Using quaternion matrices to describe the kinematics and nonlinear dynamics of an asymmetric rigid body

Abstract

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...

Kinetic energy of an asymmetric rigid body moving around a fixed point: invariant representation in terms of quaternion matrices

“…The formula for the projections of the velocity of a rigid body rotating around a fixed point onto the body axes has the following matrix form [17] V y y y t y…”

“…In the inertial frame of reference, the components of the velocity vector of the body's point are expressed in terms of the components of the angular velocity and the coordinates of the point as follows [17]:…”

“…The solution of this problem involves the principle of symmetry [10,16], Euler-Lagrange differential equations, Rodrigues-Hamilton, Cayley-Klein, and Euler parameters [13], quaternions [2, 3], Lush hypercomplex numbers [11], tensor calculus [4, 5], matrix calculus [15] and quaternion matrices [1,11,14]. These tendencies were reflected in some modern problems of aviation engineering, robotics, gyroscopy, vibroprotection [6,12,[18][19][20], where use is made of the matrix representation of mathematical models, which promotes the use of computer engineering.Here we use quaternion matrices for the invariant representation of the kinetic energy of an asymmetric rigid body undergoing spatial motion around a fixed point in the body-fixed and inertial frames of reference, which are applied in deriving the nonlinear equations of spatial motion in Euler-Lagrange symmetric matrix form [17]. Invariant representations are used in computer engineering to verify algorithms and control the precision of calculations [12].…”

“…Here we use quaternion matrices for the invariant representation of the kinetic energy of an asymmetric rigid body undergoing spatial motion around a fixed point in the body-fixed and inertial frames of reference, which are applied in deriving the nonlinear equations of spatial motion in Euler-Lagrange symmetric matrix form [17]. Invariant representations are used in computer engineering to verify algorithms and control the precision of calculations [12].…”

On determining productive capacity of EV traction battery repair area

“…Analytical methods are based on the basic laws of cars tech nical operation [7]. In this group of methods for determining the optimal frequency of maintenance, one can distinguish the fol lowing: the method determining the frequency of maintenance to the permissible level of reliability, economic method, econo metric probabilistic method, the method for determining the optimal periodicity of the inspection to the allowable value, as well as the laws of changing the parameter of technical condition.…”