Design, Optimization, and Realization of a Magnetic Multi-Layer Quasi-Zero-Stiffness Isolation Platform Supporting Different Loads

Abstract

This study presents a Multi-layer Quasi-Zero-Stiffness (ML-QZS) vibration isolation platform for variable loads in large-amplitude and low-frequency dynamic environments. In one isolation mount of the proposed ML-QZS isolation platform, Multi-layer permanent magnets are constructed to generate discontinuous Multi-layer negative-stiffness regions. The first design criterion is to achieve the low-frequency and wide-amplitude vibration isolation range for different loads. The second design criterion is carried out for the dynamic performances of transient and steady states. Since both structural design and damping determine vibration transient time and the displacement transmissibility, which often exhibit contradictions depending on system parameters, a bi-objective Pareto optimization criterion is proposed to balance the vibration transients between different layers while ensuring significant isolation effectiveness in one layer. Finally, the relevant experimental prototype is constructed, and the results verify the design principle of Multi-layer double magnetic ring construction and optimization criterions for structural parameters and damping coefficients. This study provides an advanced nonlinear isolation platform with a wide QZS range for different loads, and the optimization method to coordinate the vibration performances, which provides important theoretical and experimental guidance for the design and realization of isolation platforms in practical engineering applications for large-amplitude and low-frequency dynamic environments.

1. Introduction

Vibration is a common phenomenon in engineering applications detrimental to the accuracy and service life of equipment [1,2]. To mitigate vibrations from the environment, vibration isolators are usually installed between the excitation source and protective equipment to suppress environmental vibrations [3,4,5]. In previous studies, linear vibration isolators usually consist of a linear spring and a viscous damper. It is known that linear vibration isolators can suppress excitations with frequencies greater than $\sqrt{2}$ times the natural frequency [6,7]. As the natural frequency of the linear vibration isolator is reduced by increasing the loading mass or decreasing the spring stiffness, an increase in mass or reduction in stiffness may result in large initial installation deformation and even lead to instability. To resolve this problem, various design principles of the Quasi-Zero-Stiffness (QZS) isolators are proposed [8,9,10]. In recent years, various types of QZS isolators have been applied in different dynamic environments. Typical QZS isolators are commonly composed of a positive-stiffness component and a negative-stiffness component. The negative-stiffness structure mainly includes triple springs [11,12,13], cam-rollers [14,15,16], permanent magnets and electromagnets [17,18,19], buckled-beams [20,21,22], bionic structures [23,24,25], X-shaped structures [26,27,28], and other structures [29,30]. The positive-stiffness structure mainly includes springs [31], permanent magnets [32], flexible rods or beams [33,34], and others [35]. The connection between any positive-stiffness mechanism and any negative-stiffness mechanism in parallel can generate the QZS property. In QZS isolators, the positive elastic components are usually assembled in the direction of gravity to bear the load, and then, other elastic components are assembled at the equilibrium. Because the negative-stiffness components always result in nonlinearity, typical QZS isolators realize high loading capacity and low dynamic stiffness at equilibrium, known as the High-Static-Low-Dynamic-Stiffness (HSLDS) characteristic. Unfortunately, QZS isolators have the QZS property at the equilibrium point but exhibit a narrow QZS range around it. Since the nonlinearity cannot be ignored, the vibration characteristics depend on both structural parameters and vibration states, and the multi-steady states phenomenon occurs with increasing excitation amplitude. Also, since the system has very low dynamic stiffness around the equilibrium, a variable load can easily break the designed QZS property and increase the initial frequency required for effective isolation. Therefore, it meets two challenging requirements in one isolation system: extending the amplitude range for the QZS property and improving adaptation to increasing loads.

To address the first requirement—expanding the QZS range to isolate large-amplitude, low-frequency vibration excitations—various design methods and bionic structures are proposed. The first design method to increase the QZS range is reducing the nonlinear characteristics of negative stiffness to compensate for linear positive stiffness. Kovacic et al. [36] proposed a triple-spring mechanism consisting of two nonlinear pre-stressed oblique springs and a linear spring. Compared to typical QZS isolators consisting of one pair of pre-deformed springs, two nonlinear pre-deformed oblique springs can provide a larger range for linear negative stiffness. And, by using the cam-roller mechanism instead of elastic components, the nonlinear strength can be reduced to achieve a wide-range QZS property [37,38,39]. The second design method to increase the QZS range is to employ nonlinear positive stiffness to match the nonlinear negative stiffness. Yu et al. [40] proposed a novel nonlinear stiffness-modulated anti-vibration structure consisting of a disc spring group and a volute spring. The nonlinear positive stiffness provided by the volute spring compensates for the negative stiffness provided by the disc spring group to construct QZS characteristics. Their nonlinearities can weaken each other, which can lead to a wide-range QZS property. Yan [41] et al. utilized a pair of magnets to generate hardening positive stiffness to compensate for the negative stiffness generated by the diamond structure. The third design method to increase the QZS range is to reduce both linear and nonlinear stiffness coefficients close to zero, and the typical novel mechanism is the bio-inspired structures [42,43,44]. Unfortunately, for the three design methods above, the QZS property is still achieved around the equilibrium. Recently, a novel method was proposed, which utilizes multiple bistable negative-stiffness structures connected in parallel with positive-stiffness structures to form a multi-point QZS property. Compared to single-point zero-stiffness design, the double-point zero-stiffness design has a wider range for the QZS property. Gatti et al. [45] and Zhao et al. [46] added a pair of oblique springs to the triple-springs isolation system, which results in a stiffness curve with double zero-stiffness points. Around the double zero-stiffness points, there occurs a region with small stiffness, defined as the wide range for the QZS property. Further, by adding more pairs of pre-stressed oblique springs into the triple-spring mechanism, the stiffness curve of the QZS isolator has more zero-stiffness points [47,48]. The zero-stiffness points can be connected to form a wide-range QZS region through reasonable configuration.

For the second requirement of sufficient loading capacity, the typical QZS design usually fails under variable loads. Chen et al. [49] discovered that even a slight load variation could induce a significant impact on the vibration isolation performance of the QZS isolation system through a triple-spring mechanism, since a slight variation in load could result in a deviation of equilibrium from the original position. However, in engineering practice, loads typically vary over a wide range. Xu et al. [50] designed a magnetic isolator that can adjust the load mass within a certain range by adjusting the initial length of the pre-deformed springs. Chen et al. [51] proposed a low-frequency vibration isolator consisting of disk positive-stiffness spring and a pair of buckling Euler beams with adjustable buckling strength that provide adjustable negative stiffness. Furthermore, Jiao et al. [52] proposed a QZS isolator consisting of a cam-roller negative-stiffness system and a positive-stiffness permanent magnetic spring. The load can be increased by increasing the supported force resulting from reducing the distance between the permanent magnets. The above isolation system requires manual adjustment of the connections of components, while in practice, the components in the vibration isolation system cannot be manually adjusted frequently. To achieve automatic load adjustment, Wen et al. [53] added an electromagnetic constant force mechanism at the end of the QZS isolator to counteract the loads by electromagnetic force. Lu et al. [54] proposed an isolator composed of electromagnetic negative-stiffness and electromagnetic positive-stiffness mechanisms. The QZS isolator can achieve load adjustment by controlling the coil current of the electromagnets. Although electromagnetic control can continuously adjust the loading capacity of the isolation system, the range of the load is limited and requires continuous energy input. Ye et al. [55] and Zhou et al. [56] developed a Multi-layer QZS isolator capable of undertaking different loads at different equilibrium positions. This is composed of a linear-stiffness spring providing positive stiffness and multiple cam-roller mechanisms providing multiple negative-stiffness regions. Thus, in each layer, the QZS property can be realized and different loads can be undertaken. Zeng et al. [57] introduced stair-stepping mechanical metamaterials with a multi-step shape-restoring force curve, based on the programmable characteristics of metamaterials. The height and length of the step can be designed, and each step can undertake a load.

In conclusion, the design methods for a wide-range QZS property or variable loads can be achieved separately. However, there is less research on achieving both requirements simultaneously. In this study, we propose an isolation platform with four Multi-Layer Quasi-Zero-Stiffness (ML-QZS) isolation mounts, appropriate for variable loads. The proposed isolation support is composed of multiple layers of double-ring magnetic mechanisms. Due to the local and nonlinear characteristics of the magnetic force, discontinuous negative-stiffness characteristics can be achieved through a Multi-layer arrangement. Thus, in the proposed vibration isolator, the use of a multiple-point QZS design increases the amplitude range for effective vibration isolation, and the parallel connection of positive spring can generate an ML-QZS design capable of undertaking different loads. The design criterion, therefore, focuses on the coordination of dynamic performances, including the vibration transient attenuation and board frequency displacement transmissibility, both of which depend on all structural parameters. In order to obtain not only the static structural design but also the dynamic performances, the optimization criterions are carried out, and the verification experiment is given. The organization of the paper content is as follows. Section 2 describes the working principle of the ML-QZS with a wide QZS range for various loads and introduces the derivation of restoring force and dynamic stiffness of the ML-QZS isolator. Section 3 establishes the relationship between parameters and dynamic performances, and the bi-objective Pareto optimization criterion is proposed to coordinate vibration transient attenuation and isolation effectiveness across a broad frequency band. The experimental prototype is constructed to verify the theoretical results in Section 4. Lastly, conclusions are given in Section 5.

2. Design for ML-QZS Isolation Platform for Variable Loads

2.1. Design Method of the ML-QZS Isolation Mounts

This section provides the design principle of the isolation platform applied for low-frequency and wide-amplitude dynamic excitation. Multiple pieces of equipment can be assembled on the isolation platform, and thus, the platform should have a wide-range QZS property to accommodate variable loads, as shown in Figure 1a. The isolation platform consists of four isolation mounts. In one isolation mount, the Multi-layer magnetic rings for negative stiffness and the vertical spring for positive stiffness are constructed in parallel, as shown in Figure 1b. The load platform is fixed with the inner magnetic ring using the guide rod and multiple outer magnetic rings fixed on the frame. In the magnetic negative-stiffness construction, every magnetic ring has the same magnetization direction and the same axis around the guide rod. The guide rod can only move in the vertical direction under the limitation of a linear bearing. As a result, when the displacement excitation occurs from the bottom, the guide rod, inner ring, and load platform vibrate together vertically.

As shown in Figure 1b, in one isolation mount, the restoring force applied on the isolation platform is the sum of the elastic forces provided by the vertical spring and the Multi-layer magnetic rings. As shown in Figure 1c, the outer double magnetic rings apply magnetic force on the inner magnetic ring. For the magnetic rings, R₁, R₂, R₃, and R₄ are the radius of inner and outer magnetic rings, h₁ and h₂ are the thicknesses of the inner and the outer rings, B_r1 and B_r2 are the residual magnetic flux density of Neodymium magnetic material, and μ₀ is the air permeability.

The magnetic force between the inner ring and outer rings is generated only in the vertical direction due to the axially symmetrical configuration, as shown in Figure 2a. According to Ref. [27], using the equivalent magnetic charge method, the magnetic force applied on the inner ring (rigidly connected to the mass) of one layer is given as

$$\begin{array}{l} F_{M1}\left(z,l_j,R_1,R_2,R_3,R_4,h_1,h_2\right)\\ ={\displaystyle \frac{B_{r1}{B}_{r2}}{4\pi {\mu }_0}}{\displaystyle \sum _{k=1}^4{\displaystyle {\int }_{R_3}^{R_4}{\displaystyle {\int }_{R_1}^{R_2}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{2\pi }\frac{z_{1k}{r}_1{r}_2}{{\left[z_{1k}^2+{\left(r_1\cos {\theta }_1-r_2\cos {\theta }_2\right)}^2+{\left(r_1\sin {\theta }_1-r_2\sin {\theta }_2\right)}^2\right]}^{\frac{3}{2}}}}}}}\mathrm{d}{\theta }_1\mathrm{d}{\theta }_2\mathrm{d}{r}_1\mathrm{d}{r}_2}\end{array}$$

(1)

where z_1k (k = 1, 2, 3, 4) is the distance between the inner and outer ring surfaces as the detail diagram in Figure 2a. For the multiple layers of outer magnetic rings as shown in Figure 2b, the magnetic force applied on the inner magnetic ring is defined as F_M, given as

$$\begin{array}{l}{F}_M(z,R_1,R_2,R_3,R_4,h_1,h_2)={\displaystyle \sum _{j=1}^{i+1}{F}_{Mj}\left(z,l_j,R_1,R_2,R_3,R_4,h_1,h_2\right)}\\ ={\displaystyle \frac{B_{r1}{B}_{r2}}{4\pi {\mu }_0}}{\displaystyle \sum _{j=1}^{i+1}{\displaystyle \sum _{k=1}^4{\displaystyle {\int }_{R_3}^{R_4}{\displaystyle {\int }_{R_1}^{R_2}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{2\pi }\frac{z_{jk}{r}_1{r}_2}{{\left[z_{jk}^2+{\left(r_1\cos {\theta }_1-r_2\cos {\theta }_2\right)}^2+{\left(r_1\sin {\theta }_1-r_2\sin {\theta }_2\right)}^2\right]}^{\frac{3}{2}}}}}}}}}\mathrm{d}{\theta }_1\mathrm{d}{\theta }_2\mathrm{d}{r}_1\mathrm{d}{r}_2\end{array}$$

(2)

where j denotes the jth outer magnetic ring and i denotes the jth layer. In Equation (2), z_jk is the distance of the surfaces between the inner ring and each outer magnetic ring, as shown in Figure 2b. The first subscript j in z_jk indicates the jth layer’s outer magnetic ring, and the second subscript k indicates the four different surface distances, as shown in Figure 2a. The symbol z_jk (k = 1, 2, 3, 4) has the follwoing detailed expression:

$$\begin{array}{l}{z}_{j1}=z-h_1/2-\left(l_j+h_2/2\right),z_{j2}=z-h_1/2-\left(l_j-h_2/2\right),\\ z_{j3}=z+h_1/2-\left(l_j+h_2/2\right),z_{j4}=z+h_1/2-\left(l_j-h_2/2\right),\end{array}$$

(3)

where l_j is the location of the jth layer’s outer magnetic ring from the origin vertically in the positive direction of the Z-axis. There is a relationship between each l_j, given as l_j+1 − l_j = dis and l₁ = −dis/2.

The linear spring with positive stiffness k is constructed parallel for supporting the load. Considering the restoring force provided by the linear vertical spring as F_E = kz, the restoring force of the proposed ML-QZS isolator is defined as F_R, the sum of the magnetic force and the elastic force of the linear spring, given as

$$F_R=F_E+F_M=kz+F_M$$

(4)

Then, the magnetic forces exerted by each outer magnetic ring F_Mj and by Multi-layer outer magnetic rings F_M are shown in Figure 2c, and the sum restoring force F_R applied to the inner magnetic ring is shown in Figure 2d, with the distance between the double-layer outer rings set at dis = 40 mm.

For the original equilibrium fixed at the point O (O₁), the center of the first layer of double magnetic rings, the dashed lines in red represent the magnetic forces F_M1 for the first layer of the double outer-ring design. Since the outer rings of other layers induce little effect due to the large distance, the red dotted line—representing the sum of the magnetic force acting on the inner ring in the first layer—reveals a wide range for a negative-stiffness property. Similarly, when the inner ring locates the positions at the middle of each double magnetic ring layer as z = 0 mm (z = 40 mm, z = 80 mm, and z = 120 mm), the magnetic force has a wide-range negative-stiffness property at these equilibriums. With the positive restoring force provided by the linear vertical spring, the sum restoring force F_R displays step-shape QZS characteristics, as shown in Figure 2d. As the total number n of double magnetic ring layers increases, a wider range of load variations can be achieved.

2.2. Structural Optimization for Small-Variation Load Step and Wide-Range QZS Property

On one hand, for the Multi-layer double magnetic ring design principle as shown in Figure 2a, the restoring force F_R shown in Figure 2d demonstrates that every step of the restoring force curve can undertake variable loads with the step ΔMg. Taking the center points O_i of each double magnetic ring layer as the equilibrium for QZS vibration isolation, the loading capacity at each layer is given by the sum mass M_i, written as

$$M_i=M_1+{\displaystyle \frac{F_R}{g}}=M_1+{\displaystyle \frac{{\left.\left(F_M+kz\right)\right|}_{z=(i-1)dis}}{g}}=M_1+{\displaystyle \frac{{\left.{\displaystyle \sum _{j=1}^i{F}_{Mj}}\right|}_{z=(i-1)dis}+(i-1)kdis}{g}}=M_1+\left(i-1\right)\Delta M$$

(5)

where M₁ is the initial mass, undertaken at the equilibrium in the first double magnetic ring layer at O (O₁), and ΔM is the variable mass undertaken by the vertical spring for each layer. The variation-step load ΔM obtained by Equation (5) is given as

$$\Delta M={\displaystyle \frac{{\left.F_M\right|}_{z=0}+kdis}{g}}$$

(6)

On the other hand, from Equation (4), the sum magnetic force F_M depends on both the adjustable structural parameter dis and the stiffness coefficient k of the vertical positive spring. The two structural parameters are required to satisfy the requirement for a wide-range QZS property around each equilibrium. The stiffness induced by the magnetic field and elastic component, defined as K_R, can be obtained by the following derivation:

$$\begin{array}{l}{K}_R={\displaystyle \frac{\mathrm{d}{F}_R(z,R_1,R_2,R_3,R_4,h_1,h_2,dis)}{\mathrm{d}z}}=k+{\displaystyle \frac{\mathrm{d}{F}_M(z,R_1,R_2,R_3,R_4,h_1,h_2,dis)}{\mathrm{d}z}}\\ =k+{\displaystyle \frac{B_{r1}{B}_{r2}}{4\pi {\mu }_0}}{\displaystyle \sum _{j=1}^{n+1}{\displaystyle \sum _{k=1}^4{\displaystyle {\int }_{R_3}^{R_4}{\displaystyle {\int }_{R_1}^{R_2}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^{2\pi }\left[\frac{r_1 r_2 }{{\left(z_{jk}^2+{\left(r_1\cos {\theta }_1-r_2\cos {\theta }_2\right)}^2+{\left(r_1\sin {\theta }_1-r_2\sin {\theta }_2\right)}^2\right)}^{3/2}}\right.}}}}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left.-3{\displaystyle \frac{ z_{jk}^3 r_1 r_2 }{{\left(z_{jk}^2+{\left(r_1\cos {\theta }_1-r_2\cos {\theta }_2\right)}^2+{\left(r_1\sin {\theta }_1-r_2\sin {\theta }_2\right)}^2\right)}^{5/2}}}\right]\mathrm{d}{\theta }_1\mathrm{d}{\theta }_2\mathrm{d}{r}_1\mathrm{d}{r}_2\end{array}$$

(7)

In order to obtain the QZS property with static stability at equilibrium, the linear stiffness of the positive spring k should satisfy the following condition:

$$k=\left|\min \left({\left.{\displaystyle \frac{\mathrm{d}{F}_M}{\mathrm{d}z}}\right|}_{z=0}\right)\right|$$

(8)

Since the dynamic stiffness of the positive spring is equal to the minimum value of negative stiffness as shown in Equation (8), the QZS property can be realized. Considering the vibration around the first equilibrium O₁, the equivalent dynamic stiffness K_R for different values of dis are shown in Figure A1 in Appendix A. We define the QZS range as L_QZS, the region with equivalent stiffness in [0, 0.02 k], expressed by the following condition:

$$L_{\mathrm{QZS}}=‖z_1-z_2‖$$

(9)

where z₁ and z₂ are obtained by the condition K_M = 0.02 k and ordered from large to small value.

According to Equations (2) and (7), the magnetic force F_M and magnetic stiffness K_M are proportional to B_r1 and B_r2. The QZS length L_QZS is the region with equivalent stiffness in [0, 0.02 k], where k is the linear stiffness of the positive spring. The linear stiffness of the positive spring k is also proportional to B_r1 and B_r2, according to Equation (8). Therefore, the residual magnetic flux density of Neodymium magnetic material B_r does not affect the quasi-zero-stiffness range. The material and environment’s temperature and humidity only affect B_r1 and B_r2, so it does not affect the QZS length L_QZS.

In the structural design of the double-ring system, two indices, including the variation step of load ΔM and the range of QZS property L_QZS, have been defined and described. The optimization criterion of the key structural parameter dis is given by the loss function f_s as

$$\begin{array}{l}\left\{\left.di{s}_{\mathrm{opt}}\right|\max f_s\left(dis\right)=-\Delta M+\lambda L_{\mathrm{QZS}}\right\}\\ s.t.dis\in \left[25 \mathrm{mm},40 \mathrm{mm}\right] \end{array}$$

(10)

where λ is the weightiness and ΔM and L_QZS are defined as Eqiations (6) and (9). Figure 3a gives the optimum values dis_opt for different weightiness λ from 1 to 8. As weightiness coefficient λ is larger than 4, the range of the QZS property is predominant and the optimum distance of the double magnetic rings is dis_opt = 32.4 mm. Figure 3b shows that the comparisons of restoring force F_R and dynamic stiffness K_R are among the cases for the optimum structural parameter and two other values as fixing the weightiness λ = 5. From the comparison, the variation-step of the load is the smallest and the QZS range is widest for dis_opt.

In summary, the optimization criterion for structural parameter of the double-rings design is proposed for the ML-QZS property with small variation-step of different loads. However, the analysis and results above are based on the restoring force rather than the dynamic performances of the isolation system. Since dynamic characteristics depend on not only structural parameters but also the vibration states, the dynamic performances should be optimized by appropriate structural and damping coefficients, which could be discussed in the following section.

3. Optimization for Dynamic Performances

In this section, the dynamic performances of the proposed ML-QZS isolator are analyzed, Based on this, the optimization for dynamic performances is provided. The ML-QZS isolator is designed to be appropriate for variable loads and wide-range QZS vibration isolation, and thus, the dynamic performances, including the vibration isolation, a single-layer wide frequency band, and the transient vibration attenuation between adjacent layers, are coordinated and optimized. Both indices depend on the damping coefficient and the structural design. In this section, the optimization criterion is proposed according to nonlinear vibration analysis.

Substituting Equations (A1)–(A3) into the Lagrange’s equation ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\left({\displaystyle \frac{\text{∂}\mathrm{L}}{\text{∂}{\dot{z}}_r}}\right){\displaystyle \frac{\text{∂}\mathrm{L}}{\text{∂}{z}_r}}=\mathrm{Q}$, where L is Lagrange function as L = T − V, the dynamic equation is given as

$$\left\{\begin{array}{l}{M}_i{\ddot{z}}_r+F_R\left(z_r\right)+c{\dot{z}}_r=M_{i}g-M_i{\ddot{z}}_e\\ z_r\left(0\right)=0,{\dot{z}}_r\left(0\right)=0\end{array}\right.$$

(11)

For the dynamic Equation (11), the vibration solution contains transient vibration and steady-state vibration. Since we have assembled the initial equilibrium at z_r = 0 and the variation of load is ΔM for one time, the complete solution contains transient vibration, occurring between the adjacent layer, and steady-state vibration, occurring in one layer. For one layer, the dynamic equation around each equilibrium is given as shown in Equation (A4) in Appendix B.

3.1. Bifurcation Condition for Single-Steady State and Displacement Transmissibility Curve

For the dynamic model, as shown in Equations (11) and (A4), in order to solve the steady states in one double-ring layer, we assume that ΔM = 0 and apply the Averaging Method. Then, the derivation process for steady states is shown in Appendix B. From Equation (A6), which is the differential equation of the amplitude and phase, in one double-ring layer, the following conditions should be satisfied for steady-state response:

$$\left\{\begin{array}{l}\dot{a}={\displaystyle \frac{1}{2\pi \omega }}\left[Y\left(a,\omega \right)-F_{isp}\sin \theta \right]=0\\ \dot{\theta }={\displaystyle \frac{-1}{2\pi \omega a}}\left[P\left(a,\omega \right)+F_{isp}\cos \theta \right]=0\end{array}\right.$$

(12)

In Equation (12), the expressions of Y (a,ω), P (a,ω), and F_isp are given in Appendix A. According to the sum of squares formula of trigonometric functions, it has the following equation:

$$Y{(a,\omega )}^2+P{(a,\omega )}^2-F_{isp}^2=0$$

(13)

Equation (13) gives the relationship between frequency and vibration amplitude. The solution from Equation (13) is obtained as a₀, and the solution for phase is θ₀, given as

$$\left\{\begin{array}{l}\sin {\theta }_0=Y\left(a_0,\omega \right)/F_{isp}\\ \cos {\theta }_0=-P\left(a_0,\omega \right)/F_{isp}\end{array}\right.$$

(14)

Then, the transmissibility of the ML-QZS isolator is obtained from the ratio of the maximum response amplitude of the load and excitation amplitude of the exciter, and is given as

$$T_d=10\lg \left({\displaystyle \frac{\max \left(z_d\right)}{z_0}}\right)=10\lg \left({\displaystyle \frac{\sqrt{a_0^2+z_0^2+2a_0{z}_0\cos {\theta }_0}}{z_0}}\right)$$

(15)

Due to the nonlinearity induced by the magnetic force, the multi-steady states result from bifurcation, which causes the jumping phenomenon of vibration.

Based on the eigenvalues of Jacobian Matrix J in Equation (A10), the differential equation for the perturbation variables around the single steady state (a₀, θ₀) is derived, allowing the determination of critical parameters for both single- and multi-steady states. The eigenvalues λ of the Jacobian Matrix J can be obtained by solving the equation λ² – tr (J)λ + det (J) = 0. According to the Routh–Hurwitz criterion for the stability of the steady states, the state is asymptotic stability when tr (J) < 0 and det (J) > 0. In Equation (A10), as tr (J) is negative, the condition det (J) > 0 corresponds to the stable steady state. Because the multi-steady states include both stable and unstable solutions, the boundary separating the single steady-state range from the multi-steady-state range is given by the condition det (J) = 0, written as

$$\det \left(J\right)=c^2+{\displaystyle \frac{\left[M_i{a}_0{\omega }^2-F_R\left(a_0,{\theta }_0\right)\right]\left[-M_i{\omega }^2+K_R\left(a_0,{\theta }_0\right)\right]}{a_0{\omega }^2}}=0$$

(16)

From Equation (16), the boundary of excitation amplitude z₀ and damping coefficient c for single- or multi-steady states around the resonance is shown in Figure 4a for dis_opt = 32.4 mm. In Region I, there is only a single-steady state around the resonance, while in Region II, the multi-steady states occur. Increasing the damping coefficient c would raise the excitation amplitudes. Figure 4b shows the amplitude-frequency curves with the double-ring design for different parameters fixed in Region I and II. By comparison with the linear spring, the proposed QZS design by double magnetic rings has much lower effective isolation initial frequency and achieves the low-frequency isolation, which displays the advantages of the proposed nonlinearity. The system is stable if the excitation amplitude does not exceed the critical excitation amplitude. However, for the existence of multi-steady states, the vibration amplitude would jump to a large amplitude, which is the disadvantage of nonlinearity. Fortunately, as shown in Figure A2 in Appendix C, with the comparison of Region I between single magnetic ring (proposed in Ref. [17]) and double magnetic-ring designs, the proposed double-ring QZS isolator has a larger range for excitation amplitude for the single-steady state. Additionally, more comparisons of the isolator’s performance for different QZS isolators are shown in Appendix C, which verifies the advantages of the double-ring design principle for the wide-amplitude QZS property.

In addition, from the displacement transmissibility curves in Region I for different damping coefficients, as shown in Figure 4b, increasing the damping coefficient c would raise the displacement transmissibility in the high-frequency band. To address the requirement that the vibration is reduced to 10%, the transmissibility T_d should equal to −10 dB. The corresponding excitation frequency is around 6 Hz. Thus, we introduce the first index, which is the transmissibility at ω = 6 Hz, to evaluate the effectiveness of the vibration isolation, given as

$$T_e={\left.T_d\right|}_{\omega =6 \mathrm{Hz}}$$

(17)

3.2. Transient Vibration Spanning to the Next Layer

For the ML design of the proposed ML-QZS isolator, as the load increases, when the load drops to the next layer, the transient response is a free vibration. The transient response for variable loads can be solved using the following equation as

$$\left\{\begin{array}{l}{M}_i{\ddot{z}}_r^{\left(i\right)}+F_R\left(z_r^{\left(i\right)}\right)+c{\dot{z}}_r^{\left(i\right)}=-\Delta Mg\delta \left(t=0\right)\\ z_r^{\left(i\right)}\left(0\right)=0,{\dot{z}}_r^{\left(i\right)}\left(0\right)=0\end{array}\right.$$

(18)

According to Equation (18), the proposed ML-QZS isolator is only applicable to static loads. The variable load can only be an integer multiple of ΔM, not a real-time changing load.

Substituting ${\mathrm{z}}_{\mathrm{r}}^{\left(\mathrm{i}+1\right)}={\mathrm{z}}_{\mathrm{r}}^{\left(\mathrm{i}\right)}+\mathrm{dis}$ into Equation (18) and considering ${\mathrm{F}}_{\mathrm{R}}\left(\mathrm{dis}\right)=-\mathrm{Mg}$, the nonhomogeneous equation is transformed into a homogeneous equation as

$$\left\{\begin{array}{l}{M}_i{\ddot{z}}_r^{\left(i+1\right)}+F_R\left(z_r^{\left(i+1\right)}\right)+c{\dot{z}}_r^{\left(i+1\right)}=0\\ z_r^{\left(i+1\right)}\left(0\right)=dis,{\dot{z}}_r^{\left(i+1\right)}\left(0\right)=0\end{array}\right.$$

(19)

According to Ref. [11], the envelope curve of the transient vibration can be expressed as

$$z_r^{\left(i+1\right)}=dis\ast \exp \left[-b\left(a\right)t\right]$$

(20)

where b(a) is the attenuation function coupled with the transient vibration amplitude. The transient time t_s is defined as the time when the amplitude of z_r is reduced to less than 10% of the initial motion dis, given as

$$t_s=-\log \left(0.1\right)/b\left(0.1dis\right)$$

(21)

which is shown in Figure 5 for different values of c.

As shown in Figure 5, the envelope curves described by Equation (21) can obtain the effective vibration transient time, which obviously decreases with the increase of damping coefficient c. However, it is known that larger damping would worsen the isolation effectiveness in the high-frequency band. Thus, the optimization criterion for coordination of transient vibration attenuation and isolation effectiveness is needed in the high-frequency band.

3.3. Optimization Criterion for Dynamic Performances

According to the analysis in Section 3.2, the vibration isolation band and vibration attenuation are obtained using the Averaging Method. The two indices, including transient time and isolation effectiveness in the high-frequency band, mainly depend on the damping coefficient c. According to the practical needs, it is required to simultaneously reduce the transient time and the displacement transmissibility. As shown in Figure A3 and Figure A4 in Appendix D and Figure A5 and Figure A6 in Appendix E, the two objectives are contradictory and incompatible. To coordinate the vibration transient time for variable loads and transmissibility, the multi-objective optimization criterion is proposed as follows:

$$\begin{array}{l}\min F(\mu )=\left\{t_s(\mu ),T_e(\mu )\right\}\\ s.t.\mu =(c,dis)\in \mathbb{P}\\ \mathbb{P}=\{0.3 \mathrm{N}\cdot \mathrm{s}\cdot {\mathrm{m}}^{-1}\le c\le 1.2 \mathrm{N}\cdot \mathrm{s}\cdot {\mathrm{m}}^{-1},30 \mathrm{mm}\le dis\le 35 \mathrm{mm}\}\end{array}$$

(22)

where μ denotes the optimization parameter vector and P is the feasible domain of structural parameters c and dis. In function F(μ), t_s is the transient time for variable loads; T_e is the value of displacement transmissibility at 6 Hz. Although the optimum value of dis is dis_opt = 32.4 mm obtained using static analysis in Section 2.2, we vary it in the range of 30 mm to 35 mm to coordinate the two dynamic indices.

Figure 6 shows the feasible index solution that t_s and $T_{{\omega }_a}$ vary for different values of c and dis, marked with grey points. According to Equation (22), if there is no value of $\stackrel{⌢}{\mu }=(\stackrel{⌢}{c},\stackrel{⏜}{dis})$, so that F(μ) < $F(\stackrel{⌢}{\mu })$, then $F(\stackrel{⌢}{\mu })$ is the Pareto index solution and the parameter set $\stackrel{⌢}{\mu }=(\stackrel{⌢}{c},\stackrel{⏜}{dis})$ is called the Pareto front, as shown with red points in Figure 6a. All points for the Pareto front meet the design requirement for dynamic performance optimization. However, the variations of the two dynamic objectives, transient time t_s and transmissibility at high-frequency T_e, are contradictory, so the Knee Point is selected from all Pareto front points for coordination of the two indices. Figure 6b shows the transmissibility curve and the vibration transient at five Pareto front points from P₁ to P₅. Defining the Knee Point as the one with the longest distance from the line across the two edge points P₁ to P₅, we figure out that P₃ is the optimization indices for dynamic performances. At the Knee Point P₃, where dis = 32.2 mm and c = 0.64, the displacement transmissibility at 6 Hz is −9.68 dB, indicating that the vibration excitation is reduced to about 10% under continuous base excitation. Additionally, the vibration is effectively attenuated at 8.1 s as the system transitions from the upper layer to the lower one as the loading increases ΔM.

4. Isolation Platform and Experimental Verification for Structural Optimization

In this section, the prototype of the ML-QZS isolator platform is constructed, and the experiment processes are carried out to show the optimum design for the coordination of dynamic performances.

4.1. Experimental Prototype for One ML-QZS Isolation Mount

The experimental prototype for one ML-QZS isolation leg is shown in Figure 7a. In one leg, the inner magnetic ring is connected to the isolation platform with a rigid guide bar, supported by a linear spring in vertical direction. A linear bearing is connected between the frame and the guide bar (fixed with the platform). In one QZS layer, the double-layer outer magnetic rings are configured for the wide-amplitude QZS property as shown in the analysis in Section 2.2. In the double magnetic layer, the inner and outer diameters of the inner magnet are 5 mm and 10 mm; the inner and outer diameters of the two outer magnetic rings are 30 mm and 50 mm. The residual magnetic flux density of the magnetic rings is 0.9 T. Five standard linear bearings are utilized in the comparison of dynamic performances, with the types of KIF-LM-5LUU, PNY-LM-5UU, BKD-LM-5UU, IKO-LM-5UU, and MISIMI-LM5P. Through simple parametric regression testing, the damping coefficients for KIF, PNY, BKD, IKO, and MISIMI, are 0.35 N·s·m⁻¹, 0.44 N·s·m⁻¹, 0.65 N·s·m⁻¹, 0.89 N·s·m⁻¹, and 1.21 N·s·m⁻¹, respectively.

The experimental processes are constructed as shown in Figure 7b, typically consisting of four Steps. Step 1 is the structural design of the double-layer magnetic rings for the widest range for the QZS property and the smallest variation for different loads according to the optimization criterion (10). Step 2 is the static testing verification for the variation loading and range for the QZS property, which can provide the prediction for the excitation amplitude range for the single-steady state and avoid the jumping phenomenon. Step 3 involves optimizing the linear bearing and structure using the bio-objective optimization criterion (23). Step 4 is the dynamic testing verification for the dynamic performances including the transient vibration attenuation at the adjacent layer for increasing the loading and the vibration isolation effectiveness in one double-ring QZS layer. In the dynamic testing, the base excitation is generated by the exciter (VE-5110, Yiheng Technology Co., Ltd., Hangzhou, Zhejiang, China), which is driven by a signal generator (ECON-MI8008, Yiheng Technology Co., Ltd., Hangzhou, Zhejiang, China) with a power amplifier (ECON-H181A, Yiheng Technology Co., Ltd., Hangzhou, Zhejiang, China). The vibration frequency and amplitude of the exciter are controlled using the software (TestEngineX 4.2.40) of the signal generator. In this paper, the vibration amplitude is set to 1.2 mm for all dynamic tests. Two laser sensors (KathMaticKVD-4525L, KathMaticKVD-4525S, KathMatic Technology Co., Ltd, Nanjing, Jiangsu, China) are placed to estimate the absolute vibrations of the base and the isolation platform in the time domain, respectively. Then, the data, measured using laser sensors, are collected and sent to the computer, and analyzed using the signal processing software to obtain the amplitude.

4.2. Experiment Results

According to the optimization criterion of the structural design in Equation (10), the vertical positive-spring stiffness is k = 0.196 N·mm⁻¹ and the optimum value of dis is dis_opt = 32.0 mm. In order to show the comparison of the QZS range and loading variation for different values of structural parameter dis, we construct other three prototypes with dis_opt = 32 mm, dis = 35 mm, and dis = 40 mm. Then, through quasi-static testing using the tensile testing machine, the restoring forces F_R for different dis_opt are obtained as shown in Figure 8. For the dis_opt, the initial load mass is M₀ = 238.6 g, with 10.2 mm QZS range and variation load ΔM = 596 g. While for the cases dis = 35 mm and dis = 40 mm, the loading variations are both larger and the QZS ranges are both narrower compared to the optimum case.

Based on the above experimental results from tensile testing, the main structural parameter dis is chosen at the optimum value dis_opt. After fixing the structural parameter dis, the Pareto optimization, as shown in Equation (22), is carried out to determine the optimum damping coefficient for dynamic performances, including vibration attenuation and isolation effectiveness, as shown in Figure 9. As shown in Figure 9a, the Pareto front points based on theoretical and experimental results are given, represented by point P_i and ${\stackrel{⌢}{P}}_i$, respectively. In the experimental analysis, five Pareto front points are shown with the damping coefficients for the five different linear bearings shown in Figure 7a. For the linear bearing with the model number BKD-LM-5UU, the damping coefficient c = 0.65 N·s·m⁻¹ is the closest value to the Knee point. Then, the experimental results for displacement frequency and transient attenuation for the five prototypes with structural parameters and damping coefficients at the five Pareto front points are shown in Figure 9b.

As shown in Figure 8, the theoretical restoring force is not completely consistent with the measured value, which is caused by nonlinear damping. As shown in Figure 9a, the transient response time in the optimization test is significantly smaller than the calculation result, but has no effect on the steady-state response results and the parameter of optimization result.

By comparing the displacement transmissibility frequency curves and vibration transient attenuations with fixed parameters among the five Pareto front points, the damping coefficient at ${\stackrel{⌢}{P}}_3$ can coordinate the two dynamic indices. For structural parameter dis_opt = 32 mm and BKD-LM-5UU with c = 0.65 N·s·m⁻¹≈c_opt, the effective vibration isolation begins at 2.5 Hz, the displacement transmissibility reaches to −10 dB at 6 Hz, and the transient vibration is attenuated in 4 s. The corresponding videos are shown in Video S1. Therefore, the proposed bi-objective optimization can coordinate multiple dynamic performances, and the proposed ML-QZS isolation mount has low-frequency isolation effectiveness for variable loads.

4.3. Isolation Platform with ML-QZS Property for Variable Loads

The isolation platform, as displayed in Figure 10, is constructed using four isolation mounts with the ML-QZS property design for dis_opt = 32 mm and BKD-LM-5UU with c = 0.65 N·s·m⁻¹≈c_opt. When manufacturing the isolation platform, each leg’s damping is identified to control parameter consistency and reduce error. In the dynamic experimental prototype, as shown in Figure 10, a six-DOF motion platform simulates the low-frequency excitation (NJQK-STEWRT, Nanjing All Controller Electronic Technology Co., Ltd. Nanjing, Jiangsu, China), two accelerometers (DYTRAN-3023A, Dytran Instruments, Inc., Chatsworth, CA, USA) are fixed on the isolation platform and the base, and the signals are sent to the data acquisition and processing system. The absolute vibration displacement is obtained by integrating the acceleration data twice in time, and then the amplitude is obtained.

In the dynamic experiment for the isolation platform with optimization structural parameters, the isolation performances are demonstrated by three experiments as shown in Figure 11. The first experiment is to show the vibration attenuation effect for adding multiple equipment, the second is to show the isolation effect in different layers for low-frequency excitation, and the third is to show the vibration suppression effect for low-frequency excitation and adding load simultaneously.

As shown in Figure 11a, vibration transient attenuation times for adding the loads as multiple equipment are estimated. We apply the load ΔMg at t = 5 s, t = 13 s, and t = 19 s, and the corresponding theoretical equation for the dynamic performance can be given as

$$\left\{\begin{array}{l}{M}_0\ddot{z}+F_R\left(z\right)+c\dot{z}=\Delta Mg{\left.\delta \left(t\right)\right|}_{t=5\mathrm{s}}+\Delta Mg{\left.\delta \left(t\right)\right|}_{t=13\mathrm{s}}+\Delta Mg{\left.\delta \left(t\right)\right|}_{t=19\mathrm{s}}\\ z\left(0\right)=0,\dot{z}\left(0\right)=0\end{array}\right.$$

(23)

where the load was calibrated before the experiment, the variable load for one mount is 596 g.

For the experimental results, the attenuation times for vibration transients from the first layer to the second, third, and fourth layers are t_s1 = 3.8 s, t_s2 = 4.9 s, and t_s3 = 5.2 s, respectively, which reveals that the vibration can be effectively eliminated with the optimum damper chosen from the five model numbers. The corresponding videos are shown in Video S2.

As shown in Figure 11b, the vibration isolation in different layers with different loads are shown. We apply the excitation with amplitude z₀ = 1.2 mm and with frequency ω from 1 Hz to 6 Hz. The corresponding theoretical equation for dynamic performance is given as

$$\left\{\begin{array}{l}{M}_i{\ddot{z}}_r+F_R\left(z_r\right)+c{\dot{z}}_r=M_{i}g-M_i{\ddot{z}}_e=M_{i}g+M_i{z}_0{\omega }^2\cos \omega t\\ z_r\left(0\right)=0,{\dot{z}}_r\left(0\right)=0\end{array}\right.$$

(24)

where the definition of M_i is given by Equation (5). The displacement transmissivity T for the theoretical analysis is obtained using Equation (15). As shown in Figure 11b, the effect isolation band begins from ω_e = 2.5 Hz, ω_e = 1.3 Hz, ω_e = 1.06 Hz, and ω_e = 0.94 Hz in the first, second, third, and fourth layers with the excitation amplitude z₀ = 1.2 mm. The theoretical results are consistent with the experimental results, proving that the conclusion regarding the system’s stability is effective in the theoretical study. Due to the increase in loading, the frequency band for the subjacent layer becomes wider. Thus, the low-frequency isolation for different loads can be realized. The corresponding videos are shown in Video S2.

Finally, as shown in Figure 11c, the vibration suppression effectiveness when simultaneously applying low-frequency base excitation and variable loads demonstrated. As the base excitation z₀cos ωt with z₀ = 1.2 mm and ω = 3 Hz, for t ranging from 0 s to 11 s, the absolute vibration amplitude of the isolation platform is 50% smaller than z₀, as verified by the displacement transmissibility curves shown in Figure 11b as T = −3 dB. When applying the ΔMg at t = 10.5 s, the vibration equilibrium drops from O₁ to O₂ in the second layer. The vibration attenuation time is t_s = 3.9 s and the vibration amplitude of the isolation platform is smaller than the excitation amplitude as T = −10 dB for t from 14.5 s. The corresponding videos are shown in Video S2.

Therefore, the dynamic performances shown in Figure 11 demonstrate that the proposed isolation platform is suitable for low-frequency dynamic environment with variable loads due to the ML-QZS design. By optimizing the structural parameters and damping coefficient, low-frequency isolation effectiveness and rapid vibration attenuation can be realized.

Table 1. The comparison of isolator’s performance for different isolators.

Vibration Isolator	QZS Range/Design Parameter (kQZS < 0.02 ks)	$\mathbf{Starting} \mathbf{Isolation} \mathbf{Frequency}{\mathit{\omega }}_{\mathit{e}}$(Hz)	Vibrable Load
Linear spring damping isolator	0	6.5	Yes
QZS with triple nonlinear spring [43]	0.2	3.0	No
QZS with single magnetic ring [17]	0.16	3.0	No
QZS with X-shaped structure [28]	0.105	3.5	No
QZS with electromagnetic constant force mechanism [53]	0.112	2.5	Yes
QZS with dual electromagnetic nonlinear stiffness mechanisms [54]	0.250	3.5	Yes
QZS with Multi-layer cam-roller [55]	0.5	3.6	Yes
QZS with mechanical metamaterials [57]	0.08	17.6	Yes
The proposed ML-QZS isolator	0.32	2.5	Yes

shows the performance comparison for different vibration isolators. Since the structure size has a great influence on the QZS range, we take the ratio of the QZS range and the parameters of structure design as the evaluation index. The starting vibration isolation frequency is chosen as the second evaluation index to evaluate the first-order stiffness of the QZS system at the equilibrium point.

The comparison results show that the proposed quasi-zero-stiffness isolator not only has advantages in the QZS range and the starting vibration isolation frequency range but also can adapt to variable loads.

5. Conclusions

This study provides the design principle of an ML-QZS vibration isolation platform, constructed using multiple magnetic rings and positive-stiffness spring. The proposed isolation platform is applicable for low-frequency and large-amplitude dynamic excitation with variable loads. The main results are as follows.

a. We propose the optimization criterion of the structure for the ML-QZS vibration isolation platform applied in the low-frequency and wide-amplitude dynamic environments. The wide-range negative-stiffness property is realized using multiple magnetic rings design, which breaks the locality of the QZS property around equilibrium in traditional triple-springs design.

b. Two dynamic performances, including vibration attenuation time and isolation effectiveness in a broad frequency band, are considered for obtaining the optimum structure and damper. Considering that the variations of the two dynamic performances are contradictory, the Pareto optimization criterion is given for the coordination of different indices.

c. Different groups of experimental prototypes are constructed for comparison. The experimental results not only verify the theoretical results and design principle for the ML-QZS property but also verify the two optimization criteria proposed in this study. The isolation platform is successfully applied in low-frequency excitations with variable loads.

In summary, based on the design principle proposed, the proposed ML-QZS vibration platform has remarkable potential applications in ocean dynamic environments for assembly of variable equipment mass. But the proposed vibration isolation platform is only suitable for integer multiples of variable loads. So, in the future, the quasi-zero-stiffness vibration isolator for adaptive stepless, variable loads will be realized by controlling electromagnetic force.