Substantial condition for the fourth first integral of the rigid body problem

2017 ◽  
Vol 23 (8) ◽  
pp. 1237-1246 ◽  
Author(s):  
TS Amer ◽  
WS Amer
Keyword(s):  
Rigid Body ◽  
Body Problem ◽  
Nonlinear Differential Equations ◽  
First Integral ◽  
Industrial Applications ◽  
The Body ◽  
Important Condition ◽  
First Order ◽  
Moment Vector ◽  
Gyroscopic Motion

The aim of this article is to study the possibility of obtaining the fourth integral for the motion of a rigid body about a fixed point in the presence of a gyrostatic moment vector. This problem is governed by a system consisting of six nonlinear differential equations from first order, as well as three first integrals. A most important condition for a function F, depending on all the body variables, to be that integral is presented. This work can be considered a mainstreaming of previous works. The importance of this work lies in several applications of the rigid body problem and gyroscopic motion in different areas, such as physics, engineering and industrial applications, for example, in aircraft specially designed to use the auto-pilot function, calculating aircraft turns about various axes of operation (pitch, yaw and roll), and maintaining aircraft orientation.

Generalized Kirchhoff equations for a deformable body moving in a weakly non-uniform flow field

1994 ◽  
Vol 446 (1926) ◽  
pp. 169-193 ◽  
Keyword(s):  
Flow Field ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Surface Deformation ◽  
Nonlinear Differential Equations ◽  
Deformable Body ◽  
Equations Of Motion ◽  
First Integral ◽  
Uniform Flow ◽  
The Body

The classical Kirchhoff’s method provides an efficient way of calculating the hydrodynamical loads (forces and moments) acting on a rigid body moving with six-degrees of freedom in an otherwise quiescent ideal fluid in terms of the body’s added-mass tensor. In this paper we provide a versatile extension of such a formulation to account for both the presence of an imposed ambient non-uniform flow field and the effect of surface deformation of a non-rigid body. The flow inhomogeneity is assumed to be weak when compared against the size of the body. The corresponding expressions for the force and moment are given in a moving body-fixed coordinate system and are obtained using the Lagally theorem. The newly derived system of nonlinear differential equations of motion is shown to possess a first integral. This can be interpreted as an energy-type conservation law and is a consequence of an anti-symmetry property of the coefficient matrix reported here for the first time. A few applications of the proposed formulation are presented including comparison with some existing limiting cases.

scholarly journals The periodic rotary motions of a rigid body in a new domain of angular velocity

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Fixed Point ◽  
Angular Velocity ◽  
Rigid Body ◽  
Periodic Solutions ◽  
Nonlinear Differential Equations ◽  
First Integral ◽  
High Energy ◽  
The Body ◽  
New Technique ◽  
Large Parameter

AbstractIn the previous works, the limiting case for the motion of a rigid body about a fixed point in a Newtonian force field, which comes from a gravity center lies on Z-axis, is solved. The authors apply the small parameter technique which is achieved giving the body a sufficiently large angular velocity component ro about the fixed z-axis of the body. The periodic solutions of motion are obtained in neighborhood ro tends to $$\infty$$ ∞ . In our work, we aim to find periodic solutions to the problem of motion in the neighborhood of r0 tends to $$0$$ 0 . So, we give a new assumption that: ro is sufficiently small. Under this assumption, we must achieve a large parameter and search for another technique for solving this problem. This technique is named; a large parameter technique instead of the small one well known previously. We see the advantage of the new technique which appears in saving high energy used to begin the motion and give the solution of the problem in another domain. The obtained solutions by the new technique depend on ro. We consider that the center of mass of this body does not necessarily coincide with the fixed point O. We reduce the six nonlinear differential equations of the body and their three first integrals to a quasilinear autonomous system of two degrees of freedom and one first integral. We solve the rational case when the frequencies of the generating system are rational except $$(\,\omega = \,1,\,2,1/2,3,1/3, \ldots )$$ ( ω = 1 , 2 , 1 / 2 , 3 , 1 / 3 , … ) under the condition $$\gamma^{\prime\prime}_{0} = \cos \theta_{o} \approx 0$$ γ 0 ″ = cos θ o ≈ 0 . We use the fourth-order Runge–Kutta method to find the periodic solutions in the closed interval of the time t and to compare the analytical method with the numerical one.

scholarly journals The Motion of a Rigid Body in the Presence of a Gyrostatic : الحركات المغزلية السريعة لجسم متماسك في مجال قوة نيوتوني في حالة وجود العزم

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Ghadir Ahmed Sahli
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
First Integral ◽  
Geometric Interpretation ◽  
The Body ◽  
Parameter Method ◽  
Principal Axes ◽  
Linear Autonomous System ◽  
Newtonian Force Field

In this study، the rotational motion of a rigid body about a fixed point in the Newtonian force field with a gyrostatic momentum  about the z-axis is considered. The equations of motion and their first integrals are obtained and reduced to a quasi-linear autonomous system with two degrees of freedom with one first integral. Poincare's small parameter method is applied to investigate the analytical peri­odic solutions of the equations of motion of the body with one point fixed، rapidly spinning about one of the principal axes of the ellipsoid of inertia. A geometric interpretation of motion is given by using Euler's angles to describe the orientation of the body at any instant of time.

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.

Designing Experiments for Model Identification of the Nitrification Process

1991 ◽  
Vol 24 (6) ◽  
pp. 9-16 ◽  
Author(s):  
P. J. Ossenbruggen ◽  
H. Spanjers ◽  
H. Aspegren ◽  
A. Klapwijk
Keyword(s):  
Mechanistic Model ◽  
Nonlinear Differential Equations ◽  
Model Identification ◽  
Reaction Rates ◽  
Variable Coefficients ◽  
Model Specification ◽  
First Order ◽  
Interactive Behavior ◽  
Batch Tests ◽  
Nitrification Process

A series of batch tests were performed to study the competition for oxygen by Nitrosomonas and Nitrobacter in the nitrification of ammonia in activated sludge. Oxygen uptake rate (OUR) and dynamic (compartment) models describing the process are proposed and tested. The OUR model is described by a Monod relationship and the biogradation process by a set of first order nonlinear differential equations with variable coefficients. The results show a mechanistic model and ten reaction rates are sufficient to capture the interactive behavior of the nitrification process. Methods for model specification, calibrating, and testing the model and the design of additional experiments are described.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

Analysis of the First Order and Slowly Varying Motions of an Axisymmetric Floating Body in Bichromatic Waves

2013 ◽  
Vol 135 (1) ◽  
Author(s):  
João Pessoa ◽  
Nuno Fonseca ◽  
C. Guedes Soares
Keyword(s):  
Vertical Cylinder ◽  
Vertical Axis ◽  
Second Order ◽  
The Body ◽  
Floating Body ◽  
Drift Motion ◽  
First Order ◽  
Motion Responses ◽  
Simple Geometry ◽  
Good Agreement

The paper presents an experimental and numerical investigation on the motions of a floating body of simple geometry subjected to harmonic and biharmonic waves. The experiments were carried out in three different water depths representing shallow and deep water. The body is axisymmetric about the vertical axis, like a vertical cylinder with a rounded bottom, and it is kept in place with a soft mooring system. The experimental results include the first order motion responses, the steady drift motion offset in regular waves and the slowly varying motions due to second order interaction in biharmonic waves. The hydrodynamic problem is solved numerically with a second order boundary element method. The results show a good agreement of the numerical calculations with the experiments.

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.

Does a rigid body limit maneuverability?

2000 ◽  
Vol 203 (22) ◽  
pp. 3391-3396 ◽  
Author(s):  
J.A. Walker
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Alternative Measure ◽  
Pectoral Fin ◽  
Caudal Fin ◽  
Minimum Radius ◽  
Current Definition ◽  
Theoretical Minimum

Whether a rigid body limits maneuverability depends on how maneuverability is defined. By the current definition, the minimum radius of the turn, a rigid-bodied, spotted boxfish Ostracion meleagris approaches maximum maneuverability, i.e. it can spin around with minimum turning radii near zero. The radius of the minimum space required to turn is an alternative measure of maneuverability. By this definition, O. meleagris is not very maneuverable. The observed space required by O. meleagris to turn is slightly greater than its theoretical minimum but much greater than that of highly flexible fish. Agility, the rate of turning, is related to maneuverability. The median- and pectoral-fin-powered turns of O. meleagris are slow relative to the body- and caudal-fin-powered turns of more flexible fish.

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