Displacement Transmissibility Research Articles

Design of hyperbolic quasi-zero stiffness metastructures coupled with nonlinear stiffness for low-frequency vibration isolation

Quasi-zero stiffness (QZS) metastructures with simple structures exhibit a high low-frequency vibration isolation performance, but the bearing capacity and working regions are limited due to the linear positive stiffness elements. In this study, a novel hyperbolic QZS metastructure coupled with nonlinear stiffness is designed by combining a pair of cosine-shaped beams and arc-shaped beams for low-frequency vibration isolation. Finite element models (FEM) and experimental analysis are utilized to examine the static characteristics of the metastructure, revealing a wide QZS region and significant load-bearing capacity. The metastructure can achieve effective vibration isolation properties through reasonably designing the unit cells. The influence of geometrical parameters on the QZS region is investigated to optimize and realize desirable QZS features. The dynamic equation and response of the vibration isolation system are determined using the harmonic balance method. Moreover, the corresponding displacement transmissibility is confirmed by numerical simulation. The results of the dynamic sweep frequency experiment further verify the validity of the theoretical analysis model. The hyperbolic QZS metastructure system possesses good vibration isolation performance due to the introduction and compensation of a nonlinear positive stiffness arc beam element. The proposed hyperbolic QZS metastructure provides a significant route for the stiffness compensation of metastructures.

On the displacement control performance of base-isolated structures incorporating nonlinear negative stiffness device

Low-frequency dominant far-field harmonic-like long period ground motions (FGMs) excitations have the potential to amplify the seismic response of base-isolated structures (BIS) and may result in isolator displacements exceeding their maximum limits. The application of negative stiffness devices (NSD) in isolated structures primarily focuses on SDOF systems, which fails to capture the displacement response of the superstructure on the isolation story. To address this issue, this study proposes the application of NSD to a 2DOF system in a BIS, with root mean square value as the metric for displacement transmissibility. Firstly, the displacement transmissibility of base-isolated structures with negative stiffness devices (BIS-NSD) and BIS is derived using the triple-harmonic balance (T-HB) method and the single-harmonic balance (S-HB) method, respectively. Subsequently, the applicability of the T-HB method and the S-HB method in obtaining analytical solution for the displacement transmissibility of BIS-NSD and BIS is revealed and validated through numerical solution. Parameters are selected based on fixed-point theorem for comparative analysis of the displacement and acceleration transmissibility between BIS-NSD and BIS. Finally, a comparative analysis between BIS-NSD and BIS under FGMs confirms the application advantages of NSD. The results indicate that compared to BIS, BIS-NSD significantly reduces the displacement transmissibility of the superstructure and the isolation story in the low-frequency range. As the primary resonance frequency decreases, NSD widens the vibration bandwidth of BIS. BIS-NSD effectively mitigates the increase in displacement caused by low-frequency ground motions. The acceleration transmissibility demonstrates that, without sacrificing the acceleration of the superstructure, it effectively controls the displacement of the isolation story.

A nonlinear low frequency quasi zero stiffness vibration isolator using double-arc flexible beams

There is an increasing demand for integrated small vibration isolators in precision sensing instruments and medical equipment. Therefore, a quasi-zero stiffness (QZS) vibration isolator with double-arc flexible beams (D-AFB) is proposed. The QZS isolator consists of D-AFB units, the base, platform, fixture, guide rail, and linear bearing. The static performance of D-AFB is analyzed using the FEA method, and the QZS condition is obtained. The harmonic balance method (HBM) and the pseudo arc length method (PALM) are used to solve the dynamic equations of the nonlinear system, and the analytical solutions of displacement transmissibility and Force transmissibility are derived. In theory, the initial frequency is 77.6 % lower than that of the equivalent linear system. The QZS characteristic of D-AFB is verified by static experiment, and the low-frequency isolation performance of the QZS isolator is verified by dynamic experiment under constant amplitude and constant acceleration excitation. The D-AFB has a simple structure and can easily satisfy QZS conditions. So the D-AFB can be customized according to load requirements. Multiple vibration isolation loads can be obtained by adjusting the assembly of the D-AFB. The QZS isolator can adapt to load changes within 10 % and has an isolation effect on vibrations above 4 Hz. The idea of designing a QZS isolator with D-AFB as an array unit has a good research prospect and great application value in meeting the requirements of miniaturization and integration.

Origami-inspire quasi-zero stiffness structure for flexible low-frequency vibration isolation

As an emerging technology, origami exhibits rich nonlinear static and dynamic properties. However, simulating creases remains a key challenge hindering practice application, especially for volumetric (deformable) origami. Inspired by the bistable behavior and axial-rotational coupling mechanism of triangulated cylindrical origami (TCO), this paper proposes a novel combined truss-spring structure with additional inertial mechanism (TCTS-RIM) for low-frequency vibration isolation. Unlike pure TCO, the TCTS-RIM uses elastic linkages to simulate all creases and modifies crease positions, enhancing physical realization in engineering applications and offering tunable stiffness and inertia on demand. Static analysis shows the modified truss-spring structure in TCTS-RIM has superior quasi-linear negative stiffness compared to pure TCO, which enables a very wide quasi-zero stiffness (QZS) range via linear spring compensation. The wide QZS range for different loads can be realized by setting appropriate structural parameters. From system dynamic equations, nonlinear inertia, nonlinear quadratic force, and nonlinear damping due to RIM are identified and discussed. The Alternating frequency–time harmonic balance method is used to obtain the displacement transmissibility, and its parametric influencing investigations verify that the proposed TCTS-RIM has excellent low-frequency vibration isolation performance and rich tunability. Comparative discussions show the wide QZS range significantly reduces the resonance frequency, broadens the isolation frequency band, and yields superior dynamic performance versus pure TCO isolators. Moreover, equivalent mass, anti-resonance and softening behavior induced by the rotational inertia mechanism further enhance low-frequency vibration isolation performance. An experimental prototype is built, and experimental results validate the correctness of the theoretical analyses. This work presents a simple mechanical implementation of volumetric origami, which may progress the engineering applications of origami and provide a viable approach to low-frequency vibration control.

The nexus of sustainability and damping: A quasi-zero stiffness and pseudo-piecewise inerter damper for vehicle suspension

This paper proposes a vibration control-energy harvesting damper (VCEHD) for vehicle suspension. The principles used in the proposed VCEHD include quasi-zero stiffness (QZS), mechanical motion rectifier (MMR), and semi-active control. In the dynamic analysis, the authors found that the inherent disengagement and engagement properties of the MMR can undermine the robustness of the system, that is, the inherent properties of the MMR as an energy harvesting device are not conducive to vibration isolation. Therefore, it needs to be restrained. According to the numerical simulation results, VCEHD cannot simultaneously achieve the minimum body displacement and the maximum energy harvest, and the energy harvest performance needs to be partially sacrificed. The experimental results show that the maximum energy output is 14.70 mW/kg, and the minimum displacement transmissibility is 24.32%. Combined with the numerical simulation results, VCEHD can harvest 8.82–97.79 W of power in actual vehicle running, which can provide auxiliary power for intelligent vehicles.

Proportional-retarded control of a quasi-zero-stiffness vibration isolator

This paper explores the control design and performance analysis of an active quasi-zero-stiffness vibration isolator utilizing the proportional-retarded (PR) control architecture. The PR architecture replaces the conventional “derivative” term in the proportional-derivative (or displacement-velocity) controller with the novel “retarded (time-delayed)” term, which is a new and effective way of active vibration suppression. Comprehensive gain-delay conjoint regulation strategies are developed for the strengthened isolation performance. For impact excitations, a complete stability analysis is conducted near the equilibrium position. Furthermore, a parameterized dominant pole placement algorithm is proposed to optimize the transient response of the PR-actuated isolator. Focusing on the rectangular pulse and long-duration step impacts, the algorithm adjusts the dominant poles to optimize the transient performance metrics, including the maximum displacement overshoot, steady-state displacement overshoot, and settling time. Regarding harmonic excitations, the delay-coupled effect is scrutinized for better low-frequency isolation performance, including the resonance frequency, nonlinear steady-state amplitude, multiple steady-state bands and nonlinearity elimination, equivalent active damping and stiffness, and absolute displacement transmissibility. In the case of random excitations, we demonstrate the superiority of the PR structure over the passive system.

Displacement and Force Transmissibility of a High-Static-Low-Dynamic-Stiffness Isolator with Geometric Nonlinear Damping

Displacement and Force Transmissibility of a High-Static-Low-Dynamic-Stiffness Isolator with Geometric Nonlinear Damping

Study on the performance of a novel 1DOF isolator with QZS characteristics

A typical one-degree-of-freedom (1DOF) shock isolator that has quasi-zero stiffness (QZS) characteristics consists of two transversal springs and one portrait spring. The transversal springs can provide negative stiffness and the vertical spring is used as a load-bearing component. In this article, an improved 1DOF QZS shock isolator is obtained by utilizing the combination of lateral bars and springs. The displacement transmissibility under the harmonic excitation is obtained and the principle of minimum displacement transmissibility under the condition of system stability is proposed. Finally, numerical simulation is conducted, and the simulation results reflect that the proposed 1DOF QZS isolator significantly performs better than the previous one.

Vibration Analysis of Suspension System for 3DOF Quarter Car Model with Tire Damping

Quarter car model is the simplest way for the analysis of vehicle suspension system. The objective of this study is to propose the natural frequency and to identify the displacement analysis of 3 Degree of Freedom (3DOF) suspension system using Matlab software. Research methodology are state space modelling method and logarithmic decrement method. From the state space modelling, seat mass, sprung mass and unsprung mass displacement have been calculated. Logarithmic decrement method is a time domain technique to obtain experimental damping ratio. The novelty of this research has demonstrated the experimental approach by using Quantum X (catman AP v5 software). Natural frequency and acceleration results are obtained from the time domain response of experimental data. The value of sprung mass natural frequency for theoretical (1.23 Hz) and experimental (1.134 Hz) result are within 1Hz to 2Hz. The maximum displacement transmissibility values for theoretical (2.7) and experimental (2.54) approach are less than the reference value (3.5) at the resonance point. The findings demonstrated that the results of natural frequency and damping ratio are within the acceptable limit. So, the obtained results are validated the suspension system of Nissan Sunny model for vehicle safety and passenger comfort.

Theoretical and experimental study of a novel nonlinear quasi-zero Stiffness vibration isolator based on a symmetric link-rod-type structure

This study proposes a novel nonlinear low-frequency vibration isolator with quasi-zero stiffness (QZS) based on a symmetric link-rod-type structure. The principle of the isolator is analyzed, and the properties of the vibration isolator are investigated experimentally. Varying displacements with a restoring force and stiffness are acquired by a static model; thus, the QZS conditions and system contributing factors are elucidated. Various effects of the excitation amplitude, damping ratio, rod length ratio, and initial spring compression ratio on the displacement transmissibility, which are determined using the harmonic balance method (HBM), are discussed. Through a series of experiments on the isolator prototype, the performance of a QZS vibration-isolation system with a symmetric link-rod-type structure is validated. The results indicate that the presented symmetric link-rod-type QZS vibration isolator has a wider QZS interval, lower frequency of vibration isolation, and better isolation performance, providing a valuable reference for the design of QZS vibration isolators.

Dynamical analysis and design theory for active bistable vibration isolators considering delay effect

Bistable vibration isolators (BVIs), which can undergo intra- and inter-well motions, have recently gained much attention as different motions can lead to different low-frequency isolation properties. To fully exploit the strength of BVIs, feedback actuation is injected into a classic three-spring-two-link bistable structure, and the classic PD control is applied to tune system's stiffness and damping. Moreover, the effect of the inevitable loop delay on BVI performance is considered. Operable regions of control parameters to maintain bistable characteristics and robustness regions where active BVIs are operable regardless of delay perturbations are determined. Besides, bistable dynamics, complicated by the transcendentality caused by delay, are investigated. By taking the BVI as phase zero, the number of variables to be solved is reduced, and an efficient frequency-sweeping-based calculation procedure is proposed. The BVI is evaluated by force and displacement transmissibility, showing that both intra- and inter-well motions benefit vibration isolation. For the former, the BVI yields broadband vibration isolation similar to conventional high-static-low-dynamics-stiffness vibration isolators. For the latter, an isolation valley can occur, resulting in exclusive ultra-low-frequency vibration isolation. Furthermore, properly tuning feedback actuation enhances the vibration isolation performance in both cases, and the often-overlooked loop delay can play a very positive role. Finally, the mechanism of isolation valley is qualitatively elucidated. The presented theories, including control stability analysis, delay-coupled nonlinear dynamical analysis, and beneficial parameter tuning, establish a basic design framework for active BVIs.

Mechatronic design of a composite vibration isolation system

Composite materials have attracted researchers in vibration and noise control applications due to their significant dynamic characteristics such as high strength and high damping level. In this paper, a Glass Fiber Reinforced Composite material (GFRC) is presented as a vibration isolation system to control vibration levels in industry. In addition, the impact of integration of a mechatronic control system to improve the machining process and increase the control of vibration nature. A prototype of an industrial cam–follower machine is motorized, and the Frequency Response Function (FRF) is recorded using a B&K data acquisition analyzer at five rotational speeds. The transmitted vibrations to the machine foundation are estimated without any isolation system. Then, two optimized GFRC plates of optimum stacking sequences are used as an isolation system to reduce the transmitted vibration. The displacement transmissibility is calculated theoretically and experimentally. The results show that the use of GFRC plates as an isolator reduces the vibration level of the system by 98.46% and 98.5% for [90/90/90/0/0]s and [90/ ± 45/ ± 35/90/ ± 35]s GFRC configurations respectively.

Strategies to enhance the attenuation band in finite beams using periodic attachments

One-dimensional waveguides, such as beams are frequently found in structural engineering applications. A finite periodic structure composed of a beam with periodically spaced vibration absorbers is considered in this work. Additionally, the behaviour of the structure when a hinge condition is imposed between adjacent cells is investigated. Based on the dynamic stiffness matrix approach, the transverse displacement transmissibility is evaluated for a single cell. The effectiveness of using appropriately tuned vibration absorbers in the formation of a smooth super attenuation band is demonstrated. Prediction of the attenuation band is also discussed for a structure with several periodic cells. In addition, the effect of hinge connections between the cells on the boundary frequencies of the attenuation band and the minimum transmissibility is presented.

Bio-inspired multi-joint-collaborative vibration isolation

Inspired by the collaborative action of ankle, knee and hip joint, a novel bio-inspired multi joint structure (BIMJS) is proposed, and its favorable nonlinearity is systematically investigated for low frequency vibration isolation. Two rigid rods are adopted to imitate the femur and tibia, and three torsional springs are utilized to simulate the muscle function in each joint. Based on the principle of virtual work, a static model considering the action of three joints and leg posture is established to characterize the stiffness property. The independent action mechanism of each joint is revealed and the influence of posture adjustment on equivalent stiffness of the BIMJS is studied. By adjusting proper posture and setting reasonable stiffness of each joint, the quasi zero stiffness (QZS) can be easily obtained for low frequency vibration isolation. Dynamic model is developed and the displacement transmissibility is derived to estimate the isolation performance of the BIMJS. The experimental results confirm the advantages of realizing QZS and low-frequency vibration isolation under multiple loads. The vibrations above 2.5 Hz can be effectively isolated. The concept of utilizing the collaborative action of multi joints may provide an innovative method for the design of low frequency vibration isolators.

Further investigation and experimental study of an origami structure-based quasi-zero-stiffness vibration isolator

A quasi-zero stiffness (QZS) vibration isolator formed from a truss-spring stacked Miura-ori (TS-SMO) origami structure can provide a desired ultra-low dynamic stiffness for vibration isolation while remaining a high-static stiffness for load supporting capacity. This paper further investigates the design parameters and experimentally studies the dynamic performance of the proposed TS-SMO vibration isolation system. The effects of the spring parameters and the initial setup conditions on its static response are analysed. With the proper parameter selection, the resultant supporting force generated by the origami structure can be expressed as a polynomial containing the static force and dynamic force components which does not have the linear term. The displacement transmissibility of the proposed system is calculated to evaluate its isolation performance. Analytical and numerical results are in good agreement and both demonstrate an ultra-low resonance frequency for vibration isolation. The dynamic behaviour of the proposed system is also investigated under different conditions to enhance the vibration isolation performance. A proof-of-concept prototype is designed, fabricated and tested to verify both static and dynamic performances of the TS-SMO QZS isolator. The comparative experimental results between the corresponding linear isolation system and the proposed nonlinear QZS system validate the design of origami-inspired structure for vibration isolation and further confirm the effectiveness of the QZS vibration isolator. It is hoped that this research would provide a solid foundation for designing and modelling the TS-SMO structure adopted for vibration isolation in practical engineering applications.

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.

The X-shaped structure with nonlinear positive stiffness compensation for low-frequency vibration isolation

X-shaped structure with quasi-zero stiffness (QZS) properties are suitable for low-frequency vibration isolation because of its simple structure. The effective working range of classic X-shaped structure (CXS) is narrow for the existence of negative stiffness region, and using positive stiffness compensation to overcome this deficiency is a common way. This paper proposes a novel nonlinear positive stiffness compensation method (NPSC) to improve the load capacity and QZS range of CXS. Different from the previous studies that utilize linear or hardening stiffness compensation, the NPSC exhibits the characteristics of softening stiffness first and then hardening stiffness and is provided by torsion springs simply. The static properties of X-shaped structure with NPSC are investigated. The result indicates the NPSC possesses dual significant advantages. Firstly, the X-shaped structure with NPSC can achieve larger load capacity and wider QZS range than that with linear or hardening stiffness compensation in optimal parameter region. Secondly, the X-shaped structure with NPSC can conveniently overlap the equilibrium position with QZS point to obtain ultra-low frequency vibration isolation effect compared to that with linear stiffness compensation. The motion equation of the proposed system is derived, and the dynamic response is solved through harmonic balance method (HBM). The influence of different parameters on displacement transmissibility is investigated and the result is verified by numerical simulation. The transmissibility comparison between the proposed model and the CXS with linear compensation shows that the proposed model possesses better vibration isolation performance. Besides, the static compression test and dynamic sweep frequency experiment are implemented, and the experiment results verify the validity of proposed model. The proposed NPSC is an important inspiration for the stiffness compensation research of X-shaped structures.

On the trade-off performance curve of vibration isolation controlled by passive or adaptive dampings

This paper addresses an on-off adaptive damping isolation, in which the damping can be frequency dependent. The comparison with the passive isolation is illustrated by the trade off performance curve, which is shown the relation between the displacement and force transmissibility. The adaptive control is based on on-off dampings and a switching frequency. The analytical optimization of the adaptive damping law can be obtained. The optimized adaptive damping control showed a remarkable improvement comparing with the passive one. At last, a self-made magnetorheological fluid is introduced to show the capability to provide the variable damping.

Tetrahedron structure with nonlinear stiffness and inertia modulation for enhanced low frequency vibration isolation

A novel magnetically modulated tetrahedron structure with nonlinear inertia (MMTS-NI) is proposed and systematically investigated for broadband vibration isolation. The MMTS-NI consists of the tetrahedron structure, a pair of repulsive magnets, and the attached masses. Static analysis reveals that quasi-zero stiffness (QZS) can be realized by utilizing the harden stiffness of the magnets to modulate the negative stiffness of the tetrahedron structure. And the QZS can be obtained under different loads by setting appropriate initial angles of rods and matching initial magnet spacing. Considering the achieved QZS and the inertia of the attached masses, the dynamic model of the MMTS-NI is developed to evaluate its isolation performance when subject to base excitation. By using harmonic balance method (HBM), dynamic equation is solved and the displacement transmissibility is derived. The effects of QZS and the nonlinear inertia are clarified through parametric analysis and comparative discussions. The realization of QZS is beneficial to reduce the resonance frequency and broaden the isolation band. The introduce of nonlinear inertia can considerably suppress the hardening behavior of the transmissibility curve and further reduce the starting isolation frequency. And the inertia induced anti-resonance enhances the vibration attenuation effect in the low frequency band. The experimental prototype is manufactured and thoroughly tested under sweep, periodic and random excitation. The experimental results show that the MMTS-NI can effectively isolate the frequency components above 4 Hz. And the vibration attenuation effect can be significantly enhanced in the range of 4–10 Hz by increasing the attached masses. The results provide a novel and significant methods of improving low frequency vibration isolation performance using nonlinear stiffness and inertia modulation.

Prediction on the Performance of Polymer-Based Mechanical Low-Pass Filters for High-G Accelerometers

A polymer-based mechanical low-pass filter(m-LPF) for high-g accelerometers makes it possible to remove high-frequency transient noises from acceleration signals, thus ensuring repeatable and reliable measurement on high-g acceleration. We establish a prediction model for performance of m-LPF by combining a fundamental vibration model with the fractional derivative standard linear solid(FD SLS) model describing the storage modulus and loss modulus of polymers. Here, the FD SLS model is modified to consider the effect of m-LPF shape factor (i.e., thickness) on storage modulus and loss modulus. The prediction accuracy is verified by comparing the displacement transmissibility(or cut-off frequency) estimated using our model with that measured from 3 kinds of polymers(polysulfide rubber(PSR), silicone rubber(SR), and polydimethylsiloxane(PDMS)). Our findings will contribute a significant growth of m-LPF for high-g accelerometers.