# Modeling and Dynamic Properties of a Four-Parameter Zener Model Vibration Isolator

Authors: Wen-ku Shi; Chen Qian; Zhi-yong Chen; Yang Cao; Heng-hai Zhang
Journal: Shock and Vibration
 (2016)

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## Abstract

To install high-performance isolators in a limited installation space, a novel passive isolator based on the four-parameter Zener model is proposed. The proposed isolator consists of three major parts, namely, connecting structure, sealing construction, and upper and lower cavities, all of which are enclosed by four segments of metal bellows with the same diameter. The equivalent stiffness and damping model of the isolator are derived from the dynamic stiffness of the isolation system. Experiments are conducted, and the experiment error is analyzed. Test results verify the validity of the model. Theoretical analysis and numerical simulation reveal that the stiffness and damping of the isolator have multiple properties with different exciting amplitudes and structural parameters. In consideration of the design of the structural parameter, the effects of exciting amplitude, damp channel diameter, equivalent cylinder diameter of cavities, sum of the stiffness of the bellows at the end of the isolator, and length of damp channel on the dynamic properties of the isolator are discussed comprehensively. A design method based on the parameter sensitivity of the isolator’s design parameter is proposed. Thus, the novel isolator can be practically applied to engineering and provide a significant contribution in the field.


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## Body

## 1. Introduction

Microvibration usually refers to low-level mechanical vibration or disturbance in the microgravity environment; such vibrations can be induced by onboard mechanical moving systems, including mechanically tunable optical filters, cryocoolers, solar array drive mechanisms, antenna pointing mechanisms, and reaction wheel assemblies [1–3]. The main microvibration of onboard satellites focuses on the harmonic disturbances with a frequency range of 0.1–300 Hz [4, 5]. Microvibration in these satellites can take a long time to attenuate because environment damping in aerospace is minute [6]. This scenario can cause serious degradation, for example, downgrading of the image quality or positional accuracy of optical payloads, in the performance of high-precision payloads. Vibration isolators are often applied in vibration propagation paths to protect high-precision payloads from the effect of microvibration. Vibration isolators are designed for space application, such as advanced X-ray Astro-Physics Facility [7], James Webb Space Telescope [8], and satellite ultraquiet isolation technology experiment [9].Passive isolation techniques are generally deployed in aerospace engineering, thereby providing high-performance and stability and requiring no external power [10–13]. With regard to passive isolators, the vibrations are dissipated through passive damping and the displacement transmissibility is low at a high frequency band [10, 12]. In general, passive isolators contain a high-performance damper, such as viscous fluid damper or viscoelastic composite damper. Traditional passive isolators such as metal coil spring and rubber are generally soft for vibration isolation at relatively low frequencies, which is not expected in practice [14]. Nonlinear stiffness and nonlinear damping have been recently reported to be beneficial to vibration isolation for low and high frequencies [15–17]. For microvibration isolation, liquid damping is generally applied, in which no dead zone of friction exists, thereby benefiting small amplitude vibration isolation [18]. A typical liquid damping isolator is called D-strut [12]; the critical part of this isolator is the passive damper. D-strut is a three-parameter configuration instead of a conventional two-parameter configuration, which is called the Zener model [19]. This model, which greatly improves isolation performance, is composed of a parallel combination of a spring and an elastically supported damper.To install high-performance isolators in a limited installation space, this paper presents a novel passive isolator with compact structures based on a four-parameter Zener model. The isolator is made using bellows and viscous fluid. In the isolator, the bellows act as a seal structure and provide supporting stiffness at the same time, and the damping structure can be arranged in the bellows conveniently, thereby ensuring that the size of the isolator is small enough for the isolator to be installed in a limited space. The dynamic characters of the isolator are influenced by the mass of flowing fluid pumped through the damp channel by vibratory displacements in the isolator, which introduces a fourth parameter for the conventional Zener model. The novel isolator shows some new characteristics in its equivalent stiffness and damping because of the additional parameter. These characteristics enhance the performance of the novel isolator in vibration isolation. The stiffness and damping properties of the novel isolator are investigated comprehensively in this study to determine how the fourth parameter influences the performance of the novel isolator. The equivalent stiffness and damping model of the isolator are obtained theoretically and tested numerically in Section2. The experiments on stiffness and damping of the novel isolator under different excitation amplitudes are presented in Section 3. The effects of the designed parameters on the stiffness and damping of the novel isolator are analyzed in Section 4. To inspect the performance of the isolator comprehensively, the displacement transmission rate characteristics of the isolator are studied in Section 5. Finally, the summary and concluding remarks are presented in Section 6.


## 2. Analysis on Stiffness and Damping of the Vibration Isolator

As shown in Figure1, the vibration isolator consists of three major parts, connecting structure, sealing construction, and upper and lower cavities, all of which are enclosed by four segments of metal bellows with the same diameter. The two cavities are filled with liquid, and a damp channel exists in the intermediate connecting plate, which connects the two cavities. The stiffness of the vibration isolator is provided when the bellows stretch and compress. The pressure in the two cavities changes while load M vibrates up and down. The change in the two cavities causes a pressure difference between these cavities and forces liquid to flow through the two cavities via the damp channel. Energy loss occurs because of linear damping caused by the viscosity of the liquid and the quadratic damping caused by the eddy current, which occurs as a result of the sudden change in diameter of the liquid flow channel at the inlet and outlet. The quadratic damping performs well in suppressing resonance while having no negative effects on vibration isolation at high frequencies [20]. This energy loss results in the damping of the vibration isolator. In addition, the inertia of the liquid in the damp channel affects the stiffness and damping characteristics of the isolator, because the diameter of the damp channel is smaller than the equivalent diameter of the cavities, thereby increasing the liquid motion in the damp channel.Figure 1
Structural diagram of the novel isolator with payload.For a vibration isolation system, three or more vibration isolators are mounted. To achieve satisfactory performance of the vibration isolation system, reasonably matching the stiffness and damping properties of each vibration isolator is of importance. Therefore, this study establishes the relationship model between the structural parameters of the isolator and its stiffness and damping properties, comprehensively analyzes the stiffness and damping properties of the vibration isolator, and conducts a parameter sensitivity analysis on each key structural parameter of the isolator. This study lays the foundation for selecting the design parameters of the vibration isolator and reasonably matching the stiffness and damping properties of each vibration isolator.


### 2.1. Establishment of the Vibration Isolator Model

The working schematic diagram of the vibration isolator is shown in Figure2(a). For the bellows, the longitudinal stiffness is recorded as ka. The equivalent stiffness of the expansion and compression of liquid and radial of the bellows is recorded as kb. When kb is three to five times larger than ka, the effect of kb can be ignored in a certain range within the center of the first-order support frequency [21]. The influence of kb is ignored when modeling in the case where the longitudinal stiffness of the bellows at both ends is not very different from that of the bellows at the middle. The stiffness of the middle bellows k1 is parallel with fluid damping and with the equivalent liquid mass in the damp channel, whereas the stiffness of the bellows at both ends k2 is in series with them. Linear damping cl and square damping cq occur when the liquid flows through the damp channel. The axial deformation response of the upper and lower cavities of the vibration isolator is x2, which is the displacement response of the equivalent liquid mass in the damp channel. The vibration isolator can be equivalent to the model with two degrees of freedom, as shown in Figure 2(b), in which meq acts as the fourth parameter of the conventional Zener model. The significance of each parameter is as follows:Aeq: equivalent cross-sectional area of the bellows.k2: sum of the stiffness of bellows at both ends.k1: sum of the stiffness of the middle bellows.M: mass of the load.pu: pressure change of the upper cavity.pl: pressure change of the lower cavity.q: flow of fluid in the damp channel.x1: displacement of excitation.x2: axial deformation response of the upper and lower cavities.meq: equivalent liquid mass flow through the damp channel.cl: linear damping coefficient of damp channel.cq: square damping coefficient of damp channel.Figure 2
(a) Working schematic diagram of the isolator. (b) Equivalent system with two degrees of freedom.
(a) (b)The equivalent cylindrical cross-sectional area of the bellows can be calculated as follows [22]: (1)Aeq=πDb2+db28.In the formula,Db is the large diameter of the bellows, whereas db is the small diameter of the bellows.The flow of liquidQ between the two cavities of the isolator can be expressed as(2)Q=Aeqx˙2.The change of pressure in the two cavities shown in Figure2(a) is(3a)Δpu=-k2x1-x22Aeq,(3b)Δpl=k2x1-x22Aeq.When the fluid flows in the damp channel, the total pressure difference between the upper and lower cavities equals the sum of pressure loss caused by the damp channel and the equivalent pressure loss caused by the inertia of the liquid flowing in the damp channel; that is,(4)Δpu-Δpl+I·Q˙t+ΔP=0.In the preceding formula (4), I is the fluid inductance of the fluid in the damp channel. It can be expressed as follows: (5)I=ρlcAc.In formula (5), Ac is the cross-sectional area of the damp channel and lc is the length of the damp channel.Δ P is the sum of linear pressure loss ΔP1, inlet pressure loss ΔP2, and outlet pressure loss ΔP3 caused by the damp channel; that is,(6)ΔP=ΔP1+ΔP2+ΔP3.Δ P 1, ΔP2, and ΔP3 are calculated as follows: (7a)ΔP1=128μlcπdc4Q,(7b)ΔP2=12ρζ1QAc2,(7c)ΔP3=12ρζ2QAc2.In formula ((7a), (7b), and (7c)), dc is the diameter of the damp channel; ζ1 and ζ2 are the sudden expansion and sudden contraction resistance coefficients caused by the sudden change in the cross-sectional area of the liquid flow channel at both ends of the damp channel, which are determined by Reynolds number (be related to ω and X2) when the fluid flow through the damp channel. The damp channel can be found in the relevant manual.The following can be obtained by arranging all the preceding formulas:(8)πρlcdeq44dc2x¨2+8πμlcdeq4dc4x˙2+πρdeq6ζ1+ζ28dc4x˙2x˙2-k2x1-x2=0.In formula (8), deq is the equivalent cylindrical diameter of the bellows.As for the equivalent mass of the liquid in the damp channel in Figure2(b),(9)meqx¨2+clx˙2+cqx˙2x˙2-k2x1-x2=0.In formula (9), cl is the linear damping coefficient caused by the viscosity of the fluid, cq is the square damping coefficient of the eddy current caused by the sudden change of the cross-sectional area of the fluid flow channel and the inertia of the fluid, and meq is the equivalent mass of the fluid in the damp channel. The comparison of Formula (8) with Formula (9) shows that(10)cl=8πμlcdeq4dc4,cq=πρdeq68dc4ζ1+ζ2,(11)meq=πρlcdeq44dc2,where mc=ρlcAc is the actual mass of the fluid in the damp channel. It is plugged into formula (11); thus, the following is determined: (12)meq=deq4dc-4mc.The cross-sectional area of the damp channel is usually smaller than the equivalent cross-sectional area of the cavities of the isolator. Thus, the inertia effect of the fluid in the damp channel is enhanced.The pressure in the upper and lower cavity of the isolator cannot be transferred to the base (the connecting ring), so the reaction force can only viak1 and k2 be transferred to the base, while k2 is distorted by a displacement x1-x2. Actually, k2 is directly connected to the base, so Figure 2(b) cannot reflect the actual state of the isolator at this time. The reaction force transferred to the base can be obtained by Figure 2(a): (13)FT=k1x1+k2x1-x2.Ifx1 is harmonic excitation, then x1 and x2 can be expressed as follows:(14)x1=X1ejωt,x2=X2ejωt+θ.To simplify the research, the equivalent damping coefficient of Formula (9) is used as follows according to the principle of energy equivalence: (15)c-p=cl+8ωX23πcq,where c-p is obtained by calculating the energy dissipated in the system by considering only the dominant harmonic as an approximation.Formulas (14) and (15) are plugged into formula (9), and the following formula can be obtained: (16)X2X1=k2k2-meqω22+c-pω2,(17)ejθ=k2-meqω2-c-pjωk2-meqω22+c-pω2.According to formulas (13), (14), (16), and (17), the span dynamic stiffness of the vibration isolator is as follows: (18)ks=FTx1=k1+k2-k22k2-meqω2k2-meqω22+c-pω2+c-pk22jωk2-meqω22+c-pω2.According to SAE (Society of Automotive Engineers) 1085B standards, the real part ofks is in-phase dynamic stiffness, which denote the elastic part of the vibration isolator, whereas the imaginary part is orthogonal dynamic stiffness, which reflect the damping characteristics of the vibration isolator. Thus, the equivalent stiffness in Figure 2(b) is as follows:(19)keq=k1+k2-k22k2-meqω2k2-meqω22+c-pω2.The equivalent damping coefficient is(20)ceq=c-pk22k2-meqω22+c-pω2.


### 2.2. Analysis of the Equivalent Stiffness Characteristic

Ifk0=keqω→0 and k∞=keqω→0, then, according to Formula (19), the following can be obtained: (21)k0=k1,k∞=k1+k2.The numerical value ofk0 is equal to the sum of the stiffness of the bellows in the middle of the isolator, which is the quasistatic stiffness of the vibration isolator, and of the static support stiffness of the vibration isolator while the isolator is working. According to the demand of system stiffness, k0 should be matched first. When ω→0, the reaction force of the mounting base is FTω→0=kx1. When combined with formula (13), we easily determine that x1=x2. The axial deformation of the two cavities of the isolator equals the displacement of the external excitation. The bellows at the end of the isolator do not result in axial deformation. When ω→∞, the reaction force of the mounting base is FTω→0=kx1+k2x1; thus, x2=0. The two cavities of the vibration isolator do not result in axial deformation, and their volumes remain constant. At this time, the axial deformation of both bellows at the end of the vibration isolator equals the displacement of the external excitation. Thus, the fluid in the damp channel is no longer flowing, which results in dynamic hardening. This scenario indicates that the damp channel is equivalently blocked.According to formula (9), the natural frequency of the fluid in the damp channel is as follows: (22)ωn=k2meq.This natural frequency is plugged into formula (19). When ω=ωn, the following is obtained:(23)keqω=ωn=k∞=k1+k2.Formula (11) is plugged into formula (22): (24)ωn=dcdeq2k2ρlc.The combination of formulas (21), (23), and (24) determine that, for a vibration isolator with a specific parameter, the various curves of the equivalent stiffness of the isolator with the change of excitation frequency are always through two certain points 0,k1 and ωn,k1+k2, regardless of how the external excitation amplitude changes. When ω→∞, keq→k1+k2; thus, parameter identification can be made using the characteristics of equivalent stiffness [23].Ifc-p is constant, then the derivative of keq can be obtained with respect to ω in formula (19), and the two extreme points of keq, namely, ωv and ωp, can be obtained:(25)ξ=c-p2k2meq,(26)ωv=ωn1-2ξifξ<12,(27)ωp=ωn1+2ξ.ξ is the equivalent damping ratio of the vibration isolator, ωv is the minimum point of keq, and ωp is the maximum point of keq.Formulas (25) to (27) are plugged into formula (19), and the value of the extreme point of keq can be obtained:(28)kv=k1+k2-k24ξ1-ξifξ<12,(29)kp=k1+k2+k24ξ1+ξ,where ωvξ=1/2=0,kvξ=1/2=k0. Therefore, when ξ=1/2, points ωv,kv and 0,k1 coincide. When ξ>1/2, no minimum point exists.ω v < ω n < ω p can be obtained using formulas (26) and (27), whereas kv<k0<k∞<kp can be obtained using formulas (21), (28), and (29). Thus, the rule of change of keq with excitation frequency ω is as follows: (1)   ξ<1/2. When ω∈0,ωv, keq begins to decrease from point 0,k1 with the increase of frequency, and the minimum value kv appears at ωv. With the increase of ξ, ωv decreases while kv increases. When ω∈ωv,ωp, keq increases and goes through a certain point ωn,k1+k2 with the increase of frequency; the maximum value kp also appears at ωp. With the increase of ξ, ωp decreases while kp increases. When ω>ωp, keq decreases with the increase of frequency. When ω→∞, keq→k1+k2. (2)   ξ≥1/2. When ω∈0,ωp, keq increases and goes through a certain point ωn,k1+k2 with the increase of frequency. Moreover, the maximum value appears at ωp. With the increase of ξ, ωp increases while kp decreases, and kpξ→∞=k1+k2. When ω>ωp, keq decreases with the increase of frequency. When ω→∞, keq→k1+k2. kpξ→∞=k1+k2. Therefore, when the damping ratio of the system is sufficiently large, the maximum point ωp,kp disappears. In this case, keq∈k1,k1+k2.According to the preceding analysis, the law by whichkeq changes with ω can be summarized in Table 1.Table 1
Law by whichkeq changes with ω.
Condition k eq ω 0 k 1 ωn k 1 + k 2 ∞ k 1 + k 2 ξ < 1 2 0<ω<ωv ↓ ωv<ω<ωp ↑ ω>ωp ↓ ξ ≥ 1 2 0<ω<ωp ↑ ω>ωp ↓In Table1, “↑” stands for the equivalent stiffness increase along with the increase of frequency, whereas “↓” stands for the equivalent stiffness decrease along with the increase of frequency.The law by whichξ affects the key points of the equivalent stiffness is shown in Table 2.Table 2
Law by whichξ affects the key points of the equivalent stiffness.
Condition ωv kv ωp kp ξ ξ<12 ↓ ↑ ↑ ↓  ξ≥12 — — ↑ ↓In Table2, “↑” stands for the key point of equivalent stiffness increase along with the increase of ξ, “↓” stands for the key point of equivalent stiffness decrease along with the increase of ξ, and “—” means not existing.


### 2.3. Property Analysis of Equivalent Damping Coefficient

c 0 = c eq ω → 0 and c∞=ceqω→0 are assumed; thus, according to formulas (15) and (20):(30)c0=cl,c∞=0.For different external incentives, the equivalent damping of the isolator always goes through two certain points, namely,0,cl and ∞,0. When ω→0, the fluid in the damp channel flows slowly, and the damping of the vibration isolator mainly consists of linear damping. When ω→∞, the dynamic hardening of the fluid in the damp channel occurs, the fluid in the damp channel no longer flows, and the damping coefficient of the isolator is zero. This finding is consistent with that of the previous analysis. The vibration isolator can satisfy the small damping of the vibration isolation system at high frequency.The derivative ofceq is obtained with respect to ω in formula (20), and the frequency of the extreme point of the equivalent damping coefficient is as follows:(31)ωc=ωn1-2ξ2ifξ<22.Formula (31) is plugged into formula (20), and the peak value of the equivalent damping coefficient can be obtained: (32)cc=k2ωn12ξ1-ξ2ifξ<22.ω c ξ = 2 / 2 = 0, ccξ=2/2=2k2/ωn=cl. Therefore, when ξ=2/2, the maximum value occurs at ω=0, and when ξ≥2/2, no extreme point exists.Thus, the rule of change ofceq with excitation frequency ω is as follows: (1)   ξ<2/2. ceq increases first and then decreases with the increase of ω. The maximum value cc appears at ωc, and peak point ωc decreases with the increase of ξ. When ξ<3/3, cc increases first and then decreases with the increase of ξ. When3/3<ξ<2/2, cc increases with the increase of ξ. (2)   ξ≥2/2. ceq decreases with the increase of ω, and the maximum value is c0=cl.According to the preceding formulas above, the rule by whichceq changes with ω can be summarized in Table 3.Table 3
Rule by whichceq changes with ω.
Condition c eq ω 0 cl ∞ 0 ξ<22 0<ω<ωc ↑ ω>ωc ↓ ξ≥22 ↓In Table3, “↑” stands for the equivalent damping coefficient increase along with the increase of frequency, whereas “↓” stands for the equivalent damping coefficient decrease along with the increase of frequency.The rule by whichξ affects the key points of the equivalent damping coefficient is shown in Table 4.Table 4
Rule by whichξ affects the key points of the equivalent damping coefficient.
Condition ωc cc ξ ξ<33 ↓ ↓ 33<ξ<22 ↓ ↑ ξ≥22 — —In Table4, “↑” stands for the key point of equivalent damping coefficient increase along with the increase of ξ, “↓” stands for the key point of equivalent damping coefficient decreases with the increase of ξ, and “—” means not existing.


### 2.4. Reliability Verification on the Model of Equivalent Stiffness and Equivalent Damping Ratio of the Vibration Isolator

Formulas (9) and (13) are changed into a dimensionless form to facilitate the numerical simulation. λ=ω/ωn, τ=ωnt, x˙1t=x˙1τωn, x˙2t=x˙2τωn, x0=0.06 mm, z1=x1τ/x0, z2=x2τ/x0, ζl=clωn/k2, ζq=cqx0ωn2/k2, f0=k2x0, fT=FT/f0, and N=k1/k2 are defined, and the equations of the dimensionless forms are as follows: (33a)z¨2+ζlz˙2+ζqz˙2z˙2=z1,(33b)fT=N+1z1-z2.In formulas (33a) and (33b), λ, z1, z2, and fT are the dimensionless forms of ω, x1, x2, and FT, respectively. keq and ceq are identified on the basis of the equation ks=keq+ceqjω. The related parameters are k2=90.07 N/mm, N=0.2, deq=25 mm, dc=2 mm, and ζl=0.1172. ζq is calculated according to ζ1 and ζ2, which are queried from the simulation results in real time. The comparison of equivalent stiffness and damping coefficient results of the model with the results of the numerical simulation is shown in Figure 3.Figure 3
(a) Equivalent stiffness. (b) Equivalent damping coefficient.
(a) (b)The equivalent stiffness and damping coefficient results of the model and the numerical simulation match well; thus, the reliability of the model can be verified. However, a numerical simulation must be performed first to determine the parameterc-p of the model by obtaining the value of X2. Thus, the following quantitative analysis is based on the numerical simulation.


### 2.5. Analysis of the Effect of the Fourth Parametermeq

The numerical simulations of the equivalent stiffness and the equivalent damping coefficient results of the isolator whenmeq≠0 and meq=0 are shown in Figures 4 and 5, respectively. These simulations clearly determine how the fourth parameter meq influences the characteristic of the novel isolator.Figure 4
(a) Numerical simulations of the equivalent stiffness whenmeq≠0. (b) Numerical simulations of the equivalent stiffness when meq=0.
(a) (b)Figure 5
(a) Numerical simulations of the equivalent damping coefficient whenmeq≠0. (b) Numerical simulations of the equivalent damping coefficient when meq=0.
(a) (b)Figure4 shows that when meq=0, the valley point ωv,kv and the peak point ωp,kp disappear. Thus, the fourth parameter meq introduces a valley point ωv,kv and a peak point ωp,kp to the equivalent stiffness characteristic of the novel isolator. When meq≠0, if ζq is given a large value, such as ζq=100, then it appears at the same situation as meq=0; that is, the influence of the fluid mass in the damp channel can be neglected. This analysis indicates that meq can significantly influence the stiffness of the vibration isolation system at low frequency bands, thereby determining the resonance frequency of the vibration isolation system. Thus, we must choose the relevant design parameter of the isolator carefully to ensure its vibration isolation performance.Figure5 shows that the existence of meq has two main effects on the equivalent damping coefficient. The first effect is introducing a peak point near λ=1 when the damping parameters cl and cq are relatively small, and the other is speeding up the equivalent damping decay when λ>1. When meq≠0, if ζq is given a large value, then it will appear at the same situation as meq=0; that is, the influence of the fluid mass in the damp channel can be neglected.For a vibration isolation system, when the damping of the isolator near the resonance frequency is larger and the damping of the isolator at high frequency is smaller, the performance of the isolation system is better. Thus, introducing the fourth parametermeq improves the damping performance of the isolator.The nonlinearity makeskeq smaller near ωv and larger near ωp and makes ceq larger near ωc. For a vibration isolation system, the stiffness and the damping of a vibration isolation system should be large when at a low frequency band of working frequencies. Thus, the nonlinearity of keq and ωeq may improve the performance of the isolator at a low frequency band when the design parameter is chosen reasonably.


## 2.1. Establishment of the Vibration Isolator Model

The working schematic diagram of the vibration isolator is shown in Figure2(a). For the bellows, the longitudinal stiffness is recorded as ka. The equivalent stiffness of the expansion and compression of liquid and radial of the bellows is recorded as kb. When kb is three to five times larger than ka, the effect of kb can be ignored in a certain range within the center of the first-order support frequency [21]. The influence of kb is ignored when modeling in the case where the longitudinal stiffness of the bellows at both ends is not very different from that of the bellows at the middle. The stiffness of the middle bellows k1 is parallel with fluid damping and with the equivalent liquid mass in the damp channel, whereas the stiffness of the bellows at both ends k2 is in series with them. Linear damping cl and square damping cq occur when the liquid flows through the damp channel. The axial deformation response of the upper and lower cavities of the vibration isolator is x2, which is the displacement response of the equivalent liquid mass in the damp channel. The vibration isolator can be equivalent to the model with two degrees of freedom, as shown in Figure 2(b), in which meq acts as the fourth parameter of the conventional Zener model. The significance of each parameter is as follows:Aeq: equivalent cross-sectional area of the bellows.k2: sum of the stiffness of bellows at both ends.k1: sum of the stiffness of the middle bellows.M: mass of the load.pu: pressure change of the upper cavity.pl: pressure change of the lower cavity.q: flow of fluid in the damp channel.x1: displacement of excitation.x2: axial deformation response of the upper and lower cavities.meq: equivalent liquid mass flow through the damp channel.cl: linear damping coefficient of damp channel.cq: square damping coefficient of damp channel.Figure 2
(a) Working schematic diagram of the isolator. (b) Equivalent system with two degrees of freedom.
(a) (b)The equivalent cylindrical cross-sectional area of the bellows can be calculated as follows [22]: (1)Aeq=πDb2+db28.In the formula,Db is the large diameter of the bellows, whereas db is the small diameter of the bellows.The flow of liquidQ between the two cavities of the isolator can be expressed as(2)Q=Aeqx˙2.The change of pressure in the two cavities shown in Figure2(a) is(3a)Δpu=-k2x1-x22Aeq,(3b)Δpl=k2x1-x22Aeq.When the fluid flows in the damp channel, the total pressure difference between the upper and lower cavities equals the sum of pressure loss caused by the damp channel and the equivalent pressure loss caused by the inertia of the liquid flowing in the damp channel; that is,(4)Δpu-Δpl+I·Q˙t+ΔP=0.In the preceding formula (4), I is the fluid inductance of the fluid in the damp channel. It can be expressed as follows: (5)I=ρlcAc.In formula (5), Ac is the cross-sectional area of the damp channel and lc is the length of the damp channel.Δ P is the sum of linear pressure loss ΔP1, inlet pressure loss ΔP2, and outlet pressure loss ΔP3 caused by the damp channel; that is,(6)ΔP=ΔP1+ΔP2+ΔP3.Δ P 1, ΔP2, and ΔP3 are calculated as follows: (7a)ΔP1=128μlcπdc4Q,(7b)ΔP2=12ρζ1QAc2,(7c)ΔP3=12ρζ2QAc2.In formula ((7a), (7b), and (7c)), dc is the diameter of the damp channel; ζ1 and ζ2 are the sudden expansion and sudden contraction resistance coefficients caused by the sudden change in the cross-sectional area of the liquid flow channel at both ends of the damp channel, which are determined by Reynolds number (be related to ω and X2) when the fluid flow through the damp channel. The damp channel can be found in the relevant manual.The following can be obtained by arranging all the preceding formulas:(8)πρlcdeq44dc2x¨2+8πμlcdeq4dc4x˙2+πρdeq6ζ1+ζ28dc4x˙2x˙2-k2x1-x2=0.In formula (8), deq is the equivalent cylindrical diameter of the bellows.As for the equivalent mass of the liquid in the damp channel in Figure2(b),(9)meqx¨2+clx˙2+cqx˙2x˙2-k2x1-x2=0.In formula (9), cl is the linear damping coefficient caused by the viscosity of the fluid, cq is the square damping coefficient of the eddy current caused by the sudden change of the cross-sectional area of the fluid flow channel and the inertia of the fluid, and meq is the equivalent mass of the fluid in the damp channel. The comparison of Formula (8) with Formula (9) shows that(10)cl=8πμlcdeq4dc4,cq=πρdeq68dc4ζ1+ζ2,(11)meq=πρlcdeq44dc2,where mc=ρlcAc is the actual mass of the fluid in the damp channel. It is plugged into formula (11); thus, the following is determined: (12)meq=deq4dc-4mc.The cross-sectional area of the damp channel is usually smaller than the equivalent cross-sectional area of the cavities of the isolator. Thus, the inertia effect of the fluid in the damp channel is enhanced.The pressure in the upper and lower cavity of the isolator cannot be transferred to the base (the connecting ring), so the reaction force can only viak1 and k2 be transferred to the base, while k2 is distorted by a displacement x1-x2. Actually, k2 is directly connected to the base, so Figure 2(b) cannot reflect the actual state of the isolator at this time. The reaction force transferred to the base can be obtained by Figure 2(a): (13)FT=k1x1+k2x1-x2.Ifx1 is harmonic excitation, then x1 and x2 can be expressed as follows:(14)x1=X1ejωt,x2=X2ejωt+θ.To simplify the research, the equivalent damping coefficient of Formula (9) is used as follows according to the principle of energy equivalence: (15)c-p=cl+8ωX23πcq,where c-p is obtained by calculating the energy dissipated in the system by considering only the dominant harmonic as an approximation.Formulas (14) and (15) are plugged into formula (9), and the following formula can be obtained: (16)X2X1=k2k2-meqω22+c-pω2,(17)ejθ=k2-meqω2-c-pjωk2-meqω22+c-pω2.According to formulas (13), (14), (16), and (17), the span dynamic stiffness of the vibration isolator is as follows: (18)ks=FTx1=k1+k2-k22k2-meqω2k2-meqω22+c-pω2+c-pk22jωk2-meqω22+c-pω2.According to SAE (Society of Automotive Engineers) 1085B standards, the real part ofks is in-phase dynamic stiffness, which denote the elastic part of the vibration isolator, whereas the imaginary part is orthogonal dynamic stiffness, which reflect the damping characteristics of the vibration isolator. Thus, the equivalent stiffness in Figure 2(b) is as follows:(19)keq=k1+k2-k22k2-meqω2k2-meqω22+c-pω2.The equivalent damping coefficient is(20)ceq=c-pk22k2-meqω22+c-pω2.


## 2.2. Analysis of the Equivalent Stiffness Characteristic

Ifk0=keqω→0 and k∞=keqω→0, then, according to Formula (19), the following can be obtained: (21)k0=k1,k∞=k1+k2.The numerical value ofk0 is equal to the sum of the stiffness of the bellows in the middle of the isolator, which is the quasistatic stiffness of the vibration isolator, and of the static support stiffness of the vibration isolator while the isolator is working. According to the demand of system stiffness, k0 should be matched first. When ω→0, the reaction force of the mounting base is FTω→0=kx1. When combined with formula (13), we easily determine that x1=x2. The axial deformation of the two cavities of the isolator equals the displacement of the external excitation. The bellows at the end of the isolator do not result in axial deformation. When ω→∞, the reaction force of the mounting base is FTω→0=kx1+k2x1; thus, x2=0. The two cavities of the vibration isolator do not result in axial deformation, and their volumes remain constant. At this time, the axial deformation of both bellows at the end of the vibration isolator equals the displacement of the external excitation. Thus, the fluid in the damp channel is no longer flowing, which results in dynamic hardening. This scenario indicates that the damp channel is equivalently blocked.According to formula (9), the natural frequency of the fluid in the damp channel is as follows: (22)ωn=k2meq.This natural frequency is plugged into formula (19). When ω=ωn, the following is obtained:(23)keqω=ωn=k∞=k1+k2.Formula (11) is plugged into formula (22): (24)ωn=dcdeq2k2ρlc.The combination of formulas (21), (23), and (24) determine that, for a vibration isolator with a specific parameter, the various curves of the equivalent stiffness of the isolator with the change of excitation frequency are always through two certain points 0,k1 and ωn,k1+k2, regardless of how the external excitation amplitude changes. When ω→∞, keq→k1+k2; thus, parameter identification can be made using the characteristics of equivalent stiffness [23].Ifc-p is constant, then the derivative of keq can be obtained with respect to ω in formula (19), and the two extreme points of keq, namely, ωv and ωp, can be obtained:(25)ξ=c-p2k2meq,(26)ωv=ωn1-2ξifξ<12,(27)ωp=ωn1+2ξ.ξ is the equivalent damping ratio of the vibration isolator, ωv is the minimum point of keq, and ωp is the maximum point of keq.Formulas (25) to (27) are plugged into formula (19), and the value of the extreme point of keq can be obtained:(28)kv=k1+k2-k24ξ1-ξifξ<12,(29)kp=k1+k2+k24ξ1+ξ,where ωvξ=1/2=0,kvξ=1/2=k0. Therefore, when ξ=1/2, points ωv,kv and 0,k1 coincide. When ξ>1/2, no minimum point exists.ω v < ω n < ω p can be obtained using formulas (26) and (27), whereas kv<k0<k∞<kp can be obtained using formulas (21), (28), and (29). Thus, the rule of change of keq with excitation frequency ω is as follows: (1)   ξ<1/2. When ω∈0,ωv, keq begins to decrease from point 0,k1 with the increase of frequency, and the minimum value kv appears at ωv. With the increase of ξ, ωv decreases while kv increases. When ω∈ωv,ωp, keq increases and goes through a certain point ωn,k1+k2 with the increase of frequency; the maximum value kp also appears at ωp. With the increase of ξ, ωp decreases while kp increases. When ω>ωp, keq decreases with the increase of frequency. When ω→∞, keq→k1+k2. (2)   ξ≥1/2. When ω∈0,ωp, keq increases and goes through a certain point ωn,k1+k2 with the increase of frequency. Moreover, the maximum value appears at ωp. With the increase of ξ, ωp increases while kp decreases, and kpξ→∞=k1+k2. When ω>ωp, keq decreases with the increase of frequency. When ω→∞, keq→k1+k2. kpξ→∞=k1+k2. Therefore, when the damping ratio of the system is sufficiently large, the maximum point ωp,kp disappears. In this case, keq∈k1,k1+k2.According to the preceding analysis, the law by whichkeq changes with ω can be summarized in Table 1.Table 1
Law by whichkeq changes with ω.
Condition k eq ω 0 k 1 ωn k 1 + k 2 ∞ k 1 + k 2 ξ < 1 2 0<ω<ωv ↓ ωv<ω<ωp ↑ ω>ωp ↓ ξ ≥ 1 2 0<ω<ωp ↑ ω>ωp ↓In Table1, “↑” stands for the equivalent stiffness increase along with the increase of frequency, whereas “↓” stands for the equivalent stiffness decrease along with the increase of frequency.The law by whichξ affects the key points of the equivalent stiffness is shown in Table 2.Table 2
Law by whichξ affects the key points of the equivalent stiffness.
Condition ωv kv ωp kp ξ ξ<12 ↓ ↑ ↑ ↓  ξ≥12 — — ↑ ↓In Table2, “↑” stands for the key point of equivalent stiffness increase along with the increase of ξ, “↓” stands for the key point of equivalent stiffness decrease along with the increase of ξ, and “—” means not existing.


## 2.3. Property Analysis of Equivalent Damping Coefficient

c 0 = c eq ω → 0 and c∞=ceqω→0 are assumed; thus, according to formulas (15) and (20):(30)c0=cl,c∞=0.For different external incentives, the equivalent damping of the isolator always goes through two certain points, namely,0,cl and ∞,0. When ω→0, the fluid in the damp channel flows slowly, and the damping of the vibration isolator mainly consists of linear damping. When ω→∞, the dynamic hardening of the fluid in the damp channel occurs, the fluid in the damp channel no longer flows, and the damping coefficient of the isolator is zero. This finding is consistent with that of the previous analysis. The vibration isolator can satisfy the small damping of the vibration isolation system at high frequency.The derivative ofceq is obtained with respect to ω in formula (20), and the frequency of the extreme point of the equivalent damping coefficient is as follows:(31)ωc=ωn1-2ξ2ifξ<22.Formula (31) is plugged into formula (20), and the peak value of the equivalent damping coefficient can be obtained: (32)cc=k2ωn12ξ1-ξ2ifξ<22.ω c ξ = 2 / 2 = 0, ccξ=2/2=2k2/ωn=cl. Therefore, when ξ=2/2, the maximum value occurs at ω=0, and when ξ≥2/2, no extreme point exists.Thus, the rule of change ofceq with excitation frequency ω is as follows: (1)   ξ<2/2. ceq increases first and then decreases with the increase of ω. The maximum value cc appears at ωc, and peak point ωc decreases with the increase of ξ. When ξ<3/3, cc increases first and then decreases with the increase of ξ. When3/3<ξ<2/2, cc increases with the increase of ξ. (2)   ξ≥2/2. ceq decreases with the increase of ω, and the maximum value is c0=cl.According to the preceding formulas above, the rule by whichceq changes with ω can be summarized in Table 3.Table 3
Rule by whichceq changes with ω.
Condition c eq ω 0 cl ∞ 0 ξ<22 0<ω<ωc ↑ ω>ωc ↓ ξ≥22 ↓In Table3, “↑” stands for the equivalent damping coefficient increase along with the increase of frequency, whereas “↓” stands for the equivalent damping coefficient decrease along with the increase of frequency.The rule by whichξ affects the key points of the equivalent damping coefficient is shown in Table 4.Table 4
Rule by whichξ affects the key points of the equivalent damping coefficient.
Condition ωc cc ξ ξ<33 ↓ ↓ 33<ξ<22 ↓ ↑ ξ≥22 — —In Table4, “↑” stands for the key point of equivalent damping coefficient increase along with the increase of ξ, “↓” stands for the key point of equivalent damping coefficient decreases with the increase of ξ, and “—” means not existing.


## 2.4. Reliability Verification on the Model of Equivalent Stiffness and Equivalent Damping Ratio of the Vibration Isolator

Formulas (9) and (13) are changed into a dimensionless form to facilitate the numerical simulation. λ=ω/ωn, τ=ωnt, x˙1t=x˙1τωn, x˙2t=x˙2τωn, x0=0.06 mm, z1=x1τ/x0, z2=x2τ/x0, ζl=clωn/k2, ζq=cqx0ωn2/k2, f0=k2x0, fT=FT/f0, and N=k1/k2 are defined, and the equations of the dimensionless forms are as follows: (33a)z¨2+ζlz˙2+ζqz˙2z˙2=z1,(33b)fT=N+1z1-z2.In formulas (33a) and (33b), λ, z1, z2, and fT are the dimensionless forms of ω, x1, x2, and FT, respectively. keq and ceq are identified on the basis of the equation ks=keq+ceqjω. The related parameters are k2=90.07 N/mm, N=0.2, deq=25 mm, dc=2 mm, and ζl=0.1172. ζq is calculated according to ζ1 and ζ2, which are queried from the simulation results in real time. The comparison of equivalent stiffness and damping coefficient results of the model with the results of the numerical simulation is shown in Figure 3.Figure 3
(a) Equivalent stiffness. (b) Equivalent damping coefficient.
(a) (b)The equivalent stiffness and damping coefficient results of the model and the numerical simulation match well; thus, the reliability of the model can be verified. However, a numerical simulation must be performed first to determine the parameterc-p of the model by obtaining the value of X2. Thus, the following quantitative analysis is based on the numerical simulation.


## 2.5. Analysis of the Effect of the Fourth Parametermeq

The numerical simulations of the equivalent stiffness and the equivalent damping coefficient results of the isolator whenmeq≠0 and meq=0 are shown in Figures 4 and 5, respectively. These simulations clearly determine how the fourth parameter meq influences the characteristic of the novel isolator.Figure 4
(a) Numerical simulations of the equivalent stiffness whenmeq≠0. (b) Numerical simulations of the equivalent stiffness when meq=0.
(a) (b)Figure 5
(a) Numerical simulations of the equivalent damping coefficient whenmeq≠0. (b) Numerical simulations of the equivalent damping coefficient when meq=0.
(a) (b)Figure4 shows that when meq=0, the valley point ωv,kv and the peak point ωp,kp disappear. Thus, the fourth parameter meq introduces a valley point ωv,kv and a peak point ωp,kp to the equivalent stiffness characteristic of the novel isolator. When meq≠0, if ζq is given a large value, such as ζq=100, then it appears at the same situation as meq=0; that is, the influence of the fluid mass in the damp channel can be neglected. This analysis indicates that meq can significantly influence the stiffness of the vibration isolation system at low frequency bands, thereby determining the resonance frequency of the vibration isolation system. Thus, we must choose the relevant design parameter of the isolator carefully to ensure its vibration isolation performance.Figure5 shows that the existence of meq has two main effects on the equivalent damping coefficient. The first effect is introducing a peak point near λ=1 when the damping parameters cl and cq are relatively small, and the other is speeding up the equivalent damping decay when λ>1. When meq≠0, if ζq is given a large value, then it will appear at the same situation as meq=0; that is, the influence of the fluid mass in the damp channel can be neglected.For a vibration isolation system, when the damping of the isolator near the resonance frequency is larger and the damping of the isolator at high frequency is smaller, the performance of the isolation system is better. Thus, introducing the fourth parametermeq improves the damping performance of the isolator.The nonlinearity makeskeq smaller near ωv and larger near ωp and makes ceq larger near ωc. For a vibration isolation system, the stiffness and the damping of a vibration isolation system should be large when at a low frequency band of working frequencies. Thus, the nonlinearity of keq and ωeq may improve the performance of the isolator at a low frequency band when the design parameter is chosen reasonably.


## 3. Experiment on the Stiffness and Damping of the Vibration Isolator

A sample isolator was built to verify the correctness of the model. The key structural parameters of the sample isolator are shown in Table5.Table 5
Key structural parameters of the sample isolator.
dc 1 mm k2 100.09 N/mm N 0.181 lc 5 mmThe installation status of the isolator and schematic diagram of the experiment is shown in Figure6.Figure 6
(a) Installation status of the isolator. (b) Schematic diagram of the experiment.
(a) (b)The excitation was a sinusoidal frequency sweep displacement, and the range of excitation frequency was 5–200 Hz. The dynamic characteristics changing with the frequency of the sample isolator under different displacement excitations were tested. Figure7 illustrates the test and numerical simulation results for comparison.Figure 7
(a) Equivalent stiffness of 1# isolator with an exciting amplitude of 0.2 mm. (b) Equivalent damping coefficient of 1# isolator with an exciting amplitude of 0.2 mm. (c) Equivalent stiffness of 1# isolator with an exciting amplitude of 0.4 mm. (d) Equivalent damping coefficient of 1# isolator with an exciting amplitude of 0.4 mm. (e) Equivalent stiffness of 1# isolator with an exciting amplitude of 0.8 mm. (f) Equivalent damping coefficient of 1# isolator with an exciting amplitude of 0.8 mm.
(a) (b) (c) (d) (e) (f)Figure7 indicates that the numerical simulation results matched well with the experimental results; thus, the correctness of the theoretical analysis was verified. The simulated value of the sample isolator’s equivalent stiffness was larger than the tested value when the excitation amplitude was relatively large. The simulated value of the equivalent damping coefficient was larger than the tested value. In general, the radial stiffness of the bellows was three to five times larger than that of the axial stiffness. When the pressure in the cavity of the vibration isolator was large, the radial stiffness of the corrugated pipe could not be ignored. The radial stiffness of the bellows was in series with k2, thereby resulting in the equivalent stiffness k-2<k2 and, finally, in the decrease of k∞. The radial stiffness of the bellows was not considered; thus, the prediction error of the dynamic stiffness was produced. Moreover, the larger the magnitude of excitation, the larger the simulated error. T=2π/ω and W, respectively, represent the excitation period and energy dissipation within one period. W can be calculated as follows: (34)W=-∫0Tclx˙2+cqx˙2x˙2x˙2dt.The equivalent linear damping coefficient can also be defined as follows:(35)ceq=-∫0Tclx˙2+cqx˙2x˙2x˙2dt2π2fX12.The combination of formulas (9) and (35) determined that when X1 is constant, the equivalent damping coefficient ceq is positively correlated with k2. Thus, when k-2<k2, the simulated value of the equivalent damping coefficient is large. According to the preceding comprehensive analysis, the experimental error is mainly caused by the incorrect input of the stiffness parameters of the bellows. This error can be corrected by improving the accuracy of the parameter value of the bellows. Therefore, the model established in this study has high accuracy when applied to the microvibration isolation system.


## 4. Analysis of the Influence of the Key Parameters on the Stiffness and Damping Characteristics of the Isolator

To isolate vibration, the natural frequency of the vibration isolation system must be lower than2/2 times of the frequency of the vibration disturbance. For the same vibration disturbance, the lower the natural frequency of the vibration isolation system, the better the performance of the vibration isolation; the lower the damping ratio and the higher the resonance peak value, however, the better the performance of the vibration isolation at high frequency. The stiffness and damping characteristics of a single isolator play a decisive role in the natural frequency and damping of the vibration isolation system. Thus, the influence of vibration isolator design parameters on the stiffness and damping characteristics of the vibration isolation system must be analyzed to reasonably configure the natural frequency and damping ratio of the vibration isolation system.The design parameters that affect the stiffness and damping properties of the vibration isolator can be obtained aslc, dc, k2, deq, and k1 by combining formulas (10), (11), (15), (19), and (20). Among these parameters, k1 is parallel to the damping of the vibration isolator, which is not influenced by excitation frequency. Acting as a basic stiffness of the isolator, k1 should meet the requirement of the supporting stiffness; hence, k1 is not suitable for adjustment. Excitation amplitude X1 can likewise influence the values of ζ1 and ζ2, which also affect the stiffness and damping properties of the isolator. Therefore, the analysis is made based on the effect of X1, dc, deq, k2, and lc on the key points ωv,kv, ωp,kpωc,cc.


### 4.1. Exciting Amplitude

According to formulas (10), (11), (15), and (25), (36)ξ=24πdeq2μlc+ρdeq4ωX2ζ1+ζ23dc3πρk2lc.Formula (16) shows that X2 is positively related to X1; thus, ξ increases with the increase of X1. The combination of formula (36) and Table 2 reveals that, with the increase of X1, ωv decreases while kv increases. When X1 is increased to a certain extent, point ωv,kv disappears. With the increase of X1, kp decreases while ωp increases. The combination of formula (36) and Table 4 illustrates that ωc decreases with the increase of X1, whereas the influence of X1 variation on cc is uncertain. Figure 8 shows the numerical simulation results of the equivalent stiffness and equivalent damping coefficient under different exciting amplitudes.Figure 8
(a) Equivalent stiffness under differentX1 values. (b) Equivalent damping coefficient under different X1values.
(a) (b)


### 4.2. Damp Channel Diameter

According to formulas (24), (26), (27), (31), (32), and (36),(37)ωv=k2ρlcdc2deq4-48πμlc+2ρdeq2ωX2ζ1+ζ23dcdeq2πρk2lc,(38)ωp=k2ρlcdc2deq4+48πμlc+2ρdeq2ωX2ζ1+ζ23dcdeq2πρk2lc,(39)ωc=k2ρdc2deq4lc-224πμ+ρdeq2ωX2ζ1+ζ2/lc29πdc4ρk2,(40)cc=3πdc2k2ρlc224πμlc+deq2ρωX2ζ1+ζ21-24πdeq2μlc+ρdeq4ωX2ζ1+ζ22/9πdc6ρk2lc.Formula (37) shows that ωv decreases with the increase of dc, whereas the influence of dc variation on ωp is uncertain. In combination with formula (36), Tables 1 and 2 show that ξ decreases with the increase of dc, thereby resulting in the increase of kp and in the decrease of kv. According to the analysis in Section 2.2, when dc is decreased to a certain extent, point ωv,kv disappears, and kp→k∞. According formula (39), when dc increases, ωc appears and then increases with the increase of dc. According to formula (40), the influence of dc variation on cc is uncertain. Figure 9 shows the simulation results of the equivalent stiffness and equivalent damping coefficient under different damp channel diameter. The dotted line in Figure 9(b) represents the numerical simulation results of the equivalent damping when dp=10 mm.Figure 9
(a) Equivalent stiffness under differentdc values. (b) Equivalent damping coefficient under different dc values.
(a) (b)


### 4.3. Equivalent Cylinder Diameter of the Cavities

According to formulas (37) and (38), the influence of deq variation on ωv is uncertain, whereas ωp decreases with the increase of deq. In combination with formula (36), Tables 1 and 2 indicate that when deq increases, ξ decreases, thereby resulting in the increase of kv and the decrease of kp. According to the analysis in Section 2.2, when deq is increased to a certain extent, point ωv,kv disappears and kp→k∞. According to formula (39), ωc decreases with the increase of deq. When deq increases to a certain extent, point ωc,cc disappears. According to formula (40), the influence of deq variation on cc is uncertain. Figure 10 shows the numerical simulation results of the equivalent stiffness and the equivalent damping coefficient under different cavity equivalent cylindrical diameters.Figure 10
(a) Equivalent stiffness under differentdeq values. (b) Equivalent damping coefficient under different deq values.
(a) (b)


### 4.4. Sum of the Stiffness of the Bellows at the End of the Isolator

Formulas (37) and (38) show that ωv and ωp increase with the increase of the sum of the stiffness of the bellows at the end of the isolator k2. According to formulas (25), (28), and (29):(41)kv=k1+k2+k22meqc-p2-2c-pk2meq,(42)kp=k1+k2+k2meqc-p2/k2+2c-pmeq/k2.The influence ofk2 variation on kv is uncertain, whereas kp increases with the increase of k2. According to formula (39), ωc increases with the increase of k2. Formulas (22) and (25) are plugged into formula (32):(43)cc=4k2meq24c-pmeq-c-p3/k2.c c increases with the increase of k2. Figure 11 shows the numerical simulation results of the equivalent stiffness and equivalent damping coefficient under different k2 values.Figure 11
(a) Equivalent stiffness under differentk2 values. (b) Equivalent damping coefficient under different k2 values.
(a) (b)


### 4.5. Length of Damp Channel

According to formulas (37) and (38), the influence of lc variation on ωv and ωp is uncertain. According to formulas (28), (29), and (36),(44)kv=k1+k2+9πk22dc6ρlc4deq224πμlc+deq2ρωX2ζ1+ζ224deq2πμlc-3dc3πρk2lc+deq4ρωX2ζ1+ζ2,(45)kp=k1+k2+9πk22dc6ρlc4deq224πμlc+deq2ρωX2ζ1+ζ224deq2πμlc+3dc3πρk2lc+deq4ρωX2ζ1+ζ2.The influence oflc variation on kv and kp is uncertain. According to formulas (39) and (40), the influence of lc variation on ωc and cc is uncertain. Figure 12 shows the numerical simulation results of equivalent stiffness and equivalent damping coefficient under different damp channel lengths.Figure 12
(a) Equivalent stiffness under differentlc values. (b) Equivalent damping coefficient under different lc values.
(a) (b)


## 4.1. Exciting Amplitude

According to formulas (10), (11), (15), and (25), (36)ξ=24πdeq2μlc+ρdeq4ωX2ζ1+ζ23dc3πρk2lc.Formula (16) shows that X2 is positively related to X1; thus, ξ increases with the increase of X1. The combination of formula (36) and Table 2 reveals that, with the increase of X1, ωv decreases while kv increases. When X1 is increased to a certain extent, point ωv,kv disappears. With the increase of X1, kp decreases while ωp increases. The combination of formula (36) and Table 4 illustrates that ωc decreases with the increase of X1, whereas the influence of X1 variation on cc is uncertain. Figure 8 shows the numerical simulation results of the equivalent stiffness and equivalent damping coefficient under different exciting amplitudes.Figure 8
(a) Equivalent stiffness under differentX1 values. (b) Equivalent damping coefficient under different X1values.
(a) (b)


## 4.2. Damp Channel Diameter

According to formulas (24), (26), (27), (31), (32), and (36),(37)ωv=k2ρlcdc2deq4-48πμlc+2ρdeq2ωX2ζ1+ζ23dcdeq2πρk2lc,(38)ωp=k2ρlcdc2deq4+48πμlc+2ρdeq2ωX2ζ1+ζ23dcdeq2πρk2lc,(39)ωc=k2ρdc2deq4lc-224πμ+ρdeq2ωX2ζ1+ζ2/lc29πdc4ρk2,(40)cc=3πdc2k2ρlc224πμlc+deq2ρωX2ζ1+ζ21-24πdeq2μlc+ρdeq4ωX2ζ1+ζ22/9πdc6ρk2lc.Formula (37) shows that ωv decreases with the increase of dc, whereas the influence of dc variation on ωp is uncertain. In combination with formula (36), Tables 1 and 2 show that ξ decreases with the increase of dc, thereby resulting in the increase of kp and in the decrease of kv. According to the analysis in Section 2.2, when dc is decreased to a certain extent, point ωv,kv disappears, and kp→k∞. According formula (39), when dc increases, ωc appears and then increases with the increase of dc. According to formula (40), the influence of dc variation on cc is uncertain. Figure 9 shows the simulation results of the equivalent stiffness and equivalent damping coefficient under different damp channel diameter. The dotted line in Figure 9(b) represents the numerical simulation results of the equivalent damping when dp=10 mm.Figure 9
(a) Equivalent stiffness under differentdc values. (b) Equivalent damping coefficient under different dc values.
(a) (b)


## 4.3. Equivalent Cylinder Diameter of the Cavities

According to formulas (37) and (38), the influence of deq variation on ωv is uncertain, whereas ωp decreases with the increase of deq. In combination with formula (36), Tables 1 and 2 indicate that when deq increases, ξ decreases, thereby resulting in the increase of kv and the decrease of kp. According to the analysis in Section 2.2, when deq is increased to a certain extent, point ωv,kv disappears and kp→k∞. According to formula (39), ωc decreases with the increase of deq. When deq increases to a certain extent, point ωc,cc disappears. According to formula (40), the influence of deq variation on cc is uncertain. Figure 10 shows the numerical simulation results of the equivalent stiffness and the equivalent damping coefficient under different cavity equivalent cylindrical diameters.Figure 10
(a) Equivalent stiffness under differentdeq values. (b) Equivalent damping coefficient under different deq values.
(a) (b)


## 4.4. Sum of the Stiffness of the Bellows at the End of the Isolator

Formulas (37) and (38) show that ωv and ωp increase with the increase of the sum of the stiffness of the bellows at the end of the isolator k2. According to formulas (25), (28), and (29):(41)kv=k1+k2+k22meqc-p2-2c-pk2meq,(42)kp=k1+k2+k2meqc-p2/k2+2c-pmeq/k2.The influence ofk2 variation on kv is uncertain, whereas kp increases with the increase of k2. According to formula (39), ωc increases with the increase of k2. Formulas (22) and (25) are plugged into formula (32):(43)cc=4k2meq24c-pmeq-c-p3/k2.c c increases with the increase of k2. Figure 11 shows the numerical simulation results of the equivalent stiffness and equivalent damping coefficient under different k2 values.Figure 11
(a) Equivalent stiffness under differentk2 values. (b) Equivalent damping coefficient under different k2 values.
(a) (b)


## 4.5. Length of Damp Channel

According to formulas (37) and (38), the influence of lc variation on ωv and ωp is uncertain. According to formulas (28), (29), and (36),(44)kv=k1+k2+9πk22dc6ρlc4deq224πμlc+deq2ρωX2ζ1+ζ224deq2πμlc-3dc3πρk2lc+deq4ρωX2ζ1+ζ2,(45)kp=k1+k2+9πk22dc6ρlc4deq224πμlc+deq2ρωX2ζ1+ζ224deq2πμlc+3dc3πρk2lc+deq4ρωX2ζ1+ζ2.The influence oflc variation on kv and kp is uncertain. According to formulas (39) and (40), the influence of lc variation on ωc and cc is uncertain. Figure 12 shows the numerical simulation results of equivalent stiffness and equivalent damping coefficient under different damp channel lengths.Figure 12
(a) Equivalent stiffness under differentlc values. (b) Equivalent damping coefficient under different lc values.
(a) (b)


## 5. Displacement Transmission Rate Characteristics of the Isolator

The equivalent stiffness and equivalent damping represent the force transmission rate characteristics of the isolator. To inspect the performance of the isolator comprehensively, the displacement transmission rate characteristics of the isolator should also be studied. To study the displacement transmission rate characteristics of the isolator, the equivalent system of the isolator with a displacement excitation base is shown in Figure13.Figure 13
Equivalent system of the isolator with a displacement excitation base.In Figure13, x is the displacement excitation of the base, and x1 and x2 are the displacement response of the load M and equivalent mass of the liquid in the damp channel meq, respectively.As for the loadM(46)Mx¨1+c-px˙1-x˙2+k1x1-x=0.As for the equivalent mass of the liquid in the damp channelmeq(47)meqx¨2+c-px˙2-x˙1+k2x2-x=0.Ifx is the harmonic excitation, then x, x1, and x2 can be expressed as follows: (48)x=Xejωt,x1=X1ejωt+θ1,x2=X2ejωt+θ2.Formula (48) is plugged into formulas (46) and (47), and the following formula can be obtained:(49)x1-ω2M+jωc-p+k1=x2jωc-p+k1x,x2-ω2meq+jωc-p+k2=x1jωc-p+k2x.Assuming thatA1=-ω2M+jωc-p+k1, A2=-ω2meq+jωc-p+k2 and A3=jωc-p, From formula (49), we can obtain(50)x1x=A3k2+A2k1A1A2-A32.By pluggingA1, A2, and A3 into formula (50), we can obtain the displacement transfer function(51)T1=x1x=k1k2+jωc-pk1+k2-k1meqω2c-p2ω2+k2+jωc-p-meqω2k1+jωc-p-Mω2.For comparison, assuming thatmeq=0, then the displacement transfer function is(52)T2=x1x=k1k2+jωc-pk1+k2k1k2+jωc-pk1+k2-k2Mω2-jc-pMω3.Thus, we can obtain the amplitude-frequency characteristic of the displacement transfer functionx1/x when meq≠0 and meq=0, and it is shown in Figures 14(a) and 14(b), respectively.Figure 14
(a) Amplitude-frequency characteristic whenmeq≠0. (b) Amplitude-frequency characteristic when meq=0.
(a) (b)From Figure14, we can see that the existence of meq gives the amplitude-frequency characteristic of the displacement transfer function curve an additional peak at a high frequency band, thereby resulting in the displacement transfer function of the novel isolator decaying twice at high frequency band. By contrast, the displacement transfer function of the novel isolator decays only once at a high frequency band when meq=0. The existence of meq improves the performance of the isolator at a high frequency band. Moreover, the relevant parameter should be chosen carefully to ensure a reasonable frequency of the second curve peak, thereby preventing the performance of the isolator from being influenced at the frequency of the second peak.


## 6. Conclusion

This study conducted a theoretical analysis, experimental verification, and analysis of the influence on the dynamic characteristics of the key parameter variation of the new type of fluid damping vibration isolator with bellows. The following conclusions were drawn from the results.(1)
The model established in this study has high accuracy when applied to microvibration isolation.(2)
The fourth parametermeq improves the performance of the isolator at a high frequency band.(3)
The parameters of the vibration isolator can be identified according to the fixed points0,k1, ωn,k1+k2, and ∞,k1+k2 of the equivalent stiffness curve of the vibration isolator with different excitation amplitudes.(4)
According to the influence of the key parameter variation on the performance of the vibration isolator, the rule of the influence of key design parameters of the vibration isolator on the equivalent stiffness and damping coefficient of the isolator is as follows:ω v k v ω p k p ω c c c X 1 ↓ ↑ ↑ ↓ ↓ uncertain d c ↑ ↓ uncertain ↑ ↑ uncertain d eq uncertain ↑ ↓ ↓ ↓ uncertain k 2 ↑ uncertain ↑ ↑ ↑ ↑ l c uncertain uncertain uncertain uncertain uncertain uncertain( 5 )
When the vibration isolator is designed for a vibration isolation system,k1 can initially be determined according to the static stiffness demand of the isolation system, whereas k2 can be determined by the stiffness demand at high frequency. Then, according to the rule of dc, lc, and deq influencing the stiffness and damping of the isolator and the performance demand of the vibration isolation system, dc, lc, and deq of each vibration isolator can be designed. Thus, the design of the isolators can be accomplished.



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*Source: 5081812-2016-12-07.xml*