THE STABILITY OF A FREELY FLOATING SHIP

The paper presents the problem of calculating the righting arms (GZ-curve) for a freely floating ship, longitudinally bal- anced at each heel angle. In such cases the GZ-curve is ambiguous, as it depends on the way the ship is balanced. Three cases are discussed: when the ship is balanced by rotating her around the trace of water in the midships, around a normal to the ship plane of symmetry (PS), and around a normal to the initial waterplane, fixed to the ship, identical with the curve of minimum stability. In all these cases the direction of the righting moment in space and area under the GZ-curves, which is the lowest possible, are preserved. Angular displacements (heel and trim) are the Euler's angles related to the relevant reference axis. The most important features of the GZ-curve with free trim are provided. Exemplary calculations illustrate how the way of balancing affects the GZ-curves.

Analytical Solution of the Euler-Poinsot Problem

In the present article, an analysis was performed on the torque-free motion of a rigid body, developing Euler's analytical solution and Poinsot's geometric solution. The analytical solution for the angular velocity and Euler's angles was described given some initial conditions. Besides, an animation of Poinsot's geometric solution was elaborated and a study was carried out on the conditions in which the herpolhode forms a closed curve. Finally, an algorithm was developed that displays the obtained solutions, which also generates an animation of the geometric solution, and moreover provides an algorithm that produces closed herpolhodes.

The Motion of a Rigid Body with Irrational Natural Frequency

In this paper, we consider the problem of the rotational motion of a rigid body with an irrational value of the frequency ω . The equations of motion are derived and reduced to a quasilinear autonomous system. Such system is reduced to a generating one. We assume a large parameter μ proportional inversely with a sufficiently small component r o of the angular velocity which is assumed around the major or the minor axis of the ellipsoid of inertia. Then, the large parameter technique is used to construct the periodic solutions for such cases. The geometric interpretation of the motion is obtained to describe the orientation of the body in terms of Euler’s angles. Using the digital fourth-order Runge-Kutta method, we determine the digital solutions of the obtained system. The phase diagram procedure is applied to study the stability of the attained solutions. A comparison between the considered numerical and analytical solutions is introduced to show the validity of the presented techniques and solutions.

On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic

In this article, the study of qualitative properties of a special type of non-autonomous nonlinear second order ordinary differential equations containing Rayleigh damping and generalized Duffing functions is considered. General conditions for the stability and periodicity of solutions are deduced via fixed point theorems and the Lyapunov function method. A gyro dynamic application represented by the motion of axi-symmetric gyro mounted on a sinusoidal vibrating base is analyzed by the interpretation of its dynamical motion in terms of Euler’s angles via the deduced theoretical results. Moreover, the existence of homoclinic bifurcation and the transition to chaotic behaviour of the gyro motion in terms of main gyro parameters are proved. Numerical verifications of theoretical results are also considered.

The Effects of 3D Electrical Anisotropy on Magnetotelluric Responses: Synthetic Case Studies

Using the staggered-grid finite difference method, a numerical modeling algorithm for a 3D arbitrary anisotropic Earth is implemented based on magnetotelluric (MT) theory. After the validation of this algorithm and comparison with predecessors, it was applied to several qualitative and quantitative analyses containing electrical anisotropy and a simple 3D prism model. It was found that anisotropic parameters for ρ 1 , ρ 2 , and ρ 3 play almost the same role in affecting 3D MT responses as in 1D and 2D without considering three Euler's angles α S , α D , and α L . Significant differences appear between the off-diagonal components of the apparent resistivity tensor and also between the diagonal components in their values and distributing features under the influence of 3D anisotropy, which in turn help to identify whether the MT data are generated from 3D anisotropic earth. Considering the deflecting effects arising from the inconsistency between the anisotropy axes and the measuring axes, some strategies are also provided to estimate the deflecting angles associated with anisotropy strike α S or dip α D , which may be used as initial values for the 3D anisotropy inversion. [Figure: see text]

On the triangular points within frame of the restricted three–body problem when both primaries are triaxial rigid bodies

In the framework of the restricted three–body problem when both primaries are triaxial rigid bodies, for different cases of Euler’s angles, the locations of the triangular points, and the stability conditions of motion in the proximity of these points are constructed. The numerical solution is obtained by using a fourth order Runge–Kutta–Gill integrator with some graphical investigations.

The Stability of a Freely Floating Ship

The paper presents the problem of calculating the righting arms (GZ-curve) for a freely floating ship, longitudinally bal-anced at each heel angle. In such cases the GZ-curve is ambiguous, as it depends on the way the ship is balanced. Three cases are discussed: when the ship is balanced by rotating her around the trace of water in the midships, around a normal to the ship plane of symmetry (PS), and around a normal to the initial waterplane, fixed to the ship, identical with the curve of minimum stability. In all these cases the direction of the righting moment in space and area under the GZ-curves, which is the lowest possible, are preserved. Angular displacements (heel and trim) are the Euler's angles related to the relevant reference axis. The most important features of the GZ-curve with free trim are provided. Exemplary calculations illustrate how the way of balancing affects the GZ-curves.

Collision Detection and Trajectory Planning for Palletizing Robots Based OBB

This paper proposes an algorithm for Collision detection between two convex polyhedra (Boxes). It is assumed that the Palletizing robot’s end effector grasped the Box in accurate incremental method, by compare the distances between objects in same space . The detections proposed by ignorance of time consumption, and make comparisons between performances of theories discussed this problem, like Axes Aligned Bounded Boxes (AABB) and Sphere theories. Oriented Bounded Boxes (OBB) theory is chosen because unspecified orientation of objects which meets the requirements for detecting collisions with accuracy and handling transformations. Transformation assumptions are based on Z-Y-Z Euler’s angles representation. The key factor for detecting collision between two OBB convex polyhedra is the Separating Axes Theorem (SAT). The trajectory algorithm presents incremental distance computation by implementing translation and spherical trajectory. The checking and simulations in C++ language and Auto CAD software, attests that implementations in both show accuracy results.

Exact Axially Symmetric Solution inf(T)Gravity Theory

A general tetrad field with sixteen unknown functions is applied to the field equations off(T)gravity theory. An analytic vacuum solution is derived with two constants of integration and an angleΦthat depends on the angle coordinateϕand radial coordinater. The tetrad field of this solution is axially symmetric and the scalar torsion vanishes. We calculate the associated metric of the derived solution and show that it represents Kerr spacetime. Finally, we show that the derived solution can be described by two local Lorentz transformations in addition to a tetrad field that is the square root of the Kerr metric. One of these local Lorentz transformations is a special case of Euler’s angles and the other represents a boost when the rotation parameter vanishes.