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.

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

2015 ◽  
Author(s):  
Hidenori Murakami ◽  
Oscar Rios ◽  
Thomas J. Impelluso
Keyword(s):  
Numerical Study ◽  
Space Station ◽  
The Body ◽  
Moving Frame ◽  
Coordinate Frame ◽  
Moments Of Inertia ◽  
Tennis Racket ◽  
Angular Momenta ◽  
Principal Axes ◽  
Moving Coordinate

In this paper, we present complete explanation of the Dzhanibekov phenomenon demonstrated in a space station (www.youtube.com/watch?v=L2o9eBl_Gzw) and the tennis racket phenomenon (www.youtube.com/watch?v=4dqCQqI-Gis). These phenomena are described by Euler’s equation of an unconstrained rigid body that has three distinct values of moments of inertia. 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, especially for students, we investigate the phenomena using more precise mathematical and graphical tools than those employed previously. Following Élie Cartan [1], we explicitly write the vector basis of a body-attached, moving coordinate system. Using this moving frame method, we describe the Newton and Euler equations. The adoption of the moving coordinate frame expresses the rotation of the body more clearly and allows us to use the Lie group theory of special orthogonal group SO(3). We integrate the torque-free Euler equation using the fourth-order Runge-Kutta method. Then we apply a recovery equation to obtain the rotation matrix for the body. By combining the geometrical solutions with numerical simulations, we demonstrate that the unexpected rotations observed in the Dzhanibekov and the tennis racket experiments preserve the conservation of angular momentum.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

scholarly journals Numerical study of coupled oscillator system using the classical Euler-Lagrange equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
H. Shanak ◽  
H. Khalilia ◽  
R. Jarrar ◽  
J. Asad
Keyword(s):  
Coupled Oscillators ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Lagrangian Method ◽  
Order Equation ◽  
Kutta Method ◽  
Lagrange Equations ◽  
Coupled Oscillator ◽  
Oscillator System ◽  
Runge Kutta Method

Abstract Problems involving vibrations (mechanical or electrical) can be reduced to problems of coupled oscillators. For this, we consider the motion of coupled oscillators system using Lagrangian method. The Lagrangian of the system was initially constructed, and then the Euler-Lagrange equations (i.e., equations of motion of the system) have been obtained. The obtained equations of motion are a homogenous second-order equation. These equations were solved numerically using the ode45 code, which is based on Runge-Kutta method.

scholarly journals Real-Time Cloth Simulation for Hula Costume CAD Development

2021 ◽  
Vol 10 (3) ◽  
pp. 85-89
Author(s):  
Taketo Kamasaka ◽  
◽  
Kodai Miyamoto ◽  
Makoto Sakamoto ◽  
◽  
...  
Keyword(s):  
High Performance ◽  
Euler Method ◽  
The Body ◽  
Spring Model ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Mass Spring Model ◽  
Simulation Results ◽  
Mass Spring ◽  
Very High

In recent years, 3D computer graphics (3DCG) technology has been applied in various fields such as AR technology, VR technology, movies, games, and virtual fitting of clothes. Among them, there is the problem of contact between clothing and other objects (such as the body). In this paper, we focus on that problem. Since this research is part of the development of CAD for the design of hula costumes, and since we assume that the users of the CAD will have PCs with not very high performance, we took an approach that does not require a large amount of computation. As a representation of the cloth, we used a mass-spring model in which springs and mass points are connected in a grid. We also compared how the three methods of calculating position and velocity, Euler method, FB Euler method, and Runge-Kutta method, affect the simulation results.

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.

PHENOMENOLOGY OF NUCLEI AT VERY HIGH ANGULAR MOMENTA USING PARAMETRIZED TWO-CENTER NUCLEAR SHAPES

1995 ◽  
Vol 04 (04) ◽  
pp. 789-800
Author(s):  
RAJ K. GUPTA ◽  
SHAM S. MALIK ◽  
J.S. BATRA ◽  
PETER O. HESS ◽  
WERNER SCHEID
Keyword(s):  
Angular Momentum ◽  
Deformation Energy ◽  
Moment Of Inertia ◽  
Yrast Band ◽  
Spin States ◽  
High Spin States ◽  
Angular Momenta ◽  
Forward Bending ◽  
Very High

The nuclear shapes and variation of moment of inertia with angular momentum, as well as the limiting angular momentum carried by a nucleus at its fissioning stage, are derived from the observed data of the ground-state yrast band and quadrupole deformations of these states. The necking-in of the nuclear shapes are shown to start already at J*~14+−18+. The empirical variation of moment of inertia with angular momentum is found to include the back-bending and forward-bending effects and supports the nuclear softness model of the nucleus. The fission of nuclei is shown to occur at very high angular momenta, which is different for different nuclei. The role of deformation energy is analyzed and the possibility of predicting the quadrupole deformations, or B(E2) transitions, for very high spin states is discussed. The calculations are presented for 156Dy, 158Er, and 164Hf.

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.

Sign in / Sign up

Export Citation Format

Share Document