Variations In Moment Of Inertia Research Articles

Trajectory deflection of noncircular cross-section projectile penetrating into metal targets

To analyze the trajectory deflection of a noncircular cross-section projectile penetrating into multiple metal targets, numerical simulations with different impact velocities, projectile tip height and charge geometry are conducted. The simulation model is established using the finite element software LS-DYNA and verified based on the experiment carried out by a single-stage light gas gun. The results show that the impact velocity significantly affects the trajectory deflection of noncircular cross-section projectile. Although the deflections may be different in direction, the magnitudes exhibit decreasing trends with the increasing impact velocity. Besides, the trajectory varies from downward deflection to upward deflection with the increase of projectile tip height, and changes significantly with the tip height ranging from 0.8mm to 1.1mm. Further, as the geometry profile of projectile remains unchanged, the ones with noncircular charge exhibit better ballistic stability as compared with the circular charge projectile. This is mainly related to the variation in projectile mass instead of the variation in moment of inertia.

Design Solutions for Slender Bars with Variable Cross-Sections to Increase the Critical Buckling Force.

In large metal civil constructions (stadium roofs, bridges), slender bars can lose their stability under compression loading. There is a lack in the literature regarding design solutions and methods for increasing the critical buckling force of bars with variable cross-sections. The aim of this research is to present a numerical model with finite elements used for a comparative analysis of increasing the critical force of stability loss in cases of (i) bars with stepwise variation in the cross-sections and (ii) bars with continuous variation in the moment of inertia along the bar axis (parabolic, sinusoidal, triangular, and trapezoidal variation). Considering the large-scale applications in civil engineering, bars that were pin-connected at one end and simple-supported at the other end were analyzed. Firstly, the analytical model was described to compute the critical buckling force for bars with stepwise variation in the cross-sections. Then, a finite element model for a slender bar and the assumptions considered were presented. The results were computed using the MATLAB program based on the numerical model proposed and were validated with the analytical model for stepwise variable cross-sections of the bars. The numerical model was adapted for bars with continuous variation in the moment of inertia along the bar axis. It was shown that, by trapezoidal variation in the second moment of inertia along the axis of a bar, i.e., as buckling occurred in the elastic field, the critical buckling force could be increased by 3.556 times compared to a bar with a constant section. It was shown that there was certain bar with stepwise variation in the cross-section for which the critical buckling force was approximately equal to the one obtained for the bar with sinusoidal variation in the moment of inertia (increased by 3.427 times compared to a bar with a constant section).

Projected shell model analysis of structural evolution and chaoticity in fast-rotating nuclei

Rotationally-induced chaotic motion in fast-rotating nuclei is investigated using the projected shell model with an aggressively extended multi-quasiparticle configuration space. The two-body Hamiltonian in separable forms with quadrupole–quadrupole plus pairing forces is diagonalized in superimposed states with more than 2000 angular momentum projected configurations. By taking 164Yb as an example, it is shown that the experimental energies and moment of inertia (MoI) for the yrast band can be clearly described. The variations in MoI are explained in terms of band crossings among the 2-quasiparticle (qp), 4-qp, 6-qp, and 8-qp bands, corresponding to successive pair-breakings caused by rotation. Moreover, the chaotic motion in different spin and excitation regions is studied by quantitatively analyzing the branching number. It is found that the degree of chaoticity increases monotonically with spin and excitation energy. At the highest spins, bands built by different numbers of quasiparticles, with their own characteristic rotational behavior originally at low spins, tend to rotate uniformly with the same rotational frequency, indicating an emergent nuclear collectivity appearing from an extremely chaotic system. The important role played by nucleon pairing in increasing the chaoticity is emphasized.

Systematical study of high-spin rotational bands in neutron-deficient Kr isotopes by the extended projected shell model

We analyze the high-spin structure of the even–even 72–80Kr isotopes using the Projected Shell Model (PSM). With the help of the Pfaffian formulas, we have vigorously extended the quasi-particle (qp) basis of the PSM code and applied in this mass region for the first time. We consider a sufficiently large multi-qp configuration space in order to describe high-spin rotational behavior. The results show that the calculation can reproduce most of the known rotational bands with positive- or negative-parity. Moreover, some side bands appearing in the near-yrast region are predicted. The main structure for each band is discussed in terms of multi-qp configurations. The variations in moment of inertia with spin are explained in terms of successive band crossings among the 2-qp, 4-qp, 6-qp, and 8-qp states. The B(E2) transition probabilities in these bands are also calculated. To further understand the high-spin behavior of these neutron-deficient nuclei and to confirm predictions of the present work, good high-spin data, especially for B(E2) transitions, are called for.

Reduction of collectivity at very high spins inNd134: Expanding the projected-shell-model basis up to 10-quasiparticle states

Background: The recently started physics campaign with the new generation of $\ensuremath{\gamma}$-ray spectrometers, ``GRETINA'' and ``AGATA,'' will possibly produce many high-quality $\ensuremath{\gamma}$ rays from very fast-rotating nuclei. Microscopic models are needed to understand these states.Purpose: It is a theoretical challenge to describe high-spin states in a shell-model framework by the concept of configuration mixing. To meet the current needs, one should overcome the present limitations and vigorously extend the quasiparticle (qp) basis of the projected shell model (PSM).Method: With the help of the recently proposed Pfaffian formulas, we apply the new algorithm and develop a new PSM code that extends the configuration space to include up to 10-qp states. The much-enlarged multi-qp space enables us to investigate the evolutional properties at very high spins in fast-rotating nuclei.Results: We take $^{134}\mathrm{Nd}$ as an example to demonstrate that the known experimental yrast and the several negative-parity side bands in this nucleus could be well described by the calculation. The variations in moment of inertia with spin are reproduced and explained in terms of successive band crossings among the 2-qp, 4-qp, 6-qp, 8-qp, and 10-qp states. Moreover, the electric quadrupole transitions in these bands are studied.Conclusions: A pronounced decrease in the high-spin $B(E2)$ of $^{134}\mathrm{Nd}$ is predicted, which suggests reduction of collectivity at very high spins because of increased level density and complex band mixing. The possibility for a potential application of the present development in the study of highly excited states in warm nuclei is mentioned.

Terrestrial tidal variations in the selenopotential coefficients

The variations in the coefficients of the harmonics of second degree of the selenopotential caused by the terrestrial tides have been studied. In the paper we use analytical expressions for the tidal variations in the Stokes coefficients obtained for a model of the elastic celestial body with a concentric distribution of mass using the fundamental elastic parameter k 2 of the Moon. Taking into account the resonant properties of the Moon’s motion, the variations in the selenopotential coefficients are presented in the form of Fourier series in the arguments of the theory of lunar orbital motion: l M, l S, F and D. The variations in the polar moment of inertia of the Moon due to the terrestrial tides lead to marked variations in the Moon’s axial rotation, which also have been determined and tabulated. From the results obtained, it follows that the tide periodic variations in the gravitational coefficients of the Moon are an order larger than the corresponding tide variations in the geopotential coefficients.

Equivalent Systems for Inelastic Analysis of Prismatic and Nonprismatic Members

This paper deals with the analysis of prismatic and nonprismatic members of any arbitrary variation in moment of inertia, and where the material is permitted to be stressed well beyond its elastic limit, thus causing the modulus of elasticity to vary along their length. The stress and deflection characteristics of such members are determined by the method of the equivalent systems, which permits to replace the original member of variable stiffness with one of uniform stiffness, whose elastic line is identical to the one of the original variable stiffness member. It is proven mathematically that the inelastic analysis of members with continuously varying moment of inertia and modulus of elasticity can be carried out by using equivalent linear systems of constant stiffness and applying known handbook formulas or methods of linear elementary mechanics. The member can be analyzed for both elastic and inelastic ranges, all the way to failure, thus permitting observation of progressive deterioration of the memb...

Dealing with varying moments of inertia of girders in bridge analysis

In many slab-on-girder bridges, especially those that are continuous over two or more spans, the moment of inertia of a girder varies significantly along the length of the bridge. This paper critically examines the practice of analyzing such bridges for load distribution by methods that make the assumption of constant longitudinal torsional and flexural rigidities. It is found that this practice may not be valid for those slab-on-girder bridges in which variations of the girder moments of inertia are very large.A recommended procedure is given for cases in which the variation in moment of inertia is not too severe. The procedure involves (a) the determination of total bending moments, treating the bridge as a beam of variable moment of inertia, and (b) the determination of an equivalent constant moment of inertia for beams of varying moment of inertia. Using this procedure the load distribution properties of the bridge can be realistically analyzed by those computer-based methods (e.g., orthotropic plate, finite strip, and semicontinuum methods) or manual methods (e.g., AASHTO and Ontario methods) that cannot directly take account of the variation of longitudinal flexural rigidity.The validity of the recommended procedure is established by comparing its results with those of the grillage analysis method that does take account of the variation of the girder moment of inertia. Key words: bridge analysis, girders, load distribution, slab-on-girder bridges.

Rotor Dynamics and Helicopter Stability

The component of velocity, μ0ΩR, along the x‐axis is specified, and also X, the angle of inclination of the flight path. The concept of the ‘internal’ and ‘external’ parameters of the motion is introduced, the ‘internal’ parameters consisting of certain combinations of the control and flapping angles θ0, θ1, θ2, β0, β1, β2 and the ‘external’ parameters consisting of μ0, Λ0, Θ, X, ∈ and z*, where Θ is the inclination of the x‐axis to the horizontal, and e the inclination to the vertical of the rotor resultant force‐vector. For a given centre of gravity position of the helicopter the ‘external’ parameters are shown to be determinable independently of the ‘internal’ parameters, subject to certain assumptions regarding the fuselage drag coefficient. The fundamental ‘internal’ parameter then emerges as (β1—Θ), which physically is the fore or aft inclination to the vertical of the rotor cone axis. The value of this parameter is found as the root of a quadratic equation, and not a linear equation as hitherto. It is shown that at high values of μ0 the rotor force‐vector docs not coincide with the rotor cone axis (as it is commonly supposed to do), the difference between (β1—Θ) and ∈ amounting possibly to 5 deg., of which no more than 1½ deg. can be accounted for by the rotor mean drag force component. Particular attention is paid to the case of horizontal flight, in which the true speed is μ0ΩR see Θ, the factor sec Θ being important. It is shown that at a given true speed the following quantities are independent of the centre of gravity position of the helicopter: (i) ∈, (ii) the quantity (λ1—Λ0+λ0β1), i.e. the component of the air velocity along the rotor cone axis, (iii) the horse‐power of the engine. In addition the following ‘internal’ parameters, (iv) (β1—Θ), (v) (θ2—β1), (vi) (θ1—β2), are very nearly independent of C.G. position, their variations with C.G. position only just becoming noticeable at large values of μ0. This feature enables a much simplified solution to the trim problem to be obtained by working in terms of a fictitious C.G. position chosen to make Θ zero. The genuinely C.G.‐sensitive parameters θ1, θ2, β1, β2 and Θ are subsequently converted to their true values corresponding to the actual C.G. position. The effect of variation in blade moment of inertia is examined through variations in the associated parameter γ, and it is shown that both the ‘external’ and ‘internal’ conditions are substantially unaltered, save for (θ1+ β2) and the mean coning angle β0, which do vary considerably with γ. The analysis is accurate at least so far as μ05 and the results are shown to be consistent with an energy equation. The only sources of error in the dynamics of the problem lie in the rejection of higher powers of μ0 and of flapping harmonics of β beyond the first. In order to assess the effects of this rejection, an analysis is made of a rigid rotor system free only to feather without flapping, and an exact dynamical solution for horizontal flight is obtained, subject to the same aerodynamical assumptions as before. At a given value of μ0 see Θ the results differ slightly from the case of the flapping rotor because now the purely feathering rotor applies a pitching couple mechanically to the fuselage via the shaft, which the flapping rotor is not able to do. However, by calculating the value of this couple, and supposing that the helicopter with the flapping rotor experiences either an aerodynamic fuselage pitching moment of the same amount or alternatively an appropriate C.G. shift, the two cases are reduced to essentially equivalent physical systems, particularly if the mean coning angle, β0, is deliberately arranged to be zero by proper choice of γ. A recalculation of the flapping case is then made; if its theory were exact, its solution would coincide with the known exact solution of the purely feathering rotor case. The discrepancies are shown to be very small, and thus the validity of rejecting μ06 is established for the earlier flapping rotor analysis. Part III is concerned with the theoretical solution of the feathering‐cumflapping rotor case, and also with a geometrical account of the trim configuration. Part IV contains the solution for the purely feathering rotor together with a discussion of the large number of numerical results and related tables, obtained on a Pegasus Computer.