High-static-low-dynamic Stiffness Isolator Research Articles

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.

Nonlinear dynamical system identification using the sparse regression and separable least squares methods

This paper proposes a novel nonlinear dynamical system identification method based on the sparse regression algorithm and the separable least squares method. To effectively avoid solving the second derivative of the displacement signal and reduce the effect of noise, the Duhamel's integral is adopted to represent the dynamic relationship between the system input and output. In the expression form of Duhamel's integral, nonlinear dynamical system identification can be cast as a separable least squares problem. Thus, the separable least squares method is leveraged to separately identify the parameters of the linear subsystem and the coefficients corresponding to nonlinearities among the nonlinear dynamical system. During the identification process of nonlinear restoring forces, one complete set of nonlinear basis functions are used to represent the nonlinear restoring forces. Not all the candidate nonlinear terms are contributing, however, thus the sparse regression algorithm is adopted to select the actual contributing nonlinear components in the candidate nonlinear terms and eliminate the non-contributing nonlinear components, and then the corresponding parameters of contributing nonlinear components are estimated by the unbiased least squares method. Finally, one RKHS (Reproducing Kernel Hilbert Space)-based non-parametric de-noise method is further proposed to reduce the noise in the vibration displacement and obtain the noise-reduced velocity from the displacement signal. The numerical simulation about the identification of the rotating blade-casing system and the dynamic experiment of the HSLDS (high-static-low-dynamic stiffness) isolator system verify the effectiveness of the new identification method for nonlinear dynamical systems proposed in this paper.

Tunable negative stiffness spring using maxwell normal stress

A vibration isolator based on a tunable negative stiffness mechanism combines the advantages of high-static-low-dynamic stiffness (HSLDS) isolators to expand the isolation frequency band and variable stiffness isolators to suppress resonance. In this paper, a novel tunable negative stiffness spring using Maxwell normal stress (SMNS) is proposed. The stiffness tunable range and energy utilization efficiency are greatly improved due to the newly designed magnetic circuit. Moreover, the electromagnetic negative stiffness device has the advantages of no friction, no backlash, compact structure and easy control. An analytical model of the electromagnetic force is built based on magnetic circuit analysis, and the parameter analysis is performed. An HSLDS isolator is constructed by connecting the SMNS in parallel with a linear isolator. The stiffness and vibration isolation performance are measured. The experimental results show that the SMNS produces an online tunable negative stiffness, which expands the isolation bandwidth and significantly improves the vibration isolation performance.

Analytical and experimental investigation of a high-static-low-dynamic stiffness isolator with cam-roller-spring mechanism

A high-static-low-dynamic stiffness (HSLDS) isolator using cam-roller-spring mechanism with a specially designed cam profile is proposed in this study. The proposed HSLDS isolator is mainly composed of a cam-roller mechanism (negative mechanism) and a vertical spring (positive mechanism). Different from other studies, the cam profile in this paper can be designed specially to accommodate different working requirements. Firstly, the cam design theory is described and a cam whose force-displacement variation is a piecewise linear function is established. After that, the dynamic response of the HSLDS isolator under base excitation is acquired by the Averaging Method, at the same time, the stability and parameters effect of the dynamic system are discussed. Finally, an experimental test system is set up to verify the isolation performance of the HSLDS isolator and its corresponding linear isolator under base excitation. The results show that the HSLDS isolator starts to have isolation effect at 2.36 Hz, while the linear isolator is 3.20 Hz. Meanwhile, the resonance amplitude of the HSLDS isolator is smaller than the linear isolator. The proposed HSLDS isolator could achieve a better vibration isolation performance than the linear isolator, especially in the low-frequency range.

Design and Analysis of a High-Static-Low-Dynamic Stiffness Isolator Using the Cam-Roller-Spring Mechanism

Abstract This paper presents a new design of a high-static-low-dynamic stiffness (HSLDS) isolator with an adjustable cam profile. The interaction force between the cam and roller provides the negative stiffness force and the linear spring provides the positive stiffness force in the HSLDS isolator. Unlike previous studies, the cam profile in this paper can be individually designed to meet different working conditions. Firstly, the harmonic balance method is used to acquire the dynamic response of the HSLDS isolator. Then, the effects of the damping ratio, stiffness ratio, and external force amplitude on the frequency response amplitude and force transmissibility are discussed. Finally, the frequency responses of four designed nonlinear HSLDS isolators and a linear isolator are acquired by the numerical method. The results show that the nonlinear isolator begins to achieve vibration isolation at 0.11 Hz and the linear one is 8.9 Hz. The proposed HSLDS isolator realizes lower vibration isolation frequency than the linear isolator.

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.

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 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.

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.

Characteristic analysis of vibration isolation system based on high-static-low-dynamic stiffness

The purpose of this study is to investigate the characteristics of vibration isolation system with a single degree-of-freedom (SDOF) and a two-degree-of-freedom (2DOF) respectively based on the high-static-low-dynamic-stiffness (HSLDS). This model consists of a simple configuration connecting a vertical spring and a pair of oblique springs. The restoring force of the isolation system is approximated to linear and cubic stiffness by applying the Maclaurin series expansion. The dynamic equations of the SDOF and 2DOF are established for the harmonic force excitation. The frequency-amplitude response equation of the SDOF is obtained by employing the harmonic balance method (HBM) and is demonstrated in the classical Runge-Kutta method. The solution stability is ensured by applying the Floquet theory. Effects on the frequency response curves (FRCs) for the damping ratio and excitation amplitude are explored and discussed. The force transmissibility (FT) is defined to evaluate the vibration suppression capability. Effects on the FT of the SDOF and 2DOF for the excitation amplitude, mass ratio, and damping ratio are investigated. An experimental investigation of the SDOF is carried out to evaluate the actual attenuation performance in comparison with the equivalent linear system (ELS). The simulation and experimental results show that the HSLDS system with harmonic force excitation demonstrates hardening stiffness with multi-valued solutions. The occurrence of jump phenomenon is observed and explained by the stiffness variation. The system response and resonance frequency are affected by the excitation amplitude and damping ratio. The HSLDS system outperforms the ELS in a low frequency range if an appropriate mass is mounted. It is excited by a proper force and owns a suitable damper, which offers a theoretical guidance for the design and application of a novel HSLDS isolator.

Vibration control of a continuous rotating shaft employing high-static low-dynamic stiffness isolators

In this paper, the effects of high-static low-dynamic stiffness (HSLDS) isolators on the supports of a continuous rotating shaft for vibration control of a rotary system under mass eccentricity force are investigated. The rotating shaft is modeled using the Euler–Bernoulli beam theory. HSLDS isolators have a linear damping and linear and nonlinear (cubic) equivalent stiffness. Isolators are positioned on the supports of the rotating shaft, so that their forces are applied in radial directions. Equations of motion are extracted using the extended Hamilton principle and they are analyzed using the multiple scale method; then, the steady-state solutions and stability are studied. The effects of variations in linear and nonlinear parameters of the isolators on the static load bearing, resonant peak, frequency band of isolation and hardening nonlinearity are considered, in order to design an appropriate HSLDS isolator and to set its parameters in an optimal way. Investigating the effects of the cubic stiffness and damping values on bifurcations of the system, one may observe that inappropriate setting of these parameters causes strong or weak nonlinearity in the system and, consequently, HSLDS isolators perform less effectively than a linear one does. Then, the results are verified through analyzing the time history of the rotary system under study.

A planar shock isolation system with high-static-low-dynamic-stiffness characteristic based on cables

In this paper, a simple and designable shock isolation system with ideal high-static-low-dynamic-stiffness (HSLDS) is proposed, which is intended for the horizontal plane shock isolation application. In this system, the isolated object is suspended by several bearing cables and constrained by a number of uniformly distributed pretensioned cables in the horizontal plane, where the low dynamic stiffness of the system is main controlled by the pretension of the planar cables, whilst the high static stiffness is determined by the axial stiffness of the planar cables and their geometric settings. To obtain the HSLDS characteristic of the system, a brief theoretical description of the relationship between the restoring force and displacement is derived. By obtaining the three-order Taylor expansion with sufficient accuracy of the restoring force, influence of planar cable parameters on the low dynamic and high static stiffness is thus given, therefore, the required HSLDS isolator can be easily designed by adjusting the planar cable length, pretension and tensile stiffness. Finally, the isotropy characteristic of the restoring force of the system with different numbers of planar cables is investigated. To evaluate the performance of the system, a rigid isolated object and flexible cables coupling simulation model considering the contacts of the system is established by using multibody dynamics approach. In this model, flexible cables are simulated by 3-node cable element based on the absolute nodal coordinate formulation; the contact between cable and isolated object is simulated based on Hertz contact theory. Finally, the time-domain shock excitation is converted from the design shock spectrum on the basis of BV043/85 criterion. The design procedure of this isolator and some useful guidelines for choosing cable parameters are presented. In addition, a summary about the performance of the isolators with different numbers of cables shocking in an arbitrary direction is given in the conclusion.

Effects of stiffness and load imperfection on the isolation performance of a high-static-low-dynamic-stiffness non-linear isolator under base displacement excitation

The dynamic characteristics of a high-static-low-dynamic-stiffness (HSLDS) non-linear isolator, built by combining an Euler beams formed negative stiffness corrector and a traditional linear isolator are investigated. The system imperfection caused by detuning of the stiffness and the load are considered. The frequency response curves (FRCs) for the non-linear isolator with load and stiffness imperfection under base displacement excitation are calculated by using Harmonic Balance Method (HBM). The stability is studied by the Floquet theory. The effect of the imperfection on the dynamic response as well as the displacement transmissibility of the non-linear isolator is explored and discussed. The results show that both the stiffness and load imperfection can affect the performance of the non-linear isolator significantly. The HSLDS isolator with load and stiffness imperfection for different base excitation amplitudes can demonstrate pure softening, softening-to-hardening and pure hardening characteristics with multi-valued solutions. The occurrence of jump phenomena is observed and explained by the stiffness variation. When the isolator is subjected to load imperfection, keeping a small positive stiffness, rather than making the minimum dynamic stiffness as zero, is preferred in order to obtain the best isolation performance. The performance of the non-linear isolator can outperform the linear one provided that the base displacement excitation is not too large. The non-linear system may undergo unbounded responses for very large base displacement excitation.

Effect of the system imperfections on the dynamic response of a high-static-low-dynamic stiffness vibration isolator

The dynamic response of a high-static-low-dynamic stiffness (HSLDS) isolator formed by parallelly connecting a negative stiffness corrector which uses compressed Euler beams to a linear isolator is investigated in this study. Considering stiffness and load imperfections, the resonance frequency and response of the proposed isolator are obtained by employing harmonic balance method. The HSLDS isolator with quasi-zero stiffness characteristics can offer the lowest resonance frequency provided that there is only stiffness or load imperfection. If load imperfection always exists, there is no need to make the stiffness to zero since it cannot provide the lowest resonance frequency any longer. The reason for this unusual phenomenon is given. The dynamic response will exhibit softening, hardening, and softening-to-hardening characteristics, depending on the combined effect of load imperfection, stiffness imperfection, and excitation amplitude. In general, load imperfection makes the response exhibit softening characteristic and increasing stiffness imperfection will weak this effect. Increasing the excitation level will make the isolator undergo complex switch between different stiffness characteristics.

A tunable high-static–low-dynamic stiffness vibration isolator

In this study, a novel vibration isolator is developed. The developed isolator possesses the characteristics of high-static–low-dynamic stiffness (HSLDS) and can act passively or semi-actively. The HSLDS property of the isolator is obtained by connecting a mechanical spring, in parallel with a magnetic spring that is constructed by a pair of electromagnets and a permanent magnet. The mechanical spring is a structural beam whose stiffness exhibits a hardening behavior. The stiffness of the magnetic spring can be positive or negative, depending on the polarity of the current to the electromagnets. A passive HSLDS isolator is obtained when the electromagnet current is zero. In the stiffness characterization study, the analytical model for each of the springs is established and the tuning parameters are identified. Using the stiffness models, the design optimization issues are investigated. In the experimental study, the effectiveness of the isolator for vibration isolation is tested. The analytical natural frequencies of the isolator are validated experimentally. The relationships between the displacement transmissibility and the exciting frequency are measured both under the passive mode and under the semi-active mode. The on-line tuning capability of the isolator is investigated.