Beam Boundary Research Articles

Vibration Control and Trajectory Tracking for General In-Plane Motion of an Euler–Bernoulli Beam Via Two-Time Scale and Boundary Control Methods

In this paper, general in-plane trajectory tracking problem of a flexible beam is studied. To obtain the dynamic equations of motion of the beam, Hamiltonian dynamics is used and then Lagrange’s equations of beam dynamics and corresponding expressions for boundary conditions are derived. Resulting equations show that the coupled beam dynamics including beam vibration and its rigid in-plane motion take place in two different time domains. By using two-time scale (TTS) control theory, a control scheme is elaborated that makes the orientation and position of the mass center of the beam track a desired trajectory while suppressing its vibration. TTS composite controller has two parts: one is a tracking controller designed for the slow (rigid) subsystem, and the other one is a stabilizing controller for the fast (flexible) subsystem. For the fast subsystem, the proposed boundary control (BC) method does not require any information about vibration along the beam except at the end points, nor requires discretizing the partial differential equation of beam vibration to a set of ordinary differential equations. So, the method avoids the need for instruments to measure data from vibration of any point along the beam or designing an observer for estimating this information. Also, the proposed method prevents control spillover due to discretization. Simulation results show that fast BC is able to remove undesirable vibration of the flexible beam and the slow controller provides very good trajectory tracking with acceptable actuating forces/moments.

Formation of an ion beam in an elementary cell with inhomogeneous emission current density

A well-known Pierce solution that allows focusing a beam of charged particles using properly shaped electrodes outside the beam aperture is generalized to the case of an accelerating system with inhomogeneous emission current density. It is shown that the defocusing effect of the space charge can, in principle, be evenly compensated over the entire cross section of the beam. In contrast to the beam with a uniform emission current density, both the electric potential and the transverse electric field must be controlled along the beam boundary in order to eliminate the angular divergence. However, eliminating the angular spread evenly across the beam constitutes a mathematically ill-posed problem which needs to be solved with the use of one or another method of regularization. An alternative way of diminishing beam emittance is proposed for the beam where the emission current is uniform across the entire aperture except for a narrow beam edge layer and a simple formula for the Pierce electrodes is derived. Numerical simulation has proved the reasonable accuracy of our analytical theory.

Numerical Simulation Of The Laser Welding

The model takes into consideration thermophysical and metallurgical properties of theremelting steel, laser beam parameters and boundary conditions of the process. As a resultof heating the material, in the area of laser beam operation a weld pool is being created,whose shape and size depends on convection caused by the Marangoni force. The directionof the liquid stream depends on the temperature gradient on the surface and on the chemicalcomposition as well. The model created allows to predict the weld pool shape depending onmaterial properties, beam parameters, and boundary conditions of the sample.

On Euler–Bernoulli discontinuous beam solutions via uniform-beam Green’s functions

The bending problem of Euler–Bernoulli discontinuous beams is dealt with. The purpose is to show that uniform-beam Green’s functions can be used to build efficient solutions for beams with internal discontinuities due to along-axis constraints and flexural-stiffness jumps. Specifically, upon deriving the equilibrium equation in the space of generalized functions, first it is seen that the original bending problem may be recast as linear superposition of a principal and an auxiliary bending problem, both involving a uniform reference beam and homogeneous boundary conditions. Then, based on the Green’s functions of the reference beam, closed-form solutions are developed for the principal beam response, while the auxiliary beam response is obtained by solving, in general, ( r + 2 s) algebraic equations written at the discontinuity locations, being r the number of discontinuities due to along-axis constraints, and s the number of flexural-stiffness jumps. In this manner, an appreciable reduction of computational effort is achieved as compared to alternative analytical solutions in the literature.

Estimation of Wave Properties of a Lightly Damped Beam with a Real-Coded Genetic Algorithm

A method is presented to estimate the flexural wave number and wave amplitudes in lightly damped beams with the combination of wave decomposition technique and optimization algorithm. The determination of the wave number is regarded as a nonlinear optimization problem in which a real-coded genetic algorithm is used. The fitness function for the genetic algorithm is derived based on the minimizing of the errors between the estimated and measured responses. After the derivation of the complex wave number, the wave amplitudes can be estimated using wave decomposition method, in which the responses at a few evenly spaced locations are used. In comparison to existing approaches, the method generally requires far fewer measurement locations and does not depend on beam boundary conditions. An experimental test is carried out with a beam excited by a shaker and the results are compared with theoretical predictions. It is shown that high-accuracy estimation for the wave properties can be obtained for lightly damped beams.

Comparison of 1D and 2D Theories of Thermoelastic Damping in Flexural Microresonators

AbstractThermoelastic damping (TED) represents the lower limit of material damping in flexural mode micro- and nanoresonators. Current predictive models of TED calculate damping due to thermoelastic temperature gradients along the beam thickness only. In this work, we develop a two dimensional (2D) model by considering temperature gradients along the thickness and the length of the beam. The Green's function approach is shown to be a robust means of solving the coupled heat conduction equation in one and two dimensions. In the 1D model, curvature information is lost and, hence, the effects of structural boundary conditions and mode shapes on TED are not captured. In contrast, the 2D model retains curvature information in the expression for TED and can account for beam end conditions and higher order modes. The differences between the 1D and 2D models are systematically explored over a range of beam aspect ratios, frequencies, boundary conditions, and flexural mode shapes.

Flexural-torsional coupled vibration of slewing beams using various types of orthogonal polynomials

Dynamic behavior of flexural-torsional coupled vibration of rotating beams using the Rayleigh-Ritz method with orthogonal polynomials as basis functions is studied. Performance of various orthogonal polynomials is compared to each other in terms of their efficiency and accuracy in determining the required natural frequencies. Orthogonal polynomials and functions studied in the present work are : Legendre, Chebyshev, integrated Legendre, modified Duncan polynomials, the special trigonometric functions used in conjunction with Hermite cubics, and beam characteristic orthogonal polynomials. A total of 5 cases of beam boundary conditions and rotation are studied for their natural frequencies. The obtained natural frequencies and mode shapes are compared to those available in various references and the results for coupled flexural-torsional vibrations are especially compared to both previously available references and with those obtained using NASTRAN finite element package. Among all the examined orthogonal functions, Legendre orthogonal polynomials are the most efficient in overall CPU time, mainly because of ease in performing the integration required for determining the stiffness and mass matrices.

Effect of Boundary Conditions on Impact Stresses of Beams

A theoretical analysis based on the numerical solution of the beam impact integral equation is carried out to determine the impact force and deflection time histories, the strain energy absorbed by the beams and the maximum bending moment. Effect of beam boundary conditions on impact response of beam is also discussed. The theoretical results obtained in the present analysis are compared with experimental and theoretical works previously done. A good agreement is found between theoretical and experimental results. This indicates that the impact resistance of relatively large beams may be predicted by using the theoretical approach based on equation of undamped beam vibration. All the derivations required to predict the effect of boundary conditions are performed for both forced and free vibrations. For the same falling mass and the same applied kinetic energy (height of drop) for all cases, the maximum central deflection and the maximum impact force are affected by the boundary conditions of the beams

A Green’s function model for the analysis of laser heating of materials

An analytical model based on Green’s function method is developed to analyze the temperature distribution and heated regions in a material irradiated by a high-energy laser beam. The model is multi-dimensional, transient and incorporates different types of beam characteristics and boundary conditions. The multi-dimensional integration formulas in the Green’s function solution equation are evaluated using an adaptive numerical integration algorithm. A parametric study is conducted to show the effect of various laser beam parameters and material properties on the laser heating process.

Estimation of complex modulus using wave coefficients

The paper presents a new, efficient and accurate method for experimentally determining structural damping properties of stiff materials in flexural vibration. The estimation method seeks a wavenumber that forces data at all measurement points to conform with the general forced-vibration solution for a beam, and, thus, to have the same wave coefficients. The method does not depend on beam boundary conditions, making it relatively simple to implement in any laboratory setting. The paper shows results of using the method to estimate the complex modulus of aluminum and of polymethyl methacrylate (PMMA) beams in the frequency range from 30 to 800 Hz.Results compare very well with those obtained by more conventional methods and with those previously published over most of the frequencies of interest. Discrepancies in results for frequencies less than 80 Hz are attributed to the difficulty in measuring differences of velocity between points that are near each other at low frequency, and a technique for avoiding such discrepancies is outlined. Evidence shows that the error in the results near 350 Hz is due to a resonance in the scanning laser vibrometer that was used to take the experimental measurements.

Causes and cures for errors in the simulation of ion extraction from plasmas

For many years, computer programs have been available to simulate the extraction of positive ions from plasmas. The results of such simulations may not always agree with measurements. There are different reasons for this: the mathematical formulation must match with the simulated physics, the number of meshes must be high enough to correctly take into account the nonlinear space charge in the sheath, and ray tracing must be done in sufficiently small steps, using numerically correct field components and partial derivatives. In addition to these hidden problems the user may create errors by a wrong choice of parameters, which are not matching the assumptions of the mathematical formulation. Examples are the use of a positive ion extraction program for the extraction of negative ones, the choice of a wrong angle between the plasma electrode and the beam boundary in the vicinity of the meniscus, and the use of too few trajectories. The design of extraction electrodes generally has the aim to optimize the optical properties and the current of the ion beam. However, it is also important to take into account the surface fields in order to avoid dark currents and sparking.

Commissioning and quality assurance of a commercial stereotactic treatment‐planning system for extracranial IMRT

A 3D treatment‐planning system (TPS) for stereotactic intensity‐modulated radiotherapy (IMRT) using a micro‐multileaf collimator has been made available by Radionics. In this work, we report our comprehensive quality assurance (QA) procedure for commissioning this TPS. First, the accuracy of stereotaxy established with a body frame was checked to ensure accurate determination of a target position within the planning system. Second, the CT‐to‐electron density conversion curve used in the TPS was fitted to our site‐specific measurement data to ensure the accuracy of dose calculation and measurement verification in a QA phantom. Using the QA phantom, the radiological path lengths were verified against known geometrical depths to ensure the accuracy of the ray‐tracing algorithm. We also checked inter‐ and intraleaf leakage/transmission for adequate jaw settings. Measurements for dose verification were performed in various head/neck and prostate IMRT treatment plans using the patient‐specific optimized fluence maps. Both ion chamber and film were used for point dose and isodose distribution verifications. To ensure that adjacent organs at risk receive dose within the expectation, we used the Monte Carlo method to calculate dose distributions and dose‐volume histograms (DVHs) for these organs at risk. The dosimetric accuracy satisfied the published acceptability criteria. The Monte Carlo calculations confirmed the measured dose distributions for target volumes. For organs located on the beam boundary or outside the beam, some differences in the DVHs were noticed. However, the plans calculated by both methods met our clinical criteria. We conclude that the accuracy of the XKnife™ RT2 treatment‐planning system is adequate for the clinical implementation of stereotactic IMRT.PACS numbers: 87.53.Xd, 87.53.Ly, 87.53.Wz

Commissioning and quality assurance of a commercial stereotactic treatment-planning system for extracranial IMRT

A 3D treatment-planning system (TPS) for stereotactic intensity-modulated radiotherapy (IMRT) using a micro-multileaf collimator has been made available by Radionics. In this work, we report our comprehensive quality assurance (QA) procedure for commissioning this TPS. First, the accuracy of stereotaxy established with a body frame was checked to ensure accurate determination of a target position within the planning system. Second, the CT-to-electron density conversion curve used in the TPS was fitted to our site-specific measurement data to ensure the accuracy of dose calculation and measurement verification in a QA phantom. Using the QA phantom, the radiological path lengths were verified against known geometrical depths to ensure the accuracy of the ray-tracing algorithm. We also checked inter- and intraleaf leakage/transmission for adequate jaw settings. Measurements for dose verification were performed in various head/neck and prostate IMRT treatment plans using the patient-specific optimized fluence maps. Both ion chamber and film were used for point dose and isodose distribution verifications. To ensure that adjacent organs at risk receive dose within the expectation, we used the Monte Carlo method to calculate dose distributions and dose-volume histograms (DVHs) for these organs at risk. The dosimetric accuracy satisfied the published acceptability criteria. The Monte Carlo calculations confirmed the measured dose distributions for target volumes. For organs located on the beam boundary or outside the beam, some differences in the DVHs were noticed. However, the plans calculated by both methods met our clinical criteria. We conclude that the accuracy of the XKnife™ RT2 treatment-planning system is adequate for the clinical implementation of stereotactic IMRT. PACS numbers: 87.53.Xd, 87.53.Ly, 87.53.Wz

Emittance growth as mesh artefact

The space charge beam spreading calculations of the laser ion source group of CERN [K. Hanke, et al., Rev. Sci. Instrum. 73 (2002) 783] has been recomputed with IGUN. Instead of only looking to the total value of beam spreading, we also investigated the behaviour of the rms-emittance and found the following unexpected results: 1) Even zero emittance beams show considerable emittance growth. 2) The emittance growth of the central part depends on both the number of meshes used and the number of trajectories used. 3) The most prominent emittance growth occurs at the beam boundary, which is independent of the number of trajectories used and only depends on the number of meshes inside the beam. The more meshes used, the less this emittance growth will be. Generally, emittance growth is associated with the non-linear electric field components, integrated up along the particles trajectories. By using a fractional emittance analysis, we found two different contributions to emittance growth, which might explain the observed results: (a) the deposition of space charge and the interpolation of fields inside the meshes produces errors and provides “noise” to the particle ray tracing algorithm. (b) at the beam boundary, the filling of the space charge map depends very much on the relative location of the beam boundary to the discrete mesh structure. This mesh artefact is inherent to all programs that need meshes for the evaluation of the influence of space charge on the potential distribution. When calculating a “real” beam with given emittance, it is good advice to compare this calculation with the emittance growth, obtained for a beam of zero emittance. Furthermore, a variation of the number of trajectories and the number of meshes will allow to extrapolate to the unaccessible use of an infinite number of trajectories and meshes.

Benchmark of Damage Localisation Algorithms Using Mode Shape Data

This paper presents a comparative study of three enhanced signal processing methods to locate damage on mode shape data. The first method called curvature mode shape is used as a reference. The second tool uses wavelet transform and singularity detection theory to locate damage. Finally we introduce the windowed fractal dimension of a signal as a tool to easily measure the local complexity of a signal. Our benchmark aims at comparing the crack detection using optimal spatial sampling under different severity, beam boundary conditions (BCs) and added noise measurements.

Optimal design of an electrostatically actuated microbeam for maximum pull-in voltage

We consider optimizing the shape of a capacitive micro-beam for maximum pull-in voltage. The problem is discretized using finite elements. The normal flow algorithm is used to trace the nonlinear voltage displacement curve and find the pull-in point. Optimal width and thickness distributions are obtained for different beam boundary conditions using a resizing scheme based on optimality criterion. The results indicate that substantial increase in pull-in voltage can be achieved by varying the width and/or thickness distribution.

Adaptive Control of a Flexible Robot Using Fuzzy Logic

Operational problems with robot manipulators in space relate to several factors, one most importantly being structural flexibility and subsequently significant difficulties with the control systems, especially, for endpoint position control. Elastic vibrations of the links coupled with their large rotations and nonlinear dynamics is the primary cause. This paper presents a control scheme for tracking the endpoint of a two-link flexible robot. The dominant assumed modes of vibration for Euler-Bernoulli cantilever and pinned-pinned beam boundary conditions are coupled with the nonlinear dynamics for rigid links to form an Euler-Lagrange inverse flexible dynamics robot model. A Jacobian transpose control law actuating the robot joints is adapted by a fuzzy logic system (FLS) with link deformation inputs and a single variable output. Results obtained with an FLS adaptive control strategy show significantly diminished vibration amplitudes for both cantilever and pinned-pinned link dynamics and greatly improved control performance compared to the nonadaptive strategy.

NIGUN: A two-dimensional simulation program for the extraction of H− ions

A new theory for a multiparticle plasma sheath in the case of H− extraction recently provided the basis to modify the well-established two-dimensional (2D) computer program IGUN.© As in the case of IGUN, stability of convergence is obtained for the very nonlinear sheath region by directly inserting the analytical space charge according to the Poisson equation for each mesh point above the wall potential. For three-component plasmas (e, p, and H−) only the relation of proton birth potential to electron energy and H− energy to electron energy is needed to obtain all parameters. For four-component plasmas (e, p, H−, and Cs+) the density and energy ratios of protons and Cs+ ions are required as input for the calculations. Up to 10 different positive ion species may be declared, also requiring the input of the corresponding ratios for density and energy. The actual field strength in the extraction gap is used to determine the proper densities of all particles by the program. The numerical “proof” will be given that, as predicted by theory, an angle of 45° between the beam boundary and the wall electrode will provide rectilinear flow once the density matches the field in the extraction gap.

Self-consistent multicomponent plasma sheath theory for the extraction of H− ions (invited)

A self-consistent one-dimensional plasma sheath theory is presented to provide the basis for a correct numerical simulation of the extraction of volume produced H− ions. The plasma may consist not only of electrons and H− ions, but may also contain other positive ions such as protons, molecular ions and those of heavier elements, like cesium or tantalum. For the transition from the classical plasma sheath with a falling potential to the extraction region for H− ions with an increasing potential there exists the problem of a saddle point with adverse optical properties. This is eliminated by requiring sufficient space charge of H− ions near the extraction electrode. The formation of a virtual cathode in the extraction region by reflected positive ions is also taken into account. The integration of the Poisson equation in the extraction region establishes a criterion to avoid the creation of a nonphysical periodical sequence of potential maximums and minima. This criterion is an antithesis to the Bohm sheath criterion and has a corresponding interpretation: a virtual cathode in the extraction region can only be avoided, if the space charge of positive ions rapidly decreases. The acceptable range of parameters is thus reduced considerably. The resulting axial potential function is then used to derive the shape of the plasma wall electrode in the vicinity of the ion beam edge in order to obtain an aberration free beam boundary, this information being equivalent to the Pierce angle in the case of solid electron or ion emitters.

Repetitive learning with fuzzy logic adaptive control of a flexible robot manipulator

Operational problems with robot manipulators in space relate to several factors, one most importantly being structural flexibility and subsequently significant difficulties with the control systems, especially, position control. A control strategy is devised for positioning the endpoint of a two-link robot manipulator modeled with assumed modes flexible dynamics repetitively tracking a square trajectory. The dominant assumed modes of vibration are determined for Euler-Bernoulli cantilever beam boundary conditions then, coupled with the nonlinear dynamics for rigid links to fonn an Euler-Lagrange inverse flexible dynamics robot model. A Jacobian transpose control law actuates the robot links. While repetitive tracking alone achieves no improvement in control precision, adapting the control law by a fuzzy logic system achieves consistent tracking precision.