# Implementation and Experiment of a Novel Molecular Spring Isolator

Authors: Gaofa Nie; Xuhong Miao; Yuhang Tang; Qian Chen
Journal: Shock and Vibration
 (2022)

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## Abstract

A molecular spring isolator comprising ZIF-8 with hydrophobic micropores, water, and a cylinder/piston unit is introduced. Liquid water can be intruded into the micropores of ZIF-8 under critical pressure. In the meantime, mechanical energy is stored. And once the external pressure is withdrawn, spontaneous capillary evaporation will occur. The stored energy, meanwhile, is released. This behavior is similar to a coil spring. The Laplace–Washburn equation is used to describe the force equilibrium of a single water column in a micropore. After that, the result of water infiltrating a great deal of hydrophobic pores is revealed by theoretical analysis and an experiment. An averaging method is devoted to computing the primary resonance response and its accuracy is proved by the fourth-order Runge–Kutta method. Furthermore, the energy transmissibility is used to estimate its vibration isolation capability. Finally, a vibration isolation test indicated that the inherent frequency of the isolator is as low as 1.3 Hz.


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## Body

## 1. Introduction

Vibration isolation has been more and more demanding in various industrial areas, such as varieties of vehicles, precise instruments, and megastructures. Vibration control technologies have been a hot issue. For traditional linear isolators, the contradiction between wide frequency bandwidth vibration absorption and static deflection cannot be resolved. However, novel passive vibration isolators have been attracting engineers’ attention all the time because of their lower cost, higher reliability, and simpler structure, especially the nonlinear passive vibration isolation technique [1, 2].As one solution to the above-mentioned contradiction, kinds of high-static-low-dynamic (hereinafter referred to as HSLD) stiffness isolators have been progressed [3]. An HSLD stiffness suspension used in helicopter seats was studied to reduce the vibration in both vertical and lateral directions [4]. An ultra-low-frequency vibration isolation (quasizero-stiffness isolators) was proposed in the face of neonatal transport [5]. Based on the cam-roller-spring mechanism, an HSLDS isolator was built and proved to possess superior performance to traditional linear isolators [6]. However, negative stiffness is commonly found in these kinds of nonlinear passive vibration isolation techniques. X-shaped vibration isolation was investigated because of its nonlinear stiffness advantages and damping dissipation characteristics [7]. By imitating the principle of a molecular spring, a mechanical molecular spring isolator with HSLD stiffness was mentioned [8]. However, these technical solutions usually face some kinds of problems, such as complex structure, stability, and device’ geometrical nonlinearities. So, it is still valuable to study the new passive vibration isolation technology.Hydrophobic nanoporous materials have been extensively used for gas absorption, separation, heterogeneous catalysis, sewage treatment, etc. However, some research findings have suggested that storage or release of energy is realized during the process of liquid water intruding into or escaping from ZIF-8′ hydrophobic nanopores (see Figure1). And a stable reciprocal transformation between mechanical energy and interfacial energy is found [9–12]. During the processes of loading and unloading, the mechanical performance of the mixture comprising water and silicalites-1 zeolite presents the same function as a spiral spring. And then the mixture is named molecular spring [13]. From a microscopic viewpoint, the process of forced intrusion can be elucidated by the development of new intermolecular bonds between water molecules and zeolites’ framework [14]. Zeolitic imidazolate frameworks were considered as an attempt to realize engineering applications after understanding the underlying water intrusion and extrusion mechanisms [15]. Otherwise, many other hydrophobic zeolite materials could be used as working medium of molecular springs, too.Figure 1
Intrusion of water into hydrophobic nanopores.The Laplace–Washburn equation is used to account for the Laplace capillary pressure and verified through experiment and simulation methods [16]. For hydrophobic mesopores (MCM-41 materials), a macroscopic capillary model is proved to be applicable to the interpretation of the intrusion process. However, the model is not valid for the mechanism of the extrusion pressure [17]. Molecular dynamics simulations are employed to investigate the process of water into/out of hydrophobic microporous materials and the free volume of the zeosil unit cells is deemed to make a difference in the volume of water being intruded [18].In summary, the current research on hydrophobic nanoporous materials is mainly focused on their physicochemical characteristics, molecular simulations of water intruding, and synthesizing new materials. However, application research on vibration isolation of these materials is rare. A molecular spring isolator based on silicalite-1 is reported. And its HSLD stiffness is testified. However, the critical pressure is up to 60 MPa and the experimental analysis is also ignored [19].The research on mechanical applications of 2-methylimidazole zinc salt (ZIF-8), the same as other hydrophobic microporous materials, especially in the field of vibration isolation is not found in related literature. In this paper, a novel HSLD stiffness isolator based on ZIF-8 is brought up. First, the working mechanism of the ZIF-8/water system is explained in detail in this section. Second, a mechanical model and a quasistatic experiment are presented. Third, the primary resonance response of the ZIF-8 HSLD stiffness isolator (named ZIFI) is solved by an averaging method. Finally, a vibration isolation test is performed.


## 2. ZIFI Mechanical Model

### 2.1. Critical Pressure

Figure2 demonstrates a microforce balance between nonwetting water and solid surface. The surface tension of a solid and liquid is σS and σL, respectively. The interfacial tension between water and solid surface is denoted as γLS. θ is the equilibrium contact angle, as shown in Figure 2. Based on Young’s law, a macroscopic capillary model among them can be written as follows:(1)σL×cosθ+γLS=σS.Figure 2
A microforce balance between nonwetting liquid and solid surface.As demonstrated in Figure3, liquid water will be squeezed into microporous materials once the Pout is bigger than the sum of the two Pc and Pair. σL and θ (from 90 to 180°) are defined as the same as above. Dm is the radius of the pore. Once the Pout is withdrawn, a spontaneous extrusion occurs.Figure 3
Model of liquid water in single nanopore space.Assumed that water enters the micropore at a uniform velocity, the moving distance of the water isx, and the change of kinetic energy of the water is ignored. Then, the process can be expressed as follows:(2)γLS−σL×π×Dm×x=Pout×π×Dm×x.Pc is equal to (Pout-Pair). According to equations (1) and (2), the critical pressure Pc is given by(3)Pc=4×σL×cosθDm.During liquid water squeezed into microporous materials, the surface free energy will be generated and stored. Because microporous materials have a large surface area, the interface Ω needs to be considered. The Gibbs free energy (ΔG) is given by(4)ΔG=−σL×cosθ×ΔΩ.To remove the energy stored in the nanopore space, the spontaneous process of extrusion of the water out of the nanopore space begins once the external pressure disappears. As a result, the mechanism of hydrophobic mesopores forced to fill with and spontaneously escape liquid water works like a real spring.


### 2.2. Testing Apparatus and Stiffness Model

ZIFI’s diagram is presented as shown in Figure4. The ZIFI consists of three main parts, that is, the molecular spring chamber full of the mixture of water and ZIF8 (as shown in the red-dotted box), the container, and the piston rod.Figure 4
Schematic diagram of the ZIFI.At the beginning, as the piston rod moves downwards, the pressure in the confined space (namely, the red-dotted box) increases and only water is compressed at this stage. When the external pressure is greater than thePc, liquid water will be forced into ZIF8’s nanopores. After all of the nanopores have been filled, water is the unique compressed object again. During the whole process, the evolution of the ZIFI stiffness presents successively three stages, that is, high stiffness, low stiffness, and high stiffness.D is the diameter of the piston rod. The compressibility of water is ηW and its initial volume is Vw. Define the measured porous volume of ZIF-8 as Vpv, the critical pressure as Pc, the saturation pressure as Ps, and the material mass as m. P0 is the atmospheric pressure.In the first stage, the change in water volume (∆VW) in the container with the external pressure Pout is given by(5)ΔVW=Vw⋅ηW⋅Pout−Pair.Actually,Pc (about 18 MPa) is 180 times larger than Pair (0.1 MPa). Thus, Pair can be ignored.Let the vertical displacement of the piston rod bex. Then, the equilibrium equation can be built by(6)ΔVW=x⋅14⋅π⋅D2,that is,(7)Pout=x⋅1/4⋅π⋅D2Vw⋅ηW.According toF = PS and K = F/Δx, the force-displacement relationship and the stiffness expression in this stage can be, respectively, obtained as(8)F=x⋅1/4⋅π⋅D22Vw⋅ηW,K1=1/4⋅π⋅D22Vw⋅ηW0,1.In the second stage, the change in water volume is mainly determined by the ZIF-8′ porous volumeVpv, and the compressed water plays a secondary role. So, the stiffness expression in this stage can be written as follows:(9)K2=FΔx=3.14⋅D/22⋅Ps−PcVpv⋅m/3.14⋅D/22=3.14⋅D/222⋅Ps−PcVpv⋅m.In the last stage, because the nanopores are filled with water, the downward displacement of the piston rod is caused by the compressed water. Actually, the principle is the same as in the first stage. So, the stiffness expression in this stage is (Vw1 is the volume of water after the second stage)(10)K3=1/4⋅π⋅D22Vw1⋅ηW.The quasistatic characteristics of the ZIFI were tested by an electrohydraulic servo fatigue machine, as illustrated in Figure5. The force sensor is used to measure the force. The displacement is recorded by the actuator. According to GB/T 15168-2013 (a Chinese national standard), the downward velocity of the actuator is set to 5 mm/min. After the preloaded process, the stable static force-displacement relationship can be measured by reloading.Figure 5
Quasistatic loading test.The outstanding HSLD characteristic of the ZIFI is shown in Figure6. The test data and the theoretical prediction of the ZIFI are expressed in the blue line and the black line, respectively. The two lines prove the correctness of the theoretical model. The deviation, in the first stage, is due to residual air in the ZIFI. In the last stage, these two curves are going to be parallel. The dotted lines are guides for the eyes. The red line stands for the stiffness of the control group. As a control group, the inside of the ZIFI is only filled with water. Comparing the red line and the blue line, the phenomenon of water intruding the nanopores of ZIF-8 is proved.Figure 6
The force-displacement relationship.


## 2.1. Critical Pressure

Figure2 demonstrates a microforce balance between nonwetting water and solid surface. The surface tension of a solid and liquid is σS and σL, respectively. The interfacial tension between water and solid surface is denoted as γLS. θ is the equilibrium contact angle, as shown in Figure 2. Based on Young’s law, a macroscopic capillary model among them can be written as follows:(1)σL×cosθ+γLS=σS.Figure 2
A microforce balance between nonwetting liquid and solid surface.As demonstrated in Figure3, liquid water will be squeezed into microporous materials once the Pout is bigger than the sum of the two Pc and Pair. σL and θ (from 90 to 180°) are defined as the same as above. Dm is the radius of the pore. Once the Pout is withdrawn, a spontaneous extrusion occurs.Figure 3
Model of liquid water in single nanopore space.Assumed that water enters the micropore at a uniform velocity, the moving distance of the water isx, and the change of kinetic energy of the water is ignored. Then, the process can be expressed as follows:(2)γLS−σL×π×Dm×x=Pout×π×Dm×x.Pc is equal to (Pout-Pair). According to equations (1) and (2), the critical pressure Pc is given by(3)Pc=4×σL×cosθDm.During liquid water squeezed into microporous materials, the surface free energy will be generated and stored. Because microporous materials have a large surface area, the interface Ω needs to be considered. The Gibbs free energy (ΔG) is given by(4)ΔG=−σL×cosθ×ΔΩ.To remove the energy stored in the nanopore space, the spontaneous process of extrusion of the water out of the nanopore space begins once the external pressure disappears. As a result, the mechanism of hydrophobic mesopores forced to fill with and spontaneously escape liquid water works like a real spring.


## 2.2. Testing Apparatus and Stiffness Model

ZIFI’s diagram is presented as shown in Figure4. The ZIFI consists of three main parts, that is, the molecular spring chamber full of the mixture of water and ZIF8 (as shown in the red-dotted box), the container, and the piston rod.Figure 4
Schematic diagram of the ZIFI.At the beginning, as the piston rod moves downwards, the pressure in the confined space (namely, the red-dotted box) increases and only water is compressed at this stage. When the external pressure is greater than thePc, liquid water will be forced into ZIF8’s nanopores. After all of the nanopores have been filled, water is the unique compressed object again. During the whole process, the evolution of the ZIFI stiffness presents successively three stages, that is, high stiffness, low stiffness, and high stiffness.D is the diameter of the piston rod. The compressibility of water is ηW and its initial volume is Vw. Define the measured porous volume of ZIF-8 as Vpv, the critical pressure as Pc, the saturation pressure as Ps, and the material mass as m. P0 is the atmospheric pressure.In the first stage, the change in water volume (∆VW) in the container with the external pressure Pout is given by(5)ΔVW=Vw⋅ηW⋅Pout−Pair.Actually,Pc (about 18 MPa) is 180 times larger than Pair (0.1 MPa). Thus, Pair can be ignored.Let the vertical displacement of the piston rod bex. Then, the equilibrium equation can be built by(6)ΔVW=x⋅14⋅π⋅D2,that is,(7)Pout=x⋅1/4⋅π⋅D2Vw⋅ηW.According toF = PS and K = F/Δx, the force-displacement relationship and the stiffness expression in this stage can be, respectively, obtained as(8)F=x⋅1/4⋅π⋅D22Vw⋅ηW,K1=1/4⋅π⋅D22Vw⋅ηW0,1.In the second stage, the change in water volume is mainly determined by the ZIF-8′ porous volumeVpv, and the compressed water plays a secondary role. So, the stiffness expression in this stage can be written as follows:(9)K2=FΔx=3.14⋅D/22⋅Ps−PcVpv⋅m/3.14⋅D/22=3.14⋅D/222⋅Ps−PcVpv⋅m.In the last stage, because the nanopores are filled with water, the downward displacement of the piston rod is caused by the compressed water. Actually, the principle is the same as in the first stage. So, the stiffness expression in this stage is (Vw1 is the volume of water after the second stage)(10)K3=1/4⋅π⋅D22Vw1⋅ηW.The quasistatic characteristics of the ZIFI were tested by an electrohydraulic servo fatigue machine, as illustrated in Figure5. The force sensor is used to measure the force. The displacement is recorded by the actuator. According to GB/T 15168-2013 (a Chinese national standard), the downward velocity of the actuator is set to 5 mm/min. After the preloaded process, the stable static force-displacement relationship can be measured by reloading.Figure 5
Quasistatic loading test.The outstanding HSLD characteristic of the ZIFI is shown in Figure6. The test data and the theoretical prediction of the ZIFI are expressed in the blue line and the black line, respectively. The two lines prove the correctness of the theoretical model. The deviation, in the first stage, is due to residual air in the ZIFI. In the last stage, these two curves are going to be parallel. The dotted lines are guides for the eyes. The red line stands for the stiffness of the control group. As a control group, the inside of the ZIFI is only filled with water. Comparing the red line and the blue line, the phenomenon of water intruding the nanopores of ZIF-8 is proved.Figure 6
The force-displacement relationship.


## 3. Primary Resonance Response

For this vibration isolation system, its equilibrium position (see Figure6) locates at the center of the second stage. Concerning the piecewise nonlinear stiffness, a cubic polynomial (as shown in Figure 7) for fitting the force-displacement relationship in the second stage is adopted. The amplitude of vibration studied in this paper is limited to this stage, too. Then, an absorber with linear viscosity is appended in parallel. Finally, the kinematic equation of the ZIFI, under simple harmonic vibration, can be expressed as follows:(11)Mx¨+cx˙+fx=Fcosωt,where M and c are the load and damping coefficients, respectively. f(x) is the fitting nonlinear elastic restoring force.where a1, a2, and a3 are the stiffness coefficients of each order in the working stage. k3 and k1 are the stiffness of the other two stages. x1 and x2 are the turning points (Pc and Ps) of the different stiffness, and(12)fx=k3x+Fb1,x>x1,a1x+a2x2+a3x3,x2≤x≤x1,k1x+Fb2,x<x2,Figure 7
Fitting of the force-displacement relation.(13)Fb1=a1+a2x1+a3x12⋅x1−k3x1,Fb2=a2+a2x2+a3x22⋅x2+k1x2.Based on an averaging method, equation (11) needs to be simplified as(14)x¨+ω2x=ε−c0x˙+F0cosωt−gx,where(15)gx=ω02−σx+fb1,x>x1,−σx+r2x2+r3x3,x2≤x≤x1,ω02−σx+fb2x<x2,in which ε=k0/a1−1, c0=c/Mε, F0=F/Mε, ω0=a1/M, ω2=ω02+εσ, r2=a2/Mε, r3=a3/Mε, fb1=Fb1/Mε, and fb2=Fb2/Mε.Whenε = 0, equation (14) can be replaced by(16)x¨+ω2x=0.Then, the solution of equation (16) is given by(17)x=Acosωt−θ.And the derivative ofx is(18)x˙=−Aωsinωt−θ,where A is the amplitude. When ε ≠ 0, ε and θ are counted as functions with respect to time (t). Hence, a new equation can be introduced as follows:(19)φ=ωt−θ.Then, after differentiating with timet, equation (17) is rewritten as(20)x˙=A˙cosφ−Aωsinφ+Aθ˙sinφ.Obviously, a new expression can be deduced by combining equations (18) and (20). Namely,(21)A˙cosφ+Aθ˙sinφ=0.Equally, equation (18) is reformulated as(22)x¨=−A˙ωsinφ−Aω2cosφ+Aθ˙ωcosφ.Based on the above-mentioned equations, the differential equations ofA˙ and θ˙ can be obtained, respectively, as follows:(23)A˙=−εω−c0x˙+F0cosωt−gxsinφ,(24)θ˙=εωa−c0x˙+F0cosωt−gxcosφ.BecauseA and θ are slowly varying functions, the right-hand sides of equations (23) and (24) can be replaced approximately by the average of φ over a period. Then, their averaging equation is given by(25)A˙=−εω⋅12π∫02π−c0⋅−Aωsinφ+F0cosωt−gAcosφsinφdφ=−ε2ωΦA,ω−F0sinθ,(26)θ˙=εωA⋅12π∫02π−c0⋅−Aωsinφ+F0cosωt−gAcosφcosφdφ=ε2ωAΨA,ω+F0cosθ,in which(27)ΦA,ω=1π∫02π−−c0Aωsinφ+gAcosφsinφdφ,ΨA,ω=1π∫02π−−c0Aωsinφ+gAcosφcosφdφ.A new equation can be achieved through equations (25) and (26)as follows:(28)dAdθ=−aΦA,ω−F0sinθΨA,ω+F0cosθ.After the excitation parameters are set, the singular point in theA-θ phase plane confirmed by equation (28) is the steady-state response of equation (14). Then, this plane is evaluated as follows:(29)ΦA,ω−F0sinθ=0,ΨA,ω+F0cosθ=0.Eliminating the variable (θ), the expressions with respect to amplitude-frequency and phase-frequency can be developed. They are defined as follows:(30)Φ2+Ψ2+F02=0,tanθ=−ΦΨ.The expressions of Φ and Ψ are deduced through integral transforms of equation (27) and given as follows:(31)Φ=c0Aλω0,Ψ=−14A−4σ+3r3A2.For this ZIFI system, there may be unexcited frequency, such as sub- and superharmonic, in the nonlinear steady-state response. Hence, in this section, energy transmissibilityη of the system rather than the traditional force transmissibility is used as the evaluation index of vibration isolation performance and defined as follows:(32)η=20⋅lgP1P0,where P1 is the power of transmission force and P0 is the power of excitation force.The value ofη is an important reference for evaluating the performance of vibration isolators. If the value of η is not more than zero dB, the vibration isolator makes a beneficial difference in decreasing the energy transferred from the external input to the base. If η≤−20 dB, the vibration isolator possesses excellent performance in vibration isolation. That is, the power transferred to load is reduced by more than 90% after a certain frequency.P1 and P0 come in two forms. One of them is achieved as follows:(33)P1=ω2⋅π⋅∫02⋅π/ωFr2dt,P0=ω2⋅π⋅∫02⋅π/ωF0⋅cosωt2dt.In view of the fact that the restorative forceFr is not easy to get, this form is not the first choice. In the test system, the acceleration signal of load M is easily obtained through the acceleration sensor. So, Fr can be rewritten as follows:(34)Fr=F0⋅cosωt−M⋅x¨.Therefore, another form ofP1 and P2 is given by(35)P1=ω2⋅π⋅∫02⋅π/ωF0⋅cosωt−M⋅x¨2dt,P0=ω2⋅π⋅∫02⋅π/ωF0⋅cosωt2dt.Because the acceleration signals are discrete signals, the restorative force and the excitation force both need to be rewritten as time series. Thus, equation (35) is redefined as follows:(36)P1=ω2⋅π⋅∑i=0NF0⋅cosωt−M⋅x¨2Δt,P0=ω2⋅π⋅∑i=0NF0⋅cosωt2Δt.


## 4. Experiment Results and Discussion

An experimental system was built and tested to prove the identified model of the ZIFI. The experimental system makes up of a vibration exciter, a payload, the ZIFI, a power amplifier, and a dynamic signal analyzer HP35670A (see Figure8).Figure 8
Dynamic experiment setup.The vibration exciter provides the excitation force. The payload puts the ZIFI in the static equilibrium position. In constant current mode, the payload amplitude is adjusted by changing the voltage of the power amplifier. For convenience, the payload amplitude is not more than the above-mentioned second stage. The HP35670A is used to analyze the experimental data collected by force and displacement sensors. In this paper, the payload is 230 kg. And frequency response range and sensitivity of the sensor are 0.1 Hz–500 Hz and 10000 mV/g, respectively.According to Figure9, the important role of ZIF-8 in the ZIFI is clearly discovered. Compared to the control group (the black line) which the apparatus is only filled with pure water, the inherent frequency of the ZIFI decreases from 3.8 Hz to 1.3 Hz. The solid lines (including the black line and the blue line) are test results. These dotted lines are results of the averaging method. The others (represented by the circles) are the results of the fourth-order Runge–Kutta method (RKM). As indicated in Figure 9, the experimental results are almost the same as the theoretical results. This phenomenon proves the reliability of theoretical analysis and experimental results. And once the excitation frequency is above 5 Hz, the energy transmissibility of the ZIFI is below −20 dB, which illustrates the ZIFI possesses excellent performance in vibration isolation. Besides, according to this figure, the accuracy of the analytical method is well proved by the RKM.Figure 9
Energy transmissibility curves of the ZIFI.Figure10 investigates the effect of material mass on energy transmissibility. As indicated in Figure 10, the natural frequency of the ZIFI decreases with increasing material mass. And another useful conclusion is that the mass of ZIF-8 does not affect the accuracy.Figure 10
Energy transmissibility curves of the ZIFI with different material mass.Figure11 shows the difference among the ZIFI and two kinds of traditional linear isolators, as indicated by the force-displacement relation. The dotted lines are only guide for the eyes. Compared to the linear isolator 1 (shown by the black line) whose payload (shown as point A) is equal to the ZIFI’s, the latter’s stiffness indicated by the slope of the blue line in the working stage is lower than the stiffness of this isolator shown by the slope of the black line under the same static displacement (40 mm). The lower stiffness is conducive to vibration isolation. Compared to the linear isolator 2 whose payload is also equal to the ZIFI’s, the static displacement (40 mm) of the ZIFI is shorter than the linear isolator 2′ static displacement (90 mm) when the ZIFI′ stiffness in the working stage is equal to the stiffness of isolator 2. In limited installation space, the shorter static displacement means the ZIFI will be a better choice. In conclusion, the ZIFI, as a novel HDLS isolator, has advantages over traditional linear vibration isolators.Figure 11
Comparison with linear vibration isolators.


## 5. Conclusions

This paper expanded the application of hydrophobic nanoporous materials and reported a novel HSLD stiffness isolator (ZIFI based on ZIF-8). The feasibility of ZIF-8 used to build an HSLD vibration system was proved. Its static stiffness model was reasoned by mechanics principles and proved by a quasistatic loading test. An averaging method was used to calculate ZIFI’s primary resonance response. And then the energy transmissibility was proposed. Finally, theoretical analysis in this work was proved by the dynamic experiment design.In short, the ZIFI’s advantages over existing technologies are summarized as follows: (1) ZIF-8 is a piece of good advice for vibration isolation, especially the low-frequency vibration isolation situation; (2) thanks to its lower intrusion pressure than other hydrophobic nanoporous materials (such as the water/silicalite-1 system), engineering application of this material will be worthy of further study; and (3) compared to other HSLD stiffness isolators, the ZIFI has the advantages of simpler structure and heavier payload.



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*Source: 7174089-2022-04-07.xml*