Negative stiffness devices for vibration isolation applications: A review

Abstract

In recent years, negative stiffness vibration isolation device with nonlinear characteristic has become an emerging research area and attracted a significant amount of attentions in the community due to the promising potentials it brought into the field. Its high-static-low-dynamic stiffness property endows the capacity to realize effective vibration isolation and in the meantime to maintain the system stability. This article presents a comprehensive review of the recent research and developments on negative stiffness vibration isolation device. It begins with an introduction on the concept of negative stiffness and then provides a summary and discussion regarding the realization and characteristics of negative stiffness vibration isolation device. The article places its special interest on the principles, structure design, and device characterisation of different types of negative stiffness vibration isolation devices, including spring type, pre-bucked beam type, magnetism type, geometrically nonlinear structural type, and composite structural type. Besides, the applications of negative stiffness vibration isolation device, as well as negative stiffness damper, are summarized and discussed based on the current state-of-the-art. Finally, the conclusions and further discussion provide highlights of the investigation.

Keywords

base isolation negative stiffness negative stiffness damper seismic protection vibration isolation

Introduction

Structural vibration control has received considerable attentions from the mechanical and civil engineering research community for many decades. As an effective method to protect a structure, vibration isolation designed with negative stiffness characteristics, namely negative stiffness vibration isolation, becomes increasingly popular recently (Alabuzhev, 1989; Snowdon, 1979). Negative stiffness vibration isolation device (NSVID) has potential to significantly reduce the fundamental frequency of the protected structure to minimise the force or displacement transmission even for low- or ultra-low-frequency excitation cases, where the traditional vibration isolations are ineffective. The popularity of NSVID lays on its unique nonlinear characteristics, which enable NSVID to reduce vibration transmissibility from vibration sources and at the same time to maintain the system stability by providing sufficient load-carrying capacity.

Negative stiffness concept, as the core of NSVID, was first proposed by Molyneux (1957). As opposition to positive stiffness, the reactive force is to decrease resistance and even assist deformation progress in the case of the negative stiffness. Since it releases or losses energy along with its deformation without inducing significant amount of resistant force, it will need external energy input or pre-stored energy. Besides, the first device proposed by Molyneux was proven to be unstable since it had no load-carrying capacity resulting in large deformation at any loading (Wang and Lakes, 2004). It was that reason he connected negative stiffness elements with positive stiffness elements together to compensate the instability associated with negative stiffness in his later designs. The combination contributes to nonlinear force–displacement relationship, which addresses the dilemma for linear systems between the effective vibration isolation and small static deflection. The nonlinear characteristic endows NSVIDs with low dynamic stiffness, so-called quasi-zero stiffness (QZS), as well as high static stiffness simultaneously by carefully choosing structural parameters. Besides, negative stiffness characteristic is also verified to be able to improve the damping ratio and energy absorption capacity of a damper. An excellent review of nonlinear property was presented by Alabuzhev (1989).

To design a NSVID, positive stiffness elements are commonly in the form of coil springs, and the pre-compressed springs (Carrella et al., 2007b; Lan et al., 2014; Niu et al., 2014; Xu and Sun, 2015) are widely adopted to provide stored energy to facilitate negative stiffness. Besides, pre-buckled beams (Fulcher et al., 2014; Huang et al., 2014; Lee et al., 2007; Liu et al., 2013), magnetics (Shan et al., 2015; Wu et al., 2014; Xu et al., 2013; Zheng et al., 2016), convexity (Cheng et al., 2015; Toyooka et al., 2015; Zhou et al., 2015a, 2017), and composite structures (Correa et al., 2015; Izard et al., 2017; Virk et al., 2013) have also been verified for their capacities as negative stiffness elements. By supplementing negative stiffness device with damping, negative stiffness damper (NSD) is proposed and designed, that is also included as NSVID in this article. Further elaboration on the realization and application of NSVIDs will be discussed in the following section of this article.

In the past decades, systems and mechanisms designed with negative stiffness property have attracted many attentions and a great number of results have been presented, hence a systematic summarization of these research results is of vital importance. The growing increase of the publications on this topic since the 1990s is shown in Figure 1 and historical Scopus data also presents the distribution of these publications by countries.

Figure 1.

Historical statistics: Number of articles published in Scopus-indexed sources on the application of negative stiffness in vibration isolation by authors’ country of origin (from 1996 to 2019).

The review article looked into the recent research and developments on NSVIDs. The concept of negative stiffness will be mentioned first for the convenience of readers. Following that, the fundamentals of NSVID, including realization and characteristic are discussed. A comprehensive review on the structure design, characteristic, and evaluation of NSVIDs will be presented in five different groups based on the forms of negative stiffness elements, including springs, pre-bucked beams, magnets, geometrically nonlinear structures, as well as composite structures and metamaterials. Besides, the applications of NSVIDs and NSDs are illustrated by exemplars. Conclusive remarks and comments will be given at the end of the article.

Fundamentals of NSVIDs

Concept of negative stiffness

Negative stiffness, as opposition to positive stiffness, is with the reactive force to assist deformation progress instead of resisting, that is, the resistant force is in the same direction as deformation (Van Eijk and Dijksman, 1979). As equation (1) described, the magnitude of reactive force is linearly proportional to the deformation. The potential energy produced by the negative stiffness element is negative, which means loss of energy along deformation; the mathematical expression is depicted in equation (2)

k = - \frac{F}{D}

(1)

U = - \frac{1}{2} k D^{2}

(2)

where F is the applied force, D is the deformation, and k is the stiffness of the object.

Negative stiffness was initially observed from the buckling phenomenon of column in structural engineering, which was regarded as a failure mode since the element hereby lost its stability with sudden incremental of deformation. The buckling process of the beam can be divided into three transition states, illustrated in Figure 2. States (1) and (3) represent the bistable states of the beam, while state (2) indicates an unstable equilibrium, where the beam can hold its configuration without external lateral force and any small disturbance can push it to one of the two stable states, due to the release of restored energy. The constitutive curve for this process is shown in Figure 2(b) (Bazant et al., 1991), the force–displacement relationship of which is nonlinear. Hence, the stiffness of the beam in state (2) is negative stiffness that results in the sensitivity and instability of the buckled beam.

Figure 2.

The beam buckling process: (a) different transition states; (b) force–displacement relationship.

Molyneux (1957) first proposed the concept of negative stiffness and designed a mechanism for vibration mitigation in his pioneer work in 1957. He initially adopted two oblique coil springs as energy storage elements to facilitate negative stiffness. However, he found that the negative stiffness device was not suitable for large payload since it leads to large excursions. Hence, he introduced a vertical spring into this device to provide positive stiffness to maintain stability. The principle of his design will be introduced in detail in the “NSVIDs using springs” section. It is certain that negative stiffness element is unstable, with no load-carrying capacity, and cannot release energy from avoid. Hence, it needs to work with positive stiffness elements to keep stable and requires external energy input or pre-stored energy to assist deformation.

Besides, negative stiffness control strategy (Iemura et al., 2006; Iemura and Pradono, 2002a, 2003; Li et al., 2008, 2011; Mizuno et al., 2005; Weber et al., 2011) can realize the same effect by directly controlling actuator to generate control force to promote displacement along the same direction. Composite material with negative stiffness properties (Drugan, 2007; Lakes, 2001; Lakes and Drugan, 2002; Lakes et al., 2001) was developed by employing innovative composite materials to provide negative bulk and Young’s moduli. Limited by the scope, these research results will not be included in this article.

Realization of NSVIDs

Negative stiffness element should be connected with positive stiffness elements, either in series connection or parallel connection, to overcome its instability (Mizuno et al., 2003, 2005, 2007; Tobias, 1959; Trimboli et al., 1994). As Figure 3 shows, imaging two springs with constant stiffness of $K_{1}$ and $K_{2}$ , for series-connected springs, the total force $F_{t}$ and stiffness $K_{t}$ can be calculated by the following equations

\begin{matrix} F_{1} = F_{2} = F_{t} \\ K_{t} = \frac{K_{1} K_{2}}{K_{1} + K_{2}} \end{matrix}

(3)

Figure 3.

Connection ways of springs: (a) series connection; (b) parallel connection.

Compared with the individual, the total force is unchanged and the total stiffness is lower than that of each spring. If the stiffness of $K_{1}$ is positive and the stiffness of $K_{2}$ is negative, which satisfies

K_{1} = - K_{2}

(4)

The stiffness of the new series-connection spring mechanism is minus infinite and still negative, which means the hybrid series-connected device is unstable either.

However, on the other hand, if they are connected in parallel, the total force and stiffness will be calculated as follows

\begin{matrix} F_{t} = F_{1} + F_{2} \\ K_{t} = K_{1} + K_{2} \end{matrix}

(5)

By changing the values of $K_{1}$ and $K_{2}$ we can obtain a combined mechanism with QZS, negative stiffness, or positive stiffness in theory, that are all smaller than that of positive stiffness element. In Figure 4(a), compared with positive element, the hybrid system can realize larger resultant force. Besides, the total force represented by the solid line is nonlinear and its stiffness is changeable with displacement, which endows high static stiffness and low dynamic stiffness, so-called high-static-low-dynamic stiffness. High static stiffness can maintain sufficient load-carrying capacity in static state and safeguards excessive deformation. Besides, a large stiffness is necessary for excessive deformation to reduce the possibility of the overturn of main structure. Low dynamic stiffness contributes to low natural frequency and small transmissibility.

Figure 4.

The force–displacement relationship: (a) parallel-connection-spring force; (b) NSD force.

Furthermore, negative stiffness characteristic can also increase the damping ratio of a system without increasing mass or viscous damping. It is realized by supplementing negative stiffness element with damping behaviour to work as NSDs for vibration isolation. Figure 4(b) demonstrates the behaviour of NSD. Compared with conventional dampers, especially that with low viscous damping, negative stiffness facilitates NSD lower natural frequency and higher and even several orders of magnitude damping ratio (Antoniadis et al., 2015; Chen et al., 2015; Iemura et al., 2008). Hence, it can dissipate more vibration energy and reduce excessive displacement.

Characteristics of NSVIDs

NSVID with QZS is designed with nonlinear force–displacement characteristic. A comprehensive review of nonlinear isolation devices was presented by Ibrahim (2008). The nonlinearity endows NSVID with specific characteristics compared with linear vibration isolator. As one of the key measures of the vibration isolation, the transmissibility can be categorized into force transmissibility and displacement transmissibility, equation (6). The former is defined as the ratio of the amplitude of the force transmitted to the base and the amplitude of the excitation force. The ratio of the magnitude of the displacement transmitted to the isolated objects, to the magnitude of the excitation displacement is defined as displacement transmissibility (Carrella et al., 2009)

TR = \sqrt{\frac{1 + {(2 ζ r)}^{2}}{{(1 - r^{2})}^{2} + {(2 ζ r)}^{2}}}

(6)

where, $r$ is the frequency ratio, $r = ω / ω_{n}$ , $ζ$ is the damping ratio.

The force and displacement transmissibility of a linear system are the same that are related to frequency ratio and damping ratio, Figure 5(a). It is desirable to keep the system under isolation region, where frequency ratio $r$ is larger than $\sqrt{2}$ , otherwise, the structural responses will be amplified, that is, falling into an amplification region, especially around the resonant area $(r = 1)$ . High damping is desired for resonant region while low damping is preferred in isolation region.

Figure 5.

The response curves: (a) transmissibility; (b) amplitude-frequency response. represents stable solutions; --- represents unstable solutions.

However, the representations of force and displacement transmissibility for nonlinear system are different and variously related to its nonlinear degree, damping ratio, and amplitude of excitation (Carrella et al., 2009; Cheng et al., 2015; Liu et al., 2013; Zheng et al., 2016). In Figure 5(b), nonlinear characteristic causes the bend of amplitude-frequency response curves of NSVID, which depends on whether it is a hardening or softening system (Kolovsky, 2013; Ravindra and Mallik, 1994). The curves of softening system bent to the left, which causes a reduction in the resonant frequency, though generating relative large response. Hardening system, compared with softening system, is designed with smaller response while larger resonance frequency. Therefore, the softening system is profitable for low-frequency moderate excitation and the hardening system is desirable under relative high frequency excitation with larger amplitude.

In addition, due to the jump phenomenon, there are two segments on the amplitude-frequency response curves of NSVID. The solid line segments denote the stable solutions and the dash lines represent the unstable solutions. There are multiple solutions for system under the dashed segment frequency range, which results in the jump-up and jump-down phenomenon. The jump-up or jump-down is decided by whether the frequency is increasing or decreasing, as well as whether it is a hardening system or a softening system. A systemic and detail investigation about the jump-up and jump-down phenomenon has been done by Brennan et al. (2008). They utilized the Harmonic Balance Method to solve the Duffing oscillator to determine the expressions of jump-up and jump-down frequencies, as well as the conditions for jump phenomenon to occur.

Classification of NSVIDs

In the decades after Molyneux, various types of NSVIDs have been proposed (Dijkstra et al., 1988; Platus, 1992; Tobias, 1959; Van Eijk and Dijksman, 1979). One of the main features in designing NSVIDs is to combine negative stiffness elements with positive stiffness elements (normally in form of mechanical springs). The negative stiffness elements could be energy storage elements (spring and pre-buckled beam etc.), magnetic elements, geometrically nonlinear structures, as well as composite structures and metamaterials.

NSVIDs using springs

Pre-compressed springs are normally used in the NSVIDs to assist the deformation of the main spring element. The three-spring device proposed by Molyneux (1957) is the pioneering work and the most studied model (shown in Figure 6). Its vertical restoring spring force consists of two parts including vertical spring force and oblique spring force

f = k_{2} x + 2 k_{1} (\frac{\sqrt{h^{2} + a^{2}}}{\sqrt{a^{2} + {(h - x)}^{2}}} - 1) (h - x)

(7)

where $k_{1}$ , $k_{2}$ are the stiffness of oblique spring and vertical spring, respectively, and a and h are the installation positions of the springs.

Figure 6.

Three-spring device: (a) schematic representation; (b) non-dimensional stiffness.

Differentiating equation (7) with respect to x gives the stiffness of the system

K = k_{2} - 2 k_{1} \frac{\sqrt{h^{2} + a^{2}}}{\sqrt{a^{2} + {(h - x)}^{2}}} + 2 k_{1} {(h - x)}^{2} \frac{\sqrt{h^{2} + a^{2}}}{{(a^{2} + {(h - x)}^{2})}^{1.5}}

(8)

When the oblique springs are compressed to the static equilibrium position in horizontal direction where $h = x$ , the restoring spring force is only decided by the vertical spring, which means only the vertical spring provides load-carrying capacity. The two oblique springs do not influence the restoring force but provide negative stiffness in vertical direction (Carrella et al., 2007b, 2008; Hao and Cao, 2015). In Figure 6(b), by adjusting the stiffness of these three springs, the stiffness of three-spring device is zero $({\hat{K}}_{QZS} |_{x = h} = 0)$ at equilibrium position, hence realizing low dynamic stiffness and high static stiffness simultaneously. This characteristic enables the device with lower transmissibility without sacrificing the vertical load supporting capacity. Carrella et al. (2007a, 2007b) conducted a series of research on this model and obtained the optimal stiffness ratio by optimizing the maximum excursion.

Instead of coil springs, other springs or spring structures can also be used to provide restoring force. Lan et al. (2014) and Wu et al. (2016) developed a more compact NSVID that replaced the coil springs (Figure 6(a)) with special planar springs, which can realize arbitrary stiffness with easily redesigned planar springs. Niu et al. (2014) designed a novel NSVID by adopting one disc spring to replace the two oblique coil springs in Figure 7. Compared with three-spring NSVID, it can achieve zero stiffness, restore and produce more axial nonlinear force with small displacement at equilibrium position. Through theoretical analysis, static and vibration testing, they verified that the device could mitigate vibration from a lower frequency with wider effective frequency region.

Figure 7.

Disc-spring NSVID: (a) schematic representation; (b) stiffness–displacement characteristic (Niu et al., 2014).

In addition, spring structures can also be used to improve device design. A typical horizontal-spring NSVID is designed with sliding block-bar structures, which has been applied to the vibration isolation of vehicle seat (Le and Ahn, 2011, 2013a, 2013b; Wang et al., 2018). In the horizontal-spring device, an additional damper is installed in vertical direction and two symmetrical sliding block-bar structures are installed in horizontal direction. In Figure 8, there are two regions on its force–displacement curve, including positive stiffness region and QZS region that is regarded as working region. The dynamic stiffness value and QZS region of this device are adjustable by changing the structural parameters and the stiffness of springs. Antoniadis et al. (2015) designed a horizontal-spring passive NSVID with further two vertical springs added to improve the load-carrying capacity. Since the existing devices are designed with zero stiffness at equilibrium position, it could easily result in a statically unstable structure. To consider such case, they conducted parametric analysis to evaluate the isolation effect with the proposed oscillator and optimize the properties by introducing an engineering safety margin.

Figure 8.

Horizontal-spring NSVID: (a) schematic representation; (b) stiffness–displacement characteristic $γ_{2} = b / L_{0}$ (Le and Ahn, 2011, 2013a, 2013b; Wang et al., 2018).

As typical NSVIDs using springs, shown in Figures 6(b), 7(b), and 8(b), a small change of the displacement around equilibrium position may lead to significant performance variation. As the stroke length is even shorter for disc spring, the working region of the NSVID containing disc spring is narrow. However, the load-carrying capacity of the device with disc spring is greatly improved and the device is more stable than the device with coil springs. By contrast, the horizontal-spring NSVID with sliding block-bar structure is with wider effective negative stiffness region. Besides, it has more stable properties and advantage of easier installation.

Negative stiffness has also been proposed to be used in the base isolation of civil structures. Low lateral stiffness is preferred if only vibration isolation is considered (Reinhorn et al., 2005; Viti et al., 2006). However, it will lead to large displacement response in the base level, hence likely causing the collapse of buildings or structures under strong excitation. Negative stiffness base isolator can address this dilemma with nonlinear force–displacement relationship. Its stiffness is changeable with displacement, which is negative when displacement is smaller than a threshold value, while increases dramatically if large displacement response occurs. One of a typical negative stiffness base isolators was designed and systematically investigated by Nagarajaiah and co-workers (Attary et al., 2013, 2014a, 2015b; Nagarajaiah et al., 2013; Sarlis et al., 2013, 2016), which takes full advantages of geometric lever structure and springs to realize negative stiffness in lateral direction. The negative stiffness will be triggered only when the displacement response is larger than the yield displacement of superstructure to keep it stable under small excitations, for example, wind. Besides, its lateral stiffness increases dramatically under large displacement.

NSVIDs using pre-buckled beams

Pre-buckled beam is another negative stiffness element to design NSVIDs. It is a column of elastic material compressed beyond its buckling load, as such with initial deflection and restored energy. NSVIDs that are realized by combining pre-buckled beams with mechanical springs can be regarded as evolving three-spring NSVIDs. The critical load of Euler column at buckling for the pin-ended case is calculated by the following equation (Feynman et al., 1964)

P_{cr} = \frac{π^{2} EI}{l^{2}}

(9)

where E is the modulus of elasticity, I is the area moment of inertia, and l is the effective length of the column.

Liu et al. (2013) designed a buckled-beam NSVID shown in Figure 9(a) and comprehensively compared the proposed nonlinear structure with linear device. Similarly, its vertical restoring force includes the spring force and buckled-beam force (provides negative stiffness). The effects of stiffness ratio on non-dimensional stiffness are shown in Figure 9(b). Compared with the three-spring device in Figure 6, this device is superior in realizing wide QZS region. Besides, they investigated the influence of nonlinear characteristic on the jump phenomenon under different excitation cases and verified that nonlinear stiffness results in large response under large amplitude vibration.

Figure 9.

Buckled-beam NSVID: (a) schematic representation; (b) stiffness–displacement characteristic (Liu et al., 2013).

Most of the conducted studies about NSVIDs with QZS characteristic are built upon the assumption that these devices can maintain stability at the equilibrium position where the dynamic stiffness equals to zero. However, in practice, it is challenging to adjust the overall stiffness to zero and make the load balance at the desired equilibrium point due to the imperfections in fabrication and installation as well as the changing environment. Liu et al. (2013) verified that smaller initial imperfection could provide a larger range of negative stiffness. Similarly, Huang et al. (2014) designed a NSVID and obtained its force and stiffness variation laws with the rated mass. In Figure 10, the two-hinged joint thin-walled beams were designed with initial imperfection and the applied mass was overload. Both stiffness and load imperfection increase the total stiffness of the device that results in unwanted high natural frequency.

Figure 10.

Hinged joint pre-buckled-beam NSVID: (a) schematic representation; (b) stiffness–displacement characteristic (Huang et al., 2014).

Comparing Figures 9(b) and 10(b), two designs, that are, single buckled-beam device and two-hinged joint thin-walled beams device, can both realize wide working region. The difference is the non-dimensional stiffness of the former is with a significant wide flat region, beyond which stiffness increases sharply. In contrast, there is no obvious segment on the stiffness curves of the latter. Compared with single buckled-beam device, the force of two-hinged joint thin-walled beams device changes gently around the initial position.

Winterflood et al. (2002) employed buckled beams to form two Euler buckling springs to replace the vertical mechanical springs and established a high-performance vibration isolator that has high-static-low-dynamic stiffness properties without large deflection. Zhang et al. (2004) used a beam under axial force at its two ends to obtain similar stiffness properties. Kashdan et al. (2012) and Fulcher et al. (2014) introduced a meso-scale negative stiffness device, where single-beam, uncoupled double-beam, and coupled double-beam configurations with first-mode and third-mode buckling supports were adopted separately. Device with double-beam at third-mode buckling support outperforms in reducing stiffness and offering larger load-carrying capacity under comparable levels of axial compression. Mori et al. (2016) also demonstrated that utilizing buckled beams as negative stiffness elements could reduce the ratio of local stiffness and static stiffness compared with the cases using mechanical springs. The buckled beam can also be replaced by magnets (Xu et al., 2013), which is achieved by using the nonlinear repulsive force between magnets that will be reviewed in the next part.

NSVIDs using magnetism

Magnetism, such as permanent magnets, electromagnets, and hybrid magnets, are employed to develop NSVIDs since the magnetic force between two cylindrical magnets is inversely proportional to the square of the distance, as in equation (10) (Ravaud et al., 2010; Shi and Zhu, 2017)

F (x) \approx \frac{π μ_{0}}{4} M^{2} R^{4} [\frac{1}{s^{2}} + \frac{1}{{(s + 2 L)}^{2}} - \frac{2}{{(s + L)}^{2}}]

(10)

where, R, L, and M are the radius, length, and magnetization of the magnets, s is the distance between two magnets, $μ_{0}$ is the permeability of space, which equals 4π × 10⁻⁷ T·m/A.

When the magnet leaves the equilibrium position, a magnetic force difference will be generated in the same direction of relative displacement, which assists relative movement. The inverse force–displacement feature and sound stability make magnets great candidate to develop NSVIDs. Magnetic springs can work solely or work together with other positive stiffness elements, for example, coil springs, to compose magnetic NSVIDs.

Xu et al. (2013) designed a magnetic spring with one fixed magnet, one free magnet, and a smooth guide-rod that is utilized to develop a magnetic NSVID in Figure 11. It can realize QZS at equilibrium position and attenuate vibration starting from low frequency, that is, 1.5 Hz. Robertson et al. (2009) conducted analysis of a magnetic levitation device that was composed of a pair of fixed cuboid magnets in vertical direction. A magnetic-sensitive mass is suspended between two cuboid magnets by repelling or attracting magnetic force. As the suspending mass moves relatively from equilibrium, the reactive force will increase to assist the movement, hence realizing negative stiffness. Shi and Zhu (2015) designed two types of NSDs for stay cables including two-magnet design and three-magnet design by integrating passive eddy-current damping with magnetic negative stiffness springs. These two devices can realize softening and hardening force–displacement relationship, respectively. Compared with normal conventional damper, the NSDs can significantly shorten the vibration duration time and realize superior vibration isolation effect. They also conducted parametric optimization to achieve the largest stiffness coefficients. Wu et al. (2014) also developed a similar magnetic NSVID using tri-magnets and conducted experiments to verify its ability in reducing the inherent frequency without scarifying the load capacity.

Figure 11.

Magnetic NSVID: (a) schematic representation; (b) stiffness–displacement characteristic (Xu et al., 2013).

Mizuno et al. (2007) designed a NSVID to realize nonlinear force–displacement characteristic and high static stiffness through optimizing the combination of permanent magnets and coil springs. A NSVID composed of linear mechanical springs and magnets was designed as the axial distribution structure (Carrella et al., 2008). Two vertical coil springs provide positive stiffness to keep the device stable and improve the load supporting capacity. The smooth bar makes the set-up and alignment easier and quicker. Li et al. (2013) designed a magnetic suspension NSVID that utilized stretching forces of rubber membranes to provide negative stiffness. They conducted parametric study to determine the optimal structure and experimentally verified that the rubber membranes could reduce the natural frequency by 50%.

Shan et al. (2015) designed a miniaturized NSVID with pneumatic spring and two ring magnets. By adjusting the stiffness of pneumatic spring and magnetic spring, the total stiffness is zero at equilibrium position. The device can reduce resonance frequency (from 3.61 to 2.34 Hz) and have high load-carrying capacity simultaneously. However, the peak transmissibility is large in the resonance region due to insufficient damping. For a NSVID designed with ring magnets, its stiffness–displacement relationship is significantly depended on whether the heights of inner and outer magnets are the same or not. The stiffness fluctuates more significantly when these two magnets are with different heights (Zheng et al., 2016). Actually, the permanent magnets can be replaced by electromagnets. For example, hybrid-magnet NSVIDs were designed by combining permanent magnets with electromagnets. Compared with passive controlled permanent-magnet system, the hybrid-magnet systems are more stable and can realize controllable properties (Easu and Siddharthan, 2013; Mizuno et al., 2003; Zhou and Liu, 2010).

NSVIDs using geometrically nonlinear structures

Geometrically nonlinear structures can be used to develop NSVIDs as the acceleration of sliding mass changes constantly along the curved surface. When a mass slides in the lateral direction along a convex friction interface, the vertical acceleration of the mass is variable in different place with the presence of gravity, which is the foundation of negative stiffness.

Zhou et al. (2015a) and Cheng et al. (2016) presented a high-static-low-dynamic stiffness vibration isolator that is composed of rollers, cams and springs, and so on, in Figure 12. During the dimensionless process, the final relationship of non-dimensional restoring force and non-dimensional displacement is expressed by equation (11)

\hat{f} (\hat{x}) = {\begin{matrix} (1 - 2 β (1 + \frac{\hat{δ} - 1}{\sqrt{1 - {\hat{x}}^{2}}})) \hat{x} \\ \hat{x} \end{matrix}, \begin{matrix} (| \hat{x} | \leq {\hat{x}}_{d}) \\ (| \hat{x} | > {\hat{x}}_{d}) \end{matrix}

(11)

where $\hat{x} = \frac{x}{(r_{1} + r_{2})}$ , $β = \frac{k_{h}}{k_{v}}$ , $\hat{δ} = \frac{δ}{(r_{1} + r_{2})}$ , ${\hat{x}}_{d} = \frac{\sqrt{r_{2} (2 r_{1} + r_{2})}}{(r_{1} + r_{2})}$ . $k_{h}$ , $k_{v}$ are the stiffness of horizontal and vertical springs, $r_{1}, r_{2}$ are the radius of roller and cam, $δ$ is the pre-compression of horizontal springs.

Figure 12.

Convex NSVID: (a) schematic presentation; (b) stiffness–displacement characteristic (Cheng et al., 2016; Zhou et al., 2015a).

The equation (11) shows the device can realize two working states including contact state and detached state, which are suitable for both small and large amplitude excitations. Static analysis results in Figure 12(b) indicate the isolator can achieve positive, negative, and zero stiffness at equilibrium position and the stiffness changes in a steady way.

Zhou et al. (2015b) designed another novel NSVID using nonlinear structure, which was used for reducing torsional vibration along shaft. By adjusting the stiffness of rubber and the configuration parameters to satisfy a certain condition, it can realize QZS. They also investigated the torque transmissibility under different structural parameters to demonstrate the superiority of this device compared with linear system. Zhou et al. (2017) subsequently proposed a novel six-degree-of-freedom vibration isolation platform to mitigate vibration in multiple directions with broad bandwidth and high effectiveness in low-frequency range. Sun et al. (2017) designed a compact negative stiffness base isolator to isolate lateral vibration with spring, wheel, and curved template. Matsumoto et al. (1999), Iemura et al. (2008, 2009) and Toyooka et al. (2015) designed two NSDs based on convex surface to realize negative stiffness. Both of them can significantly reduce the maximum force response without the demerit of conventional bearings, that is, increasing displacement response. Compared with Iemura’s device, Toyooka’s device successfully avoided the dependence on gravity by adopting mechanical springs to provide restoring force. The device is applicable to various structures without sufficient pressurizing weight. The nonlinear device is capable of reducing response under both sinusoidal and hybrid loading excitation.

NSVIDs using composite structures and metamaterials

Various types of novel NSVIDs incorporating composite structures and metamaterials have been designed. A novel honeycombs NSVID fabricated in nylon 11 by selective laser sintering was designed with negative stiffness characteristic and could recover to its original shape. Besides, it was compact and easily mountable, especially for the applications with limited space. Another advantage is that it triggered the energy absorption only when the force exceeded a predefined threshold (Correa et al., 2015). Similar structures have been proposed by Virk et al. (2013) and Izard et al. (2017) which are called cellular structures and also fully recoverable. The device was designed by implementing the three-spring model to work as a unit cell of a periodic architected material. Numerical and experimental investigations were also conducted to optimize the structure to realize extremely high combinations of Young’s modulus and damping.

Haghpanah et al. (2017) proposed a novel design that consists of elastic elements with non-convex strain energy and one convex elastic element. The architected materials can realize extremely high damping ψ = 6.02 that was larger than any reported experimental results. Hence, it enhanced the energy absorption capacity under dynamic loading. Ren et al. (2018) employed prefabricated curved beams into multidimensional rigid frames to design a NSVID. They conducted systematic finite element analysis on the configurations to verify the outstanding mechanical properties of these metamaterials. Besides, Lakes and his co-workers conducted a series of investigation on the composite material with negative stiffness properties (Drugan, 2007; Lakes, 2001; Lakes and Drugan, 2002; Lakes et al., 2001). It was developed by employing innovative composite materials to provide negative bulk and Young’s moduli.

NSVIDs used for vibration isolation are categorized into five groups based on their key negative stiffness elements. The comparison among the representatives of different types of NSVIDs shown in Figure 8 (device 1), Figure 10 (device 2), Figure 11 (device 3), and Figure 12 (device 4) are summarized below. The properties of NSVIDs in the “NSVIDs using composite structures and metamaterials” section are decided by their shapes and materials, which are quite distinct from the other four types of NSVIDs. Hence, it is not compared with the other four devices.

A desirable NSVID should be designed with low stiffness to attain low natural frequency and transmissibility during a large displacement range around its equilibrium position. According to their stiffness–displacement curves, obtaining wide QZS range and in the meantime realizing high degree negative stiffness are not contradictory for all these four devices. Besides, stiffness curves of devices 1 and 4 change smoothly during the effective QZS region, while increase dramatically outside this region. The non-dimensional stiffness curves of devices 2 and 3 increase gradually without obvious flat and steep region. Therefore, devices 1 and 4 perform more effectively than devices 2 and 3 under moderate excitation with small amplitude and low frequency because of their lower natural frequencies. However, with displacement increasing, the vibration isolation effect of devices 1, 2, and 3 will be undesirable due to large stiffness outside the QZS region results in high natural frequency and transmissibility. Nevertheless, there is a threshold stiffness value for device 4 due to its design combined with two working states, which make it suitable for both small and large amplitude excitation.

Applications of NSVIDs

NSVIDs have been demonstrated as promising vibration isolation devices for engineering application. Although the expectation is high, the solid evidence or comprehensive experimental verifications on the vibration isolation or seismic isolation are yet to be shown to the community. Until now, only limited researches have extended to the stage of scaled structures or systems including vehicle, train suspension, cable-stayed bridges, buildings, and highway bridges. This section looks into the current state-of-the-art of NSVIDs in the experimental testing of device performance and lab-scale evaluation in engineering structures.

Mechanical engineering

In mechanical engineering, various types of NSVIDs for vertical vibration isolation have been investigated and fabricated. It is yet to see the standard characterisation testing on the device alone in the open literatures, such as using standard sinusoidal loadings to interpolate the device performance and to establish phenomenological model. The effectiveness of some NSVIDs has been demonstrated for various structures, including land vehicles, aircraft, vibration tables, and motors.

A typical application for mechanical engineering is the horizontal-spring type NSVID, which is used for vehicle seat suspension and has been proven to significantly improve the vibration isolation effect (Le and Ahn, 2011, 2013a, 2013b). Both numerical and experimental results confirmed its great vibration isolation performance and anti-resonance phenomenon under 0.1–10 Hz excitation. This device can reduce about 67.2% of displacement response, while the response without it increased to 268.54%. Based on that passive device, Danh and Ahn (2014) designed an active control pneumatic device with the aforementioned NSVID for vehicle seat suspension. Its superior effect was experimentally verified under sinusoidal excitation, bump excitation, and random excitation. The result indicates that it can work under a wide frequency range. Lee and Goverdovskiy (2012) and Lee et al. (2016) conducted experiments to verify the effectiveness of the “negative” stiffness springs for its application in a high-speed train. The vibration isolation efficiency increases from 25 to 30 times and 100 times in the infra-low frequency range and low-frequency range, respectively. Shi et al. (2018, 2019) applied re-centering NSDs to high-speed trains, which can isolate vibration forces and in the meantime control vibration deflection, hence improving the ride comfort significantly.

Huang et al. (2014) conducted comparative experiments between the pre-buckled-beam NSVID and linear isolator. Compared with linear isolator, the transmissibility value and the frequency of maximum transmissibility of device under both sinusoidal signal and sinusoidal sweeping signal base excitation reduced significantly. Lan et al. (2014, 2016) conducted numerical and experimental investigations of a NSVID, which could carry 0 to 6 kg weight and is effective under frequency excitation as low as 2.3 Hz. Carrella et al. (2008) manufactured the prototype and carried out experimental research on a spring-magnet NSVID. They completed the comparative experiments by replacing magnets with non-magnetic elements, hence demonstrating the effect of negative stiffness, that is, reducing the natural frequency from 14 to 7 Hz. The novel NSVID designed by Xu et al. (2013) was realized by employing magnets to the traditional three-spring device, which is verified to be able to attenuate vibration from a very small frequency (1.5 Hz) and attain lower transmissibility than linear device by a series of prototype experiments.

Zhou et al. (2015a) built a cam-roller-spring mechanism prototype described in Figure 12 and conducted experimental testing to investigate its performance in comparison with a linear device. Its capabilities on reducing force transmissibility and widening the operational frequency region were significantly improved. On this basis, they applied an improved design to six-degree-of-freedom vibration isolation platform, which was with broader bandwidth of working frequency in all six degrees of freedom (Zhou et al., 2017). Besides, Zhu et al. (2012) and Li et al. (2013) designed and fabricated a suspension NSVID with non-metal material and magnets. An experimental table was used to illustrate the performance of this device, which was made up of four couples of NSVIDs. Based on the comparative testing result, the utilization of rubber membrane was verified to be able to reduce the natural frequency of the device by 50%.

Civil engineering

In civil engineering, the potential applications are in the protection of civil structures, such as buildings and bridges. Nagarajaiah et al. applied the negative stiffness base isolator to seismic isolation of civil structures (Nagarajaiah et al., 2013; Pasala et al., 2013; Sarlis et al., 2012). Two negative stiffness base isolators were mounted under a three-story structure and comparative testing was conducted using a shake table (Pasala et al., 2014; Sarlis et al., 2011, 2016). The results confirmed that this device could reduce considerable base shear force, acceleration response, and inter-story drift as well as displacement when it was with sufficient damping. In addition, further investigations have been conducted by implementing the device on a quarter-scale highway bridge model. The responses of the case with negative stiffness base isolator are significantly smaller than the original design (Attary et al., 2013, 2014a, 2014b, 2015a, 2015b). In civil engineering, the passive NSD designed by Iemura et al. (2008) has been proposed for the vibration isolation of bridges and the testing result demonstrates clear reduction in absolute acceleration without significant displacement increment.

Shi et al. experimentally investigated the vibration effect of two types of NSDs in a stable cable testbed under harmonic load, sweep sine load, and random load. Cables with NSD demonstrated obvious better performance in controlling displacement response (Shi et al., 2016, 2017; Shi and Zhu, 2015, 2017). Li et al. theoretically investigated the necessity of employing NSDs to the cable-stayed bridge based on Iemura and Pradono (2002b) model. Zhou and Li (2016) designed a NSD consisting of two symmetric pre-compressed springs and oil damper. Compared with oil damper, the new hybrid NSD could significantly increase the damping ratio under both single-mode and multiple-mode vibration. In addition to passive control NSD, Iemura et al. and Li et al. also conducted a series of researches on the active and semi-active control NSDs for the seismic response control (Iemura et al., 2006; Iemura and Pradono, 2009; Li et al., 2008).

Conclusion

This article conducted a comprehensive survey on the state-of-the-art of the research and developments on NSVIDs with nonlinear characteristic for the vibration isolation of engineering structures.

To keep the integrity, the article started with the fundamentals of NSVIDs, including the concept of negative stiffness, realization, and characteristics of NSVIDs. The nature of negative stiffness is that the pre-stored force assists deformation accompanied by energy release. Moreover, due to the inherently unstable properties, the NSVIDs are designed by combining negative stiffness elements with positive stiffness elements. This article compared two common connection approaches for designing negative stiffness devices, namely series-connected and parallel-connected methods. It was concluded that the parallel-connected system is desirable since it can realize QZS by adjusting the values of negative stiffness and positive stiffness. Hence, it can achieve low dynamic stiffness and enhance the load-carrying capacity of system simultaneously.

The characteristics of NSVID were presented by comparing the nonlinear system with linear system. The nonlinear force–displacement characteristic results in the shifting of transmissibility curves under force excitation and displacement excitation cases. Both hardening system and softening system can realize low transmissibility since nonlinear characteristic endows the system with low dynamic system. In addition, the NSVIDs also can achieve high static stiffness to provide sufficient load-carrying capacity. It addresses the dilemma of linear system that it cannot maintain low transmissibility and keep system stable simultaneously. Besides, the structural damping will be improved significantly, which is resulted by the rearrangement effect of nonlinear characteristic on stiffness and damping.

Based on the types of negative stiffness elements, the existing NSVIDs were categorized into five types, named as springs type, pre-bucked beams type, magnetism type, geometrically nonlinear structures type, as well as composite structures and metamaterials type. The positive stiffness is normally provided by coil springs. This article also presented the principle, characteristics, and performance of each type of devices. In addition, NSVIDs with negative stiffness in lateral direction designed for seismic isolation were also included.

The potential applications of NSVIDs were presented by several experimental investigations and applications to the engineering project at the end of the article. There were limited applications of NSVIDs to the practical structures or systems. Most of current research and developments were conducted with numerical simulations and laboratory validations. Although the applications of NSVIDs are limited, the results of experimental investigations show their promising potential, that is, the systems with NSVIDs perform much better than that with linear vibration isolators or without vibration isolators in reducing the resonance frequency, transmissibility, and vibration response of system. Besides, the applications of NSDs have shown its significant superiority in suppressing the vibration of cables and bridges compared with normal dampers.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the research funding from Australian Research Council under Discovery Project Scheme (DP150102636) to support PhD scholarship in conducting this research.

ORCID iD

Jianchun Li

References

Alabuzhev

(1989) Vibration Protection and Measuring Systems with Quasi-Zero Stiffness. Boca Raton, FL: CRC Press.

Antoniadis

Chronopoulos

Spitas

, et al. (2015) Hyper-damping properties of a stiff and stable linear oscillator with a negative stiffness element. Journal of Sound and Vibration 346: 37–52.

Attary

Symans

Nagarajaiah

, et al. (2013) Performance assessment of a highway bridge structure employing adaptive negative stiffness for seismic protection. Structures Congress 2013: 1736–1746.

Attary

Symans

Nagarajaiah

, et al. (2014a) Performance evaluation of negative stiffness devices for seismic response control of bridge structures via experimental shake table tests. Journal of Earthquake Engineering 19: 249–276.

Attary

Symans

Nagarajaiah

, et al. (2014b) Shake table testing of a highway bridge structure with passive negative stiffness devices for seismic response control. Available at: https://s3-amazonaws-com-443.web.bisu.edu.cn/academia.edu.documents/44548133/Shake_Table_Testing_of_a_Passive_Negativ20160408-31426-1i2na4l.pdf?response-content-disposition=inline%3B%20filename%3DShake_Table_Testing_of_a_Passive_Negativ.pdf&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAIWOWYYGZ2Y53UL3A%2F20191228%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20191228T091236Z&X-Amz-Expires=3600&X-Amz-SignedHeaders=host&X-Amz-Signature=69edde1de49d65024cd678b1b6c944c2057af784bca09fb8c36d5fc62f68aca8

Attary

Symans

Nagarajaiah

, et al. (2015a) Numerical simulations of a highway bridge structure employing passive negative stiffness device for seismic protection. Earthquake Engineering & Structural Dynamics 44: 973–995.

Attary

Symans

Nagarajaiah

, et al. (2015b) Experimental shake table testing of an adaptive passive negative stiffness device within a highway bridge model. Earthquake Spectra 31: 2163–2194.

Bazant

Cedolin

Tabbara

(1991) New method of analysis for slender columns. ACI Structural Journal 88: 391–401.

Brennan

Kovacic

Carrella

, et al. (2008) On the jump-up and jump-down frequencies of the Duffing oscillator. Journal of Sound and Vibration 318: 1250–1261.

10.

Carrella

Brennan

Waters

(2007a) Optimization of a quasi-zero-stiffness isolator. Journal of Mechanical Science and Technology 21: 946.

11.

Carrella

Brennan

Waters

(2007b) Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. Journal of Sound and Vibration 301: 678–689.

12.

Carrella

Brennan

Kovacic

, et al. (2009) On the force transmissibility of a vibration isolator with quasi-zero-stiffness. Journal of Sound and Vibration 322: 707–717.

13.

Carrella

Brennan

Waters

, et al. (2008) On the design of a high-static–low-dynamic stiffness isolator using linear mechanical springs and magnets. Journal of Sound and Vibration 315: 712–720.

14.

Chen

Sun

Nagarajaiah

(2015) Cable with discrete negative stiffness device and viscous damper: passive realization and general characteristics. Smart Structures and Systems 15: 627–643.

15.

Cheng

Wang

, et al. (2015). On the analysis of a piecewise nonlinear-linear vibration isolator with high-static-low-dynamic-stiffness under base excitation. Journal of Vibroengineering 17: 3453–3470.

16.

Cheng

Wang

, et al. (2016) On the analysis of a high-static-low-dynamic stiffness vibration isolator with time-delayed cubic displacement feedback. Journal of Sound and Vibration 378: 76–91.

17.

Correa

Klatt

Cortes

, et al. (2015) Negative stiffness honeycombs for recoverable shock isolation. Rapid Prototyping Journal 21: 193–200.

18.

Danh

Ahn

(2014) Active pneumatic vibration isolation system using negative stiffness structures for a vehicle seat. Journal of Sound and Vibration 333: 1245–1268.

19.

Dijkstra

Videc

Huizinga

(1988) Mechanical spring having negative spring stiffness useful in an electroacoustic transducer. Google Patents US-4722517-A.

20.

Drugan

(2007) Elastic composite materials having a negative stiffness phase can be stable. Physical Review Letters 98: 055502.

21.

Easu

Siddharthan

(2013) Theoretical and experimental analysis of a vibration isolation system using hybrid magnet. Procedia Engineering 64: 1139–1146.

22.

Feynman

Leighton

Sands

(1964) The Feynman Lectures on Physics (Exercises). Reading, MA: Addison-Wesley Publishing Company.

23.

Fulcher

Shahan

Haberman

, et al. (2014) Analytical and experimental investigation of buckled beams as negative stiffness elements for passive vibration and shock isolation systems. Journal of Vibration and Acoustics 136: 031009.

24.

Haghpanah

Shirazi

Salari-Sharif

, et al. (2017) Elastic architected materials with extreme damping capacity. Extreme Mechanics Letters 17: 56–61.

25.

Hao

Cao

(2015) The isolation characteristics of an archetypal dynamical model with stable-quasi-zero-stiffness. Journal of Sound and Vibration 340: 61–79.

26.

Huang

Liu

Sun

, et al. (2014) Vibration isolation characteristics of a nonlinear isolator using Euler buckled beam as negative stiffness corrector: a theoretical and experimental study. Journal of Sound and Vibration 333: 1132–1148.

27.

Ibrahim

(2008) Recent advances in nonlinear passive vibration isolators. Journal of Sound and Vibration 314: 371–452.

28.

Iemura

Pradono

(2002a) Passive and semi-active seismic response control of a cable-stayed bridge. Structural Control and Health Monitoring 9: 189–204.

29.

Iemura

Pradono

(2002b) Passive and semi-active seismic response control of a cable-stayed bridge. Journal of Structural Control 9: 189–204.

30.

Iemura

Pradono

(2003) Application of pseudo-negative stiffness control to the benchmark cable-stayed bridge. Structural Control and Health Monitoring 10: 187–203.

31.

Iemura

Pradono

(2009) Advances in the development of pseudo-negative-stiffness dampers for seismic response control. Structural Control and Health Monitoring 16: 784–799.

32.

Iemura

Igarashi

Pradono

, et al. (2006) Negative stiffness friction damping for seismically isolated structures. Structural Control and Health Monitoring 13: 775–791.

33.

Iemura

Kouchiyama

Toyooka

, et al. (2008) Development of the friction-based passive negative stiffness damper and its verification tests using shaking table. In: Proceedings of the 14th world conference on earthquake engineering, Beijing, China, 12–17 October.

34.

Izard

Alfonso

McKnight

, et al. (2017) Optimal design of a cellular material encompassing negative stiffness elements for unique combinations of stiffness and elastic hysteresis. Materials & Design 135: 37–50.

35.

Kashdan

Conner Seepersad

Haberman

, et al. (2012) Design, fabrication, and evaluation of negative stiffness elements using SLS. Rapid Prototyping Journal 18: 194–200.

36.

Kolovsky

(2013) Nonlinear Dynamics of Active and Passive Systems of Vibration Protection. Dordrecht: Springer Science+Business Media.

37.

Lakes

(2001) Extreme damping in compliant composites with a negative-stiffness phase. Philosophical Magazine Letters 81: 95–100.

38.

Lakes

Drugan

(2002) Dramatically stiffer elastic composite materials due to a negative stiffness phase? Journal of the Mechanics and Physics of Solids 50: 979–1009.

39.

Lakes

Lee

Bersie

, et al. (2001) Extreme damping in composite materials with negative-stiffness inclusions. Nature 410: 565–567.

40.

Lan

C-C

Yang

S-A

Y-S

(2014) Design and experiment of a compact quasi-zero-stiffness isolator capable of a wide range of loads. Journal of Sound and Vibration 333: 4843–4858.

41.

Ahn

(2011) A vibration isolation system in low frequency excitation region using negative stiffness structure for vehicle seat. Journal of Sound and Vibration 330: 6311–6335.

42.

Ahn

(2013a) Experimental investigation of a vibration isolation system using negative stiffness structure. International Journal of Mechanical Sciences 70: 99–112.

43.

Ahn

(2013b) Fuzzy sliding mode controller of a pneumatic active isolating system using negative stiffness structure. Journal of Mechanical Science and Technology 26: 3873–3884.

44.

Lee

C-M

Goverdovskiy

(2012) A multi-stage high-speed railroad vibration isolation system with “negative” stiffness. Journal of Sound and Vibration 331: 914–921.

45.

Lee

C-M

Goverdovskiy

Sim

C-S

, et al. (2016) Ride comfort of a high-speed train through the structural upgrade of a bogie suspension. Journal of Sound and Vibration 361: 99–107.

46.

Lee

C-M

Goverdovskiy

Temnikov

(2007) Design of springs with “negative” stiffness to improve vehicle driver vibration isolation. Journal of Sound and Vibration 302: 865–874.

47.

Liu

(2011) Seismic response control of a cable-stayed bridge using negative stiffness dampers. Structural Control and Health Monitoring 18: 265–288.

48.

Liu

(2008) Negative stiffness characteristics of active and semi-active control systems for stay cables. Structural Control and Health Monitoring 15: 120–142.

49.

Zhu

, et al. (2013) A negative stiffness vibration isolator using magnetic spring combined with rubber membrane. Journal of Mechanical Science and Technology 27: 813–824.

50.

Liu

Huang

Hua

(2013) On the characteristics of a quasi-zero stiffness isolator using Euler buckled beam as negative stiffness corrector. Journal of Sound and Vibration 332: 3359–3376.

51.

Matsumoto

Okano

Arita

, et al. (1999) Seismic behaviour of railway viaducts with steel dampers and braces. Journal of Structural Engineering 45: 1411–1422.

52.

Mizuno

Murashita

Takasaki

, et al. (2005) Pneumatic three-axis vibration isolation system using negative stiffness. In: Proceedings of the 44th IEEE conference on decision and control, Seville, 12–15 December, p. 8254. New York: IEEE.

53.

Mizuno

Takasaki

Kishita

, et al. (2007) Vibration isolation system combining zero-power magnetic suspension with springs. Control Engineering Practice 15: 187–196.

54.

Mizuno

Toumiya

Takasaki

(2003) Vibration isolation system using negative stiffness. JSME International Journal Series C Mechanical Systems, Machine Elements and Manufacturing 46: 807–812

55.

Molyneux

(1957) Supports for Vibration Isolation. ARC/CP-322. Available at: http://naca.central.cranfield.ac.uk/reports/arc/cp/0322.pdf

56.

Mori

Waters

Saotome

, et al. (2016) The effect of beam inclination on the performance of a passive vibration isolator using buckled beams. Journal of Physics Conference Series 744: 012229.

57.

Nagarajaiah

Pasala

DTR

Reinhorn

, et al. (2013) Adaptive negative stiffness: a new structural modification approach for seismic protection. Advanced Materials Research 639–640: 54–66.

58.

Niu

Meng

, et al. (2014) Design and analysis of a quasi-zero stiffness isolator using a slotted conical disk spring as negative stiffness structure. Journal of Vibroengineering 16: 1769–1785.

59.

Pasala

DTR

Sarlis

Nagarajaiah

, et al. (2013) Adaptive negative stiffness: new structural modification approach for seismic protection. Journal of Structural Engineering 139: 1112–1123.

60.

Pasala

DTR

Sarlis

Reinhorn

, et al. (2014) Simulated bilinear-elastic behavior in a SDOF elastic structure using negative stiffness device: experimental and analytical study. Journal of Structural Engineering 140: 04013049.

61.

Platus

(1992) Negative-stiffness-mechanism vibration isolation systems. In: Vibration control in microelectronics, optics, and metrology. International Society for Optics and Photonics, pp. 44–55. Available at: https://wp.optics.arizona.edu/optomech/wp-content/uploads/sites/53/2016/10/Platus-1992.pdf

62.

Ravaud

Lemarquand

Babic

, et al. (2010) Cylindrical magnets and coils: fields, forces, and inductances. IEEE Transactions on Magnetics 46: 3585–3590.

63.

Ravindra

Mallik

(1994) Performance of non-linear vibration isolators under harmonic excitation. Journal of Sound and Vibration 170: 325–337.

64.

Reinhorn

Viti

Cimellaro

(2005) Retrofit of structures: strength reduction with damping enhancement. In: Proceedings of the 37th UJNR panel meeting on wind and seismic effects, Tsukuba, Japan, 16–21 May.

65.

Ren

Yang

Qin

(2018) Mechanical performance of multidirectional buckling-based negative stiffness metamaterials: an analytical and numerical study. Materials 11: 1078.

66.

Robertson

Kidner

MRF

Cazzolato

, et al. (2009) Theoretical design parameters for a quasi-zero stiffness magnetic spring for vibration isolation. Journal of Sound and Vibration 326: 88–103.

67.

Sarlis

Pasala

DTR

Constantinou

, et al. (2012) Negative stiffness device for seismic protection of structures. Journal of Structural Engineering 139: 1124–1133.

68.

Sarlis

Pasala

DTR

Constantinou

, et al. (2011) Negative stiffness device for seismic protection of structures -an analytical and experimental study. In: Proceedings of 3rd ECCOMAS thematic conference on computational methods in structural dynamics and earthquake engineering (COMPDYN 2011), Corfu, Greece, 25–28 May.

69.

Sarlis

Pasala

DTR

Constantinou

, et al. (2013) Negative stiffness device for seismic protection of structures. Journal of Structural Engineering 139: 1124–1133.

70.

Sarlis

Pasala

DTR

Constantinou

, et al. (2016) Negative stiffness device for seismic protection of structures: shake table testing of a seismically isolated structure. Journal of Structural Engineering 142: 04016005.

71.

Shan

Chen

(2015) Design of a miniaturized pneumatic vibration isolator with high-static-low-dynamic stiffness. Journal of Vibration and Acoustics 137: 045001.

72.

Shi

Zhu

(2015) Magnetic negative stiffness dampers. Smart materials and structures 24(7): 072002.

73.

Shi

Zhu

(2017) Simulation and optimization of magnetic negative stiffness dampers. Sensors and Actuators A: Physical 259: 14–33.

74.

Shi

Xing

(2019) Performance analysis of vehicle suspension systems with negative stiffness. Smart Structures and Systems 24: 141–155.

75.

Shi

Zhu

Spencer

Jr (2017) Experimental study on passive negative stiffness damper for cable vibration mitigation. Journal of Engineering Mechanics 143: 04017070.

76.

Shi

Zhu

J-Y

, et al. (2016) Dynamic behavior of stay cables with passive negative stiffness dampers. Smart Materials and Structures 25: 075044.

77.

Shi

Zhu

, et al. (2018) Vibration suppression in high-speed trains with negative stiffness dampers. Smart Structures and Systems 21: 653–668.

78.

Snowdon

(1979) Vibration isolation: use and characterization. Journal of the Acoustical Society of America 66: 1245–1274.

79.

Sun

Lai

Nagarajaiah

, et al. (2017) Negative stiffness device for seismic protection of smart base isolated benchmark building. Structural Control and Health Monitoring 24: e1968.

80.

Tobias

(1959) Design of small isolator units for the suppression of low-frequency vibration. Journal of Mechanical Engineering Science 1: 280–292.

81.

Toyooka

Motoyama

Kouchiyama

, et al. (2015) Development of autonomous negative stiffness damper for reducing absolute responses. Quarterly Report of RTRI 56: 284–290.

82.

Trimboli

Wimmel

Breitbach

(1994) Quasi-active approach to vibration isolation using magnetic springs. In: Smart structures and materials 1994: passive damping, Orlando, FL, 1 May, pp. 73–84. Bellingham, WA: International Society for Optics and Photonics.

83.

Van Eijk

Dijksman

(1979) Plate spring mechanism with constant negative stiffness. Mechanism and Machine Theory 14: 1–9.

84.

Virk

Monti

Trehard

, et al. (2013) SILICOMB PEEK Kirigami cellular structures: mechanical response and energy dissipation through zero and negative stiffness. Smart Materials and Structures 22: 084014.

85.

Viti

Cimellaro

Reinhorn

(2006) Retrofit of a hospital through strength reduction and enhanced damping. Smart Structures and Systems 2: 339–355.

86.

Wang

Liu

Chen

, et al. (2018) Beneficial stiffness design of a high-static-low-dynamic-stiffness vibration isolator based on static and dynamic analysis. International Journal of Mechanical Sciences 142: 235–244.

87.

Wang

Lakes

(2004) Extreme stiffness systems due to negative stiffness elements. American Journal of Physics 72: 40–50.

88.

Weber

Boston

Maślanka

(2011) An adaptive tuned mass damper based on the emulation of positive and negative stiffness with an MR damper. Smart Materials and Structures 20: 015012.

89.

Winterflood

Blair

Slagmolen

(2002) High performance vibration isolation using springs in Euler column buckling mode. Physics Letters A 300: 122–130.

90.

T-H

Lan

C-C

(2016) A wide-range variable stiffness mechanism for semi-active vibration systems. Journal of Sound and Vibration 363: 18–32.

91.

Chen

Shan

(2014) Analysis and experiment of a vibration isolator using a novel magnetic spring with negative stiffness. Journal of Sound and Vibration 333: 2958–2970.

92.

Zhou

, et al. (2013) Theoretical and experimental analyses of a nonlinear magnetic vibration isolator with quasi-zero-stiffness characteristic. Journal of Sound and Vibration 332: 3377–3389.

93.

Sun

(2015) A multi-directional vibration isolator based on Quasi-Zero-Stiffness structure and time-delayed active control. International Journal of Mechanical Sciences 100: 126–135.

94.

Zhang

J-Z

Dong

, et al. (2004) Study on ultra-low frequency parallel connection isolator used for precision instruments. China Mechanical Engineering 15: 69–71.

95.

Zheng

Zhang

Luo

, et al. (2016) Design and experiment of a high-static–low-dynamic stiffness isolator using a negative stiffness magnetic spring. Journal of Sound and Vibration 360: 31–52.

96.

Zhou

Wang

, et al. (2015a) Nonlinear dynamic characteristics of a quasi-zero stiffness vibration isolator with cam–roller–spring mechanisms. Journal of Sound and Vibration 346: 53–69.

97.

Zhou

Xiao

, et al. (2017) A novel quasi-zero-stiffness strut and its applications in six-degree-of-freedom vibration isolation platform. Journal of Sound and Vibration 394: 59–74.

98.

Zhou

Bishop

(2015b) A torsion quasi-zero stiffness vibration isolator. Journal of Sound and Vibration 338: 121–133.

99.

Zhou

Liu

(2010) A tunable high-static–low-dynamic stiffness vibration isolator. Journal of Sound and Vibration 329: 1254–1273.

100.

Zhou

(2016) Modeling and control performance of a negative stiffness damper for suppressing stay cable vibrations. Structural Control and Health Monitoring 23: 764–782.

101.

Zhu

, et al. (2012) Modeling and analysis of a negative stiffness magnetic suspension vibration isolator with experimental investigations. Review of Scientific Instruments 83: 095108.