Moments of inertia of triaxial nuclei in covariant density functional theory

The covariant density functional theory(CDFT)and five-dimensional collective Hamiltonian(5DCH)are used to analyze the experimental deformation parameters and moments of inertia(MoIs)of 12 triaxial nuclei as extracted by Allmond and Wood[J.M.Allmond and J.L.Wood,Phys.Lett.B 767,226(2017)].We find that the CDFT MoIs are generally smaller than the experimental values but exhibit qualitative consistency with the irrotational flow and experimental data for the relative MoIs,indicating that the intermediate axis exhibites the largest MoI.Additionally,it is found that the pairing interaction collapse could result in nuclei behaving as a rigid-body flow,as exhibited in the^(186-192)Os case.Furthermore,by incorporating enhanced CDFT MoIs(factor of f≈1.55)into the 5DCH,the experimental low-lying energy spectra and deformation parameters are reproduced successfully.Compared with both CDFT and the triaxial rotor model,the 5DCH demonstrates superior agreement with the experimental deformation parameters and low-lying energy spectra,respectively,emphasizing the importance of considering shape fluctuations.

Moments of Inertia, Magnetic Dipole Moments, and Electric Quadrupole Moments of the Lithium Isotopes

The single-particle Schrödinger fluid model is designed mainly to calculate the moments of inertia of the axially symmetric deformed nuclei by assuming that each nucleon in the nucleus is moving in a single-particle potential which is deformed with time t, through its parametric dependence on a classical shape variable α(t). Also, the Nilsson model is designed for the calculations of the single-particle energy levels, the magnetic dipole moments, and the electric quadrupole moments of axially symmetric deformed nuclei by assuming that all the nucleons are moving in the field of an anisotropic oscillator potential. On the other hand, the nuclear superfluidity model is designed for the calculations of the nuclear moments of inertia and the electric quadrupole moments of deformed nuclei which have no axes of symmetry by assuming that the nucleons are moving in a quadruple deformed potential. Furthermore, the cranked Nilsson model is designed for the calculations of the total nuclear energy and the quadrupole moments of deformed nuclei which have no axes of symmetry by modifying the Nilsson potential to include second and fourth order oscillations. Accordingly, to investigate whether the six p-shell isotopes 6Li, 7Li, 8Li, 9Li, 10Li, and 11Li have axes of symmetry or not, we applied the four mentioned models to each nucleus by calculating their moments of inertia, their magnetic dipole moments, and their electric quadrupole moments by varying the deformation parameter β and the non-axiality parameter γ in wide ranges of values for this reason. Hence for the assumption that these isotopes are deformed and have axes of symmetry, we applied the single-particle Schrödinger fluid model and the Nilsson model. On the other hand, for the assumption that these isotopes are deformed and have no axes of symmetry, we applied the cranked Nilsson model and the nuclear super fluidity model. As a result of our calculations, we can conclude that the nucleus 6Li may be assumed to be deformed and has an axis of symmetry.

Earth＇s Temporal Principal Moments of Inertia and Variable Rotation

Based on the gravity field models EGM96 and EIGEN-GL04C, the Earth＇s time-dependent principal moments of inertia A, B, C are obtained, and the variable rotation of the Earth is determined. Numerical results show that A, B, and C have increasing tendencies; the tilt of the rotation axis increases 2.1×10^ 8 mas/yr; the third component of the rotational angular velocity, ω3 , has a decrease of 1.0×10^ 22 rad/s^2, which is around 23% of the present observed value. Studies show in detail that both 0 and ω3 experience complex fluctuations at various time scales due to the variations of A, B and C.

Microscopic Mechanism of Large Fluctuation in Odd-Even Differences in Moments of Inertia for Actinide Nuclei

The experimental large fluctuation in odd-even differences in moments ofinertia of deformed actinide nuclei is investigated using the particle-number conserving (PNC)method for treating the cranked shell model with monopole and quadrupole pairing interactions. PNCcalculations show that the large odd-even difference in moments of inertia mainly comes from theinterference contributions j(μv) from particles in high j intruder orbitals μ and v quite near theFermi surface, which have no counterpart in the BCS formalism. The effective monopole andquadrupole pairing interaction strengths are determined to fit the experimental odd-even differencesin binding energies and bandhead moments of inertia. The experimental results for the variation ofmoments of inertia with rotational frequency ω are reproduced well by the PNC calculation. Thenearly identical experimental moments of inertia between ~(236)U(gsb) and ~(238)U(gsb) at lowfrequencies hω ≤ 0.20 MeV are also reproduced quite well.

Torque Free Axi-Symmetric Gyros with Changing Moments of Inertia

The properties and characteristics of torque free gyros with rotational symmetry and changing moments of inertia are the subject of the subsequent discussion. It shall be understood that the symmetry can be expressed by the notation (A=B) which does not presuppose geometric symmetry, where A and B are the principle moments of inertia about x and y axes respectively. We study the case of a torque free gyro upon which no external torque is acting. The equations of motion are derived when the origin of the xyz-coordinate system coincides with the gyro’s mass center c. This study is useful for the satellites, which have rotational symmetry and changed inertia moments, the antennas and the solar power collector systems.

The principal moments of inertia calculated with the hydrostatic equilibrium figure of the Earth

As an indication of the Earth＇s mass distribution, the principal moments of inertia （PMOI, i.e., A, B, C） of the Earth are the basic parameters in studies of the global dynamics of the earth, like earth nutation, and the geophysics. From the aspect of observation, the PMOI can be calculated from the spherical coefficients of observed gravity field. In this paper, the PMOI are calculated directly according to its definition with the figures of the Earth＇s interior derived by a generalized theory of the hydrostatic equilibrium figure of the Earth. We obtain that the angle between the principal axis of the maximum moment of PMOI and the rotational axis is 0.184~, which means that the other two principal axes are very closely in the equatorial plane. Meanwhile, B-A is 1.60 x 10-5 MR2, and the global dynamical flattening （H） is calculated to be 3.29587 ~ 10-3, which is 0.67% different from the latest observation derived value Hobs（3.273795 × 10 ^-3） （Petit and Luzum, 2010）, and this is a significant improvement from the 1.1% difference between the value of H derived from traditional theories of the figure of the Earth and the value of Hobs. It shows that we can calculate the PMOI and H with an appropriate accuracy by a gener- alized theory of the hydrostatic equilibrium figure of the Earth.

Nonadditivity in moments of inertia of high-K multiquasiparticle bands

The experimental high-K 2- and 3-quasiparticle bands of well deformed rare-earth nuclei are analyzed. It is found that there exists significant nonadditivity in moments of inertia （MOIs） for these bands. The microscopic mechanism of the rotational bands is investigated by the particle number conserving （PNC） method in the frame of cranked shell model with pairing, in which the blocking effects are taken care of exactly. The experimental rotational frequency dependence of these bands is well reproduced in PNC calculations. The nonadditivity in MOIs originates from the destructive interference between Pauli blocking effects.

Determining the moment of inertia of triaxial Mars with updated global gravity models

The principal moments of inertia(PMIs)with the principal axes are usually taken as the dynamic figure parameters of Mars;they can be deduced from satellite-observed degree-two gravitational potentials in recent global gravity models and from the dynamic ellipticities resulting from precession observations.These PMIs are natural and significant for the geodetic,geophysical,and geodynamic problems of Mars,which are functions of internal density distributions.In this study,a closed and concise formula for determining the PMIs of the entire planet and its core was developed based on the second invariants of gravity and a multipole expansion.We deduced the polar oblateness J^(2)and the equatorial ellipticity J_(22)of Mars to be 1.9566×10^(−3)and 6.3106×10^(−5),respectively.The preferred principal moments of inertia of Mars are A=2.66589×1036 kg·m^(2),B=2.66775×10^(36)kg·m^(2),and C=2.68125×10^(36)kg·m^(2).These values indicate that Mar is slightly triaxial.The equatorial principal moment of inertia of the Martian core is 1.46008×10^(35)kg·m^(2),accounting for~5.47%of the planet’s PMI;this result is critical for investigating the density and size of the core of Mars,and the planet’s free core nutation.

Systematic study of kinematic and dynamic moments of inertia of superdeformed bands with NpNn scheme

The systematics of kinematic moment of inertia J and dynamic moment of inertia J of superdeformed（SD） bands in A -130, 150, 190 mass regions have been studied. We have obtained the values of J^（1） and J^（2）for all the SD bands observed in the A-130, 150, 190 mass regions by using the experimental intraband E2 transition energies. The result of this work includes the variation of J with the product of valence proton and neutron numbers（NpNn）. The phenomenon of band mixing has been observed in the A -130, 150 mass regions and band crossing has been observed in the A-190 mass region. The systematics also includes the variation of J with the product of valence proton and neutron numbers（NpNn）. Evidence of staggering behaviour has been observed in all three A-130, 150 and 190 mass regions. We present for the first time the variation of J^（1）and J^（2）of SD bands in the A-130, 150, 190 mass regions with NpNn.

Moments of inertia of neutron stars in relativistic mean field theory:The role of the isovector scalar channel

With the inclusion of the isovector scalar channel in the meson-nucleon couplings, taking DD-MEδ as an effective interaction, the moments of inertia of neutron stars possessing various stellar masses are studied within the density dependent relativistic mean field （RMF） theory. The isovector scalar channel contributes to the softening of the neutron-star matter equation of state （EOS） and therefore the reduction of the maximum mass and radius of neutron stars. Smaller values of the total moment of inertia 1 and the crustal moment of inertia AI are then obtained in DD-ME~ via numerical procedure in comparison with those in other selected RMF functionals. In addition, the involvement of the isovector scalar channel lowers the thickness of the neutron star crust and its mass fi＇action as well. The sensitivity to both the crustal mass and stellar radius causes the crustal moment of inertia to be more obviously reduced than the total one, eventually leading to a suppression on the fraction of crustal moment of inertia △I/I in DD-MFδ. The results indicate the crustal moment of inertia as a more sensitive probe of the neutron-star matter EOS than the total one, and demonstrate that the isovector scalar meson-nucleon couplings in the RMF theory could exert influence over the physics of pulsar glitches.

Improved Units of Measure in Rotational Mechanics

The SI system of units in rotational mechanics yields correct numerical results, but it produces physically incorrect units of measure in many cases. SI units also violate the principle of general covariance—the general rule for defining continuous coordinates and units in mathematics and mathematical physics. After 30+ years of wrestling with these problems, the ultimate authority on units of measure has declared that Newton–meter and Joule are not equivalent in rotational mechanics, as they are in the rest of physics. This article proposes a simple modification to SI units called “Nonstandard International units” (“NI units”) until a better name is agreed upon. NI units yield correct numerical results and physically correct units of measure, and they satisfy the principle of general covariance. The main obstacle to the adoption of NI units is the consensus among users that the radius of rotation should have the unit meter because the radius can be measured with a ruler. NI units assigned to radius should have units meter/radian because the radius is a conversion factor between angular size and circumferential length, as in arclength = rθ. To manage the social consensus behind SI units, the author recommends retaining SI units as they are, and informing users who want correct units that NI units solve the technical problems of SI units.

Uncertainty analysis of ship model vertical center of gravity and transverse moment of inertia test

The usability of test results of ship model vertical center of gravity and transverse moment of inertia is generally depends on its uncertainty. Referring to the guidelines for uncertainty analysis in examination of liquid dynamic recommended by International Towing Tank Conference （ ITTC）, the results were analyzed, bias limits and precision limits were calculated and total uncertainty was estimated. The total uncertainty of six tests on ship model vertical center of gravity is is 0. 16% of the mean value, and the total uncertainty of six tests on ship model transverse moment of inertia is 5.66% of the mean value. The test results show that the total uncertainty of both the multiple tests and the single test is from the precision limits of ship model vertical center of gravity and transverse moment of inertia tests. Thus, the improved measurement system stability can enormously decrease the total uncertainty of multiple tests and the single test.

Spin Cut-off Parameter of Nuclear Level Density and Effective Moment of Inertia

The spin cut-off parameter of the nuclear level density and effective moment of inertia for a large number of nuclei have been determined from analysis of the experimental data on S-wave neutron resonances and spins of lowlying levels. Contrary to claims made before, it is shown the spin cut-off parameter differs considerably from their corresponding rigid body values, and the energy dependence of the effective moment of inertia confirms the interacting fermion model prediction.

The Steiner Formula and the Polar Moment of Inertia for the Closed Planar Motions in Complex Plane

In this paper, the Steiner area formula and the polar moments of inertia were expressed during one-parameter closed planar motions in complex plane. The Steiner points or Steiner normal concepts were described according to whe-ther rotation number was different from zero or equal to zero. The moving pole point was given with its components and its relation to the Steiner point or Steiner normal which was specified. The Steiner formula and the polar moments of inertia were expressed for the inverse motion. The fixed pole point was calculated for the inverse motion. The sagittal motion of a tele-scopic crane was considered as an example. This motion was described by a double hinge consisting of the fixed control panel of the telescopic crane and its moving arm. The results obtained in the first section of this study were applied to this motion.

Theoretical and Practical Studies of Determining the Force of the Lifting-Lowering of the Roll Box

The lifting and lowering mechanism of the roll box of the saw gin, being an integral part of the ginning process, is used when starting the electric motor of the saw cylinder until its rated speed is reached to prevent an increase in the load current on it, as well as to start the ginning process. The article defines the kinematics of the movement, and also studies the regularity of the movement of the roll box of the saw gin when lifting it with the help of a pneumatic drive. The kinematics of the roll box movement was determined through experimental studies. Knowing the equation of motion, using the second-order Lagrange equation, the regularity of the change in the lifting force of the roll box was determined. Here the system is closed, its degree of freedom is equal to one. The angle of rotation of the roll box is taken as a generalized coordinate. Numerical results were obtained using the Matlab software environment. The theoretical analysis and graphs for determining the lifting force of the roll box are presented. To study the kinematics of lifting the roll box, a method and a test bench have been developed. According to him, when the roll box was raised, its movement was recorded on video, the image was processed using the After Effect and CorelDraw programs, and the movement of the rod attachment point to the roll box was determined. And also the equation for changing the angle of rotation of the roll box is determined using the Matlab Curve Fiting program.

Description of Rotation Bands in ^(157,158)Er at Ultrahigh Spin

Properties of the four rotation bands, ^157Er（1,2） and ^15SEr（1,2）, at ultrahigh spin are investigated within the supersymmetry scheme including many-body interactions and possessing the SO（5） （or SU（5）） symmetry on the rotational symmetry. Quantitatively good results of the 7-ray energies and the dynamical moments of inertia in the rotation bands in ^157 Er and ^158 Er at ultrahigh spin are obtained. We theoretically predict that the competition between the anti-pairing and pairing effects may exist in ^157 Er（1,2） and ^158Et（2） bands states. In ^158Er（1） band state, the favourepairing effects may exist and the SO（5） （or SU（5）） symmetry play a dominant role. There may be sphere coexisting with headecupole deformed in ^158Et（1） rotation band state.

Systematic Study on Triaxial Superdeformed Bands of Hf Isotopes

Properties of the triaxiai superdeformed （TSD） bands of Hf isotopes are investigated systematicaily within the supersymmetry scheme including many-body interactions and a perturbation possessing the SO（5） （or SU（5）） symmetry on the rotational symmetry. Quantitatively good results of the γ-ray energies, the dynamical moments of inertia, and the spin of the TSD bands in Hf isotopes are obtained. It shows that this approach is quite powerful in describing the properties of the triaxial superdeformation in Hf isotopes.

Description of Properties of Triaxial Superdeformed Bands in Odd-A Lu Isotopes

Properties of the triaxial superdeformed （TSD） bands of odd-A Lu isotopes are investigated systematically within the supersymmetry scheme including many-body interactions and a perturbation possessing the S0（5） （or SU（5）） symmetry on the rotational symmetry. Quantitatively good results of the γ-ray energies, the dynamical moments of inertia and the spin of the TSD bands in odd-A Lu isotopes are obtained. The calculation shows that the competition between the pairing and anti-pairing effects exists in these TSD bands. Meanwhile, the SU（3） symmetry in TSD bands are broken more seriously than in superdeformed （SD） bands.

Description of the Superdeformed Bands of Odd-Odd Nuclei in A～80 Region

Properties of the superdeformed bands of odd-odd nuclei in A ～ 80 massregion are investigated system-atically within the supersymmetry scheme including many-bodyinteractions and a perturbation possessing the SO(5) (or SU(5)) symmetry on the rotational symmetry.The obtained γ-ray energies, and the dynamical moments of inertia agree with experimental data. Itshows that this approach is quite powerful in describing odd-odd nuclei in A ～ 80 mass region.

Equivalent bending stiffness of simply supported preflex beam bridge with variable cross-section

In order to understand mechanical characters and find out a calculating method for preflex beams used in particular bridge engineering projects, two types of simply supported preflex beams with variable crosssection, preflex beam with alterative web depth and preflex beam with aherative steel flange thickness, are dis- cussed on how to achieve the equivalent moment of inertia and Young＇ s modulus. Additionally, methods of cal- culating the equivalent bending stiffness and post-cracking deflection are proposed. Results of the experiments on 6 beams agree well with the theoretical analysis, which proves the correctness of the proposed formulas.