High Static Low Dynamic Stiffness Research Articles

Integrated vibration isolation and energy harvesting via a bistable piezo-composite plate

In passive vibration isolation, a bistable composite plate can be used to generate high-static low-dynamic stiffness, and in vibratory energy harvesting, when used with piezoelectric film flexing, the bistable composite plate can enhance the performance via snap-through motion. In the present work, we designed a bistable piezo-composite plate for both vibration isolation and energy harvesting. The analytical model for performance prediction is derived from the virtual work principle and the composite elasticity. Displacement transmissibility and output voltage generation were analytically determined for different mechanical and electrical parameters. The results illustrate that the bistable piezo-composite plate improves the feasibility and effectiveness of the integration design. Hardening and softening nonlinear phenomena coexist in both the displacement transmissibility and the output voltage plot. Approximate analytical solutions and numerical simulations are in good agreement. Both analytical and numerical results demonstrate that, at a certain frequency bands, enhanced energy harvesting is accompanied by a vibration transmissibility reduction. Compared with a linear system with the bistable piezo-composite plate removed, it achieved a reduction in the displacement transmissibility of about 13 dB at 100 Hz, simultaneously, and produced a considerable output voltage of about 0.05 V.

High-Static–Low-Dynamic Stiffness Isolator With Tunable Electromagnetic Mechanism

A high-static-low-dynamic stiffness (HSLDS) vibration isolator based on a tunable nesting-type electromagnetic negative stiffness (NS) mechanism is presented. The stiffness characteristics of the HSLDS system can be tuned by regulating the current in the coil. When the current is zero, the HSLDS system may degenerate into a passive isolator. The distinctive features of this proposed isolation system are demonstrated from two aspects: 1) In contrast to passive isolators, the proposed isolator can further widen the bandwidth of vibration isolation and improve the isolation performance near the natural frequency of the passive system; and 2) compared with certain other electromagnetic isolation systems, the proposed system has a more compact structure, heavier load capacity, and higher NS-generating efficiency. The proposed electromagnetic NS mechanism consists of four permanent magnets and four coil windings. Analytical models of the electromagnetic force and the displacement transmissibility are established. To select an appropriate structural parameter for the prototype, the effects of geometric parameters on transmissibility are discussed from a theoretical viewpoint. The vibration isolation performance of the proposed isolator system is verified, and a tuning strategy is investigated through experiments.

Tuned mass damper and high static low dynamic stiffness isolator for vibration reduction of beam structure

In this work, the combination effect of tuned mass damper and high static low dynamic stiffness (HSLDS) isolator is investigated in reducing the vibration amplitude of Euler–Bernoulli beam with a nonlinear attachment. The performance of the absorber is studied in two cases; the first case, HSLDS isolator is one degree of freedom and the second case, two degree of freedom isolator is combined of HSLDS isolator and tuned mass damper absorber. By comparing the performance of these two isolators, it is revealed the two degree of freedom isolator has much better performance in direct force excitation and also improves the system performance in the base excitation. This isolator reduces the system amplitude at all frequencies, especially ultra-low frequencies, which is the main advantage to this isolator with respect to other isolators and reduces the natural frequency until the phenomenon of resonance occurs at a lower frequency. Moreover, decreasing the natural frequency increases the damping and in quasi zero stiffness and negative stiffness structure, the system has supercritical damping. This isolator is studied for positive, quasi zero and negative stiffness. The results show that the system with quasi zero stiffness has the best performances. Also, by increasing the excitation amplitude, the isolator loses its effectiveness.

The design of negative stiffness spring for precision vibration isolation using axially magnetized permanent magnet rings

A negative stiffness element is always employed to generate high-static–low-dynamic stiffness characteristic of the vibration isolator, reduce the resonance frequency of the isolator, and improve the vibration isolation performance under low and ultra-low frequency excitation. In this paper, a new compact negative stiffness permanent magnetic spring (NSPMS) that is composed of two axial-magnetized permanent magnetic rings is proposed. An analytical expression of magnetic negative stiffness of the NSPMS is deduced by using the Coulombian model. After analyzing the effect of air-gap width, air-gap position, height difference between the inner ring and outer ring on the negative stiffness characteristic, a design procedure is proposed to realize the negative stiffness characteristic with a global minimum linear component and uniformity stiffness near the equilibrium position. Finally, an experimental prototype is developed to validate the effectiveness of the NSPMS. The experimental results show that combining a vibration isolator with the NSPMS in parallel can lower the natural frequency and improve the isolation performance of the isolator.

A Novel Passive Quasi-Zero Stiffness Isolator for Ultra-Precision Measurement Systems

In the paper, a novel passive vibration isolator is proposed for an ultra-precision sensing system, utilizing a quasi-zero stiffness (QZS) mechanism. The QZS mechanism implements the high static low dynamic stiffness, which effectively reduces the dynamic force transmission while minimizing the static deflection under the natural frequency of conventional passive isolator. Furthermore, it does not need any electric components; the mechanism is suitable for the ultra-precision sensing systems measuring extremely weak electromagnetic fields. However, nonlinear stiffness and hysteresis caused from the negative stiffness elements degrade the system performance. A vertical spring with a pre-tension and eight horizontal plate springs with nonlinear buckling characteristics constitute the proposed system to solve these problems. The mathematical model compares the negative stiffness design with previous QZS research. The buckled plate spring with ball joint design reduces stiffness variation. Transmissibility of the proposed system for low frequency range is investigated experimentally.

Mathematical modeling and analysis of a meta-plate for very low-frequency band gap

• A meta-plate with high-static-low-dynamic stiffness local resonators is proposed. • Mathematical modeling of the meta-plate is established and solved by plane-wave expansion method. • The band gaps are predicted analytically and validated by numerical simulations. • The meta-plate with HSLD resonators can generate very low-frequency band gaps. Aiming to attenuate very low-frequency flexural wave in a thin plate, this paper proposes a meta-plate model by periodically attaching high-static-low-dynamic-stiffness (HSLDS) resonators onto the thin plate. The HSLDS resonator consists of a linear spring with a negative stiffness mechanism (NSM) in parallel, and thus its stiffness can be adjusted to any low values within a range from zero to the stiffness of the linear spring. Using the plane-wave expansion (PWE) method, and considering the linearized stiffness of the resonator, the dispersion relation of the meta-plate can be derived and the band structure can be obtained. A dynamic model of the meta-plate with the original nonlinear stiffness under external excitations is also established. The numerical simulations are carried out by the Galerkin method to evaluate the band structure and study the propagation of the flexural wave along the meta-plate with different azimuth angles. The analytical results exhibit good agreement with the numerical ones, which indicates that the proposed meta-plate can create a very low-frequency band gap.

Lower band gaps of longitudinal wave in a one-dimensional periodic rod by exploiting geometrical nonlinearity

Abstract In this paper, a local resonant (LR) rod with high-static-low-dynamic-stiffness (HSLDS) resonators is proposed to create a very low-frequency band gap for longitudinal wave propagation along the rod. The HSLDS resonator is devised by employing geometrical nonlinearity, and attached onto a periodic rod composed of rigid frames and rubbers to construct a LR rod. To reveal the band gaps, the LR rod is modeled as a lumped mass-spring chain. The effects of damping and nonlinearity of the HSLDS resonator on the dispersion relation is studied analytically by the Harmonic Balance method. The analytical results indicate that the damping mainly affects the width and depth of the band gap, while the nonlinearity can influence the central frequency and width of the band gap. In addition, both multi-body dynamic analyses and numerical simulations are conducted to predict longitudinal wave propagation along the LR rod, and thus to validate the very low-frequency band gap. The results show that the periodic rod with HSLDS resonators can create a very low-frequency band gap for longitudinal waves propagating along the rod.

Effect of negative stiffness mechanism in a vibration isolator with asymmetric and high-static-low-dynamic stiffness

Abstract In this paper, a high-static-low-dynamic stiffness (HSLDS) vibration isolator with a novel parabolic-cam-roller negative stiffness mechanism is proposed to study the effectiveness of negative stiffness on asymmetric spring supporting structures. Based on the physical model, the intrinsic geometrical nonlinearity of the isolator is analyzed and the differential equation of motion is derived by the generalized Lagrange's equation. Here, the multi-term incremental harmonic balance method (MT-IHBM) is employed and extended to capture both primary and subharmonic resonances in nonlinear response. By means of arc-length increment, the adaptability and reliability of this method are enhanced under strong nonlinear vibration. Vibration transmissibility and frequency response relationships of various frequency components are demonstrated in order to better explain the properties of asymmetry vibration. The analysis results indicate that the performance of the isolation system is affected by different parametric excitation. Under some parameters, the introduction of negative stiffness will cause obvious 1/2 subharmonic resonance. The existence of the bias term in response makes the peak and frequency of resonances shift vertically and horizontally, respectively. Due to the asymmetry, the proposed HSLDS isolator does not present quasi-zero stiffness at the static equilibrium position, but it has superior ability to suppress vibration. Meanwhile, comparisons show that adding negative stiffness mechanism can still significantly improve the vibration isolation performance when a supporting structure has the characteristic of asymmetric stiffness.

A nonlinear resonator with inertial amplification for very low-frequency flexural wave attenuations in beams

Although elastic metamaterials in a subwavelength scale can control macroscopic waves, it is still a big challenge to attenuate elastic waves at very low frequency (a few tens Hz). The main contribution of this paper is to develop a high-static-low-dynamic-stiffness (HSLDS) resonator with an inertial amplification mechanism (IAM), which is able to create a much lower band gap than a pure HSLDS resonator. The nonlinear characteristics of a locally resonant (LR) beam attached with such new resonators are also explored. The band gap of this LR-IAM beam is revealed by employing transfer matrix method and validated by numerical simulations using Galerkin discretization. It is shown that a very low-frequency band gap can be formed by tuning the net stiffness of the resonator towards an ultra-low value. In addition, the nonlinearity, arising from the restoring force of the resonator, the damping force and effective inertia of the IAM, gives rise to an intriguing feature of amplitude-dependent wave attenuation, which could potentially act as a switch or filter to manipulate flexural waves.

Multi-layer electromagnetic spring with tunable negative stiffness for semi-active vibration isolation

Abstract Semi-active systems have demonstrated excellent performance in vibration isolation, However, the tunable range of conventional variable stiffness elements is limited by the mount stiffness required to support a load. To improve the performance of low-frequency vibration isolation, a novel compact and contactless multi-layer electromagnetic spring (MES) with tunable negative stiffness is proposed in this paper. The negative stiffness is generated by the electromagnetic force between the coils and the magnets and can be tuned online by controlling the current. Based on a stiffness analysis model, a design procedure for configuring the MES is presented to expand the region where the stiffness is negative and to increase its tunable range. As a result, with only one type of magnet and coil, various negative stiffness elements with different stiffness characteristics can be manufactured, thereby reducing costs. To verify the dynamic behavior of the MES, a tunable high-static-low-dynamic stiffness (HSLDS) vibration isolator based on the MES was modeled and evaluated by applying it to a lidar. The experimental results demonstrate the following: the MES tuned the isolator online and broadened the isolation frequency band such that the isolator attenuated the vibration of the lidar, and it significantly improved the lidar modeling performance.

Low frequency vibration control of railway vehicles based on a high static low dynamic stiffness dynamic vibration absorber

In order to control the low frequency vibration of railway vehicles, a vertical two degrees of freedom (2DOF) low frequency dynamic vibration absorber (DVA) based on acceleration is proposed. Parameters of the dynamic vibration absorber are put forth to control the low frequency vibration of car body bouncing and pitching. Next, the acceleration power spectrum density (PSD) and ride quality of the car body are calculated based on the pseudo excitation method (PEM) and covariance algorithm, respectively. According to the requirement of 2DOF low frequency DVA, the isolators with high static low dynamic stiffness (HSLDS) are designed. A high-speed train dynamic model containing HSLDS isolators is established to validate the effects on the car body vibration. The results reveal that the 2D low frequency DVA can significantly reduce the vibration of the car body bouncing and pitching. Thus, the ride quality of the vehicle is increased, and passenger comfort is improved.

Design of a nonlinear energy harvester based on high static low dynamic stiffness for low frequency random vibrations

Most of the vibration sources, which could be used for energy harvesting, are vertical and very low frequency (e.g. human movement, vehicle transportation, etc…). Under those conditions and while considering the size constraints, usual vibration energy harvesters (VEHs) underperform due to the combined effect of gravity, mechanical damping and the necessity to be tuned to very low frequencies. In order to overcome these limitations, the concept of High Static Low Dynamic (HSLD) stiffness is proposed and validated for VEHs. To do so, a theoretical study is performed to optimize the electromagnetic structure and design a folded-beam suspension of the moving mass allowing a low mechanical damping. This leads to the design of an original VEH, which is experimentally characterized under harmonic and white noise excitations. Compared to the literature, it demonstrated high performances with up to 41.3 mWcm−3 g−2 of normalized harvested power at an operating low frequency of 5.3 Hz.

High-static-low-dynamic-stiffness vibration isolation enhanced by damping nonlinearity

High-static-low-dynamic-stiffness (HSLDS) nonlinear isolators have proven to have an advantage over linear isolators, because HSLDS nonlinear isolators allow low-frequency vibration isolation without compromising the static stiffness. Previously, these isolators have generally been assumed to have linear viscous damping, degrading the performance of the isolator at high frequencies. An alternative is to use nonlinear damping, where the nonlinear behavior is achieved by configuring linear dampers so they are orthogonally aligned to the excitation direction. This report compares the performances of single-stage and two-stage isolators with this type of damping with the corresponding isolators containing only linear viscous damping. The results show that both isolators with linear viscous damping and nonlinear damping reduce the transmissibility around the resonance frequencies, but the results show that the isolators with nonlinear damping perform better at high frequencies.

Bifurcation and singularity analysis of HSLDS vibration isolation system with elastic base

In order to reveal the bifurcation mechanism and optimize the system design for high-static-low-dynamic-stiffness (HSLDS) vibration isolation system (VIS) with elastic base, the local bifurcation analyses both in unfolding parameter space and physical parameter space were carried out theoretically and numerically. Firstly, the restoring force of the HSLDS-VIS was approximated to linear and cubic stiffness by applying the Maclaurin series expansion and the motion equations of HSLDS-VIS with elastic base were established. Subsequently, the motion equations of HSLDS-VIS with elastic base were formulated to transform the system into a standard form and the averaging method was applied to obtain the single-variable bifurcation equation for the HSLDS-VIS with elastic base in case of primary resonance and 1:2 internal resonance. Furthermore, the transition sets and bifurcation diagrams in the unfolding parameter space were studied by means of singularity theory. Finally, for the engineering application, the transition sets were transferred back to the physical parameter space, thus to obtain the bifurcation diagrams of the amplitude with respect to the external force. The numerical simulation results show that the local bifurcations of HSLDS-VIS with elastic base in case of 1:2 internal resonance are considerable complex and need to be analyzed in six two-parameters spaces, meanwhile, the necessary condition of multiple solutions lies in some physical parameters, which can provide a theoretical basis and reference for design and application of the HSLDS-VIS with elastic base.

A high-static–low-dynamic-stiffness vibration isolator with the auxiliary system

High-static–low-dynamic-stiffness (HSLDS) vibration isolators seek to widen the frequency range of isolation by decreasing the dynamic stiffness so as to reduce the natural frequency, while maintaining the same static displacement as equivalent linear isolators. However, in many cases especially under light damping or large excitations, the peak transmissibility is very large and the hardening nonlinearity causes the transmissibility curve to bend to the right seriously or in other words causes the jump phenomenon, resulting in a greatly reduced isolation region. In this paper, an auxiliary system is added to the HSLDS isolator to overcome these disadvantages, with the static displacement of the isolation object remaining unchanged. Coefficient-varying harmonic balance method is proposed in this paper to find the dynamic response and most importantly analyze the stability of the steady-state response. The isolation performance of the HSLDS-AS isolator, which is evaluated by displacement transmissibility, is compared with that of the equivalent HSLDS isolator. The effects of system parameters on the isolation performance are investigated. It is shown that the auxiliary system can lower the peak transmissibility and eliminate the jump phenomenon, resulting in a wide isolation region, and the HSLDS-AS isolator has strong designability with many parameters tunable.

High performances low frequency vibration energy harvester with HSLD stiffness

The contributions put forth by this paper first include the design and testing of a new high performance electromagnetic vibration energy harvester based on an optimized electromagnetic structure and a friction-less compliant folded-beam suspension. Then, High Static Low Dynamic (HSLD) stiffness principle is used to reduce the resonance frequency of the harvester while maintaining small static displacement.

Modeling and analysis of a high-static-low-dynamic stiffness vibration isolator with experimental investigation

In order to attenuate low-frequency vibration, a novel nonlinear vibration isolator with high-static-low-dynamic stiffness (HSLDS) is developed in this paper by combining the negative stiffness corrector in parallel with a vertical linear spring. The force and stiffness characteristics are first derived by the static analysis. Then, the displacement transmissibility of the HSLDS system is obtained to evaluate the isolation performance using the harmonic balance method. The parametric analysis shows that the proposed HSLDS system can outperform the equivalent linear one in some aspects. Besides, the initial isolation frequency is defined and further investigated with the purpose of providing some useful guidelines for choosing parameter combinations conveniently. Finally, a prototype is developed and the experimental test is conducted to verify the isolation performance of the proposed HSLDS system.

Enhanced isolation performance of a high-static–low-dynamic stiffness isolator with geometric nonlinear damping

To enhance the low-frequency vibration isolation performance of the high-static–low-dynamic stiffness (HSLDS) isolator, a novel design of the geometric nonlinear damping (GND) comprising semi-active electromagnetic shunt damping is proposed. The GND is dependent on the vibration displacement and velocity, which can make the HSLDS isolator attain different damping characteristics in different frequency bands. Firstly, the configuration of the HSLDS isolator assembled with GND is presented, and then the restoring force, stiffness, and damping are derived. The dynamics of the mount under both base and force excitations are investigated based on the harmonic balance method, which are then verified by numerical simulations. After that, the effects of GND on the displacement and force transmissibility are studied, and the excellent performance caused by GND is analyzed based on the equivalent viscous damping mechanism. Finally, the comparison between the GND and cubic nonlinear damping is performed. The results demonstrate that the HSLDS isolator assembled with GND can realize the requirements of an isolation system under both base and force excitations of broadband vibration isolation performance and a low resonance peak with the high-frequency attenuation unaffected. Moreover, the GND outperforms the linear damping no matter the base excitation or force excitation is applied. For base excitation, the GND exhibits some desirable properties that the cubic nonlinear damping does not have at high frequencies.

Beneficial stiffness design of a high-static-low-dynamic-stiffness vibration isolator based on static and dynamic analysis

A high-static-low-dynamic-stiffness (HSLDS) isolator has good application prospects in low-frequency vibration and micro-vibration isolation. The nonlinear stiffness of the vibration isolation system strongly influences its dynamic responses and vibration isolation performance. In this paper, for a HSLDS isolator model, a stiffness range parameter d and a static equilibrium position stiffness Ksep are introduced to indicate the different stiffnesses of the nonlinear isolator. The averaging method is employed to solve the nonlinear dynamic equation of the proposed system at steady state for each excitation frequency. The effects of the parameters d and Ksep on the dynamic responses and isolation effectiveness of the HSLDS isolator are clearly analysed. The results show that an increase in d and a reduction in Ksep improve the isolation performance of the system. Considering the application of the HSLDS isolator in micro-vibration environments, the beneficial stiffness design under various excitation amplitudes and damping conditions is discussed on the basis of the limited maximum response amplitude. Several useful guidelines for the stiffness design of the HSLDS isolator are developed, which help to improve the vibration isolation performance of the system. The results provide useful insight for the design, analysis and application of HSLDS vibration isolators.

Analytical study of the low frequency multi-direction isolator with high-static-low-dynamic stiffness struts and spatial pendulum

By now, existing isolators with quasi-zero stiffness (QZS) or high-static-low-dynamic stiffness (HSLDS) have been designed to migrate the vibration in single direction, but the vibration isolation in multiple directions is more expected. Hence a passive low frequency multi-direction isolator based on the HSLDS struts and spatial pendulum is developed in this study. The remarkable feature of the proposed mount is that the dynamic coupling is so weak that the responses in different directions do not interact with each other, which is desirable for low frequency vibration isolation in multiple directions. Firstly, the design of the low frequency multi-direction isolator is presented. With the Lagrange principle, the equations of motion of the isolator are established, and then solved via harmonic balance method to obtain the amplitude-frequency relationships. The vibration isolation performance is evaluated in terms of the displacement transmissibility, and the parameter studies are carried out as well. The results show that: (a) the isolator can realize the high-static-low-dynamic stiffness in the vertical direction and quasi-zero stiffness in the other two horizontal directions; (b) with the spatial ball hinges, the displacement-velocity-dependent nonlinear damping is introduced in two horizontal directions, which is the function of the displacement and velocity; (c) compared with the linear counterpart, the combination of the HSLDS struts and spatial pendulum enables the proposed passive isolator to achieve much better low frequency vibration isolation performance in multiple directions, especially in the vertical direction.