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 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 Solving a Problem of Rotary Motion for a Heavy Solid Using the Large Parameter Method

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Angular Velocity ◽  
Small Parameter ◽  
Initial Conditions ◽  
Geometric Interpretation ◽  
Rigid Bodies ◽  
Parameter Method ◽  
Large Parameter ◽  
Small Parameter Method ◽  
Rotational Motions ◽  
The Stability

The small parameter method was applied for solving many rotational motions of heavy solids, rigid bodies, and gyroscopes for different problems which classify them according to certain initial conditions on moments of inertia and initial angular velocity components. For achieving the small parameter method, the authors have assumed that the initial angular velocity is sufficiently large. In this work, it is assumed that the initial angular velocity is sufficiently small to achieve the large parameter instead of the small one. In this manner, a lot of energy used for making the motion initially is saved. The obtained analytical periodic solutions are represented graphically using a computer program to show the geometric periodicity of the obtained solutions in some interval of time. In the end, the geometric interpretation of the stability of a motion is given.

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 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 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.

Overstability of a Viscoelastic Fluid Layer Oscillating in a Vertical Periodic Motion and Heated From Below

1979 ◽  
Vol 46 (2) ◽  
pp. 454-456
Author(s):  
S. O. Onyegegbu
Keyword(s):  
Natural Frequency ◽  
Viscoelastic Fluid ◽  
Periodic Motion ◽  
Oscillation Frequency ◽  
Numerical Solutions ◽  
Fluid Layer ◽  
The Stability ◽  
Oscillation Frequencies

This Note examines the effect of vertical periodic motion on the stability characteristics of a viscoelastic fluid layer in a classical Benard geometry. Numerical solutions show that a resonant type behavior which enhances stability occurs at oscillation frequencies near the convective natural frequency of the viscoelastic fluid, while the effect of the periodic motion vanishes as the oscillation frequency gets very large.

scholarly journals A unified approach to rigid body rotational dynamics and control

2011 ◽  
Vol 468 (2138) ◽  
pp. 395-414 ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Aaron D. Schutte
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Equations Of Motion ◽  
Fundamental Equation ◽  
Rotational Dynamics ◽  
Constrained Motion ◽  
Dynamics And Control ◽  
Common Framework ◽  
The Stability ◽  
And Control

This paper develops a unified methodology for obtaining both the general equations of motion describing the rotational dynamics of a rigid body using quaternions as well as its control. This is achieved in a simple systematic manner using the so-called fundamental equation of constrained motion that permits both the dynamics and the control to be placed within a common framework. It is shown that a first application of this equation yields, in closed form, the equations of rotational dynamics, whereas a second application of the self-same equation yields two new methods for explicitly determining, in closed form, the nonlinear control torque needed to change the orientation of a rigid body. The stability of the controllers developed is analysed, and numerical examples showing the ease and efficacy of the unified methodology are provided.

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.

scholarly journals TO THE QUESTION OF ANALYSIS OF EQUATIONS OF MOTION OF A RIGID BODY DURING THE MECHANICAL OSCILATIONS

2021 ◽  
Author(s):  
Igor Dobrov ◽  
Andriy Semichev
Keyword(s):  
Differential Equation ◽  
Rigid Body ◽  
Natural Frequency ◽  
Equations Of Motion ◽  
Oscillation Amplitude ◽  
Current Position ◽  
Homogeneous Differential Equation ◽  
Inhomogeneous Differential Equation ◽  
Oscillation Process ◽  
Undamped Oscillations

 Depending on the current position of the mass in different areas of the spring deformation during the oscillation process the values that determines the natural frequency of free continuous oscillations have opposite signs. It is defined by the change in the direction of acceleration of the mass in these areas, which makes it possible to determine a single inhomogeneous differential equation of the oscillation process in different areas of the movement of the mass. When the oscillation amplitude is much less than the static position of the mass, this inhomogeneous differential equation represents a homogeneous differential equation of free undamped oscillations.

Dynamics of Axially Accelerating Beams With an Intermediate Support

2011 ◽  
Vol 133 (3) ◽  
Author(s):  
S. M. Bağdatli ◽  
E. Özkaya ◽  
H. R. Öz
Keyword(s):  
Natural Frequency ◽  
Axial Velocity ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Intermediate Support ◽  
Approximate Analytical Solutions ◽  
Fluctuation Frequency ◽  
Simple Supports ◽  
The Stability ◽  
Euler Bernoulli Beam

The transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on simple supports are investigated. The supports are at the ends, and there is a support in between. The axial velocity is a sinusoidal function of time varying about a constant mean speed. Since the supports are immovable, the beam neutral axis is stretched during the motion, and hence, nonlinear terms are introduced to the equations of motion. Approximate analytical solutions are obtained using the method of multiple scales. Natural frequencies are obtained for different locations of the support other than end supports. The effect of nonlinear terms on natural frequency is calculated for different parameters. Principal parametric resonance occurs when the velocity fluctuation frequency is equal to approximately twice of natural frequency. By performing stability analysis of solutions, approximate stable and unstable regions were identified. Effects of axial velocity and location of intermediate support on the stability regions have been investigated.

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