Moving Coordinate Frame Research Articles

Theoretical Analysis of Sheet Beam Electron Gun for Terahertz Vacuum Electron Devices

In the terahertz band, the vacuum electron devices with planar and miniaturized structures require the ultrahigh-precision description of high-density electron beams for formation and transportation with the theoretical analysis and simulation. Thus, the absence of the available theory for the sheet beam electron is put forward to investigate the 3-D spatial potential using a differential geometry and tensor analysis. The equation of electron beam potential can be derived by establishing a moving coordinate frame, carrying out the numerical calculations, and doing series expansion. To solve the complex sheet beam description of the existence of four corner singularities, the Riemann method and conformal mapping are adopted to overcome this difficulty. Then, a 0.22-THz vacuum electron tube using a sheet beam electron gun and corresponding optical system are designed. The beam voltage and current are 16.5 kV and 0.5 A, respectively, with a beam channel size of 1.4 mm <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times0.14$ </tex-math></inline-formula> mm and a uniform magnetic field of 5980G. With the 3-D simulation, the high-density sheet electron beam can transport 18 mm successfully. The spatial potential of numerical calculation with the obtained sheet beam theoretical function and the simulation model are compared to verify the theoretical description. Both the results are with good consistency, which preliminarily proves the effectiveness of the theoretical function. This work will give the deep physical mechanics vision for the development of high-power and high-efficiency sheet beam terahertz tube in the future.

Theoretical Analysis and Simulated Verification of Circular Beam Electron Optical System for Terahertz Vacuum Electron Devices

The micro-structure miniaturization of vacuum electron devices (VEDs) in terahertz band requires the high precision description of formation and transportation for the ultrahigh density electron beams with the novelty theoretical analysis and simulation. In this article, the 3-D spatial potential of circular electron beam formation is presented by adopting differential geometry and tensor analysis theory. The specific numerical solution of the circular electron beam potential can be obtained by establishing a moving coordinate frame, carrying out numerical calculation, series expansion with adopting of Riemann method. In order to verify the complex theory, a 0.34 THz vacuum electron tube using thin circular electron beam gun and corresponding optical system are designed and analyzed. The beam voltage and current are 23.8 kV and 0.28 A respectively, with beam channel radius of 0.16 mm and magnetic field of 5250 Gs. Using 3-D simulation of CST, the ultrahigh density electron beam can transport 30 mm with good stability. The spatial potential of numerical calculation with the obtained theoretical functions and the simulation model are compared to verify our theoretical description. Both the results are with good consistent which preliminary proves the effectiveness of the theoretical function. Furthermore, we combine the theoretical function potential with the electron beam envelope under the same coordinate frame, the detailed characteristics of ultrahigh density beam formation with the potential function can be directly understood easily, which will give the deep physical mechanics vision for the development of high power and high-efficiency terahertz tube in the future.

Experimental-numerical approach to the estimation of uncertainty in robot pose measurement

Accurate pose measurements are crucial for the reliability of the kinematic calibration and performance evaluation of a robotic manipulator. Spherical coordinate measurement systems such as laser trackers and total stations are commonly used to collect sets of 3D points from which position and orientation of a moving coordinate frame attached to the end-effector in relation to a reference coordinate frame are calculated. Estimating the uncertainty of this specific measurement task is necessary to provide traceability to these measurements. In this paper we propose combing measurements of calibrated artefacts with Monte Carlo simulations to estimate this task-specific measurement uncertainty. The application of this method has been demonstrated for the performance evaluation of a Stewart Platform using a robotic total station. Expanded uncertainties between 0.40 mm and 0.63 mm have been estimated for position. Expanded uncertainty was 0.10° for all orientation results.

Slant Helix Curves and Acceleration Centers in Minkowski 3-Space Ε31

In this study, some basic concepts (e.g., instant screw axis (ISA), instantaneous pole points, acceleration pole points) will be given and analyzed about an alternative one-parameter motion of a rigid-body in 3-dimensional Minkowski space Ε31 obtained by moving coordinate frame {N, C, W} along a non-null unit speed curve α = α(t), where N, C and W correspond to unit principal normal vector field, derivative vector field of unit principal normal vector field and Darboux vector field (or angular-velocity vector field) of the non-null unit speed curve α, respectively.

Bandwidth Maximization Using Vehicle Arrival Functions

We revisit the offset optimization problem for maximization of two-way progression bands. A new formulation is proposed relying on the concepts of relative offset and vehicle arrival functions. Vehicle arrival functions represent the probability that a vehicle reaches a given intersection at a given time. Relative offsets are the displacement of the arrival functions with respect to a moving coordinate frame. An explicit formula for the bandwidth is derived based on these two quantities. The bandwidth maximization problem is then formulated as an unconstrained nonlinear program. The cases of pulse and Gaussian arrivals are considered in detail. Numerical techniques are proposed for both that return globally optimal solutions with small computational cost.

Variational approximations for traveling solitons in a discrete nonlinear Schrödinger equation

Traveling solitary waves in the one-dimensional discrete nonlinear Schrödinger equation with saturable onsite nonlinearity are studied. A variational approximation (VA) for the solitary waves is derived in analytical form. The stability is also studied by means of the VA, demonstrating that the solitons are stable, which is consistent with previously published results. Then, the VA is applied to predict parameters of traveling solitons with non-oscillatory tails (embedded solitons, ESs). Two-soliton bound states are considered too. The separation distance between the solitons forming the bound state is derived by means of the VA. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton–Raphson method. In general, good agreement between the analytical and numerical results is obtained. In particular, we demonstrate the relevance of the analytical prediction of characteristics of the embedded solitons.

Analysis of peristaltic two-phase flow with application to ureteral biomechanics

Fluid flow in the ureter is sometimes accompanied by solid particles that are produced in the kidneys or result from the breakup of larger kidney stones; ureteral peristalsis is affected by the presence of these solids. Peristaltic flow is analyzed for a solitary traveling wave in an axisymmetric tube with an incompressible, Newtonian fluid in which identical, solid spherical particles are distributed. A two-phase flow model is used in conjunction with a perturbation method based on a small radius to length ratio of the wave to obtain a closed-form solution of the flow and particle velocities. The phenomenon of trapping in which closed fluid recirculation streamlines in a moving coordinate frame occurs is discussed. Peristaltic pumping is affected as particle volume fraction is increased. The pressure drop diminishes as the amplitude ratio (wave amplitude/wave radius) decreases. The pressure in the contracted part of the ureter increases as the particle volume fraction is increased. It is suggested that certain pathological and physiological manifestations on the ureter can be related to these findings. The results may also be relevant to the transport of other physiological fluids and industrial applications in which peristaltic pumping is used.

移动坐标系FDTD方法在脉冲波传播中的应用

The moving coordinate frame flnite-difierence time-domain (FDTD) method is developed to simulate the propagation of electromagnetic pulses over long distances. In this paper, this method is extended to three dimensional cases. The discretized form of moving coordinate frame FDTD in three dimensions is given, and is applied to calculate propagation of difierent pulses in rectangular waveguides. The widths of pulses in propagation direction increase when pluses propagate in waveguide. Dispersion phenomena appear, and slow parts of the pulses move to the tail of pulses. By comparing the results of both FDTD method and moving coordinate frame FDTD method, this method is numerically validated. Relative difierence between two methods is less than 0.1%, the simulation results are shown to be accurate. With the increasing applications on communication and detection, there is an ever-increasing in- terest in the propagation of pulses over long distance. The flnite-difierence time-domain (FDTD) method (1) is widely used in electromagnetic simulation. Because the FDTD method is a full- wave method, when simulating propagation of electromagnetic pulses over very long distance, the computational burden quickly becomes too large to afiord.This is the major di-culty why FDTD method can not be applied to simulate long distance propagation of pulses directly. In order to overcome this di-culty, moving coordinate is combined with FDTD method. Models of one di- mension as electromagnetic pulse propagation over large distance in medium (2) and acoustic wave propagation in waveguide has been studied (3) approach this method. Coordinate of the direction in which pulse's propagation is transformed into moving coordinate, which is 'frozen' with pulses, and the pulses are in small calculation zones when simulating (as shown in Fig. 1). Moving grid whose size is limited to the order of the pulse length is created, and moves along the approach of the center-of-mass of the wavepacket. This method takes advantage of the fact that when an electromagnetic pulse propagates over long distance, the signiflcant pulse energy exists only over a small part of the propagation path at any time. The FDTD meshes (in the direction of the propagation path) need only to be large enough to contain the pulses while using this method to simulate the propagation of pulses. This method is useful in particular for tracking well-collimated short pulse beam, referred to as pulsed beams or localized waves. In this paper, the models are extended to simulate electromagnetic pulses propagating in waveg- uide over large distance and results are compared with the results using FDTD approach in order to testify the validity and feasibility of the moving coordinate FDTD method.

Spectral Stability of Small-Amplitude Viscous Shock Waves in Several Space Dimensions

A planar viscous shock profile of a hyperbolic–parabolic system of conservation laws is a steady solution in a moving coordinate frame. The asymptotic stability of viscous profiles and the related vanishing-viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions. It is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile. The Evans function method provides a geometric dynamical-systems framework to study the eigenvalue problem. In this approach eigenvalues correspond to zeros of an essentially analytic function \({\mathcal{E}(\rho\lambda,\rho\omega)}\) which detects nontrivial intersections of the so-called stable and unstable spaces, that is, spaces of solutions that decay on one (“−∞”) or the other side (“ + ∞”) of the shock wave, respectively. In a series of pioneering papers, Kevin Zumbrun and collaborators have established in various contexts that spectral stability, that is, the non-vanishing of \({\mathcal{E}(\rho\lambda,\rho\omega)}\) and the non-vanishing of the Lopatinski–Kreiss–Majda function Δ(λ,ω), imply nonlinear stability of viscous shock profiles in several space dimensions. In this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions. This is done by exploiting the slow-fast nature of the small-amplitude limit, which was used in a previous paper by the authors to prove spectral stability of small-amplitude shock waves in one space dimension. Geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior. Three qualitatively different regimes are distinguished that relate the small strength \({\epsilon}\) of the shock wave to appropriate ranges of values of the spectral parameters (ρλ, ρω). Various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality.

Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets

We consider the approximation of multiparticle quantum dynamics in the semiclassical regime by Hagedorn wavepackets, which are products of complex Gaussians with polynomials that form an orthonormal $L^2$ basis and preserve their type under propagation in Schrödinger equations with quadratic potentials. We build a fully explicit, time-reversible time-stepping algorithm to approximate the solution of the Hagedorn wavepacket dynamics. The algorithm is based on a splitting between the kinetic and potential part of the Hamiltonian operator, as well as on a splitting of the potential into its local quadratic approximation and the remainder. The algorithm is robust in the semiclassical limit. It reduces to the Strang splitting of the Schrödinger equation in the limit of the full basis set, and it advances positions and momenta by the Störmer–Verlet method for the classical equations of motion. The algorithm allows for the treatment of multiparticle problems by thinning out the basis according to a hyperbolic cross approximation and of high-dimensional problems by Hartree-type approximations in a moving coordinate frame.

Simulation on the thermal cycle of a welding process by space–time convection–diffusion finite element analysis

Welding plays an important role in manufacturing. But difficulties still exist for simulation of the welding process of large welded structures, due to the limitation of computer capacity, mathematical models and software. This paper is devoted to developing an algorithm that tries to simulate the thermal cycle during welding efficiently and accurately. A space–time finite element method (FEM) is proposed to solve the transient convection–diffusion thermal equation. The method has been applied to the steady-state thermal analysis of welds. A moving coordinate frame (Eulerian frame), in which the heat source is stationary, is used to improve the spatial resolution of a numerical analysis for the thermal cycle of welds effectively, as well as to incorporate the addition of the filler metal naturally. This method is suitable for the thermal analysis in the weld pool or/and weld joint region including starting and stopping transients.

波浪中非定常運動する浮体に作用する非線形波浪流体力 第一報

There exist nonlinear wave loads such as wave drift force, wave drift damping and wave drift added mass, when the floating body is slowly oscillating in waves. In the present study, wave drift added mass acting on an arbitrarily shaped body is evaluated based on the potential theory. A moving coordinate frame following the low frequency motion of the body is adopted to express this interaction model. Two small parameters which express wave slope(ε) and the frequency of slow oscillations(σ) are used for the perturbation expansion. The boundary value problem of each order is solved by the hybrid method. The fluid domain is divided into two regions. The potentials are expressed by the eigen function expansion in the outer region while the inner region is solved by the boundary element method (BEM) with the Rankine source as the Green's function. Element functions with higher order are utilized to improve the accuracy. Wave drift added mass is contributed from five components of potentials. The following potentials are solved in this work. First is the wave potential (Ο(ε)). Second is the disturbance potential for slowly oscillation of the body (Ο(σ)). The last one is the linear potential referring to the incident wave but affected by disturbance of the body (Ο(ε σ)). In the case of the Ο(ε σ) problem, it is important to treat the inhomogeneous free surface boundary condition. In the present paper, the Emmerhoff solution and a new approach about integral over the free surface are expanded for it. The wave drift added mass is calculated. Result is compared with the semi-analytical one. Good agreement is observed.

Freezing Multipulses and Multifronts

We consider nonlinear time dependent reaction diffusion systems in one space dimension that exhibit multiple pulses or multiple fronts. In an earlier paper two of the authors developed the freezing method that allows us to compute a moving coordinate frame in which, for example, a traveling wave becomes stationary. In this paper we extend the method to handle multifronts and multipulses traveling at different speeds. The solution of the Cauchy problem is decomposed into a finite number of single waves, each of which has its own moving coordinate system. The single solutions satisfy a system of partial differential algebraic equations coupled by nonlinear and nonlocal terms. Applications are provided to the Nagumo and the FitzHugh–Nagumo systems. We justify the method by showing that finitely many traveling waves, when patched together in an appropriate way, solve the coupled system in an asymptotic sense. The method is generalized to equivariant evolution equations and is illustrated by the complex Ginzburg–Landau equation.

Simulating optimal structure of the magnetic field of electrostatic plasma lens

The paper considers the motion of electrons in the curvilinear magnetic field of a plasma lens. This motion is modeled mathematically on the assumption that the force acting on the electron is central in a moving coordinate frame. The existence conditions for optimal and stable equipotentials are studied.

Applying envelope theory and deviation function to tooth profile design

In this work, a mathematical model of an elastic conjugate element is presented, using envelope theory and a deviation function. A basic deformation curve in a two-dimensional system is defined, and determined by the maximum deformation of the originally generated curve. The set of these points of maximum deformation in the moving coordinate frame determines the deviation of the originally generated curve. The degree of deviation is described by a deviation function. A deviation function is chosen to reshape the originally generated curve. The reshaped curve is called a basic deformation curve. If the basic deformation curve is known, an envelope associated with them can be determined. The results are applied to rotary gear pump design. The reshaped curve is superior to the originally generated curve. The reshaped curve has lower deformation, von-Mises stress, and greater bending strength than the originally generated curve. This investigation indicates that envelope theory and deviation function can avoid singular points of conjugate elements, using an example.

On-vehicle visual tracker with disturbance rejection

This paper presents the synthesis and implementation of a passivity based controller designed for visual tracking of moving objects in a disturbance rejection scheme. The purpose of this application is to control the line of sight of a 2-DOF sensor platform in a moving coordinate frame, when the movement of the vehicle carrying the platform acts as a disturbance of finite energy. External torques due to vehicle movement downgrade the visual tracking performance and can cause instability and total loss of sight of the objective. These disturbances can be compensated for in an effectively manner introducing a classical L2-gain disturbance rejection scheme, widely deployed in inertial frame robotic systems. The proposed controller has been tested in a real 2-DOF platform mounted on a moving testbench that emulates different types of non-inertial vehicle surfaces.

A new method to calculate B-matrix elements for external rotations with application to the B-matrix internal coordinate molecular dynamics formalism

B-matrix elements for external rotations are conventionally computed by imposing the Casimir–Eckart condition (CEC). However, this results in matrix elements that not only depend on atomic masses but also on the choice of a moving coordinate frame. A new method to directly calculate the B-matrix elements for external rotations, which avoids these problems, is therefore introduced. An underlying symmetry relation is used to reduce the number of required computations. This method is applied to the dynamics equation in nonredundant generalized coordinates, which enhances the efficiency of the B-matrix internal coordinate molecular dynamics formalism.

The Casimir–Eckart condition and the transformation of dipole moment derivatives revisited

Thus far, the Casimir–Eckart condition (CEC) has been considered to be the general requirement in a rigorous transformation of differential tensor quantities like dipole moment derivatives or polarizability derivatives from Cartesian coordinates to internal coordinates. It is well known that this conventional transformation matrix ( A-matrix) based on the Wilson method, which always guarantees the CEC, depends on the atomic masses and provides atomic mass-dependent differential quantities. Since this conventional method may give abnormal isotope effects in dipole derivatives or polarizability derivatives even in mass-free internal coordinates, we have re-examined the validity of the CEC and propose a new transformation method that does not impose this condition. We show that the A-matrix obtained by the method used in internal coordinate molecular dynamics (ICMD) formalisms is mass-free for mass-free generalized (internal and external) coordinates and does not cause such an abnormal isotope effect. Since the CEC itself depends on the choice of a moving coordinate frame, the transformed internal coordinate dipole derivatives by the Wilson method also depend on the choice of the moving coordinate frame, while those by the ICMD method are independent of the choice of the moving coordinate frame. As Eckart comments, the Casimir condition is not valid in general but its validity depends on the type of internal coordinates used. All the additional degrees of freedom in defining a moving coordinate frame can be satisfactorily removed by relations from the orthogonality of internal coordinates to external translations and rotations. Moreover, an analogy with normal coordinates for internal modes provides the very equivalent to Eckart's original derivation.

Moving Coordinate Frame FDTD Analysis of Long Range Tracking of Pulsed Fields in Graded Index Waveguides

Modeling of long range propagation of pulsed fields along guiding structures poses some major difficulties with the conventional FDTD scheme, arising from the vast computer resources needed to discretize the entire domain and the accumulation of numerical dispersion error. The moving frame FDTD approach, presented in this work, overcomes these difficulties by limiting the computational grid to a window centered about the propagating pulse and moving along with it at the local wavespeed of the medium. In this work, we first present an FDTD solution of the field equations as formulated in the moving coordinate frame, and then explore such issues as the numerical dispersion, which is shown to be reduced substantially compared with the stationary formulation, the numerical stability, and the absorbing boundary conditions at the leading, trailing and side boundaries of the moving window. Numerical results of pulsed field propagation along graded index waveguides are shown, and the capability of the method is d...

Moving Coordinate Frame FDTD Analysis of Long Range Tracking of Pulsed Fields in Graded Index Waveguides- Abstract

Modeling of long range propagation of pulsed fields along guiding structures poses some major difficulties with the conventional FDTD scheme, arising from the vast computer resources needed to discretize the entire domain and the accumulation of numerical dispersion error. The moving frame FDTD approach, presented in this work, overcomes these difficulties by limiting the computational grid to a window centered about the propagating pulse and moving along with it at the local wavespeed of the medium. In this work, we first present an FDTD solution of the field equations as formulated in the moving coordinate frame, and then explore such issues as the numerical dispersion, which is shown to be reduced substantially compared with the stationary formulation, the numerical stability, and the absorbing boundary conditions at the leading, trailing and side boundaries of the moving window. Numerical results of pulsed field propagation along graded index waveguides are shown, and the capability of the method is demonstrated with propagation distances exceeding the order of 104 pulse lengths. We choose initial field distributions that give rise to pulsed beam fields. The propagation characteristics of these fields along nonuniform guides are explored analytically in the Appendix. The numerical results are also checked against frequency domain adiabatic mode solutions developed in the Appendix.