Research on dynamic characteristic analysis and parameter optimization design of metal-rubber vibration isolator

Abstract

In this study, a novel metal-rubber hanger vibration isolator is proposed to enhance vibration isolation performance by utilizing the nonlinear hysteresis characteristics of metal-rubber. A hyperelastic–viscoelastic constitutive model capable of accurately simulating the static and dynamic mechanical behavior of metal-rubber was established by combining a three-parameter Mooney–Rivlin model with a Prony series. Finite element analysis shows that increasing the outer diameter and height of the metal-rubber block reduces the first natural frequency and vibration transmissibility of the isolator. In addition, the isolator exhibits significant frequency-dependent damping characteristics: energy dissipation decreases as the excitation frequency increases. This behavior ensures effective suppression of resonance in the low-frequency range while maintaining high isolation efficiency in the high-frequency range. Experimental validation confirms the accuracy of the developed hyperelastic–viscoelastic constitutive model and the established numerical model. These findings provide an important basis for the design of high-performance metal-rubber-based vibration isolators.

Keywords

metal-rubber vibration isolation finite element simulation transmissibility

Introduction

Metal-rubber has been widely used in different engineering applications because of its strong nonlinear hysteresis characteristics. In the semi-active control scheme, the adaptive stiffness and damping of isolators play an important role in improving the control effect. Considering that the natural rubber is too soft to bear the load, metal-rubber based isolators were designed and applied in many industrial cases, and a good compatibility of economy and working efficiency has been proved.^1,2

To suppress random vibrations induced by uneven ground excitations, Cao et al. proposed a type of metal-rubber structure, and the geometric parameters of the structures were optimized by analyzing the dynamical behavior of the integrated system.³ In Ma’s study, broadband vibration efficient isolation was achieved by using a nonlinear meta-rubber isolator, and high vibration isolation efficiency in three directions was realized.⁴ Yu et al. demonstrated a parametric design of metal-rubber isolation platform. In the analysis, five different types of vibration properties were found, and the critical conditions for the transformation were determined. The results provided the uniform reference for choosing an MR isolation platform.⁵ Ma et al. established a surrogate model for metal-rubber isolators which was used for fast optimizing the structure, and lightweight design requirements could be well satisfied without loss of precision.⁶ Banić et al. introduced a procedure to select the rubber compound for designing rubber-metal springs in railway engineering, and a numerical simulation of spring deflection was displayed to verify the used procedure.⁷ The marine environment greatly influences the properties of vibration isolators, such as variations of temperature and seawater corrosion. Experimental tests were carried out to verify the vibro-isolator characteristics considering the overload and impact of petroleum products.⁸

Lee designed a novel isolation system taking advantage of rubber and wire isolators, and the dynamic properties of the system were investigated by a pure compressive test, pure-shear test, shear-strain dependence test, and compressive-stress dependence test. The following deformation was obtained, including shear modulus, shear stiffness, energy dissipation and damping ratio.⁹ The dynamics of a laminated rubber-metal spring in the vibration isolator was discussed while the excitation coming from the axial direction was taken into account. A simple mechanical model was developed and the influence of the parameters was analyzed by calculating the vibration transmissibility.¹⁰ Furthermore, Zheng et al. investigated the influence of the pot-shaped metal-rubber on the nonlinear dynamic model of the wire rope shock absorber. This study gave a theoretical reference and experimental verification for the dynamic design of pot-shaped metal-rubber in three-dimensional vibration reduction.¹¹ In seismic isolation, steel reinforced elastomeric isolators are widely used, and the adhesion influences the property of the isolator significantly. Using the tensile tests, Pauletta et al. investigated the adhesion behavior between elastomer and steel layers.¹²

Xia et al. proposed a novel metal-rubber bearing made of porous metal wire, and the effects of shear strain, compressive-stress, loading frequency, and repeated loading cycles on the hysteretic behavior were experimentally measured. Meanwhile, it was found that the Bouc–Wen model was a good choice to simulate the hysteretic curves for this metal-rubber bearing with appropriate input parameters.¹³ Zhou et al. used a high-order nonlinear friction model to describe the inclination angle and contact form of the spatial distribution of metal wires in metal-rubber, showing a very high prediction accuracy and providing a reference for modeling the dynamic system of MR materials with a complex network structure.¹⁴ Ren et al. developed a constitutive model for flexible microporous metal-rubber, and then the nonlinear mechanical behavior of meta-rubber was analyzed by investigating the influence of the macroscopic parameters, such as shape, density, wire diameter, helix pitch, and wire elastic modulus, and microstructure parameters, such as spatial distribution of wire turns, contact shape, and friction coefficient.¹⁵

Seismic isolation is an efficient strategy to protect structures from the effects of moderate to severe earthquake shaking. Conventional isolation devices can reduce the structural response in horizontal directions only; however, three-dimensional (3D) ground motions have been observed in earthquake records. In this paper, Liu et al. proposed an innovative 3D seismic isolator and introduced a new asymmetric oblique hysteretic model.¹⁶ Hu et al. investigated the compressive mechanical behavior of multiple wire metal-rubber experimentally by designing and measuring three batches of MW-MRs with different weight percentage ratios.¹⁷ The natural rubber bearing system could be greatly improved by structural steel core, wires and shaped memory alloy.¹⁸ Lei et al. studied the mechanical property of two types of rubber blends, one was made of nitrile rubber and brominated butyl rubber, another was made of nitrile rubber, brominated butyl rubber and ethylene-vinyl acetate copolymer.¹⁹ Koo et al. inserted the small-sized lead into laminated rubber bearing to improve the vibration isolation ability in seismic engineering and the influence of design performance parameters was verified. The numerical results showed a good agreement with experimental tests.²⁰

The isolation of high importance in long-time vibration is necessary in nuclear equipment and large structures.²¹ Sliding bearings incorporating a hydrostatic scheme were used by Villaverde to minimize the friction between bearings, and this isolation scheme was proved to be feasible, effective, and easy to build and install.²² Chen et al. focused on the base-isolation using lead-rubber-bearing and negative stiffness springs devices. The proposed LRB-NS device effectively improved the post-earthquake function because the device could alleviate the nonlinearity and damage in piers and prevent failure during catastrophic excitations.²³

In seismic isolation, the soils were also taken into consideration as a parameter in the design of the isolators. Pistolas et al. examined the utilization of granular soils mixed with tire-derived aggregates, and discussed the effectiveness and capabilities of the proposed isolation scheme.²⁴

In applications, rubber bearings with multi-level design usually show a very good performance vibration isolation.²⁵ Menga et al. investigated the dynamic behavior of a Rubber-Layer Roller Bearing, which had a nonlinear damping behavior.²⁶ The nonlinear damping resulted in a significant reduction of the vibration and constraint forces. Liu et al. investigated the high-frequency property of a rubber isolator while the excitation frequency exceeded 2 kHz. The influence of the inertial force on high-frequency dynamic stiffness was considered in the theoretical model, illustrating that the inertial force played a non-negligible role in high-frequency response.^27,28 Zhao et al. developed a constitutive model for rubber-cord composites and the mechanical behaviors of rubber-cord composites could be predicted, agreeing well with test data and finite element simulations. Besides, the proposed model could be extended in finite element analysis easily.²⁹

In conclusion, metal-rubber offers several advantages over conventional rubber as a material for vibration isolators. Metal-rubber can show an excellent vibration isolation effect in a wide frequency range. Secondly, its high and low temperature resistance and corrosion resistance enable it to maintain its physical properties and vibration isolation effect in extreme temperatures and harsh chemical environments. In addition, metal-rubber has a long service life, and good elastic recovery performance, and even after being pressurized, it can quickly recover to its original state and maintain a good vibration isolation effect. The purpose of this study is to propose a novel vibration isolator design that utilizes metal-rubber as the core isolation material.

Compared with existing metal-rubber isolators,^3–6 the proposed hanger-type isolator adopts a unique compression–shear combined deformation mode, significantly improving low-frequency isolation while maintaining a compact structure (overall height <90 mm). Traditional designs rely solely on compression or shear,^7–9 whereas the present configuration subjects the metal-rubber element to both compressive and shear forces simultaneously, markedly enhancing energy dissipation over each operating cycle. Furthermore, this paper develops a hyperelastic–viscoelastic constitutive model for metal-rubber under dynamic loading, enabling accurate prediction of nonlinear hysteretic behavior within the 10–100 Hz band and overcoming the limitations of previous linear¹⁰ or quasi-static models.¹¹

In addition to the introduction, this study consists of the following sections: Section 2 describes the design of the metal-rubber hanger vibration isolator and the identification of the parameters of the constitutive model; Section 3 evaluates the performance of the metal hanger vibration isolator and discusses the effect of the metal-rubber parameters on the vibration isolation performance; Section 4 compares the simulation results with the experimental results; and the final section summarizes the conclusions.

A novel vibration isolator design

Metal-rubber vibration isolator design

Previous studies have demonstrated that vibration isolators incorporating special materials and structural designs can effectively reduce noise and mitigate vibration-induced damage to equipment, thereby enhancing operational stability and performance. In particular, metal-rubber materials have gained attention in recent years due to their lower sensitivity to temperature variations and more stable performance compared to conventional vulcanized rubber. The main energy dissipation mechanism of metal-rubber arises from the dry friction between the entangled metal wires and the internal relative sliding of its spatial network structure. Under cyclic loading, the inherent hysteresis of the material leads to significant energy dissipation through Coulomb friction and internal damping, converting mechanical energy into heat. This nonlinear damping mechanism is less sensitive to temperature changes compared to traditional viscoelastic rubbers, contributing to its stable performance across a wide temperature range.

Since the metal-rubber isolator presented in this study is designed as a hanger-type vibration isolator, the equipment is suspended from one end, subjecting the isolator to tensile loads during operation. For numerical simulation, the model is appropriately simplified, retaining only essential components. It primarily consists of a metal outer shell, a metal-rubber element, and a central metal connecting rod. Figure 5(a) shows a physical model of the metal-rubber vibration isolator, while a simplified representation is provided in Figure 1.

Figure 1.

Simplified model of a metal hanger vibration isolator.

During operation, the metal-rubber block experiences significant shear deformation, and due to the inherent damping properties of the metal-rubber material, vibrational energy is dissipated, thereby achieving effective vibration attenuation. An additional advantage of this spring-connected isolator design is its structural simplicity, enabling low-frequency vibration isolation without the need for complex linkage mechanisms or additional springs.

Damping material

Metal-rubber materials are classified as viscoelastic materials due to their dual elastic and viscous characteristics. Viscoelasticity is a rate-dependent behavior, and its mechanical response can be characterized as a combination of elastic and viscous elements. When a load is applied, the elastic deformation is instantaneous (rate-independent) and the part of the deformation that is recoverable, while the viscous deformation is the part that will occur over time (rate-dependent) and dissipate energy during the deformation. Consequently, metal-rubber material can be used as an ideal damping material in this design.

To obtain the mechanical performance parameters of the metal-rubber used in the metal hanger vibration isolator, relevant mechanical experiments need to be carried out. It is worth noting that although the experimental specimens used in material testing have a square cross-section for ease of gripping and testing consistency, the actual metal-rubber component in the isolator is designed as a ring to better fit the assembly and loading conditions. The material properties, however, are characterized by stress-strain relationships that are independent of the macroscopic specimen geometry. First of all, compression experiments were carried out on the metal-rubber specimens, and the metal-rubber specimens and Suns electronic testing machine. The test setup is shown in Figure 2, and the corresponding stress-strain parameters are presented in Figure 3 obtained for the metal-rubber. We can usually describe the intrinsic relationship of hyperelastic materials in terms of the strain potential energy function. There have been many different hyperelastic intrinsic models. Among them, the Mooney–Rivlin model has a good fit at smaller strains, a slightly worse fit at larger strains, and an overall good fit. The metal-rubber block of the vibration isolator has a small strain during operation, so the Mooney–Rivlin model is considered first in this study.

Figure 2.

Mechanical property test of metal-rubber: (a) metal-rubber specimen; (b) Suns electronic universal tensile testing machine. Mooney-Rivlin.

Figure 3.

Mooney–Rivlin fitting curve of hyperelastic model.

Parametric model elastic strain energy for

W = C_{10} ({\bar{I}}_{1} - 3) + C_{01} ({\bar{I}}_{2} - 3) + C_{11} ({\bar{I}}_{1} - 3) ({\bar{I}}_{2} - 3) + \frac{1}{D_{1}} {(J - 1)}^{2} .

(1)

where C₁₀, C₀₁, C₁₁, and D₁ are the material parameters of the model; J is the elastic volume ratio;

{\bar{I}}_{1}

is the first-order strain invariant;

{\bar{I}}_{2}

is the second-order strain invariant. For the material parameters C₁₀, C₀₁, and C₁₁, the stress-strain data obtained from the tests were processed, and Figure 3 shows the Mooney–Rivlin curves obtained from the fitting.

Metal-rubber is a viscoelastic material, and its mechanical response is related to the deformation rate and time. At the moment t, the total stress in the viscoelastic cell is

τ^{ve} (t) = \int_{0}^{t} G_{R} (t - s) γ (s) d s .

(2)

In equation (2), γ(s) is the shear strain; G_R(t) is the shear modulus; τ^ve(t) is the viscoelastic stress, and the normalized shear relaxation function as g_R(t) can be expressed as

g_{R} (t) = \frac{G_{R} (t)}{G_{R} (0)} = 1 - \sum_{i = 1}^{N} {g_{i}}^{ve} (1 - e^{- \frac{t}{t_{r_i}}}),

(3)

{G_{i}}^{ve} = G_{R} (0) \cdot {g_{i}}^{ve}, i = 1, 2, \cdot \cdot \cdot, N .

(4)

In equation (3): t_{r_i}、g_i^ve are the intrinsic parameters to be identified, which represent the relaxation time and elastic modulus scaling factor corresponding to each Maxwell viscoelastic unit, respectively; G_R(0) is the instantaneous shear modulus; G_i^ve is the elastic modulus corresponding to each Maxwell unit.

The metal-rubber specimen of the length a = 110 mm, the width b = 110 mm, and the thickness c = 14 mm was prepared for the experiment, and the tests were conducted using an electronic universal tensile testing machine. During the test, a shear deformation was applied to the specimen and held constant. The resulting reaction force, F(t), was recorded as a function of time. The processed experimental results are presented in Figure 4.

Figure 4.

Shear relaxation modulus fitting curve.

At the moment t, the total stress τ^ve in the viscoelastic unit is

τ^{ve} (t) = \int_{0}^{t} G_{R} (t - s) γ (s) d s = G_{R} (t) γ .

(5)

In equation (5), γ(s) is the relaxation strain; G_R(t) is the relaxation shear modulus.

τ^{ve} (0) = G_{R} (0) γ .

(6)

where the normalized shear relaxation function g_R(t) can be expressed as

τ^{ve} (0) = G_{R} (0) γ,

(7)

In equation (7), G_R(0) is the instantaneous shear modulus, where

G_{R} (t) = F (t) / 2 A / γ .

(8)

In equations (8), F(t) is the force versus time measured by the relaxation experiment; 2A is the total shear area of the metal-rubber. Then, the normalized shear relaxation function g_R(t) can be expressed as

g_{R} (t) = 1 - \sum_{i = 1}^{N} {g_{i}}^{ve} (1 - e^{- t / t_{r_i}}) .

(9)

The parameters_{_i} 、g_i^ve” are the relaxation time and elastic modulus scale factor corresponding to each Maxwell viscoelastic unit, respectively, then the elastic modulus given corresponding to each Maxwell unit can be expressed as (Table 1).

Table 1.

The viscoelastic intrinsic parameters were obtained from the relaxation experiment.

Parameters	g₁^ve/MPa	g₂^ve/MPa	g₃^ve/MPa	t_r_₁/s	t_r_₂/s	t_r_₃/s
Numerical results	0.752	0.125	0.00322	0.0136	0.867	86.3

For the viscoelastic metal-rubber material applied in the vibration isolator, increasing its deformation in the vibration isolation process will increase the energy dissipation. In this structure, the rubber blocks are stretched in both directions during operation. At the same time, the shear deformation applied to the block significantly enhances the energy dissipation of the vibration isolator, achieving high damping characteristics.

Experimental test of vibration isolator

In order to optimize the design of metal-rubber vibration isolators, it is necessary to clarify the influence of metal-rubber parameters on the performance of vibration isolators. In this work, before analyzing the effect of metal-rubber on the vibration isolation performance of the vibration isolator, the mechanical properties of the metal-rubber vibration isolator are first tested. The test results are compared with those analyzed by the finite element model, which is based on the material parameters obtained from the previous work. The aim of this study is to validate the accuracy of the established finite element model and to conduct preliminary evaluations of the static and dynamic characteristics of the isolator.

The physical model of the metal hanger vibration isolator and the test machine are shown in Figure 5(a) and (b), respectively.

Figure 5.

Experimental testing of the metal hanger vibration isolator: (a) physical model of the metal hanger vibration isolator; (b) static and dynamic testing of the vibration isolator.

To validate the finite element model of the metal-rubber hanger vibration isolator, standard mechanical tests were performed on the prototype. The test equipment was a SANS standard electronic tensile testing machine operating at a constant crosshead speed of 2 mm/min. In the experiments, the shell of the metal hanger isolator was held in place by a special fixture, and the testing machine pulled the other end of the metal hanger isolator. The measured longitudinal tension versus displacement graph of the structure is shown in Figure 6. The experimental results are in good agreement with the finite element simulation results. The measured longitudinal tension–displacement curve is plotted in Figure 6. Excellent agreement is observed between experimental and simulation results. To quantify the agreement between experiment and simulation, the root-mean-square error (RMSE) between the two curves is calculated to be 4.2 %. The results show that the metal hanger vibration isolator has nonlinear characteristics, and the stiffness of the metal hanger vibration isolator is initially large, and the stiffness of the metal hanger vibration isolator decreases with the increase of the tensile displacement. In the main working load of the metal hanger vibration isolator about 200 kg, the stiffness of the metal hanger vibration isolator is approximately linear stiffness.

Figure 6.

Static tensile test of the metal hanger vibration isolator.

Dynamic characteristics are crucial in the design of vibration isolators. In this study, both finite element simulations and laboratory experiments were conducted on the metal hanger vibration isolator under controlled conditions. All tests were performed at a laboratory temperature of 25°C ± 1°C using a 100 kN servo-hydraulic testing machine with a displacement accuracy of 0.01 mm. A constant loading rate of 20 mm/min was applied, and the dynamic displacement amplitude was set to 2 mm.

Figure 7 compares the resulting load-displacement hysteresis loops obtained from the experiments and the finite element analyses. The root-mean-square error between the experimental and FEA curves is 6.8 %. Despite minor discrepancies, the principal characteristics of the hysteresis loops are in close agreement, confirming that the isolator exhibits pronounced energy-dissipative behavior.

Figure 7.

Dynamic tensile test of a metal hanger vibration isolator.

Finite element analysis of metal hanger vibration isolator

Analysis of vibration isolation performance of metal-rubber vibration isolator under sinusoidal excitation

Vibration transmissibility, as one of the most critical indicators to describe the performance of vibration isolators, reflects the level of vibration transmission at different frequencies. The stiffness and damping of the metal-rubber hanger vibration isolator proposed in this study are mainly provided by two sets of metal-rubber blocks. The geometry of the metal-rubber blocks will determine their stiffness and damping. In this work, finite element simulation is used to optimize the stiffness and damping parameters of the isolator. Firstly, the effect of the outer diameter and height of the metal-rubber block on the isolator’s performance is studied.

In order to systematically study the influence of geometrical parameters on the performance of the metal-rubber vibration isolator, a numerical simulation was conducted. In the finite element pre-processing stage, the material parameters are set as follows: structural steel is used for all components except the metal-rubber, and its modulus of elasticity is set to 210 GPa and Poisson’s ratio to 0.3; the material parameters of the metal-rubber are obtained using the identification method described previously, and a nonlinear constitutive model is constructed through the coupling of hyperelasticity and plastic deformation theory. The mesh is constructed primarily using hexahedral elements, and the element size of the key areas is controlled within 1.0 mm to ensure the accuracy of the calculation of the contact interface and stress concentration areas.

The boundary conditions are as follows: full constraints are applied to the mounting surface of the metal shell, and a point mass of 200 kg is directly attached to the bottom to simulate the actual working load; a displacement excitation of s₁ = 0.1 mm is applied to the center metal shaft, and the displacement of the metal shell in the direction of the load, s₂, is recorded by post-processing. The transmissibility was calculated as the ratio s₂/s₁. A total of 16 models were created by combining the outer diameter (86, 82, 78, and 74 mm) and height (25, 26, 27, and 28 mm). The results of the analysis are presented as displacement-load curves in Figures 8–11.

η = s_{2} / s_{1} .

(10)

Analysis results indicate that with the increase of the height of the metal-rubber block, the peak frequency of the metal hanger vibration isolator decreases gradually, and its vibration transmissibility is also reduced. The main reasons for this result are as follows: the metal-rubber hanger vibration isolator can be mechanically simplified as a single-degree-of-freedom vibration system, in which the metal-rubber block in the mechanical model can be simplified as a spring with specific stiffness and damping. Increasing the height of the metal-rubber block is equivalent to adding a spring segment in series, which reduces the overall stiffness of the system, thereby lowering the peak frequency. This is beneficial for vibration isolator design, since a lower peak frequency helps broaden the effective isolation frequency range.

In addition, the outer diameter of the metal-rubber block also affects the peak frequency and vibration transmission rate of the metal hanger vibration isolator. As shown in Figures 8–11, the peak frequency of the metal-rubber hanger isolator decreases with the increase of the outer diameter of the rubber block. At the same time, the vibration transmission rate of the metal hanger isolator also decreases. However, the change in the diameter of the metal-rubber block has a greater effect on the peak frequency of the isolator compared to the change in the height of the metal-rubber block. The significance of this work for our design of metal hanger vibration isolators is to make the diameter and height of the metal-rubber blocks as large as possible to obtain a vibration isolator with better vibration isolation performance, if the conditions permit.

Figure 8.

Vibration transmission rate of the vibration isolator with different heights of metal-rubber elements (diameter = 74 mm).

Figure 9.

Vibration transmission rate of the vibration isolator with different heights of metal-rubber elements (diameter = 78 mm).

Figure 10.

Vibration transmission rate of the vibration isolator with different heights of metal-rubber elements (diameter = 82 mm).

Figure 11.

Vibration transmission rate of the vibration isolator with different heights of metal-rubber elements (diameter = 86 mm).

Analysis of vibration isolation performance of metal-rubber vibration isolator under random vibration excitation

Compared with the vibration caused by sinusoidal waves, random vibration is a non-deterministic vibration whose frequency, acceleration, and amplitude are random and cannot be described by a deterministic function, but has certain statistical laws. The study of random vibration usually relies on the methods of probability theory and statistics to characterize its properties through the statistical analysis of a large number of samples. Random vibration can better simulate the vibration environment in real situations and has a wider range of applications.

In the random vibration experiments, the main characteristic parameters of the metal hanger vibration isolator include the root-mean-square acceleration response, the vibration isolation efficiency, and the resonance frequency of the system. The power spectral density (PSD) is a physical quantity that describes the energy distribution of random vibration and quantitatively characterizes the distribution of vibration energy in the frequency domain in units of g²/Hz. This parameter provides a key basis for evaluating the vibration characteristics by revealing the energy distribution of different frequency components through frequency domain analysis. The power spectral density curve of random vibration set in this study is shown in Figure 12, which visualizes the energy-frequency distribution relationship under specific conditions.

Figure 12.

Target spectrum of random vibration analysis.

In the finite element analysis, the input is the power spectrum shown in Figure 12. Figures 13–16 present the power spectral density output curves of the metal hanger vibration isolators with different parameters under random vibration tests of the same intensity. According to the obtained power spectral density output curves, the random vibration resonance frequency of the metal hanger isolator decreases with the increase of the diameter and height of the metal-rubber block, indicating that the stiffness of the system decreases with increasing metal-rubber element parameters. The vibration isolation efficiency of the metal hanger vibration isolator increases with the increasing parameters of the metal-rubber element, and this trend is consistent with the results obtained from the previous deterministic vibration analysis.

Figure 13.

Output curve of power density spectrum of rubber block (diameter 74 mm) vibration isolator with different heights.

Figure 14.

Output curve of power density spectrum of rubber block (diameter 78 mm) vibration isolator with different heights.

Figure 15.

Output curve of power density spectrum of rubber block (diameter 82 mm) vibration isolator with different heights.

Figure 16.

Output curve of power density spectrum of rubber block (diameter 86 mm) vibration isolator with different heights.

Hysteresis characteristics at different frequencies

To verify the excellent performance of the new metal-rubber hanger isolator proposed in this study across different frequencies, a numerical simulation was carried out using commercial finite element software. The load-displacement curves at different frequencies are obtained. The enclosed load-displacement curve represents the energy difference between absorption during deformation and release during recovery, forming a hysteresis loop. The hysteresis loop of the metal-rubber is caused by the inherent viscoelasticity of metallic rubber and is dependent on the loading rate.

In the numerical simulation analysis, a fixed constraint was applied to the metal shell of the metal hanger isolator. Since this metal hanger isolator is used with a load of 200 kg, a preload of 2000 N needs to be applied to the intermediate shaft first before applying the displacement to the intermediate shaft boundary, and then the cyclic load d = 2.5 sin(2πω) mm is applied to the metal hanger isolator, with ω denoting the frequency of the excitation frequency.

The dynamic performance test results of the proposed vibration isolator under 30 Hz, 50 Hz, 70 Hz, and 90 Hz harmonic excitation are shown in Figure 17. Due to the significant differences in the coordinate scales of the original load-displacement curves, the data were processed using a normalization method in order to accurately assess the energy dissipation characteristics. The analysis results show that the hysteresis return area of the load-displacement curves of the vibration isolator under harmonic excitation at 30 Hz, 50 Hz, 70 Hz, and 90 Hz are 0.153, 0.144, 0.131, and 0.121, respectively, after normalization. The energy dissipated by the vibration isolator in one operating cycle decreases with the increase of the harmonic excitation frequency. This frequency-dependent damping characteristic has important engineering significance: it maintains high damping in the low-frequency region to effectively suppress resonance, while reducing damping in the high-frequency region to improve isolation efficiency.

Figure 17.

Hysteresis curves at different vibration frequencies.

Conclusion

In this paper, a new metal-rubber vibration isolator structure is designed, which achieves excellent vibration isolation performance by optimizing the geometrical configuration of the metal-rubber block. A constitutive relationship integrating the Mooney–Rivlin hyperelastic model and the Prony series viscoelastic model is established, which can accurately describe the nonlinear mechanical behavior of the metal-rubber material under static and dynamic loading. Numerical simulation studies reveal that the geometrical parameters of the metal-rubber blocks have a significant effect on the vibration isolation performance. With the increase of the diameter and height of the metal-rubber block, the vibration isolation effect is significantly improved and the damping ratio increases. This increase in performance is mainly due to the enhanced dry friction effect generated by the wire mesh structure inside the metal-rubber during the deformation process, which improves the energy dissipation capacity. The study reveals that the metal-rubber vibration isolators have obvious frequency-dependent damping characteristics. The test results show that the energy dissipated by the vibration isolator in one operating cycle gradually decreases as the excitation frequency increases. This characteristic enables the isolator to maintain high damping in the low-frequency region to effectively suppress resonance, and to reduce damping in the high-frequency region to enhance the isolation efficiency, thus realizing optimized isolation performance in a wide frequency range.

These research results provide an important theoretical basis and design guidance for the engineering application of metal-rubber vibration isolators. The design methods and analysis techniques established in this study can also be extended to the development of other types of damped vibration isolators.

Footnotes

ORCID iD

Tianyan Liu

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

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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