scholarly journals NEW CASE OF INTEGRABILITY IN DYNAMICS OF MULTI-DIMENSIONAL BODY

2017 ◽  
Vol 18 (9) ◽  
pp. 136-150
Author(s):  
N.V. Pokhodnya ◽  
M.V. Shamolin
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Dynamical Symmetry ◽  
Jet Flow ◽  
Rigid Bodies ◽  
The Body ◽  
Phase Variable ◽  
Resisting Medium ◽  
Servo Constraints ◽  
Typical Point

In this chapter the new results are systematized on study of the equations of motion of dynamically symmetrical four-dimensional (4D—) rigid body which residing in a certain nonconservative field of forces in case of special dynamical symmetry. Its type is unoriginal from dynamics of the real smaller-dimensional rigid bodies of interacting with a resisting medium on the laws of a jet flow, under which the nonconservative tracing force acts onto the body and forces both the value of velocity of a certain typical point of the rigid body and the certain phase variable to remain as constant in all time, that means the presence in system nonintegrable servo-constraints.

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 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.

A Computational Model of Scapulo-Humeral-Clavicle Complex via Multibody Dynamics

2009 ◽  
Author(s):  
Shanzhong Shawn Duan ◽  
Keith M. Baumgarten
Keyword(s):  
Rigid Body ◽  
Soft Tissues ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
The Body ◽  
Constraint Forces ◽  
Upper Arm ◽  
Body Model ◽  
Rigid Body Model ◽  
Concentric Ball

The shoulder-upper arm complex has the most mobile joint in the body and is composed of three main bones: the collarbone (clavicle), the shoulder blade (scapula), and the upper arm bone (humerus). The shoulder joint is a non-concentric ball and socket joint. It differs from the hip, a highly stabilized, concentric ball and socket joint, that is constrained mostly by its osseous anatomy. Thus, the shoulder has more flexibility and less inherent stability than the hip because it is mainly stabilized by muscles, tendons, and ligaments. The relative decrease in stability of the shoulder compared to other joints puts the shoulder at increase risk of damage by disease or injury. The constraints added by muscles, tendons, and ligaments make modeling of the shoulder a challenge task. This paper presents a multi rigid body model to describe dynamical properties of the scapulo-humeral-clavicle complex. The bones are represented by rigid bodies, and the soft tissues (tendons, ligaments and muscles) are represented by springs and actuators attached to the rigid bodies. The rigid bodies are connected by ideal kinematic joints and have fixed centers of gravity. Equations of motion of the multi rigid body model are derived via Kane’s methods. Combination of springs and actuators includes independent variables for both motion and constraint forces, the sum of which determine the activation level.

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.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

scholarly journals Perturbations of Small Moons Orbits due to their rotation: The Model Problem

1999 ◽  
Vol 172 ◽  
pp. 379-380
Author(s):  
K. Goździewski ◽  
A.J. Maciejewski
Keyword(s):  
Rigid Body ◽  
Dynamical Symmetry ◽  
The Body ◽  
Polar Coordinates ◽  
Spherically Symmetric ◽  
Body Frame ◽  
Principal Moments Of Inertia ◽  
Relative Orbit ◽  
The Mean ◽  
Qualitative Estimation

We consider here the spin—orbit coupling influence on the relative orbital motion of two bodies interacting gravitationally. We assume that one of the bodies is spherically symmetric and the other possesses a plane of dynamical symmetry. In the full non-linear settings, this problem permits coplanar motion when the mass center of the spherically symmetric body moves in the plane. We used this simple model for a qualitative estimation of the changes of the relative orbit in two cases: A) the Sun-asteroid case (the fast rotating rigid body), B) a small satellite of a big planet in resonant rotation.The motion is described in the rigid body fixed frame. An appropriate change of physical units (Goźdiewski,1998a) leads to nondimensional dynamical variables and parameters. After that the Hamiltonian of the problem, written in polar variables, is the followingwhere (I1, I2, I3) are the principal moments of inertia, (r, φ) are the relative polar coordinates of the point mass in the body frame, (Pr, pφ) are the canonical momenta, (G3 represents the constant of total angular momentum, ε = (ro/r)2, and ro is the mean radius of the body.

Die Pauli-Gleichung mit Differentialoperatoren für den Spin/ The Pauli Equation with Differential Operators for the Spin

1978 ◽  
Vol 33 (10) ◽  
pp. 1133-1150
Author(s):  
Eberhard Kern
Keyword(s):  
Angular Momentum ◽  
Rigid Body ◽  
Differential Operators ◽  
Spin Operator ◽  
Rigid Bodies ◽  
The Body ◽  
Wave Functions ◽  
Pauli Equation ◽  
Spin Eigenfunctions ◽  
Cartesian Coordinate

The spin operator s = (ħ/2) σ in the Pauli equation fulfills the commutation relation of the angular momentum and leads to half-integer eigenvalues of the eigenfunctions for s. If one tries to express s by canonically conjugated operators Φ and π = (ħ/i) ∂/∂Φ the formal angular momentum term s = Φ X π fails because it leads only to whole-integer eigenvalues. However, the modification of this term in the form s = 1/2 {π + Φ(Φ π) + Φ X π} leads to the required result.The eigenfunction system belonging to this differential operator s(Φ π) consists of (2s + 1) spin eigenfunctions ξm (Φ) which are given explicitly. They form a basis for the wave functions of a particle of spin s. Applying this formalism to particles with s = 1/2, agreement is reached with Pauli’s spin theory.The function s(Φ π) follows from the theory of rotating rigid bodies. The continuous spinvariable Φ = ((Φx , Φy, Φz) can be interpreted classically as a “turning vector” which defines the orientation in space of a rigid body. Φ is the positioning coordinate of the rigid body or the spin coordinate of the particle in analogy to the cartesian coordinate x. The spin s is a vector fixed to the body.

scholarly journals Equivalence of Euler equations and torque-angular momentum relation

2021 ◽  
Vol 18 (1) ◽  
pp. 136
Author(s):  
V. Tanriverdi
Keyword(s):  
Angular Momentum ◽  
Rigid Body ◽  
Reference Frame ◽  
Euler Equations ◽  
Equations Of Motion ◽  
The Body ◽  
Moments Of Inertia ◽  
Body Reference ◽  
Stationary Reference Frame ◽  
Momentum Relation

Euler derived equations for rigid body rotations in the body reference frame and in the stationary reference frame by considering an infinitesimal part of the rigid body.Another derivation is possible, and it is widely used: transforming torque-angular momentum relation to the body reference frame.However, their equivalence is not shown explicitly.In this work, for a rigid body with different moments of inertia, we calculated Euler equations explicitly in the body reference frame and in the stationary reference frame and torque-angular momentum relation.We also calculated equations of motion from Lagrangian.These calculations show that all four of them are equivalent.

Evolution of Rotations of a Rigid Body Under the Action of Restoring and Control Moments

2005 ◽  
Author(s):  
L. D. Akulenko ◽  
D. D. Leshchenko ◽  
T. A. Kozachenko
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
The Body ◽  
System Of Equations ◽  
Slow Time ◽  
Case Examples ◽  
First Approximation ◽  
Nutation Angle ◽  
Regular Precession ◽  
And Control

Perturbed rotations of a rigid body close to the regular precession in the Lagrangian case under the action of a restoring moment depending on slow time and nutation angle, as well as a perturbing moment slowly varying with time, are studied. The body is assumed to spin rapidly, and the restoring and perturbing moments are assumed to be small with a certain hierarchy of smallness of the components. A first approximation averaged system of equations of motion for an essentially nonlinear two-frequency system is obtained in the nonresonance case. Examples of motion of a body under the action of particular restoring, perturbing, and control moments of force are considered.

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