Application of a Large-Parameter Technique for Solving a Singular Case of a Rigid Body

In this paper, the motion of a rigid body in a singular case of the natural frequency ( ω = 1 / 3 ) is considered. This case of singularity appears in the previous works due to the existence of the term ω 2 − 1 / 9 in the denominator of the obtained solutions. For this reason, we solve the problem from the beginning. We assume that the body rotates about its fixed point in a Newtonian force field and construct the equations of the motion for this case when ω = 1 / 3 . We use a new procedure for solving this problem from the beginning using a large parameter ε that depends on a sufficiently small angular velocity component r o . Applying this procedure, we derive the periodic solutions of the problem and investigate the geometric interpretation of motion. The obtained analytical solutions graphically are presented using programmed data. Using the fourth-order Runge-Kutta method, we find the numerical solutions for this case aimed at determining the errors between both obtained solutions.

The Slow Spinning Motion of a Rigid Body in Newtonian Field and External Torque

In this paper, the problem of the slow spinning motion of a rigid body about a point O, being fixed in space, in the presence of the Newtonian force field and external torque is considered. We achieve the slow spin by giving the body slow rotation with a sufficiently small angular velocity component r 0 about the moving z-axis. We obtain the periodic solutions in a new domain of the angular velocity vector component r 0 ⟶ 0 , define a large parameter proportional to 1 / r 0 , and use the technique of the large parameter for solving this problem. Geometric interpretations of motions will be illustrated. Comparison of the results with the previous works is considered. A discussion of obtained solutions and results is presented.

An Approximate Solution for Flow between Two Disks Rotating about Distinct Axes at Different Speeds

The flow of a linearly viscous fluid between two disks rotating about two distinct vertical axes is studied. An approximate analytical solution is obtained by taking into account the case of rotation with a small angular velocity difference. It is shown how the velocity components depend on the position, the Reynolds number, the eccentricity, the ratio of angular speeds of the disks, and the parameters satisfying the conditionsu=0andν=0in midplane.

Large Deformations of a Rotating Solid Cylinder for Non-Gaussian Isotropic, Incompressible Hyperelastic Materials

The purpose of this research is to investigate the steady rotation of a solid cylinder for a class of strain-energy densities that are able to describe hardening phenomena in rubber. It is well known that use of the classic neo-Hookean strain energy gives rise to physically unrealistic response in this problem. In particular, solutions exist only for a sufficiently small angular velocity. As the velocity approaches this limiting value, the analysis predicts that the rotating cylinder collapses to a disk. It is shown here that this nonphysical behavior does not occur when generalized neo-Hookean models, which exhibit hardening at large deformations, are used.

Second-order horizontal steady forces and moment on a floating body with small forward speed

In a recent work, a simple formula was derived for the ‘wave drift damping’ in a two-dimensional floating body and the obtained expression is exact within the context of the related theory, where only leading-order terms in the forward speed are retained. This formula is now generalized for a three-dimensional problem and the coefficients of the ‘wave drift damping matrix’ are given explicitly in terms of the standard second-order steady forces and moment in the horizontal plane; Munk's yaw moment, related with the steady second-order potential and discussed in Grue & Palm (1993), is not analysed in this paper and the effect of an eventual small angular velocity around the vertical axis is also not considered.Numerical results agree in general with the proposed formula although in a specific case a consistent disagreement has been observed, as discussed in §5.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

Kelvin–Helmholtz problem in a rotating Hall plasma

The Kelvin–Helmholtz problem in a Hall plasma, including the effects of rotation, is studied. In contrast to previous results, it is shown that if the angular velocity is normal to the wave vector that describes perturbations of the interface, the rotation does not affect the stability of a shear plane. The special case of very small angular velocity is studied and it is shown that the rotation has either a stabilizing or a destabilizing effect, according as the gravitational speed is greater or less than the Alfvén speed.

Dynamical Friction Effects on the Motion of Stars in Rotating Spherical Clusters

Effects of dynamical friction on star orbits in a spherical cluster uniformly rotating with small angular velocity about a fixed axis are considered, deformations of the cluster due to the rotation being neglected. The test star is supposed to move in a noncircular restricted orbit under the influence of both the attraction of the cluster with the smoothed-out distribution of stellar matter and dynamical friction due to random encounters of the test star with other stars of the cluster.The approximate formula for dynamical friction has been deduced, the encounters being supposed to be the binary ones. The differential equations for the osculating elements of the star orbit have been obtained for the two cases of the density distribution - the uniform and the exponential ones. The numerical results demonstrate the complicated character of dynamical friction effects on the evolution of the orbit. The orbit tends to become circular, and its inclination decreases. These effects are proportional to the mass of the test star. This leads to the conclusion that dynamical friction contributes noticeably to the concentration of massive stars near the center of the cluster.

On almost rigid rotations

In order to answer some of Proudman's questions (1956) concerning shear layers in rotating fluids, a study is made of the flow between two coaxial rotating discs, each having an arbitrary small angular velocity superposed on a finite constant angular velocity. It is found that, if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layer on the other at the same value of r. At a discontinuity in the angular velocity of either disc a shear layer parallel to the axis occurs. If the angular velocity of the main body of the fluid is continuous, according to the theory given below the purpose of this shear layer is solely to transfer fluid from the boundary layer on one disc to the boundary layer of the other. It has a thickness O(v1/3), where v is the kinematic viscosity, and in it the induced angular velocity is O(v1/6) of the perturbation angular velocity of the discs. On the other hand, if the angular velocity of the main body of fluid is discontinuous, according to the theory given below the thickness of the shear layer is O(v1/4). A secondary circulation is also set up in which fluid drifts parallel to the axis in this shear layer and is returned in an inner shear layer of thickness O(v1/3).The theory is also applied to the motion of fluid inside a closed circular cylinder of finite length rotating about its axis almost as if solid.