High Static Low Dynamic Stiffness Research Articles

A high-static-low-dynamic-stiffness delayed resonator vibration absorber

Delayed resonator (DR), which enables complete vibration suppression through loop delay tuning, has been extensively investigated as a linear active dynamic vibration absorber since its invention. Besides, the nonlinear high-static-low-dynamic stiffness (HSLDS) has been widely used in vibration isolators for broadband (yet incomplete) vibration reduction. This work combines the benefits of DR and the nonlinear HSLDS characteristics, thus creating a nonlinear DR (NDR). Three unexplored topics are considered: (1). To address the nonlinear dynamics that are made complicated by the introduction of delay and the nonlinearity coupled between the NDR and the primary structure; (2). To tune the control parameters to seek possible complete vibration suppression in the nonlinear case, and accordingly, to determine the operable frequency band; (3). To evaluate how the HSLDS characteristics can enhance the performance of the linear DR (LDR) and how to design an NDR to maximize its benefits. Without loss of generality, a classic nonlinear HSLDS structure with three springs and two links is considered, and mathematical tools and computational algorithms are introduced or proposed to efficiently address theoretical analysis. Using the parameters of an experimental setup, we show that a properly tuned NDR suppresses the vibrations on the primary structure to a sufficiently low level. Besides, the HSLDS characteristics extend the operable frequency band compared with the LDR, while strong nonlinearity limits such extension. The proposed analysis procedures for delay-coupled nonlinear dynamics, alongside the control parameter tuning, determination of operable frequency bands, and structural design rules, establish a basic theoretical framework for the NDR design.

Tunable High-Static-Low-Dynamic Stiffness Isolator under Harmonic and Seismic Loads

High-Static-Low-Dynamic Stiffness (HSLDS) mechanisms exploit nonlinear kinematics to improve the effectiveness of isolators, preserving controlled static deflections while maintaining low natural frequencies. Although extensively studied under harmonic base excitation, there are still few applications considering real seismic signals and little experimental evidence of real-world performance. This study experimentally demonstrates the beneficial effects of HSLDS isolators over linear ones in reducing the vibrations transmitted to the suspended mass under near-fault earthquakes. A tripod mechanism isolator is presented, and a lumped parameter model is formulated considering a piecewise nonlinear–linear stiffness, with dissipation taken into account through viscous and dry friction forces. Experimental shake table tests are conducted considering harmonic base motion to evaluate the isolator transmissibility in the vertical direction. Excellent agreement is observed when comparing the model to the experimental measurements. Finally, the behavior of the isolator is investigated under earthquake inputs, and results are presented using vertical acceleration time histories and spectra, demonstrating the vibration reduction provided by the nonlinear isolator.

Subharmonic response suppression of a quasi-zero stiffness system

Quasi-zero stiffness (QZS) isolators can effectively suppress low-frequency vibration due to high-static-low-dynamic stiffness (HSLDS) characteristics. The existence of nonlinearity induces intricate nonlinear phenomena in QZS isolators, particularly under conditions of small damping and large excitation. These nonlinear characteristics can seriously affect the vibration isolation performance, which has rarely been focused on in existing studies. This study examines a typical QZS isolator with nonlinear damping. The modified incremental harmonic balance (IHB) method is applied to directly analyze its dynamic characteristics and investigate the suppression of the subharmonic response. The dynamic analysis results indicate multiple subharmonic vibration intervals, including the 1/2, the 1/3, and the 1/9 subharmonic vibration. The basin of attraction at different excitation frequencies shows that subharmonic vibration is highly likely to occur. The effect of different damping on the transmissibility proves to be different. Simultaneously, rational parameter matching can guarantee the absence of subharmonic vibration while maintaining the primary resonance frequency and amplitude unchanged. The research results of this study provide theoretical support for the parameter selection of QZS isolators. In addition, the higher-order approximations of most isolators have the same form, so these results can guide other QZS isolators in suppressing subharmonic vibrations and improving their operating range.

Nonlinear wire rope isolator with magnetic negative stiffness

The high-static-low-dynamic-stiffness (HSLDS) isolators have been widely employed to achieve low-frequency vibration isolation. However, traditional HSLDS isolators are prone to nonlinear jump phenomena due to the introduction of nonlinear stiffness characteristics, which will cause the performance deterioration under high excitation levels. To overcome this issue, inspired by the dry friction characteristic of wire ropes, the wire ropes are employed to design HSLDS isolators to suppress nonlinear jump phenomenon by introducing hysteretic damping in this paper. Therefore, a nonlinear wire rope isolator with magnetic negative stiffness is proposed to achieve HSLDS with hysteretic damping. The elastic force of wire rope and the magnetic force are modeled to obtain nonlinear restoring force, and then the dynamic model of the proposed isolator is established to investigate isolation performance. The equivalent beam element method is proposed to model wire ropes for obtaining dynamic characteristics by considering tensile and bending stiffnesses. Then, numerical simulation is performed to investigate the isolation performance of the wire rope isolators with different magnetic negative stiffnesses, and demonstrates that the proposed isolator can work in a higher excitation level than the isolator without dry friction of wire ropes. Moreover, experiments are carried out to show that the onset isolation frequency of the proposed isolator can be reduced by 66.7% compared with the wire rope isolator without magnetic negative stiffness. Therefore, it is demonstrated that the proposed wire rope isolator with magnetic negative stiffness has the preferable performance in low frequency isolation, and can improve the adaptability under a higher excitation level.

Analytical optimization of inerter-based dynamic vibration absorbers with negative stiffness (INDVAs) using a reduced 2-degree-of-freedom (2DOF) model

The incorporation of an inerter, with its mass amplification capability, along with a negative stiffness spring characterized by high static-low dynamic stiffness, can significantly enhance the vibration control performance of traditional tuned mass dampers (TMDs). The previous analytical solution for the optimal parameters of inerter-based dynamic vibration absorbers with negative stiffness (INDVAs) is a generalized single-degree-of-freedom (GSDOF) model with INDVAs attached. When utilizing INDVAs for controlling wind-induced responses in high-rise buildings, the accuracy of the GSDOF model is compromised. Therefore, a multi-degree of freedom (MDOF) model is initially reduced to a simplified 2-degree of freedom (2DOF) model using a clustering algorithm in this study. The H-∞ optimization method is then employed to derive simplified analytical solutions for the optimal parameters of the three types of INDVAs. The robustness of the parameters and the control effect of these optimal INDVAs are subsequently compared with those of numerical solutions generated through a multiobjective genetic algorithm and with those of analytical solutions based on the GSDOF model. The results demonstrate that for the same type of INDVA, compared with the multiobjective genetic algorithm, the optimal INDVA calculated by the 2DOF model can achieve the same control effect on the structural wind-induced response, while the time cost is lower. In addition, the optimal INDVA obtained by the 2DOF model has a superior control effect on the wind-induced displacement and acceleration response compared to those of the GSDOF model, especially when the INDVA has a substantial inertance ratio and spans multiple floors.

Mechanical characteristics analysis of high dimensional vibration isolation systems based on high-static-low-dynamic stiffness technology

Large floating raft vibration isolation systems (FRVISs) based on high-static-low-dynamic stiffness (HSLDS) technology offer excellent low frequency vibration isolation performance with broad application prospects. However, the design process for these complex high-dimensional coupled nonlinear systems remains poorly developed, particularly when applied for ocean-going vessels that experience rolling and pitching motions. The present work addresses this issue by establishing a six-degree-of-freedom HSLDS vibration isolation model for FRVISs composed of eight isolators, and the model is applied to fully analyze the swing stability and multidimensional vibration isolation performance of these systems. The influence of nonlinearity on the mechanical properties of the vibration isolators is analyzed more clearly by assuming that each vibration isolator realizes nonlinear HSLDS characteristics in the z direction and linear characteristics in the x and y directions. The results demonstrate that the swing displacement responses of the system are greatly reduced under weak nonlinearity, which reflects the high static stiffness and high static stability characteristics of an HSLDS system. The multidimensional vibration isolation performance of the system is evaluated according to the impacts of nonlinearity, the installation height Hz of the isolators, and the relative position Dr of the two middle isolators. The results of analysis demonstrate that applying a value of Hz = 0 produces the best vibration isolation performance overall under strong nonlinearity by avoiding unnecessary secondary peaks in the force transmission rate under harmonic mechanical excitation and ensuring a maximum high-frequency vibration isolation effect. However, applying a weak nonlinearity is better than a strong nonlinearity if Hz is not zero. In contrast, Dr has little effect on the vibration isolation effect of the raft in the x, y, and z directions. Therefore, an equidistant installation with Dr = 0.5 would be considered ideal from the standpoint of installation stability.

A Low-Frequency Vibration Isolator with Cross-Ring Structure

Low-frequency vibration isolation is essential for ultra-precision manufacturing and measurement equipment. Low stiffness is beneficial for low-frequency isolation, while leading to a degradation of the load capacity of the isolator. To tackle the problem, a nonlinear isolator is proposed with a cross-ring structure, composed of a circular ring and a semi-circular ring. Experimental studies validate that the isolator is applicable to different payload mass, and a cut-off frequency of 1.1[Formula: see text]Hz is achieved for an effective isolation. The isolator is theoretically investigated to reveal the mechanism leading to the low-frequency isolation performance. An elliptic integral method is adopted to characterize the stiffness characteristics of the ring structures under compression. The whole compression process of a semi-circular ring is divided into five stages according to the deformed shapes, exhibiting quasi-linear stiffness, stiffness-softening, negative stiffness, and stiffness-hardening characteristics in sequence. Together with the stiffness-softening circular ring, the cross-ring structure demonstrates a growing high-static-low-dynamic-stiffness (HSLDS) characteristic in a wide load range, and the result is verified by a restoring force measurement test. A harmonic balance analysis is performed to predict the frequency responses of the proposed isolator. It is shown that a low-frequency isolation can be achieved with the structure compressed to the HSLDS region by the payload, and an ultra-low-frequency isolation is achieved with the dynamic-to-static stiffness ratio below 0.1. A numerical investigation is performed to further reveal the frequency responses of the isolator with lightweight/overweight payloads and excessive excitation amplitudes. Jump phenomena are presented. This work provides a prototype and the theoretical basis for low-frequency vibration isolation via a cross-ring structure.

Bandgap regulations of longitudinal wave for a nonlinear metastructure isolator with high-static-low-dynamic stiffness

Linear structures are inherently difficult to exhibit mechanical properties with high-static-low-dynamic stiffness (HSLDS), which means that linear isolators with these structures struggle to have both high-quality load-bearing capacity and low-frequency isolation ability at the same time. To address this issue, we have designed a high-load-capacity local resonance (LR) nonlinear metastructure isolator for low-frequency vibration isolation of target objects. We reveal the regulation mechanism of longitudinal wave attenuation in such metastructures and conduct comparative study by finite element method (FEM) and experiments for a 3D printed prototype metastructure. The results show that in specific excitation frequencies, the LR nonlinear metastructure with a cantilever beam oscillator having a natural frequency of 11.7 Hz triggers a LR mechanism. Under the action of the nonlinear isolator, the vibration transmissibility is as low as 10 % at the center frequency of the bandgap under the base excitation. Furthermore, the dispersion equation of such nonlinear metastructures is derived, and through the analysis of the regulation of bandgaps, the variation trend of the transmissibility of nonlinear metastructure isolators is effectively predicted. The calculation method and design idea of this metastructure isolators provide a new approach for vibration isolation.

Modular quasi-zero-stiffness isolator based on compliant constant-force mechanisms for low-frequency vibration isolation

To effectively isolate low-frequency vibrations, we present a rigid–flexible coupling quasi-zero-stiffness (QZS) vibration isolator with high-static-low-dynamic stiffness (HSLDS) characteristics. Specifically, the QZS isolator is realized by the development of a compliant constant-force mechanism, formed by parallelly combining a diamond-shape mechanism and a nonlinear bi-stable beam in parallel. To evaluate performance of the QZS isolator, we derived an analytical force–displacement model and dynamic model based on pseudo-rigid body method and Lagrange’s equations. Then, finite element analysis was performed in Workbench to verify theoretical analysis and identify the optimal design parameters. Furthermore, the dynamic responses of the QZS isolator are established with the harmonic balance method. Finally, the relationships among displacement transmissibility and factors including damping, BSB, payload mass, and material property are discussed. The results have shown that our QZS isolator design can effectively isolate vibrations in low frequency.

Vibration isolation performance of a rectangular panel with high-static-low-dynamic stiffness supports

In this paper, the nonlinear vibration of a corner-supported rectangular thin panel with high static low dynamic stiffness (HSLDS) isolator supports is investigated. The dynamic model of the panel clamped at corners is established and the frequency equations are derived by employing the Rayleigh-Ritz procedure. The characteristic orthogonal polynomial series is introduced as the assumed modes and the equations of motions are obtained. The dynamic response of the system is analytically calculated for the first time by applying the complexification-averaging method. And the approximation solutions are verified with the numerical and finite element method. The isolation performance of the system is evaluated by the displacement transmissibility and the effects of the parameters are investigated for optimal design. The comparison with the isolation performance of linear elastic-supported panel illustrates that introducing the HSLD isolators has a superior effect on reducing the peak transmissibility in the low-frequency band and provides a wide isolation range for the system.

A semi-active quasi-zero stiffness vibration isolator based on magnetorheological elastomer with a fast convergence switch control

Quasi-zero stiffness (QZS) isolators have attracted extensive attention due to their high-static–low-dynamic stiffness. The effectiveness of the traditional QZS (T-QZS) isolators to attenuate low-frequency vibration has been extensively studied. The T-QZS isolator, however, cannot usually provide satisfactory vibration mitigation performance under ultra-low frequency excitation. To tackle this challenge, we propose a semi-active QZS isolator featuring magnetorheological elastomer (SAQZS-MRE isolator). The prominent merit of the proposed design is that the applied MRE material can provide varying stiffness and damping to the proposed SAQZS-MRE isolator, making the isolator suitable for vibration isolation under wide-band low-frequency range. The dynamic viscoelasticity properties of the MRE are experimentally characterized, to formulate the dynamic modeling of the SAQZS-MRE isolator. The open-loop vibration isolation characteristics of the SAQZS-MRE isolator are assessed using the harmonic balance method (HBM). Subsequently, two semi-active control strategies including skyhook and fast convergence switch (FCS) are effectively utilized in the proposed SAQZS-MRE isolator to evaluate their closed-loop performance. The results demonstrate that the SAQZS-MRE isolator with FCS control can eliminate the resonance peak and reduce the initial isolation frequency. In the higher frequency range, it can also improve the vibration isolation performance compared with the SAQZS-MRE isolator with passive control.

Low-frequency vibration suppression of metastructure beam with high-static–low-dynamic stiffness resonators employing magnetic spring

The low/ultra-low-frequency structural vibrations exist widely in some engineering structures, for example, aerospace, naval, and building structures and so on. They could seriously affect the working performance and even cause the destruction of these engineering structures. In order to restrain the low/ultra-low-frequency structural vibration, a locally resonant (LR) metastructure beam equipped with high-static–low-dynamic stiffness (HSLDS) resonators employing negative stiffness magnetic spring is proposed. The analytical model of magnetic negative stiffness spring is firstly derived and the design method of HSLDS resonators is presented based on parametric optimization. Then, the dynamic model of the LR metastructure beam with HSLDS resonators is established using the wave finite element method. The effects of the number of local resonator units on low-frequency band gap are analyzed. Finally, the low-frequency vibration control performance of LR metastructure beam is validated experimentally. The experimental result is in agreement with the theoretical analysis, which demonstrates that the HSLDS resonators are conductive to suppress vibration of the metastructure beam in the low-frequency region (f < 10 Hz). The results also showed that the more the HSLDS resonators are used, the better the vibration suppression effect is.

Experiment investigation and dynamic responses of three-dimensional isolation system with high-static-low-dynamic stiffness

The application of three-dimensional (3D) isolators was limited due to the demands of long isolation period and strong supporting capacity. This paper introduces the configuration and mechanical behavior of 3D isolation system with high-static-low-dynamic stiffness (HSLDS). The proposed 3D isolator consists of the negative stiffness device and pre-deformed inclined rubber bearing. The mechanical model combining negative and positive stiffness is developed. Furthermore, experiments on both the separated and combined negative stiffness device and the pre-deformed rubber bearing are performed to characterize the behavior of HSLDS isolator. Subsequently, a numerical study of different isolated structures is performed and the variations on the structural responses as well as the hysteretic loops are illustrated. By comparing the vertical accelerations and the isolator behavior, the efficiency of HSLDS isolator on mitigating the structural acceleration as well as the vertical static displacement is confirmed.

A novel multi-directional vibration isolation system with high-static–low-dynamic stiffness

A novel multi-directional vibration isolation system with high-static–low-dynamic stiffness

Rolling stability analysis of high-static-low-dynamic stiffness floating raft vibration isolation systems

Floating raft vibration isolation (FRVI) systems employing isolators with high-static-low-dynamic stiffness (HSLDS) characteristics offer excellent large-scale and multidirectional vibration attenuation performance ideally suited to stealth applications in ocean faring vessels. However, the rolling motions of vessels in heavy seas can produce large displacements and induce instability in HSLDS-FRVI systems, and our present state of understanding of these systems is insufficient to facilitate the design of more stable systems. The present work addresses this issue by establishing an accurate analytical model of a double-layer FRVI system with 12 degrees of freedom to facilitate detailed rolling stability analyses when adopting HSLDS isolators. This model is also adapted to obtain a simplified single-layer HSLDS-FRVI system model and a corresponding single-layer equivalent linear system (ELS) model for investigating the factors contributing to FRVI system stability. The results demonstrate that the single-layer HSLDS-FRVI system obtains increasingly superior rolling stability compared to that of the corresponding ELS as the system loading increases. Also, good rolling stability can be maintained by applying proper mounting locations for the isolators. Additional factors contributing to FRVI system stability are deduced by comparing the results obtained for double-layer HSLDS-HSLDS, HSLDS-ELS, ELS-HSLDS, and ELS-ELS systems. The results demonstrate that the use of isolators with HSLDS characteristics is most beneficial for improving the rolling stability of double-layer FRVI systems when applied between the ground and bottom layers, and maximum rolling stability is obtained when the mass of the lower level is close to that of the upper level.

An Experimental Investigation of the Displacement Transmissibility for a Two-Stage HSLD Stiffness System

Vibration isolation across the frequency spectrum is a challenge in many applications, particularly at low frequencies where linear oscillators amplify excitation forces. To overcome this, nonlinear high static low dynamic (HSLD) stiffness oscillators have been proposed with the aim of reducing the resonant frequency while maintaining the high load capacities of much stiffer linear systems. A two-degree of freedom (2DOF) HSLD stiffness system is proposed to investigate the effectiveness of such systems. Experiments reveal that a 2DOF non-linear HSLD stiffness system outperforms a similar single-degree of freedom (SDOF) HSLD stiffness system, as well as similar SDOF and 2DOF linear systems. Three performance criteria are used to assess these systems, including (1) minimizing the resonant frequency and maximizing the isolation zone, (2) minimizing the magnitude of amplification at resonance, and (3) maximizing the ability to isolate large input frequencies. Exact numerical and approximate analytical simulations are validated using these experimental data. A sensitivity analysis of system parameters reveals that it is necessary to incorporate adjustability into the geometry of a design to counteract unavoidable manufacturing tolerances. Changes of less than 2% to the stiffness or geometry of a system can drastically change its dynamic response.

A state-of-the-art review on low-frequency nonlinear vibration isolation with electromagnetic mechanisms

Vibration isolation is one of the most efficient approaches to protecting host structures from harmful vibrations, especially in aerospace, mechanical, and architectural engineering, etc. Traditional linear vibration isolation is hard to meet the requirements of the loading capacity and isolation band simultaneously, which limits further engineering application, especially in the low-frequency range. In recent twenty years, the nonlinear vibration isolation technology has been widely investigated to broaden the vibration isolation band by exploiting beneficial nonlinearities. One of the most widely studied objects is the “three-spring” configured quasi-zero-stiffness (QZS) vibration isolator, which can realize the negative stiffness and high-static-low-dynamic stiffness (HSLDS) characteristics. The nonlinear vibration isolation with QZS can overcome the drawbacks of the linear one to achieve a better broadband vibration isolation performance. Due to the characteristics of fast response, strong stroke, nonlinearities, easy control, and low-cost, the nonlinear vibration with electromagnetic mechanisms has attracted attention. In this review, we focus on the basic theory, design methodology, nonlinear damping mechanism, and active control of electromagnetic QZS vibration isolators. Furthermore, we provide perspectives for further studies with electromagnetic devices to realize high-efficiency vibration isolation.

On the Characteristics and Practical Parameters Selection of a Quasi-Zero-Stiffness Isolator

Quasi-zero-stiffness (QZS) isolators, characterized by high-static-low-dynamic stiffness (HSLDS), have attracted interest in the field of low-frequency vibration isolation. An existing QZS isolator, using horizontal springs in series with rigid oblique rods as a negative stiffness mechanism, has been proposed to obtain desirable low-frequency isolation performance. However, it is unknown if the rigid oblique rods will obtain the optimal isolation performance for the existing QZS isolation system. In this research, considering that the oblique rods are flexible rather than rigid, the effects of the stiffness of the springs and the rods on mechanical characteristics and isolation performance of this QZS isolator are investigated to answer the above question and to guide design of the QZS isolator. The QZS conditions ensuring the QZS characteristics of the proposed QZS isolator are derived, and the parametric analysis on QZS characteristics is conducted. Further, amplitude-frequency response of the QZS isolation system is obtained by the average method and the numerical methods; the following parametric analysis indicates that the optimal stiffness ratio of horizontal spring to vertical spring should be slightly greater than 0.5 to obtain a reasonable compromise between the peak amplitude control and peak force transmissibility reduction. Meanwhile, when the stiffness ratio of the oblique rod to the vertical spring is greater than 2, the QZS isolator with finite-stiffness oblique rods has similar isolation performance as that of the existing QZS isolator with rigid oblique rods. Therefore, designers could select oblique rods with finite stiffness to avoid excessive size or non-negligible weights of the rods.

Lever-type high-static-low-dynamic-stiffness vibration isolator with electromagnetic shunt damping

Owing to the ultra-low frequency vibration isolation performance without compromising static stiffness, high-static-low-dynamic-stiffness (HSLDS) vibration isolators (VIs) have advantages over linear vibration isolators. This paper presents a lever-type HSLDS vibration isolator (L-HSLDS-VI) by employing negative resistance electromagnetic shunt damping (EMSD) together with eddy current damping to eliminate the jump phenomenon and thus to improve the stability of L-HSLDS-VIs. The lever inerter system can amplify the mass effect to broaden the isolation band. The theoretical model of L-HSLDS-VIs with EMSD (L-HSLDS-VI-EMSD) was established. The effects of negative resistance and lever ratio on the vibration isolation performance of L-HSLDS-VI-EMSD were investigated analytically and experimentally. The L-HSLDS-VI can provide significant nonlinear stiffness, which can realize the quasi-zero stiffness (QZS), and thus broaden the isolation band. EMSD produces a considerable damping effect to enhance the vibration mitigation performance of L-HSLDS-VI. The combination of the EMSD and nonlinear ECD damping is an efficient approach to improve the vibration isolation performance to overcome the jump phenomenon. This paper utilizes the lever effect to amplify damping effects, which could provide a guideline to modify the performance of HSLDS-VIs.

Design and Analysis of a Vibration Isolator with Adjustable High Static–Low Dynamic Stiffness

A new adjustable high static–low dynamic stiffness vibration isolator is presented to improve the effective vibration isolation frequency bandwidth. Firstly, the finite element simulation of electromagnetic positive stiffness device and negative one is carried out. Through static analysis, the mathematical expressions of force–displacement–current and stiffness–displacement–current are derived. The requirements that the system works at quasi-zero-stiffness condition near the equilibrium position are obtained. The influences of different parameters on the force transmission rate of the high static–low dynamic stiffness vibration isolation system are investigated with harmonic balance method. Finally, the experimental platform is built, and the vibration isolation performance experiment is conducted. Results show that the adjustable high static–low dynamic stiffness vibration isolator has a wider effective vibration isolation frequency range and lower peak transmissibility than the traditional one.