Vibration isolation via a scissor-like structured platform

More and more attentions are attracted to the analysis and design of nonlinear vibration control/isolation systems for better isolation performance. In this study, an isolation platform with n-layer scissor-like truss structure is investigated to explore novel design of passive/semi-active/active vibration control/isolation systems and to exploit potential nonlinear benefits in vibration suppression. Due to the special scissor-like structure, the dynamic response of the platform has inherent nonlinearities both in equivalent damping and stiffness characteristics (although only linear components are applied), and demonstrates good loading capacity and excellent equilibrium stability. With the mathematical modeling and analysis of the equivalent stiffness and damping of the system, it is shown that: (a) the structural nonlinearity in the system is very helpful in vibration isolation, (b) both equivalent stiffness and damping characteristics are nonlinear and could be designed/adjusted to a desired nonlinearity by tuning structural parameters, and (c) superior vibration isolation performances (e.g., quasi-zero stiffness characteristics etc.) can be achieved with different structural parameters. This scissor-like truss structure can potentially be employed in different engineering practices for much better vibration isolation or control.     

分数阶时滞反馈对Duffing振子动力学特性的影响

研究了含分数阶时滞耦合反馈的Duffing自治系统, 通过平均法得到了系统周期解的一阶近似解析形式, 定义了以反馈系数、分数阶阶次、时滞参数表示的等效刚度和等效阻尼系数, 发现分数阶时滞耦合反馈同时具有速度时滞反馈和位移时滞反馈的作用. 比较了三种参数条件下近似解析解与数值积分的结果, 二者的吻合精度都很高, 证明了近似解析解的正确性和准确性. 分析了反馈系数、分数阶阶次和非线性刚度系数等参数对系统分岔点、周期解稳定性、周期解的存在范围、零解的稳定性以及稳定性切换次数等系统动力学特性的影响.     

Dynamic stiffness vibration analysis of an elastically connected three-beam system

An exact dynamic stiffness method is developed for predicting the free vibration characteristics of a three-beam system, which is composed of three non-identical uniform beams of equal length connected by innumerable coupling springs and dashpots. The Bernoulli-Euler beam theory is used to define the beams’ dynamic behaviors. The dynamic stiffness matrix is formulated from the general solutions of the basic governing differential equations of a three-beam element in damped free vibration. The derived dynamic stiffness matrix is then used in conjunction with the automated Muller root search algorithm to calculate the free vibration characteristics of the three-beam systems. The numerical results are obtained for two sets of the stiffnesses of springs and a large variety of interesting boundary conditions.     

Robust vibration control based on identified models

In active vibration control, model accuracy of a vibration field is crucial to the stability and performance of closed-loop systems, especially multiple-input–multiple-output feedback control systems. A state-space model is popular for the design of vibration controllers. Its accuracy may be affected by mode truncation, errors in eigenfunctions for a modal model or errors in mass/stiffness coefficients of finite elements for a finite element model. There are few analytical results on controller stability margins with respect to these errors. This paper proposes a controller based on transfer matrices identified from the measurement data, on the ground that the accuracy of transfer matrices is manageable by identification algorithms. The proposed controller is able to introduce active damping to vibration fields. An analytical link is available between the stability margin and identification errors for the proposed controller. These are important features analyzed theoretically and verified numerically and experimentally here.     

Chains of oscillators with negative stiffness elements

Negative stiffness is not allowed by thermodynamics and hence materials and systems whose global behaviour exhibits negative stiffness are unstable. However the stability is possible when these materials/systems are elements of a larger system sufficiently stiff to stabilise the negative stiffness elements. In order to investigate the effect of stabilisation we analyse oscillations in a chain of n linear oscillators (masses and springs connected in series) when some of the springs? stiffnesses can assume negative values. The ends of the chain are fixed. We formulated the necessary stability condition: only one spring in the chain can have negative stiffness. Furthermore, the value of negative stiffness cannot exceed a certain critical value that depends upon the (positive) stiffnesses of other springs. At the critical negative stiffness the system develops an eigenmode with vanishing frequency. In systems with viscous damping vanishing of an eigenfrequency does not yet lead to instability. Further increase in the value of negative stiffness leads to the appearance of aperiodic eigenmodes even with light damping. At the critical negative stiffness the low dissipative mode becomes non-dissipative, while for the high dissipative mode the damping coefficient becomes as twice as high as the damping coefficient of the system. A special element with controllable negative stiffness is suggested for designing hybrid materials whose stiffness and hence the dynamic behaviour is controlled by the magnitude of applied compressive force.     

Sensor fusion for active vibration isolation in precision equipment

Sensor fusion is a promising control strategy to improve the performance of active vibration isolation systems that are used in precision equipment. Normally, those vibration isolation systems are only capable of realizing a low transmissibility. Additional objectives are to increase the damping ratio of internal vibration modes and to provide a high support stiffness. It is shown that these three objectives cannot be realized simultaneously if only acceleration or force feedback is used. An active hard mount suspension with a feedback strategy based on sensor fusion is proposed that uses the acceleration signal at low frequencies and the force signal at high frequencies. Using sensor fusion, the three objectives can be achieved simultaneously. Experiments on a single-axis setup show that this feedback strategy provides an excellent performance.     

Vibration control for an overhung roller in textile machine considering the stiffness of control device stand

This paper treats a vibration control method that can be used in textile machinery to reduce the unbalanced vibration of an overhung roller–motor system. To control the vibration of the overhung roller, a drive motor with a hybrid type vibration control device consisted with rubber springs and electromagnets is used. When the vibration control system is set up in the textile machinery for industrial use, the stand supporting the control system to the base may be assumed not too rigid but elastic. For a certain value of the elastic stand stiffness, the vibration control performance of the overhung roller becomes very low. In order to prevent this deterioration, a stiffness control achieved by a positive feedback of the displacement signal of the rubber spring is proposed and the effectiveness of the stiffness control is confirmed by simulations and experiments.     

Analysis of a damped pneumatic spring

The complex dynamic stiffness of a damped spring is determined. The damping is produced by transient pressure feedback from an auxiliary tank connected by a capillary to the spring cylinder. From the complex stiffness, the damping and stiffness are determined as functions of excitation frequency. The behavior of a compound spring, consisting of a damped pneumatic spring in parallel with a stiffer linear spring, is also examined. The analysis shows that the damping loss factor depends only on the tank/cylinder volume ratio, and that the capillary dimensions affect only the frequency at which maximum damping occurs. The compound spring is shown to have a maximum loss factor which quickly reaches an asymptotic value as the tank/cylinder volume ratio increases. From this presentation a clearer understanding of the behavior of a damped air spring, and a better sense of how design parameters affect the component's characteristics are obtained.     

Self-tuning control systems of decentralised velocity feedback

This paper is concerned with decentralised velocity feedback for the control of vibration on a flexible structure. Previous studies have shown that a direct velocity feedback loop with a collocated force actuator produces a damping action. Multiple velocity feedback control loops thus reduce the vibration and sound radiation of structures at low frequency resonances, where the response is controlled by damping. However, if the control gains are too high, so that the response of the structure at the control point is close to zero, the feedback control loops will pin the panel at the control positions and thus no damping action is generated. Therefore, in order to maximise the active damping effect, the feedback gains have optimum values and the loops need to be properly tuned.In this paper, a numerical investigation is performed to investigate the possibility of self-tuning the feedback control gains to maximise the power absorbed by the control loops and hence maximise the active damping. The tuning principle is first examined for a single feedback loop for different excitation signals. The tuning of multiple control loops is then considered and the implementation of a practical tuning algorithm is discussed.     

弹性基础隔振系统的简化性能指标和有源控制力

系统研究了基础弹性对单层隔振系统、双层隔振系统及浮筏隔振系统隔振性能的影响。分析了不同隔振系统与不同弹性基础间的振动耦合特性,讨论了不同隔振系统的振级落差和力传递率特性,给出了振级落差和力传递率的简化计算方法。针对不同隔振系统的有源隔振问题,比较了不同作动器安装方式所需的控制力。研究表明,对于所有隔振系统,增加基础的刚度和阻尼有利于提高振级落差和力传递率;对于浮筏隔振系统,增加筏架的刚度和阻尼有利于提高隔振性能和减少有源隔振所需的控制力。      

Stability and performance limits for active vibration isolation using blended velocity feedback

This paper presents a theoretical study of active vibration isolation on a two degree of freedom system. The system consists of two lumped masses connected by a coupling spring. Both masses are also attached to a firm reference base by a mounting spring. The lower mass is excited by a point force. A reactive control force actuator is used between the two masses in parallel with the coupling spring. Both masses are equipped with an absolute velocity sensor. The two sensors and the actuator are used to implement velocity feedback control loops to actively isolate the upper mass from the vibrations of the lower mass over a broad range of frequencies. The primary concern of the study is to determine what type of velocity feedback configuration is suitable with respect to the five parameters that characterise the system (the three spring stiffnesses and the two masses). It is shown analytically that if the ratio of the lower mounting spring stiffness to the lower mass is larger than the ratio of the upper mounting spring stiffness to the upper mass (supercritical system), feeding back the absolute upper mass velocity to the reactive force actuator results in an unconditionally stable feedback loop and the vibration isolation objective can be fully achieved without an overshot at higher frequencies. In contrast, if the ratio of the lower mounting spring stiffness to the lower mass is smaller than the ratio of the upper mounting spring stiffness to the upper mass (subcritical system), the upper mass velocity feedback is conditionally stable and the vibration isolation objective cannot be accomplished in a broad frequency band. For subcritical systems a blended velocity feedback is proposed to stabilise the loop and to improve the broad-band vibration isolation effect. A simple inequality is introduced to derive the combinations between the two error velocities that guarantee unconditionally stable feedback loops.     

Simplified performance indices and active control force of vibration isolation systems with elastic base

Influence of the elasticity of the base on vibration isolation performances of single layer, double layer and floating raft vibration isolation systems is investigated systematically.Characteristics of vibration coupling between different vibration isolation systems and different elastic bases are analyzed. Moreover the characteristics of vibration acceleration level difference and force transmissibility of different vibration isolation systems are discussed and their simplified expressions are given. In addition the required control forces of active vibration isolation under different installations of actuators for different vibration isolation systems are compared.The results show that for all vibration isolation systems, the addition of the stiffness and damping of the base can enhance their vibration acceleration level difference and force transmissibility.Moreover for floating raft vibration isolation system, the addition of the stiffness and damping of the raft can enhance its vibration isolation performance and reduce the control force required by active vibration isolation.     

A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic

A vibration isolator consisting of a vertical linear spring and two nonlinear pre-stressed oblique springs is considered in this paper. The system has both geometrical and physical nonlinearity. Firstly, a static analysis is carried out. The softening parameter leading to quasi-zero dynamic stiffness at the equilibrium position is obtained as a function of the initial geometry, pre-stress and the stiffness of the springs. The optimal combination of the system parameters is found that maximises the displacement from the equilibrium position when the prescribed stiffness is equal to that of the vertical spring alone. It also satisfies the condition that the dynamic stiffness only changes slightly in the neighbourhood of the static equilibrium position. For these values, a dynamical analysis of the isolator under asymmetric excitation is performed to quantify the undesirable effects of the nonlinearities. It includes considering the possibilities of the appearance of period-doubling bifurcation and its development into chaotic motion. For this purpose, approximate analytical methods and numerical simulations accompanied with qualitative methods including phase plane plots, Poincaré maps and Lyapunov exponents are used. Finally, the frequency at which the first period-doubling bifurcation appears is found and the effect of damping on this frequency determined.     

Partial eigenvalue assignment for structural damage mitigation

In partial eigenvalue assignment, not all eigenvalues of the open loop system matrix are modified through a multiple input state or output feedback controller. This freedom available to assign selected eigenvalues of the closed loop system matrix has been widely used in design contexts such as to eliminate spillover effects in structural control problems. Similar approach is also required to modify damping and/or stiffness characteristics in selected eigenmodes of a damaged structure. When an external force acts on the damaged structure, partial eigenvalue assignment in this fashion will attempt to use minimal control effort and keep the structure active with safe operation. In this paper, a new approach to partial eigenvalue assignment and its application to structural damage mitigation are presented. A three mass spring-damper model with damage in one of the springs is illustrated with damping modifications at specific eigenmodes. The procedure is repeated for a second example, which is a cantilever beam modeled using two inputs and 10 state variables.     

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.     

Design and experiment of a compact quasi-zero-stiffness isolator capable of a wide range of loads

This paper proposes the design and experiment of a vibration isolator capable of isolating a wide range of loads. The isolator consists of two oblique springs and one vertical spring to achieve quasi-zero stiffness at the equilibrium position. The quasi-zero-stiffness characteristic makes the isolator attenuate external disturbance more at low frequencies, when compared with linear isolators. Unlike previous studies, this paper focuses on the analysis of the effect of different loads and the implementation of an adjustment mechanism to handle a wide range of loads. To ensure zero stiffness under imperfect stiffness matching, a lateral adjustment mechanism is also proposed. Instead of using coil springs, special planar springs are designed to realize the isolator in a compact space. Static and dynamic models are developed to evaluate the effect of key design parameters so that the isolator can have a wide isolation range without sacrificing its size. A prototype and its associated experiments are presented to validate the transmissibility curves under three different loads. The results clearly show the advantage of quasi-zero-stiffness isolators against linear isolators.     

FINITE ELEMENT FORMULATION AND ACTIVE VIBRATION CONTROL STUDY ON BEAMS USING SMART CONSTRAINED LAYER DAMPING (SCLD) TREATMENT

This work deals with the active vibration control of beams with smart constrained layer damping (SCLD) treatment. SCLD design consists of viscoelastic shear layer sandwiched between two layers of piezoelectric sensors and actuator. This composite SCLD when bonded to a vibrating structure acts as a smart treatment. The sensor piezoelectric layer measures the vibration response of the structure and a feedback controller is provided which regulates the axial deformation of the piezoelectric actuator (constraining layer), thereby providing adjustable and significant damping in the structure. The damping offered by SCLD treatment has two components, active action and passive action. The active action is transmitted from the piezoelectric actuator to the host structure through the viscoelastic layer. The passive action is through the shear deformation in the viscoelastic layer. The active action apart from providing direct active control also adjusts the passive action by regulating the shear deformation in the structure. The passive damping component of this design eliminates spillover, reduces power consumption, improves robustness and reliability of the system, and reduces vibration response at high-frequency ranges where active damping is difficult to implement. A beam finite element model has been developed based on Timoshenko's beam theory with partially covered SCLD. The Golla-Hughes-McTavish (GHM) method has been used to model the viscoelastic layer. The dissipation co-ordinates, defined using GHM approach, describe the frequency-dependent viscoelastic material properties. Models of PCLD and purely active systems could be obtained as a special case of SCLD. Using linear quadratic regulator (LQR) optimal control, the effects of the SCLD on vibration suppression performance and control effort requirements are investigated. The effects of the viscoelastic layer thickness and material properties on the vibration control performance are investigated.     

Pole assignment of friction-induced vibration for stabilisation through state-feedback control

Friction-induced self-excited linear vibration is often governed by a second-order matrix differential equation of motion with an asymmetric stiffness matrix. The asymmetric terms are product of friction coefficient and the normal stiffness at the contact interface. When the friction coefficient becomes high enough, the resultant vibration becomes unstable as frequencies of two conjugate pairs of complex eigenvalues (poles) coalesce (when viscous damping is low).This short paper presents a receptance-based inverse method for assigning complex poles to second-order asymmetric systems through (active) state-feedback control of a combination of active stiffness, active damping and active mass, which is capable of assigning negative real parts to stabilise an unstable system.     

Self-powered active vibration control using a single electric actuator

The authors have proposed self-powered active vibration control systems that achieve active vibration control using regenerated vibration energy. Such systems do not require external energy to produce a control force. This paper presents a self-powered system in which a single actuator realizes active control and energy regeneration.The system proposed needs to regenerate more energy than it consumes. To discuss the feasibility of this system, the authors proposed a method to calculate the balance between regenerated and consumed energies, using the dynamical property of the system, the feedback gain of the active controller, the specifications of the actuator, and the power spectral density of disturbance. A trade-off was found between the performance of the active controller and the energy balance. The feedback gain of the active controller is designed to have good suppression performance under conditions where regenerated energy exceeds consumed energy.A practical system to achieve self-powered active vibration control is proposed. In the system, the actuator is connected to the condenser through relay switches, which decide the direction of the electric current, and a variable resistor, which controls the amount of the electric current. Performance of the self-powered active vibration was examined in experiments; the results showed that the proposed system can produce the desired control force with regenerated energy, and that it had a suppression performance similar to that of an active control system using external energy. It was found that self-powered active control is attainable under conditions obtained through energy balance analysis.     

Nonlinear analysis,design and vibration isolation for a bilinear system with time-delayed cubic velocity feedback

This paper combines cubic nonlinearity and time delay to improve the performance of vibration isolation. Nonlinear dynamics properties, design methodology and isolation performance are studied for a piecewise bilinear vibration isolation system with the time-delayed cubic velocity feedback control. By the multi-scale perturbation method, the equivalent stiffness and damping are first defined to interpret the effect of feedback control loop on dynamics behaviours, such as frequency island phenomenon. Then, a design criterion is proposed to suppress the jump phenomenon induced by the saddle-node bifurcation. With the purpose of obtaining the desirable vibration isolation performance, stability conditions are obtained to find appropriate feedback parameters including gain and time delay. Last, the influence of the feedback parameters on vibration transmissibility is assessed. Results show that the strategy developed in this paper is practicable and feedback parameters are significant factors to alter dynamics behaviours, and more importantly, to improve the isolation effectiveness for the bilinear isolation system.