Effect of an impact damper on a multi-degree of freedom system

Theoretical analysis of the steady state vibrational motion of a multi-degree of freedom system equipped with an impact damper is presented. The analysis is based on the assumption that two generally distributed impacts occur in each cycle. The theory is applied to the special case of a single degree of freedom main system and the effects of various parameters are investigated. The theoretically possible modes of steady state motion with two impacts/cycle and with no impacts are predicted. The non-linear behaviour of the damper is manifested by the existence of as many as three modes of steady state motion for a given exciting frequency. The conditions leading to more or less than two impacts/cycle are predicted although the system response under such conditions is not studied. Experimental results are presented and compared with theoretical predictions.     

Vibration suppression of a cantilever beam using magnetically tuned-mass-damper

Eddy currents are induced by the movement of a conductor through a stationary magnetic field or a time varying magnetic field through a stationary conductor. These currents circulate in the conductive material and are dissipated, causing a repulsive force between the magnet and the conductor. These electromagnetic forces can be used to suppress the vibrations of a flexible structure. A tuned mass damper is a device mounted in structures to reduce the amplitude of mechanical vibrations and is one of the effective vibration suppression methods. In the present study, an improved concept of this tuned mass damper for the vibration suppression of structures is introduced. This concept consists of the classical tuned mass damper and an eddy current damping. The important advantages of this magnetically tuned mass damper are that it is relatively simple to apply, it does not require any electronic devices and external power, and it is effective on the vibration suppression. The proposed concept is designed for a cantilever beam and the analytical studies on the eddy current damping and its effects on the vibration suppression. To show the effectiveness of the proposed concept and verify the eddy current damping model, experiments on a cantilever beam are performed. It is found that the proposed concept could significantly increase the damping effect of the tuned mass damper even if not adequately tuned.     

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.     

Extending the dynamic range of an energy harvester using nonlinear damping

This paper introduces the use of nonlinear damping for extending the dynamic range of vibration energy harvesters. A cubic nonlinear damper is initially considered and the average harvested power and the throw are obtained for different sinusoidal base excitation amplitudes and frequencies, both numerically and analytically. It is demonstrated that when excited at resonance, at an amplitude below its maximum operational limit, the harvested power using a nonlinear damper can be significantly larger than that of a linear energy harvester, therefore expanding its dynamic range. A potential approach to implementing cubic nonlinearity using a shunted electromagnetic device is also presented.     

SENSITIVITY ANALYSIS AS A METHOD OF ABSORBER TUNING FOR REDUCTION OF STEADY STATE RESPONSE OF LINEAR PARAMETRIC SYSTEMS

The aim of this paper is to determine whether one dynamic absorber can reduce the amplitude of the steady state vibration of a parametric system for natural and parametric resonance frequencies simultaneously. The efficiency of both the conventional dynamic absorber and the parametric absorber is analyzed. The first order sensitivity analysis of parametric periodic systems in the time domain is applied to obtain logarithmic sensitivity functions in the frequency domain. The first order sensitivity logarithmic functions are used to tune the conventional absorber and the parametric absorber.     

Analysis of energy dissipation in an elastic moving string with a viscous damper at one end

In this paper transverse vibration of an axially moving viscoelastic string with a viscous damper at one end is investigated analytically. The string is assumed to be travelling with constant velocity and the length of string is constant or time varying. The linear and nonlinear mathematical models are derived using the Lagrangian function and implemented using a finite element method. The method considers a time varying state space function applied to the linear model, the Newmark-Beta method is used to solve the response for the nonlinear problem numerically. The case of energy dissipated by a viscoelastic damper at one end of the string for different axial string velocities is considered. When a disturbance arrives at the boundary an exact value for the damper which provides maximum energy dissipation is investigated. Finally, numerical simulations are presented to establish the feasibility of the method.     

Non-linear dynamics of flexible multi-bearing rotors

A general analysis has been developed to computer simulate steady state and transient vibration phenomena of complex rotor-bearing-support systems. A central feature of this analysis is a proper handling of various highly non-linear effects (most notably journal bearings) which dominate the dynamic phenoména encountered during large amplitude rotor-bearing vibrations. There are a number of potential causes of large amplitude rotor vibration, such as high rotor imbalance (e.g., loss of turbine blades at running speed), critical speed operation, journal bearing dynamic instability (oil whip), earthquakes, and shock. Failure mode analysis requires the evaluation and understanding of such potentially large dynamic forces and displacements. The paper presents development of the analysis, comparison with experiment and examples of its use in industrial applications.     

磁流变液阻尼器在转子振动控制中的应用

设计了一种转子振动控制用的剪切式磁流变液阻尼器,建立了磁流变液阻尼器－悬臂转子系统的分析模型,理论和实验研究了转子系统的不平衡响应特性。研究表明,随着施加磁场强度的增加,磁流变液阻尼和刚度增大,转子系统的临界振幅明显下降,系统的临界转速也明显提高。通过简单的开关控制,可抑止转子通过临界转速过程中的振动。     

基于混沌和随机共振的微弱信号检测

推导了分数阶线性振子系统响应的一阶稳态矩的频率不变性和相移特性, 并通过理论分析得出, 在随机共振机制下, 分数阶线性振子对系统响应一阶稳态矩的幅值具有放大作用. 构造Duffing混沌振子检测器, 利用混沌系统对参数摄动的敏感性以及对噪声的免疫能力实现弱信号检测. 数值模拟证实, 该方法可以有效地从噪声背景中将微弱正弦信号检测出来, 并且相对传统的混沌检测方法能显著降低信噪比检测门限.     

MEASUREMENT AND ANALYSIS OF MODULATED DOUBLET MODE RESPONSE IN MOCK BLADED DISKS

The steady state response of spatially modulated doublet modes that occur in low count flexible bladed disks is investigated for the case in which the structure is driven by a harmonic travelling wave excitation source. Finite-element simulation and modal testing of prototypical bladed-disk structures demonstrate the presence of particular wavenumbers, beyond the base number of nodal diameters, which contaminate and distort the appearance of certain doublet modes. The manner in which the natural frequency and wavenumber content of such modes shift and split as functions of the number of blades and their span angle is discussed in the light of a companion perturbation analysis for rotationally periodic structures. Resonance conditions are established and verified through simultaneous measurements made with a spin test stand using sensors that are placed in the rotating (structure) and stationary (excitation) frames of reference. The travelling wave response components of a repeated frequency doublet mode are shown to propagate either in the same or opposite direction as the excitation source, depending on whether certain algebraic relationships between the excitation order, the base number of nodal diameters, and the contamination wavenumbers are satisfied. To the extent that such components can travel at different phase speeds and directions relative to one another, the placement of sensors on the structure can be optimized to best measure the response amplitude. Conversely, other placements can result in submaximal measurement of peak vibration amplitude over the structure.     

Steady state response of optical feedback in orthogonally polarized microchip Nd:YAG laser based on optical feedback rate equation

We present a rate equation model to explain and simulate the steady state response of optical feedback in orthogonally polarized microchip Nd:YAG lasers, which is observed in our experiment: When the external-cavity length is tuned, the two orthogonal polarization states perform an opposite-phase power modulation. With the increase of feedback light intensity, the power modulation of each orthogonal polarization state will present an amplitude increase, and finally become a polarization-state hopping; when the frequency difference of the two orthogonal polarization states is tuned, the power modulation amplitude performs a periodic variation. The theoretical analysis and numerical simulation are in good agreement with the experimental results.     

Hysteresis-induced bifurcation and chaos in a magneto-rheological suspension system under external excitation

The magneto-rheological damper(MRD) is a promising device used in vehicle semi-active suspension systems, for its continuous adjustable damping output. However, the innate nonlinear hysteresis characteristic of MRD may cause the nonlinear behaviors. In this work, a two-degree-of-freedom(2-DOF) MR suspension system was established first, by employing the modified Bouc–Wen force–velocity(F –v) hysteretic model. The nonlinear dynamic response of the system was investigated under the external excitation of single-frequency harmonic and bandwidth-limited stochastic road surface.The largest Lyapunov exponent(LLE) was used to detect the chaotic area of the frequency and amplitude of harmonic excitation, and the bifurcation diagrams, time histories, phase portraits, and power spectrum density(PSD) diagrams were used to reveal the dynamic evolution process in detail. Moreover, the LLE and Kolmogorov entropy(K entropy) were used to identify whether the system response was random or chaotic under stochastic road surface. The results demonstrated that the complex dynamical behaviors occur under different external excitation conditions. The oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations, and chaotic oscillations was observed in detail. The chaotic regions revealed that chaotic motions may appear in conditions of mid-low frequency and large amplitude, as well as small amplitude and all frequency. The obtained parameter regions where the chaotic motions may appear are useful for design of structural parameters of the vibration isolation, and the optimization of control strategy for MR suspension system.     

Nonlinear amplitude approximation for bilinear systems

An efficient method to predict vibration amplitudes at the resonant frequencies of dynamical systems with piecewise-linear nonlinearity is developed. This technique is referred to as bilinear amplitude approximation (BAA). BAA constructs a single vibration cycle at each resonant frequency to approximate the periodic steady-state response of the system. It is postulated that the steady-state response is piece-wise linear and can be approximated by analyzing the response over two time intervals during which the system behaves linearly. Overall the dynamics is nonlinear, but the system is in a distinct linear state during each of the two time intervals. Thus, the approximated vibration cycle is constructed using linear analyses. The equation of motion for analyzing the vibration of each state is projected along the overlapping space spanned by the linear mode shapes active in each of the states. This overlapping space is where the vibratory energy is transferred from one state to the other when the system switches from one state to the other. The overlapping space can be obtained using singular value decomposition. The space where the energy is transferred is used together with transition conditions of displacement and velocity compatibility to construct a single vibration cycle and to compute the amplitude of the dynamics. Since the BAA method does not require numerical integration of nonlinear models, computational costs are very low. In this paper, the BAA method is first applied to a single-degree-of-freedom system. Then, a three-degree-of-freedom system is introduced to demonstrate a more general application of BAA. Finally, the BAA method is applied to a full bladed disk with a crack. Results comparing numerical solutions from full-order nonlinear analysis and results obtained using BAA are presented for all systems.     

Two-dimensional friction oscillator: group-preserving scheme and handy formulae

With vibration isolation of buildings and turbomachinery blades in mind, we study the dynamic behaviour of a single-mass two-degree-of-freedom oscillator with dry friction damper, viscous damper and elastic spring connected in parallel. The mass is mounted on an elastic supporting plate allowing movement in two directions on a plane. We formulate a multi-dimensional friction model, from which the sliding conditions and the sticking conditions of the mass are derived. For calculations we develop a group-preserving scheme, which preserves the projective proper orthochronous Lorentz group PSO_o(2,1) symmetry of the model in the sliding phase so as to satisfy automatically the sliding conditions at each time step without iteration at all. The oscillator is then subjected to simple harmonic excitations, and the responses are displayed. According to the simple harmonic balance method together with a circular orbit assumption on displacements, we derive closed-form formulae for handily estimating the steady state responses, which are then compared with the results calculated by the group-preserving scheme to confirm the applicability of the formulae. We also derive formulae specifically for a two-dimensional friction oscillator with rigid base support, which include an exact formula of the magnification factor and a simple formula for estimating the minimum driving force amplitude (or the maximum friction force bound) to avoid sticking.     

An inverse model of MR damper using optimal neural network and system identification

Magnetorheological (MR) damper is one of the more promising new devices for vibration control of structures. External energy required by the adjustable fluid damper is minuscule while speed of its response is in the order of milliseconds. The MR damper is a semi-active control device and has been characterized by a set of non-linear differential equations which represent a forward model of the MR damper, i.e., the model can generate a force to a given displacement and applied voltage.This paper presents an inverse model of the MR damper, i.e., the model can predict the required voltage so that the MR damper can produce the desired force for the requirement of vibration control of structures. The inverse model has been constructed by using a multi-layer perceptron optimal neural network and system identification, which are Gauss-Newton-based Levenberg-Marquardt training algorithm, optimal brain surgeon strategy and autoregressive with exogenous variables (ARX) model. Based on the data from numerical simulation of the MR damper, the trained optimal neural networks can accurately predict voltage. If the inverse model is used in a control system, the semi-active vibration control can be implemented easily by using the semi-active MR damper.     

H-infinity optimization of a variant design of the dynamic vibration absorber—Revisited and new results

The H_∞ optimum parameters of a dynamic vibration absorber (DVA) with ground-support are derived to minimize the resonant vibration amplitude of a single degree-of-freedom (sdof) system under harmonic force excitation. The optimum parameters which are derived based on the classical fixed-points theory and reported in literature for this non-traditional DVA are shown to be not leading to the minimum resonant vibration amplitude of the controlled mass. A new procedure is proposed for the H_∞ optimization of such a dynamic vibration absorber. A new set of optimum tuning frequency and damping of the absorber is derived, thereby resulting in lower maximum amplitude responses than those reported in the literature. The proposed optimized variant DVA is also compared to a ground-hooked damper of the same damping capacity of the damper in the DVA. It is proved that the proposed optimized DVA has better suppression of the resonant vibration amplitude of the controlled system than both the traditional DVA and also the ground-hooked damper if the proposed design procedure of the variant DVA is followed.     

Impacting under harmonic excitation

The steady state response of a one dimensional linear oscillator impacting under harmonic excitation on to rigid obstructions with a coefficient of velocity restitution is investigated. Numerical experiments are discussed and an expression of the steady state velocity for one sided impacting is derived. The variation of impacting velocity with excitation frequency for symmetric two sided impacting steady states is analyzed. It is shown that steady states cannot exist below a minimum frequency whilst for frequencies in the band for which the steady state amplitude exceeds the clearance, a unique steady state can exist. For greater frequencies, two possible steady states exist for a given excitation frequency.     

The magneto-elastic subharmonic resonance of current-conducting thin plate in magnetic filed

Based on Maxwell equations and corresponding electromagnetic constitutive relations, the electrodynamic equations and electromagnetic force expressions of a current-conducting thin plate in electromagnetic field are deduced. Nonlinear magneto-elastic vibration equations of the thin plate are given. In addition, nonlinear subharmonic resonances of the thin plate with two opposite sides simply supported which is under the mechanic live loads and in constant transverse magnetic field are studied. The corresponding vibration differential equation of Duffing type is deduced by the Galerkin method. The method of multiple scales is used to solve the equation, and the frequency-response equation of the system in steady motion under subharmonic responses is obtained, and the stability of solution is analyzed. According to the Liapunov stability theory, the critical conditions of stability are obtained. By the numerical calculation, the curves of resonance amplitude changing with the detuning parameter, the excitation amplitude and the magnetic intensity and corresponding state planes are obtained. The existing regions of nontrivial solutions and the changing law of stable and unstable solutions are analyzed. The time history response plots, the phase charts and the Poincare mapping charts are plotted. And the effect of the magnetic intensity on the system is discussed, and some complex dynamic performances as period-doubling motion and quasi-period motion are analyzed.     

Analytical method for steady state vibration of system with localized non-linearities using convolution integral and Galerkin method

The analytical method using transfer function or impulse response is very effective for analyzing non-linear systems with localized non-linearities. This is because the number of non-linear equations can be reduced to that of the equations with respect to points connected with the non-linear element. In the present paper, analytical method for the steady state vibration of non-linear system including subharmonic vibration is proposed by utilizing convolution integral and the impulse response. The Galerkin method is introduced to solve the non-linear equations formulated by the convolution integral, and then the steady state vibration is obtained. An advantage of the present method is that stability or instability of the steady state vibration can be discriminated by the transient analysis from convolution integral. The three-degree-of-freedom mass-spring system is shown as a numerical example and the proposed method is verified by comparing with the result by Runge-Kutta-Gill method.     

A comparison of two nonlinear damping mechanisms in a vibration isolator

The vibration transmissibility characteristics of a single-degree-of-freedom (SDOF) passive vibration isolation system with different nonlinear dampers are investigated in this paper. In one configuration, the damper is assumed to be linear and viscous, and is connected to the mass so that it is perpendicular to the spring (horizontal damper). The vibration is in the direction of the spring. The second configuration is one in which the damper is in parallel with the spring but the damping force is proportional to the cube of the relative velocity across the damper (cubic damping). Both configurations are studied for small amplitudes of excitation, when some analysis can be conducted based on analytical expressions, and for large amplitudes of excitation, where the analysis is based on numerical simulations. It is found that the two nonlinear systems can outperform the linear system when force transmissibility is considered. However, for displacement transmissibility, the system with the horizontal damper exhibits some desirable properties, but the system with cubic damping does not.