Principal Axes Of Inertia Research Articles

Stability of Nonlinear Oscillations of an Elastic Rod

The stability of plane steady oscillations of a slender elastic rod subjected to lateral harmonic excitation is investigated. For certain values of excitation amplitudes and frequencies the planar response is unstable, and nonplanar motions are parametrically excited. Steady nonplanar motions are characterized by each point on the rod centerline tracing an elliptical path perpendicular to the rod axis. The investigation is restricted to cantilevered rods with nearly equal principal moments of inertia of the cross-section, and excitation is always in the direction of one of the principal axes of inertia. The effects of damping and of slight variations of the natural frequencies associated with motions in the two principal planes are investigated. Supporting experimental evidence is presented. Finally, the principal features of the transversely driven rod response are compared with the response for the same type rod excited axially.

Geometrical Approach to Torque Free Motion of a Rigid Body Having Internal Energy Dissipation

A geometrical approach, which differs from the classical Poinsot solution, is developed for the moment free motion of a rigid body in order to discuss the case when energy is dissipated internally. Conservation of angular momentum and conservation of energy are used, respectively, to construct two surfaces, a sphere S and an ellipsoid E, which always intersect along a closed curve C; some typical intersections are shown. The motion of the body is such that the lip of the angular momentum vector remains on a curve C. Although less complete than Poinsot solution, this geometrical description provides a check on some classical results and is particularly useful in characterizing the motion when the angular momentum is kept constant while the energy decreases; then the sphere S remains the same while the ellipsoid E shrinks. The final state, corresponding to the minimum possible energy configuration, is a pure rotation about the principal axis of maximum inertia. Many interesting problems (e.g., the rotational stability of artificial satellites, the axial alignment of asteroids, and the damping of the Chandler wobble for the Earth and pulsars) are discussed in the context of this solution.

High resolution rotational spectra of formamide

Hyperfine structure on the 1 01 → 0 00 and 5 14 → 5 15 rotational transitions in NH 2CHO was measured using a molecular beam maser spectrometer with 8 kHz resolution. The nitrogen quadrupole coupling tensor in the principal inertial axis system was determined. The values are eQq aa = 1960 ± 2 kHz; eQq bb = 1888 ± 3 kHz and eQq cc = −3848 ± 4 kHz. Accurate values for the center frequencies for the rotational transitions were found to be 21 207 432 kHz for the 1 01 → 0 00 transition and 23 081 205 kHz for the 5 14 → 5 15 transition. One of the nitrogen spin-rotation parameters was also determined: C(5 14) – C(5 15) = 1.2 ± 0.6 kHz.

Determination of the earth’s pole of rotation from laser range observations to satellites

The methods of using earth satellites for determining the motion of the earth’s axis of rotation and of the earth’s principal axis of maximum inertia are discussed.

Shape of a Random-Flight Chain

An analysis is made of the distribution function and its moments for a linear combination of three randomly chosen orthogonal components of the so-called radius of gyration of an unrestricted random-flight chain, and certain averages of moments are obtained of the three-dimensional distribution W (L12L22L32), where L1 ≤ L2 ≤ L3 are the orthogonal components of the radius of gyration along the principal axes of inertia of the chain. The strong departures of the chain shape from spherical symmetry indicated by these results are confirmed and complemented by Monte Carlo studies of unrestricted random walks on a simple cubic lattice. A surprisingly high ratio of principal components is found for chains with 50 and 100 bonds, 〈L32〉: 〈L22〉: 〈L12〉 ≃ 11.7:2.7:1.

The microwave spectrum, barrier to internal rotation of the methyl group, and 14N nuclear quadrupole coupling constants in trans-crotononitrile

The microwave spectrum of trans-crotononitrile was investigated in the 36–40 and 8–13 GHz frequency regions. Several J 0 J - J 1, J-1 transitions were assigned and the results lead to the following rotational constants for the A torsional state: A A = 38 053.97 ± 0.4 MHz; B = 2297.06 ± 0.02 MHz; and C = 2195.21 ± 0.02 MHz. The barrier to internal rotation of the methyl group was determined to be V 3 = 1900 ± 20 cal/mole. The 14N nuclear quadrupole coupling was observed under high resolution leading to the following values of the coupling constants along the principal inertial axes: χ aa = −3.7 ± 0.1 MHz and χ bb = 1.7 ± 0.05 MHz. The methyl group barrier in trans-crotononitrile is discussed in light of the barriers in isoprene, trans-crotonoaldehyde, 2-methyl-1-buten-3-yne, and the cis- and trans-pentadienes.

Stability of Nonlinear Oscillations of an Elastic Rod

The stability of plane steady oscillations of a slender elastic rod subjected to lateral harmonic excitation is investigated. For certain values of excitation amplitudes and frequencies, the planar response is unstable, and nonplanar motions are parametrically excited. Steady nonplanar motions are characterized by each point on the rod centerline tracing an elliptical path perpendicular to the rod axis. The investigation is restricted to cantilevered rods with nearly equal principal moments of inertia of the cross section, and excitation is always in the direction of one of the principal axis of inertia. The effects of damping and of slight variations of the natural frequencies associated with motions in the two principal planes are investigated. Supporting experimental evidence is presented. Finally, the principal features of the transversely driven rod response are compared with the response for the same type rod excited axially.

Microwave spectrum, structure, dipole moment and quadrupole coupling constants of cis and trans nitrous acids

Microwave spectra for eight isotopic species of cis and trans nitrous acids are reported. The planarity of both isomers has been accurately established on the basis of the inertial defects. Accurate structures, electric dipole moments and quadrupole coupling constants at nitrogen have been determined for both isomers. The structure of cis nitrous acid is found to be: O—H = 0.982, N—O(H)= 1.392, N—O = 1.185 A, ∠ NOH = 104.0 and ∠ ONO = 113.6°, and for trans nitrous acid: O—H = 0.958, N—O(H)= 1.432, N—O = 1.170 A, ∠ NOH = 102.1 and ∠ ONO = 110.7°. Stark effect measurements yield the following values for the components of the dipole moment for cis nitrous acid: µa= 0.306, µb= 1.389 and the total dipole moment µ= 1.423±0.005 D. Previous investigations of the Stark effect in the trans isomer have been refined to give µa= 1.378, µb= 1.242 and µ= 1.855±0.016 D. Orientation of the dipole moment has been established for both isomers from isotopic measurements, despite a moderately large vibrational change in µa for trans DNO2. Quadrupole coupling constants in the principal inertial axis system have been determined for cis nitrous acid to be: χaa= 2.05, χbb=–5.83 and χcc= 3.78 MHz, and for trans nitrous acid: χaa= 1.73, χbb=–5.28 and χcc= 3.55 MHz. The structural parameters are markedly different in the two isomers and have been discussed, together with the torsional barrier and relative stability of the isomers, in terms of a cis interaction. The structures and charge properties of the nitrous acids are compared with a range of related compounds.

Microwave spectrum of dibromomethane. Part 2.—Nuclear quadrupole coupling tensor

The nuclear quadrupole hyperfine structure in the spectra of four isotopic species of dibromomethane, CH279Br2,CH281Br2,CD279Br2 and CD281Br2, has been analyzed to give the quadrupole coupling tensor directly, using second-order perturbation theory. Coupling constants (MHz) in the principal inertial axes system are as follows, CH279Br2: χaa= 334.9 ± 12, χbb=–25.2 ∓ 6, χcc=–309.7 ∓ 6, χab= 395.9 ± 13; CH281Br2: χaa= 280.6 ± 6, χbb=–21.4 ∓3, χcc=–259.2 ∓ 3, χab= 332.5 ± 10. Consistent values are reported for CD279Br2 and CD281Br2. Diagonalization of the quadrupole coupling tensor yields, CH279Br2: χzz= 591 ± 20, χxx=–281∓14, χyy=–310 ∓ 6 and CH281Br2: χzz= 495 ± 12, χxx=–236∓9, χyy=–259.2∓3. It is concluded that the Br-bond projection coincides with the internuclear axis within the experimental error of 1°; if bent bonding is involved in the C—Br bond the bend cannot be more than the experimental error of 2°. The analysis of the nuclear quadrupole hyperfine structure of a dibromine molecule is outlined and a comprehensive list of assignments presented in an appendix.

Minimum fuel attitude control of a non-linear satellite system

The problem here is the computation of fuel-optimal controls for the large angle attitude motions of a satellite system in which the control is obtained by three sets of gas jets with bounded thrusts, each generating torques about one of the principal axes of inertia. Using a result from optimal control theory, an algorithm is developed that iteratively improves on an initial guess (nominal) for the control history which does not meet terminal constraints and/or does not minimize the fuel cost. In using the algorithm, which is based on linear programming, it is necessary to express the variation of the fuel cost and variations of the components of the terminal state constraint vector as linear functions of variations in the control. The algorithm is tested on two sets of satellite differential equations. In one case, all dynamical effects are considered. In the other case, because control torque bounds are large enough, it is possible to neglect gravity gradient torque effects and orbital motion effects. A method to recursively approach minimum time control solutions by using this minimum fuel algorithm is described and illustrated. Numerical results are compared with the results of others who have worked identical examples.

The microwave spectrum, dipole moment, and nuclear quadrupole coupling constants of pyrimidine

The microwave spectrum of pyrimidine has been observed between 8000 MHz and 18,000 MHz. The rotational constants are A = 6276.86 MHz, B = 6067.18 MHz, and C = 3084.49 MHz. These results agree with earlier work on this molecule. The dipole moment measured by the Stark effect is 2.334 ± 0.010 D ((7.785 ± 0.030) × 10 −30 C.m). From an analysis of the hyperfine structure on a number of rotational transitions, the three diagonal elements of the nuclear quadrupole coupling tensor in the principal inertial axis system were obtained. They are χ aa = −3.107 ± 0.014 MHz, χ bb = −0.223 ± 0.016 MHz, and χ cc = 3.330 ± 0.014 MHz. The success with which nitrogen coupling constants in other aza-derivatives of aromatic hydrocarbons can be predicted by using the experimental data of pyridine is illustrated.

Active damping of free-rotor gyroscopes during initial spin-up

The free-rotor gyro is so named because the nearly spherical gyrorotor is not mechanically constrained to rotate about a particular axis. Therefore, it may be initially spun up about any rotor fixed axis. As seen in the rotor, the subsequent motion of the angular velocity vector after spin-up is the well-known classical polhode motion about either the maximum or the minimum principal axes of inertia. The maximum axis of inertia is the desired spin axis direction because it is the only stable free-rotor spin direction in the presence of minute energy losses during flexures of the rotor. Consequently, the rotor-fixed markings used for spin axis readout of many of these gyros are referenced to the maximum axis of inertia. The principle delays in activating typical gyros are 1) waiting for a passive d.c. field to damp the gyro spin axis into the maximum axis of inertia of the rotor after it has been spun up (this requires typically five times the spin-up time) and 2) waiting for the thermal transient (of spin-up and damping) to subside. This paper describes a method of actively controlling the spin axis in the rotor, while leaving it unchanged with respect to a case-fixed reference. A control algorithm, called Hemispheric Torquing is used to actively damp the spin axis to the proper rotor fixed direction. It offers the advantage of being able to reduce damping times to less than is required for initial spin-up.

The microwave spectrum, barrier to internal rotation of the methyl group, and molecular dipole moment of 2-methyl-1-buten-3-yne

The mirowave spectrum of 2-methyl-1-buten-3-yne is recorded and the ground vibrational state spectrum is assigned by the observation of μ a and μ b type transitions. The ground state rotational constants are A = 9359.58 ± 0.02, B = 4013.52 ± 0.01, and C = 2854.80 ± 0.01, all in units of MHz. The moments of inertia show that only two hydrogen atoms are out of the molecular plane. The barrier to internal rotation of the methyl group is found to be V 3 = 2010 ± 70 cal/mole. This surprisingly low barrier is interpreted in light of a recently proposed model. The molecular dipole moment components in the principal inertial axis system are μ a = 0.448 ± 0.015 and μ b = 0.248 ± 0.01 in Debye units. The total dipole moment is orientated 29 ± 1° in a direction toward the methyl group from the a axis with a magnitude of μ = 0.513 ± 0.020 D.

Molecular g Values, Magnetic Susceptibilities, Molecular Quadrupole Moments, and Second Moments of the Electronic Charge Distribution in OF2, O3, and SO2

The rotational Zeeman effect has been observed in several transitions in OF2, O3, and SO2. The absolute values of the molecular g values (gaa, gbb, and gcc) and the signs and magnitudes of the magnetic susceptibility anisotropies [(2χaa − χbb − χcc) and (2χbb − χaa − χcc)] are obtained by high-resolution microwavespectroscopy with the sample in the presence of high magnitude fields. Arguments are presented to show that all of the molecular g values are negative leading to the molecular quadrupole moments, (Qaa, Qbb, and Qcc) also listed below. The a, b, and c axes are the principal inertial axes in each molecule. The b axis bisects the interatomic angle and the a axis is also in the molecular plane. The units of χ are 10−6erg/G2·mole and the units of Q are 10−26 esu·cm2.OF2O3SO2gaa− (0.213 ± 0.005)− (2.968 ± 0.035)− (0.6037 ± 0.0005)gbb− (0.058 ± 0.002)− (0.228 ± 0.007)− (0.1161 ± 0.0002)gcc− (0.068 ± 0.002)− (0.081 ± 0.006)− (0.0882 ± 0.0004)(2χaa − χbb − χcc)− (8.8 ± 1.4)+ (98.0 ± 5.6)+ (6.39 ± 0.53)(2χbb − χaa − χcc)− (4.4 ± 0.7)− (17.4 ± 4.4)+ (3.07 ± 0.34)Qaa− (1.6 ± 1.4)− (17.1 ± 5.2)− (5.3 ± 0.4)Qbb+ (2.1 ± 1.1)+(9.2 ± 4.7)+(1.3 ± 0.3)Qcc− (0.5 ± 1.9)+ (7.9 ± 7.9)+ (4.0 ± 0.6)The diagonal elements in the paramagnetic susceptibility tensor and the anisotropies in the second moment of the electronic charge distributions (〈a2〉 − 〈b2〉 and 〈b2〉 − 〈c2〉) are computed from the above results and the known molecular structures. The bulk susceptibilities are known for O3 and SO2 which leads to a direct determination of the diagonal elements of the total and diamagnetic susceptibilities and the individual values of 〈a2〉, 〈b2〉, and 〈c2〉 in these two molecules. The bulk susceptibility for OF2 is estimated from a reliable estimate of 〈c2〉 for this molecule.

Frequency spectrum of the depolarized light scattered by a rigid molecule in solution

The frequency spectrum of the depolarized light scattered from a dilute solution of large, rigid molecules can give information about their principal axis of inertia and therefore their symmetries. It is shown that, for spherical molecules, one finds a single Lorentzian line. In the case of axial symmetry, the spectrum is the sum of two Lorentzian lines. Linewidths and relative intensities of lines are calculated as a function of the optical polarizability tensor and rotational diffusion coefficients of the molecule.

Molecular g Values, Magnetic Susceptibility Anisotropies, Molecular Quadrupole Moments, and the Second Moment of the Electronic Charge Distribution in OCF2, CH2CF2, cis-CHFCHF, and CH2F2

The rotational molecular Zeeman effect is observed under high resolution and high magnetic fields in OCF2, CH2CF2, cis-CHFCHF, and CH2F2. The absolute values of the molecular g values (gaa, gbb, and gcc) and the signs and magnitudes of the magnetic susceptibility anisotropies [(2χaa − χbb − χcc) and (2χbb − χaa − χcc)] in the principal inertial axes are obtained and listed. Arguments are presented to show that all molecular g values are negative leading to the molecular quadrupole moments. For ease of comparison, the values are given below in an arbitrary x, y, and z axis system. The x and y axes are in the heavy-atom molecular plane, and the x axis bisects the FCF angle in all of the molecules. The units of χ are 10−6 erg/G2·mole and the units of Qxx are 10−26 esu·cm2. OCF2CH2=CF2cis-CHF=CHFH2CF2gxx− (0.0568 ± 0.0006)− (0.0421 ± 0.0005)− (0.0296 ± 0.0003)− (0.0411 ± 0.0004)gyy− (0.0747 ± 0.0004)− (0.0466 ± 0.0004)− (0.1015 ± 0.0009)− (0.0725 ± 0.0006)gzz− (0.0328 ± 0.0006)− (0.0119 ± 0.0004)− (0.0158 ± 0.0002)− (0.0398 ± 0.0004)(2χxx − χyy − χzz)+ (1.6 ± 0.9)− (2.3 ± 0.6)− (1.6 ± 0.3)+ (0.8 ± 0.4)(2χyy − χxx − χzz)+ (5.3 ± 0.6)+ (7.7 ± 0.5)+ (5.7 ± 0.4)− (3.9 ± 0.5)Qxx− (3.7 ± 0.7)+ (2.4 ± 0.5)+ (3.0 ± 0.3)+ (1.9 ± 0.3)Qyy− (0.2 ± 0.5)− (0.9 ± 0.4)− (1.7 ± 0.4)− (4.1 ± 0.4)Qzz+ (3.9 ± 1.1)− (1.5 ± 0.8)− (1.3 ± 0.5)+ (2.2 ± 0.6) The diagonal elements in the paramagnetic susceptibility tensor and the anisotropies in second moment of the electronic charge distributions are computed from the above results and the known molecular structures. The above results are used along with additivity relations for the second moment of the electronic charge distribution to give estimates of the diamagnetic and total magnetic susceptibility tensor elements.

Molecular Magnetic Susceptibility Tensor in Sulfur Dioxide

The diagonal elements in the magnetic susceptibility tensor in the principal molecule-fixed inertial axis system in SO2 have been determined by observing the molecular rotational Zeeman effect with a high-resolution microwave spectrometer. The signs of gaa, gbb, gcc have been found to be negative from theroetical considerations. The principal values of the magnetic susceptibility tensor, paramagnetic susceptibility tensor, and diamagnetic susceptibility tensor are, respectively: χaa = − 14.4 ± 1.2, χbb = − 19.6 ± 0.5, χcc = − 20.6 ± 0.6; χaap = 38.9 ± 0.3, χbbp = 129.3 ± 1.4, χccp = 139.5 ± 1.2; χaad = − 53.3 ± 1.2, χbbd = − 148.9 ± 1.5, χccd = − 160.1 ± 1.3 (all in units of 10−6 erg/G2·mole). The ground-state averages of the sum of the squares of the electronic coordinates, and approximate values of the diagonal components of the quadrupole moment tensor have also been determined.

Optimal configurations for hinged, two-body satellites.

Optimal configurations for hinged, two-body, four-degree-of-freedom satellite systems in both circular and elliptical orbits are determined by a parameter optimization in which the geometrical and physical properties of the system are used as parameters. Specifically, the moments of inertia of the control body, the angles specifying the orientation of the principal axes of inertia of the bodies and the hinge rotation axis, and the spring and viscous damping constants are considered as the optimizing variables. An algorithm for the optimization is developed which uses a gradient method, the step size being selected with a Fibonacci search technique. A quadratic cost function is used which is physically the mean-square error of the main body in pitch and roll from its nominal earth-pointing attitude. The results of the optimization show that the parameter values that give good transient performance give poor steady-state performance, and vice versa. This dichotomous condition suggests the new concept of an adaptive system for use in elliptical orbits which would have the ability to change the control-body configuration in orbit.

Zur rotatorischen Bewegung eines Systems von n Massenpunkten. II

In part I, the problem of the separation of a collective rotational motion from the internal motion in a system of point masses was discussed. It was shown that in the Hamiltonian H = H1+ Hk + H2, where H1 describes the internal, H2 the collective motion, the coupling term Hk in general cannot be eliminated by any choice of the internal frame of reference. This result applies to classical as well as to quantum mechanics. In this paper, the problem is further discussed on the basis of wave mechanics, especially for the case of the three-body-problem. Starting from the dependence of Hk on the connection of the internal frame with the configuration, we choose this frame in such a way that Hk Φ0 = 0, where Φ0 is the separated internal part of the wave function. Our procedure, which we call the method of specific decoupling, ensures that for the considered wave function and the corresponding frame of reference, the internal and collective motions are fully decoupled in that degree of perturbation which corresponds to the concept of rotational energies proportional to L2. It turns out that the condition Hk Φ0 = 0 is equivalent to a minimum principle for 〈H2〉. Thus the frame of reference in question can be found by solution of a differential equation as well as by variational methods. The Eulerian angles of that frame of reference respective to a fixed system of coordinates are the variables which enter the functions D1 mn (α, β, γ) describing the rotational motion of our system of point masses. In the course of our procedure, we immediately get effective moments of inertia. These are markedly smaller than the normal expectation values. Some simple models are quantitatively discussed. It is shown that the normally defined principal axes of inertia are no suitable starting point for the problem, because their connection with the configuration becomes undefined, when two principal moments are equal.

Anisotropic Molecular Rotation in Liquid N,N-Dimethylformamide by Nuclear Magnetic Resonance

The anisotropy in the molecular rotation of N,N-dimethylformamide (DMF) in the liquid is investigated by considering the NMR relaxation times of the 14N, 17O, and formyl deuteron nuclei in DMF, DMF17O, and DMF-d1, respectively. Assuming that the molecular rotation is diffusional and that one principal diffusion axis (the z axis) is perpendicular to the molecular plane, a range of possible orientations for the diffusion axes in the plane and a corresponding range of values for each of the three diffusion constants are calculated. The range of possible orientations of the x diffusion axis, which varies little with temperature, extends over about 26°, from the direction of the x principal inertial axis toward and past the direction of the molecular dipole moment. At 280°K the diffusion constants are Dx = (2.0 ± 0.4) × 1011sec−1, Dy = (6 ± 3) × 109sec−1, and Dz = (2.9 ± 0.7) × 1010sec−1. At 390°K, Dx = (4.0 ± 0.7) × 1011sec−1, Dy = (10 ± 8) × 109sec−1 and Dz = (2.1 ± 0.7) × 1011sec−1. The ratio of the rotation rate about the ith axis in the free gas to Di is a direct measure of the extent to which rotation about the ith axis is hindered in the liquid by intermolecular interactions. Defining this ratio as χi (where larger values of χi indicate a more hindered rotation), it is found that at low temperatures χy, χz ≫ χx and at high temperatures χy ≫ χz, χx. Once the rotation of the molecular framework is known, the relaxation times of the methyl deuterons in DMF-d7 can be used to calculate the internal rotation rates of the deuterated methyl groups. These rotation rates are shown to be comparable to the rate of the molecular rotation.