An expansion of system with time delayed feedback control into spatio-temporal state space

Time delayed feedback control is well known as an effective continuous control method for stabilizing the unstable periodic orbit embedded in chaotic attractors. As for the system with time delay, the solution is in a function space and shows characteristics governed by an infinite dimension. Therefore it is difficult to understand the system behavior analytically. In this paper, it is shown that, when the state space for the system with time delay is expanded into the spatio-temporal state space, the solution propagates in the space as a wave theoretically and numerically. The dynamic behavior experimentally obtained in the sinusoidally excited magneto-elastic beam system under time delayed feedback control is also discussed by the transformation into the expanded spatio-temporal state space. (c) 1999 American Institute of Physics.     

Vibration of electro-elastic versus magneto-elastic circular/annular plates using the Chebyshev–Ritz method

The aim of this paper is to analyze three-dimensional free vibration of magneto-elastic/electro-elastic circular/annular plates with different boundary conditions using the Chebyshev–Ritz method, in which a set of duplicate Chebyshev polynomial series multiplied by the boundary function satisfying the boundary conditions are chosen as the trial functions of the displacement components, the electric potential and the magnetic potential. Convergence of the method is checked using various Chebyshev polynomial terms. The effect of geometrical parameters and material properties of magneto-elastic/electro-elastic circular/annular plates on the eigenfrequencies of free vibration is considered.     

CONTROL OF MULTI-SPAN BEAM VIBRATION BY A RANDOM WAVE REFLECTOR

In this paper, a new wave reflector called random wave reflector (RWR) is introduced for the control of transverse vibration and wave propagation in an infinite, multi-span, simple-support beam. In order to illustrate the theory, RWR is first tested in a simple configuration of controlling plane wave propagation through layers of gas media. Results demonstrate that RWR has great advantages over other types of noise abatement methods. RWR is then applied to control the vibration of the multi-span beam in which the support locations are given in a random manner. Two types of external excitations, an incident vibration wave and an external point force, are considered separately. Transmission loss, localization factor, mode shape and input power flow are used to investigate the effectiveness of RWR. The results show that no vibrational power flow can be tapped into or propagate through a random system at any frequency. The passbands, which always exist in traditional systems, are eliminated in a random system for which much better performance is obtained over a broad frequency range.     

多振子梁弯曲振动中的局域共振带隙

从梁的弯曲振动方程出发,利用传递矩阵法,给出了无限周期结构的一维多振子声子晶体梁的弯曲振动能带结构,并利用有限元方法计算了有限周期结构梁的弯曲振动频率响应.建立了多振子声子晶体梁的简化模型,推导出带隙起始截止频率公式.结果表明:一维多振子声子晶体梁具有比单振子声子晶体梁更宽更丰富的振动带隙,可应用于呈倍频关系的减振降噪中;振动在带隙频率范围内频率响应具有明显的衰减;所建立的简化模型与理论模型结果符合较好.研究工作为梁类结构的减振提供一种新的思路.     

X形超阻尼局域共振声子晶体梁弯曲振动带隙特性

以声子晶体理论为基础,设计了一种具有超阻尼特性的X形局域共振结构,分析了周期性附加X形局域共振的梁弯曲振动传播特性.利用拉格朗日方程分析了X形局域共振结构动力学等效特性,揭示了该结构的阻尼放大的机理,分析了几何结构参数对于带隙特性的影响,并利用有限元法验证了X形局域共振结构的超阻尼特性.研究结果表明,周期性附加X形局域结构能够有效地抑制低频弯曲振动在梁中的传播,产生超阻尼特性,实现低频、宽带的减振效果,为结构的低频减振提供了一个新的设计方案.     

液体对微振动的放大效应

根据表面波声光效应的原理,实验上建立了固体表面微振动的激光衍射测量系统.当激光斜入射到微振动引起的液体表面波上,观察到了清晰、反衬度非常高的衍射图样.利用MATLAB软件对拍摄的衍射图样进行扫描,得到了衍射光斑的光强分布图,并根据衍射图样宽度与表面波振幅的解析关系式,求出了液体表面波的振幅,其大小在微米量级.改变样品池中液体的深度,测得不同深度下液体表面波的振幅,给出了表面波振幅与液体深度之间的解析关系,并发现了液体对微振动的放大效应.利用液体对微振动的放大效应,求出了固体表面微振动的振幅,实现了固体表面微振动的探测.     

液体对微振动的放大效应

根据表面波声光效应的原理,实验上建立了固体表面微振动的激光衍射测量系统.当激光斜入射到微振动引起的液体表面波上,观察到了清晰、反衬度非常高的衍射图样.利用MATLAB软件对拍摄的衍射图样进行扫描,得到了衍射光斑的光强分布图,并根据衍射图样宽度与表面波振幅的解析关系式,求出了液体表面波的振幅,其大小在微米量级.改变样品池中液体的深度,测得不同深度下液体表面波的振幅,给出了表面波振幅与液体深度之间的解析关系,并发现了液体对微振动的放大效应.利用液体对微振动的放大效应,求出了固体表面微振动的振幅,实现了固体表面微振动的探测.     

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.     

Experimental study of coupled vibration in a finite periodic dual-layered structure with transverse connection

The analytical equations of the transfer matrix method are further derived for the multi-coupled vibration of flexural and longitudinal waves in a periodic dual-layered beam structure with connection branches, with full consideration given to the flexural and longitudinal motions that are tri-coupled at each connection. Measurements of mobilities at the junctions on the uni-layered beam and the cross-layered beam are made. The numerical results agree well with the experimental results at all frequencies from 10 to 2000 Hz, which verifies the theoretical methodology for the multi-coupled vibration in a finite dual-layered beam. The cross-layer energy transmission is calculated, which reveals that the transmitted longitudinal energy is enhanced not only at the longitudinal resonant modes but also at the flexural resonant modes of the connection branches due to the structural wave coupling. The flexural energy is excited by wave coupling and becomes stronger at the longitudinal resonant modes and the flexural resonant modes of the connection branches. The cross-layer vibration motions from coupled waves in the branches can be effectively controlled by the attached cantilevers with mass at the resonance modes. This method can be used to control the structure-borne sound transmission in multi-layer beam structures.     

Three methods for analyzing forced vibration of a fluid-filled cylindrical shell

The forced vibration of an infinite elastic circular cylindrical shell filled with fluid is studied. Three methods are employed to analyze the forced vibration problem of this shell-fluid coupled system, that is, wave propagation approach (wave mode superposition), theorem of residues and a numerical integral method. In order to explain these methods more explicitly, before being used to investigate the vibration of an infinite fluid-filled elastic circular cylindrical shell, all these three methods are employed firstly to analyze the forced vibration problem of an infinite beam and an infinite elastic circular cylindrical shell in vacuo. Advantage and disadvantage of these three methods are discussed and their interesting relationship is revealed. That is, to any circumferential wavenumber and frequency of the external force, there is an unchangeable relationship between the general coordinates of various waves in the wave propagation approach and the residuals in the theorem of residues.     

Magneto-elastic dynamics and bifurcation of rotating annular plate

In this paper, magneto-elastic dynamic behavior, bifurcation, and chaos of a rotating annular thin plate with various boundary conditions are investigated. Based on the thin plate theory and the Maxwell equations, the magneto-elastic dynamic equations of rotating annular plate are derived by means of Hamilton's principle. Bessel function as a mode shape function and the Galerkin method are used to achieve the transverse vibration differential equation of the rotating annular plate with different boundary conditions. By numerical analysis, the bifurcation diagrams with magnetic induction, amplitude and frequency of transverse excitation force as the control parameters are respectively plotted under different boundary conditions such as clamped supported sides, simply supported sides, and clamped-one-side combined with simply-anotherside. Poincare′ maps, time history charts, power spectrum charts, and phase diagrams are obtained under certain conditions,and the influence of the bifurcation parameters on the bifurcation and chaos of the system is discussed. The results show that the motion of the system is a complicated and repeated process from multi-periodic motion to quasi-period motion to chaotic motion, which is accompanied by intermittent chaos, when the bifurcation parameters change. If the amplitude of transverse excitation force is bigger or magnetic induction intensity is smaller or boundary constraints level is lower, the system can be more prone to chaos.     

原子力显微镜微探针与光束偏转法在纳米振动检测中的应用

本文探讨了表面弹性波传播的机制,提出了一种利用原子力显微镜微探针与光束偏转法检测纳米振动的新方法,并利用这种方法对表面波质点的纳米振动进行了实时检测.当LiNbO₃晶片被施以13MHz的超高频激振电压时,在原子力显微镜示波器监视端观察到了质点的纳米振动,幅值为6.7nm,频率和驱动电压的频率一致.     

Non-stationary response of a beam to a moving random force

The non-stationary random vibration of a beam is investigated. The beam is subjected to a random force with constant mean value which is moving with constant speed along the beam. The statistical characteristics of the first and second order for the deflection and bending moment of the beam are computed by using the correlation method. The numerical results of the coefficient of variation of the deflection at beam span mid-point are given for five basic types of convariances of the force (white noise, constant, exponential cosine, exponential, and cosine wave). The effect of the speed of the movement of the force along the beam as well as the effect of the beam damping is investigated in detail. It is concluded that the resulting beam vibration turns out to be a non-stationary process even though the motion considered is that of a stationary random force.     

Reflection of spin and spin-elastic waves at the interface of a ferromagnetic half-space

This paper considers the reflection of pure spin and spin-elastic (or magneto-elastic) waves at the interface of a ferromagnetic half-space and a vacuum. For pure spin waves two cases are considered, with exchange effects, and without. It is shown that when exchange effects are taken into account, volume spin waves in the ferromagnetic half space incident at the boundary with the vacuum generate a reflected volume spin wave, and an accompanying compound surface wave propagating along the boundary and consisting of two partial inhomogeneous spin waves in the ferromagnetic half-space and a partial magneto-static inhomogeneous surface wave in the vacuum. When exchange effects are neglected the incident wave generates only a reflected volume wave in the ferromagnetic half-space.

Reflection and transmission of spin-elastic (or magneto-elastic) waves has been considered only in the case of the absence of exchange effects. An incident volume wave generates a volume spin-elastic reflected wave and one inhomogeneous magneto-static accompanying surface wave.

Excitations of the magnetic field are not transmitted into the vacuum in both cases when the exchange effect is neglected. In all cases the reflection of a spin wave has the character of a full internal reflection.     

Generating ultra wide low-frequency gap for transverse wave isolation via inertial amplification effects

Low-frequency transverse wave propagation plays a significant role in the out-of-plane vibration control. To efficiently attenuate the propagation of transverse waves at low-frequency range, this letter proposed a new type phononic beam by attaching inertial amplification mechanisms on it. The wave propagation of the beam with enhanced effective inertia is analyzed using the transfer matrix method. It is demonstrated that the low-frequency gap within inertial amplification effects can possess much wider bandwidth than using the local resonance method, thus is more suitable for designing applications to suppress transverse wave propagation.     

测量水的振动频率与水波传播速度

利用激光器、硅光电池等器件,采用光折射方法设计了激光测量水的振动频率与水波传播速度实验.本文介绍了实验原理和装置,测量了水的振动频率和水波向外传播的速度,观察了小块物体落水激起的水波情况.     

声表面波梁式加速度传感器的优化设计

为获得传感器的优化设计,对一种声表面波梁式加速度传感器敏感机理进行了研究。从声波波动方程出发,结合有限元分析以及微扰理论对加速度作用力作用下声表面波传播特性进行分析,以此构建梁式声表面波加速度传感器敏感机理的理论模型,特别分析了压电梁材料及几何结构、振子质量对传感响应的贡献以确定传感器优化的几何参数。为验证理论分析结果,实验研制了基于ST-X石英悬臂梁结构的差分振荡式声表面波加速度传感器,并利用精密振动台对所研制传感器性能进行评价。实验结果显示,在给定加速度测试范围内,采用ST-X石英梁并延长梁长度、降低梁厚度以及采用较大的阵子质量将有效的改善传感器检测灵敏度,在±2 g范围内加速度灵敏度可达27 k Hz/g,且实验结果很好的验证了理论模型。     

In-plane free vibration analysis of combined ring-beam structural systems by wave propagation

This paper presents a systematic, wave propagation approach for the free vibration analysis of networks consisting of slender, straight and curved beam elements and complete rings. The analysis is based on a ray tracing method and a procedure to predict the natural frequencies and mode shapes of complex ring/beam networks is demonstrated. In the wave approach, the elements are coupled using reflection and transmission coefficients, and these are derived for discontinuities encountered in a MEMS rate sensor consisting of a ring supported on an array of folded beams. These are combined, taking into account wave propagation and decay, to provide a concise and efficient method for studying the free vibration problem. A simplification of the analysis that exploits cyclic symmetry in the structure is also presented. The effects of decaying near-field wave components are included in the formulation, ensuring that the solutions are exact. To illustrate the effectiveness of the approach, several numerical examples are presented. The predictions made using the proposed approach are shown to be in excellent agreement with a conventional FE analysis.     

洛匈棱镜的正反向特性

给出了洛匈棱镜正、反向使用时分束角的精确表达式。从理论上证明,对任意合理的结构角及主折射率值来说,洛匈棱镜反向使用时的分束角小于正向使用时的分束角。研究了在锥光入射条件下洛匈棱镜中的e光和o光的传播方向、折射率、主平面、偏振方向等问题。证明了洛匈棱镜正向使用时第二块晶体中无偏折的光波只含有o光,参考圆圆周上各点光波的主平面相互平行,通过一个检偏器后可以实现消光;洛匈棱镜反向使用时,第二块晶体无偏折的光波同时包含了o光和e光,参考圆圆周上各点光波的主平面沿参考圆的径向,e光和o光的偏振方向分别沿径向和垂直于径向,通过检偏器后形成锥光干涉现象,不能消光。     

Theoretical and experimental study on the transverse vibration properties of an axially moving nested cantilever beam

An axially moving nested cantilever beam is a type of time-varying nonlinear system that can be regarded as a cantilever stepped beam. The transverse vibration equation for the axially moving nested cantilever beam with a tip mass is derived by D’Alembert?s principle, and the modified Galerkin?s method is used to solve the partial differential equation. The theoretical model is modified by adjusting the theoretical beam length with the measured results of its first-order vibration frequencies under various beam lengths. It is determined that the length correction value of the second segment of the nested beam increases as the structural length increases, but the corresponding increase in the amplitude becomes smaller. The first-order decay coefficients are identified by the logarithmic decrement method, and the decay coefficient of the beam decreases with an increase in the cantilever length. The calculated responses of the modified model agree well with the experimental results, which verifies the correctness of the proposed calculation model and indicates the effectiveness of the methods of length correction and damping determination. Further studies on non-damping free vibration properties of the axially moving nested cantilever beam during extension and retraction are investigated in the present paper. Furthermore, the extension movement of the beam leads the vibration displacement to increase gradually, and the instantaneous vibration frequency and the vibration speed decrease constantly. Moreover, as the total mechanical energy becomes smaller, the extension movement of the nested beam remains stable. The characteristics for the retraction movement of the beam are the reverse.