Design and performance analysis of a low-frequency vibration isolator with dual negative stiffness: Roller–cam,steel wire rope,and symmetric tetrahedron

Abstract

Low-frequency vibration isolation is essential for precision systems; however, conventional linear isolators are limited by the inherent coupling between static stiffness and natural frequency. To overcome this limitation, a dual negative-stiffness quasi-zero-stiffness (DNS-QZS) isolator is developed by integrating a roller–cam mechanism, steel wire-rope elements with hysteretic damping, and a symmetric tetrahedral support structure. An improved Bouc–Wen hysteresis model is formulated to characterize the nonlinear restoring force behavior. Experimental and numerical results show that the proposed isolator begins to attenuate transmitted vibration energy in the low-frequency range and achieves effective vibration isolation when the excitation frequency exceeds approximately 7.7 Hz, where the transmissibility falls below 0 dB and remains suppressed over a broad frequency band. The DNS-QZS isolator exhibits low equivalent stiffness near equilibrium and enhanced damping, enabling stable and broadband low-frequency isolation while retaining sufficient load-bearing capability. These findings demonstrate the practical applicability of the proposed design in advanced vibration control scenarios.

Keywords

nonlinear vibration isolation dual negative stiffness quasi-zero stiffness wire-rope isolator roller–cam mechanism geometric nonlinearity tetrahedral structure

1. Introduction

Low-frequency vibration isolation plays a vital role in maintaining the stability of precision instruments, aerospace payloads, and metrological equipment.^1–4 Conventional linear isolators cannot simultaneously achieve high load capacity and low natural frequency, leading to poor performance in low-frequency environments.^5,6 To overcome this limitation, nonlinear isolators featuring high-static–low-dynamic-stiffness (HSLDS) and quasi-zero-stiffness (QZS) characteristics have been developed.^7–10 These isolators reduce the effective stiffness near equilibrium by integrating positive and negative stiffness elements, thereby extending the isolation bandwidth while preserving structural stability.^11–13

Positive stiffness is typically supplied by linear or geometric springs, whereas negative stiffness is generated through mechanical or magnetic mechanisms.^14,15 Carrella et al.¹⁶ first analyzed a system composed of a vertical spring and two inclined springs, revealing how geometric parameters affect the quasi-zero-stiffness effect. Later studies investigated buckled beams,¹⁷ shell-type elastic members,¹⁸ and linkage mechanisms¹⁹ to improve negative-stiffness performance and tunability. Although these approaches achieved wide isolation bandwidths, they often suffered from frictional degradation and limited adjustability.

Mechanical QZS isolators using cam–roller assemblies have attracted great attention due to their simple geometry and passive tunability.^20–22 Yao et al.²³ developed a roller–cam–spring isolator that achieved ultra-low-frequency vibration attenuation. Zhang et al.²⁴ optimized a parabolic cam–roller geometry, demonstrating that cam curvature significantly influences negative-stiffness strength and bandwidth. Multi-stage cam–roller systems have also been proposed to enhance isolation stability and broaden frequency response.²⁵

Geometric negative stiffness can also be realized through symmetric tetrahedral mechanisms.^26,27 Zhang et al.²⁸ designed a magnetically modulated tetrahedral structure capable of achieving quasi-zero stiffness under varying loads, illustrating the tunability of geometric nonlinearity. Such structures provide adjustable stiffness characteristics that can complement other mechanical elements. In addition, the tetrahedral frame ensures structural symmetry, improving system balance and load capacity.^29,30

Energy dissipation is another critical factor for nonlinear isolation systems. Active or semi-active damping, such as magnetorheological or electromagnetic control, provides tunable damping but requires complex control circuits and external power.^31–33 Passive hysteretic damping, particularly that found in steel wire-rope isolators, provides broadband energy dissipation without the need for control systems.^34,35 The internal strand friction of wire ropes introduces nonlinear hysteresis, producing amplitude-dependent damping that stabilizes the system response.^36,37 Niu and Chen³⁸ demonstrated that a flexible isolation system using wire ropes combined with a compliant mechanism achieved high energy dissipation and improved stability. These findings confirm that wire ropes can act as both elastic and damping elements in vibration isolation structures.^39,40

Based on these principles, this study proposes a dual negative-stiffness quasi-zero-stiffness (DNS-QZS) isolator combining a roller–cam mechanism, steel wire ropes, and a symmetric tetrahedral structure. The roller–cam assembly and the tetrahedral frame jointly generate mechanical negative stiffness, while the wire ropes provide nonlinear elasticity and hysteretic damping. A coupled restoring-force model incorporating the improved Bouc–Wen formulation for the wire ropes is developed to describe the static and dynamic characteristics of the system. Both simulations and experiments confirm that the DNS-QZS isolator achieves a significant reduction in the onset isolation frequency while maintaining high load-bearing capacity and stable broadband performance. This configuration provides a practical, compact, and efficient solution for low-frequency vibration isolation applications requiring enhanced reliability and tunability.

2. System design and modeling

This section presents the concept design of the QZS isolator based on a DNS architecture. We describe the topology and the operating mechanism. First, the spatial layout of each subsystem is introduced. Then, we explain how negative stiffness is produced by the cam–roller–wire rope module and by the symmetric tetrahedral linkage.

2.1. Overview of system architecture and working principle

Figure 1 shows the structure of the proposed DNS-based quasi-zero-stiffness (QZS) isolator. The load is mounted on the upper platform, which is guided by a vertical guide shaft and a linear bearing to ensure purely vertical motion. Two vertical springs are installed between the platform and the base to carry the static load. Four negative-stiffness correctors are symmetrically arranged around the springs. Each corrector consists of a cam, a roller, and a wire rope, forming a roller–cam–wire-rope module that produces a controllable negative-stiffness effect. The rods and connecting links constitute a symmetric tetrahedral linkage that constrains lateral motion and maintains alignment during deformation. The combined action of the vertical springs and the negative-stiffness modules enables high static load-bearing capacity and low dynamic stiffness, achieving the quasi-zero-stiffness characteristic. Base excitation is applied at the base, and the dynamic response is measured at the payload on the platform.

Figure 1.

DNS-based QZS isolator. (a) Isometric view. (b) Unloaded configuration. (c) Loaded configuration showing the negative-stiffness mechanism.

The isolator combines two vertical linear springs with four negative-stiffness correctors arranged symmetrically around them. In each corrector, the cam, roller, and wire-rope ring generate negative stiffness in the vertical direction, while the vertical springs provide positive stiffness. The symmetric tetrahedral linkage constrains the motion of the rollers and wire ropes to the horizontal direction, ensuring vertical translation of the platform. Under external loading, the cam–roller–wire-rope assemblies introduce nonlinear negative stiffness, and the linkage contributes additional negative stiffness due to geometric nonlinearity within a specific displacement range. In parallel, the vertical linear springs supply an approximately constant positive stiffness across the entire stroke. The interaction between these components yields a high-static–low-dynamic-stiffness (HSLDS) characteristic, achieving a quasi-zero-stiffness (QZS) behavior that allows effective isolation at low and very low frequencies.

From a force-composition perspective (Figure 2), the total restoring force results from the superposition of the elastic and hysteretic restoring force of the wire rope, the contribution of the tetrahedral linkage, and the force generated by the roller–cam–wire-rope module. The vertical springs provide the positive-stiffness component (Figure 2(a)), whereas the roller–cam–wire-rope and geometric-nonlinearity modules contribute the negative-stiffness component (Figure 2(b)). Their combination produces the total restoring-force curve (Figure 2(c)), allowing the isolator to maintain high static load capacity and low dynamic stiffness with hysteretic restoring behavior.

Figure 2.

Static behavior of the parabolic cam–roller mechanism. (a) Configuration at equilibrium. (b) Roller engaged with the cam surface. (c) Roller separated from the cam.

2.2. Cam–roller mechanism design

The parabolic cam–roller mechanism introduces geometric nonlinearity that produces a controllable negative stiffness, thereby reducing the equivalent stiffness near static equilibrium and realizing a quasi-zero-stiffness (QZS) effect. Under a given static load, the structure maintains stable equilibrium, and when employed as a nonlinear auxiliary element, it effectively lowers both the onset frequency and the force transmissibility of the system.

Because the cam–roller interface operates through rolling contact, frictional resistance is negligible compared with the horizontal correction force $F_{h}$ generated by the negative-stiffness unit. The key geometric parameters include the cam base radius $r_{1}$ , the roller radius $r_{2}$ , and the parabolic coefficient $α_{0}$ , which defines the contour of the cam and the trajectory of the roller center.

Under a static vertical load $M g$ , the equilibrium condition of the system is $F = M g$ , and the corresponding vertical spring compression satisfies $Δ x = M g / K_{v}$ , where $K_{v}$ denotes the equivalent stiffness of the vertical spring. The parabolic cam contour is defined as $Y = α_{0} X^{2}$ , and the contact–separation threshold displacement is determined by $X_{0} = \sqrt{r_{1} / α_{0}}$ . During motion, the roller follows the parabolic profile until separation occurs when $∣ X ∣ = X_{0}$ .

The vertical coordinate $Y$ of the roller center can be expressed as

Y = {\begin{array}{c} - α_{0} X^{2} + r_{1} + r_{2} & | X | \leq X_{0} \\ - r_{1}, & | X | > X_{0} \end{array}

(1)

which represents the geometric relationship between the roller center and the cam surface.

The mechanism operates in two distinct regimes.

When the displacement satisfies $∣ X ∣ \leq X_{0}$ , the roller remains in continuous contact with the cam surface, producing a vertical negative-stiffness effect that counteracts the spring force and reduces the overall stiffness of the system.

Once the displacement exceeds the contact limit $∣ X ∣ > X_{0}$ , the roller separates from the cam, and the negative-stiffness contribution vanishes, leaving only the positive stiffness provided by the vertical spring. The overall restoring force of the dual negative-stiffness quasi-zero-stiffness (DNS–QZS) isolator can be expressed piecewise as

F = {\begin{array}{c} K_{v} X + 4 F_{h} \tan φ + F_{1}, & | X | \leq X_{0} \\ K_{v} X + F_{1}, & | X | > X_{0} \end{array}

(2)

where

F_{1}

is the restoring force contribution from the symmetric-tetrahedral structure. The horizontal correction force is given by

F_{h} (X) = 4 F_{rope} (Δ_{0} - α_{0} X^{2})

(3)

and the geometric inclination of the cam surface satisfies

\tan φ = 2 α_{0} X

Here, $F_{rope}$ represents the equivalent restoring force contributed by the wire rope and vertical springs, projected in the vertical direction through the cam geometry.

This piecewise behavior allows the isolator to sustain static load capacity through the vertical spring while maintaining low dynamic stiffness during vibration. Consequently, the cam–roller mechanism serves as an effective negative-stiffness source, providing broadband vibration isolation and improved stability near equilibrium.

2.3. Wire-rope modeling

As the core elastic and energy-dissipative component of the proposed dual negative-stiffness (DNS) isolator, the steel wire rope exhibits inherent nonlinear mechanical behavior resulting from the combined effects of strand bending, inter-wire friction, and geometric nonlinearity. This combination provides the rope with high static stiffness and strong hysteretic damping, which are two critical properties for achieving effective low-frequency vibration isolation.

2.3.1. Selection and mechanical characteristics

The wire rope acts as the primary elastic support for the DNS–QZS isolator, operating in parallel with the cam–roller mechanism to achieve the negative-stiffness effect. Drawing on prior experimental and theoretical investigations on multi-strand wire ropes,^17,18 the 7×7 strand configuration was selected for its stable mechanical performance, smooth hysteretic response, moderate stiffness nonlinearity and strong frictional damping. All these characteristics align with the isolator dynamic requirements.

A 4 mm diameter 7×7 stainless-steel wire rope was selected for the DNS–QZS isolator, considering its geometric constraints and dynamic performance goals. The wire rope is formed into a 90 mm diameter loop and fixed symmetrically between the upper and lower load-bearing plates. Together with the parabolic cam–roller unit, it creates a parallel negative-stiffness mechanism. In this system, the cam–roller unit introduces geometric nonlinearity, and the wire rope provides frictional-elastic coupling. Their combined action offers a tunable restoring force and broadband energy dissipation. This design enables effective low-frequency vibration isolation without compromising static load capacity.

The equivalent tensile modulus $E_{t}$ and bending stiffness $A_{b}$ of the 7×7 wire rope were calculated using empirical formulas validated by Zhao et al.¹⁷ and Liu et al..¹⁸ This ensures the elastic-damping behavior of the rope aligns with QZS design requirements, balancing stiffness and damping to suppress resonance amplitudes and broaden the effective isolation bandwidth.

The nonlinear restoring force of the wire rope arises from both elastic deformation and inter-strand frictional slip. To describe this behavior, an improved Bouc–Wen hysteretic model^17,18 is adopted, introducing amplitude-dependent stiffness and loop asymmetry to capture the experimentally observed softening–hardening transition. The constitutive relationships are expressed as

F_{rope} = F_{2 a} (z + F_{1 a})

(4)

F_{1 a} = k_{1} x + k_{2} sign (x) x^{2} + k_{3} x^{3}

(5)

F_{2 a} = b^{c x}

(6)

\dot{z} = \dot{x} (α - (γ + β sign (\dot{x}) sign (z)) | z |^{n})

(7)

where

F_{rope}

denotes the total restoring force,

F_{1 a}

is the elastic contribution, and

F_{2 a}

is the nonlinear amplification factor. The coefficients

k_{1}, k_{2}, k_{3}

represent the linear and higher-order stiffness terms, while

α, β, γ, n, b, c

define the hysteretic loop shape.

The identified parameters, obtained from harmonic loading tests, are summarized in Table 1. The results confirm that the designed 7×7 wire rope ring with a 90 mm loop diameter provides sufficient flexibility, high energy dissipation, and a progressive stiffness-hardening trend at large displacements, which enhances broadband vibration attenuation and ensures mechanical stability of the DNS–QZS isolator.

Table 1.

Identified parameters of the wire-rope hysteretic model.

Category	Parameter description	Value
Hysteretic loop parameters	Hysteretic model parameter α	2000
	Hysteretic model parameter β	1500
	Hysteretic model parameter γ	300
	Hysteretic model parameter n	0.9
	Hysteretic model parameter b	0.92
	Hysteretic model parameter c	90
	Linear stiffness k₁ [N·mm^-1]	3.14125
	Quadratic stiffness k₂ [N·mm^-2]	0.2472
	Cubic stiffness k₃ [N·mm^-3]	3.913×10^-3

2.3.2. Model verification and parametric analysis

The accuracy of the identified hysteretic model was verified by comparing simulated and experimental restoring-force loops Figure 3. The results show excellent agreement, with relative RMS errors of 5.59 % (at ±5 mm) and 7.33 % (at ±10 mm). The improved Bouc–Wen model effectively reproduces the nonlinear stiffness and hysteretic damping characteristics required for dynamic analysis.

Figure 3.

Comparison between calculated and experimental restoring forces of the wire rope. (a) ±5 mm; (b) ±10 mm.

A coupled model integrating the wire rope, symmetric tetrahedral frame, and vertical springs was then established. This configuration combines high static stiffness with nonlinear damping and quasi-zero stiffness behavior, achieving both strong load-bearing capacity and excellent vibration isolation.

To examine sensitivity, two key parameters were varied: the parabolic coefficient $α_{0}$ and ring diameter $d$ .

Figure 4 shows that decreasing $α_{0}$ reduces force amplitude and broadens the roller–cam contact region, enhancing stability. The smoothest and most stable curve occurs at $α_{0} = 0.01$ .

Figure 4.

Influence of the parabolic coefficient on the restoring-force characteristics.

Figure 5 demonstrates that increasing the ring diameter lowers stiffness but enlarges hysteresis. A 70 mm ring yields higher stiffness and weaker damping, while a 110 mm ring is overly compliant. The 90 mm ring provides the best balance between stiffness and energy dissipation.

Figure 5.

Influence of the ring diameter on the restoring-force characteristics.

In summary, a smaller parabolic coefficient ( $α_{0} = 0.01$ ) and a moderate ring diameter (90 mm) enhance the negative-stiffness effect and stability, yielding superior broadband vibration isolation.

2.4. Symmetric tetrahedral support structure

The symmetric-tetrahedral structure, illustrated in Figure 6(a)–(d), consists of six rigid rods forming a vertically symmetric spatial frame. The upper and lower joints are connected to the top and bottom plates, respectively, while vertical linear springs link the mechanism to the external system. A linear guiding pair constrains the entire assembly to vertical motion, thereby providing a single degree of freedom.

Figure 6.

Symmetric-tetrahedral structure. (a) Overall configuration; (b) deformation geometry; (c) projection view; (d) geometric constraints.

Three identical linear springs are mounted between adjacent rods to generate restoring forces. When a downward load $F_{1}$ acts on the upper platform, the coordinated motion of the rods causes the springs to stretch or compress simultaneously. The combined spring forces yield an upward restoring force that balances the external load and maintains stable vertical motion of the platform.

In Figure 6(a), the overall configuration under a vertical load is presented, where $H_{0}$ denotes the initial height of the platform and $H$ is the instantaneous distance between the center point $D$ and the base point $O$ . Each connecting rod has a fixed length $L$ , and the spring length is $l$ when the height is $H$ Figure 6(b) shows the geometric relationship during deformation, where $A^{'} A^{″}$ and $D^{'} D^{″}$ represent the incremental displacements $Δ α$ and $Δ x$ , respectively. Figure 6(c) presents the projection of the mechanism on the plane containing point $O$ , illustrating the horizontal displacement $Δ α$ and half-length variation $Δ l / 2$ of the connecting rods Finally, Figure 6(d) summarizes the geometric constraints among the principal structural parameters based on the isosceles-triangle configuration, which provides the foundation for the subsequent kinematic and stiffness analyses.

l = \sqrt{3} a = \sqrt{3 (L^{2} - H^{2})}

(8)

l_{0} = \sqrt{3} a_{0} = \sqrt{3 (L^{2} - H_{0}^{2})}

(9)

Given the spring length l and the initial length l0, the extension is

δ = l - l_{0} = \sqrt{3 (L^{2} - H^{2})} - \sqrt{3 (L^{2} - H_{0}^{2})}

(10)

The platform displacement at height H is:

X = 2 H_{0} - 2 H

(11)

As shown in Figure 6(b), a vertical force F1drives the platform downward by Δx, producing a spring extension $Δ l$ . Treating the three springs as an equivalent stiffness keq, energy balance gives:

\int_{H}^{H + Δ H} F_{1} d H = \frac{1}{2} k_{e q} [{(δ + Δ l)}^{2} - δ^{2}]

(12)

In the quasi-static limit ΔH→0, F1 can be regarded as a static restoring force, and the integral reduces to

\lim_{Δ h \to 0} F_{1} Δ H = \frac{1}{2} k_{eq} \lim_{Δ h \to 0} [2 δ Δ l + Δ l^{2}]

(13)

From this, the force F1 is derived as:

F_{1} = k_{e q} \cdot δ \cdot (- \frac{d l}{d H})

(14)

By substituting Equations (8) and (10) into Equation (14), the force F₁ at height H can be obtained as follows.

F_{1} = k_{eq} (3 H - \frac{3 H \sqrt{L^{2} - H_{0}^{2}}}{\sqrt{L^{2} - H^{2}}})

(15)

Expressing F1 in terms of the platform displacement X, the equation becomes: By substituting Equation (11) into Equation (15), the expression can be rewritten as follows.

F_{1} = k_{eq} [3 (H_{0} - \frac{1}{2} X) - \frac{3 (H_{0} - \frac{1}{2} X) \sqrt{L^{2} - H_{0}^{2}}}{\sqrt{L^{2} - {(H_{0} - \frac{1}{2} X)}^{2}}}]

(16)

When the stiffness of the spring is denoted as $k$ , the equivalent stiffness can be expressed as $k_{eq} = 3 k$ .

The symmetric-tetrahedral structure exhibits distinct nonlinear restoring-force characteristics arising from geometric nonlinearity. As the displacement amplitude increases, the tangent stiffness transitions from positive to negative within the large-displacement region, resulting in complex mechanical behavior. To investigate these nonlinear effects, a half-model of the tetrahedral structure was developed and analyzed.

Figure 7 illustrates the effect of geometric parameters on the restoring force–displacement behavior of the symmetric-tetrahedral structure. In Figure 7(a), an increase in rod length leads to reductions in both the peak restoring force and overall stiffness, resulting in a more compliant structural response. When displacement exceeds approximately 30 mm, the stiffness transitions to negative, marking the onset of the negative-stiffness region. Shorter rods exhibit higher peak stiffness and sustain a broader positive-stiffness range, while longer rods trigger the shift to negative stiffness at smaller displacements.

Figure 7.

Restoring force–displacement characteristics of the symmetric-tetrahedral structure. (a) Effect of rod length; (b) effect of initial platform height.

Figure 7(b) illustrates the effect of the initial platform height. A larger initial height increases the peak restoring force and overall stiffness, extends the range of positive stiffness, and shifts the onset of negative stiffness toward larger displacements, beyond approximately 30 mm. Conversely, a smaller initial height reduces the peak restoring force and causes negative stiffness to appear at smaller displacements. These results indicate that adjusting the initial platform height provides an effective means to control the critical displacement at which negative stiffness emerges, thereby improving the dynamic performance of the isolator.

For the parabolic coefficient $α_{0} = 0.01$ and the corresponding displacement range, the negative-stiffness region begins near 30 mm, consistent with the transitions observed in Figure 7 The geometric nonlinearity of the symmetric tetrahedral structure, combined with the roller and cam configuration, governs the nonlinear restoring force characteristics and directly influences the is tion performance. The interaction between structural geometry and the cam and roller mechanics provides a reliable basis for the design and optimisation of low-frequency quasi-zero-stiffness isolation systems.

Analysis of Figure 7 demonstrates how the structural geometry leads to a local reduction of stiffness near the equilibrium position. However, negative stiffness arising from geometric nonlinearity alone is often insufficient to achieve a practical quasi-zero-stiffness effect. A controlled combination with additional negative-stiffness elements is necessary to form a usable low-stiffness region for effective vibration isolation.

2.5. Composite system stiffness and restoring force modeling

The restoring force characteristic in Figure 8 is obtained by combining the cam–roller–wire-rope negative-stiffness unit with a parallel linear spring. The negative-stiffness branch is determined from the identified geometric parameters ( $α_{0} = 0.01$ , $d = 90 mm$ , $L = 55 mm$ , $H_{0} = 40 mm$ ), and the positive-stiffness component is provided by a supporting spring with $K_{v} = 3.14907 N / mm$ . The overall restoring force of the assembled system is fitted by a quintic polynomial $F (x) \approx K_{1} x + K_{3} x^{3} + K_{5} x^{5},$ with identified coefficients $K_{1} = 0, K_{3} = 4.383 \times 10^{- 3} {N / mm}^{3}, K_{5} = 4.836 \times 10^{- 7} {N / mm}^{5} .$ The black curve in Figure 8 shows the resulting $F (x)$ . Near $x = 0$ , the curve becomes nearly flat, indicating extremely low incremental stiffness and confirming the quasi-zero-stiffness (QZS) condition. The effective stiffness $K (x) = \frac{d F}{d x}$ (red curve) reveals a distinct negative-stiffness region around equilibrium, while outside this region $K (x)$ increases rapidly, ensuring global stability and adequate load capacity. Overall, Figure 8 demonstrates that the geometric nonlinearity and roller–wire-rope coupling collaboratively generate a low-stiffness plateau, enabling broadband low-frequency vibration isolation.

Figure 8.

Restoring force and effective stiffness of the QZS system.

3. Dynamic analysis

This section investigates the dynamic characteristics and vibration isolation performance of the proposed DNS-QZS isolator under harmonic excitation. The restoring-force model derived in Section 3 is employed to formulate the equations of motion and analyze steady-state and transmissibility behaviors.

3.1. Harmonic response

The dynamic equation of the DNS–QZS isolator under external and base excitations is expressed as

M \ddot{X} + C_{v} \dot{X} + F = F_{0} (t)

(17)

And

M \ddot{X} + C_{v} \dot{X} + F = - M {\ddot{Z}}_{0} (t)

(18)

where

M

is the supported mass,

C_{v}

is the viscous damping coefficient,

X (t)

and

Z (t)

represent the platform and base displacements, and

F (X)

is the nonlinear restoring force.

Introducing the dimensionless parameters

f_{0} = \frac{F_{0}}{K_{v} (r_{1} + r_{2})}, x = \frac{X}{r_{1} + r_{2}}, z_{0} = \frac{Z_{0}}{r_{1} + r_{2}}, Ω = \frac{ω}{ω_{0}}, ω_{0} = \sqrt{\frac{K_{v}}{M}}, τ = ω_{0} t, ζ = \frac{C_{v}}{2 M ω_{0}}, {(\cdot)}^{'} = \frac{d (\cdot)}{d τ}

the motion equations are rewritten as

x^{″} + 2 ζ x^{'} + f = f_{0} \cos (Ω τ)

(19)

for external excitation, and

x^{″} + 2 ζ x^{'} + f = z_{0} Ω^{2} \cos (Ω τ)

(20)

for base excitation. where

x

and

f

denote the dimensionless displacement and restoring force, respectively.

Considering the nonlinear restoring characteristic, the equivalent Duffing-type form is expressed as

M \ddot{X} + C \dot{X} + K_{1} X + K_{3} X^{3} + K_{5} X^{5} = F_{0} \cos (ω t)

(21)

where

K_{1}

K_{3}

, and

K_{5}

denote the equivalent linear, cubic, and quintic stiffness coefficients, respectively.

By introducing the equivalent stiffness scale $K_{v}$ and characteristic geometric length $r_{1} + r_{2}$ , the nondimensional parameters are defined as

δ_{1} = \frac{K_{1}}{K_{v}}, δ_{3} = \frac{K_{3} {(r_{1} + r_{2})}^{2}}{K_{v}}, δ_{5} = \frac{K_{5} {(r_{1} + r_{2})}^{4}}{K_{v}}

Accordingly, Eq. (21) can be transformed into its dimensionless form:

x^{″} + 2 ζ x^{'} + δ_{1} x + δ_{3} x^{3} + δ_{5} x^{5} = f_{0} \cos (Ω τ)

(22)

Assuming a first-harmonic steady-state solution $x (τ) = A \cos (Ω τ)$ , and applying the harmonic balance method, the steady-state amplitude satisfies

{[- A Ω^{2} + δ_{1} A + \frac{3}{4} δ_{3} A^{3} + \frac{5}{8} δ_{5} A^{5}]}^{2} + {(2 ζ A Ω)}^{2} = {f_{0}}^{2}

(23)

Equation (23) represents the nonlinear backbone curve of the DNS–QZS isolator, where the system exhibits either hardening or softening behavior depending on the sign of $δ_{3}$ .By tuning $δ_{1}, δ_{3},$ and $δ_{5}$ , the stiffness characteristics and isolation bandwidth can be optimized.

3.2. Vibration isolation performance analysis

The vibration isolation performance of the DNS-QZS isolator is analyzed based on its steady-state response and the corresponding force and displacement transmissibility characteristics under harmonic excitation.

3.2.1. Steady-state response analysis

For the DNS–QZS isolator, the nonlinear restoring force adopts the quintic form $F (x) = K_{1} x + K_{3} x^{3} + K_{5} x^{5}$ , where the stiffness coefficients are taken from the fitted results in Section 3. Combined with the viscous damping ratio $ζ$ , the dynamic response is governed by Eq. (22), and the steady-state amplitude is obtained from Eq. (23).

The steady-state frequency responses are calculated using the harmonic balance method (HBM) and further validated through fourth-order Runge–Kutta (RK4) integration, showing excellent agreement and confirming the reliability of the nonlinear model.

Figure 9 illustrates the steady-state frequency responses derived via the harmonic balance method (HBM) and validated by fourth-order Runge–Kutta (RK4) integration.

Figure 9.

Steady-state frequency response of the nonlinear isolator. (a) Force excitation; (b) base-displacement excitation.

In Figure 9(a), under force excitation conditions ( $f_{0} = 0.002$ , $ζ = 0.02$ ), the results show strong agreement between HBM and RK4. A pronounced resonance peak and a multivalued region are observed, reflecting typical nonlinear hysteresis during forward and backward frequency sweeps.

In Figure 9(b), under base-displacement excitation ( $z_{0} = 0.05$ , $ζ = 0.02$ ), the response amplitude decreases markedly, and the multivalued region narrows, indicating improved dynamic stability and reduced sensitivity to resonance.

In both cases, the backbone curve shifts toward higher frequencies with increasing amplitude, illustrating a stiffness hardening effect that confirms the quasi-zero-stiffness (QZS) behavior. The consistent results from both methods validate the robustness and accuracy of the proposed nonlinear dynamic model.

3.2.2. Force transmissibility analysis

For the quasi-zero-stiffness system, the total transmitted force—including both elastic and damping components—is expressed as

f_{T} = (δ_{1} A + \frac{4}{3} δ_{3} A^{3} + \frac{8}{5} δ_{5} A^{5}) \cos (Ω τ + ϕ) - 2 ζ A Ω \sin (Ω τ + ϕ)

(24)

The corresponding maximum transmitted force and transmissibility are given by

| f_{T} {| \sqrt{{(δ_{1} A + \frac{4}{3} δ_{3} A^{3} + \frac{8}{5} δ_{5} A^{5})}^{2} + {(2 ζ A Ω)}^{2}}}_{\max}

(25)

T_{f} = \frac{1}{f_{0}} \sqrt{{(δ_{1} A + \frac{4}{3} δ_{3} A^{3} + \frac{8}{5} δ_{5} A^{5})}^{2} + {(2 ζ A Ω)}^{2}}

(26)

Figure 10 illustrates the sensitivity of the DNS–QZS isolator steady-state response to changes in excitation amplitude and damping ratio under force excitation. In Figure 10(a), the damping ratio is fixed at ζ = 0.02, while the excitation amplitude is varied (F₀ = 0.002, 0.004, and 0.006). As F₀ increases, the resonance peak height and the width of the multivalued region both grow significantly, indicating stronger hardening-type nonlinear behavior and enhanced hysteresis. This response underscores the amplitude-dependent stiffness characteristic inherent in quasi-zero-stiffness (QZS) systems.

Figure 10.

Influence of excitation amplitude and damping on the force-transmissibility response: (a) Variation with excitation amplitude; (b) variation with damping ratio.

In Figure 10(b), with a constant excitation amplitude of F₀ = 0.002, the damping ratio is increased from ζ = 0.02 to 0.04. As damping increases, the resonance peak is progressively suppressed, and the multivalued region becomes narrower. These results confirm that higher damping improves dynamic stability and enhances isolation performance, particularly in the high-frequency regime.

3.2.3. Displacement transmissibility analysis

Under base-displacement excitation, the base motion can be expressed as $z (τ) = z_{0} \cos (Ω τ)$ and the relative response of the isolator is $x (τ) = A \cos (Ω τ + ϕ)$

Hence, the absolute displacement of the isolated body is given by

x_{a} = x + z = A \cos (Ω τ + ϕ) + z_{0} \cos (Ω τ)

(27)

The maximum amplitude of the absolute displacement can be obtained as

| x_{a} {| \sqrt{A^{2} + z_{0}^{2} + 2 A z_{0} \cos ϕ}}_{\max}

(28)

and the displacement transmissibility is

T_{d} = \frac{| x_{a} |_{\max}}{z_{0} \sqrt{{(\frac{A}{z_{0}})}^{2} + 1 + 2 (\frac{A}{z_{0}}) \cos ϕ}}

(29)

where the phase angle

ϕ

is obtained from

\cos ϕ = \frac{z_{0} Ω^{2} - A Ω^{2} + δ_{1} A + \frac{4}{3} δ_{3} A^{3} + \frac{8}{5} δ_{5} A^{5}}{f_{0}}

(30)

Figure 11 presents the displacement transmissibility $T_{d} (Ω)$ over the normalized frequency range $0 \leq Ω \leq 0.5$ . When the excitation amplitude increases from $z_{0} = 0.02$ to $z_{0} = 0.05$ at a fixed damping ratio of ζ = 0.02, the overall transmissibility rises slightly, indicating a weak dependence on amplitude. Conversely, increasing the damping ratio from ζ = 0.02 to 0.03 at $z_{0} = 0.05$ significantly reduces the resonance peak and improves high-frequency attenuation. In all scenarios, the transmissibility remains below 0 dB in the low-frequency range, confirming the broadband isolation capability of the QZS mechanism.

Figure 11.

Displacement transmissibility $20 \log (T_{d})$ under base excitation for different excitation amplitudes $z_{0}$ and damping ratios $ξ$ .

3.2.4. Comparative analysis with linear isolator

For comparison, Figure 12 contrasts the performance of the DNS–QZS isolator with a conventional linear isolator characterized by equivalent stiffness coefficients $K_{1} = 3.14907 N / mm, K_{3} = 0, K_{5} = 0$ . The DNS–QZS system exhibits a substantially lower resonance peak and a broader isolation bandwidth where $20 \log_{10} (T) < 0$ . This superior performance arises from the dual negative-stiffness (DNS) configuration, which reduces the effective dynamic stiffness near equilibrium and delays the onset of resonance. In contrast, the linear isolator shows a sharp resonance and narrower isolation bandwidth. These results demonstrate the enhanced low-frequency and broadband vibration attenuation enabled by the DNS–QZS design.

Figure 12.

Comparison of force transmissibility between the DNS–QZS isolator and a linear isolator under base excitation (f₀ = 0.002, z₀ = 0.02).

4. Experimental validation

To evaluate the vibration isolation performance of the proposed dual negative stiffness quasi-zero-stiffness (DNS–QZS) isolator, a dynamic test platform was constructed and a prototype was fabricated according to the nonlinear restoring force model. Preliminary static tests confirmed that the isolator exhibits reduced effective stiffness near equilibrium, enabling a low onset isolation frequency.

As shown in Figure 13, the experimental setup comprises a signal generator, power amplifier, electromagnetic shaker (RT-30T/RTZ200/RTP300), controller, and WTVB01-485 acceleration sensors. Figure 13(a) displays the overall configuration, where sinusoidal excitations are applied to the base and both input and output accelerations are captured. The data are processed in real-time via a dynamic signal analyzer and a DELL G15 5530 workstation. Figure 13(b) shows the prototype in its unloaded state, while Figure 13(c) presents a top view after installing the rigid outer shell. This shell provides structural reinforcement, maintains consistent cam–roller contact, and prevents radial and axial displacement during motion, thereby ensuring stable force transmission and enhancing dynamic robustness.

Figure 13.

Experimental setup and configuration of the DNS-QZS isolator.(a) Experimental platform and control system; (b) Structural model of the isolator; (c) Top view with outer shell installed.

The time-domain responses under steady-state sinusoidal excitation at 8, 10, 12, and 15 Hz with a 2 mm amplitude are presented in Figure 14. In Figure 14(a)–(d), the blue and orange curves represent the base and upper platform accelerations, respectively. Across all frequencies, the upper platform exhibits markedly reduced acceleration, demonstrating efficient vibration suppression. For instance, at 12 Hz, as shown in Figure 14(c), the base acceleration reaches approximately 4.5 m/s², while the platform response is about 2.0 m/s², a reduction of approximately 56%. This successful suppression of vibration across the frequency spectrum validates the accuracy of the DNS–QZS dynamic model.

Figure 14.

Time-domain acceleration responses of the DNS-QZS isolator under sinusoidal excitations at 8, 10, 12, and 15 Hz, showing effective high-frequency vibration attenuation. (a) 8 Hz; (b) 10 Hz; (c) 12 Hz; (d) 15 Hz.

To further explore the frequency-dependent behavior, a sinusoidal swept-frequency excitation test was conducted with a 2 mm input amplitude, spanning the range from 1 Hz to 15 Hz over 240 seconds. Figure 15 presents the corresponding results. As shown in Figure 15(a), the upper platform exhibits significantly reduced acceleration compared to the base, highlighting effective vibration attenuation. The transmissibility curve in Figure 15(b) indicates that vibration isolation begins when the excitation frequency exceeds approximately 7.7 Hz, at which point the transmissibility drops below 0 dB. Above this threshold, the isolator provides stable broadband attenuation, aligning with the designed quasi-zero-stiffness behavior.

Figure 15.

Experimental response of the DNS-QZS isolator under sinusoidal sweep excitation.(a) Time-domain acceleration responses; (b) acceleration transmissibility showing onset of isolation above 7.729 Hz.

This enhanced isolation performance is attributed to the DNS mechanism, which effectively lowers the equivalent dynamic stiffness near equilibrium and shifts the resonance frequency downward. The dual negative stiffness configuration enables energy redistribution and resonance suppression, thereby broadening the isolation bandwidth. Minor discrepancies between experimental and simulation results are primarily due to frictional effects and contact imperfections in the 3D-printed cam–roller mechanism. Nonetheless, these deviations are minor and do not compromise the overall isolation trend.

To capture the dynamic evolution of the isolation effect during the frequency sweep, Figure 16 presents the time–frequency energy distribution of the input and output signals, along with the derived time–frequency transmissibility. In Figure 16(a), the transmissibility remains near 0 dB below 4 Hz, indicating that the platform closely follows the base motion. Between 4 Hz and 10 Hz, transmissibility gradually declines, marking the transition into the isolation regime. Beyond 10 Hz, transmissibility drops sharply, with most values falling below –20 dB, signifying strong attenuation of mid-to-high frequency components. A clear reduction in vibration energy is observed in the output spectrogram Figure 16(b), particularly above 10 Hz, which further validates the broadband suppression capability of the isolator.

Figure 16.

Time–frequency analysis of the DNS–QZS isolator under swept-frequency excitation: (a) time–frequency transmissibility; (b) output spectrogram indicating frequency-dependent attenuation characteristics.

Overall, the experimental findings demonstrate a high level of agreement with theoretical simulations, confirming that the proposed DNS–QZS isolator effectively reduces vibration transmissibility across a broad frequency range. The attenuation within the isolation band reaches up to –20 dB, verifying the reliability of the nonlinear dynamic model. The minor differences observed are well explained by physical imperfections inherent in prototype fabrication, particularly those related to surface friction and cam–roller contact fidelity. These results collectively validate the feasibility and robustness of the DNS–QZS isolation concept for low-frequency and broadband vibration mitigation applications.

5. Conclusions

This study has developed a dual negative-stiffness quasi-zero-stiffness (DNS-QZS) isolator by integrating a roller–cam mechanism, steel wire-rope elements with hysteretic damping, and a symmetric tetrahedral support structure. An improved Bouc–Wen hysteresis model is formulated to characterize the nonlinear restoring force behavior. Based on the mathematical model, the dynamic characteristics and vibration isolation performance of the proposed DNS-QZS isolator is comprehensive investigated. A prototype was fabricated, and the experiment results show that the proposed isolator begins to attenuate transmitted vibration energy in the low-frequency range and achieves effective vibration isolation when the excitation frequency exceeds approximately 7.7 Hz, where the transmissibility falls below 0 dB and remains suppressed over a broad frequency band. The DNS-QZS isolator exhibits low equivalent stiffness near equilibrium and enhanced damping, enabling stable and broadband low-frequency isolation while retaining sufficient load-bearing capability. Compared with conventional linear isolation systems, the DNS-QZS isolator achieves improved low-frequency isolation performance without compromising load-bearing capacity, owing to its quasi-zero-stiffness characteristic and nonlinear energy dissipation mechanism. The experimental and numerical results are consistent, confirming the feasibility and engineering applicability of the proposed design.

Footnotes

ORCID iD

Liuzhe Xu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Key Area Special Project of Guangdong Provincial Department of Education (Grant Nos. 6022210111K, 2022ZDZX3071), Research Projects of Department of Education of Guangdong Provice- 2024GCZX014.

Declaration of conflicting interests

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

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