative stiffness occurs in a large displacement range, as shown in Fig. 4b, which confirms the idea of using a pair of mutually repelling magnet rings to realize the negative stiffness along the axial direction. 3.3. Parametric analysis of the NS mechanism According to Ref. [44], the main parameters affecting the axial magnetic force and stiffness of the NS mechanism are the height and radial thickness of the magnet ring, and the air gap between two magnet rings. The influences of these parameters on the stiffness of the NS mechanism are revealed by using the FEM, when the geometrical parameters of the inner ring and the intensity of residual magnetic induction of the permanent magnet are kept unchanged, as listed in Table 2. The effects of the air gap between two magnet rings on the restoring force and stiffness are demonstrated in Fig. 5. Obviously, with the reduction in air gap, the peak and valley values of the restoring force increase, and the snap-through

Table 1 Physical parameters of NS mechanism simulation. Parameter

Br (T)

lr

R1(m)

R2(m)

R3(m)

R4(m)

l1(m)

l2(m)

Value

1.298

1

0.0150

0.0394

0.0454

0.0810

0.034

0.034

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Q. Wang et al. / Mechanical Systems and Signal Processing 139 (2020) 106633

Fig. 3. (a) Axisymmetric finite element model of the magnet rings for calculating the magnetic force, and (b) distribution of magnetic flux lines.

Fig.4. (a) Restoring force and (b) stiffness of the magnet rings (NS mechanism) obtained by analytical method and finite element method.

Table 2 Unchanged parameters during the parametric analysis. Parameter

Br (T)

lr

R1(m)

R2(m)

l1(m)

Value

1.35

1

0.01

0.02

0.01

instability becomes stronger. Accordingly, the absolute value of negative stiffness in the vicinity of static equilibrium increases steeply, however the displacement range of negative stiffness decreases. Fig. 6 shows the influence of the thickness of the outer ring on the restoring force and stiffness. It can be seen that, with increase in the thickness, the absolute value of the negative stiffness increases, but the growth rate becomes slow, and the displacement range of negative stiffness is widened slightly. The effects of the height of the outer ring on the restoring force and stiffness can be seen from Fig. 7a and b, respectively, which are similar to those of the thickness. The negative stiffness (absolute value) is increased and the displacement range of negative stiffness is slightly widened, with increase in the height. In practice, the height of the inner ring is always selected to be the same as the outer ring, and the influence of the height is shown in Fig. 7c and d. One can see both the negative stiffness (absolute value) and the displacement range of negative stiffness increase with the height. In summary, the height, thickness and air gap of the magnet rings have a significant impact on the restoring force and stiffness of the NS mechanism. From the perspective of engineering application, the QZS isolator should be compact and easy to install. Therefore, the thickness and air gap should be as small as possible, and thus adjusting the height of the magnet rings is a proper way to design the NS mechanism.

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Fig. 5. Influence of air gap on (a) restoring force (b) stiffness.

Fig. 6. Influence of thickness on (a) restoring force and (b) stiffness.

3.4. Parametric design of the QZS isolator As shown in Fig. 1, the infant housing unit is supported by four QZS isolators evenly, and thus the payload for each isolator is a quarter of the total weight of the housing unit, including the weight of the infant. For the experimental prototype of the incubator, the maximum allowable static deflection is selected as about 20 mm and the payload supported by each isolator is 

about 35 N, and then the stiffness of the coil spring can be determined as ks ¼ 1:7N=mm. However, due to possible manufacturing errors, the actual stiffness of the coil spring is selected as ks ¼ 1:65N=mm. In order to achieve quasi-zero stiffness at the static equilibrium position, the coil spring is connected with the pair of magnet rings (NS mechanism) in parallel, as shown in Fig. 1c. The parametric design of the NS mechanism is conducted to obtain a negative stiffness close to the stiffness of the coil spring, and the results are listed in Table 3. The restoring force and stiffness of the QZS isolators are represented as the triangle dotted lines in Fig. 8. Obviously, in the vicinity of the static equilibrium, the restoring face changes slightly against displacement (Fig. 8a), which implies very low stiffness, and the stiffness even tends towards zero at the static equilibrium (Fig. 8b). Additionally, the restoring force at the static equilibrium is equal to the payload 33 N, and in such a case, the payload is totally supported by the coil spring. Therefore, it is an ideal stiffness feature for low-frequency vibration isolation, due to the capability of supporting payload with any weight and ultra-low stiffness during oscillation around the static equilibrium.

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Fig. 7. Influence of height on (a) restoring force (b) stiffness.

Table 3 Parameters of the magnet rings for the QZS isolator. Parameter Value

Br(T)

l0(T)

1.4

4p  10

7

R1(m)

R2(m)

R3(m)

R4(m)

l1(m)

l2(m)

kN(N/mm)

0.010

0.016

0.03

0.04

0.005

0.005

1.63

More importantly, the other advantage of this QZS isolator is that the position of the coil spring can be changed by the adjusting screw nut (Fig. 1c) to handle payloads different from the designated value. For example, when the payload is heavier than the designated value, the coil spring is raised to ensure the center of the inner magnet ring aligns with that of the outer magnet ring. On the other hand, the coil spring is lowered when the payload is lighter than the designated value. The restoring force and stiffness are depicted as square dotted lines and circular dotted lines in Fig. 8 for the cases of payload deviation. It can be seen that the restoring force curve just translates along the longitudinal coordinate for different payload deviations, while the stiffness is independent of the payload deviation.

3.5. Dynamics of the QZS vibration isolation system Considering the payload m supported by the QZS isolator, the QZS vibration isolation system can be simplified as a singledegree-of-freedom spring-damper model, as shown in Fig. 9, where kP and kN denote the stiffness of PS mechanism and NS mechanism, respectively. The expressions of the magnetic force and stiffness represented by Eqs. (17) and (19) are too complicated to analyze the QZS vibration isolation system theoretically. Therefore, the restoring force of the QZS isolator is fitted as a cubic polynomial

Q. Wang et al. / Mechanical Systems and Signal Processing 139 (2020) 106633

Fig. 8. (a) Restoring force and (b) stiffness of the QZS isolator.

Fig. 9. Equivalent dynamic model of the QZS vibration isolation system.

Fig. 10. Restoring force of the QZS isolator. (a) Comparison between the analytical and polynomial fitting results, and (b) error of fitting polynomial.

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with respect to the axial displacement z. As shown in Fig. 10a, the black solid line and red dotted line represent the analytical result and fitting polynomial, respectively. The difference between the analytical restoring force and the fitting one is depicted in Fig. 10b, which shows good agreement especially in the vicinity of the equilibrium position. The fitted polynomial of the restoring force can be given by

F QZS ¼ k1 z þ k3 z3

ð19Þ 6

where k1 = 178.92, k3 = 4.912  10 . For the QZS vibration isolation system in Fig. 9, assuming that the base displacement excitation is z1 ¼ Z 1 cosxt, the dynamic equation can be given by

m€z þ cz_ þ k1 z þ k3 z3 ¼ mZ 1 x2 cosxt

ð20Þ

where z ¼ z2  z1 is the relative displacement between the payload mass and the base. By introducing the following parameters

^z ¼

z x c ;X ¼ ;f ¼ ; xn ¼ xn zs 2mxn

rffiffiffiffiffi k1 x k1 k3 z2s ^ Z1 dðÞ ;a¼ ;b ¼ ; Z 1 ¼ ; s ¼ xn t; ðÞ0 ¼ ;X ¼ xn ds m zs 3k1 zs

^z00 þ 2f^z0 þ a^z þ b^z3 ¼ Z^ 1 X2 cosðXsÞ

ð21Þ

Note that zs is the static displacement of the linear spring kp under the payload. ^ is the relative displacement amplitude, and u is ^ ðXs þ uÞ, where Z Assuming that the response of the system is ^z ¼ Zcos the phase between the excitation and the response. By using the harmonic balance method, one can yield

3 3 X2 Z^ þ aZ^ þ bZ^ ¼ X2 Z^ 1 cosu 4

ð22Þ

2fZ^ X ¼ X2 Z 1 sinu

ð23Þ

Squaring and adding Eqs. (22) and (23), the amplitude-frequency equation can be given by

 2  2  2 3 3 X2 Z^ þ aZ^ þ bZ^ þ 2fXZ^ ¼ X2 Z^ 1 4

ð24Þ

By solving Eq. (24), the relationship between the response amplitude and the driving frequency can be yielded

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  2    u ^ b2  48bf2 Z^ 4 þ 24Z^ 2 ab  64af2 þ 64f4 Z^ 2 þ 16Z^ 2 a2 u4Z^ 2 a  3Z^4 b þ 8Z^ 2 f2 þ Z^ 9 Z 1 1 1 u  2  X1 ¼ u t ^ ^ 2 4 Z1  Z vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  2    u ^ b2  48bf2 Z^ 4 þ 24Z^ 2 ab  64af2 þ 64f4 Z^ 2 þ 16Z^ 2 a2 u4Z^ 2 a  3Z^4 b þ 8Z^ 2 f2  Z^ 9 Z 1 1 1 u  2  X2 ¼ u t 4 Z^  Z^2

ð25Þ

ð26Þ

1

Let X1 ¼ X2 , the maximum displacement response amplitude can be obtained

Z^ max

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  2 u 2 4 ^ 4 3 Z a b þ 8 a f  8f  4 4a2 f4 þ 3Z 21 abf4  8af6 þ 4f8 u 1 u   ¼t 3 3Z 21 b2  16bf2

ð27Þ

The corresponding resonant frequency at this amplitude is

Xmax

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u4Z^ 2 a  3Z^ 4 b þ 8Z^2 f2 max  2 max  max ¼u t 4 Z^1  Z 2max

ð28Þ

The non-dimensional absolute displacement of the mass can be given by

z2 ¼ z þ z1 ¼ ZcosðXs þ uÞ þ Z 1 cosðXsÞ Therefore, the displacement transmissibility of the QZS vibration isolation system can be formulated as

ð29Þ

Q. Wang et al. / Mechanical Systems and Signal Processing 139 (2020) 106633

  z2  T ¼ 20log  ¼ 20log z1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 þ Z 21 þ 2Z 1 Zcosu Z1

11

ð30Þ

where cosu can be obtained from Eq. (22). Substituting cosu into Eq. (30), the displacement transmissibility is then expressed as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u   u 2 2Z aZ þ 34 bZ 3  X2 Z z2  Z t T ¼ 20log  ¼ 20log 1 þ 2 þ z1 Z1 Z 21

ð31Þ

The displacement amplitude and transmissibility of the QZS vibration isolation system for different base excitation are shown in Fig. 11. From Fig. 11, it is clear that as the base excitation amplitude increases, the peak displacement amplitude and transmissibility increase significantly. When the base excitation amplitude is large, such as Z1 = 0.05, the peak displacement amplitude and transmissibility would appear unbounded, as revealed by Carrella et al. [45]. Therefore, this is an inherent limitation of the QZS vibration isolation system under base excitation. However, in engineering practices, stoppers are always used to limit the displacement amplitude in a reasonable range. In the normal working condition, the QZS vibration isolation system performs a good function of low-frequency vibration isolation, and the stopper just need to work in an unexpected situation of large-amplitude vibration. In the case of small-amplitude vibration, the degree of nonlinearity coefficient b is so weak that the dynamic Eq. (20) can be linearized to become a linear one [46]. In this case, the transmissibility can be given by

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1 þ ð2fXÞ2 T ¼ 20logu 2 t 1  X2 þ ð2fXÞ2

ð32Þ

Fig. 12 shows the displacement transmissibility under excitations with different amplitudes, compared with that of the linearized system. It can be seen that the displacement transmissibility calculated by Eq. (32) is in good agreement with the transmissibility calculated by Eq. (31) under small-amplitude excitation, such as Z1 = 0.04, which firmly demonstrates the effectiveness of the linearized model. Hence, the linearized model will be used to predicate the transmissibility theoretically of the QZS isolator in the following Section 4.2.

4. Experimental investigation The prototype of the QZS isolator is built by using the parameters listed in Table 3. The static and dynamic characteristics of the QZS isolators are tested. And then, the prototype of the QZS incubator is constructed, and the vibration isolation performance is measured under both harmonic and random excitations.

Fig. 11. (a) Displacement amplitude and (b) displacement transmissibility of the QZS vibration isolation system for different base excitation amplitude (a = 1, b = 100, f = 0.2).

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Fig. 12. Displacement transmissibility under excitations with different amplitudes (a = 1, b = 12, f = 0.2).

4.1. Static feature of the QZS isolator The prototype of the QZS isolator is shown in Fig. 13a, and the quasi-static testing setup is illustrated in Fig. 13b. The restoring force vs displacement relationship of the QZS isolator is measured by the compression testing machine. The compressive loading is reflected by the deformation it produces and this starts from 0 mm until to 25 mm, and the loading rate is 5 mm/min, which can be regarded as a quasi-static test. Note that, when the compression arrives at about 15 mm, the center of the inner ring aligns to that of the outer ring, and the QZS isolator reaches its static equilibrium position. The measured relationship between the restoring force and displacement is presented in Fig. 14. Note that the displacement shown on the horizontal ordinate is a relative displacement to the static equilibrium, which is obtained by subtracting the ordinate of the static equilibrium position, 15 mm, from the measured displacement. Obviously, in the displacement range from 5mm to 5 mm, the curves are nearly flat, which implies quasi-zero stiffness. Moreover, the cases of payload deviations are taken into account. In the experiment, the adjusting screw is turned to lift the coil spring by 0 mm, 2 mm and 4 mm, and the corresponding compressions of the coil spring at the static equilibrium position are 15 mm, 17 mm, and 19 mm, which represent payload deviations of 0 N, 3 N, and 6 N, respectively. In order to evaluate vibration isolation performance theoretically, the restoring force–displacement curve is fitted by a 0 first-order polynomial f z ¼ a þ bz in displacement range from 5mm to 5 mm, as shown in Fig. 14b, and the coefficients a and b for three cases are listed in Table 4. According to the fitting polynomial, the approximate stiffness ka of the isolator is equal to the coefficient b. For these three cases, the approximate stiffnesses are 0.1315 N/mm,0.1335 N/mm and 0.1321 N/ mm, respectively, which can be regarded as very low stiffness, compared with the stiffness of the coil spring, 1.63 N/mm.

Fig. 13. Static test of the QZS isolator: (a) prototype, (b) testing setup.

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Fig. 14. (a) Experimental measured restoring force–displacement curves for different payload deviations and (b) fitting force–displacement curves.

Table 4 Parameters of the QZS isolator. Payload Deviation (N)

DW = 0 DW = 3 DW = 6

Parameter a (N)

b (N/mm)

ka (N/mm)

W(N)

22.7 24.5 27.7

0.1315 0.1335 0.1321

0.1315 0.1335 0.1321

22 25 28

4.2. Transmissibility of the QZS isolator The experimental setup for measuring acceleration transmissibility of the QZS isolator is shown in Fig. 15. The isolator is installed upon the shaker through a fixture, and a payload is supported by the isolator. The shaker is driven by a signal generator embedded in the data collecting instrument and signal generator (ECON MI-7008) through a power amplifier. The excitation signal inputted into the shaker is a sinusoidal sweep from 0 Hz to 8 Hz with amplitude 2 mm. Two accelerometers are tied onto the payload and fixture, respectively, to measure accelerations of the payload and fixture. The transmissibility is defined by the ratio of the acceleration of the payload to that of the fixture. This is a single-degree-of-freedom vibration isolation system under base excitation, and its equation of motion can be given by

m€z þ cðz_  y_ Þ þ f z ¼ 0

ð33Þ

where m is the mass of the payload, c is the damping coefficient, z is the displacement of the payload, y ¼ YsinðxtÞ is the harmonic base excitation, and fz is the restoring force of the isolator. Since the excitation amplitude is not large (2 mm in the experiment), as stated in Section 3.5, the linear fitting polynomial can be used as the expression of the restoring force fz, i.e. f z ¼ ka ðz  yÞ, and the transmissibility can be obtained by Eq. (32). For three cases of payload 2.2 kg, 2.5 kg, and 2.8 kg, the experimental tests are conducted. The acceleration transmissibility is shown in Fig. 16, which is also compared with the theoretical results. Note that the isolator performs a function of vibration isolation only when the acceleration of the payload is smaller than that of the fixture, i.e. the transmissibility is below 1. From Fig. 16, it can be seen that, for the case of 2.2 kg, the beginning frequency of vibration isolation is about 2 Hz, and when the payload becomes heavier, the beginning frequency slightly decreases. As reported in Figs. 8 and 14, the stiffness of the QZS isolator can be allowed to stay unchanged as the payload alternates. Hence, the natural frequency would be reduced by the increasing payload, leading to lower beginning frequency of vibration isolation. More importantly, as reported by Shenai et al. [2], the frequency range of the mechanical vibration during neonatal transport is mainly from 3 Hz to 20 Hz. In this experiment, the beginning frequency of vibration isolation is lower than 3 Hz. Therefore, this QZS isolator is qualified for vibration isolation during neonatal transport. The analytical transmissibility based on the linear fitting expression of the restoring force is also shown in Fig. 16. For these three cases of payload, the approximately linear stiffness ka can be found in Table 4, and the transmissibility is calculated by using Eq. (32). Note that the damping coefficient of the isolator is determined by matching the peak value of the

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Fig. 15. Experimental setup for transmissibility of the QZS isolator.

Fig. 16. Experimental and analytical acceleration transmissibility of the QZS isolator for three cases of payload 2.2 kg, 2.5 kg and 2.8 kg.

experimental transmissibility [15]. It can be seen the curves obtained by the theoretical analysis matches well with experimental tests in the low-frequency range. However, the theoretical transmissibility is higher than the measured one in the frequency range of vibration isolation, since the QZS vibration isolation system is a nonlinear system rather than a linear one, and the actual damping is heavier than the equivalent one for the linear system.

4.3. Vibration isolation performance of the QZS incubator The prototype of the incubator is built, as shown in Fig. 17. The infant housing unit is supported by four QZS isolators evenly, and an infant model is placed in the housing unit. One end of the isolator is fixed into the housing unit and another end is installed onto the box for auxiliary equipment. The whole incubator is placed on a shake table that is excited by a large shaker. The shaker is driven by harmonic sweep and random excitation signals, respectively, through a power amplifier. Three accelerometers are attached onto the infant model, the bottom plate of the housing unit, and the upper plate of the box for auxiliary equipment, separately. Similarly, the vibration isolation performance is evaluated by acceleration transmissibility, which is defined by the ratio of the acceleration of the infant (Accelerometer 1) to that of the top plate of the auxiliary box (Accelerometer 3). In order to present the advantage of the incubator with QZS isolators, a counterpart experiment on the incubator with linear isolators is conducted. Note that the linear isolator is made by removing the pair of magnet rings from the QZS isolator.

Q. Wang et al. / Mechanical Systems and Signal Processing 139 (2020) 106633

15

Fig. 17. Experimental setup for measuring vibration isolation performance of the QZS incubator. The housing unit is supported by QZS isolators, and the whole incubator is placed on the vibration table.

The weight of the infant model can be adjusted by filling grains into the model. In this experiment, the mass of the infant model is selected as 1.5 kg to simulate a premature infant, and the total mass of the housing unit and the infant model is 10.5 kg. Therefore, the weight support by each isolator is 26.25 N (g = 10 m/s2), which is larger than the designated payload 22 N. According to the design procedure in Section 3.4 and the static experimental study in Section 4.1, one can easily tune up the isolator to maintain the QZS feature by just turning the adjusting screw in the isolator. The acceleration transmissibility is shown in Fig. 18 under harmonic excitation sweeping from 2 Hz to 15 Hz with different amplitudes 2 mm, 2.2 mm and 2.4 mm, separately. The transmissibility of the incubator with the QZS and linear isolators are denoted by red solid lines and black dashed lines, respectively, in Fig. 18a–c. Obviously, the QZS incubator notably outperforms the linear one, especially in low frequency range from 3 Hz to 8 Hz. Specifically, the beginning frequency of vibration isolation of the QZS incubator is 3.2 Hz, which is about a half of the linear one (5.8 Hz); and the maximum peak transmissibility is 2.25, which is about a quarter of the linear one (8.9). It indicates remarkable reduction in the beginning frequency of vibration isolation and substantial attenuation of the peak resonance, which are desired characteristics for ultra-low frequency vibration isolation. In addition, Fig. 18d shows a comparison among the three cases for the QZS incubator. One can see the peak transmissibility increases with the excitation amplitude, and the beginning frequency of vibration isolation slightly decreases. In the frequency range from 3.5 Hz to 7 Hz, the second peak of transmissibility occurs. Moreover, compared Fig. 18d with Fig. 16, it also can be observed that the beginning frequency of vibration isolation of the QZS incubator is larger than that of a single QZS isolator, which indicates a reduction in vibration isolation performance after assembling all parts into a whole incubator. This can be attributed to a fact that the housing unit does not vibrate only along the vertical direction, and weak pitch and yaw motions might happen due to the manufacturing and assembly errors. In order to simulate the case of neonatal transport by using road ambulance, the prototype incubator is subjected to a random base excitation with frequency bandwidth from 2 Hz to 15 Hz and with amplitude 2 mm. The acceleration responses of the incubator are acquired by accelerometers (Fig. 17), and illustrated in time domain and frequency domain (Fig. 19), respectively. In Fig. 19, the response of the infant is recorded by Accelerometer 3 and denoted by red solid lines, that of the bottom plate of housing unit by Accelerometer 2 and represented by blue dashed lines, and the response of the top plate of the auxiliary box by Accelerometer 1 and denoted by black dotted lines. The infant is placed onto the bottom plate of the

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Fig. 18. Acceleration transmissibility of the incubator supported by QZS isolators and linear ones under excitation with different amplitudes: (a) 2 mm, (b) 2.2 mm, (c) 2.4 mm, and (d) comparison among three cases.

Fig. 19. Acceleration response of the incubator with QZS vibration isolators (a) in time domain and (b) in frequency domain.

Q. Wang et al. / Mechanical Systems and Signal Processing 139 (2020) 106633

17

housing unit. The auxiliary box is located upon the vibration table, and thus its acceleration acquired by Accelerometer 1 is equal to the base excitation outputted by the shake table. From Fig. 19, the difference between the responses of the infant and the housing unit can be observed, especially for the low-frequency components. These can be attributed to the elastic body of the infant model, and hence the stiffness between the infant and the housing unit should be taken into account when one tries to establish an analytical lumped-mass-spring model for a neonatal transport system. More importantly, from Fig. 19a, it can be seen that the response of the infant is much lower than the excitation, and the transmissibility of mean square root (RMS) acceleration is about 0.3230, which implies considerable vibration attenuation produced by the QZS isolators. From the frequency-amplitude relations (Fig. 19b), one also can see a remarkable reduction in acceleration amplitude from about 3.5 Hz, which is close to the beginning frequency of vibration isolation of the QZS incubator. Although the acceleration amplitude would be amplified in ultra-low frequency from 2 Hz to 3.5 Hz, the total responses felt by the infant are substantially lower than the excitation (Fig. 19a). Moreover, in the proposed incubator, the QZS isolators are installed between the box of auxiliary equipment and the infant house unit. Hence, both the vibrations from the ambulance floor and auxiliary equipment can be reduced by the QZS isolators. The vibration from auxiliary equipment is mainly generated by the air conditioner, and the frequency is dominated in the bandwidth from 60 Hz to 90 Hz [47], which is much higher than the frequency of the vibration from the ambulance floor (3–18 Hz). According to the vibration isolation theory [48], the QZS isolators can effectively isolate the vibration from the auxiliary equipment within the frequency bandwidth from 60 Hz to 90 Hz. Therefore, the incubator with QZS isolators is capable of mitigating mechanical vibration effectively during neonatal transport.

5. Conclusions A new incubator containing quasi-zero-stiffness (QZS) vibration isolators is proposed to mitigate mechanical vibration transmitted from ambulance vehicle body to the infant. Each of the QZS isolator is fulfilled by combining a pair of mutually repealing permanent magnet rings and a coil spring. The coil spring is designed to meet requirement on weight capability. The magnet rings are engineered to realize negative stiffness, whose absolute value should be equal to the stiffness of the coil spring, and the space constraint is also taken into account in the design. Both prototypes of the QZS isolator and the incubator with QZS isolators are built, and their vibration isolation performances are evaluated by experimental tests. The following conclusions can be drawn. (1) The height, thickness and air gap of the magnet rings have a notable impact on the negative stiffness, which should be carefully designed to fulfil the QZS feature. (2) The beginning frequency of vibration isolation of the proposed QZS isolator is about 2 Hz under 2.2 kg payload, and thus it is qualified for ultra-low frequency vibration isolation. (3) The beginning frequency of vibration isolation of the incubator with four QZS isolators is about 3.2 Hz under 10.5 kg payload. The transmissibility of mean square root (MSR) acceleration is 0.3230 under random excitation with frequency bandwidth from 2 Hz to 15 Hz and with amplitude 2 mm. Therefore, the incubator with the QZS isolators should be a good solution to mitigate mechanical vibration during neonatal transport.

CRediT authorship contribution statement Qiang Wang: Software, Visualization, Investigation, Validation, Writing - original draft. Jiaxi Zhou: Conceptualization, Methodology, Writing - original draft, Project administration, Funding acquisition. Daolin Xu: Writing - review & editing. Huajiang Ouyang: Writing - review & editing.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements This research work was supported by National Natural Science Foundation of China (11972152, 11572116). The initial idea of this vibration-isolation incubator was conceived during the second author’s visit to the University of Liverpool, sponsored by the China Scholarship Council.

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