A Linear Algebra Approach to the Analysis of Rigid Body Velocity From Position and Velocity Data

1983 ◽  
Vol 105 (2) ◽  
pp. 92-95 ◽  
Author(s):  
A. J. Laub ◽  
G. R. Shiflett
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Matrix Theory ◽  
The Body ◽  
Instantaneous Velocity ◽  
Body Motion ◽  
Translational Velocity ◽  
Velocity Data ◽  
Algebra Approach ◽  
Body Velocity

The instantaneous velocity of a rigid body in space is characterized by an angular and translational velocity. By representing the angular velocity as a matrix and the translational component as a vector the velocity of any point in the rigid body may be found if the position of the point and the parameters of the angular and translational velocities are known. Alternatively, the parameters of the rigid body velocity may be determined if the velocity and position of three points fixed in the body are known. In this paper, a new matrix-theory-based method is derived for determining the instantaneous velocity parameters of rigid body motion in terms of the velocity and position of three noncollinear points fixed in the body. The method is shown to possess certain advantages over traditional vectoral solutions to the same problem.

A Velocity Metric for Rigid-Body Planar Motion

2010 ◽  
Author(s):  
Joseph M. Schimmels ◽  
Luis E. Criales
Keyword(s):  
Rigid Body ◽  
Average Distance ◽  
The Body ◽  
Body Motion ◽  
Length Scale ◽  
Translational Velocity ◽  
Body Velocity ◽  
Assembly Problem ◽  
Instantaneous Center ◽  
Selection Of

A planar rigid-body velocity metric based on the instantaneous velocity of all particles that constitute a rigid body is developed. A measure based on the discrepancy in the translational velocity at each particle for two different planar twists is introduced. The calculation of the measure is simplified to the calculation of the product of: 1) the discrepancy in angular velocity, and 2) the average distance of the body from the instantaneous center associated with the twist discrepancy. It is shown that this measure satisfies the mathematical requirements of a metric and is physically consistent. It does not depend on either the selection of length scale or the frames used to describe the body motion. Although the metric does depend on body geometry, it can be calculated efficiently using body decomposition. An example demonstrating the application of the metric to an assembly problem is presented.

scholarly journals The equations of motion for a rigid body using non-redundant unified local velocity coordinates

2019 ◽  
Vol 48 (3) ◽  
pp. 283-309 ◽  
Author(s):  
Stefan Holzinger ◽  
Joachim Schöberl ◽  
Johannes Gerstmayr
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Motion Vector ◽  
The Body ◽  
Body Motion ◽  
Integration Scheme ◽  
Exponential Map ◽  
Local Velocity ◽  
Translational Velocity ◽  
Angular Velocity Vector

Abstract A novel formulation for the description of spatial rigid body motion using six non-redundant, homogeneous local velocity coordinates is presented. In contrast to common practice, the formulation proposed here does not distinguish between a translational and rotational motion in the sense that only translational velocity coordinates are used to describe the spatial motion of a rigid body. We obtain these new velocity coordinates by using the body-fixed translational velocity vectors of six properly selected points on the rigid body. These vectors are projected into six local directions and thus give six scalar velocities. Importantly, the equations of motion are derived without the aid of the rotation matrix or the angular velocity vector. The position coordinates and orientation of the body are obtained using the exponential map on the special Euclidean group $\mathit{SE}(3)$SE(3). Furthermore, we introduce the appropriate inverse tangent operator on $\mathit{SE}(3)$SE(3) in order to be able to solve the incremental motion vector differential equation. In addition, we present a modified version of a recently introduced a fourth-order Runge–Kutta Lie-group time integration scheme such that it can be used directly in our formulation. To demonstrate the applicability of our approach, we simulate the unstable rotation of a rigid body.

scholarly journals SOME PROBLEMS ABOUT THE MOTION OF A RIGID BODY IN A RESISTIVE MEDIUM

2021 ◽  
Vol 3 (2) ◽  
pp. 6-17
Author(s):  
D. Leshchenko ◽  
◽  
T. Kozachenko ◽  
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Modern Technology ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
The Body ◽  
Body Motion ◽  
Body Rotation ◽  
Angular Velocity Vector ◽  
Resistive Medium

The dynamics of rotating rigid bodies is a classical topic of study in mechanics. In the eighteenth and nineteenth centuries, several aspects of a rotating rigid body motion were studied by famous mathematicians as Euler, Jacobi, Poinsot, Lagrange, and Kovalevskya. However, the study of the dynamics of rotating bodies of still important for aplications such as the dynamics of satellite-gyrostat, spacecraft, re-entry vehicles, theory of gyroscopes, modern technology, navigation, space engineering and many other areas. A number of studies are devoted to the dynamics of a rigid body in a resistive medium. The presence of the velocity of proper rotation of the rigid body leads to the apearance of dissipative torques causing the braking of the body rotation. These torques depend on the properties of resistant medium in which the rigid body motions occur, on the body shape, on the properties of the surface of the rigid body and the distribution of mass in the body and on the characters of the rigid body motion. Therefore, the dependence of the resistant torque on the orientation of the rigid body and its angular velocity can de quite complicated and requires consideration of the motion of the medium around the body in the general case. We confine ourselves in this paper to some simple relations that can qualitative describe the resistance to rigid body rotation at small angular velocities and are used in the literature. In setting up the equations of motion of a rigid body moving in viscous medium, we need to consider the nature of the resisting force generated by the motion of the rigid body. The evolution of rotations of a rigid body influenced by dissipative disturbing torques were studied in many papers and books. The problems of motion of a rigid body about fixed point in a resistive medium described by nonlinear dynamic Euler equations. An analytical solution of the problem when the torques of external resistance forces are proportional to the corresponding projections of the angular velocity of the rigid body is obtain in several works. The dependence of the dissipative torque of the resistant forces on the angular velocity vector of rotation of the rigid body is assumed to be linear. We consider dynamics of a rigid body with arbitrary moments of inertia subjected to external torques include small dissipative torques.

scholarly journals Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Thomas Eiter ◽  
Mads Kyed
Keyword(s):  
Rigid Body ◽  
Sobolev Spaces ◽  
Periodic Motion ◽  
Viscous Incompressible Fluid ◽  
Body Motion ◽  
Translational Velocity ◽  
Ill Posed ◽  
Homogeneous Sobolev Spaces ◽  
The Mean ◽  
Time Periodic

AbstractThe equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

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