Components Of Displacement Vector Research Articles

Scanning tunneling microscopy of RF oscillating surfaces

AbstractAmplitude and phase of high frequency surface acoustic wave (SAW) fields are investigated by a novel scanning tunneling microscopy technique. The gap voltage is modulated at a slightly detuned high frequency. Due to the nonlinearity of the tunneling process a frequency mixing appears. For scanned areas with dimensions much smaller than the wavelength of the SAW a remarkable local variation of amplitude and phase of the tunneling current at the difference frequency is observed. Depending on the local morphology different components of the particle displacement vector are detected. Model calculations of amplitude and phase images are presented for a real topography.

Remarks on symmetric Lamb waves with dominant longitudinal displacements

Proof of the vanishing of the normal component of the particle displacement vector on the free surfaces of an isotropic, homogeneous plate for nonzero-order symmetric Lamb waves is given. Such a conclusion was stated without proof in the book by Viktorov [A. I. Viktorov, Rayleigh and Lamb Waves: Physical Theory and Applications (Plenum, New York, 1967), 1st ed., Chap. II, p. 121]. An expression for the frequency-thickness products where this occurs is also given. This enables practical (e.g., experimental) selection of such modes. Other interesting features of such modes, such as the independence of their group velocity on the mode’s order or its nondispersitivity, are also addressed. The features discussed here are of great practical importance, e.g., in the nondestructive testing (NDT) of plates loaded by liquids and/or in the ultrasonic tensometry of a thin plate.

Corotational total Lagrangian formulation for three-dimensional beamelement

A corotational total Lagrangian formulation of beam element is presented for the nonlinear analysis of three-dimensional beam structures with large rotations but small strains. The nonlinear coupling among the bending, twisting, and stretching deformations is considered. All of the element deformations and equations are defined in body-attached element coordinates. Three rotation parameters are proposed to determine the orientation of the element cross section. Two sets of element nodal parameters termed explicit nodal parameters and implicit nodal parameters are introduced. The explicit nodal parameters are used in the assembly of the system equations from the element equations and chosen to be three components of the incremental translation vector and three components of the incremental rotation vector. The implicit nodal parameters are used to determine the deformations of the beam element and chosen to be three components of the total displacement vector and nodal values of the three rotation parameters. The element internal nodal forces corresponding to the implicit nodal parameters are obtained from the virtual work principle. Numerical examples are presented and compared with the numerical and experimental results reported in the literature to demonstrate the accuracy and efficiency of the proposed method. HREE-DIMENSIONAL beams are very important structural elements in all types of engineering systems. In many applications, these beam elements undergo finite rotations that require a nonlinear formulation to their structural analysis. The development of new and efficient formulations for nonlinear analysis of beam structures has attracted the study of many researchers in recent years. Based on different kinematic assumptions, different alternative formulation strategies and procedures to accommodate large rotation capability during the large deformation process have been presented.120 The kinematic assumptions used in Refs. 4, 5, 7, 11, and 17-19 are based on Timoshenko's hypothesis: the effect of stretching, bending, torsion, and transverse shear are taken into account.

Developments and applications of conjugate-wave holographic interferometry

Conjugate-wave holographic interferometry is an optical technique which was recently proposed by one of the authors for the measurement, in real time, of in-plane deformation. Basically, the technique consists of projecting, from two symmetrical directions, two real images of an object upon the object itself. If the object deforms, interference fringes are observed on its surface. These fringes give a representation of one in-plane component of the point displacement vector and are independent of the out of plane component. On this paper some aspects of the optical principles underlying the technique are analyzed in deeper detail, and more advanced solutions for their practical implementation are proposed. In addition, a few examples of applications on-plane and not-plane specimens are presented.

ELASTIC EXTRAPOLATION OF PRIMARY SEISMIC P‐ AND S‐WAVES1

AbstractThe elastic Kirchhoff‐Helmholtz integral expresses the components of the monochromatic displacement vector at any point A in terms of the displacement field and the stress field at any closed surface surrounding A. By introducing Green's functions for P‐ and S‐waves, the elastic Kirchhoff‐Helmholtz integral is modified such that it expresses either the P‐wave or the S‐wave at A in terms of the elastic wavefield at the closed surface. This modified elastic Kirchhoff‐Helmholtz integral is transformed into one‐way elastic Rayleigh‐type integrals for forward extrapolation of downgoing and upgoing P‐ and S‐waves. We also derive one‐way elastic Rayleigh‐type integrals for inverse extrapolation of downgoing and upgoing P‐ and S‐waves. The one‐way elastic extrapolation operators derived in this paper are the basis for a new prestack migration scheme for elastic data.

Deformations of elastic earth and station displacements

We give the complete formulae for calculating the station displacements caused by the earth tide and the earth pole tide. For cm order of accuracy, the luni-solar tide raising potential should be calculated to the third order and the rotational centrifugal potential, to the second order. Results of our calculation of baseline correction are 1) if the radial component of the station displacement vector is positive (negative), then the baseline correction is greatest when the baseline is in the opposite (same) direction as the displacement vector. 2) When the radial component of the station displacement vector is larger than the horizontal component, the correction increases with increasing baseline length. 3) Long baselines are more sensitive for the determination of the Love number h, and short baselines, the Love number l. Our Shanghai-Ulumuqi baseline is favourable for determining l.

Processing of acoustic signals in the auditory system of bony fish.

In order to determine unambiguously the bearing of a sound source, a fish must be able to resolve acoustic pressure and the components of the acoustic displacement vector from the signals detected by the otolithic organs. A new hypothesis for the processing of acoustical information by bony fish is presented. It is demonstrated that much of the processing required to do this may be implicit in the structure of the ear and its associated neural innervation. Possible algorithms are presented that the central nervous system might use to further process the derived information to localize a sound source and discriminate frequency and range. The hypothesis is shown to be consistent with much of what is known of the morphology and physiology of the auditory system of bony fishes.

Effect of substrate on the efficiency of an arbitrarily shaped microstrip patch antenna

A general approach is presented for the determination of power radiated via the space wave and surface wave from the aperture of an arbitrarily shaped microstrip antenna. The magnetic current model is used for this, and the analysis is carried out in the Fourier domain to determine the effect of the substrate. It has been shown that, in the Fourier domain, the longitudinal components of electric and magnetic displacement vectors follow the transmission line equation and can be solved by inspection. An expression for total radiated power has been derived. The singularities of the integral for power radiation indicate the presence of surface wave modes, and the associated power has been obtained using a singularity extraction technique. The effect of the substrate on space wave power has also been determined. This theory has been applied to a rectangular patch antenna. The results are in conformity with those reported in the literature. It has been found that for the frequency range ( h/\lambda, ), the effect of dielectric substrate can be neglected.

A three axis tactile sensor-probe for robotic use

SUMMARYThe paper deals with a three axis reaction force and displacement sensor. The design procedure of the strain gauges sensor for its intended application as a tactile proble is described. The deformable sensor body with strain gauges arrangement allows all three output voltage readings to be directly related with rectangular components of the force displacement vector in a remote contact reference system. The structural orthogonalisation that results in minimal cross component couplings is briefly outlined. The practical use of the designed probe is exemplified by seam following in a robotic welding system.

Multiple aperture holographic interferometry and its application to the measurement of plant movement

Multiple aperture holographic interferometry is a simple method of detecting several types of displacement pattern by using small apertures in a rotatable mask in front of a holographic plate. This paper shows the possibility of applying the method to the observation of growing plant movement. The measurements of temporal change of plant and three components of displacement vector are made and the measuring accuracy is discussed.

On an axisymmetric boundary value problem for an elastic dielectric half-space

Hankel transforms are used to construct a closed-form solution for the axisymmetric boundary value problem of a point charge forced normal to the surface of an isotropic elastic dielectric semi-space. Exact closed form expressions are obtained for the components of displacement and polarization vectors in terms of the Bessel functions and the fundamental solutions (1/R), (e−mR/R), R being the distance from the source point. The electric potential fields are determined both inside and outside the elastic dielectric semi-space. In the absence of electric polarization effects, the problem reduces to the classical axisymmetric Boussinesq problem of a point force applied normal to the surface of an isotropic elastic semi-space. The expressions of the components of displacements derived from this particular case are found to agree with known results.

A geometrical approach to holographic interferometry

Geometrical considerations are used to obtain quantitative data for the components of the displacement vector. Use is made of a dual-beam illumination and a variant is described, using point light sources and a single point of observation. Contour lines for the displacement vector in the viewing direction and in a perpendicular direction are obtained as the algebraic sum and difference of two interference patterns. Densification of the initial pattern is used to obtain moire patterns for the out-of-plane and in-plane displacements of flat Surfaces and for the derivatives of these displacements. To determine the complete displacement field of objects of arbitrary shape, one holographic plate is sufficient, using two times two-point light sources.

Interpretation of the d-parameters, occurring in the Mayants-Averbukh theory of infrared intensity, in terms of the bond dipole moment concept : Equivalence with the valence-optical theory in the zeroth approximation for linear and planar molecules

From a comparison with the valence-optical theory of infrared intensity in the zeroth approximation, it is suggested that the d-parameters (elements of the effective charge tensor D n 0 of a bond) occurring in the Mayants-Averbukh theory can be expressed in terms of bond dipole moment components and their derivatives with respect to components of the Cartesian bond displacement vector. With this interpretation, which may be suitable for practical application, the Mayants-Averbukh theory formally appears as a combination of the “special” and the generalized valence-optical theory. In particular, it is found that for linear molecules the Mayants-Averbukh theory is equivalent to the “special” valence-optical theory in the zeroth approximation. For planar molecules the Mayants-Averbukh theory is equivalent to the generalized valence-optical theory (zeroth approximation) for in-plane modes and to the “special” valence-optical theory for out-of-plane modes.

Kinetic theory of surface wave propagation along a hot plasma column

In this paper we propose an exact solution of Vlasov and Maxwell's equations for a bounded hot plasma in order to derive the dispersion relation of the axially-symmetric surface waves propagating along a plasma column. Assuming specular reflexion of plasma particles from the boundary, expressions for the components of the electric displacement vector are obtained on the basis of the Vlasov equation. Their substitution in Maxwell's equations, neglecting the spatial dispersion in the transverse plasma dielectric function, allows us to determine the plasma impedance. The equating of plasma and dielectric impedances gives the wave dispersion relation which, in different limiting cases, coincides with the well-known results.

Structural reanalysis by iteration

Reanalysis based on an iteration process is expressed as an infinite series. It is shown that for large changes in the design, convergence is poor or not possible. A method to improve convergence is presented, based on an expression of the matrix of changes in stiffness as a linear combination of two matrices. The method is then modified to further improve convergence separately for each component of the unknown displacement vector. The advantage of the proposed procedure is demonstrated by numerical examples.

On the Problem of Large-Deflection Analysis of Space Fibers

A general theory is presented to analyze the geometric free configuration of statically loaded space fiber. The external loads acting along the center line of the fiber are both distributed and concentrated forces and moments. The generaliza tion is confined largely to the formulation of the problem, with the assumption that the fiber is inextensible and plane cross sections remain plane and perpendicular to the center line after deformation. The dependent variables in the governing equations are the components of the linear displacement vector u and the angle of twist ψ of the fiber cross section. The relation between the linear displacement vector u and the angular displacement vector ϕ of the fiber cross section is also presented. A closed solution of the governing equations, in most cases, is difficult to obtain. An analog-computer block diagram is proposed to simulate the equations.

The Treatment Of Crack Propagation InInhomogeneous Materials Using TheBoundary Element Method

The purpose of this paper consists in the implementation of a boundary element model for which the crack propagation is easily studied thanks to an automatic remeshing process. The traction singular quarter point element was used in order to improve the accuracy of the values of stress intensity factors. The Erdogan and Sih criterion allows us to determine the new growth direction after each step. Validity of this implement is demonstrated, with respect to experimental results, by the example of a holed plate. In this study, we can show the manner in which the crack may propagate when the distance between the initial crack's axis and the tangent hole is changed. Two distinct behaviours can be pointed out with respect to this distance : in some cases the crack may be simply deflected by the hole, and propagates across the plate until a complete fracture is reached. In other ones, the crack propagates toward and reaches the hole. In the framework of fatigue under cyclic loading, the manner in which growth rate is affected by the proximity of a hole is shown. INTRODUCTION The use of the Boundary Element Method for the solution of crack problems has been of interest to many investigators over recent years. However, for non-symetric problems where the two crack surfaces have the same geometrical coordinates, the direct application of this method leads to a singular system of equations. Some special techniques have been established to overcome this difficulty. The most general are the subregions method introduced by Blandford et al [1] and the dual bondary element method established by Portela et al [2]. In our study we have used the first method for the modelisation of the crack problems. We have shown [3] that the use of this method does not alter the automatization of crack propagation. Transactions on Engineering Sciences vol 6, © 1994 WIT Press, www.witpress.com, ISSN 1743-3533 376 Localized Damage THE BOUNDARY ELEMENT METHOD The boundary integral equation relates displacements i% and t< on the boundary F of an elastic homogeneous isotropic body. It can be written as : , Q)uj(Q)dT(Q) = f 0y (P, Q)ti(Q)dT(Q) (1) Jr In which the tensors T^ and Uij are the components number j of the tractions and displacements at Q due to a unit i direction load at P. The term Cij represents the behaviour of the singularity in Ty. When P is on a smoothpart of T, Cij = &,/2. Uj and tj denote components of the boundary displacement vector and the traction vector respectively. The boundary is now divided into N isoparametric elements. By using a collocation method, the system of equations is produced on the form : M {«}=[«]« (2) The matrices [A] and [J3] contain coefficients of either Tij and U . METHOD FOR THE CALCULATION OF STRESS INTENSITY FACTOR 'SIP' In order to evaluate correctly and accuratly the stress intensity factor, we have compared several methods like : the J Integral, Traction Singular Quarter Point Element (TSQP) and the Crack Opening Displacement method in which Kj is calculated for different formulations : Figure 1. Quarter point element /c = (3 — 4f/) for plane strain and =(3 — %/)/(! + %/) for plane stress In figure 2, we denote as follows : Quarter point elements and one-point displacement formula Quarter point elements and two point displacement formula Traction singular quarter-point elements and one-point displacement formula Traction singular quarter-point elements and two-point displacement formula Traction singular quarter-point elements and nodal value formula The extrapolation method of the displacement The crack size and the lenght of the element at crack tip KIUIS

Rectilinear dislocation in an anisotropic plate

A solution has been found to the problem of calculating the stress and displacement fields caused by a rectilinear dislocation in an anisotropic elastic plate. Special cases of anisotropy have been found with solutions represented by elementary functions. Certain problems in describing crystal plastic deformation phenomena make it vital to know the fields of the elastic stresses and displacements caused by an individual dislocation in a bounded crystal. It is interesting to study the effect of crystal boundaries on these fields with a simple model which approximates fairly closely to experimental conditions. The model selected is shown in Fig, 1. A dislocation with a Burgers vector β (b1, b2, b3) is situated in an infinite elastic anisotropic plate of thickness 2h. The dislocation line is parallel to the plate boundaries. The following restriction is introduced in relation to the plate's elastic properties: the medium has a plane of elastic symmetry perpendicular to the dislocation line. The selection of the coordinate system and position of the dislocation are shown in Fig. 1. The requirement is to find the stresses and displacements at an arbitrary point in the plate. One limited special form of this problem has been solved by Kroupa [1]. The limitations which he introduced are as follows: the medium is isotropic, the dislocation is at the precise center of the band and the Burgers vector has only one component b2 differing from zero (the same coordinates were chosen in [1] as in Fig. 1). Thus Kroupa's results can be obtained from the results of the present work as a special case. Other special cases arising from this problem are those concerning the elastic stress and displacement fields caused by a dislocation in anisotropic semi-bounded [2] and bounded [3] media. It is immediately apparent that the problem is a plane one, in the sense that the fields to be found do not depend on coordinate z. Since the medium has a plane of elastic symmetry perpendicular to the dislocation line, it is clear from [4] that the system of stresses and strains in such a medium can be divided into two independent subsystems. The first of these is plane deformation with stress components σxx, σyy and σxy differing from zero and displacement vector components ux and uy, the second is “antiplane” deformation with stress components σxz and σyz differing from zero and the displacement vector component uz. In the case under examination, the plane deformation is caused by the Burgers vector edge components bx and by and the antiplane deformation by the screw component bz. The solution is therefore divided into two stages, corresponding to edge and screw dislocations.

Transformation of the Linear Equations of Magnetoelastic Interactions for an Infinitely Long Prism

Brown's linear equations of magnetostriction in a homogeneous cubic crystal and for an infinitely long prism along the applied external magnetic field are transformed into a new system of equations. In the transformed system, the two components of the magnetostatic scalar potential, the elasticity potential, and the three components of the elastic displacement vector, satisfy separate differential equations. The general solution is a linear combination of a finite number of functions, each one being a solution of the Helmholtz equation in two independent variables.

Longitudinal Versus Lateral Eddy Length-Scale (Second Note on a Vorticity-Transfer Hypothesis of Turbulence Theory)

Abstract A previously introduced vorticity-transfer-and-adaptation hypothesis is briefly summarized and supplemented by consideration of certain spatial derivatives of the components of the eddy displacement vector. The conventional length-scale of turbulence, redefined in the preceding note as a lateral co-variance = (x′z′¯)½, is supplemented by the new concept of a longitudinal length-scale, defined as the variance L = (x′2¯)½). In correspondence to the Karman constant (which concerns the rate of change of l in the direction lateral to the mean flow) there is a new constant which concerns the rate of change of L in the longitudinal direction. It is proposed to refer to this as the Reichardt constant, because the supplements are of direct importance for turbulence in free flows (such as submerged jets), and because the new hypothesis offers an improvement in comparison with “Reichardt's momentum transfer law” or “inductive theory” of jet-diffusion. It is shown that the structure of mean profiles in two-d...