A set of four quaternion matrices is used to represent the equations of finite rotation theory and to describe the kinematics and nonlinear dynamics of an asymmetric rigid body in space. The results obtained are tested in setting up direction-cosine matrices, calculating three-index symbols, establishing the relationship between the components of angular velocity in body-fixed and space-fixed frames of reference, and using a set of three independent rotations. Euler-Lagrange equations and a set of four quaternion matrices are used to construct a block-matrix model describing the nonlinear dynamics of a free asymmetric rigid body in three-dimensional space. The model gives the matrix Euler's equations of motion and other special cases. Algorithms adapted to use in a numerical experiment are developed Keywords: asymmetric rigid body in space, four quaternion matrices, block-matrix model, numerical experiment Introduction.A computational experiment is a decisive factor in complex dynamic design of various engineering systems [5,10,15,19,23,[28][29][30][31]. A methodology of computational experiment was pioneered by Glushkov [5, etc.]. Of importance is the development of separate stages of a computational experiment:-development of mathematical models that would adequately describe physical processes in engineering systems and that could be unified and modified;-development of compact algorithms and associated software that could be easily and promptly verified. The central element of this concept is symmetry [27] understood by Weyl as order, perfection. Searching for a symmetric form of the equations of motion for holonomic and nonholonomic systems of rigid bodies, Lurie [20] paid attention to the Euler-Lagrange equations of motion and highly recognized their capability of facilitating the derivation of simpler symmetric equations of motion compared with Newton's equations (coordinate method), Lagrange's equations (method of generalized coordinates), etc. [20]. Original mathematical methods developed to describe the kinematics and dynamics of rigid bodies widely use, along with conventional vector calculus [14,20], matrix and tensor calculus [10,11,24], algebra of quaternions and hypercomplex numbers [2,3,6,7,17,19,26], and quaternion matrices [1,12,19,22,24,27]. Numerical implementation of mathematical models necessitates specific frames of reference, index notation, unknown variables and appropriate matrix or tensor calculus tools. Matrix methods are associated with such advantages as direct and simple numerical implementation of algorithms, compact notation and comprehension of algorithms, decreased labor intensity of software, and decreased probability of errors [1,9,16,23,24,28]. According to Letov [13], the reason for adapting the equations to a matrix form is appropriate not only for the sake of the mathematical elegance achieved here by introducing a symmetric and compact notation, but also to capture the heart of the matter.The present paper improves methods for describing the kinematics and nonlinear dynam...

No abstract

No abstract

High-temperature reaction zone, steel-smelting bath, oxygen purge, oxygen converter, Gibbs energy, chemical potential, phase diagram.

The work proposes a thermodynamic model of a multicomponent condensed phase applicable to metal and slag melts. The integral and partial thermodynamic mixing functions: Gibbs energy, enthalpy, entropy, heat capacity have been obtained using configurational statistical sum. Methods for estimating of model parameters using available data on activity coefficients and heats of mixing, Wagner interaction parameters have been suggested. Using binary Fe-Mn system as an example, the high accuracy of the calculation of the mixing thermodynamic functions of the metal phase in the entire concentration range has been shown. Mixing heat capacity has been shown to comply 3rd law of thermodynamics. Suggested thermodynamic model is already being used in steelmaking control system «Master» at metallurgical plant «Zaporizhstal», and in heat projecting and control system «DesigningMelt» at XuanSteel metallurgical works (PRC).