High Static Low Dynamic Stiffness Research Articles

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.

Magnetically modulated sliding structure for low frequency vibration isolation

Magnetically modulated sliding structure (MMSS) is proposed and systematically investigated in this paper. The MMSS consists of the sliding beam and a pair of repulsive magnets. The sliding beam can provide negative stiffness, which can be modulated by a pair of repulsive magnets (hardening positive stiffness). The static analysis is completed to fully clarify the modulation mechanism. With the modulation of magnets, the sliding beam can exhibit favorable high-static-low-dynamic stiffness (HSLDS). The dynamic model of the MMSS is developed to evaluate its response when subject to base excitation. The displacement transmissibility is derived by the harmonic balance method (HBM). Transmissibility curves show that the MMSS isolator possesses a low resonant frequency and can isolate vibration in a wide frequency range. The principle prototype is fabricated and tested. The effectiveness of the modulation mechanism in realizing quasi-zero stiffness (QZS) is verified by static experiment. Dynamic tests under periodic, sweep and random excitation demonstrate that the isolator exhibits excellent low frequency isolation performance and the vibration exceeding 4 Hz can be effectively isolated. The MMSS system provides a new approach to solve the problem of low frequency vibration isolation. The modulation mechanism, that is, utilizing hardening stiffness to modulate negative stiffness, is of reference significance for the design of HSLDS isolators.

Flexural wave bandgap properties in metamaterial dual-beam structure

In this paper, a metamaterial dual-beam structure is firstly constructed by periodically adding pillars between two identical homogeneous beams. By simplifying the pillar as an equivalent spring-mass-spring system, another metamaterial dual-beam structure carrying periodic spring-mass-spring resonators is then established. This equivalent representation method is validated through both a vibration transmission analysis and a band structure analysis. Two bands corresponding to the antisymmetric and symmetric flexural vibration modes respectively open a flexural narrow bandgap in the low frequency range. Then, the improved structure with high-static-low-dynamic stiffness (HSLDS) resonators is proposed in the need of widening the bandgap and lowering the beginning frequency. Results show that with the decrease of the proportion of the residual stiffness, the beginning frequency of the bandgap can be reduced, while the ending frequency keeps unchanged. This results in a relative bandgap as high as 1.465, which is about 4.4 times of the original structure.

An Air Spring Vibration Isolator Based on a Negative-Stiffness Structure for Vehicle Seat

This research introduces an air spring vibration isolator system (ASVIS) based on a negative-stiffness structure (NSS) to improve the vehicle seat’s vibration isolation performance at low excitation frequencies. The main feature of the ASVIS consists of two symmetric bellows-type air springs which were designed on the basis of a negative stiffness mechanism. In addition, a crisscross structure with two straight bars was also used as the supporting legs to provide the nonlinear characteristics with NSS. Moreover, instead of using a vertical mechanical spring, a sleeve-type air spring was employed to provide positive stiffness. As a result, as the weight of the driver varies, the dynamic stiffness of the ASVIS can be easily adjusted and controlled. Next, the effects of the dimension parameters on the nonlinear force and nonlinear stiffness of ASVIS were analyzed. A design process for the ASVIS is provided based on the analytical results in order to achieve high static–low dynamic stiffness. Finally, numerical simulations were performed to evaluate the effectiveness of the ASVIS. The results obtained in this paper show that the values of the seat displacement of the ASVIS with NSS were reduced by 77.16% in comparison with those obtained with the traditional air spring isolator without NSS, which indicates that the design of the ASVIS isolator with NSS allows the effective isolation of vibrations in the low-frequency region.

Design and Theoretical Analysis of High-Static-Low-Dynamic Stiffness Torsion Vibration Isolator Based on Oblique Springs

A torsion vibration isolator composed of oblique springs with high-static-low-dynamic stiffness (HSLDS) is proposed to attenuate the transmission of torsion vibration along the shipping shaft in this paper. It is good at in low frequency vibration isolation as it can significantly reduce the resonance frequency of the system with the same load capability. Firstly, the model of HSLDS torsion vibration isolator is introduced in this paper. Secondly, the non-dimensional torsion stiffness is formulated using mechanics theory, and the HSLDS characteristic of designed torsion vibration isolator is verified. Finally, the torque transmissibility is analyzed using the Increment Harmonic Balance (IHB) method, and the effects of the system parameters on it are analyzed. The results show that the resonant frequency increases accordingly as the stiffness ratio and the excitation torque are increased. However, the peak value of the torsion transmissibility is decreased as the damper ratio increasing.

A nonlinear seat suspension with high-static low-dynamic stiffness based on negative stiffness structure for helicopter

One of the most important issues for the helicopter pilots is the health risk due to the vibration transmitted to the pilot through the seat. In this article, a seat suspension based on negative stiffness structure is presented to decrease the vibration transmitted to the pilot in both vertical and lateral directions without losing the loading capacity of the system. Here, an integrated model of the suspension–cushion–occupant is derived. To generalize the results of system analysis and its usability in other applications, the impact of parameters on the system performance is studied in dimensionless form. Despite coupling between the lateral and vertical directions, the design parameters of the seat suspension are determined in such a way that the system responds simultaneously as a negative stiffness structure in both directions. The system efficiency in vibration damping is assessed by seat effective amplitude transmissibility and transmissibility criteria. In addition, the whole body vibration and impact of the vibration on the pilot body are evaluated using ISO-2631. To verify the system efficiency in more realistic situation, the simulations are performed using the measured vibration data of a Bell-412 helicopter. The results indicate that the vibration amplitude is decreased by about 45% and 48% in the lateral and vertical directions, respectively. The frequency spectrum comparison of the seat and cabin floor reveals 80% reduction of amplitude in fundamental frequency in the vertical direction, whereas it is about 93% in the lateral direction. Furthermore, the level of pilot’s comfort and perception is improved that demonstrates better riding quality and reduced vibration environment.