System of differential equations of aircraft spatial motion dynamics with arbitrary tensor of inertia and center of gravity position

Derivation of analytic expressions making up the basis of a mathematical model of aircraft flight dynamics for the differential equations describing the change in the rate of roll, yaw and pitch, as well as flight velocity components in projections on the body-fixed coordinate axes is presented. The origin of the coordinate system does not in general coincide with the center of mass of the plane, and the axes are not the same as its main central axes of inertia. The differential equations for angular and linear velocities are reduced to the form convenient for the use of numerical methods and computer systems and make it possible to get consistent results of simulating the dynamics of aircraft spatial motion with an arbitrary tensor of inertia and center of gravity position.

Relative Pose and Inertia Determination of Unknown Satellite Using Monocular Vision

The paper proposes a two-stage algorithm for autonomous relative motion determination of noncooperative and unknown object flying in space. The algorithm is based on image processing and can be applied to motion determination of space debris with unknown geometry and dynamic characteristics. The first stage of the algorithm is aimed at forming a database of possible reference points of the object during continuous observation. Tensor of inertia, initial velocity, and angular velocity of the object are also estimated. Then, these parameters are used in the second stage of the algorithm to determine the relative motion in real time. The algorithm is studied numerically and tested using the video of the Chibis-M microsatellite separation.

Principal axes for clusters in crystals investigated by the tensor of inertia: a case study with BH4in Mg(BH4)2

Knowledge of how chemically predefined clusters of atoms are arranged in a crystal and how they can reorientate under external excitations is of great importance for structure determination and a description of structural transformations. In general, the probability of cluster reorientation depends on the activation energy and thus on the symmetry of the cluster and its environment. In addition to the many experimental methods available for studying molecular reorientations in a crystal, there is also a theoretical method, known for many years, that can be used to describe the reorientation processes, namely the determination of the principal axes of the cluster's tensor of inertia (TI). By such calculations, insight may be gained into the effective shape of the cluster and the orientations of the distinct reorientation axes. In the present work, such a methodology has been applied to an analysis of borohydride, BH4, clusters in several structures proposed by theoretical calculations as well as experimental studies of magnesium borohydride, Mg(BH4)2. The calculation of orientations for the TI principal axes as well as pseudo-twofold axes of the cluster revealed a strong correlation between these orientations and the Mg—B—Mg angle for the two Mg neighbours of the cluster. The exceptions from that well defined dependence are observed for the principal TI axes in situations when the cluster deformations are small and the symmetry is close to spherical, when the orientations of the principal axes are prone to fluctuations.

BH4 cluster ordering in complex metal borohydrides

Complex borohydride compounds attract an increasing interest as hydrogen storage materials. Although they have been investigated for more than fifty years, there is still a lot of unknown physical phenomena awaiting to be revealed regarding, for example the dynamics and ordering of BH4 clusters in different borohydrides. The orientation of BH4 unit with respect to the surrounding metallic atoms is unique for a particular metal and determines e.g. the crystal space group, total energy of a system and the potential freedom of BH4 movement, like rotations, within the crystal structure. The information about rotations is of great importance for understanding how the hydrogen atoms behave in such systems [1,2]. To gain a closer look on how the borohydride clusters can rotate, their geometrical arrangement has been studied using analysis of tensor of inertia (TI). Free BH4 takes the form of a regular tetrahedron, but when it is placed among metal atoms, a compound is formed, in which certain deformation of this tetrahedral atomic arrangement may take place. Here the TI analysis turns out to be an efficient tool, since it might reveal how the BH4 deformation influences its rotational freedom, by analyzing the principal moments of inertia and principal eigenvectors corresponding to them, which are in fact, the most probable rotation axes. In the paper a comprehensive study is presented and the possible axes of rotation in complex metal borohydrides MBH4 (M=Li, Na, Ca, K, Mn, Rb, Cs, Sr, Zr), are discussed. For a given metal borohydride TI is performed on both experimentally obtained structures and predicted by theoretical calculations. A comparison of these results gives an insight into how the theory reflects real measurements and what discrepancies or similarities are observed.

Use of Equimomental Systems in Calculating Inertial and Dynamic Quantities of Plane Plates

The paper proposes a method for the reduction of an arbitrary-shape plane plate to a symmetrical discrete system of material points, with the same center of mass and the same tensor of inertia. It is shown, first, that an elliptical plate can be reduced to a symmetrical system of five material points, with a spatial distribution directly related to the shape of the plate. The results are subsequently generalized for plane plates of arbitrary shape. By using the reduction procedure, various mechanical quantities can be calculated more simply, as compared to the traditional methods. The proposed method finds application in the dynamic, vibration and structural analysis of complex mechanical systems that include rigid plane plates, such as industrial robots.

On the Problem of post-Newtonian Rotational Motion

The problems of modeling of the rotational motion of the Earth are considered in the framework of general relativity. Both, rigid and deformable bodies are discussed. Rigorous definitions of the tensor of inertia, Tisserand-like axes and the angular velocity of rotation of an extended deformable body moving and rotating in external gravitational fields are proposed in the first post-Newtonian approximation. The implications of these post-Newtonian definitions on modeling of Earth rotation are analyzed.