Negative Stiffness Behavior Research Articles

Design of broad quasi-zero stiffness platform metamaterials for vibration isolation

Adaptability and reliability are challenges in designing vibration isolation structures, and mechanical metamaterials featuring broad quasi-zero stiffness (QZS) platforms are among the most promising candidates for addressing this issue. This paper proposes a novel design of vibration isolation metamaterials featuring a broad QZS platform to achieve vibration control in complex environments. The metamaterial unit cells are designed by integrating horizontal and diagonal beams based on the mechanism combining Euler buckling and flexural deformation. Herein, the component made of diagonal beams is configured to exhibit negative stiffness behavior, while the designed support components aim to relax the boundary constraints of the diagonal beam component, thereby mitigating the negative stiffness effect. By tuning the synergistic effects between horizontal and diagonal beams, QZS features can be achieved over a broad range of displacements. A combination of theoretical analysis, finite element method and experiment is employed to investigate the payload and QZS features of metamaterials comprehensively. Notably, the designed unit cell maintained a considerably broad QZS platform, with static experiments revealing that this platform accounts for approximately 55 % of the total loading range. Furthermore, the designed metamaterials exhibit excellent vibration isolation performance in the low-frequency range, with vibration experiments demonstrating that the unit cell can effectively shield vibrations across almost the entire range when the support mass corresponds to the QZS payload. The geometric parameters of the metamaterial configuration that influence the range of the QZS platform are also explored. In conclusion, the proposed mechanical metamaterials have a tunable and broad QZS platform with considerable potential for use in customized low-frequency vibration isolation applications.

Near-zero stiffness accelerometer with buckling of tunable electrothermal microbeams

Pre-shaped microbeams, curved or inclined, are widely used in MEMS for their interesting stiffness properties. These mechanisms allow a wide range of positive and negative stiffness tuning in their direction of motion. A mechanism of pre-shaped beams with opposite curvature, connected in a parallel configuration, can be electrothermally tuned to reach a near-zero or negative stiffness behavior at the as-fabricated position. The simple structure helps incorporate the tunable spring mechanism in different designs for accelerometers, even with different transduction technologies. The sensitivity of the accelerometer can be considerably increased or tuned for different applications by electrothermally changing the stiffness of the spring mechanism. Opposite inclined beams are implemented in a capacitive micromachined accelerometer. The measurements on fabricated prototypes showed more than 55 times gain in sensitivity compared to their initial sensitivity. The experiments showed promising results in enhancing the resolution of acceleration sensing and the potential to reach unprecedent performance in micromachined accelerometers.

Numerical and experimental investigation of negative stiffness beams and honeycomb structures

Metamaterial structures that exhibit negative stiffness (MSNS) behavior have shown the potential to be used as energy dissipation systems in different applications. Negative stiffness honeycomb structures consist of multiple pre-buckled beams that snapping through from one pre-buckled mode to another buckled mode, when subjected to transverse loads, displaying negative stiffness, and dissipating a significant portion of the input energy. This paper presents finite element models (FEMs) developed to investigate the performance of negative stiffness pre-buckled beams and honeycombs. The FEMs were validated using experimental results of single pre-buckled beams and honeycomb structures. The experimental testing of the pre-buckled beams was presented in the current study while the results of the honeycomb were available in the literature. The FEMs have the ability to capture the transition of the elements from one mode of buckling to another mode as well as the force thresholds, energy dissipation values, and residual displacements. The FEMs were extended to investigate the effect of different parameters such as boundary condition, apex height-to-beam thickness ratio, and constraining the second mode of buckling on the behavior of negative stiffness honeycombs.

Mechanical Response of Applying Different Parameters On Negative Stiffness Honeycomb Structure

It has become apparent that negative stiffness behavior may have potential applications in vibration isolation mechanisms, vibro-acoustic dampening materials, and mechanical switches. Unlike traditional honeycombs, due to these properties, a negative honeycomb can absorb substantial amounts of mechanical energy whilst maintaining a stable stress. The force threshold under displacement loading was investigated of three variables applied on different models of negative-stiffness honeycomb (NSH) structures. The three variables are material applied, honeycomb unit cell, and beam thickness of the negative honeycomb structure. Accordingly, nine models were developed, and the three varied materials were assigned repeatably to each model and then force threshold were studied after validating the model. The Finite element analysis (FEA) for formed model was validated and shows force value of 289 N with an error of 5% compared to the referenced model. In the 4- unit cell model, the highest force threshold of approximately 240 N was noticed during loading phase at the beam thickness of 19.05 mm for both nylon 11 and 12 material. Finally, the force threshold of 550 N during loading and unloading phases was observed for nylon 6/6 material at beam thickness of 19.05 mm. The results obtained confirm the negative stiffness behavior on the models and shows that the force threshold applied is reduced comparing to forces required in the conventional honeycombs models.

The negative stiffness electromagnetic tuned inerter damper for damping enhancement of stay cables

Long stay cables may be subject to vortex-and rain-wind-induced vibrations. To suppress the two types of vibrations simultaneously, integrating the advantages of negative stiffness and inerter-based damper, a new negative stiffness electromagnetic tuned inerter damper (NSETID) is proposed in this study. The equivalent mechanical model of the NSETID is presented according to its working principle. On this basis, non-dimensional analytical equations of the cable with NSETID are developed. Subsequently, the optimum parameters of the NSETID are identified by optimizing the damping ratio of the tuned cable mode. The effects of the cable sag and flexural stiffness on the control performance and optimum parameters of the NSETID are also taken into consideration. In addition, the benefits and damping enhancement of the NSETID for mitigating single-mode and multi-mode cable vibrations are illustrated by comparing with tuned inerter damper (TID), negative stiffness damper (NSD), and negative stiffness inerter damper (NSID). Finally, the damping enhancement mechanism of the NSETID for cable multi-mode is explored through its stiffness and damping properties. The results highlight the superior damping enhancement capabilities of the NSETID in mitigating single-mode and multi-mode cable vibrations. The better control performance of the NSETID can be attributed not only to its negative stiffness behavior but also to the adaptability of its equivalent damping and negative stiffness coefficients to the cable mode order.

A load cell with adjustable stiffness and zero offset tuning dedicated to electrical micro- and nanoprobing

The tasks of electrical micro- and nanoprobing require precision that goes beyond human perception and therefore, depend on sensors providing real-time feedback. Efforts towards automation in micro- and nanoprobing and emerging areas such as branched nanowire networks, has resulted in a growing number of applications where electrical probing without simultaneous force control is insufficient. This article presents the design of a novel mesoscale flexure-based load cell dedicated to micro- and nanoprobing. By integrating systems for stiffness adjustment and a zero offset tuning, the force-displacement characteristic of the device can be adapted to suit a wide range of applications, from the measurement of large forces up to 60 mN to a resolution as high as 10.1 nN, and even negative stiffness (bistable) behavior. By controlling the tuning during measurements, a virtual dynamic range of 6.38 ⋅ 106 is achievable, which is one order of magnitude greater than existing commercial products. We validated the results experimentally on a titanium alloy prototype fabricated by electrical discharge machining (EDM). Experimental and finite element results were also used to validate the analytical model of the load-cell. Additionally, the device allows for probe tip replacement, which is a significant advantage compared to existing MEMS load cells, and comprises a gravity compensation system to accommodate a wide range of commercially available probe tips.

Implications of Spatially Constrained Bipennate Topology on Fluidic Artificial Muscle Bundle Actuation

In this paper, we investigate the design of pennate topology fluidic artificial muscle bundles under spatial constraints. Soft fluidic actuators are of great interest to roboticists and engineers, due to their potential for inherent compliance and safe human–robot interaction. McKibben fluidic artificial muscles are an especially attractive type of soft fluidic actuator, due to their high force-to-weight ratio, inherent flexibility, inexpensive construction, and muscle-like force-contraction behavior. The examination of natural muscles has shown that those with pennate fiber topology can achieve higher output force per geometric cross-sectional area. Yet, this is not universally true for fluidic artificial muscle bundles, because the contraction and rotation behavior of individual actuator units (fibers) are both key factors contributing to situations where bipennate muscle topologies are advantageous, as compared to parallel muscle topologies. This paper analytically explores the implications of pennation angle on pennate fluidic artificial muscle bundle performance with spatial bounds. A method for muscle bundle parameterization as a function of desired bundle spatial envelope dimensions has been developed. An analysis of actuation performance metrics for bipennate and parallel topologies shows that bipennate artificial muscle bundles can be designed to amplify the muscle contraction, output force, stiffness, or work output capacity, as compared to a parallel bundle with the same envelope dimensions. In addition to quantifying the performance trade space associated with different pennate topologies, analyzing bundles with different fiber boundary conditions reveals how bipennate fluidic artificial muscle bundles can be designed for extensile motion and negative stiffness behaviors. This study, therefore, enables tailoring the muscle bundle parameters for custom compliant actuation applications.

Frequency-dependency/independency analysis of damping magnification effect provided by tuned inerter absorber and negative stiffness amplifying damper considering soil-structure interaction

As passive control techniques, both inerter device (ID) and negative stiffness device (NSD) produce forces that have a phase delay of π from common springs, i.e., negative stiffness behavior, when subjected to harmonic excitations. Thus, these two are both able to assist the motion of viscous damper in typical tuned inerter-based or negative stiffness-based absorbers, like tuned viscous mass damper (TVMD) or negative stiffness amplifying damper (NSAD), leading to a damping magnification effect. However, the negative stiffness value of NSD is frequency-independent; while that value of ID is frequency-dependent. In this regard, ID and NSD will perform differently due to structural frequency variation caused by soil-structure-interaction (SSI), which in turn further affects the energy dissipation capabilities and the seismic performance of TVMD and NSAD. Based on these effects, this paper systematically compares and discusses the optimal design and seismic performance of TVMD and NSAD from the point of SSI.

Real-time tunable negative stiffness mechanical metamaterial

Real-time tunable mechanical metamaterial is an emerging field attracting great attention. It enables a wide range of mechanical properties but is hard to design and fabricate due to its complexity. Here, a pneumatically actuated tunable negative stiffness mechanical metamaterial with real-time tunable properties is presented. Numerical simulations and experiments were both employed to investigate the proposed real-time tunable negative stiffness mechanical metamaterial. Research results demonstrate that multistage pattern-transformation can be realized through the pneumatic actuation. Uniaxial compression and vibration tests reveal the influence of the inner air pressure on the metamaterial’s basic mechanical properties and the vibration isolation performance, respectively. Additionally, real negative stiffness behavior and high energy absorption efficiency are achievable for the proposed metamaterial. This study can be a reference to relevant researches on tunable mechanical metamaterials, and presents a new path for designing real-time tunable negative stiffness mechanical metamaterials.

Simplified analysis of bilinear elastic systems exhibiting negative stiffness behavior

SummaryThis work studies the dynamics of the Negative Stiffness Bilinear Elastic (NSBE) oscillator. Such a mathematical idealization can be used to describe deformable rocking systems equipped with restraining tendons or with curved extensions of their bases. First, this paper establishes the characteristic quantities of the bilinear system to make it equivalent to the actual rocking structures. Then, it proceeds by proposing a simpler “equivalent” system that can be used to study the behavior of the NSBE. The equivalent system is not some linear elastic oscillator but a bilinear elastic system with zero stiffness of the second branch: the Zero Stiffness Bilinear Elastic (ZSBE) system. ZSBE is useful because it needs one parameter less than NSBE to be defined. Next, “Equal Displacement” and “Equal Energy” rules that provide estimates of the maximum displacement of the NSBE based on the response of the ZSBE are defined. The concept is similar to the RμΤ relations that provide estimates of the response of bilinear yielding systems based on the response of an equivalent linear elastic system, with one major difference: it does not resort to a linear elastic system but to the ZSBE. The proposed methodology is applied on the FEMA P695 ground motions scaled at three different levels. The results show that ZSBE is a good proxy of NSBE and, hence, indicate that an exhaustive study of the ZSBE is useful for the design of rocking structures.

A passive negative stiffness damper in series with a flexible support: Theoretical and experimental study

Dampers with negative stiffness behavior have been widely investigated for structural vibration control due to their superior performance. In practical applications, a support in series with these dampers might have some flexibility, that is, the stiffness is positive but finite. This paper investigates the effects of that flexibility on the performance of a passive negative stiffness damper (PNSD). First, a system consisting of a PNSD in series with a flexible support is presented, and the lower limit of the support stiffness needed to keep the system stable is derived. Second, a mechanical model for the proposed system is analyzed theoretically. An equivalent model of the damping force is established. The deformation of the flexible link is illustrated, followed by a demonstration of its effect on the hysteresis loops. Finally, an experimental investigation is performed to evaluate the performance of the PNSD support system. A device that contains an oil damper and two compressed springs is utilized to provide negative stiffness and Coulomb friction behavior. One of the set of springs is then incorporated to act as a positive but finite stiffness support in series with the device. Tests are conducted to assess the hysteretic behavior of the overall system. Theoretical and experimental results indicate that a range of equivalent negative stiffness and Coulomb friction level can be achieved simply, depending upon the positive stiffness of the flexible support in series with the PNSD.

A monolithic tunable symmetric bistable mechanism

A tunable symmetric bistable micro-machined mechanism with monolithic structure is presented. The mechanism consists of two sets of pre-shaped beams of similar properties and dimensions; however with one set is curved opposite to the other. The bistability is induced by creating a negative stiffness behavior at the initial position of the moving part to break its stability. This is possible at a certain electrothermal axial loading of both beam sets. The active bistability is symmetric with respect to the initial position. The distance between stable positions is tunable with the applied electrothermal voltages. The mechanism can be used as a tunable bi-directional actuator with different applied voltages. The design and concept are demonstrated based on finite-element simulations and experimental measurements.

Novel multi-stable mechanical metamaterials for trapping energy through shear deformation

A novel shear induced multi-stable mechanical metamaterial (SMMM), consisting of magnets systems in periodic arrangement, is proposed to trap mechanical energy from impact. The metamaterial's multi-stable and negative stiffness behaviors are determined by the repulsive force between magnets, and the input mechanical energy from impact is transformed to the magnetic potential energy and locked in the magnets systems. Through the experiment and theoretical methods, the basic mechanical properties, deformation mechanisms, and the energy trapping capacity of the metamaterial under end shear loading are investigated. Research results indicate that the metamaterials are reusable and rate independent. The metamaterial's mechanical properties are tailorable through the adjustment of the geometric parameters or the change of the magnets system's configuration. Moreover, in order to overcome the metamaterial's direction dependence, a concept, bi-directional SMMM, is proposed with a detailed design philosophy. To our knowledge, this work is first to design SMMM using the magnets system, and report the bi-directional SMMM.

Adaptive Negative Stiffness Device based nonconventional Tuned Mass Damper for seismic vibration control of tall buildings

Adaptive Negative Stiffness Device (ANSD) has been developed recently for passive vibration control of structures. ANSD emulates negative stiffness behavior (by exerting force toward the direction of motion) to a structure, so that, the combined structure-ANSD system shows smooth elasto-plastic force-deformation behavior with virtual yielding at target (“engaging”) displacement. The device has been studied analytically and its behavior has been demonstrated experimentally in simply supported bridge model and in Single Degree of Freedom (SDOF) structure considering limited input excitations. However, efficient and cost-effective exploitation of true potential of ANSD deserves further studies. With this being the eventual goal, in this study, we explore the performances of ANSD in seismic vibration control of a tall building by implementing a nonconventional Tuned Mass Damper (TMD) with drift control. In absence of a methodology for choice of ANSD parameters in literature, reported till date, we employ a tuning criteria for nonconventional TMD, on which the drift control is imposed as a constraint, given the special significance of drifts in tall buildings. The criteria of tuning and drift control are shown to be mutually conflicting and an algorithm is presented to address the same. Simplifying assumptions are made to develop the tuning criteria in presence of nonlinearity in ANSD and P−δ nonlinearity in tall building. A Reduced Order Modeling (ROM) approach for the building-ANSD system is developed to facilitate the algorithmic computations. Robustness of our design approach is shown under an acceptable level of detuning of TMD and near fault pulse type motions. Comparative assessment with a conventional design approach is presented to show the improvements. A step by step illustration of the developed methodology is briefly touched on in context of modern performance based approach. The study is a first step toward the possible utility of ANSD in controlling seismic vulnerability of tall buildings.

Refined damper design formula for a cable equipped with a positive or negative stiffness damper

Due to their high lateral flexibility and low inherent damping, stay cables are prone to dynamic excitations. Application of dampers to improve the energy dissipation capacity of stay cables and mitigate their excessive vibrations has been extensively studied, and design tools have been proposed to select the optimum damper size and predict the maximum achievable damping ratio of a cable-damper system. In this study, the effectiveness of external viscous dampers in controlling stay cable vibrations is investigated by considering the negative stiffness behavior of passive dampers. An analytical model is developed to include the damper stiffness effect for further refinement of existing damper design tools, of which the influence of cable sag, cable flexural stiffness, and damper support stiffness has already been considered. The performance of passive negative stiffness dampers (NSDs) and conventional zero or positive stiffness dampers (PSDs) is investigated in detail via parametric studies using the refined design formula. In particular, a criterion is defined for selecting the negative stiffness in NSD based on the stability limits. Two design examples are presented to illustrate the application of the proposed refined damper design tool to the selection of optimum damper size and evaluation of damper performance for a passive viscous PSD and NSD. Results show that compared with the conventional viscous dampers, a passive NSD demonstrates superior performance in stay cable vibration control. Results are also compared and verified with the numerical solution of the proposed analytical model.

Vibration energy harvesting from impulsive excitations via a bistable nonlinear attachment—Experimental study

Abstract A bistable nonlinear electromagnetic energy harvester coupled to an impulsively excited primary linear oscillator has been experimentally investigated. The design of the energy harvesting system was guided by a preliminary numerical study, which predicted a favorable dynamic regime for harvesting purposes. The harvesting system consisted of a lightweight permanent magnet, moving within a stationary coil, nonlinearly coupled to a grounded, weakly damped linear oscillator through a prebuckled, bistable slender steel beam which exhibited both cubic nonlinear and negative linear stiffness behaviors in transverse response along its weak axis. This paper provides the results of an experimental investigation, illustrating the capacity of the bistable element to dramatically enhance harvesting efficiency when subjected to broadband, low-amplitude vibration, executing large amplitude oscillations between the two stable equilibrium positions. Single and repeated impulses of varying amplitude applied to the LO are employed as system inputs, and it is shown, both experimentally and through numerical simulations, that robust harvesting efficiency is achieved primarily through periodic cross-well oscillations, particularly at low levels of input energy.

KDamper concept in seismic isolation of bridges with flexible piers

Contemporary seismic isolation systems for bridge applications provide (a) horizontal isolation from the effects of earthquake shaking, and (b) an energy dissipation mechanism to reduce displacements. Throughout the years many kinds of seismic isolation mechanisms have been developed, with two concepts being the most promising ones: the introduction of negative stiffness elements and the incorporation of an additional mass. The latter has led to the development of Tuned Mass Dampers (TMDs), engineering devices consisting of a mass, a spring and a viscous damper commonly used to suppress any undesirable vibrations induced by wind and earthquake loads. The negative stiffness behavior is primarily achieved by special mechanical designs involving conventional positive stiffness pre-stressed elastic mechanical elements, such as post-buckled beams, plates, shells and pre-compressed springs, arranged in appropriate geometrical configurations. Combining the beneficial characteristics of both concepts, a novel passive vibration isolation and damping concept is introduced, the KDamper. The KDamper is based on the optimal combination of appropriate stiffness elements, which include a negative stiffness element. The presence of the additional mass enables structural vibrating energy to be transferred from the structure to the device. The main advantage of the KDamper over other similar concepts including negative stiffness elements is that no reduction in the overall stiffness of the system is required.In this paper, an initial approach towards the implementation of the KDamper to the absorption of seismic excitation of structures is considered, by applying the KDamping concept to a typical single-pier bridge subjected to three different types of dynamic loading. The contribution of pier is taken into account during the analysis. A comparison with a non-isolated bridge with similar characteristics confirms that KDamper based seismic absorption designs can provide a promising alternative to the conventional seismic isolation bearings, offering numerous advantages, such as increased damping and simple technological implementations.

Vibration damping materials with enhanced loss due to microstructural nonlinearity

One conventional approach to attenuating structure-borne waves or reduce the ringdown time of modes in structural elements is to attach damping layers or patches to vibrating components. Common patch and layer materials are specially formulated polymers or engineered polymeric composites that demonstrate elevated viscoelastic loss factors in the frequency range of interest. Recent research has shown that small volume fractions of negative stiffness inclusions embedded in a lossy material generate effective loss factors that exceed that of the host material. The ability to generate negative stiffness behavior, however, is often the result of nonlinear inclusion material response. Presented here is a multiscale model of a particulate composite material consisting of a nearly incompressible host material containing small-scale heterogeneities with a nonlinear elastic stress-strain response. We investigate the nonlinear dynamic behavior of the heterogeneous medium for small harmonic perturbations about several pre-strain states to demonstrate the influence of the microscale dynamic response on the macroscopically observable mechanical loss. Of primary interest is the energy dissipation capabilities of the composite which can be tuned using inclusion pre-strain. Loss for composites with nonlinear inclusions is compared to conventional composites containing air voids or steel inclusions. [Work supported by ONR.]

Design and analysis of a small-scale magnetically levitated energy harvester utilizing oblique mechanical springs

In this work, we present an electromagnetic, vibration-powered energy harvester utilizing hardening and negative stiffness mechanisms simultaneously. Since most vibrational energy sources’ frequency–response spectrum are broadband, the energy harvester is designed to generate electrical power across a wider frequency range. This nonlinear energy harvester consists of a magnetic mass in-between two stationary magnets. However, instead of using a guiding structure or tight-fit container, oblique mechanical springs are used to align the moving magnet and eliminate Coulomb dry friction. The hardening effect of the magnetic springs increases the resonant frequency of the system, and the negative stiffness behavior of the mechanical springs improves the harvester’s response towards the lower frequency spectrum. A nonlinear dynamic model of the device is developed using conservation laws, and a proof-of-concept prototype is fabricated. Simulations and experiments show good agreement and exhibit broad frequency spectra. For a base excitation of 0.75 g, the proposed prototype generates a peak voltage and normalized power density of approximately 2.95 V and 0.136 mW/cm3g2, respectively. Although the normalized power density is not optimized, the prototype performs well with respect to other state-of-the-art electromagnetic energy harvester designs. Optimization of the proposed hand-held energy harvester prototype allows for the replacement of chemical batteries used in wireless sensor networks and mobile applications.

Experimental Shake Table Testing of an Adaptive Passive Negative Stiffness Device within a Highway Bridge Model

The implementation of a mechanical negative stiffness device (NSD) within a reduced-scale highway bridge model and its performance under seismic loading conditions is evaluated via shaking table tests. Four different isolation system configurations are considered: isolated bridge (IB), IB with viscous dampers, IB with NSDs, and IB with viscous dampers and NSDs. In addition, two bridge pier configurations were considered: one with flexible piers (mimicking a middle span of a multi-span bridge) and one with braced piers (mimicking a single span bridge supported on abutments). The main feature of the NSD is a large pre-compressed spring, which can push the structure away from its initial undeformed position and thus induce negative stiffness behavior. The experimental results clearly demonstrate the effectiveness of the NSDs in limiting the seismic response of the bridge and provide validation of numerical simulation results wherein numerical models of the bridge model components were calibrated via system identification testing.