SYMMETRY-BREAKING BIFURCATIONS IN STOPPELLI’S PROBLEM FOR PSEUDO-RIGID BODIES

1995 ◽  
Vol 05 (05) ◽  
pp. 683-724
Author(s):  
JOHN F. PIERCE
Keyword(s):  
Symmetry Breaking ◽  
Rigid Body ◽  
Rigid Bodies ◽  
The Body ◽  
Natural State ◽  
Group Orbit

The work examines what changes can occur to the orbit of trivially equilibrating configurations for a pseudo-rigid body possessing a natural state when perturbing loads are applied. The question is analyzed by formulating it as a problem of bifurcation on a group orbit to which the theory of singularities applies. The analysis indicates how alterations of the orbit depend upon features of the perturbing load, and of the material composing 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.

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 Which Correspond to the Motion of a Pendulum in the Three-dimensional Space

2021 ◽  
Vol 16 ◽  
pp. 73-84
Author(s):  
Maxim V. Shamolin
Keyword(s):  
Rigid Body ◽  
Force Fields ◽  
Characteristic Point ◽  
Dimensional Space ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
The Body ◽  
Spatial Motion ◽  
Free Rigid Body

We systematize some results on the study of the equations of spatial motion of dynamically symmetric fixed 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 a spatial motion of a free 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, or the center of mass of the body moves rectilinearly and uniformly; this means that there exists a nonconservative couple of forces in the system

Wave Propagation in Strings with Rigid Bodies

1995 ◽  
Vol 117 (4) ◽  
pp. 493-500 ◽  
Author(s):  
Li Li ◽  
J. Kim Vandiver
Keyword(s):  
Wave Propagation ◽  
Rigid Body ◽  
Input Impedance ◽  
Impedance Matching ◽  
Rigid Bodies ◽  
Wave Transmission ◽  
The Body ◽  
Cable Systems ◽  
Zero Frequency ◽  
Infinite String

This paper studies wave propagation in strings with rigid bodies using the method of transfer matrices. The transmission property of a single rigid body is investigated. It is found that when the size of a rigid body is included, a symmetrically defined rigid body will transmit wave energy completely at a non-zero frequency defined by the tension, the length of the body, the mass of the string replaced by the body, and the mass of the body. Using the concept of impedance matching, the effect of a discontinuity on wave transmission in an infinite string system is revealed. The same idea is extended to the study of wave propagation in a string with multiple, equally-spaced rigid bodies (a periodic structure). The input impedance of such a system and the conditions of complete transmission are expressed in terms of the transfer matrix. The input impedance is used to identify the frequencies at which there is complete wave transmission. These frequencies are related to the natural frequencies of the corresponding finite system and constitute the so-called propagation zones. The results of this work may be applied to the propagation of vibration in complex cable systems such as oceanographic moorings.

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.

On Singularity of Rigid-Body Dynamics Using Quaternion-Based Models

2012 ◽  
Vol 79 (2) ◽  
Author(s):  
Homin Choi ◽  
Bingen Yang
Keyword(s):  
Dynamic Response ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Degrees Of Freedom ◽  
Rigid Bodies ◽  
Rigid Body Dynamics ◽  
The Body ◽  
Applied Torque ◽  
Body Dynamics

It is well known that use of quaternions in dynamic modeling of rigid bodies can avoid the singularity due to Euler rotations. This paper shows that the dynamic response of a rigid body modeled by quaternions may become unbounded when a torque is applied to the body. A theorem is derived, relating the singularity to the axes of the rotation and applied torque, and to the degrees of freedom of the body in rotation. To avoid such singularity, a method of equivalent couples is proposed.

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

2017 ◽  
Vol 22 (3-4) ◽  
pp. 75-97 ◽  
Author(s):  
M. V. Shamolin
Keyword(s):  
Rigid Body ◽  
Force Fields ◽  
Characteristic Point ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
The Body ◽  
Spatial Motion ◽  
Free Rigid Body ◽  
Three Dimensional Space

In this article, we systemize some results on the study of the equations of spatial motion of dynamically symmetric fixed 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 a spatial motion of a free 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. The obtained results are systematized and served in the invariant form. We also show the nontrivial topological and mechanical analogies.

scholarly journals ON INERTIAL MOTION OF AN ABSOLUTELY RIGID BODY ON A THREE-DEGREE SUSPENSION WITH LINKS OF FINITE LENGTH

2020 ◽  
Vol 2 (2) ◽  
pp. 6-17
Author(s):  
L Akulenko ◽  
◽  
N Bolotnik ◽  
D Leshchenko ◽  
E Palii ◽  
...  
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Exact Analytical Solution ◽  
Rigid Bodies ◽  
The Body ◽  
Theoretical Research ◽  
Double Pendulum ◽  
Two Degrees Of Freedom ◽  
Practical Applications ◽  
Wide Range

Papers on the dynamics of an absolutely rigid body with a fixed point generally assume that the mechanical system has three degrees of freedom. This is the situation when the body is attached to a fixed base by a ball-and-socket joint. On engineering systems one often encounters rigid bodies attached to a base by a two-degrees-of-freedom joint, consisting of a fixed axis and a movable one, which are mutually perpendicular. Such systems have two degrees of freedom, but the set of kinematically possible motions is quite rich. Dynamic analysis of the motion of a rigid body with a two-degree hinge in a force field is an integral part of the description of the action of mechanical actions of robotic systems. In recent decades, an increasingly closed role in the dynamics of rigid body systems has been played by manipulation robots consisting of a sequential chain of rigid links and controlled by means of torque drives in articulated joints. The same class of objects can be attributed to many biological systems that imitate, for example, the movements of a person or animal (walking, running, jumping). Two-link systems have a variety of practical applications and an almost equally wide range of areas of theoretical research. We note, in particular, the analysis of free and forced plane-parallel motion of a bundle of two rigid bodies connected by an ideal cylindrical hinge and simulating a composite satellite in outer space, a two-link manipulator, and an element of a crushing machine. The dynamic behavior of a rigid body in the gimbal suspension is a system, which can be interpreted as two-degree manipulator and used an element of more complex robotic structures. The linear mathematical model of two-link manipulator free oscillations with viscous friction in both its joints is a system, which reduces to the calculation scheme of double pendulum and allows the construction of exact analytical solution in the partial case. According to the research methodology, the proposed paper is close to works, where the motion by inertia of a plane two–rigid body hinged system was studied and devoted to the study of the motion of an absolutely rigid body on a power-to-power joint.

scholarly journals Chaotic Dynamics in the Planar Gravitational Many-Body Problem with Rigid Body Rotations

2018 ◽  
Vol 28 (05) ◽  
pp. 1830013 ◽  
Author(s):  
James A. Kwiecinski ◽  
Attila Kovacs ◽  
Andrew L. Krause ◽  
Ferran Brosa Planella ◽  
Robert A. Van Gorder
Keyword(s):  
Rigid Body ◽  
Chaotic Dynamics ◽  
Body Problem ◽  
Melnikov Method ◽  
Rigid Bodies ◽  
The Body ◽  
Many Body ◽  
Tidal Dissipation ◽  
Arbitrary Mass ◽  
Orbital Resonance

The discovery of Pluto’s small moons in the last decade has brought attention to the dynamics of the dwarf planet’s satellites. With such systems in mind, we study a planar [Formula: see text]-body system in which all the bodies are point masses, except for a single rigid body. We then present a reduced model consisting of a planar [Formula: see text]-body problem with the rigid body treated as a 1D continuum (i.e. the body is treated as a rod with an arbitrary mass distribution). Such a model provides a good approximation to highly asymmetric geometries, such as the recently observed interstellar asteroid ‘Oumuamua, but is also amenable to analysis. We analytically demonstrate the existence of homoclinic chaos in the case where one of the orbits is nearly circular by way of the Melnikov method, and give numerical evidence for chaos when the orbits are more complicated. We show that the extent of chaos in parameter space is strongly tied to the deviations from a purely circular orbit. These results suggest that chaos is ubiquitous in many-body problems when one or more of the rigid bodies exhibits nonspherical and highly asymmetric geometries. The excitation of chaotic rotations does not appear to require tidal dissipation, obliquity variation, or orbital resonance. Such dynamics give a possible explanation for routes to chaotic dynamics observed in [Formula: see text]-body systems such as the Pluto system where some of the bodies are highly nonspherical.

scholarly journals Software-Engineering Optimization with Mathematical Method Improvements for Numerical Reuleaux Method

2020 ◽  
pp. 447-499 ◽  
Author(s):  
Francisco Casesnoves
Keyword(s):  
Rigid Body ◽  
Theoretical Calculations ◽  
Rigid Bodies ◽  
Optimization Techniques ◽  
The Body ◽  
Theoretical Development ◽  
Mathematical Framework ◽  
Software Optimization ◽  
Monte Carlo Techniques ◽  
Reduction Techniques

This paper presents an extension both in software optimization with simulations and detailed mathematical theory of Numerical Reuleaux Method based on previous publications. In the literature, there are a number of papers in Numerical Reuleaux Method and its applications (Aerospace, Helicopter Dynamics in Turbulence Conditions, Biomedical Engineering, Biomechanics, etc) since 2007. This contribution is a detailed presentation of the mathematical framework that constituted the basis for those articles along 2007-2020. The Classical Reuleaux Method (<strong>CRM</strong>) is frequently used in Physical Dynamics, Engineering Mechanics and Bioengineering to determine the Instantaneous Rotation Center (<strong>IRC</strong>) of a rigid body in arbitrary movement. The generic mathematical <strong>CRM </strong>only can be applied on rigid bodies, whose shape remains constant during the movement. If the solid in movement is a Pseudo-Rigid Body (<strong>PRB</strong>), the <strong>CRM </strong>has to be modified numerically to conform the shape changes and adapt on the density distribution variations of the <strong>PRB </strong>(we denominate it, in this case<sup>1</sup>, <strong>The Numerical Reuleaux Method</strong>, <strong>NRM</strong>). This Geometrical-Numerical Approximation Method is based on the division of the Pseudo-rigid body into small volume parts called voxels (roughly speaking parallelepipedic), namely, voxelization of the body subject to dynamics. The theoretical basis of the method is strictly shown in complementary details, with the necessary Theorems and Propositions of the model. Nonlinear Optimization Techniques that support the initial theory have been developed, and the Error boundaries with Error reduction techniques are determined. Computational Simulations have been carried out to prove the <strong>NRM </strong>Theoretical Model feasibility and numerical veracity of the Propositions, Theorems, and Error Boundaries. Appropriate software was made to carry out these simulations conveniently. The initial results agree to the theoretical calculations, and the IRC computation for 2 voxels shows to be simple and easy. Some initial guidelines for a theoretical development of this algorithm applied on large pseudo-rigid bodies, by using Monte-Carlo techniques, are sketched. Recent applications, Aerospace and Biomechanics, are also shown.

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