Principal Axes Of Inertia Research Articles

Properties of New Coordinates for the General Three-Body Problem

This is an analysis of the features of the new coordinate system given by the principal axes of inertia, as determined by Euler angles, and twodistances related to the inertia principal moments and an auxiliar angleas coordinates, for studying the general three-body problem, interactingthrough gravitational forces.

A non-linear hydrogen bond F⋯H–Br in vinyl fluoride ⋯HBr characterised by rotational spectroscopy

The ground-state rotational spectra of four isotopomers C2H3F⋯H79Br, C2H3F⋯H81Br, C2H3F⋯D79Br and C2H3F⋯D81Br of a complex formed between vinyl fluoride and hydrogen bromide have been observed with a pulsed-nozzle, Fourier-transform microwave spectrometer. Rotational constants (A0, B0, C0), all five quartic centrifugal distortion constants (ΔJ, ΔJK, ΔK, δJ, δK), the Br nuclear quadrupole coupling tensor χαβ(α,β = a,b,c) and the Br spin-rotation tensor M were determined in each case. A detailed interpretation of the χαβ tensor leads to the conclusion that the complex is planar, or nearly so. The direction cosine Φaz associated with the HBr axis z and the principal inertial axis a was determined together with the principal components χzz, χxx, χyy of the Br nuclear quadrupole coupling tensor. The geometry of the complex was obtained by fitting the rotational constants of its isotopomers under the constraint that the resulting geometry must reproduce Φaz. The HBr molecule was found to form a non-linear hydrogen bond F⋯H–Br [deviation from collinearity θ = 20.0(1)°] to the F atom of vinyl fluoride, with r(F⋯H) = 2.205(9) A and the angle C1F⋯H = 123.8(5)°, when the nuclei were assumed coplanar. A comparison of the geometry with that of vinyl fluoride ⋯HCl determined in the same way shows that the angular geometries of the HBr and HCl complexes of vinyl fluoride are isomorphous. This is the case even for the extent of the non-linearity θ of the hydrogen bond, which has the values θ = 20.0(1)° and 18.3(1)° in C2H3F⋯HBr and C2H3F⋯HCl, respectively. A small extent of non-planarity is also consistent with the observations for both complexes but the conclusions are not significantly affected by relaxing the assumption of coplanar nuclei.

コンピュータグラフィックス 慣性モメントの包絡線を用いた３次元モデルの形状類似検索

Effective storage and retrieval of three-dimensional (3D) models has become an important issue. In this paper, we proposes a method for shape-similarity search of 3D polygonal-mesh models. As the feature vector, we employed a set of envelopes of statistical distributions of mass concentrated on the surface. The envelopes are parameterized along principal axes of model inertia. As a distance measure among, we experimented with theEuclidean norm as well as a distance measure based on elastic matching by using the dynamic programming technique. Experiments showed that our approach could identify models having similar shapes fainy well. As a shot-coming feature vectors, the query results did not necessarily reflect subjective measures used by human beings.

RELATIVE EQUILIBRIA OF A GYROSTAT SATELLITE WITH INTERNAL ANGULAR MOMENTUM ALONG A PRINCIPAL AXIS

A new analytical solution to the problem of determining all equilibria of a gyrostat satellite in a circular orbit is given in the case where internal angular momentum of the gyrostat satellite is collinear to its principal axis of inertia. The number of isolated equilibria is shown to be no less that eight and no more than 24. Sufficient conditions for equilibria stability are obtained.

The conformational structures and dipole moments of ethyl sulfide in the gas phase

The pure rotational spectrum of ethyl sulfide has been measured from 12 to 21 GHz in a 1 K jet-cooled expansion using a Fourier-transform microwave (FTMW) spectrometer. Prominent features in the spectrum are assigned to transitions from three conformational isomers. Additional assignments of the 13C and 34S isotopomer spectra of these conformers effectively account for all of the remaining transitions in the spectrum. Accurate “heavy-atom” substitution structures are obtained via a Kraitchman analysis of 14 rotational parameter sets, permitting definitive identification of the molecular structures of the three conformers. Two of the structures designated as the gauche–gauche (GG) and trans–trans (TT) conformers have symmetric forms with C2 and C2v symmetries, respectively, and the third trans–gauche (TG) configuration is asymmetric. The components of the electric dipole moment along the principal inertial axes have been determined from Stark measurements and are consistent with these structural assignments. Detailed comparisons are made with the calculated geometries, dipole moments, and energy-level ordering at both the HF (Hartree–Fock)/6-31* and MP2 (second-order Møller–Plesset)/6-311** levels of theory. Significant discrepancies are found, which are mainly attributed to errors in the calculated dihedral angles that define the different conformations. A graphical-user-interface computer program has aided in the identification and assignment of entangled hybrid-band spectra from the different conformers and isotopomers in this study. The program includes features that enable real-time refinement of rotational constants and hybrid band intensities through visual comparisons of the experimental data with simulated spectra. Capacities also exist to rapidly assign quantum number labels for least-squares fitting purposes.

Cramer-Rao bounds for estimating range, velocity, and direction with an active array

We derive Cramer-Rao bound (CRB) expressions for the range (time delay), velocity (Doppler shift), and direction of a point target using an active radar or sonar array. First, general CRB expressions are derived for a narrowband signal and array model and a space-time separable noise model that allows both spatial and temporal correlation. We discuss the relationship between the CRB and ambiguity function for this model. Then, we specialize our CRB results to the case of temporally white noise and the practically important signal shape of a linear frequency modulated (chirp) pulse sequence. We compute the CRB for a three-dimensional (3-D) array with isotropic sensors in spatially white noise and show that it is a function of the array geometry only through the moments of of the array. The volume of the confidence region for the target's location is proposed as a measure of accuracy. For this measure, we show that the highest (and lowest) target location accuracy is achieved if the target lies along one of the principal axes of inertia of the array. Finally, we compare the location accuracies of several array geometries.

On certain integrable motions of a rigid body acted upon by gravity and magnetic fields

The motion of a gyrostat about a fixed point in uniform gravity and magnetic fields is considered. The gyrostatic moment is directed along a principal axis of inertia while the centre of mass and the magnetic moment lie in the orthogonal principal plane. We show that the conditions imposed by Dragovic (Mat. Vesnik 49 (1997) 279–281) on the motion in the integrable case pointed out by him and considered as a generalisation of the famous Goryachev–Chaplygin case (Mathematical Reviews: MR #99d: 70014) lead only to a pendulum-like motion about a fixed axis, in which the presence of the magnetic field is not significant. We show also that this motion remains possible even when the Goryachev conditions on the moments of inertia are completely removed.

Dynamic Monte Carlo Simulation on the Polymer Chain with One End Grafted on a Flat Surface

A linear polymer chain in good solvent condition with one end grafted on a infinitely large, impenetrable flat surface is investigated using dynamic Monte Carlo simulation on a simple cubic lattice. Chain shape and dimension, angular correlation between the direction of the end-to-end vector and that of the longest principal axis of inertia are studied and discussed. Results reveal that the asphericity of end-grafted polymer chains is greater than that of free ones, the limit ratio 〈L12〉 : 〈L22〉 : 〈L32〉 is about 1 : 3.0 : 14.9. The limit of mean angle 〈θ〉 of end-grafted chains is about 22°, smaller than that of free chains, indicating angular correlation between the direction of the end-to-end vector and that of the longest principal axis is reinforced.

Problems of Evolution of Rotations of a Rigid Body under the Action of Perturbing Moments

The rotatory motion of a nearly dynamically spherical rigidbody, which contains a viscoelastic element, is considered. This elementis simulated by a moving mass, connected by a spring and damper to thepoint, situated on one of a principal axis of inertia. The smallparameters caused by the proximity of moments of inertia and thepresence of moving mass are considered to be of the same order. Thespherical coordinates defining the position of the angular velocityvector are introduced. The system of differential equations is obtainedand investigated, the special cases of motion are considered. Theauthors investigate perturbed rotational motions of a rigid body,similar to the regular precession in the Lagrange case, under the actionof the moment that is slowly changing in time and the restoring momentdepending on the angle of nutation. In two problems it is assumed thatthe angular velocity of the body is large and its direction is close tothe axes of dynamic symmetry. In the first problem it is assumed thattwo projections of the vector of the perturbing moment onto principalaxes of inertia of the body are small as compared to the restoringmoment, while the third one is of the same order of the magnitude as themoment in question. In the second problem it is assumed that theperturbing moments are small as compared to the restoring one. Averagedsystems of equations of motion are obtained and investigated in thefirst and the second approximations. Examples are considered.

Microwave spectra, density functional theory calculations and molecular structure of acetylenemethyldioxorhenium

The molecular structure for acetylenemethyldioxorhenium, (C2H2)(CH3)ReO2, was obtained by measuring and analyzing the rotational spectra for 14 isotopomers. This appears to be the first gas-phase measurement for a rhenium metallacyclopropene, and the first structural data on this compound. This complex is closely related to reaction intermediates in methyltrioxorhenium and osmium tetroxide catalyzed oxidation reactions, which are important in industrial chemical production and syntheses of chiral products. The microwave spectra were measured in the 4–11 GHz range using a Flygare–Balle-type pulsed-beam microwave spectrometer. The structural parameters obtained using the 42 measured rotational constants are in very good agreement with results from the new density functional theory calculations. The structure of the acetylene ligand is modified through interaction with the metal atom, and exhibits partial sp2 hybridization in the complex. The C–C bond length is increased by 0.09 Å to 1.29 Å. The H–C–C interbond angles are reduced from 180° to 146°, and 147°. The measured rhenium, methyl–carbon bond length is 2.116(2) Å, and rhenium–acetylene carbon bond lengths are 2.043(2) Å and 2.067(2) Å. Large quadrupole splitting patterns arising from the two rhenium isotopes [χcc(185 Re)=812.62 MHz, and χcc(187 Re)=769.023 MHz], with large off-diagonal components [|χab(185 Re)|=725.40 MHz, and (|χab(187 Re)|=686.384 MHz] caused some initial difficulty in assigning transitions. When the assignments and analysis were completed, the systematic changes in the measured quadrupole coupling tensor components due to rotations of the principal inertial axes on isotopic substitution were obtained, and helped to confirm assignments for the various isotopomers.

Conformational Analysis of Chain Molecules in Liquid Crystalline Phases by a Rotational Isomeric State Scheme with Maximum Entropy Method I. 1H-1H Dipolar Couplings from n-Alkanes Dissolved in a Nematic Solvent

For conformational analysis of chain molecules incorporated in nematic fields, a rotational isomeric state (RIS) scheme was revised for use with the maximum entropy method. Analysis thus based proceeds as follows. 1) All possible conformations are enumerated within the RIS approximation. 2) The principal Axes of inertia, determined for each conformer, are considered molecular axes, and orientational order parameters are evaluated from dimensions of a rectangular parallelepiped closely fitted to the conformer. 3) Conformational statistical weight factors of first- and second-order interactions are adjusted and order parameters are scaled so as to reproduce experimental observation and maximize the information entropy regarding conformer populations. By this method, 1H-1H dipolar couplings observed from n-hexane, n-heptane, n-octane, n-nonane, and n-decane dissolved in a nematic liquid crystal Kodak EK11650 p-pentylphenyl-2-chloro(4-benzylbenzoyloxy)-benzoate [M.E. Rosen, S.P. Rucker, C. Schmidt, and A. Pines, J. Phys. Chem. 97, 3858 (1993)] were analyzed. For all n-alkanes, good agreement between theory and experiment was attained. Anisotropic conformers were slightly more populated than in the free state. The present results were quantitatively comparable to those of analysis using the mean-field theory.

Numerical and analytical study on deep biaxial bending collapse of thin-walled beams

The objective of the paper is to get a fundamental understanding of large rotation response of thin-walled prismatic beams subjected to biaxial bending. Square and rectangular section beams were subjected to a cantilever bending with hinge rotation up to 40°. A general purpose nonlinear FE code PAM-CRASH was used to generate results. Altogether 94 computer runs were made by changing aspect ratio, condition of the loading and bending plane with respect to principal inertia axes of the cross section. The main emphasis was placed on the determination of initial and subsequent yield loci. It was also shown that the direction of the vector of generalized strain rates obeys the normality rule. Calculated in the paper was also a critical angle defining a preferable direction of bending. The present finding can be used to better interpret results of FE calculation of automotive bodies and to develop simplified crash-oriented design tools.

Radiative precession of an isolated neutron star

Euler's equations of motion are derived exactly for a rigid, triaxial, internally frictionless neutron star spinning down electromagnetically in vacuo. It is shown that the star precesses, but not freely: its regular precession relative to the principal axes of inertia couples to the component of the radiation torque associated with the near-zone radiation fields and is modified into an anharmonic wobble. The wobble period τ1 typically satisfies τ1≲10^−2τ0, where τ0 is the braking time-scale; the wobble amplitude evolves towards a constant non-zero value, oscillates, or decreases to zero, depending on the degree of oblateness or prolateness of the star and its initial spin state; the (negative) angular frequency derivative ω˙ oscillates as well, exhibiting quasi-periodic spikes for triaxial stars of a particular figure. In light of these properties, a young, Crab-like pulsar ought to display fractional changes of order unity in the space of a few years in its pulse profile, magnetic inclination angle and ω˙. Such changes are not observed, implying that the wobble is damped rapidly by internal friction, if its amplitude is initially large upon crystallization of the stellar crust. If the friction is localized in the inner and outer crusts, the thermal luminosity of the neutron star increases by a minimum amount ΔL≈3×10^31(e/10^−12)(ω/10^3 rad s^−1)^2(τd/1 yr)^−1 erg s^−1, where e is the ellipticity, and τd is the damping time-scale, with the actual value of ΔL determined in part by the thermal conduction time τcond. The increased luminosity is potentially detectable as thermal X-rays lasting for a time ≈ max(τd,τcond) following crystallization of the crust.

A relation between the principal axes of inertia and ligand binding.

The principal axes of inertia are eigenvectors that can be calculated for any rigid body. We report studies of the position of the principal axes in crystallographically solved protein molecules. We find with high frequency that at least one principal axis penetrates the surface of the respective protein in a region used for ligand binding. In antibody variable regions, an axis goes through the third hypervariable loop of the heavy chain. In major histocompatibility complex proteins, an axis goes through the peptide-binding groove. In protein-protein heterodimers, a principal axis of one subunit will often penetrate the interface formed with the other subunit. In many of these protein-protein complexes, the axis specifically intersects residues known to be critical for molecular recognition.

The motion of the Earth's principal axes of inertia, caused by tidal and rotational deformations

Abstract Perturbations of the pole position of the Earth's axes of inertia due to its tidal and rotational deformations have been found.

Microwave Fourier transform spectrum of t-butyl mercaptan

The microwave spectrum of t-butyl mercaptan, (CH3)3CSH, in its vibrational ground state has been measured in the range from 8 to 26GHz, and 542 transitions were newly assigned. The spectrum was analysed using the combined axis method. The three rotational constants, four quartic centrifugal distortion constants according to Watson's S-reduction, the threefold hindering potential V3 for the CSH torsion, the moment of inertia of the t-butyl group Iα, and the angle between the internal rotation axis i and the principal inertia axis a ∠(i, a) were accurately determined. However, the standard deviation of the fit, 200kHz, is about two orders of magnitude larger than the experimental uncertainty and the residuals rapidly increase with J indicating that the rigid top–rigid frame assumption is not a perfect one and that a more sophisticated model should be used.

Rotational Spectroscopy and Ring-Puckering Conformation of 3-Hydroxytetrahydrofuran

The microwave spectrum of 3-hydroxytetrahydrofuran was recorded using a Fourier transform microwave spectrometer; the rotational constants (MHz) are: A = 5581.8230(7), B = 3638.8316(7), and C = 2924.7410(7). Microwave spectra for four 13C isotopic species were recorded in natural abundance. Stark effect measurements were used to determine the projections of the dipole moment on the principal inertial axes: μa = 0.956(1) D, μb = 0.875(2) D, μc = 1.050(2) D, and μtot = 1.667(2) D. The internal coordinates of the carbon and oxygen atoms were least-squares fit to the 15 moments of inertia, and the best-fit structure is a C4′-endo envelope ring conformation, in good agreement with ab initio calculations (MP2/6-31G††).

The role of the inertia tensor in kinesthesis.

We report experiments directed at the ability of humans to perceive the spatial orientation of occluded objects, to position an occluded limb relative to targets or directions in the environment, and to match the spatial orientation of occluded contralateral limbs. Results suggest that each of these abilities is tied to the inertial eigenvectors of each object or limb, which correspond to the object's or limb's principal axes of rotational inertia. It is suggested that the mechanisms supporting the perception of intact limbs, neuropathic or anesthetized limbs, prosthetic devices, and hand-held tools and implements via kinesthesis may be one and the same--the detection of movement-produced physical invariants such as the inertia tensor. While the research reported presently has focused on the perception of intact limbs and hand-held objects, future research should be directed at possible generalizations of this work to a variety of clinical populations, including those involving peripheral neuropathies and prosthetic devices.

Space Curve Representation and Recognition Based on Wavelet Transform Zero-Crossings

A method to construct a representation of space curves based on the zero-crossings of the dyadic wavelet transform is introduced. The principal axes of inertia of these space curves, referred to as objects, are considered as the reference system. The representation is translation, rotation and size invariant. Instances of objects in images are recognised by matching their representations with those of the models. A string-matching technique is adapted and used for this purpose. Experimental results show that the representation is robust and efficient in extracting and matching object information.

Optimal control of a rotational motion of a gyrostat on circular orbit

A control scheme is proposed to guarantee an optimal stabilization of a given rotational motion of a symmetric gyrostat on circular orbit. The gyrostat controlled by the control action generated by rotating internal rotors. In such study the asymptotic stability of this motion is proved using Barbachen and Krasovskii theorem's and the optimal control law is deduced from the conditions that ensure the optimal asymptotic stability of the desired motion. As a particular case, the equilibrium position of the gyrostat, which occurs when the principal axes of inertia coincide with the orbital axes, is proved to be asymptotically stable. The present method is shown to more general than previous ones.