Hidden symmetries and Killing tensors

Just as a Killing vector of the kinetic energy metric of a classical mechanical system can generate a symmetry of the system (an “obvious” symmetry), and thus a constant of the motion, so can a Killing tensor generate a symmetry and a constant of the motion. In the latter case the symmetry is “hidden”: for example the well-known hidden symmetries for a single particle moving under an inverse-square central force, and for the harmonic oscillator, arise in this way.     

Characteristics of an active feedback system for the control of plate vibrations

In the interests of improving airborne insulation of panels and of controlling room reverberation, a technique is studied for establishing control of the transverse vibrations of a thin plate by the application of active energy feedback. A localized point control force is derived from the sensed motion of some point on the plate surface. The superposition principle is applied in the form of a mobility analysis which shows that open loop gain conditions cannot result in a specific motion, including that of complete damping, of any arbitrary point on the plate surface but can be effective for particular points and for control of resonant modal motions under conditions of light damping. With velocity sensing, the characteristics for stable operation under the convenient condition of constant gain depend on maintenance of like symmetry, in the sense of an identity of velocity magnitude and sign, in the relative motion of sensing and control-force points. Bandwidth limitations are avoidable only by closure of the loop between these points. Two varieties of control force generator are involved: namely, where the generator mass is rigidly mounted and again where a spring mounting on the plate provides a self-supporting role.This is the first of two companion papers on active control of plate vibrations. Systems in which an array of multiple control units is used will be described in the second paper.     

Free wave propagation in two-dimensional periodic plates

The propagation of flexural waves in a two-dimensional periodic plate which rests on an orthogonal array of equi-spaced simple line supports has been investigated. A type of plane wave motion has been considered. An energy method has been developed to predict the frequency of wave propagation in terms of the propagation constants. A Galerkin type of analysis has been used, incorporating assumed complex modes of wave motion for the identical rectangular elements of the periodic plate. Expressions for the frequency have been obtained firstly by using simple polynomial modes for the plate displacements, and then (alternatively) by using characteristics beam function modes. The use of these different modes has first been demonstrated by applying them to the analysis of wave propagation in periodic beams. A single polynomial mode which satisfies the geometric and wave-boundary conditions of the periodic plate element leads to an elegant expression relating the frequency and the wave propagation constants in the first propagation band. The frequencies so obtained compare well with those found from a multi-mode, characteristic beam function analysis. The latter involves much more algebra, is solved as an eigenvalue problem, and yields the frequencies in as many propagation bands as are desired. The bounding frequencies and corresponding wave motions in the first and higher propagation bands have been identified, and it has been shown that the propagation bands can overlap. Consideration has been given to one-dimensional “strip” structures which are equivalent to the two-dimensional plate when a plane wave in a general direction is propagating. Furthermore, it is shown that the natural frequencies of finite rectangular periodic plates can be obtained very simply from the results of the wave propagation analysis.     

The anisorrhopic guiding-center fluid

A study is made of so-called “finite-orbit effects” in a two-dimensional guiding-center plasma. The macroscopic mass motion of the plasma is represented on the basis of a simple incompressible one-fluid model (so-called “representative fluid”), and the guiding-center motions of single particles are then referred to a Lagrangian coordinate network comoving with the representative fluid. The fluid motion defines the network motion. It turns out, however, to have no effect on the guiding-center motion relative to the network (autonomy theorem). It is found, in other words, that the relative trajectories of guiding centers are determinable in advance independently of the network motion (or the fluid motion), and this provides the necessary information to determine all the state parameters of the representative fluid (density of mass, density of gyrational angular momentum, etc.) as functions of the time, t, at any given point of the network. Once this information is available, the fluid motion is then completely determined by the remaining hydrodynamic equations (equation of motion, equation of incompressibility). The so-called “finite-orbit effects” take the form of gyroscopic-quasielastic forces in the equation of motion. No special isorrhopy condition is assumed. (This refers to a special initial condition assumed in an earlier work, for the sake of analytical simplicity. Here, the special initial condition is dropped.) Much attention is devoted to problems of wave propagation and stability. There are two independent sets of wave modes (if a nonvanishing anisorrhopy is allowed): so-called fluid modes, and so-called drift modes, respectively defined as first-order perturbations in the network motion (or the fluid motion) relative to the fixed coordinate frame, and in the guiding-center motion relative to the network. The stability conditions against both sets of modes are found to be quite stringent, much more so than in the earlier isorrhopic case. Nonetheless, a reasonably extensive class of stable solutions is shown to exist.     

Sound radiation from point-excited structures: Comparison of plate and sphere

Acoustic radiation from a structure can be expressed in terms of “modal radiation” and “modal coefficients”. This paper investigates the contributions of these two modal properties to radiation excited by a point force. Sound radiation from two basic structures is considered: a baffled rectangular plate and a closed spherical shell. The plate behaviour is familiar, and governed by the relation between the natural frequency of a mode and its coincidence frequency. For a closed spherical shell, there are either zero or two “critical frequencies”, depending on the radius and thickness. When there are two the shell radiates well both above and below the two frequencies, and poorly in the frequency range between them.     

Guided modes in asymmetric graphene waveguides

An asymmetric quantum well in graphene can act as a slab waveguide for electron waves in a manner analogous to the electromagnetic waves in dielectrics. Guided modes and the probability current density are analyzed in the graphene electron waveguide induced by asymmetric electrostatic potential. The modes in an asymmetric graphene waveguide include guided modes, “cover modes”, “substrate modes” and “radiation modes”. The conditions for a guided mode are quantified. It is found that the fundamental mode is absent when both the Klein tunneling and classical motion are present. The confinement of electrons for lower order mode is stronger than for higher order mode. We hope that these characteristics in asymmetric graphene waveguide can provide potential applications in graphene-based waveguide devices.     

Love waves excited by a moving source

The study analyzes the characteristics of surface Love waves excited by the moment of an oscillating torsional force with a point of action that moves uniformly and rectilinearly along the free flat boundary of a medium having the structure of a “layer on a half-space.” The azimuthal–angular distribution of the amplitude and Doppler shift in frequency of the wave modes is studied as a function of the motion velocity of a vibrating source and the parameters of the medium.     

Mode-splitting and quasi-degeneracies in circular plate vibration problems: The example of free vibrations of the stator of a traveling wave ultrasonic motor

In systems with rotational symmetry, bending modes occur in doubly-degenerate pairs with two independent vibration modes for each repeated natural frequency. In circular plates, the standing waves of two such degenerate bending modes can be superposed with a 1/4 period separation in time to yield a traveling wave response. This is the principle of a traveling wave ultrasonic motor (TWUM), in which a traveling bending wave in a stator drives the rotor through a friction contact. The stator contains teeth to increase the speed at the contact region, and these affect the rotational symmetry of the plate. When systems with rotational symmetry are modified either in their geometry, or by spatially varying their properties or boundary conditions, some mode-pairs split into singlet modes having distinct frequencies. In addition, coupling between some pairs of distinct unperturbed modes also causes quasi-degeneracies in the perturbed modes, which leads their frequency curves to approach and veer away in some regions of the parameter space. This paper discusses the effects of tooth geometry on the behavior of plate modes under free vibration. It investigates mode splitting and quasi-degeneracies and derives analytic expressions to predict these phenomena, using variational methods and a degenerate perturbation scheme for the solution to the plate’s discrete eigenvalue problem; these expressions are confirmed by solving the discrete eigenvalue problem of the plate with teeth.     

Semi-analytic solutions for the free in-plane vibrations of confocal annular elliptic plates with elastically restrained edges

A two-dimensional analytical model is developed to describe the free extensional vibrations of thin elastic plates of elliptical planform with or without a confocal cutout under general elastically restrained edge conditions, based on the Navier displacement equation of motion for a state of plane stress. The model has been simplified by invoking the Helmholtz decomposition theorem, and the method of separation of variables in elliptic coordinates is used to solve the resulting uncoupled governing equations in terms of products of (even and odd) angular and radial Mathieu functions. Extensive numerical results are presented in an orderly fashion for the first three anti-symmetric/symmetric natural frequencies of elliptical plates of selected geometries under different combinations of classical (clamped and free) and flexible boundary conditions. Also, the occurrences of “frequency veering” between various modes of the same symmetry group and interchange of the associated mode shapes in the veering region are noted and discussed. Moreover, selected 2D deformed mode shapes are presented in vivid graphical form. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature. The set of data reported herein is believed to be the first rigorous attempt to obtain the in-plane vibration frequencies of solid and annular thin elastic elliptical plates for a wide range of plate eccentricities.     

Dynamics and integrability property of the chiral string model

The effect of fermionic string conductivity by purely right (or purely left) moving “zero modes” is shown to be governed by a simple Lagrangian characterizing a certain “chiral” (null current carrying) string model whose dynamical equations of motion turn out to be explicitly integrable in a flat spacetime background.     

Self-sustained oscillations of turbulent pipe flow terminated by an axisymmetric cavity

Turbulent flow through a long pipe terminated by an axisymmetric cavity can give rise to self-sustained oscillations exhibiting a very strong coherence, as evidenced by the narrow-band character of corresponding amplitude spectra. These oscillations, associated with the turbulent axisymmetric jet passing through the cavity, are strongly influenced by the acoustic modes of the pipe. The frequencies of oscillation lie within or near the range of most “unstable” frequencies of the turbulent jet previously predicted by using concepts of inviscid hydrodynamic stability theory; consequently, these experiments show truly self-excited and strongly coherent “instability” of a fully turbulent, low Mach number (～10^?2), axisymmetric flow undergoing separation, corroborating previous experiments involving the external forcing of free turbulent jets. As flow velocity or cavity length is varied, both upward and downward jumps in oscillation frequency are observed; the sign (up or down) of these jumps tends to systematically alternate with increase of velocity or length. The role of these frequency jumps is, in effect, to allow the oscillation of the flow to remain “locked-on” to a pipe mode over a wide range of impingement length or flow velocity. Moreover, these jumps exhibit two types of behavior: for the first kind, the predominant frequency makes a relatively continuous transition between stages and the frequency of the neighboring stage appears as a secondary component; for the second kind, there is a dead zone (where no oscillation occurs) between stages. The consequence of externally exciting the system is strongly dependent on whether the self-sustaining oscillation is relatively near, or well away from, a frequency jump. During excitation, the amplitudes of pressure fluctuations in the cavity substantially exceed the corresponding no-flow values only in regions away from the frequency jumps; at locations of jumps, there can be significant attenuation of the no-flow excitation amplitude. For the type of frequency jump involving a “dead zone”, enhancement of a given mode of oscillation can be achieved by externally exciting not only the given mode, but also neighboring modes. For the other type of jump, involving a relatively continuous transition from one stage to the next, the predominant mode of oscillation following the jump is that mode giving maximum amplitude response to excitation before the jump.     

1-D and 2-D NMR of broad-line solids: an application to quasicrystals

Nuclear magnetic resonance (NMR) techniques for measuring one-dimensional absorption spectra and two-dimensional exchange spectra of solids with extremely inhomogeneously broadened lines are discussed. Among various “broad-line” solids, quasicrystals represent alloys of metallic elements, the structures of which include “forbidden” symmetry elements. NMR absorption lines of quasicrystals exhibit a strong electric-quadrupole-induced inhomogeneous broadening that originates from the lack of translational periodicity of the otherwise perfectly long-range-ordered quasiperiodic lattice. Recording an NMR spectrum of a quasicrystalline sample requires a magnetic field-sweep technique. The two-dimensional exchange experiment on quasicrystals can be performed on selectively excited portions of the NMR spectrum only. Due to the off-resonance effects in a selective excitation, the use of a simple three-pulse stimulated-echo exchange sequence is preferred. The²⁷Al spectra of the Al-Pd-Mn and Al-Pd-Re families show interesting features like temperature-dependent frequency shifts and exchange effects due to atomic motion.     

On the interpretation of fermionic zero-modes

We examine the effect of fermionic zero modes on tunneling amplitudes within some simple quantum mechanical models. It is shown that the fermionic zero modes do not cause a total suppression of the tunneling, although it may be reduced. With a θ term present, due to a non-trivial topology, it is shown that the θ dependence is not eliminated by the zero modes. Instead the non-trivial topology introduces a “twist” in the fermionic coordinates which breaks a symmetry of the hamiltonian.     

基于编码超表面的太赫兹宽频段雷达散射截面缩减的研究

本文设计了一种柔性, 非定向低散射的1bit编码超表面, 实现了太赫兹宽频带雷达散射截面的缩减. 这种设计基于对“0”和“1”两种基本单元进行编码, 其反射相位差在很宽的频段范围内接近180°, 为一种非周期的排列方式, 该电磁超表面使入射的电磁波发生漫反射, 从而实现雷达散射截面的缩减. 全波仿真结果表明, 在垂直入射条件下, 编码超表面的镜像反射率低于-10 dB的带宽频段范围为1.0-1.4 THz, 该带宽内超表面相对同尺寸金属板可将雷达散射截面所减量达到10 dB以上, 最大缩减量达到19 dB. 把柔性编码表面弯曲在直径为4 mm的金属圆柱面上, 雷达散射截面的所减量高于10 dB以上的带宽频段范围为0.9-1.2 THz, 仍然可实现宽频带缩减特性. 总之, 编码超表面为调控太赫兹波提供一种新的途径, 将在雷达隐身、成像、宽带通信等方面具有重要的意义.     

Structural waveguide behaviour of a beam–plate system

Structures which have a constant cross-section normal to a longitudinal axis can be considered as waveguides in which vibration can propagate in the form of various waves in the longitudinal direction. The dynamic behaviour of such systems can be found by using a Fourier transform approach in terms of wavenumbers in the longitudinal direction. Analytical solutions are available for simple, infinitely long waveguides, whereas for more complex waveguides numerical approaches have been developed using finite element techniques to describe the cross-section. In the present paper an analytical Fourier transform approach is used to find the dynamic behaviour of a system consisting of two parallel beams coupled by a plate, when a point force is applied to one of the beams. Multiple waves occur in the longitudinal direction, the number of waves depending on the number of modes of the equivalent cross-section. However, the motion of the driven beam is shown to be dominated by the contribution from only one or two waves at each frequency, these having wavenumbers closest to that of the uncoupled beam. The motion of the plate is also shown to be dominated by these wavenumbers for excitation on the beam. Experimental results are obtained on beam–plate–beam systems with identical and non-identical beams, which show good agreement with the predictions. In particular, these confirm that the plate response is dominated by waves with wavenumbers in the beam direction that follow those of the excited beam.     

Dynamic peierls stress in a crystal model with slip anisotropy

The analysis of screw dislocation motion in a lattice is extended to crystals with a preferred slip direction and to force laws of the “dangling bond” type. The external strain required for uniform motion, or the corresponding Peierls stress depends critically on the shape of the interatomic potential. For particular combinations of velocity, crystal anisotropy, and force law, the external strain drops sharply—indicating some modes of dislocation motion that are almost loss-free. Although the relation between dynamic Peierls stress and dislocation width is not monotonie, it shows a general exponential decrease.     

An approximate method of predicting the response of periodically supported beams subjected to random convected loading

The space-averaged response of an infinite, elastically supported, periodic beam subjected to convected random loading has been studied by using an approximate “assumed mode” method. The complex wave motion in the beam is represented by any number of suitably chosen complex modes. With a good, yet simple, choice of mode which satisfies certain boundary conditions on one periodic beam element, a “single mode approximation” can yield very accurate values of the average response. This has been verified for a wide range of the support stiffnesses and loading convection velocities. Consideration has also been given to the ratio of the maximum response in the beam to the space-averaged response. The method has been applied only to uniform beams in this paper, but it should be readily applicable to periodic systems consisting of non-uniform beam elements.     

聚变堆遥操作上层协同控制系统设计和测试

为了解决聚变堆遥操作维护上层控制系统普遍存在的软件功能模块过于集中、拓展性差、不具备中层运动规划能力等问题,设计了一种基于机器人操作系统(ROS)的上层协同控制系统。该系统采用分布式架构,基于市场成熟的开发平台进行设计,集成了中层运动规划与动力学虚拟仿真功能。通过“轴销抓取”的实验场景测试的结果表明：该系统满足设计需求,运行稳定。     

基于DSP的机器人末端力控制系统设计

工业机器人的运动分为自由运动和受力约束运动两种不同的运动类型。受力约束运动不仅要进行精确的位置控制,而且要对接触力进行精确控制。文章对机械臂的末端力控制系统进行了研究。采用了高精度电动滑台作为力控制系统的执行机构,在电动滑台末端增加了柔性单元,从而克服了系统柔性不足的缺点,并且依据Lyapunov稳定性理论确定系统的二阶参数,使用Narendra提出的稳定自适应设计方法构建了电动滑台的数学模型。对机械臂末端柔顺系统的系统设计及算法进行了研究,消除了外界噪声对系统的干扰,解决了力控制的基本控制策略问题,优化了力控制时系统的响应特性。