Second-order Displacement Research Articles

Relative Satellite Motion About an Oblate Planet

A solution to the relative motion problem for satellites is constructed that is based on nearly circular reference periodic orbits and Floquet theory for the relative motion. This solution conceptually resembles the Clohessy-Wiltshire solution for relative motion, but includes all zonal harmonics of the Earth's gravitational field. Accuracy of this solution is at least three orders of magnitude better than the Clohessy-Wiltshire solution. We treat the effect of second-order displacements from the periodic solution via a semi-analytic perturbation theory. The effects of sectoral and tesseral gravity terms, as well as air drag, make their appearance as a particular solution to the linear Floquet problem. Comparisons are made between numerical integrations in the inertial frame and predictions from the periodic orbit/Floquet/modal perturbation theory, including all gravitational harmonics through order 14, as well as air drag. Submeter-10-m accuracy is achieved for representative clusters in a 1.1 Earth radii, one radian inclination orbit, over one half-day.

Two-dimensional model for tapered pulse tubes: Part 3: unsteady components of second-order mass flux and temperature

Unsteady components of the second-order axial velocity and temperature within a tapered pulse tube were obtained by using a hybrid method of solution. In this method, the unsteady components of the second-order axial velocity and temperature consist of an analytical solution and a numerical solution. The effects of the operating frequency and the taper angle on the unsteady components of the second-order mass flux and temperature were shown. The unsteady second-order mass flux and the time-averaged enthalpy flow by the unsteady second-order mass flux increase as the second-order displacement volume at the cold end increases.

Asymptotic modelling of a nonsymmetric beam

In this paper we present a limit model for nonsymmetric elastic beams obtained from the three-dimensional problem using an asymptotic method in the framework of the linear theory of elasticity. We completely characterize the first- and second-order displacements and the first-order stress field, giving results related to existence, uniqueness and convergence for solution of the limit models.

Deformation measurements by digital image correlation: Implementation of a second-order displacement gradient

This paper outlines the procedure for refining the digital image correlation (DIC) method by implementing a second-order approximation of the displacement gradients. The second-order approximation allows the DIC method to directly measure both the first- and second-order displacement gradients resulting from nonlinear deformation. Thirteen unknown parameters, consisting of the components of displacement, the first- and second-order displacement gradients and the gray-scale value offset, are determined through optimization of a correlation coefficient. The previous DIC method assumes that the local deformation in a subset of pixels is represented by a first-order Taylor series approximation for the displacement gradient terms, so actual deformations consisting of higher order displacement gradients tend to distort the infinitesimal strain measurements. By refining the method to measure both the first- and second-order displacement gradients, more accurate strain measurements can be achieved in large-deformation situations where second-order deformations are also present. In most cases, the new refinements allow the DIC method to maintain an accuracy of ±0.0002 for the first-order displacement gradients and to reach ±0.0002 per pixel for the second-order displacement gradients.

New Models for Bending and Torsion of Variable Cross Section Rods

In this paper we propose new models for variable cross section rods in both symmetric and nonsymmetric cases. These models have been obtained from the three-dimensional linear elasticity model by using an asymptotic method where the small parameter is the area of the cross section. In this way we succeed in characterizing the first- and second-order displacements and the first-order stress field. We shall also prove existence, uniqueness, and convergence results for the solution of both limit models.

Study of the chemical dependence of the effective pair potentials of Co–Pt alloy

Effective pair interaction potentials of Co–Pt alloy have been obtained from X-ray diffuse scattering measurements on short-range ordered Co3Pt and Co65Pt35single crystals. The present work presents the evolution of the interaction energies in the Co–Pt alloy when the Pt concentration varies from 25 to 35 at. %. The aim of this work was to provide data in order to improve the computation of the phase diagram by allowing a concentration dependence of the effective pair potentials. Different methods have been proposed for separating the contributions of local order and first-order displacements from the total diffuse intensity. A comparison of different methods of analysis of Co3Pt diffuse-scattering intensity data is reported in the Appendix of this article; it shows that the second-order displacement terms are almost zero near the transition temperature.

Study of the chemical dependence of the effective pair potentials of Co–Pt alloy

Effective pair interaction potentials of Co–Pt alloy have been obtained from X-ray diffuse scattering measurements on short-range ordered Co3Pt and Co65Pt35 single crystals. The present work presents the evolution of the interaction energies in the Co–Pt alloy when the Pt concentration varies from 25 to 35 at. %. The aim of this work was to provide data in order to improve the computation of the phase diagram by allowing a concentration dependence of the effective pair potentials. Different methods have been proposed for separating the contributions of local order and first-order displacements from the total diffuse intensity. A comparison of different methods of analysis of Co3Pt diffuse-scattering intensity data is reported in the Appendix of this article; it shows that the second-order displacement terms are almost zero near the transition temperature.

The Second-Order Doppler Displacement of the M�ssbauer ?-Radiation in Amorphous Solid States

The influence of two-level tunnelling systems and bosonic peak on the second-order Doppler displacement of Mossbauer γ-radiation in disordered materials is investigated. It is shown, that the Mossbauer frequency temperature displacement at temperatures 10 -4 K < T < ω m ∞/k B is determined by bosonic peak and at T < 10 -4 K by two-level tunnelling systems.

Nonlinear acoustics of bounded solid

Reflection and refraction are investigated when the incidental angles of transverse waves are larger than the critical one. An interesting result is obtained that at this angle all the second-order displacements on the stress-free surface have a very strong maximum module and a phase leap; thereupon an appropriate second-order harmonic surface wave could be produced.

The Second-Order Doppler Displacement of the Mössbauer γ-Radiation in Amorphous Solid States

The Second-Order Doppler Displacement of the Mössbauer γ-Radiation in Amorphous Solid States

Primary and secondary 5-(alkyloxy)thianthrenium perchlorates. Characterization with1H NMR spectroscopy, reactions with iodide and bromide ion, and thermal decomposition in solution

A series of 5-(alkyloxy)thianthrenium perchlorates has been made in which the alkyl group is primary (1a-1p) and secondary (2a-2g). Preparations were carried out by reaction of the corresponding alkanol with thianthrene cation radical perchlorate in CH2Cl2solution followed by precipitation of the perchlorate salt with dry ether.1H NMR spectroscopy reveals that the presence of a stereogenic center in the alkyl group causes inequivalence in the ordinarily paired protons (e.g., H-4, H-6) of the thianthrenium ring. Reaction of iodide and bromide ion with primary alkyl-group compounds (e.g., methyl, ethyl, propyl, butyl) gave the alkyl halide in very good yield and by a second-order kinetic displacement. The second product was thianthrene 5-oxide (ThO). Rate constants for some of these reactions are reported. Reaction of secondary alkyl group compounds (e.g., 2-propyl, 2-pentyl, 2-hexyl, and 3-hexyl) with iodide ion gave good yields of alkyl iodide but also increasing evidence for a side reaction at the sulfonium sulfur, leading to I2, thianthrene, and secondary alkanol. Decomposition of some compounds at 100°C in solution (acetonitrile or 1,2-dichloroethane) was studied and gave alkene(s) and ThO.Key words: thianthrene cation radical, 5-(alkyloxy)thianthrenium perchlorates.

The ordered structure of at 70 - 75 K

The ordered structure of at 70 - 75 K has been studied by time-resolved neutron powder diffraction. The ordered superlattice has a structure with approximately four full D sites and one vacant D site in successive (420) planes, which is consistent with previous observations. Several superlattice lines not reported in previous single-crystal studies were observed. Quantitative information about the ordered structure was obtained by careful Rietveld analysis of the data, including anisotropic line broadening and diffuse peak background corrections. The structure refinement confirms a relaxation of D atoms towards the vacant planes and reveals a related second-order displacement of the D atoms in the octahedral sites towards a preferred vacant nearest-neighbour site. The thermal vibration along the c-axis is anisotropic as a result of the preferential relaxation.

Numerical algorithms and results for SC-cut quartz plates vibrating at the third harmonic overtone of thickness shear

Finite element matrix equations, derived from two-dimensional piezoelectric high frequency plate theory are solved to study the vibrational behavior of the third overtone of thickness shear in square and circular SC-cut quartz resonators. The mass-loading and electric effects of electrodes are included. A perturbation method which reduces the memory requirements and computational time significantly is employed to calculate the piezoelectric resonant frequencies. A new storage scheme is introduced which reduces memory requirements for mass matrix by about 90% over that of the envelope storage scheme. Substructure techniques are used in eigenvalue calculation to save storage. Resonant frequency and the mode shapes of the harmonic third overtone thickness shear vibrations for square and circular plates are calculated. A predominant third overtone thickness shear displacement, coupled with the third overtone of thickness stretch and thickness twist, is observed. Weak coupling between the third order thickness shear displacement and the zeroth-, first-, and second-order displacements is noted. The magnitudes of the lower order displacements are found to be about two orders smaller than that of the third overtone thickness shear displacement.

Tertiary paleomagnetism of North and South China and a reappraisal of late mesozoic paleomagnetic data from Eurasia: Implications for the cenozoic tectonic history of Asia

To contribute to the understanding of the Cenozoic tectonic history of China, and eastern Eurasia in general, we collected paleomagnetic samples from Tertiary rocks in both North and South China. Our Miocene paleomagnetic data for northern China are from four widely separated basalt sections in the Inner Mongolia Autonomous Region and Shandong Province. Both normal and reversed polarities of the characteristic magnetization are present in the stratigraphie sequence, and the mean normal and reversed directions are closely antiparallel to each other. The fact that the same characteristic direction was carried by both magnetite and hematite suggests that the hematite was probably formed by auto-oxidation processes during primary cooling of the basalts. The paleomagnetic data from these four areas also pass a regional generalized fold test. The corresponding mean Miocene paleomagnetic pole for northern China is longitude 261.6°E, latitude 88.3°N (A 95 = 6.4°). Our Early Tertiary paleomagnetic results for south China are from two sandstone sections around the city of Nanning (22.7°N, 108.3°E) in Guangxi Zhuang Autonomous Region. High-temperature characteristic directions isolated in 31 samples from five sites pass both tilt and reversal tests. The paleomagnetic pole position determined from the five sites lies at longitude 228.0°E, Latitude 83.6°N (A 95 = 8.5°). The Tertiary paleomagnetic poles for North and South China are statistically indistinguishable from each other and thus confirm that North and South China were fully assembled at least by the Tertiary. Our new Tertiary paleomagnetic data are consistent with the hypothesis that the major continental blocks of North and South China were in the same position as today during the Tertiary. A critical evaluation and selection of paleomagnetic results from the Late Jurassic onward for Siberia, Europe and North and South China further corroborate this hypothesis. Most previously existing Jurassic and Cretaceous paleomagnetic poles from Europe/Siberia are of uncertain reliability and should not be used for tectonic reconstruction. Using poles derived from transferring reliable data from other continents, we show that the previous paleomagnetic inferences of large eastward displacements between these Chinese blocks during the Cenozoic are no longer warranted. The available paleomagnetic data that meet our acceptance criteria are also of interest concerning the boundary conditions of the propagating extrusion model for Asian tectonics. Additional Mesozoic and Cenozoic paleomagnetic data from Siberia and North and South China are still needed to resolve the smaller, second-order displacements that appear to characterize the Cenozoic tectonic history of this part of Asia.

High-temperature partitioning of 181Hf-probe impurities between Li and group-V sites in LiNbO3 and LiTaO3.

Perturbed-angular-correlation (PAC) spectroscopy was used to measure nuclear-electric-quadrupole interactions at high temperatures in ceramic ternary metal oxides, ${\mathrm{LiNbO}}_{3}$ and ${\mathrm{LiTaO}}_{3}$. In these ferroelectric ceramics, the $^{18}\ensuremath{\rightarrow}^{181}$Ta PAC probe was carried by approximately 0.03 at. % Hf as an impurity dopant, and the PAC measurements were made over a temperature range from approximately 1300 to 1700 K, which included the ferroelectric-to-paraelectric transition for ${\mathrm{LiNbO}}_{3}$. As we recently reported, at temperatures below \ensuremath{\approxeq}1100 K, the probe substitutes primarily into the Li sites in both compounds. At higher temperatures, we find that the $^{181}\ensuremath{\rightarrow}^{181}$Ta probe partitions between the Li and group-V sites. At these higher temperatures, the Li-site electric-field gradients (efg's) are characterized by high frequencies that are similar to those observed at lower temperatures. The group-V site efg's are characterized by frequencies that are approximately one tenth the magnitude of those corresponding to the Li sites. The occupancy of the Li site decreases and that of the group-V sites increases monotonically with increasing temperature. This partitioning is an equilibrium, reversible process that can be described by a simple thermodynamic model of second-order displacement. The temperature dependence of the associated equilibrium constant indicates enthalpies of several eV for the process.

Propagation and trapping of obliquely incident wave groups over a trench with currents

The analytical solutions of secord-order long waves generated by the diffraction of short wave groups by a combination of depth and current velocity discontinuities are presented. The incident wave groups consist of two co-linear short waves with slightly different frequencies. These incident short wave groups are reflected partially by the abrupt changes in depth and current velocity. The incident, reflected, and transmitted wave groups are accompanied by the second-order locked long waves, which propagate together with the wave envelopes of the short waves at their group velocity. Because of the discontinuity in the second-order free surface displacement of the locked long waves, free long waves are generated, which propagate at the shallow-water phase speed and in a direction different from those of short waves and wave envelopes. With certain combinations of angle of wave incidence, depth ratios and current velocity, the free long waves can be trapped in the vicinities of depth and current discontinuities. Several examples are given to illustrate the resonance of trapped free long waves.

Barkhausen noise in thin-film recording heads

Barkhausen noise in thin-film heads was studied by exciting the heads with an external magnetic field and with a head-coil-generated field. Observed pulses show an exponential growth and decay, resulting in a second-order wall energy versus displacement model. Analytical expressions are given for the observed pulse shapes. Most pulses are essentially associated with unstable regions around attractive interaction sites where a wall in equilibrium cannot exist. A method is presented which can detect in which pole piece (P/sub 1/ or P/sub 2/) a wall jump occurs. It also differentiates wall jumps located close to the pole faces from those occurring deeper in the yoke. >

A note on long waves induced by short-wave groups over a shelf

The generation of long waves due to the refraction of wave groups over a shelf is reexamined. The appropriate boundary conditions for calculating the second-order mean free-surface displacement are discussed. Numerical results for a plane shelf are presented, which are different from those given by Mei &amp; Benmoussa (1984).

A versatile model for interactive buckling of columns and beam-columns

A new formulation has been developed to study the interactive buckling of thin-walled columns having arbitrary cross-sections. The emphasis in this paper is, however, on columns with a single axis of symmetry. The formulation is designed to take into account the simultaneous interaction of the purely flexural and flexural-torsional overall modes of buckling with local buckling. The local buckling deformations are described in terms of a primary local mode together with two secondary local modes of the same wavelength. The latter are triggered by the interaction of bending in two perpendicular planes with the primary local mode. The three eigenmodes and the six second-order in-plane displacement fields are all computed using a finite-strip technique. The modulation of the amplitudes of the local modes and the overall displacements are described in terms of a one-dimensional finite element model. Thus a new beam element which has embedded in it the local buckling information is developed. It appears that the present analytical model is very versatile being applicable to members of arbitrary cross-section and end conditions. For columns with a single axis of symmetry, it is seen that there exists a non-linear coupling between the purely flexural and the flexural-torsional modes of buckling via local buckling deformation. Typical examples of channel section columns are presented. It is shown that the channel section columns of commonly used proportions arc highly imperfection sensitive in the context of combined interaction of the enumerated modes of buckling. This sensitivity remains even for columns with well separated overall and local critical stresses—a feature which is in stark contrast with the behavior of the Tvergaard panel.

Entry Flow in a Narrow Channel of Varying Gap With One Sliding Surface

An upstream potential flow analysis is performed for the entrance region of a narrow channel in the presence of fluid inertia forces. Existing thin film solutions for slider bearings have an anomalous singularity in the pressure gradient, due to the parabolic nature of the thin film equations. Although the geometry of the present paper is somewhat different than that of the slider bearing, the problem of the singularity is resolved and some information can be obtained regarding inlet flow in the more practical slider bearing case. The potential core flow solution consists of a uniform flow first-order solution, which accelerates due to the narrowing of the bearing, and a second-order displacement solution which decelerates due to the growth of boundary layers. The resulting pressure and velocity behavior are illustrated. To form a uniformly valid solution, the present upstream analysis must be matched to the downstream (thin film) result.