scholarly journals Application of a Large-Parameter Technique for Solving a Singular Case of a Rigid Body

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
A. I. Ismail
Keyword(s):  
Rigid Body ◽  
Numerical Solutions ◽  
Geometric Interpretation ◽  
The Body ◽  
Kutta Method ◽  
Singular Case ◽  
Large Parameter ◽  
Runge Kutta Method ◽  
Small Angular Velocity ◽  
Newtonian Force Field

In this paper, the motion of a rigid body in a singular case of the natural frequency ( ω = 1 / 3 ) is considered. This case of singularity appears in the previous works due to the existence of the term ω 2 − 1 / 9 in the denominator of the obtained solutions. For this reason, we solve the problem from the beginning. We assume that the body rotates about its fixed point in a Newtonian force field and construct the equations of the motion for this case when ω = 1 / 3 . We use a new procedure for solving this problem from the beginning using a large parameter ε that depends on a sufficiently small angular velocity component r o . Applying this procedure, we derive the periodic solutions of the problem and investigate the geometric interpretation of motion. The obtained analytical solutions graphically are presented using programmed data. Using the fourth-order Runge-Kutta method, we find the numerical solutions for this case aimed at determining the errors between both obtained solutions.

scholarly journals The Motion of a Rigid Body with Irrational Natural Frequency

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
A. I. Ismail
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Geometric Interpretation ◽  
Minor Axis ◽  
The Body ◽  
Large Parameter ◽  
Small Component ◽  
Runge Kutta Method ◽  
The Stability ◽  
Euler’S Angles

In this paper, we consider the problem of the rotational motion of a rigid body with an irrational value of the frequency ω . The equations of motion are derived and reduced to a quasilinear autonomous system. Such system is reduced to a generating one. We assume a large parameter μ proportional inversely with a sufficiently small component r o of the angular velocity which is assumed around the major or the minor axis of the ellipsoid of inertia. Then, the large parameter technique is used to construct the periodic solutions for such cases. The geometric interpretation of the motion is obtained to describe the orientation of the body in terms of Euler’s angles. Using the digital fourth-order Runge-Kutta method, we determine the digital solutions of the obtained system. The phase diagram procedure is applied to study the stability of the attained solutions. A comparison between the considered numerical and analytical solutions is introduced to show the validity of the presented techniques and solutions.

scholarly journals The Slow Spinning Motion of a Rigid Body in Newtonian Field and External Torque

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
The Body ◽  
Slow Rotation ◽  
Large Parameter ◽  
Angular Velocity Vector ◽  
External Torque ◽  
Comparison Of The Results ◽  
Small Angular Velocity ◽  
Newtonian Force Field

In this paper, the problem of the slow spinning motion of a rigid body about a point O, being fixed in space, in the presence of the Newtonian force field and external torque is considered. We achieve the slow spin by giving the body slow rotation with a sufficiently small angular velocity component r 0 about the moving z-axis. We obtain the periodic solutions in a new domain of the angular velocity vector component r 0 ⟶ 0 , define a large parameter proportional to 1 / r 0 , and use the technique of the large parameter for solving this problem. Geometric interpretations of motions will be illustrated. Comparison of the results with the previous works is considered. A discussion of obtained solutions and results is presented.

scholarly journals New Treatment of the Rotary Motion of a Rigid Body with Estimated Natural Frequency

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
A. I. Ismail
Keyword(s):  
Rigid Body ◽  
Natural Frequency ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Geometric Interpretation ◽  
Parameter Method ◽  
Large Parameter ◽  
Regular Precession ◽  
New Treatment ◽  
The Stability

In this paper, the problem of the motion of a rigid body about a fixed point under the action of a Newtonian force field is studied when the natural frequency ω = 0.5 . This case of singularity appears in the previous works and deals with different bodies which are classified according to the moments of inertia. Using the large parameter method, the periodic solutions for the equations of motion of this problem are obtained in terms of a large parameter, which will be defined later. The geometric interpretation of the considered motion will be given in terms of Euler’s angles. The numerical solutions for the system of equations of motion are obtained by one of the well-known numerical methods. The comparison between the obtained numerical solutions and analytical ones is carried out to show the errors between them and to prove the accuracy of both used techniques. In the end, we obtain the case of the regular precession type as a special case. The stability of the motion is considered by the phase diagram procedures.

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.

On the Harmonic Oscillations for the Motion of a Dynamical System

2017 ◽  
Vol 13 (2) ◽  
pp. 4657-4670
Author(s):  
W. S. Amer
Keyword(s):  
Dynamical System ◽  
Numerical Solutions ◽  
Equation Of Motion ◽  
Parameter Method ◽  
Kutta Method ◽  
Small Parameter Method ◽  
Harmonic Oscillations ◽  
Pivot Point ◽  
Runge Kutta Method ◽  
High Consistency

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

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.

Numerical Solution of the Flow Between Two Disks

1997 ◽  
Author(s):  
Wahid S. Ghaly ◽  
Georgios H. Vatistas
Keyword(s):  
Shooting Method ◽  
Static Pressure ◽  
Numerical Solutions ◽  
Kutta Method ◽  
Initial Value ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Pressure Distributions ◽  
Runge Kutta Method

Abstract This paper deals with the numerical solutions of converging and diverging flows, between two disks. The results are obtained by solving a nonlinear third order ordinary differential equation using a modified shooting method. The governing equation is written as a system of three nonlinear first order ODE’s and the resulting system is solved as an initial value problem via the Runge-Kutta method. The results are given in terms of velocity profiles and static pressure distributions. These are compared with previously reported experimental data obtained by others.

scholarly journals Study on Banded Implicit Runge–Kutta Methods for Solving Stiff Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
M. Y. Liu ◽  
L. Zhang ◽  
C. F. Zhang
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Coefficient Matrix ◽  
Kutta Method ◽  
Stiff Differential Equations ◽  
Banded Matrix ◽  
Fully Implicit ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Cost

The implicit Runge–Kutta method with A-stability is suitable for solving stiff differential equations. However, the fully implicit Runge–Kutta method is very expensive in solving large system problems. Although some implicit Runge–Kutta methods can reduce the cost of computation, their accuracy and stability are also adversely affected. Therefore, an effective banded implicit Runge–Kutta method with high accuracy and high stability is proposed, which reduces the computation cost by changing the Jacobian matrix from a full coefficient matrix to a banded matrix. Numerical solutions and results of stiff equations obtained by the methods involved are compared, and the results show that the banded implicit Runge–Kutta method is advantageous to solve large stiff problems and conducive to the development of simulation.

A Theoretical and Numerical Study of the Dzhanibekov and Tennis Racket Phenomena1

2016 ◽  
Vol 83 (11) ◽  
Author(s):  
Hidenori Murakami ◽  
Oscar Rios ◽  
Thomas Joseph Impelluso
Keyword(s):  
Angular Momentum ◽  
Numerical Study ◽  
The Body ◽  
Moment Of Inertia ◽  
Kutta Method ◽  
Principal Moment ◽  
Tennis Racket ◽  
Angular Momenta ◽  
Runge Kutta Method ◽  
Principal Axes

This paper presents a complete explanation of the Dzhanibekov and the tennis racket phenomena. These phenomena are described by Euler's equation for an unconstrained rigid body that has three distinct moment of inertia values. In the two phenomena, the rotations of a body about the principal axes that correspond to the largest and the smallest moments of inertia are stable. However, the rotation about the axis corresponding to the intermediate principal moment of inertia becomes unstable, leading to the unexpected rotations that are the basis of the phenomena. If this unexpected rotation is not explained from a complete perspective which accounts for the relevant physical and mathematical aspects, one might misconstrue the phenomena as a violation of the conservation of angular momenta. To address this, the phenomenon is investigated using more precise mathematical and graphical tools than those employed previously. The torque-free Euler equations are integrated using the fourth-order Runge–Kutta method. Then, a recovery equation is applied to obtain the rotation matrix for the body. By combining the geometrical solutions with numerical simulations, the unexpected rotations observed in the Dzhanibekov and the tennis racket experiments are shown to preserve the conservation of angular momentum.

scholarly journals Static Kirchhoff Rods under the Action of External Forces: Integration via Runge-Kutta Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Ademir L. Xavier Jr.
Keyword(s):  
Reference System ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Kutta Method ◽  
Kirchhoff Equations ◽  
Runge Kutta Method ◽  
System Equations ◽  
Kirchhoff Rods ◽  
Runge Kutta ◽  
External Forces

This paper shows how to apply a simple Runge-Kutta algorithm to get solutions of Kirchhoff equations for static filaments subjected to arbitrary external and static forces. This is done by suitably integrating at once Kirchhoff and filament reference system equations under appropriate initial boundary conditions. To show the application of the method, we display several numerical solutions for filaments including cases showing the effect of gravity.

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