Tuneable vibration absorber using acceleration and displacement feedback

This study is concerned with the analysis and design of a tuneable vibration absorber, which is composed by a flexible beam with a clamping block in the middle and two masses symmetrically mounted at the two ends. The free length of the beam is used to accommodate piezoelectric strain actuators. The two masses at the ends are equipped with inertial accelerometers. This arrangement is used to generate two independent acceleration feedback control loops that produce virtual mass effects, which shift the absorbing frequency of the device. Another arrangement is also studied where the two accelerometer outputs are time-integrated twice in order to implement displacement feedback loops that change the beam stiffness to shift the characteristic frequency of the device. The two feedback approaches are first analysed theoretically, using a mobility-impedance model, and then experimentally on a prototype absorber unit. The stability of the feedback loops is studied using the Nyquist criterion in order to estimate the limits on the tuneable range of frequencies which are set by the maximum stable feedback gains. The study indicates that the stability margins for the acceleration feedback loops substantially depend on the application of an appropriate low-pass filter. On the contrary, the implementation of displacement feedback gives better stability margins.     

VIBRATION SUPPRESSION OF A BEAM STRUCTURE BY INTERMEDIATE MASSES AND SPRINGS

A method based on the dynamic Green function has been proposed to determine the optimum values of masses and/or springs and their locations on a beam structure in order to confine the vibration at an arbitrary location. In the analysis, the beam is driven by a harmonic external excitation. The added masses on the beam and the springs attached are modelled as simple reactions that provide transverse forces to the beam. These forces act as secondary forces that reduce the response caused by the external force. Numerical simulation shows that the vibration of the beam can be confined in a certain region by the presence of masses and springs in best arrangement. This method is demonstrated for both a simply supported and a cantilever beam. An experimental set-up was designed in which a simply supported beam is excited by an electrodynamic shaker and the response of the beam is measured using an He-Ne laser system. This assures very accurate measurements and avoids any additional loading effects as in the case of accelerometers. Comparisons of the theoretical and the experimental results show good agreement.     

A new approach for free vibration of axially functionally graded beams with non-uniform cross-section

This paper studies free vibration of axially functionally graded beams with non-uniform cross-section. A novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density. For various end supports including simply supported, clamped, and free ends, we transform the governing equation with varying coefficients to Fredholm integral equations. Natural frequencies can be determined by requiring that the resulting Fredholm integral equation has a non-trivial solution. Our method has fast convergence and obtained numerical results have high accuracy. The effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non-homogeneity. Moreover, fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties. The effects of the geometrical and gradient parameters are elucidated. The present results are of benefit to optimum design of non-homogeneous tapered beam structures.     

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.     

Micro-vibration suppression of equipment supported on a floor incorporating magneto-rheological elastomer core

In this study, magneto-rheological elastomers (MREs) are adopted to construct a smart sandwich beam for micro-vibration control of equipment. The micro-vibration response of a smart sandwich beam with MRE core which supports mass-concentrated equipment under stochastic support-motion excitations is investigated to evaluate the vibration suppression capability. The dynamic behavior of MREs as a smart viscoelastic material is characterized by a complex modulus dependent on vibration frequency and controllable by external magnetic fields. A frequency-domain solution method for the stochastic micro-vibration response of the smart sandwich beam supporting mass-concentrated equipment is developed based on the Galerkin method and random vibration theory. First, the displacements of the beam are expanded as series of spatial harmonic functions and the Galerkin method is applied to convert the partial differential equations of motion into ordinary differential equations. With these equations, the frequency-response function matrix of the beam–mass system and the expression of the velocity response spectrum are then obtained, with which the root-mean-square (rms) velocity response in terms of the one-third octave frequency band can be calculated. Finally, the optimization problem of the complex modulus of the MRE core is defined by minimizing the velocity response spectrum and the rms velocity response of the sandwich beam, through altering the applied magnetic fields. Numerical results are given to illustrate the influence of MRE parameters on the rms velocity response and the response reduction capacity of the smart sandwich beam. The proposed method is also applicable to response analysis of a sandwich beam with arbitrary core characterized by a complex shear modulus and subject to arbitrary stochastic excitations described by a power spectral density function, and is valid for a wide frequency range.     

黏弹细棒中直达波提取及其动态力学参数的宽频测试

黏弹性材料的动态力学参数可通过对材料样品中纵波测试并结合波速法计算得到。由于测试样品尺寸的限制,除了高损耗材料在高频范围外,不管是在样品端头或侧面都难以获得准确的直达波信号,导致参数测量不准确。针对这个问题,对一端固定一端自由的黏弹性细棒中的纵波传播过程进行了分析,提出了黏弹性细棒直达波提取的方法,并实现了黏弹性材料动态力学参数的宽频测试。采用宽带脉冲作为测试激励信号,利用两台激光测振仪对细棒的固定端和自由端的振动速度进行测试,计算得到细棒自由端处的直达波。然后,利用提取出的直达波信号进行波速法计算,得到了黏弹性细棒宽频范围内的储能模量和损耗因子,宽频脉冲测试结果与黏弹仪测试结果相吻合,验证了该方法的有效性。基于直达波提取的宽频测试方法不仅效率高,而且可以使波速法应用于更低的频段。      

Free vibration analysis of planar curved beams by wave propagation

In this paper, a systematic approach for the free vibration analysis of a planar circular curved beam system is presented. The system considered includes multiple point discontinuities such as elastic supports, attached masses, and curvature changes. Neglecting transverse shear and rotary inertia, harmonic wave solutions are found for both extensional and inextensional curved beam models. Dispersion equations are obtained and cut-off frequencies are determined. Wave reflection and transmission matrices are formulated, accounting for general support conditions. These matrices are combined, with the aid of field transfer matrices, to provide a concise and efficient method for the free vibration problem of multi-span planar circular curved beams with general boundary conditions and supports. The solutions are exact since the effects of attenuating wave components are included in the formulation. Several examples are presented and compared with other methods.     

黏弹材料动力学参数的宽频测试

利用改进的波速法,对黏弹性材料连续宽频范围的动力学参数进行测试。利用有限元方法,对波速法的测试过程进行了仿真,验证了利用长短棒波速法测量宽频动力学参数的有效性。采用可控脉冲生成技术,在激振器上产生了用于宽频测试的短脉冲信号。黏弹性长短棒在上述宽带短脉冲激励下作纵向强迫振动,利用激光测振仪测量长短两棒自由端的纵向振动速度,从而可得两振动信号在连续宽频范围的幅值比和相位差,进而可根据波速法原理计算得到材料在连续宽频范围的储能模量和损耗因子。测试结果表明,该系统通过较少次数测试,可直接计算得到1~5 kHz连续宽频范围的动力学参数,测试结果与黏弹仪数据吻合。通过对幅值比和相位差进行线性最小二乘拟合,可以进一步拓展测量的频率范围。该方法准确可靠、简便快速,具有实际应用价值。      

On the vibration and stability of a free-free beam subjected to end-loads

In this paper the vibration and stability of a free-free beam subjected to direction-controlled axial loads at its ends are investigated. The eigencurves and mode shapes of the beam are presented for various values of the directional control parameter. It is found that the behaviour of the free-free beam subjected to compressive axial loads is unstable for any direction parameter—except for the follower loading case. However, the same beam subjected to tensile loads is stable.     

Free vibration and stability of a cantilever beam attached to an axially moving base immersed in fluid

Free vibration and stability are investigated for a cantilever beam attached to an axially moving base in fluid. The equations of motion of the slender cantilever beam affiliated to an axially moving base at a known rate while immersed in an incompressible fluid are derived first. An “axially added mass coefficient” is taken into account in the obtained equations. Then, a coordinate transformation is introduced to fix the boundaries. Based on the Galerkin approach, the natural frequencies of the beam system are numerically analyzed. The effects of moving speed of the base and several other system parameters on the dynamics and stability of the beam are discussed in detail. It is found that when the moving speed exceeds a certain value the beam becomes unstable and the instability type is sensitive to the system parameters. When the values of system parameters, such as mass ratio and axially added mass coefficient, are big enough, however, no instabilities are detected. The variations of the lowest unstable critical moving speed with respect to several key parameters are also investigated.     

Free vibration of FGM Timoshenko beams with through-width delamination

Free vibration of functionally graded beams with a through-width delamination is investigated.It is assumed that the material property is varied in the thickness direction as power law functions and a single through-width delamination is located parallel to the beam axis.The beam is subdivided into three regions and four elements.Governing equations of the beam segments are derived based on the Timoshenko beam theory and the assumption of‘constrained mode’.By using the differential quadrature element method to solve the eigenvalue problem of ordinary differential equations governing the free vibration,numerical results for the natural frequencies of the beam are obtained.Natural frequencies of delaminated FGM beam with clamped ends are presented.Effects of parameters of the material gradients,the size and location of delamination on the natural frequency are examined in detail.     

A study on radial directional natural frequency and damping ratio in a vehicle tire

The measurement of radial directional natural frequency and damping ratio in a vehicle tire has been studied. Natural frequencies and damping ratios in the radial direction of various tires, from passenger car tires to truck bus tires, are reported. The radial direction modal parameters of tires subjected to different levels of inflation pressure, have been determined by using a frequency response function method. To obtain the theoretical natural frequency and mode shape, the plane vibration of a tire has been modeled as though it were that of a circular beam. By using the Tielking method that is based on Hamilton’s principle, theoretical results have been determined by considering the rotational velocity, tangential and radial stiffness, radial directional velocity and tension force which is due to tire inflation pressure. The results show that experimental conditions can be considered as the parameters that shift the natural frequency and damping ratio.     

A conserved quantity and the stability of axially moving nonlinear beams

Free nonlinear transverse vibration is investigated for an axially moving beam modeled by an integro-partial-differential equation. Based on the equation, a conserved quantity is defined and confirmed for axially moving beams with pinned or clamped ends. The conserved quantity is applied to demonstrate the Lyapunov stability of the straight equilibrium configuration in transverse nonlinear of beam with a low axial speed.     

FREE VIBRATION ANALYSIS OF NON-UNIFORM BEAMS WITH AN ARBITRARY NUMBER OF CRACKS AND CONCENTRATED MASSES

An exact approach for free vibration analysis of a non-uniform beam with an arbitrary number of cracks and concentrated masses is proposed. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the beam. Using the fundamental solutions and recurrence formulas developed in this paper, the mode shape function of vibration of a non-uniform beam with an arbitrary number of cracks and concentrated masses can be easily determined. The main advantage of the proposed method is that the eigenvalue equation of a non-uniform beam with any kind of two end supports, any finite number of cracks and concentrated masses can be conveniently determined from a second order determinant. As a consequence, the decrease in the determinant order as compared with previously developed procedures leads to significant savings in the computational effort and cost associated with dynamic analysis of non-uniform beams with cracks. Numerical examples are given to illustrate the proposed method and to study the effect of cracks on the natural frequencies and mode shapes of cracked beams.     

微振动激励作用下编组站镜架对光束指向的影响

 编组站是靶场光路传输系统的重要组成部分,编组站镜架的稳定性对光路的传输有着直接的影响。为了分析微振动对光束指向性的影响,采用有限元分析软件建立镜架的有限元模型,将数字式地震仪测得的镜架安装平台的速度功率谱密度函数作为载荷施加到分析模型上,计算得到了编组站光学元件（A,B,C,D）在基座微振动激励作用下的转角漂移分别为0.338,0.327,0.289,0.241 mrad,均小于稳定性指标0.460 mrad的要求;采用加速度传感器对光学元件A的转角漂移测试结果为0.340 mrad,与分析结果的误差为0.6%,说明所采用的计算分析方法是有效的,为精密镜架的设计分析提供了有效的方法。     

Vibration and stability of elastically supported multi-span beams under conservative and non-conservative loads

This paper deals with the vibration and stability of multi-span beams elastically supported against translation and rotation at several intermediate points as well as both ends. The beam is subjected to an axial or tangential load at the ends. The problem is studied on the basis of the Timoshenko beam theory. The influence of the support stiffness on the natural frequencies and the divergence and flutter instability loads are studied in detail.     

一种新型全光纤速度干涉仪

 基于传统速度干涉仪(VISAR)和光纤速度干涉仪(AFVISAR)的特点，提出了一种由光纤和光纤耦合器组成的工作波长为532 nm的新型全光纤速度干涉仪（NAFVISAR）。该干涉仪采用多模光纤器件构成分离系统,单模光纤器件组成核心部分。由于有两路携带不同信息的光束经不同路径传输到耦合器中，当这两路光束满足干涉条件时，可利用它们的干涉场信息来调解出被测靶的信息，从而区分波面的加减速变化。用该系统进行了Hopkinson森杆一维应力加载下的入射杆端面的速度剖面测试，实测速度最大值为49.36 m/s，与理论速度的最大值50.16 m/s基本符合，实现了全光纤速度干涉仪的实用化。     

Time-resolved measurement technique for pulsed electron beam envelope basing on framing and streaking principle

The time-resolved electron beam envelope parameters, including cross sectional distribution and beam centroid position, are very important for the study of beam transmission characteristics in a magnetic field and for verifying the rationality of the magnetic field parameters employed. One kind of high time-resolved beam envelope measurement system has recently been developed, constituted of a high-speed framing camera and a streak camera.It can obtain three panoramic images of the beam and time continuous information along the given beam profile simultaneously. Recently obtained data has proved that several fast vibrations of the beam envelope along the diameter direction occur during the front and the tail parts of the electron beam. The vibration period is several nanoseconds. The effect of magnetic field on the electron beam is also observed and verified. Beam debugging experiments have proved that the existing beam transmission design is reasonable and viable. This beam envelope measurement system will establish a good foundation for beam physics research.     

Vibration and stability of a non-uniform Timoshenko beam subjected to a follower force

An analysis is presented for the vibration and stability of a non-uniform Timoshenko beam subjected to a tangential follower force distributed over the center line by use of the transfer matrix approach. For this purpose, the governing equations of a beam are written in a coupled set of first-order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the eigenvalues of vibration and the critical flutter loads are obtained. The method is applied to beams with linearly, parabolically and exponentially varying depths, subjected to a concentrated, uniformly distributed or linearly distributed follower force, and the natural frequencies and flutter loads are calculated numerically, from which the effects of the varying cross-section, slenderness ration, follower force and the stiffness of the supports on them are studied.     

光学系统受迫振动的激光测量方法

将激光信号与高帧频CCD结合,解决了光学系统中振动信号和激光信号之间的转换问题,不仅能够测量振动对系统光束指向稳定性的影响,而且能够得到振动信号本身的频率特性。利用该方法对振源为150 Hz和200 Hz两种条件下的光学系统受迫振动进行测量,得到了与输入信号相吻合的振动信号属性。通过实验与分析得知:时域振幅测量精度为6.25 m,频域分辨力为2 Hz,方法简便高效,测量结果准确,已应用于角多路准分子激光主振荡功率放大器系统打靶试验平台光束指向稳定性的研究中。