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...

The dynamic load on the wheelset of a high-speed railroad vehicle on a track with local irregularities is determined. A block-matrix formula is given to analyze the inertial forces and moments generated during lurching along a spatial trajectory on a track with vertical and horizontal curvature. The local irregularities of the track are measured with well-known methods and means for controlling the geometry of railtracks such as inertial laser-gyro strapdown systems. Given a railtrack trajectory, a hodograph of wheelset motion is plotted versus dimensionless time and the geometrical parameters are reduced to typical dimensions. The formulas and hodograph are used to calculate the kinematical parameters needed to evaluate the dynamic load. The results obtained allow correct quantitative description of wheelset-rail interaction and resolution of one of the traffic safety issues for high-speed railroads Keywords: high-speed railroad vehicle, wheelset, local irregularities, wheelset-rail interaction, Euler-Lagrange equations, quaternion matrixIntroduction. The Ukrainian high-speed rail program provides that the speed of passenger trains on the existing Ukrainian railroads should gradually be increased to 160 km/h, followed by construction of specialized high-speed railroads [5]. This raises a number of new engineering issues associated with high-speed railroad traffic safety [6,7,14]. Increase in the travel speed of trains leads to increase in the angular speed of wheels and square-law increase in the gyroscopic forces and moments. The horizontal and vertical curvature of the track necessitates correct evaluation of centrifugal forces. Local irregularities of the track cause vertical and lateral vibrations of a fast-rotating wheelset about the truck, which induces Coriolis, gyroscopic, and centrifugal forces and moments.The purpose of the present study is to evaluate the inertial forces and moments on a wheelset interacting with a rail, depending on the speed of travel and shape of the track. The problem posed is of current importance [15][16][17][18][19] and is directly related to high-speed traffic safety [7].1. Problem Formulation. The D'Alembert principle [12] suggests that inertial forces are manifested in noninertial frames of reference. Whether there are inertial frames of reference [3] that satisfy Einstein's principle of relativity [13] is an open question [11]. Einstein stated that inertia and gravity are equivalent and are due to the field of the universe acting on a material body [13]. Unlike the forces of interaction between bodies, inertial forces do not obey Newton's third law of motion [11]. Frames of reference used in engineering problems are noninertial in some sense. In problems of dynamic loading on high-speed vehicles, the traditional frame of reference fixed to the Earth surface with the origin at the point of departure cannot be considered inertial because it rotates with an angular velocity of 7.3×10 -5 sec -1 . Also noninertial is any frame of reference fixed to a high-speed railroad vehicle....

A symmetric mathematical model is developed to describe the spatial motion of a system of space vehicles whose structure is represented by regular geometrical figures (Platonic bodies). The model is symmetrized by using the Euler-Lagrange equations of motion, the Rodrigues-Hamilton parameters, and quaternion matrix mathematics. The results obtained enable us to model a wide range of dynamic, control, stabilization, and orientation problems for complex systems and to solve various problems of dynamic design for such systems, including estimation of dynamic loading on the basic structure during maneuvers in space Introduction. Dynamic models in the form of discrete mechanical systems of asymmetric rigid bodies connected by elastic bars arise in dynamic problems for space vehicles (SVs) joined by standard adapter modules into open ( Fig. 1à) or closed chains. Closed SV chains may be either plane (triangle, square, or any other regular polygon (Fig. 1b)) or spatial (Platonic bodies: tetrahedron, octahedron, icosahedron, etc. (Fig. 1c)). The nodes of such geometrical figures are space vehicles and their edges are adapter modules. The space vehicles are modeled by asymmetric rigid bodies that experience follower forces generated by the control, stabilization, and orientation system. The adapter modules are modeled by inertialess rectilinear elastic bars of certain length and cross section embedded in the bodies. Such dynamic models are of importance since they allow us to analyze the dynamics of an SV system with allowance for the elastic interaction between them during maneuvers and to estimate the dynamic loading on both the space vehicle [1] and the adapter module [2].Design models in the form of connected rigid bodies are widely used to solve applied dynamic problems for complex engineering structures [3,4,[11][12][13][14]. Such dynamic problems can mainly be solved by conducting a computational experiment [5]. To this end, it is necessary to set up adequate mathematical models of physical processes. These models should be easily implemented on state-of-the-art computers. The key element in this model design concept is the symmetry of a model, which was understood by H. Weyl as order, perfection [6].Following the concept of symmetrization, we will develop a mathematical model to describe the spatial motion of a system of asymmetric rigid bodies connected by elastic bars into open or closed ordered chains in the form of regular polygons or polyhedrons. The model will be symmetrized by using the Euler-Lagrange equations of motion, the Rodrigues-Hamilton parameters, and quaternion matrix mathematics [7,8].1. Problem Formulation. According to the principle of removing constraints (and replacing them by their reactions), the whole variety of structures of SV systems under consideration can be reduced to a standard unified substructure consisting of an isolated spacecraft and an adapter module. This substructure is taken to be a superelement and is modeled by an inertialess elastic bar embedded in an asymmetric rigid bod...

A set of direct and inverse elements are examined and compared with a four-dimensional orthonormal basis. The aggregate of even substitutions of fourth power as a product of two transpositions are formed on this finite set. The finite set of substitutions is represented by monomial (1, 0, –1)-matrices of fourth order. An isomorphism of quaternion group and two noncommutative subgroups of eighth order is determined. Properties of four aggregates of basic matrices, corresponding to quaternion matrices, are examined.

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