Principal Axes Of Inertia Research Articles

Steady motions of gyrostat satellites and their stability

The steady motions of a rigid body rotating about its maximum principal axis of inertia, while the radius vector lies in the direction of its minimum principal axis of inertia, is known to be stable in the sense of Lyapunov. Due in part to their stowed configuration in launch vehicles, however, satellites typically have an initial rotation about their minimum principal axis of inertia. Such rotation may be unstable in the presence of some dissipations. This paper investigates the effect of momentum wheels on the stability of steady motions. It is proved that the momentum wheels increase the effective moment of inertia of the gyrostat-satellite system about some desired axis. Stability of the steady rotation about the desired axis can be established only for the case when the moment of inertia of the axis aligned with the radius vector is smaller than that of the axis of linear momentum. A new set of stability criteria is obtained which includes the effects of the coupling between the orbital and attitude dynamics and may be useful in the design of attitude control systems for large spacecraft in low Earth orbit. >

Dynamic Stiffness Matrix for a Beam Element with Shear Deformation

A method for calculating the dynamic transfer and stiffness matrices for a straight Timoshenko shear beam is presented. The method is applicable to beams with arbitrarily shaped cross sections and places no restrictions on the orientation of the element coordinate system axes in the plane of the cross section. These new matrices are needed because, for a Timoshenko beam with an arbitrarily shaped cross section, deflections due to shear in the two perpendicular planes are coupled even when the coordinate axes are chosen to be parallel to the principal axes of inertia.

Dynamic Stiffness Matrix for a Beam Element with Shear Deformation

A method for calculating the dynamic transfer and stiffness matrices for a straight Timoshenko shear beam is presented. The method is applicable to beams with arbitrarily shaped cross sections and places no restrictions on the orientation of the element coordinate system axes in the plane of the cross section. These new matrices are needed because, for a Timoshenko beam with an arbitrarily shaped cross section, deflections due to shear in the two perpendicular planes are coupled even when the coordinate axes are chosen to be parallel to the principal axes of inertia.

Flow Induced Vibrations in Engineering Systems and the Control of Flow Induced Vibrations. Determination of Equilibrium Position of Liquid Contained in Tear-Drop Tank System in a Spinning Satellite.

The equilibrium position of liquid in a tear-drop tank system is determined for the case in which fairly large displacement of the center of gravity is caused by the detachment of objects from a satellite. A double-loop iteration algorithm is applied to determine the unknown center of gravity under the prescribed total liquid mass. To calculate the mass, inertia tensors and center of gravity of the liquid, volume integration throughout the liquid domain surrounded by conical and spherical tank walls and parabolic liquid surfaces in the tanks is performed accurately by using partial spherical coordinates. The numerical results show that some of the tanks become empty when the total liquid mass is smaller than a certain critical value and that tilt of the principal inertia axis is maximum at the critical condition. A method for decreasing the critical liquid mass and the tilt of the principal inertia axis is presented.

Study of the shape of random walks

A comprehensive study of the shape of discrete random walks, considering the general case of arbitrary length and any space dimension, is presented. The probability distributions for several magnitudes such as the principal inertia moments, the asphericity and the angle between the principal axis of inertia and the end-to-end vector are evaluated numerically. In most cases, and especially in low-dimensional spaces, these probability distributions spread widely and possess a non-zero skewness. This implies that any description of the shapes of random walks which is based only on mean values of related magnitudes is incomplete. This situation does nor vary when random walks of large lengths are considered since the relative fluctuations (ratio between the standard deviation and the mean value) do not go to zero in that limit. On the other hand, when the space dimension grows, the distributions become more peaked, reaching the expected asymptotic limit for infinite dimension. The present study also indicates that the probability distributions for the principal inertia moments have approximately the form of chi-squared distributions. Analysing the probability distribution for the asphericity, it is shown that the distributions for the different moments are not completely independent.

Swerve During Three-Dimensional Impact of Rough Rigid Bodies

For collisions between rough bodies, dry friction can be represented by Coulomb’s law; this relates the normal and tangential components of contact force by a coefficient of limiting friction if the contact is sliding. The friction force acts in a direction opposed to sliding. For a collision with planar changes in velocity, sliding is in either one direction or the other; the direction can reverse before separation only if the impact configuration is eccentric or noncollinear and the initial velocity of sliding is small. In general, however, friction results in nonplanar changes in velocity; for free bodies the velocity changes are three-dimensional or nonplanar unless the initial sliding velocity lies in the same plane as two principal axes of inertia for each body. Nonplanar velocity changes give a direction of sliding that continually changes or swerves during an initial phase of contact in an eccentric impact configuration. The present paper obtains changes in relative velocity during “rigid” body collisions as a function of impulse Pn of the normal component of reaction force. The method of resolving changes in relative velocity as a function of impulse is demonstrated by obtaining the solution for a spherical pendulum colliding with a rough half-space. The solution depends on two independent material parameters—the coefficient of friction and an energetic coefficient of restitution.

Can the nonlinearity of hydrogen bonds B…HX be measured precisely? A method from rotational spectroscopy and its application to 2,5-dihydrofuran…HCl

Rotational constants, centrifugal distortion constants and the complete Cl-nuclear quadrupole coupling tensor referring to the principal inertial axis system have been determined for the ground state of 2,5-dihydrofuran…HCl from its rotational spectrum. The complex has C s symmetry, with the hydrogen bond completing a pyramidal arrangement at the oxygen atom of 2,5-dihydrofuran. The coupling tensor allows a good approximation to the equilibrium angle α az between the HCl( z) axis and the a axis, and thence the nonlinearity θ=9.5° of the hydrogen bond, to be determined. The results are rationalised in terms of a refinement of a simple model of the hydrogen bonds.

A Quantitative Structure-Activity Relationship (QSAR) for prediction of α 2μ-globulin nephropathy

A number of chemicals induce a toxic syndrome in male rats—referred to as α 2μ-globulin nephropathy—that is characterized by an accumulation of the urinary protein α 2μ-globulin in renal lysosomes, subsequent cytotoxicity and cell death. Binding affinity to α 2μ-globulin has been identified as one of the determinants for α 2μ-globulin nephropathy. A quantitative structure-activity relationship (QSAR) has been derived by multiple regression analysis relating this binding to negative charge density of the binding molecule and its molecular volume. Previous data, correlating aliphatic and alicyclic hydrocarbon structures with ability to induce renal lysosome accumulation in male rats, were analysed by the technique of principal components analysis applied to the molecular volumes and principal inertial axes of alcohols predicted to be derived from the hydrocarbons. Mapping of the first and second principal components showed clustering of molecules relative to their biological activity. A combination of the two QSARs is useful in identifying molecules with a potential to cause α 2μ-globulin nephropathy.

Non-linear hydrogen bonds and rotational spectroscopy: measurement and rationalisation of the deviation from linearity

When the rotational spectra of hydrogen-bonded dimers B⋯HX (X = Cl or Br) of suitable symmetry are investigated under high resolution by the pulsed-nozzle, Fourier-transform technique, it is possible to determine the complete halogen nuclear quadrupole coupling tensor referred to the principal inertial axis system of the molecule. This tensor can then be diagonalised to give the components χxx, χyy and χzz in the HX axis system and the angle αaz between the direction of the principal inertial axis a and the direction of the HX axis z. A consideration of the values of χxx and χyy establishes that the HX angular oscillation is close to isotropic with respect to the x and y directions. It can then be shown that the angle αaz obtained is unaffected by the zero-point angular oscillation of the HX subunit. This result allows the deviation θ of the hydrogen bond B⋯H—X from linearity to be determined.Values of θ determined for the series of hydrogen-bonded dimers B⋯HX, where B is furan, sulfur dioxide, 2,5-dihydrofuran, oxirane, thiirane and formaldehyde and X is either 35Cl or 79Br, are considered. θ is found to vary significantly along this series and this variation is discussed in terms of a simple model of the complex involving a secondary interaction between the X atom and the subunit B.

A Quantitative Structure‐ Activity Relationship (QSAR) for Prediction of α‐2μ‐Globulin Nephropathy

AbstractA number of chemicals induce a toxic syndrome in male rats – referred to as α‐2‐μ‐globulin nephropathy – which is characterized by an accumulation of the urinary protein α‐2‐μ‐globulin in renal lysosomes, subsequent cytotoxicity and cell death. Borghoff et al. [1] measured the binding affinities to α‐2μ‐globulin for a number of molecules and metabolites recognised as causing α‐2μ‐nephropathy and suggested that binding is dependent on both hydrophobic interactions and hydrogen bonding. Binding affinity to α‐2μ‐globulin has been identified as one of the determinants for α‐2μ‐globulin nephropathy. A QSAR based on these data has been derived by multiple regression analysis relating the binding to negative charge density of the binding molecule and its molecular volume. Data of Bomhard et al. [2] correlating aliphatic and alicyclic hydrocarbon structures with ability to induce renal lysosome accumulation in male rats, were analysed by the technique of principal components analysis applied to the molecular volumes and principal inertial axes of alcohols predicted to be derived from the hydrocarbons. Mapping of the first and second principal components showed clustering of molecules relative to their biological activity. A combination of the two QSARs is useful in identifying molecules with a potential to cause α‐2μ‐globulin nephropathy.

Planar Orbits about a Triaxial Body: Application to Asteroidal Satellites

We have studied the dynamical behavior of a small satellite of a strongly triaxial primary rotating about a principal axis of inertia, restricting ourselves to the planar case (satellite orbits lying in the "equatorial" symmetry plane of the primary). Since we were mainly interested in applying the results to the dynamics of(natural and artificial) asteroidal satellites, we have studied in detail the case of a triaxial ellipsoidal primary, with axial ratios 2:1: 1/2 and rotation periods ranging from 5 to 40 hr. We have employed the classical method of Poincaré's surfaces of section to display the results of numerical integration experiments, showing in particular how the triaxial shape of the primary "perturbs" the dynamics of the Kepler problem seen in the rotating reference frame where the primary is fixed. We have found that a sizeable chaotic zone appears near the 1/1 resonance between the rotation period of the primary and the orbital period of the satellite. Another smaller chaotic zone includes the retrograde orbits passing close to the primary and corresponding to initial velocities close to the escape velocity. The chaotic zones become larger and larger for shorter rotation periods of the primary. However, regular orbits staying close to the primary do also exist, some of them being locked by resonant effects. Finally, we have identified the most suitable orbits for an artificial satellite of an asteroid (or a comet), subject to the requirements of being close (though non-collisional), of avoiding strongly chaotic behavior, and of being weakly affected by the (a priori unknown) asteroidal mass distribution.

Flexural Buckling of Pretwisted Columns

The present work deals with the problem of flexural buckling of a thin column supported at its ends. The column has a constant cross section, but its principal axes of inertia rotate as a function of its axial coordinate. The problem is solved with a direct variational method using Fourier series and shows that this particular geometry improves the critical load of the column with respect to the case in which the orientation of the section of the column remains constant. The flexural buckling load increases on increasing the rotation between the two end cross sections of the column (pretwisting). The algorithm converges rather rapidly. However, on increasing the pretwisting of the column a greater number of Fourier series terms must be considered to reach the necessary precision. The calculations have been Carried out hypothesizing that the rotation angle of the principal axes of inertia varies linearly with the axial coordinate. The problem can be easily extended to other modes of varying this angle.

The Millimeter-Wave Spectrum of Dimethylether-D 6 in Excited Torsional States

The rotational spectrum of dimethylether-D 6 in the first two excited torsional states between 40 and 100 GHz has been measured, and the rotational and quartic distortion Constants for both states are given. The analysis of the internal rotation gives the potential parameters V 3 and V′ 12 and the angles between the principal axes of inertia and the internal rotation axes. Also, the moment of inertia I α of the CD 3-group has been determined.

Theoretical and experimental structures of vinyl chloride and vinyl bromide

The structures of vinyl chloride and vinyl bromide have been determined by ab initio molecular orbital calculations, and corrections applied to obtain theoretical r0 values. The theoretical r0 structures agree very well with experimental r0 structures derived from recent microwave spectral data. This contrasts with the situation for vinyl alcohol and vinyl fluoride for which significant discrepancies have been found between theoretical and experimental estimates of the CCHu angles involving the hydrogen atom geminal to the substituted center. The present results reinforce our earlier conclusion that the theoretical values for this parameter for vinyl alcohol and vinyl fluoride are more reliable than the experimental values. Difficulties in the experimental structural determinations for vinyl fluoride and vinyl alcohol are attributed to the unusual insensitivity of the experimental moments of inertia to the values of the CCHu angles, resulting in turn from the close proximity to a principal inertial axis of two of the atoms that define the CCHu angles in these molecules. In contrast, all three atoms defining the CCHu angles in vinyl chloride and vinyl bromide are far removed from the inertial axes. As a consequence, the experimental moments of inertia are sensitively dependent on the value of the CCHu angle, and theoretical and experimental structures agree well. We conclude that suggestions that substituted ethenes present unusual difficulties for ab initio structural determinations are unfounded. Sensitivity curves are suggested as valuable indicators of potential problems in experimental structural determinations.

On the dynamics of rods in the theory of Kirchhoff and Clebsch

We discuss here the dynamical equations of a theory of elastic rods that is due to Kmcm-ioFF [1, 2] and CL~Bsc~ [3, 4]. This properly invariant theory is applicable to motions in which the strains relative to an undistorted configuration remain small, although rotations may be large. It is constructed to be a first-order theory, i.e., a theory that is complete to within an error of order two in an appropriate dimensionless measure of thickness, curvature, twist, and extension. In a first-order theory of thin rods, one can treat the rod as inextensible, and we do so here at the outset. Thus, at each time t, the arc-length parameter s for the axial curve ~(t) is employed as a material coordinate, i.e., a parameter whose value at a material point is constant in time, and not only the resultant of the shearing forces on a cross section, but also the tension in the rod, are reactive quantities not given by constitutive equations. Consider for a moment a rod that is naturally prismatic and dynamically symmetric, i.e., a rod that in an undistorted stress-free configuration is a cylinder whose directrix, although not necessarily a circle, bounds a figure with equal principal moments of inertia. A motion of the rod is said to be planar and twist-flee, i.e,, a motion of pure flexure, if it is such that ~(t) lies at all times in a fixed plane ~ which contains a principal axis of inertia of each cross section. For such a motion we employ a fixed Cartesian coordinate system on ~, and, for consistency with a discussion of more general motions to be given later in this paper, we call the abscissa z and the ordinate x. We may write FZ(s, t), FX(s, t) for the z- and x-components of the resultant force F at time t on the cross section with arc-length coordinate s. The motion of the rod may be described by giving the (z, x)-coordinates of the points on ~(t) as functions of s and t. With O(s, t) the counterclockwise angle from the z-axis to the tangent of ~(t) at s, we have

Structure-activity relationship between the 3D distribution of the electrophilicity of sugar derivatives and their cytotoxic and antiviral properties.

The cytotoxic activities of a series of sugar derivatives bearing electrophilic groups (1-cyanovinyl, 4-cyanochromen-2-yl and 3-nitrochromen-2-yl) have been correlated with their electrophilic properties. To this end, an electrophilic index was defined as an isovalue surface where the interaction energy with an incoming model nucleophile (H-) was equal to a predefined value. This index, calculated from extended Hückel wave functions, allows one to quantify the electrophilic character of the substrates and to describe its spatial localization within the molecular volume (at Michael acceptor sites or on other parts of the molecules). Only sugars for which Michael acceptor reactivity was predicted were retained, and they were subdivided into two groups: those showing antiviral activity against a retrovirus and those devoid of such activity. Under these conditions, good correlations between cytotoxic activity and electrophilic reactivity--positive for the first group, negative for the second--were found. In addition, the ratio electrophilicity/sum of the absolute value of the dipole plus its projection along the principal axis of inertia, Z, of the molecule allows one to predict to which of these groups a sugar derivative belongs.

Angular correlations within coiled polymer molecules

AbstractMonte Carlo techniques were used to investigate angular correlations between the principal axes of inertia and the end‐to‐end vector as well as the first two normal coordinates. On the average, the first normal coordinate is nearly parallel to the longest principal axis. The parallelism is less distinct in the case of the end‐to‐end vector and the longest principal axis. The assumption made by some authors that the second normal coordinate is coplanar with the two longest principal axes could not be confirmed.

Rotational spectrum of a short-lived dimer of oxirane and hydrogen chloride: Evidence for a bent hydrogen bond

Ground state rotational spectra of the three isotopomers (CH2)2O...H35Cl, (CH2)2O...H37Cl, and (CH2)2O...D35Cl of a short-lived hydrogen-bonded dimer have been detected in the reactive mixture of oxirane and hydrogen chloride by using a fast-mixing nozzle in conjunction with a Balle–Flygare Fourier-transform microwave spectrometer. Rotational constants, centrifugal distortion constants and Cl-nuclear quadrupole coupling constants were determined for each isotopomer. In particular, all four components χaa, χbb, χcc, and χac of the coupling tensor were obtained. A detailed analysis of the rotational constants allows the conclusion that the dimer has Cs symmetry, with a steeply pyramidal arrangement completed at oxygen by the hydrogen bond with HCl. Diagonalization of the complete Cl-nuclear quadrupole coupling tensor leads to the principal axis components χxx, χyy, and χzz (where z is the HCl direction in the dimer). The angle of rotation α is the angle between the HCl (z) direction and the a-axis direction in the equilibrium conformation of the dimer. It is larger by ∼10° than the angle γ between the O...Cl internuclear line and the principal inertial axis a in each case and implies that the hydrogen bond is bent by 180-θ=∼16.5° from the collinear arrangement O...H–Cl (θ=0). The angle 180-θ and the angle φ=76.2° made by the O...Cl internuclear line with the extension of the oxirane local C2 axis are interpreted in terms of a simple model of the hydrogen bond.

Force-free precession of a spinning plate

The dynamics of a force-free spinning plate with a small wobble angle is formulated exclusively in terms of forces. It is shown that the wobbling can be understood as the interplay between rotations about the two principal axes of inertia lying on the plate.

Librational dynamics of water in terms of angular velocity correlation functions and orientational structure

The high frequency (librational) dynamics of water is studied by molecular dynamics simulation on the TIP4P model at 245 K. The single-molecule part of the hydrogen current is described by a combination of autocorrelation functions of components of the angular velocity along the principal axes of inertia of the molecules. The distinct part of the hydrogen current is represented by a sum of cross correlation functions of the same components. The spectrum of the self current is almost quantitatively reproduced by the sum of the spectra of two of these contributions. The first is that relative to the component along the axis normal to the molecular plane and the second along the axis normal to the dipole vector, in the molecular plane, both multiplied by a geometrical parameter of the molecule. In the case of the distinct part, the interparticle correlation function of the projection of the angular velocity over the same two axes is weighted by the dot product of the dipole moments of each pair of molecules....