scholarly journals Integrability of Differential Equations of Motion of an n-Dimensional Rigid Body in Nonconservative Fields for n = 5 and n = 6

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Rigid Body ◽  
Integrable Systems ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Elementary Functions ◽  
Finite Dimensional ◽  
Transcendental Functions ◽  
Classification Of Singularities ◽  
Tangent Bundles

In this review, we discuss new cases of integrable systems on the tangent bundles of finite-dimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

scholarly journals INTEGRABLE SYSTEMS ON TANGENT BUNDLE OF MULTI-DIMENSIONAL SPHERE

2017 ◽  
Vol 20 (7) ◽  
pp. 60-69
Author(s):  
N.V. Pokhodnya ◽  
M.V. Shamolin
Keyword(s):  
Force Field ◽  
Integrable Systems ◽  
Tangent Bundle ◽  
First Integrals ◽  
Dimensional Sphere ◽  
Elementary Functions ◽  
Finite Dimensional ◽  
Transcendental Functions

The systems which have finite-dimensional spheres as the space of positions, are arising in many problems of multi-dimensional dynamics. Accordingly, tan- gent bundles of those spheres become phase spaces of such systems. In the article activity of inductive transition in the system on tangent bundle of low-dimen- sional sphere under increase of its dimension and absence of force field is ana- lyzed. At that, nonconservative fields of forces are presented with the presence of which the systems possess the complete choice of first integrals expressing in terms of finite combination of elementary functions and are, in general, the transcendental functions of its variables.

scholarly journals Dynamic model of multi-rigid-body systems based on particle dynamics with recursive approach

2005 ◽  
Vol 2005 (4) ◽  
pp. 365-382 ◽  
Author(s):  
Hazem Ali Attia
Keyword(s):  
Rigid Body ◽  
Dynamic Model ◽  
Equations Of Motion ◽  
Particle Dynamics ◽  
Rigid Bodies ◽  
Constrained System ◽  
Open Chain ◽  
Chain System ◽  
Closed Chain ◽  
Serial Chains

A dynamic model for multi-rigid-body systems which consists of interconnected rigid bodies based on particle dynamics and a recursive approach is presented. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. For the open-chain system, the equations of motion are generated recursively along the serial chains. A closed-chain system is transformed to open-chain by cutting suitable kinematical joints and introducing cut-joint constraints. An example is chosen to demonstrate the generality and simplicity of the developed formulation.

Vibration Modes and Frequencies of Timoshenko Beams With Attached Rigid Bodies

1995 ◽  
Vol 62 (1) ◽  
pp. 193-199 ◽  
Author(s):  
M. W. D. White ◽  
G. R. Heppler
Keyword(s):  
Boundary Conditions ◽  
Rigid Body ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Rigid Bodies ◽  
The Body ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
First Moment

The equations of motion and boundary conditions for a free-free Timoshenko beam with rigid bodies attached at the endpoints are derived. The natural boundary conditions, for an end that has an attached rigid body, that include the effects of the body mass, first moment of mass, and moment of inertia are included. The frequency equation for a free-free Timoshenko beam with rigid bodies attached at its ends which includes all the effects mentioned above is presented and given in terms of the fundamental frequency equations for Timoshenko beams that have no attached rigid bodies. It is shown how any support / rigid-body condition may be easily obtained by inspection from the reported frequency equation. The mode shapes and the orthogonality condition, which include the contribution of the rigid-body masses, first moments, and moments of inertia, are also developed. Finally, the effect of the first moment of the attached rigid bodies is considered in an illustrative example.

scholarly journals Dynamics of constrained many body problems in constant curvature two-dimensional manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170425 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Maria Przybylska
Keyword(s):  
Constant Curvature ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Rigid Bodies ◽  
The Body ◽  
Two Dimensional ◽  
Many Body ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In this paper, we investigate systems of several point masses moving in constant curvature two-dimensional manifolds and subjected to certain holonomic constraints. We show that in certain cases these systems can be considered as rigid bodies in Euclidean and pseudo-Euclidean three-dimensional spaces with points which can move along a curve fixed in the body. We derive the equations of motion which are Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, we have found several integrable cases of described models. For one of them, we give the necessary and sufficient conditions for the integrability. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

On the Euler–Lagrange Equation for Planar Systems of Rigid Bodies or Lumped Masses

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
R. Wiebe ◽  
P. S. Harvey
Keyword(s):  
Rigid Body ◽  
Reference Frames ◽  
Dynamic Equilibrium ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Double Pendulum ◽  
Governing Equations ◽  
Lumped Mass ◽  
Planar Systems

The Euler–Lagrange equation is frequently used to develop the governing dynamic equilibrium expressions for rigid-body or lumped-mass systems. In many cases, however, the rectangular coordinates are constrained, necessitating either the use of Lagrange multipliers or the introduction of generalized coordinates that are consistent with the kinematic constraints. For such cases, evaluating the derivatives needed to obtain the governing equations can become a very laborious process. Motivated by several relevant problems related to rigid-body structures under seismic motions, this paper focuses on extending the elegant equations of motion developed by Greenwood in the 1970s, for the special case of planar systems of rigid bodies, to include rigid-body rotations and accelerating reference frames. The derived form of the Euler–Lagrange equation is then demonstrated with two examples: the double pendulum and a rocking object on a double rolling isolation system. The work herein uses an approach that is used by many analysts to derive governing equations for planar systems in translating reference frames (in particular, ground motions), but effectively precalculates some of the important stages of the analysis. It is hoped that beyond re-emphasizing the work by Greenwood, the specific form developed herein may help researchers save a significant amount of time, reduce the potential for errors in the formulation of the equations of motion for dynamical systems, and help introduce more researchers to the Euler–Lagrange equation.

Geometric Algorithms for Robot Dynamics: A Tutorial Review

2018 ◽  
Vol 70 (1) ◽  
Author(s):  
Frank C. Park ◽  
Beobkyoon Kim ◽  
Cheongjae Jang ◽  
Jisoo Hong
Keyword(s):  
Rigid Body ◽  
Inverse Dynamics ◽  
Group Structure ◽  
Equations Of Motion ◽  
Variable Stiffness ◽  
Rigid Bodies ◽  
Robot Dynamics ◽  
Motion Optimization ◽  
Geometric Formulation ◽  
Variable Stiffness Actuators

We provide a tutorial and review of the state-of-the-art in robot dynamics algorithms that rely on methods from differential geometry, particularly the theory of Lie groups. After reviewing the underlying Lie group structure of the rigid-body motions and the geometric formulation of the equations of motion for a single rigid body, we show how classical screw-theoretic concepts can be expressed in a reference frame-invariant way using Lie-theoretic concepts and derive recursive algorithms for the forward and inverse dynamics and their differentiation. These algorithms are extended to robots subject to closed-loop and other constraints, joints driven by variable stiffness actuators, and also to the modeling of contact between rigid bodies. We conclude with a demonstration of how the geometric formulations and algorithms can be effectively used for robot motion optimization.

scholarly journals CASES OF INTEGRABILITY CORRESPONDING TO THE PENDULUM MOTION IN FOUR-DIMENSIONAL SPACE

2017 ◽  
Vol 23 (1) ◽  
pp. 41-58
Author(s):  
M. V. Shamolin
Keyword(s):  
Rigid Body ◽  
Force Fields ◽  
Characteristic Point ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
The Body ◽  
Pendulum Motion ◽  
Nonconservative Force ◽  
Similar Force

In this article, we systemize some results on the study of the equations of motion of dynamically symmetric fixed four-dimensional rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free four-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. We also show the nontrivial topological and mechanical analogies.

scholarly journals On the integrability of Birkhoff billiards

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170419 ◽  
Author(s):  
Vadim Kaloshin ◽  
Alfonso Sorrentino
Keyword(s):  
Integrable Systems ◽  
Theme Issue ◽  
Finite Dimensional

In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Modeling of Multibody Flexible Articulated Structures With Mutually Coupled Motions: Part I — General Theory

1992 ◽  
Author(s):  
Jiechi Xu ◽  
Joseph R. Baumgarten
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Local Level ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Moving Frame ◽  
Element Analysis ◽  
Highly Nonlinear ◽  
Geometric Boundary ◽  
System Equations

Abstract In the present paper a general systematic modeling procedure has been conducted in deriving dynamic equations of motion using Lagrange’s approach for a spatial multibody structural system involving rigid bodies and elastic members. Both the rigid body degrees of freedom and the elastic degrees of freedom are considered as unknown generalized coordinates of the entire system in order to reflect the nature of mutually coupled rigid body and elastic motions. The assumption of specified rigid body gross motion is no longer necessary in the equation derivation and the resulting differential equations are highly nonlinear. Finite element analysis (FEA) with direct stiffness method has been employed to model the flexible substructures. Nonlinear coupling terms between the rigid body and elastic motions are fully derived and are explicitly expressed in matrix form. The equations of motion of each individual subsystem are formulated based on a moving frame instead of a traditional inertial frame. These local level equations of motion are assembled to obtain the system equations with the implementation of geometric boundary conditions by means of a compatibility matrix.

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