A New Form of Equations for Rigid Body Rotational Dynamics (original) (raw)

Rigid Body Dynamics Using a Novel Approach Based on Lie Group Theory

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2013), 2014

A systematic theoretical approach is presented, in an effort to provide a complete and illuminating study on motion of a rigid body rotating about a fixed point. Since the configuration space of this motion is a differentiable manifold possessing group properties, this approach is based on some fundamental concepts of differential geometry. Α key idea is the introduction of a canonical connection, matching the manifold and group properties of the configuration space. This is sufficient for performing the kinematics. Next, following the selection of an appropriate metric, the dynamics is also carried over. The present approach is theoretically more demanding than the traditional treatments but brings substantial benefits. In particular, an elegant interpretation is provided for all the quantities with fundamental importance in rigid body motion. It also leads to a correction of some misconceptions and geometrical inconsistencies in the field. Finally, it provides powerful insight and a strong basis for the development of efficient numerical techniques in problems involving large rotations.

On the representation of rigid body rotational dynamics in Hertzian configuration space

International Journal of Engineering Science, 2011

In a previous paper by the author, a geometrical procedure was presented for deriving Lagrange's equations for a rigid body. The rigid body was represented by an abstract particle moving in a 12-dimensional Euclidean space, called Hertzian configuration space, the metric of which is determined by the radius of gyration of the body. The present paper focuses on the representation of the underlying rotational dynamics in Hertzian space.

A new look into the kinematics and dynamics of finite rigid body rotations using Lie group theory

International Journal of Solids and Structures, 2013

A systematic theoretical approach is presented, in an effort to provide a complete and illuminating study on kinematics and dynamics of rigid bodies rotating about a fixed point. Specifically, this approach is based on some fundamental concepts of differential geometry, with particular reference to Lie group theory. This treatment is motivated by the form of the configuration space corresponding to large rigid body rotation, which is a differentiable manifold possessing group properties. First, the basic steps of the classical approach on the subject are briefly summarized. Then, some geometrical tools are presented, which are essential for supporting and illustrating the steps and findings of the new approach. Finally, the emphasis is placed on a thorough investigation of the problem of finite rotations. A key idea is the introduction of a canonical connection, matching the manifold and group properties of the configuration space. This proves to be sufficient and effective for performing the kinematics. Next, following the selection of an appropriate metric, the dynamics is also carried over. The present approach is theoretically more demanding than the traditional treatments in engineering but brings substantial benefits. In particular, an elegant interpretation is provided for all the quantities with fundamental importance in both rigid body kinematics and dynamics. Most importantly, this also leads to a correction of some misconceptions and geometrical inconsistencies in the field. Among other things, the deeper understanding of the theoretical concepts provides powerful insight and a strong basis for the development of efficient numerical techniques in problems of solid and structural mechanics involving large rotations.

THEORETICAL MATRIX STUDY OF RIGID BODY GENERAL MOTION

Greener Journal Physics and Natural Sciences, 2017

In this paper, a general motion of free asymmetrical rigid body to an absolute coordinate system is studied. The rotation component of body motion is described by using of Cardan angles. A new kind of theorem is formulated. It is called Theorem of change of generalized body impulse. New kinds of differential Lagrange equations of second gender are formulated. These are called Condensed Lagrange equations. Using that theorem and those equations, the general motion of the rigid body is successfully studied. The paper is theoretical, but it gives a base for a number of applications, for example, applications in the field of body overflow in fluid area and in the field of body vibrations. Moreover, the obtained formulas are appropriate for computer numerical integrations by contemporary mathematical programs.

Post-Newtonian treatise on the rotational motion of a finite body

Symposium - International Astronomical Union, 1986

The definition of the angular momentum of a finite body is given in the post-Newtonian framework. The non-rotating and the rigidly rotating proper reference frame(PRF)s attached to the body are introduced as the basic coordinate systems. The rigid body in the post-Newtonian framework is defined as the body resting in a rigidly rotating PRF of the body. The feasibility of this rigidity is assured by assuming suitable functional forms of the density and the stress tensor of the body. The evaluation of the time variation of the angular momentum in the above two coordinate systems leads to the post-Newtonian Euler's equation of motion of a rigid body. The distinctive feature of this equation is that both the moment of inertia and the torque are functions of the angular velocity and the angular acceleration. The obtained equation is solved for a homogeneous spheroid suffering no torque. The post-Newtonian correction to the Newtonian free precession is a linear combination of the seco...

The Study on Motion of a Rigid Body Carrying a Rotating Mass

Journal of Applied Mathematics and Physics

The free motion of a rigid body carrying a rotating mass without change of the centroid (this system may be called one-rotor gyrostat) is discussed. Equations of motion are derived: first integrals as a vectorial equation which contained the right vector of an angular velocity of the given rotor with respect to the carrier body and the turn-tensor of this body; a scalar relation between rotation angle of the given rotor with respect to the carrier body and the angular velocity of the carrier body. Only two of these parameters are independent variables. To get equations and to exclude the singular points in the solutions, it is necessary to determine the turn-tensor of the carrier body in the most suitable form. To this end the representation theorem of the turn-tensor and some additional arguments are used. As a final result, we enabled to get two complicated differential equations of the first order. In particular case, the exact solution is represented. Excluding the singular points numerical solutions are determined.

Equivalent problems in rigid body dynamics — I

Celestial Mechanics & Dynamical Astronomy, 1987

The general problem of motion of a rigid body about a fixed point under the action of stationary non-symmetric potential and gyroscopic forces is considered. The equations of motion in the Euler-Poisson form are derived. An interpretation is given in terms of charged, magnetized gyrostat moving in a superposition of three classical fields. As an example, the problem of motion of a satellite — gyrostat on a circular orbit with respect to its orbital system is reduced to that of its motion in an inertial system under additional magnetic and Lorentz forces. When the body is completely symmetric about one of its axes passing through the fixed point, the above problem is found to be equivalent to another one, in which the body has three equal moments of inertia and the forces are symmetric around a space axis. The last problem is well-studied and the given analogy reveals a number of integrable cases of the original problem. A transformation is found, which gives from each of these cases a class of integrable cases depending on an arbitrary function. The equations of motion are also reduced to a single equation of the second order.

A thesis about Newtonian mechanics rotations and about differential operators

This work dates from January 2016 as a result of my remarks related to physics made during my high studies. I try in this work to explain the cause behind the inability of Newtonian mechanics to describe correctly many phenomena where the studied object rotates at a very high linear speed. I proved that, in this case, the velocity field is not equiprojective and that the famous formula for changing the reference frame is not correct. I made an application to the case of the GPS system satellites, then I presented a new method for studying a rotating system velocity without needing the conventional steps of changing reference frames. I finished my work by demonstrating the formulas of the main differential operators and I presented them with all the related steps and calculations by using the elementary surfaces. I am eager to discuss the results of this work further with physics and mathematics specialists, and I hope that my formulas will help to simplify the study of many difficult physics phenomena.