Phase Portraits of Dynamical Equations of Motion of a Rigid Body in a Resistive Medium

2018 ◽  
Vol 233 (3) ◽  
pp. 398-425
Author(s):  
M. V. Shamolin
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Phase Portraits ◽  
Dynamical Equations ◽  
Resistive Medium

Nature of the Requirements for Reference Coordinate Systems

1975 ◽  
Vol 26 ◽  
pp. 49-62
Author(s):  
C. A. Lundquist
Keyword(s):  
Rigid Body ◽  
Coordinate System ◽  
Equations Of Motion ◽  
Inertial Reference Frame ◽  
Earth System ◽  
Coordinate Systems ◽  
Dynamical Equations ◽  
The Earth ◽  
Measurement Uncertainties ◽  
Rotational Motions

AbstractThe current need for more precisely defined reference coordinate systems arises for geodynamics because the Earth can certainly not be treated as a rigid body when measurement uncertainties reach the few centimeter scale or its angular equivalent. At least two coordinate systems seem to be required. The first is a system defined in space relative to appropriate astronomical objects. This system should approximate an inertial reference frame, or be accurately related to such a reference, because only such a coordinate system is suitable for ultimately expressing the dynamical equations of motion for the Earth. The second required coordinate system must be associated with the nonrigid Earth in some well defined way so that the rotational motions of the whole Earth are meaningfully represented by the transformation parameters relating the Earth system to the space-inertial system. The Earth system should be defined so that the dynamical equations for relative motions of the various internal mechanical components of the Earth and accurate measurements of these motions are conveniently expressed in this system.

VARIETY OF THE CASES OF INTEGRABILITY IN DYNAMICS OF A SYMMETRIC 2D-, 3D- AND 4D-RIGID BODY IN A NONCONSERVATIVE FIELD

2013 ◽  
Vol 13 (07) ◽  
pp. 1340011 ◽  
Author(s):  
MAXIM V. SHAMOLIN
Keyword(s):  
Rigid Body ◽  
Force Field ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
General System ◽  
Rigid Bodies ◽  
Body Motion ◽  
Dynamical Equations ◽  
Vast Number ◽  
Nonconservative Force

A vast number of papers are devoted to studying the complete integrability of equations of four-dimensional rigid-body motion. Although in studying low-dimensional equations of motion of quite concrete (two- and three-dimensional) rigid bodies in a nonconservative force field, the author arrived at the idea of generalizing the equations to the case of a four-dimensional rigid body in an analogous nonconservative force field. As a result of such a generalization, the author obtained the variety of cases of integrability in the problem of body motion in a resisting medium that fills the four-dimensional space in the presence of a certain tracing force that allows one to reduce the order of the general system of dynamical equations of motion in a methodical way.

scholarly journals A brief review of E theory

2016 ◽  
Vol 31 (26) ◽  
pp. 1630043 ◽  
Author(s):  
Peter West
Keyword(s):  
Infinite Number ◽  
Equations Of Motion ◽  
Space Time ◽  
Unified Theory ◽  
Dynamical Equations ◽  
Supergravity Theory ◽  
Maximal Supergravity ◽  
Abdus Salam ◽  
Strings And Branes ◽  
Time Lead

I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of nonlinear realisations and Kac–Moody algebras, I explain how to construct the nonlinear realisation based on the Kac–Moody algebra [Formula: see text] and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a space–time with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of space–time, lead to precisely the equations of motion of 11-dimensional supergravity theory. By taking different group decompositions of [Formula: see text] we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the nonlinear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the [Formula: see text] conjecture given many years ago.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

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.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

Seismic Analysis of Structures With Coulomb Friction

1983 ◽  
Vol 105 (2) ◽  
pp. 171-178 ◽  
Author(s):  
V. N. Shah ◽  
C. B. Gilmore
Keyword(s):  
Rigid Body ◽  
Coulomb Friction ◽  
Seismic Event ◽  
Seismic Analysis ◽  
Equations Of Motion ◽  
Relative Displacement ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Superposition Method ◽  
Modal Superposition Method

A modal superposition method for the dynamic analysis of a structure with Coulomb friction is presented. The finite element method is used to derive the equations of motion, and the nonlinearities due to friction are represented by pseudo-force vector. A structure standing freely on the ground may slide during a seismic event. The relative displacement response may be divided into two parts: elastic deformation and rigid body motion. The presence of rigid body motion necessitates the inclusion of the higher modes in the transient analysis. Three single degree-of-freedom problems are solved to verify this method. In a fourth problem, the dynamic response of a platform standing freely on the ground is analyzed during a seismic event.

Quasi-optimal deceleration of rotations of a rigid body with a moving mass in a resistive medium

2017 ◽  
Vol 56 (2) ◽  
pp. 186-191 ◽  
Author(s):  
L. D. Akulenko ◽  
D. D. Leshchenko ◽  
Yu. S. Shchetinina
Keyword(s):  
Rigid Body ◽  
Moving Mass ◽  
Resistive Medium
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