# A Novel Low Stiffness Air Spring Vibration-Isolation Mounting System

Authors: Chang-geng Shuai; Bu-yun Li; Jian-guo Ma; Zhao-hao Yang
Journal: Shock and Vibration
 (2022)

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## Abstract

Quasi-zero-stiffness (QZS) structures offer substantial practical benefits for facilitating the essential support and isolation of vibrational source loads aboard modern marine vessels. In order to design a compact QZS system of high bearing capacity, the article proposes a low stiffness air spring (LSAS) vibration-isolation mounting system composed of both vertical and lateral air springs. The vertical air springs support the load and isolate vibrations in the vertical direction, while the lateral air springs support the load and isolate transversal vibrations. Theoretical analyses based on a simple two-dimensional, single degree of freedom model demonstrate that the proposed novel LSAS design decreases the degree of stiffness in the support structure that would otherwise be induced by introducing lateral air springs and accordingly increases the vibration isolation effect. Moreover, optimization of the air spring parameters enables the lateral air springs to provide negative stiffness and thereby realize QZS characteristics. Experimental testing based on prototypes of a standard air spring mounting system and the proposed LSAS mounting system demonstrates that the stiffness of the proposed system is about 1/5 that of the standard system. Accordingly, the proposed structure design successfully alleviates the undesirable influence of lateral air springs on the stiffness of the mounting system.


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## Body

## 1. Introduction

Vibration control is an essential operation in modern marine vessels for supporting numerous factors, such as the acoustic stealth of vessels, the living comfort of passengers, the working performance of on-board precision instruments, and the service life of the equipment. Vibration isolation has been the most widely adopted method for this purpose [1]. The general aim of vibration isolation systems is to reduce the level of vibrations transmitted to primary ship structures from vibration sources, such as marine propulsion units (MPUs), generators, ventilation units, and pumps. The most common vibration isolation systems employed in marine vessels include single-stage vibration isolation systems, double-stage vibration isolation systems, and floating raft vibration isolation systems [2].The primary components increasingly employed in the above-discussed vibration isolation systems are air springs. Accordingly, the design of high-performance air springs has been the subject of intense interest for developing advanced vibration isolation systems. The development of air springs has greatly contributed to the effective isolation of vibrations arising from large-scale on-board equipment. These efforts have been further supported by the development of a new type of air spring with numerous advantages, such as high working pressure, large load capacity, compact structure, and low natural oscillation frequency [3–5].In addition to designing advanced air springs, efforts have been applied to developing more advanced vibration isolation systems, such as quasi-zero-stiffness (QZS) structures. A QZS structure has the advantage of high-static-low-dynamic stiffness, which can provide enhanced vibration isolation performance relative to linear systems. The important research part of the QZS structure is to use the benefit of nonlinearity [6]. A typical approach applied for designing a QZS structure involves the use of oblique springs. For example, Carrella et al. [7] proposed a simple QZS isolator model comprising two oblique springs in parallel with a vertical spring and studied the restoring force and force transmissibility characteristics of the system. Subsequently, a more in-depth dynamic analysis of this nonlinear mechanism with three springs was conducted [8, 9]. On this basis, researchers have used a linkage and two noninclined springs to realize QZS characteristics under small deflection [10], while large deflection conditions have been accommodated using a combination of two inclined links, one noninclined spring, and two orientating springs [11]. Another typical approach applied for designing QZS isolators involves using X-shaped, bioinspired, and ring shape structures. For example, Sun and Jing [12] evaluated the vibration isolation performance of X-shaped isolators. Other studies proposed bioinspired X-shaped structures appropriate for different engineering practices [13–15]. Lu et al. [16] investigated the dynamical characteristics of a circular ring shape isolator, and the results demonstrated that this shape exhibited a nonlinear stiffness that enhanced its vibration isolation efficiency. Magnetic springs have also been used to design nonlinear QZS vibration isolators. For example, Xu et al. [17] applied magnetic springs to design a nonlinear vibration isolator for low-frequency applications. Wu et al. [18] proposed a vibration isolator that combines a coil spring in parallel with a magnetic spring with negative stiffness. Zhang et al. [19] utilized the variable reluctance stress of a magnetic spring to generate a negative stiffness. Yuan et al. [20] designed an electromagnetic spring with negative stiffness over a long stroke by combining three toroidal coils arranged coaxially with a ring magnet. Finally, special materials have been employed to design nonlinear QZS vibration isolators. For example, Lu et al. [21] used a bistable carbon fiber-metal composite plate for realizing low-frequency vibration isolation. Li et al. [22] designed a mechanical molecular spring isolator to realize high-static-low-dynamic stiffness.The above discussion demonstrates that QZS structures offer substantial practical benefits for providing excellent low-frequency vibration isolation performance. The vibration isolation system of vessels should have a compact structure and large bearing capacity, which the current air spring vibration isolation system can satisfy. Inspired by the nonlinear structures discussed above, this article proposes a novel low stiffness air spring (LSAS) vibration-isolation mounting system based on a standard air spring vibration isolation system to further improve the vibration isolation ability. The LSAS system is composed of vertical and lateral air springs that, respectively, support the weight of large-scale loads in the vertical and lateral directions while also isolating the respective vibrations in those directions. The proposed LSAS mounting structure is modeled as a simple two-dimensional (2D), single degree of freedom system with displacements limited to the vertical direction. While displacements in the horizontal direction can certainly affect the vibration transmissibility of the structure, the applied simplifications facilitate detailed theoretical analyses for determining the optimal air spring parameters that enable the lateral air springs to provide negative stiffness to the system and thereby realize QZS characteristics. Therefore, the simplifications are quite useful. Finally, prototypes of a standard air spring mounting system and the proposed LSAS mounting system are constructed, and experimental testing demonstrates that the vertical stiffness of the proposed system is about 1/5 that of the standard system. Accordingly, the proposed structure design successfully alleviates the undesirable influence of lateral air springs on the stiffness of the mounting system.


## 2. Standard Air Spring Vibration-Isolation Mounting Systems

The theoretical analysis is based on the standard three-dimensional (3D) air spring mounting system schematically illustrated in Figure1. The model is composed of a mass (C), six vertical air springs (A) between C and the floor boundary, and six lateral air springs (B) between C and the wall boundaries. It should be noted that the number of vertical and lateral air springs employed depends entirely on the actual setting and may therefore vary considerably.Figure 1
Three-dimensional (3D) schematic illustrating a representative standard air spring mounting system: (A) vertical air springs; (B) lateral air springs; (C) supported mass. Please note that the number of air springs applied depends entirely on the actual setting.Analysis of the mounting system is greatly simplified by adopting the 2D model illustrated in Figure2(a), along with the corresponding free body diagram given in Figure 2(b). In addition, we assume that no displacements occur in the horizontal direction. Therefore, all forces in the y-direction arising from the lateral air springs are always equal and opposite. Accordingly, the 2D standard air spring mounting system can be viewed as a simple single degree of freedom vibration isolation system.Figure 2
The simplified standard air spring mounting system: (a) two-dimensional (2D) schematic and (b) free body diagram.
(a)(b)Under the assumption of small displacementsx, the vertical restoring force (F0x) of the single vertical air spring and the transversal restoring force (F1x) of a single lateral air spring can be approximately given as the following first-order functions of deformation:(1)F0x=kxxx+Fvv0.(2)F1x=kyyx.Here,kxx and kyy are the vertical stiffness and transversal stiffness coefficients of the air springs, respectively, and Fvv0 is the initial vertical force associated with the weight of C. According to (1) and (2), the total restoring force and stiffness of the standard air spring mounting system illustrated in Figure 2(b) can be, respectively, given as follows:(3)Fr−s=F0x+2F1x=kxxx+2kyyx,kr−s=kxx+2kyy.Similarly, the force transmissibility of the standard air spring mounting system can be given as follows:(4)Ts=kr−s+4ξ2Ω2kvvkr−s1−Ω2kxx/kr−s2+4ξ2Ω2kxx.Here,Ω represents the excitation frequency and ξ is the damping coefficient of the system. Finally, the force transmissibility of a single vertical standard air spring mounting structure can be given as follows [23]:(5)Ta=1+4ξ2Ω21−Ω22+4ξ2Ω2.The values ofTs and Ta are plotted in Figure 3 as a function of excitation frequency Ω applied in the vertical direction for all air springs with uniform internal pressures of 1.7 MPa. The relationships between the internal air pressure and the stiffness coefficients of the air springs are presented in Table 1. These results demonstrate that the introduction of lateral air springs makes the natural oscillation frequency of the standard system (Ω2) 2 times greater than that of the single vertical air spring structure (Ω1) and accordingly reduces the vibration isolation effect of the system.Figure 3
Comparison of force transmissibility values obtained by the standard air spring mounting system and a single vertical air spring mounting structure as a function of the excitation frequency.Table 1
Relationships between the internal air pressure of the air spring and the values of the stiffness coefficients used in the mathematical models, and the weight of a supported mass (Fvv0).
Pressure (MPa)kxx (kN·mm−1)kyy (kN·mm−1)kθθ (kN·mmrad−1)Fvv0 (kN)μ21.31.81204.17463500.218.15642.31.51.98054.33763998.420.96732.21.72.12924.43954512.524.26362.11.92.28454.51375001.129.96522.02.12.39354.55395101.329.87262.0


## 3. Low Stiffness Air Spring Mounting Design

### 3.1. Model

The proposed LSAS mounting system is illustrated schematically in Figure4. The primary difference between this model and the standard model in Figure 2(a) is that connectors are rigidly connected to the cover plates of the lateral air springs, flexibly connected by universal joints, and then rigidly connected to C to address the undesirable effect of the lateral air springs on the vibration isolation system. Here, Figure 4 represents a static equilibrium condition, where the vertical spring supports the load of the mass, the universal joints are in a nonrotated state, and the connectors lie in the horizontal direction. Accordingly, it is assumed that the initial installation height is equivalent for both lateral air springs.Figure 4
2D schematic of the proposed low stiffness air spring (LSAS) mounting system at equilibrium.


### 3.2. Static Analysis

A nonequilibrium free body diagram corresponding to the schematic illustrated in Figure4 is given in Figure 5. Here, only the universal joints and lateral air springs are able to rotate, while all other components remain rigidly connected. Therefore, the length of the connectors between the universal joints and the supported mass can be neglected in the analysis because these are merely integral parts of the supported mass. Therefore, the rotation of the lateral air springs is accommodated by their irregular deformation along the cover plates. As such, the supported mass is subjected to a static force Pe composed of F0x and other components of force in the vertical x-direction arising from the linkage between the lateral air springs and C. Again, we assume that no displacements occur in the horizontal direction, and all forces in the horizontal y-direction arising from the lateral air springs are always equal and opposite. Therefore, the vertical air spring deforms only in the x-direction and supports the mass.Figure 5
Free body diagram under a non-equilibrium condition.We first consider the behavior of the lateral springs in isolation from the vertical air spring. This decoupling of the system is acceptable under the assumption that no displacements occur in the horizontal direction. Therefore, the vertical forces of the vertical air spring and lateral air springs on the supported mass are merely additive. Here, the restoring forces of a single lateral air spring perpendicular to the cover plates (Fv) and along the cover plates (Fh), as well as the torsional moment (M), can be defined according to displacements dx and dy as follows:(6)Fv=−kxxdx+Fv0,Fh=kyydy,M=kθθθ.Here,kxx, kyy, and kθθ are the vertical, horizontal, and torsional stiffness coefficients of the lateral air springs, respectively.The relationships betweendx, dy, and θ under deformations in a lateral air spring are illustrated in Figure 6, where Point A and Point B represent the positions of the universal joint in equilibrium and under deformation associated with a vertical displacement x, respectively. Under deformation, the cover plate of the lateral air spring has rotated by an angle θ. For clarification, we also assume that Point A′ also exists that is associated with the same rotation angle θ. Therefore, the relationship between dx, dy, and x can be obtained based on the length L of the connector as follows:(7)dycosθ+Ltanθ=x,b=dytanθ,a=Lcosθ−L.Figure 6
Schematic analysis of lateral air spring deformations.These relationships yield the following two expressions that can be employed to solvedx and dy:(8)dx=dytanθ+Lcosθ−L,dy=x−Ltanθcosθ.To facilitate the solution ofθ, we adopt the following moment balance analysis of the connector:(9)M=FhL=kθθθ.Accordingly, the total restoring force of the two lateral air springs in thex-direction is defined as follows:(10)Fr−lateral=2Fhcosθ−Fvsinθ.This expression is simplified by applying the following third-order Taylor series expansion of (10) at x = 0 [24]:(11)Fapp,r−lateral=2−Fv0Lkyy+kyykθθxL2kyy+kθθ+Fv0L3kyy3+3L4kxxkyy3−2L4kyy4+6L2kxxkyy2kθθ−6L2kyy3kθθx33L2kyy+kθθ3.The values of Fr–lateral obtained numerically from (10) and Fapp,r–lateral obtained from the analytical approximation in (11) are presented in Figure 7 as functions of displacement in the x-direction. The results demonstrate that the analytical solution of the cubic truncation is closer to the numerical solution than that of the quadratic truncation. Therefore, the applied analytical cubic truncation approximation is more acceptable. The overall stiffness contribution of the dual lateral air spring system can be expressed by the derivative of (11) with respect to x as follows:(12)kr−lateral=∂Fqpp,r−lateral∂x.Figure 7
Comparison between the restoring force of a lateral air spring system obtained from the numerical solution and the analytical solution as a function of vertical displacement.The dynamic stiffness coefficient for the lateral air springs under the equilibrium condition illustrated in Figure4 (i.e., x = 0) can be given as follows:(13)kr−lateralx=0=−2kyyFv0L−kθθL2kyy+kθθ.The equation provides a simple analytical expression for defining the contribution the lateral air springs to the overall stiffness of the LSAS system at equilibrium. As can be seen, kr–lateral(x = 0) > 0 when kθθ > Fv0L, such that the lateral air springs increase the stiffness of the system. Meanwhile, kr–lateral (x = 0) = 0, when kθθ = Fv0L, such that the lateral air springs have no effect on the overall stiffness of the system. However, we also find that kr–lateral(x = 0) < 0, when kθθ < Fv0L, which indicates that the lateral air springs introduce a negative stiffness to the LSAS system at equilibrium when the torsional stiffness coefficient falls within this range of values. The effects of these different ranges of kθθ on kr–lateral(x = 0) are demonstrated in Figure 8 for different values of L when Fv0 = 18.1564 kN and kyy = 4.518 kN/mm. The figure marks the points A, B, and C, where kr–lateral(x = 0) = 0 for the three different values of L. As can be seen, the range of kθθ over which kr–lateral(x = 0) < 0 increases significantly with increasing L. However, the space required by the LSAS system also increases with increasing L. Therefore, maintaining a sufficiently small value of kθθ will provide negative stiffness to the system while minimizing the space the system occupies.Figure 8
Contribution of the lateral air springs to the vertical stiffness of the LSAS system at equilibrium versus the torsional stiffnesskθθ for different connector lengths (L).We further note from (13) by setting L = 0 that kr–lateral(x = 0) = 2kyy for the standard mounting system illustrated in Figure 2(a). Therefore, we can establish the following performance criteria for the proposed LSAS mounting system:(14)kr−lateralx=0<2kyy.Replacingkr–lateral(x = 0) in (14) with the corresponding expression presented in (13) yields the following condition:(15)Lkyy+Fv0>0.This indicates that the proposed LSAS system is guaranteed to at least provide a lower overall stiffness at equilibrium than the standard mounting system, and can potentially function as a QZS system under appropriate conditions involvingkθθ, Fv0, and L.Then, we extend our analysis of the lateral air springs to include the vertical air spring in the system illustrated in Figure5. Here, the restoring force of the vertical air spring over small displacements x is(16)F0x=kvvx+Fvv0,where kvv is the vertical stiffness of the air spring and Fvv0 is its initial force applied to the load, which is equal and opposite to the force of the load. Therefore, the total restoring force of the LSAS system is(17)Fr−LSAS=F0x+Fapp,r−lateral−mg.where m is the mass of the supported load and g is the acceleration due to gravity. Introducing nondimensional parameters F^v0=Fv0/kxxL, μ1=kθθ/kyyL2, μ2=kxx/kyy, k^r−lateral=kr−lateral/kxx, α=kvv/kxx, x^=x/L yields the following dimensionless form of (17):(18)F^r−LSAS=αx^+F^v0x^331+μ13+x^31+μ13+2x^3μ11+μ13−2x^33μ21+μ13−2x^3μ1μ21+μ13−2F^v0x^1+μ1+2x^μ1μ21+μ1.The nondimensional stiffness of the system can be obtained by differentiating (18) with respect to x^ as follows:(19)k^r−LSAS=α+F^v0x^21+μ13+3x^21+μ13+6x^2μ11+μ13−2x^2μ21+μ13−6x^2μ1μ21+μ13−2F^v01+μ1+2μ1μ21+μ1.Finally, the condition of zero stiffness for the LSAS mounting system can be obtained by settingk^r−LSASx^=0=0, which yields the following expression:(20)α=−2μ1−F^v0μ21+μ1μ2.Equation (20) provides a simple analytical condition for ensuring that the LSAS system achieves QZS performance. For instance, after carefully designing the lateral air spring to introduce negative stiffness at equilibrium, a QZS state can be ensured to satisfy (14). This example is illustrated by the effects of small dimensionless vertical displacements on Fr–LSAS demonstrated in Figure 9(a) for different values of α when μ1=0.03, μ2=0.4, and the lateral air springs have been designed to introduce negative stiffness. The results demonstrate that the LSAS system behavior can be strongly nonlinear, and a value of αQZS=0.175 provides QZS behavior over a relatively wide region of displacements about the equilibrium point. Therefore, the value of α should be greater than αQZS to avoid negative stiffness near the equilibrium point.Figure 9
Restoring force of the proposed LSAS system as a function of small dimensionless vertical displacements for different values ofα: (a) when the lateral air springs are designed to introduce negative stiffness and (b) when the lateral air springs are designed to introduce positive stiffness.
(a)(b)A counterexample was also considered when the lateral air springs have been designed to introduce positive stiffness (i.e.,kθθ > Fv0L). The same conditions employed for Figure 9(a) were employed again with αQZS=0.175, μ2=0.4, and the effects of small dimensionless vertical displacements on Fr–LSAS are demonstrated in Figure 9(b) for different values of µ1 under the positive stiffness setting. The results demonstrate that the restoring force of the LSAS system behaves linearly under this setting. The observed linearity indicates that the nonlinear components of equation (22) are small when the lateral air springs have been designed to introduce positive stiffness. And as µ1 decreases, the slope of the force-displacement curve also decreases.


### 3.3. Dynamic Analysis

It is demonstrated in Section3.2 that as the lateral air springs provide negative stiffness, the vertical air spring can be adjusted to satisfy the QZS condition. The dynamic equation of the system can be illustrated by the Duffing equation [25]. However, current air springs can only provide positive stiffness for the system. Therefore, we make use of the fact demonstrated above that the nonlinear terms in (18) are small to simplify the analysis by applying a Taylor series expansion.A first-order Taylor series is defined as follows:(21)f=f0+f′0x^.This simple first-order form is acceptable for small displacements about the equilibrium pointx^=0. Accordingly, (18) can be approximated as follows:(22)F^r−app=α−2F^v01+μ1+2μ11+μ1μ2x.This can be simplified further by settingk^=α−2Fv0/1+μ1+2μ1/1+μ1μ2. Then, the nondimensional equation of motion for the LSAS system under harmonic excitation is given as follows:(23)x^¨+2ξx^˙+k^x^=f^cosΩ1τ.Here,ω02=kxx/m, τ=ω0t, ξ=c/2mω0, Ω1=ω/ω0, and f^=f0/kxxL. where ω0 is the fundamental frequency, ω is the excitation frequency, t is time, c is the damping coefficient, and f0 is the excitation amplitude. Finally, the force transmissibility of the LSAS system can be given as follows:(24)TLSAS=1+4ξ2Ω121−Ω122+4ξ2Ω12.We compare the force transmissibility values obtained for a single vertical air spring ((5)), a standard air spring mounting system ((4)), and the proposed LSAS mounting system ((24)) in Figure 10. These results demonstrate that the decreased vibration isolation effect obtained when adding lateral air springs within the air spring system has been mitigated to a large extent by the proposed LSAS system, where the excitation response of the LSAS system is very similar to the response observed for a single vertical air spring. The results further demonstrate that the vibration isolation effect of the LSAS system increases with decreasing µ1.Figure 10
Comparison of force transmissibility values obtained by the proposed LSAS mounting system, the standard air spring mounting system, and a single vertical air spring mounting structure as a function of the excitation frequency.


## 3.1. Model

The proposed LSAS mounting system is illustrated schematically in Figure4. The primary difference between this model and the standard model in Figure 2(a) is that connectors are rigidly connected to the cover plates of the lateral air springs, flexibly connected by universal joints, and then rigidly connected to C to address the undesirable effect of the lateral air springs on the vibration isolation system. Here, Figure 4 represents a static equilibrium condition, where the vertical spring supports the load of the mass, the universal joints are in a nonrotated state, and the connectors lie in the horizontal direction. Accordingly, it is assumed that the initial installation height is equivalent for both lateral air springs.Figure 4
2D schematic of the proposed low stiffness air spring (LSAS) mounting system at equilibrium.


## 3.2. Static Analysis

A nonequilibrium free body diagram corresponding to the schematic illustrated in Figure4 is given in Figure 5. Here, only the universal joints and lateral air springs are able to rotate, while all other components remain rigidly connected. Therefore, the length of the connectors between the universal joints and the supported mass can be neglected in the analysis because these are merely integral parts of the supported mass. Therefore, the rotation of the lateral air springs is accommodated by their irregular deformation along the cover plates. As such, the supported mass is subjected to a static force Pe composed of F0x and other components of force in the vertical x-direction arising from the linkage between the lateral air springs and C. Again, we assume that no displacements occur in the horizontal direction, and all forces in the horizontal y-direction arising from the lateral air springs are always equal and opposite. Therefore, the vertical air spring deforms only in the x-direction and supports the mass.Figure 5
Free body diagram under a non-equilibrium condition.We first consider the behavior of the lateral springs in isolation from the vertical air spring. This decoupling of the system is acceptable under the assumption that no displacements occur in the horizontal direction. Therefore, the vertical forces of the vertical air spring and lateral air springs on the supported mass are merely additive. Here, the restoring forces of a single lateral air spring perpendicular to the cover plates (Fv) and along the cover plates (Fh), as well as the torsional moment (M), can be defined according to displacements dx and dy as follows:(6)Fv=−kxxdx+Fv0,Fh=kyydy,M=kθθθ.Here,kxx, kyy, and kθθ are the vertical, horizontal, and torsional stiffness coefficients of the lateral air springs, respectively.The relationships betweendx, dy, and θ under deformations in a lateral air spring are illustrated in Figure 6, where Point A and Point B represent the positions of the universal joint in equilibrium and under deformation associated with a vertical displacement x, respectively. Under deformation, the cover plate of the lateral air spring has rotated by an angle θ. For clarification, we also assume that Point A′ also exists that is associated with the same rotation angle θ. Therefore, the relationship between dx, dy, and x can be obtained based on the length L of the connector as follows:(7)dycosθ+Ltanθ=x,b=dytanθ,a=Lcosθ−L.Figure 6
Schematic analysis of lateral air spring deformations.These relationships yield the following two expressions that can be employed to solvedx and dy:(8)dx=dytanθ+Lcosθ−L,dy=x−Ltanθcosθ.To facilitate the solution ofθ, we adopt the following moment balance analysis of the connector:(9)M=FhL=kθθθ.Accordingly, the total restoring force of the two lateral air springs in thex-direction is defined as follows:(10)Fr−lateral=2Fhcosθ−Fvsinθ.This expression is simplified by applying the following third-order Taylor series expansion of (10) at x = 0 [24]:(11)Fapp,r−lateral=2−Fv0Lkyy+kyykθθxL2kyy+kθθ+Fv0L3kyy3+3L4kxxkyy3−2L4kyy4+6L2kxxkyy2kθθ−6L2kyy3kθθx33L2kyy+kθθ3.The values of Fr–lateral obtained numerically from (10) and Fapp,r–lateral obtained from the analytical approximation in (11) are presented in Figure 7 as functions of displacement in the x-direction. The results demonstrate that the analytical solution of the cubic truncation is closer to the numerical solution than that of the quadratic truncation. Therefore, the applied analytical cubic truncation approximation is more acceptable. The overall stiffness contribution of the dual lateral air spring system can be expressed by the derivative of (11) with respect to x as follows:(12)kr−lateral=∂Fqpp,r−lateral∂x.Figure 7
Comparison between the restoring force of a lateral air spring system obtained from the numerical solution and the analytical solution as a function of vertical displacement.The dynamic stiffness coefficient for the lateral air springs under the equilibrium condition illustrated in Figure4 (i.e., x = 0) can be given as follows:(13)kr−lateralx=0=−2kyyFv0L−kθθL2kyy+kθθ.The equation provides a simple analytical expression for defining the contribution the lateral air springs to the overall stiffness of the LSAS system at equilibrium. As can be seen, kr–lateral(x = 0) > 0 when kθθ > Fv0L, such that the lateral air springs increase the stiffness of the system. Meanwhile, kr–lateral (x = 0) = 0, when kθθ = Fv0L, such that the lateral air springs have no effect on the overall stiffness of the system. However, we also find that kr–lateral(x = 0) < 0, when kθθ < Fv0L, which indicates that the lateral air springs introduce a negative stiffness to the LSAS system at equilibrium when the torsional stiffness coefficient falls within this range of values. The effects of these different ranges of kθθ on kr–lateral(x = 0) are demonstrated in Figure 8 for different values of L when Fv0 = 18.1564 kN and kyy = 4.518 kN/mm. The figure marks the points A, B, and C, where kr–lateral(x = 0) = 0 for the three different values of L. As can be seen, the range of kθθ over which kr–lateral(x = 0) < 0 increases significantly with increasing L. However, the space required by the LSAS system also increases with increasing L. Therefore, maintaining a sufficiently small value of kθθ will provide negative stiffness to the system while minimizing the space the system occupies.Figure 8
Contribution of the lateral air springs to the vertical stiffness of the LSAS system at equilibrium versus the torsional stiffnesskθθ for different connector lengths (L).We further note from (13) by setting L = 0 that kr–lateral(x = 0) = 2kyy for the standard mounting system illustrated in Figure 2(a). Therefore, we can establish the following performance criteria for the proposed LSAS mounting system:(14)kr−lateralx=0<2kyy.Replacingkr–lateral(x = 0) in (14) with the corresponding expression presented in (13) yields the following condition:(15)Lkyy+Fv0>0.This indicates that the proposed LSAS system is guaranteed to at least provide a lower overall stiffness at equilibrium than the standard mounting system, and can potentially function as a QZS system under appropriate conditions involvingkθθ, Fv0, and L.Then, we extend our analysis of the lateral air springs to include the vertical air spring in the system illustrated in Figure5. Here, the restoring force of the vertical air spring over small displacements x is(16)F0x=kvvx+Fvv0,where kvv is the vertical stiffness of the air spring and Fvv0 is its initial force applied to the load, which is equal and opposite to the force of the load. Therefore, the total restoring force of the LSAS system is(17)Fr−LSAS=F0x+Fapp,r−lateral−mg.where m is the mass of the supported load and g is the acceleration due to gravity. Introducing nondimensional parameters F^v0=Fv0/kxxL, μ1=kθθ/kyyL2, μ2=kxx/kyy, k^r−lateral=kr−lateral/kxx, α=kvv/kxx, x^=x/L yields the following dimensionless form of (17):(18)F^r−LSAS=αx^+F^v0x^331+μ13+x^31+μ13+2x^3μ11+μ13−2x^33μ21+μ13−2x^3μ1μ21+μ13−2F^v0x^1+μ1+2x^μ1μ21+μ1.The nondimensional stiffness of the system can be obtained by differentiating (18) with respect to x^ as follows:(19)k^r−LSAS=α+F^v0x^21+μ13+3x^21+μ13+6x^2μ11+μ13−2x^2μ21+μ13−6x^2μ1μ21+μ13−2F^v01+μ1+2μ1μ21+μ1.Finally, the condition of zero stiffness for the LSAS mounting system can be obtained by settingk^r−LSASx^=0=0, which yields the following expression:(20)α=−2μ1−F^v0μ21+μ1μ2.Equation (20) provides a simple analytical condition for ensuring that the LSAS system achieves QZS performance. For instance, after carefully designing the lateral air spring to introduce negative stiffness at equilibrium, a QZS state can be ensured to satisfy (14). This example is illustrated by the effects of small dimensionless vertical displacements on Fr–LSAS demonstrated in Figure 9(a) for different values of α when μ1=0.03, μ2=0.4, and the lateral air springs have been designed to introduce negative stiffness. The results demonstrate that the LSAS system behavior can be strongly nonlinear, and a value of αQZS=0.175 provides QZS behavior over a relatively wide region of displacements about the equilibrium point. Therefore, the value of α should be greater than αQZS to avoid negative stiffness near the equilibrium point.Figure 9
Restoring force of the proposed LSAS system as a function of small dimensionless vertical displacements for different values ofα: (a) when the lateral air springs are designed to introduce negative stiffness and (b) when the lateral air springs are designed to introduce positive stiffness.
(a)(b)A counterexample was also considered when the lateral air springs have been designed to introduce positive stiffness (i.e.,kθθ > Fv0L). The same conditions employed for Figure 9(a) were employed again with αQZS=0.175, μ2=0.4, and the effects of small dimensionless vertical displacements on Fr–LSAS are demonstrated in Figure 9(b) for different values of µ1 under the positive stiffness setting. The results demonstrate that the restoring force of the LSAS system behaves linearly under this setting. The observed linearity indicates that the nonlinear components of equation (22) are small when the lateral air springs have been designed to introduce positive stiffness. And as µ1 decreases, the slope of the force-displacement curve also decreases.


## 3.3. Dynamic Analysis

It is demonstrated in Section3.2 that as the lateral air springs provide negative stiffness, the vertical air spring can be adjusted to satisfy the QZS condition. The dynamic equation of the system can be illustrated by the Duffing equation [25]. However, current air springs can only provide positive stiffness for the system. Therefore, we make use of the fact demonstrated above that the nonlinear terms in (18) are small to simplify the analysis by applying a Taylor series expansion.A first-order Taylor series is defined as follows:(21)f=f0+f′0x^.This simple first-order form is acceptable for small displacements about the equilibrium pointx^=0. Accordingly, (18) can be approximated as follows:(22)F^r−app=α−2F^v01+μ1+2μ11+μ1μ2x.This can be simplified further by settingk^=α−2Fv0/1+μ1+2μ1/1+μ1μ2. Then, the nondimensional equation of motion for the LSAS system under harmonic excitation is given as follows:(23)x^¨+2ξx^˙+k^x^=f^cosΩ1τ.Here,ω02=kxx/m, τ=ω0t, ξ=c/2mω0, Ω1=ω/ω0, and f^=f0/kxxL. where ω0 is the fundamental frequency, ω is the excitation frequency, t is time, c is the damping coefficient, and f0 is the excitation amplitude. Finally, the force transmissibility of the LSAS system can be given as follows:(24)TLSAS=1+4ξ2Ω121−Ω122+4ξ2Ω12.We compare the force transmissibility values obtained for a single vertical air spring ((5)), a standard air spring mounting system ((4)), and the proposed LSAS mounting system ((24)) in Figure 10. These results demonstrate that the decreased vibration isolation effect obtained when adding lateral air springs within the air spring system has been mitigated to a large extent by the proposed LSAS system, where the excitation response of the LSAS system is very similar to the response observed for a single vertical air spring. The results further demonstrate that the vibration isolation effect of the LSAS system increases with decreasing µ1.Figure 10
Comparison of force transmissibility values obtained by the proposed LSAS mounting system, the standard air spring mounting system, and a single vertical air spring mounting structure as a function of the excitation frequency.


## 4. Experiments

Four air spring mounting system prototypes were employed for experimental testing. The prototypes are defined as follows:Prototype I: the proposed LSAS mounting system (μ1 = 0.3)Prototype II: the proposed LSAS mounting system (μ1=0.03)Prototype III: standard system, where the connector and the universal joint assemblies have been removed and the lateral air springs are directly connected to the supported massPrototype IV: single vertical air spring, where the lateral air springs and the connector and universal joint assemblies have been removedImages of the general layout of the test system and the connector and universal joint assemblies are presented in Figures11(a) and 11(b), respectively. All testing was conducted on an MTS Landmark® 370-50 servohydraulic test system, and the test data were recorded using an MTS software program. The vertical displacement amplitude of the servohydraulic system was set as ±4 mm, and the displacement frequency was set as 1/16 Hz, resulting in a displacement speed of 0.1 mm/s. A uniform test mass of 1000 kg and an internal vertical air spring pressure of 1.7 MPa was employed for all tests. Various internal lateral air spring pressures were employed during testing, including values in the range of 1.3, 1.5, 1.7, 1.9, and 2.1 MPa. As indicated in Table 1, these changes in lateral air spring pressure naturally affected the values of kθθ and μ2 = kxx/kyy, and the value of the nondimensional parameter μ1 = kθθ/(kyyL2) was fixed at 0.3 or 0.03 by setting the value of L as 60 mm and 165 mm, respectively. However, we further note from Table 1 that these changes in lateral air spring pressure had little effect on the value of the nondimensional parameter µ2.Figure 11
Prototype LSAS mounting system: (a) general layout and system components and (b) structure of the connector and universal joint assemblies.
(a)(b)The experimental vertical stiffness results obtained for the four prototype systems with different lateral air spring pressures P are listed in Table2. Naturally, the vertical stiffness results obtained for prototype IV were constant because the system included no lateral air springs. However, all other prototype systems exhibited an increasing vertical stiffness with increasing P. Moreover, these vertical stiffness values were greater than that of prototype IV under all internal pressures considered, demonstrating that the introduction of lateral air springs increased the vertical stiffness of the mounting systems. However, the fact that the vertical stiffness of prototype II was uniformly less than that of prototype I for all values of P considered confirms the positive effect of applying a decreasing value of µ1, as demonstrated by the theoretical analysis. Moreover, the vertical stiffness of prototype II was approximately 1/5 of that obtained by prototype III for all values of P considered, which again clearly confirms the results of theoretical analysis. Finally, the experimental results demonstrate that the vertical stiffness of prototype II was only about 7% greater than that of prototype IV at P = 1.3 MPa, which confirms the results of theoretical analysis demonstrating that the proposed connector and universal joint assemblies mitigate the increased stiffness arising from the incorporation of lateral air springs.Table 2
Experimental vertical stiffness results obtained with different lateral air spring pressures P.
P (MPa)Prototype I (kN·mm−1)Prototype II (kN·mm−1)Prototype III (kN·mm−1)Prototype IV (kN·mm−1)1.33.1172.19410.502.0471.53.2012.24610.732.0471.73.2902.26810.962.0471.93.3672.29011.182.0472.13.4662.31311.432.047The validity of the theoretical analysis is further demonstrated in Figure12 by comparing the experimental vertical stiffness results with the corresponding theoretical results obtained at the equilibrium point. We note that the agreement between the observed dependencies on P is excellent, which supports the validity of both the theoretical analyses and the experimental results.Figure 12
Comparison of experimental and theoretical results obtained for the vertical stiffness of the various vibration isolation mounting systems at the equilibrium point as a function of the internal lateral air spring pressure P.


## 5. Conclusion

The present work proposed a novel LSAS vibration-isolation mounting system composed of both vertical and lateral air springs with connectors and universal joints. Theoretical analyses based on a simple 2D, single degree of freedom model demonstrated that the proposed novel LSAS design decreases the degree of stiffness in the support structure that would otherwise be induced by introducing lateral air springs and accordingly increases the vibration isolation effect. Moreover, optimization of the air spring parameters enables the lateral air springs to provide negative stiffness, thereby realizing QZS characteristics. Finally, the results of experiments employing prototypes of a standard air spring mounting system and the proposed LSAS mounting system demonstrated that the vertical stiffness of the proposed system is about 1/5 that of the standard system. Accordingly, the proposed structure design successfully alleviates the undesirable influence of lateral air springs on the stiffness of the mounting system. We must also note the limitations of the present study associated with an omission of lateral displacements. While displacements in the horizontal direction can certainly affect the vibration transmissibility of the structure, the simplified 2D, single degree of freedom model applied in the present work facilitated detailed theoretical analyses for determining the optimal air spring parameters that enable the lateral air springs to provide negative stiffness to the system and thereby realize QZS characteristics. The clarity obtained by these simplifications was extremely beneficial for illustrating the specific factors affecting the QZS characteristics of the proposed mounting structure. These simplifications also provided a clear means of verifying the theoretical analyses via experimental testing and can be applied in later work considering a more rigorously complete model for redesigning the lateral air springs to provide negative stiffness and thereby realize QZS characteristics.



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*Source: 5598689-2022-02-21.xml*