Kravets'Tamila . Using quaternion matrices to describe nonlinear dynamics of transport vehicles in three dimensional space

The thesis is devoted to the development of untraditional method of composition of mathematical models for nonlinear problems of dynamics of discrete mechanical systems in three-dimensional space starting from the symmetry principle in order to adapt algorithms to a computational experiment. The symmetrisation conception of description of mathematical models has been realized on the basis of complex application of equations of motion in the form of Euler-Lagrange and mathematical apparatus of quaternion matrices. The Euler-Lagrange equations of motion for a free solid body have been reduced to the ordered block-matrix structure, which elements are quaternion matrices composed on quasi-velocities, Rodrigues-Hamilton's parameters, coordinates of centre of mass and other characteristic points. The quaternion-matrix representation algorithms for operations of vector multiplication, addition of final independent rotations, composition of direction cosine matrices, transformation of matrix of inertia during rotation and transition in space, calculation of three-index Boltzmann's symbols etc. have been suggested. The matrix representation of dependence of absolute and relative time derivatives of vector function and matrix of inertia has been obtained. The formulae for angular and linear velocity of a body in the bound and inertial reference systems have been determined on the basis of quaternion matrices and Rodrigues-Hamilton's parameters. The matrix formulae of generalized forces and moments adequate to the gravity, elastic-viscous forces, a follower force, aerodynamic forces and moments have been composed. The algorithms developed have been applied to construction of block-matrix mathematical models of nonlinear dynamics of the high-speed transport vehicle in three-dimensional space, the bridge span in order to determine the structure dynamic loading, the synthesis of control influence according to the given programme trajectory, the angular stabilization in space using the Gibbs vector, the dynamic interaction of a high-speed railway vehicle and a rail track with the fixed plan and profile.