Nonlinear Vibration Isolation System Research Articles

Research and Application of Vibration Isolation Platform Based on Nonlinear Vibration Isolation System

In order to weaken the ground vibration caused by mechanical equipment, a vibration isolation platform of nonlinear vibration isolation system is proposed. The influence of flexible foundation and nonlinear isolator parameters on the performance of nonlinear vibration isolation platform is studied. By optimizing the nonlinear parameters of the vibration isolation platform, the vibration response of equipment vibration is effectively blocked. The research results show that the structural parameters such as stiffness and damping of the nonlinear vibration isolation system affect the vibration mode of the vibration isolation platform, thus affecting the vibration response of the whole system. Applying the nonlinear vibration isolation system to the project, the impact of the vibration of the refiner on the vibration of the floor is greatly reduced. It shows that this method can be used to reduce the foundation vibration caused by the equipment, improve the operation status of the equipment, and increase the safety factor of the building.

Shock Isolation of an Orthogonal Six-DOFs Platform With High-Static-Low-Dynamic Stiffness

Abstract A novel approach to enhance the shock vibration environment of multi-directions using a high-static-low-dynamic stiffness supported orthogonal six-degrees-of-freedom (DOFs) nonlinear vibration isolation (OSNVI) system is presented in this paper. By combining spring positive stiffness and magnetic negative stiffness, the proposed system achieves high-static-low-dynamic stiffness. Under the multi-directions half-sine vibration, the dynamic equation of the OSNVI is obtained. Both dynamic and static analysis methods are utilized to explore the effect of various parameters on the shock isolation performance of the OSNVI from both the time and frequency domains. The results indicate that the proposed OSNVI can efficiently suppress multi-direction shocks at the cost of only one second. Although a nonlinear jump is usually not expected, the nonlinear jump of the OSNVI could improve the load capacity by increasing the spring stiffness without changing the shock isolation frequency band. Finally, a shock experiment is employed through a three-axis shaker platform to validate the shock isolation performance of the orthogonal six-DOF nonlinear vibration isolator. The proposed OSNVI provides a promising approach to suppress the multi-directional shock vibrations.

Analysis of super-harmonic resonance and periodic motion transition of fractional nonlinear vibration isolation system

The precision instruments and equipment are often utilized in low-frequency and micro-amplitude vibration systems, in which many vibration isolators of rubber materials are widely used. Ignoring the low-frequency amplitude will result in errors in the fatigue life design of the vibration isolators and predicting the dynamic response of each frequency band accurately becomes necessary. However, integer-order models cannot describe the frequency dependence of rubber materials, while the fractional-order models can describe it instead. On the other hand, the elastic restoring force is strongly nonlinear under large deformation, and the vibration of the nonlinear system contains multiple harmonic components. In order to solve those issues, the fractional nonlinear Nishimura model is used to characterize the constitutive relation of vibration isolators such as air springs, which are made of carbon black filled natural rubber. The high-order harmonic balance method is used to obtain the steady-state response of the vibration system, while the fourth-order Runge–Kutta method is applied to simulate the dynamic response of the system in the low-frequency region, and the Lyapunov exponent is used to determine the stability of the system. Furthermore, the influence of parameters on the amplitude–frequency characteristics of different frequency bands is also studied, and a method to solve the optimal damping coefficient is proposed based on the primary resonance amplitude–frequency curves. The results show that there is a diversity of periodic motions in the process of adjacent super-harmonic resonance transition. Numerical simulations also demonstrate that multi-periodic motions coexist in the system. The motion transition law between the polymorphic coexistence region and its adjacent regions is summarized.

Computational method of underwater acoustic radiation from a spherical shell coupled with nonlinear systems

During the continuous research and practical application of nonlinear vibration isolation systems, new problems arise for the vibro-acoustic calculation of underwater targets. The present frequency-domain analysis method can only deal with linear problems, so the acoustic radiation calculation of underwater targets coupled with nonlinear systems can only be performed in the time domain. When it comes to the real-time calculation, the huge computation cost will definitely bring challenges to the practical application. To solve this problem, a new calculation method is established for the underwater vibro-acoustic problems of structures coupled with nonlinear systems, taking elastic spherical shells as research objects in this paper. This method realizes the time-frequency domain conversion of the fluid-structure coupled vibration equation of an elastic spherical shell through the Fourier transform. The explicit difference method is further used to calculate the vibration response of the spherical shell, while the Newmark method is used to calculate the vibration response of the internal nonlinear system. An efficient integrated calculation of the two coupling parts is then achieved by considering boundary conditions at the connection point. Without the time-domain acoustic field solution, this method is able to calculate the vibration response of both the internal nonlinear system and the spherical shell, as well as the radiated acoustic power generated by the spherical shell vibration in real time. The calculation complexity and efficiency are radically improved. The nonlinear system discussed in this paper is a nonlinear mass-spring oscillator system installed at the bottom of a spherical shell. Finally, the correctness of the method described in this paper is verified through specific numerical examples, and a series of calculations and analyzes are carried out to explore the influence of nonlinear effects on the acoustic radiation of underwater structures. The calculation method described in this article is not limited to spherical shells, and is still applicable to general underwater targets.

Orthogonal six-DOFs vibration isolation with tunable high-static-low-dynamic stiffness: Experiment and analysis

In order to suppress the low-frequency vibration of a high-precision instrument, an orthogonal six-DOFs vibration isolation system with tunable high-static-low-dynamic stiffness characteristics is studied using experimental and analytical methods. The current-controlled magnetic spring is used to obtain the tunable orthogonal six-DOFs HSLDS, which adapts to different suspended masses. A dynamical model of electromagnetic nonlinear vibration isolation system with orthogonal six-DOFs is developed to investigate the experimental results. Moreover, the harmonic balance analysis together with arc-length continuation are used to analyze the frequency response function of force transmissibility. Both experimental and analytical results are shown that as a controlled current increases, the isolation frequency range is widened. Parametric studies of this study demonstrate that increasing the electromagnetic coefficient and decreasing the onset distance can decrease the isolation start frequency. It is obtained that proper parameter is necessary to achieve broadband frequency range, and this article also provides a relevant explanation of these results.

Attenuation of Line Spectra and Vibration Based on a Chaotification Method and Global Analysis

The chaotification method can be used to weaken the line spectra component of the ship's radiated noise. The main obstacle that plagues the engineering realization of this innovative concept is how to realize the small-amplitude chaotic motion of the nonlinear vibration isolation system (NVIS) in a large parameter range and how to maintain the chaotic state under variable operating conditions. In this paper, the dimensionless dynamic equation of the NVIS with a flexible base is derived. A control method similar to generalized synchronization is used to realize the continuous chaos of the system. The maximum Lyapunov exponent, the conditional Lyapunov exponent, and disturbances are introduced to verify the effectiveness of the control method and its stability and robustness in a large range of parameters. Aiming at the coexistence of multiple attractors, a new global analysis method is proposed, which is used to study the distribution of basin of attraction (BA). According to the changes of BA before and after the system is controlled, the BA is partitioned. Initial conditions from different partitions are selected to study the attenuation effect of line spectra and vibration, and the favorable initial conditions for realizing the small-amplitude chaotic motion of the system are given. The method may be extended to the global analysis of other controlled systems.

The Elliptic Harmonic Balance Method for the Performance Analysis of a Two-Stage Vibration Isolation System with Geometric Nonlinearity

This study develops a modified elliptic harmonic balance method (EHBM) and uses it to solve the force and displacement transmissibility of a two-stage geometrically nonlinear vibration isolation system. Geometric damping and stiffness nonlinearities are incorporated in both the upper and lower stages of the isolator. After using the relative displacement of the nonlinear isolator, we can numerically obtain the steady-state response using the first-order harmonic balance method (HBM1). The steady-state harmonic components of the stiffness and damping force are modified using the Jacobi elliptic functions. The developed EHBM can reduce the truncation error in the HBM1. Compared with the HBM1, the EHBM can improve the accuracy of the resonance regimes of the amplitude-frequency curve and transmissibility. The EHBM is simple and straightforward. It can maintain the same form as the balancing equations of the HBM1 but performs better than it.

Nonlinear Vibration Control Device for a Vehicle Suspension Using Negative Stiffness Mechanism

Vibration suspension system is used in the automobiles to mitigate the excitations that can come from the travel over various road profiles. The conventionally used suspension systems are linear devices that are effective against high frequencies; however, they have limitations in isolation ability in the low-frequency ranges. Hence, nonlinear isolation system can be used to overcome such limitations. The quasi-zero stiffness (QZS) characteristics are efficient in the vibration isolation and can provide isolation efficiently against low-frequency excitations as well. This concept is taken into consideration and has designed a nonlinear vibration isolation system with QZS behaviour. The negative stiffness is achieved through the inclined spring arrangement, and the positive stiffness is obtained from the pneumatic system. The force–displacement model is used to develop the approximate dynamic modelling for small displacement. The performance is studied against various road profiles as per ISO 8608. The results showed that the isolation system effectively isolates the external excitations from the road profile and shows a high isolation capability in the higher velocities. The study shows the proposed system reduced on an average of 92% and 88% of external excitations in road class B–D and E–H, respectively. Based on the computational study conducted in this work on the developed design, the isolation system is found to be effective in vibration isolation. The QZS system parameters highly influence the performance of the isolation system. The system developed in this work can be used to test for real-time automobile applications.

Dynamic effect of constant inertial acceleration on vibration isolation system with high-order stiffness and Bouc–Wen hysteresis

Nonlinear vibration isolation provides with an effective way for vibration reduction with broad band and high efficiency. This investigation focuses on the dynamic effects of a constant inertial acceleration on a nonlinear vibration isolation system with high-order stiffness and Bouc–Wen hysteresis. A wire rope-based isolator subject to the inertial acceleration is analyzed to illustrate the uncertainty of the static equilibrium state and the diverse dynamic effects of the acceleration. The frequency responses are obtained through a semianalytical method based on the harmonic balance method. A recursive method is proposed to deal with the polynomial function derived from the nonlinear stiffness. An alternating frequency/time domain technique is adopted to treat the implicit function of the Bouc–Wen hysteresis. The calculation results are verified by a numerical simulation with the Runge–Kutta method. The analysis and the simulations reveal that the inertial acceleration leads to different static equilibrium states according to the paths to the equilibriums. The inertial acceleration, respectively, affects the static equilibrium displacement and the residual hysteretic force, and they change the dynamic responses differently. The acceleration effect on the static equilibrium displacement leads to additional coupling stiffness making the system harder/softener with positive/negative high-order stiffness. Furthermore, the acceleration has a significant effect on the dynamic equilibrium location, and the equilibrium reaches the minimum around the resonance and increases gradually at higher frequencies. The acceleration effect on the residual hysteretic force changes the amplitude–frequency responses indirectly through additional coupling stiffness. A positive residual hysteretic force leads to a positive overall deviation of the dynamic equilibrium location regardless of the positive or negative high-order stiffness.

The analysis and design of nonlinear vibration isolators under both displacement and force excitations

In this study, a novel criterion known as the comprehensive transmissibility is proposed to estimate the isolation performance of nonlinear isolation systems under complex loading excitations. A combined quasi-zero-stiffness (QZS) isolator is enumerated, and the comprehensive performance of other common isolator systems and the existing QZS system is analyzed. Based on the normalized differential equations of the isolation system, the isolation performances of different isolation systems are studied by using the comprehensive transmissibility criterion, showing that the combined QZS isolator has better isolation performance than others. In addition, the output frequency response function (OFRF) representation of the comprehensive transmissibility is derived, and the optimal design of the QZS isolation system is developed by using the OFRF. The simulation results demonstrate that the design can meet the requirements of the vibration isolator with both the force and displacement excitations and is promised to be applied to the design of nonlinear isolation systems in engineering practice. To solve the issue of the analysis and design of nonlinear vibration isolation systems under multi-excitation, a new criterion known as comprehensive transmissibility is proposed. Based on the comprehensive transmissibility, the isolation performance of the combined vibration isolation system is analyzed, and the output frequency response function (OFRF) method is applied to conduct the design of the isolation system.

Planar Dynamics of a Motorcycle: Influence of Vibration Isolation System Nonlinearity

This paper investigates the influence of the nonlinearities of a vibration isolation system on the planar dynamics of a motorcycle. The use of a nonlinear isolation system is often necessitated by design and packaging constraints. Although the use of a vibration isolation system is uncommon in motorcycles, it is used in some cases to enhance ride comfort by mitigating vibrations transmitted to the rider due to shaking forces. In such cases, the handling of the motorcycle can be influenced due to the coupled dynamics of the rear unsprung mass and the swing arm. In this paper, a stochastic analysis has been performed by using the statistical linearization method to specifically examine nonlinearities associated with the vibration isolation system. An eight degree-of-freedom planar model has been developed, and each isolator is represented by a modified multi-axial Kelvin-Voigt model. It has been observed that the model developed in this study can capture the coupled dynamics between the rear suspension and the vibration isolation system. Results indicate that the nonlinear design of the vibration isolation system can be useful in enhancing ride comfort in the lower frequency range without an adverse impact on handling. Furthermore, it has been observed that the parameters associated with the nonlinear vibration isolation system can be tuned to enhance ride comfort while meeting the design requirements of spatial dynamics.

DYNAMICAL ANALYSIS ON A KIND OF SEMI-ACTIVE VIBRATION ISOLATION SYSTEMS WITH DAMPING CONTROL 1)

A series of single-degree-of-freedom semi-active nonlinear vibration isolation system with cubic stiffness based on relative velocity feedback on-off control is studied. By using the averaging method, the approximate analytical solutions of the primary resonance of the system are obtained, which is controlled by the acceleration and relative velocity based acceleration drive damping control, the velocity and relative velocity based sky-hook damping control, and the displacement and relative velocity based ground-hook damping control, respectively. A further comparison between the numerical solutions and the approximate analytical solutions under different control strategies is fulfilled. And such comparison results suggest that the approximate analytical results are quite consistent with the numerical solutions, which verifies the effectiveness of the approximate analytical solutions. Moreover, the stability conditions of the steady-state solution of the system under different control strategies are analyzed by Lyapunov theory, and the influences of system parameters on the control effect are discussed. In order to compare the control performances in the three different control approaches, the amplitude-frequency response equation is conducted. The results show that the similar expressions can be found in the switching conditions in the process of the analytical analysis of the three relative velocity feedback control strategies. In the aspect of suppressing resonance response amplitude, the sky-hook damping control strategy based on velocity and relative velocity feedback has the best damping effect in low frequency band, while the acceleration drive damping control strategy based on acceleration and relative velocity feedback has the best damping effect in high frequency band. The sky-hook damping control strategy also shows good performance in reducing the amplitude response of transient response. This analytical research method can also be applied to the system with other semi-active on-off control strategies, and it provides an effective way for the control strategy research of semi-active vibration isolation system.

Modeling and controlling a semi-active nonlinear single-stage vibration isolator using intelligent inverse model of an MR damper

Semi-active systems with variable stiffness and damping have demonstrated excellent performance. The aim of this study is to investigate the new configuration of a semi-active single-stage nonlinear vibration isolation system with a Magneto-Rheological (MR) damper to reduce the magnitude of force transmissibility over the both resonant and non-resonant regions. The magnitude of force transmissibility is widely used to performance measurement for the isolation system. In current study, to achieve this reduction, two horizontal springs and one MR damper were added to the isolator. Theoretical analysis reveals that the nonlinear system with MR damper can produce ideal vibration isolation. However, due to the nonlinear characteristics behavior of the MR damper, conventional control algorithms to reach the desired are cumbersome. To address this issue, an artificial intelligent strategy using wavelet network and fuzzy logic controller is considered to be constructed to copy the inverse dynamics of the MR damper and the nonlinear isolator. Accordingly, simulation results demonstrated that the intelligent algorithm has acceptable performance.

Methods to Obtain Approximate Responses of a Non-Linear Vibration Isolation System

Methods to Obtain Approximate Responses of a Non-Linear Vibration Isolation System

Research on linear/nonlinear viscous damping and hysteretic damping in nonlinear vibration isolation systems

A nonlinear vibration isolation system is promising to provide a high-efficient broadband isolation performance. In this paper, a generalized vibration isolation system is established with nonlinear stiffness, nonlinear viscous damping, and Bouc-Wen (BW) hysteretic damping. An approximate analytical analysis is performed based on a harmonic balance method (HBM) and an alternating frequency/time (AFT) domain technique. To evaluate the damping effect, a generalized equivalent damping ratio is defined with the stiffness-varying characteristics. A comprehensive comparison of different kinds of damping is made through numerical simulations. It is found that the damping ratio of the linear damping is related to the stiffness-varying characteristics while the damping ratios of two kinds of nonlinear damping are related to the responding amplitudes. The linear damping, hysteretic damping, and nonlinear viscous damping are suitable for the small-amplitude, medium-amplitude, and large-amplitude conditions, respectively. The hysteretic damping has an extra advantage of broadband isolation.

WADA FRACTAL BASINS OF ATTRACTION IN A ZERO-STIFFNESS VIBRATION ISOLATION SYSTEM

A fractal basin boundary is a Wada fractal basin boundary if it contains at least three different basins. The corresponding basin is called a Wada fractal basin. Previous results show that some oscillators possess Wada fractal basins with common basin boundaries. Here we find that a nonlinear vibration isolation system can possess abundant coexisting basins and every basin is a Wada fractal basin. These Wada fractal basin boundaries separate different basins in the different regions. A proper classification of these Wada fractal basins is provided according to the order of saddles and Wada numbers. Basin organization is systematic and all basins spiral outward toward the infinity. The entangled basin boundaries are described by the manifolds of saddles and basins (tongues) accumulation.

1/2 sub-harmonic resonance in bistable structure and its effect on vibration isolation characteristics

<sec>In the extensive modern applications, the low-frequency and heavy-load isolators are needed to reduce the vibration transmissions. The unique properties of nonlinear systems, such as jumping, bifurcation and chaos, provide new ideas for designing the new functional structures. Bistable system is a typical non-linear system, features highly static and low dynamic stiffness, which promises to realize a low-frequency isolator with ensuring heavy load capacity. However, more studies are necessary to clarify the sub-harmonic resonance and its generation process, parameter influences, vibration isolation characteristics of the bistable structure.</sec><sec>By adopting the equivalent, analytical, numerical and experimental methods, we study the 1/2 sub-harmonic resonance, evolution process and its influence on the vibration isolation characteristics of the bistable structure in this paper. When the amplitude or nonlinear stiffness coefficient <i>k</i><sub>n</sub> increases to a certain extent, 1/2 sub-harmonic resonance appears, where the response contains high-amplitude <i>ω</i>/2 component under the excitation frequency <i>ω</i>, so the energy is transferred from high frequency to low frequency. We study the bifurcation and varying processes of the fundamental and 1/2 sub-harmonic transmission by increasing the amplitude. At critical bifurcation amplitude, the sub-harmonic transmission rapidly increases from 0 to a large peak value. And then, it decreases gradually when the damping is absent. However, the peak value of 1/2 sub-harmonic does not cause the fundamental transmission to change suddenly. When considerable damping appears with the increase of the amplitude, 1/2 sub-harmonic does not always exist, instead, it follows an interesting “generation-enhancement-degeneration-disappearance” process. This process possesses great significance in applying the 1/2 sub-harmonic to vibration manipulation or avoiding the resonant enhancement induced by it. </sec><sec>Moreover, in this process, both the peak frequency and the peak transmission of the bistable isolation system descend first. The optimal combination of the parameters can reduce the resonance frequency by 17.8% through increasing the driving amplitude. However, they jump to large values when 1/2 sub-harmonic plays a dominant role. </sec><sec>Additionally, the negative stiffness <i>k</i><sub>0</sub> has a significant effect on the primary resonance characteristics: as |<i>k</i><sub>0</sub>| increases under a specified excitation amplitude, the resonance peak shifts toward higher frequency and the transmission increases. Besides the main effect on the sub-harmonic resonance and the equilibrium point, the nonlinear coefficient <i>k</i><sub>n</sub> also affects the peak and resonance frequency of the system, but the effect is much less than the influence caused by <i>k</i><sub>0</sub>.</sec><sec>Furthermore, the sub-harmonic resonances, bifurcations and vibration isolation characteristics of the bistable bulking beam structure are demonstrated experimentally. The experimental results show that: 1) the 1/2 sub-harmonic resonance can appear in a certain bandwidth and it is not monochromic; 2) the increase of the driving amplitude can reduce the transmission of the fundamental wave; 3) the transmission of 1/2 sub-harmonic jumps from 0 upward to a large value at a critical amplitude, and then it decreases gradually. The experimental results are consistent with the analytical and numerical results. The experiment also demonstrates the law of frequency shifting and the transmission reduction of peak values. Therefore, the appropriate increase of the amplitude can improve the vibration isolation capacity. However, sub-harmonic resonance will reduce the isolation effect. In practical engineering, the strong sub-harmonic resonance should be avoided in a nonlinear vibration isolation system.</sec>

A Design Method of Chaotic Synchronous Multi-stable Manifold

Line spectra in the low frequency band of the underwater radiated noise are the main factors affecting the acoustic stealth performance of submarines. The chaotification technique for line spectra reduction can realize spectra reconstruction and reduce line spectra intensity by converting line spectra in the low frequency band into chaotic spectra in the broadband frequency. Previous studies have shown that the method of the generalized synchronization of chaos can generate continuous chaos of the system under variable operating conditions, and can reduce the amplitude significantly in some cases, which provides a new idea for the theory of the chaotification technique for line spectra reduction. However, this method can only select the appropriate synchronization control strategy of chaos through “trial and error” method to achieve chaos with small amplitude. Therefore, a design method of multiple stable synchronous manifolds including stable chaotic attractors with small amplitude is proposed. This method is verified and illustrated by the coupled Duffing system, and applied to the two degree-of-freedom (DOF) nonlinear vibration isolation system(VIS).

Noise- and delay-enhanced stability in a nonlinear isolation system

In this paper, we present a detailed study of time-delayed feedback control of the nonlinear vibration isolation system driven by stochastic environmental fluctuations. We investigate the effects of time-delayed control force and nonlinear damping on the mean first passage time from the periodic attractors to reveal the complicated nonlinear stochastic dynamics of this system. Varying the intensities of noise sources, we find that the mean lifetime of the lower amplitude periodic attractor is characterized by nonmonotonic behavior as a function of the noise intensities. One remarkable behavior is the phenomenon of noise-enhanced stability has been carried out to evaluate the isolation performance of the controlled system. Moreover, the proper time delay value is obtained based on the consideration for the noise- and delay-enhanced stability. Finally, we observe that these transient properties are connected with the dynamics of time-delayed active control. Our analysis suggests that one form of the control force outperforms the other by many reasonable measures of control isolation effectiveness, which provides the guidance for optimizing time-delayed active control for stochastic fluctuations in engineering practices.

Approximate Response of a Non-linear Vibration Isolation System Using the Harmonic Balance Method

Approximate Response of a Non-linear Vibration Isolation System Using the Harmonic Balance Method