Direction Of Symmetry Axis Research Articles

The coalescence and oscillation of eutectic gallium indium alloy droplets

The coalescence and oscillation of eutectic gallium indium alloy droplets

Inversion formula for an integral geometry problem over surfaces of revolution

AbstractAn integral geometry problem is considered on a family of ‐dimensional surfaces of revolution whose vertices lie on a hyperplane and directions of symmetry axes are fixed and orthogonal to this plane, in . More precisely, the reconstruction of a function , , , from the integrals of the form extended over a chosen side of all surfaces of revolution of a given family is investigated. Unlike the usual Radon transform, the integrals considered here are not taken with respect to the surface area element. A Fourier slice identity and a backprojection‐type inversion formula are obtained with a method based on the Fourier and Hankel transforms. The reconstruction procedure and some analytical and numerical implementations of the obtained inversion formulas in the cases of and are provided.

On the inversion of generalized Radon transforms over n-dimensional paraboloids

On the inversion of generalized Radon transforms over n-dimensional paraboloids

Flexión compuesta esviada en secciones de hormigón armado doblemente simétricas. Dirección de capacidad máxima

El uso de modelos constitutivos simplificados para caracterizar el comportamiento del hormigón, (parábola rectángulo, bloque rectangular), conduce a que la capacidad de agotamiento de una sección doblemente simétrica solicitada por un axil excéntrico se encuentre en la dirección de simetría principal. Si se evalúa la capacidad de la sección empleando un modelo hiperbólico para la descripción del hormigón se obtiene que para axiles elevados, la dirección de resistencia máxima presenta un cierto desvío respecto a la dirección de simetría principal tal como se obtiene con el uso de modelos simplificados. En este trabajo se determinan y comparan una serie de diagramas de interacción en flexión compuesta esviada (Mx, My) de dos secciones con cuantías y axiles crecientes y dos modelos constitutivos de hormigón (parábola rectángulo e hiperbólico) a fin de determinar la dirección en la que se encuentra la resistencia máxima de las secciones en los casos analizados.

Elastic properties of two VTI shale samples as a function of uniaxial stress: Experimental results and application of the porosity-deformation approach

Shale is a complex medium composed of clay, other mineral phases, and a pore space. The combined elastic properties of these components control the effective (anisotropic) properties of the composite solid. The factor that is the most dependent on the stress field is the structure of the pore space, which greatly influences the elastic properties of the medium. We have further developed and experimentally validated the porosity deformation approach (PDA) for understanding and modeling stress-dependent changes of the elastic properties of sedimentary rocks. PDA separates the pore space into stiff and compliant parts. The load dependencies of the elastic properties have linear contributions due to the former and exponential contributions due to the latter. We evaluate data sets of elastic properties of two vertical transverse isotropic shale samples measured under uniaxial stress. Then we apply the PDA and our optimization algorithm to the measured data sets to model the stress dependency of the seismic velocities and validate the modeling with experimentally obtained results. We have developed for the first time the constant anellipticity approach (CAN), which estimates the off-axis velocity (in an inclined direction relative to the symmetry axis direction) as a function of stress. Measurements of off-axis velocities are often missing information, and CAN permits us to fill this gap. This provides further background for the reconstruction of the stress dependency of the compliance tensor from acoustic log data.

Displacement Analytical Solution of a Circular Hole in Layered Composite Materials considering Shear Stress Effect

Layered composite materials can be treated as transverse isotropy in mechanics. According to basic equations of transverse isotropy, the compatibility equation in polar coordinates is firstly formulated. Then the analytical solution for a deep circular hole considering shear stress and two unequal normal stresses in layered composite materials is derived. The solution shows that the radial displacement is affected by the elastic modulus and Poisson ratio in both the isotropic plane and the direction of symmetry axis in spite of the hypothesis that the hole transect is parallel to the isotropic plane.

Azimuthal AVO for tilted TI medium

With the incident P-wave, we derive approximate formulas for amplitudes and polarizations of waves reflected from and transmitted through a planar, horizontal boundary between an overlying isotropic medium and an underlying tilted transversely isotropic (TTI) medium assuming that the directions of the phase and group velocities are consistent. Provided that the velocities in the isotropic medium are equal to the velocities along the symmetry axis direction, we derive the relational expression between the propagation angle in the TTI medium and the propagation angle in the hypothetical isotropic medium, under the condition that the horizontal slowness is the same, and then we update the approximate formula of the polarization in the TTI medium. Provided that the slow and fast transverse waves (qS and SH) are generated simultaneously in the anisotropic interface, we linearize for a six-order Zoeppritz equation, derive the azimuthal formula of longitudinal and S-waves, and determine their detailed expressions within the symmetry axis plane. According to the derived azimuthal AVO formula, we establish medium models, compare the derived AVO with the precision, and obtain the following conclusions: (1) The dip angle for the symmetry axis with respect to the vertical may have a sufficiently large impact on AVO, and the vertical longitudinal wave can generate an S-wave. (2) For the derived AVO formula, within the symmetry axis plane, the fitting effect of the approximate and exact formulas is good; however, within the other incident planes, taking the azimuth angle 45° as an example, the approximation is suitable for the large impedance contrast if the anisotropic parameters are set properly. (3) The error between the approximation and precision is mainly caused by the difference between the reflected and transmitted angles, the velocities’ derivation with respect to azimuth, and the division of approximation into isotropic and anisotropic parts.

Investigation of dynamic characteristics of tall industrial chimney based on GPS measurements using Random Decrement Method

Abstract The paper is concerned with the investigation of dynamic characteristics, i.e. the natural frequencies and structural damping ratios, of a tall industrial chimney, located in the power station of Belchatow in Poland, excited by the wind, based on GPS measurements using Random Decrement Method (RDM). The GPS receiver was able to measure only the first natural frequency of the chimney. The basic characteristics of a background noise of GPS technology based on the result of a static test were examined thoroughly. In order to increase the accuracy of GPS measurements as well as the dynamic characteristics estimation the filtering procedure using Type 1 Chebyshev band-pass digital filter was adopted. Then, the principles of RDM taking into account the influence of GPS measurement random noise were considered. The influence of filter parameters on the stability of the RD signature was also examined. Based on the filtered record of chimney dynamic response, the dynamic characteristics of the chimney were investigated using RDM for response considered in ten various directions in the horizontal plane. The results delivered some important practical information about two symmetry axes of the chimney. The effective investigation of a damping ratio should be made based on the analysis of the chimney response considered only in the directions of symmetry axes of the chimney. The number of time segments of the chimney response taken into account in averaging of the RD signature as well as the time interval of the RD signature was also recommended. The obtained results proved that the dynamic characteristics of tall slender structures under wind excitations can be effectively determined based on GPS measurements using RDM.

Wave Spring Dynamic Characteristics Analysis

The natural vibration frequency and forced vibration amplitude of the spring decides the stability of the whole of the spring system. Take the monolayer wave spring as the research object, do dynamics analysis for the three different kinds of wave spring with the same structure parameters. Obtain the natural vibration frequencies of the three kinds of spring through modal analysis, the result shows that the natural vibration frequency of the integral type wave spring is the highest, then is the wave spring with opening, the natural vibration frequency of the wave spring with lapping is the lowest. Do the harmonic response analysis for the springs based on the modal analysis result, the result shows that the main vibration direction of the wave spring is the symmetry axis direction of the spring, all the three kinds of the spring is much easier resonate in this direction. Through the above provides certain theoretical guidance for the design of the wave spring

Finite element modeling of impulsive excitation and shear wave propagation in an incompressible, transversely isotropic medium

Elastic properties of materials can be measured by observing shear wave propagation following localized, impulsive excitations and relating the propagation velocity to a model of the material. However, characterization of anisotropic materials is difficult because of the number of elasticity constants in the material model and the complex dependence of propagation velocity relative to the excitation axis, material symmetries, and propagation directions. In this study, we develop a model of wave propagation following impulsive excitation in an incompressible, transversely isotropic (TI) material such as muscle. Wave motion is described in terms of three propagation modes identified by their polarization relative to the material symmetry axis and propagation direction. Phase velocities for these propagation modes are expressed in terms of five elasticity constants needed to describe a general TI material, and also in terms of three constants after the application of two constraints that hold in the limit of an incompressible material. Group propagation velocities are derived from the phase velocities to describe the propagation of wave packets away from the excitation region following localized excitation. The theoretical model is compared to the results of finite element (FE) simulations performed using a nearly incompressible material model with the five elasticity constants chosen to preserve the essential properties of the material in the incompressible limit. Propagation velocities calculated from the FE displacement data show complex structure that agrees quantitatively with the theoretical model and demonstrates the possibility of measuring all three elasticity constants needed to characterize an incompressible, TI material.

Wavefield propagation characteristics in fracture-induced TTI double-porosity medium

The direction of symmetry axis of parallel fracture set in fractured hydrocarbon reservoir affects the transmission of seismic waves markedly, so a medium named fracture-induced TTI (tilted transverse isotropy) double-porosity medium is studied here to discuss the effect of different dip and azimuth angles of a fracture system. Based on the theories of fracture-induced HTI (horizental transverse isotropy) double-porosity medium, the softness and dispersion matrixes of fracture-induced TTI double-porosity medium are derived with the application of Bond transform, and finally, single-order velocity-stress equations are obtained. Furthermore, numerical simulations in xoz plane of 2.5 dimensional vector wavefield are carried out by the method of high-order staggered-grid finite-difference under perfect matched layer (PML) boundary conditions. The results show that the dipand azimuth angles of fractures have great impacts on seismic wave propagation, since the angles can cause the phenomena of shear wave splitting and, in the two-layer model of fracture-induced TTI double-porosity, converted shear wave splitting and shear wave sub-splitting. All of these increase the complexity of seismic wavefield and will lay a foundation of further studies on seismic wave propagation in actual earth layers.

Seismic anisotropy of the shallow crust in the Umbria–Marche (Italy) region

Small magnitude earthquake data, recorded during the 1997 Umbria–Marche (Italy) seismic crisis, have been used to study the shear wave splitting in the central Apennine. Using the method of diagonalization of the covariance matrix, a set of 1234 S waves impinging the Earth at an angle less than the critical angle has been selected. The cross-correlation method was used to estimate the splitting time between the two horizontal S components and the direction of the symmetry axis of the anisotropic structure. A further analysis, based on visual inspection of corrected traces, allowed us to estimate the anisotropic parameters for 884 S waves. An average splitting time 〈τ〉=32.3∓4ms has been estimated. Splitting times do not generally show a clear pattern with distance, which is an indication of possible spatial variations of seismic anisotropy. A minimum 2 km depth of the anisotropic layer is estimated. The trend of normalized splitting time vs. depth roughly indicates a maximum 4–5 km depth of the anisotropic layer. The distribution of the fast symmetry axis indicates a main symmetry axis oriented along the direction of the maximum compressive stress in the area and a secondary symmetry axis almost orthogonal to the first one. The stereographic plots of fast axis direction seem exclude a depth dependence of anisotropic parameters. A clear spatial correlation between the directions of the fast symmetry axes at the considered stations and the orientations of the nearest fault systems and thrust fronts is found, which confirms the spatial variability in the anisotropic properties of the area, as inferred by the analysis of splitting times. Based also on previous studies, results are interpreted in terms of the extensive dilatancy anisotropy (EDA) model. The estimated crack density is always smaller than the critical coalescence value, enforcing the hypothesis that fluid-filled separated crack systems control the polarization of S waves.

Multiparameter joint tomography for TTI model building

Model building for tilted transversely isotropic media has commonly been performed by a single parameter tomography that updates the velocity in the symmetry direction, while the orientation of the symmetry axis and Thomsen parameters [Formula: see text] and [Formula: see text] are typically estimated from the migration stack and well data. Unfortunately, well data are often not available. In addition, when they are available, their lateral sampling is typically very sparse and their vertical sampling usually spans only a limited range of depths. In order to obtain spatially varying anisotropic models, with or without well data, we developed a multiparameter joint tomographic approach that simultaneously inverts for the velocity in the symmetry axis direction, [Formula: see text] and [Formula: see text]. We derived a set of reflection tomography equations for slowness in the symmetry axis direction and Thomsen parameters [Formula: see text] and [Formula: see text]. In order to address the nonuniqueness of the tomography, we developed a regularization strategy that uses an independent regularization operator and regularization factor for each individual anisotropy parameter. Synthetic tests found that ambiguity exists between the anisotropy parameters and that velocity has a better resolution than [Formula: see text] and [Formula: see text]. They also confirmed that joint tomography provides a better data fit than single parameter tomography. The field example was used to test a way to incorporate the sonic data in the model building process and limit the tomographic updates on certain anisotropy parameters by adjusting the regularization.

Stable P-wave modeling for reverse-time migration in tilted TI media

We present an approach for P-wave modeling in inhomogeneous transversely isotropic media with tilted symmetry axis (TTI media), suitable for anisotropic reverse-time migration. The proposed approach is based on wave equations derived from first principles — the equations of motion and Hooke’s law — under the acoustic TI approximation. Consequently, no assumptions are made about the spatial variation of medium parameters. A rotation of the stress and strain tensors to a local coordinate system, aligned with the TI-symmetry axis, makes it possible to benefit from the simple and sparse form of the TI-elastic tensor in that system. The resulting wave equations can be formulated either as a set of five first-order or as a set of two second-order partial differential equations. For the constant-density case, the second-order TTI wave equations involve mixed and nonmixed second-order spatial derivatives with respect to global, nonrotated coordinates. We propose a numerical implementation of these equations using high-order centered finite differences. To minimize modeling artifacts related to the use of centered first-derivative operators, we use discrete second-derivative operators for the nonmixed second-order spatial derivatives and repeated discrete first-derivative operators for the mixed derivatives. Such a combination of finite-difference operators leads to a stable wave propagator, provided that the operators are designed properly. In practice, stability is achieved by slightly weighting down terms that contain mixed derivatives. This has a minor, practically negligible, effect on the kinematics of wave propagation. The stability of the presented scheme in inhomogeneous TTI models with rapidly varying anisotropic symmetry axis direction is demonstrated with numerical examples.

Building tilted transversely isotropic depth models using localized anisotropic tomography with well information

Tilted transverse isotropy (TTI) is increasingly recognized as a more geologically plausible description of anisotropy in sedimentary formations than vertical transverse isotropy (VTI). Although model-building approaches for VTI media are well understood, similar approaches for TTI media are in their infancy, even when the symmetry-axis direction is assumed known. We describe a tomographic approach that builds localized anisotropic models by jointly inverting surface-seismic and well data. We present a synthetic data example of anisotropic tomography applied to a layered TTI model with a symmetry-axis tilt of 45 degrees. We demonstrate three scenarios for constraining the solution. In the first scenario, velocity along the symmetry axis is known and tomography inverts for Thomsen’s [Formula: see text] and [Formula: see text] parame-ters. In the second scenario, tomography inverts for [Formula: see text], [Formula: see text], and velocity, using surface-seismic data and vertical check-shot traveltimes. In contrast to the VTI case, both these inversions are nonunique. To combat nonuniqueness, in the third scenario, we supplement check-shot and seismic data with the [Formula: see text] profile from an offset well. This allows recovery of the correct profiles for velocity along the symmetry axis and [Formula: see text]. We conclude that TTI is more ambiguous than VTI for model building. Additional well data or rock-physics assumptions may be required to constrain the tomography and arrive at geologically plausible TTI models. Furthermore, we demonstrate that VTI models with atypical Thomsen parameters can also fit the same joint seismic and check-shot data set. In this case, although imaging with VTI models can focus the TTI data and match vertical event depths, it leads to substantial lateral mispositioning of the reflections.

Analysis of Phase Velocity and Polarization Features for Elastic Waves in TTI Media

Phase velocity and polarization direction are the important parameters for studying seismic wave propagation and describing medium properties, which play essential roles in theory and application. We assume that the symmetry axis of tilted transverse isotropy (TTI) media lies in XOZ plane in the observation coordinates system and directly derive an analytic expression for phase velocity and polarization direction in TTI media in this system. Based on the Thomson dimensionless anisotropic parameters and weak anisotropy approximation theory, we also derive the approximate expressions of phase velocity for elastic waves, polarization direction for qP wave and qSV wave. Theoretical analysis and numerical examples indicate that the change of velocity for qP wave and qSH wave is gentle, and the change of velocity for qSV wave is dramatic. The errors of approximate values of phase velocity are small, so the approximate expression is a good approximation for precise phase velocity. The polarization direction of SH wave is the function of propagation angle and tilted angle of symmetry axis instead of anisotropic parameters. The polarization directions of SH wave is not only perpendicular to propagation direction, but also perpendicular to the direction of symmetry axis. The polarization directions of qP wave and qSV wave are neither parallel nor perpendicular to propagation direction except some special directions, and changes with the angle of symmetry axis in TTI media. For both precise and approximate expressions, the polarization directions of qP wave and qSV wave are always perpendicular. In a given range of errors, the weak anisotropy approximation and analytic expressions match well.

Attenuation anisotropy in the linear-slip model: Interpretation of physical modeling data

This study attempts to validate a mathematical formalism of introducing attenuation into Schoenberg’s linear slip model. This formalism is based on replacing the real-valued weaknesses by complex-valued ones. During an ultrasonic experiment, performed at a central frequency of [Formula: see text] on a plate-stack model with [Formula: see text]-thick Plexiglas™ plates, the velocity and attenuation (inverse of the quality factor [Formula: see text]) of P-, SH-, and SV-waves are measured in directions from 25° to 90° with the symmetry axis for dry and oil-saturated models and loading uniaxial pressures of 2 and [Formula: see text]. The velocity and attenuation data are fitted by the derived theoretical functions. The values of the real and imaginary parts of the complex-valued weaknesses are estimated. Thereal parts of the weaknesses, which have a clear physical meaning (they affect the weakening of the material), are three times larger for the dry model than for the oil-saturated one. The imaginary parts of the weaknesses are responsible for attenuation; their values are an order of magnitude smaller than the real parts. The derived expressions for angle-dependent velocities and attenuations can be used to distinguish between dry and oil-saturated fractures. In particular, the P-wave attenuation function in the symmetry-axis direction (normal to fracture planes) is different in dry and saturated media. The experiment shows that the plate-stack model is inhomogeneous because of the nonuniform pressure distribution, which degrades the experimental results and creates difficulties in the inversion for the complex-valued weaknesses — particularly in joint inversion of P- and S-wave data.

Adaptive Robust Posture Control of Parallel Manipulator Driven by Pneumatic Muscles With Redundancy

This paper presents an adaptive robust posture controller for a parallel manipulator driven by pneumatic muscles (PMDPM) with a redundant DOF. Rather severe parametric uncertainties and uncertain nonlinearities exist in the dynamics of the PMDPM. To deal with these uncertainties effectively, the recently developed adaptive robust control strategy is applied. Furthermore, the developed control strategy explicitly takes into account the particular physical properties of the system studied. Specifically, the symmetric geometric structure of the parallel manipulator driven by identical pneumatic muscles and no external moments around symmetry-axis direction of the parallel manipulator make the rotation angle of the parallel manipulator around its symmetry axis direction negligible and a nonfactor during normal operations. As such, the axial rotation angle is not measured and controlled when the PMDPM are used in practice, leading to a single-DOF redundancy in synthesizing the precise posture controller for rotation angles around other axes. To make full use of this redundancy, an equivalent average-stiffness-like desired constraint is introduced in the development of the adaptive robust posture controller to achieve precise posture tracking while reducing control input chattering caused by measurement noise. Experimental results are obtained to verify the validity of the proposed controller for the redundant PMDPM.

A novel microstream injection molding method for thermotropic liquid crystalline polymers to promote mechanical isotropy: A matrixing microbeam X-ray study

The effects of flow-altering inserts and mold cavity geometry on the mechanical properties of an injection molded liquid crystalline polymer were studied to produce parts with properties approaching macroscopically isotropic state. By inserting fine metal mesh barriers to the gates of the mold cavities, a large number of highly oriented microstreams are produced. After their creation these highly oriented streams of differing flow vectors intertwine and this texture remains reasonably intact even after substantial shearing and extension history imparted on them during ensuing flow into the cavity. This method is effective in the interior away from the skin regions formed under the shearing flow during injection. The local molecular orientation was determined using a matrixing microbeam WAXS technique that allows precision movement of the sample in the microbeam X-ray. Samples produced with the 1.0 mm 2 mesh showed large variations in the local symmetry axis with respect to the machine as measured by microbeam X-ray diffraction incrementally from the edge to the core of the parts. In comparison, samples with no mesh insert showed only gradual changes in the tilt angle (angle between local symmetry axis and flow direction). The modulus and tensile strengths of all samples with the 1.0 mm 2 mesh inserts were found to approach virtual global mechanical isotropy.

NCS-constrained exhaustive search using oligomeric models

The efficiency of the cross-rotation function step of molecular replacement (MR) is intrinsically limited as it uses only a fraction of the Patterson vectors. Along with general techniques extending the boundaries of the method, there are approaches that utilize specific features of a given structure. In special cases, where the directions of noncrystallographic symmetry axes can be unambiguously derived from the self-rotation function and the structure of the homologue protein is available in a related oligomeric state, the cross-rotation function step of MR can be omitted. In such cases, a small number of yet unknown parameters defining the orientation of the oligomer and/or its internal organization can be optimized using an exhaustive search. Three difficult MR cases are reported in which these parameters were determined and the oligomer was positioned according to the maximal value of the correlation coefficient in a series of translation searches.