scholarly journals On New Modifications of Some Perturbation Procedures

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
A. I. Ismail
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Velocity Vector ◽  
The Body ◽  
Large Parameter ◽  
Angular Velocity Vector ◽  
Rigid Body Dynamic ◽  
Motion Problem ◽  
Slow Motions ◽  
Body Dynamic

In this paper, we present new modifications for some perturbation procedures used in mathematics, physics, astronomy, and engineering. These modifications will help us to solve the previous problems in different sciences under new conditions. As problems, we have, for example, the rotary rigid body problem, the gyroscopic problem, the pendulum motion problem, and other ones. These problems will be solved in a new manner different from the previous treatments. We solve some of the previous problems in the presence of new conditions, new analysis, and new domains. We let complementary conditions of such studied previously. We solve these problems by applying the large parameter technique used by assuming a large parameter which inversely proportional to a small quantity. For example, in rigid body dynamic problems, we take such quantity to be one of the components of the angular velocity vector in the initial instant of the rotary body about a fixed point. The domain of our solutions will be depending on the choice of a large parameter. The problem of slow (weak) oscillations is considered. So, we obtain slow motions of the bodies instead of fast motions and find the solutions of the problem in present new conditions on both of center of gravity, moments of inertia, and the angular velocity vector or one of these parameters of the body. This study is important for aerospace engineering, gyroscopic motions, satellite motion which has the correspondence of inertia moments, antennas, and navigations.

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.

Sensor Placement for Angular Velocity Determination

2005 ◽  
Vol 128 (3) ◽  
pp. 543-547 ◽  
Author(s):  
Guy M. Genin ◽  
Joseph Genin
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Velocity Vector ◽  
Constraint Equation ◽  
Complete Characterization ◽  
Angular Velocity Vector ◽  
Algebraic Constraint ◽  
Transducer Placement

Velocity transducer placement to uniquely determine the angular velocity of a rigid body is investigated. The angular velocity of a rigid body can be determined with no fewer than five properly placed velocity transducers, if no other types of sensors are present and no algebraic constraint equation involving the angular velocity vector can be written. Complete characterization of the velocity of a rigid body requires six transducers. Choice of transducer placement and orientation requires care, as suboptimal transducer placement can result in data from which the determination of a unique angular velocity vector is impossible. Conditions for successful transducer placement are established, and two examples of adequate transducer placement are presented: an Earth-penetrating projectile, and a bioengineering device for the measurement of head motion.

Algorithm of the Time-Optimal Reorientation of an Axially Symmetric Spacecraft in the Class of Conical Motions

2018 ◽  
Vol 19 (12) ◽  
pp. 10-17587/mau.19.797-805
Author(s):  
Ya. G. Sapunkov ◽  
A. V. Molodenkov ◽  
T. V. Molodenkova
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Velocity Vector ◽  
Space Shuttle ◽  
Control Function ◽  
Axially Symmetric ◽  
Bounded Control ◽  
Angular Velocity Vector ◽  
Time Optimal ◽  
Optimal Turn

The problem of the time-optimal turn of a spacecraft as a rigid body with one axis of symmetry and bounded control function in absolute value is considered in the quaternion statement. For simplifying problem (concerning dynamic Euler equations), we change the variables reducing the original optimal turn problem of axially symmetric spacecraft to the problem of optimal turn of the rigid body with spherical mass distribution including one new scalar equation. Using the Pontryagin maximum principle, a new analytical solution of this problem in the class of conical motions is obtained. Algorithm of the optimal turn of a spacecraft is given. An explicit expression for the constant in magnitude optimal angular velocity vector of a spacecraft is found. The motion trajectory of a spacecraft is a regular precession. The conditions for the initial and terminal values of a spacecraft angular velocity vector are formulated. These conditions make it possible to solve the problem analytically in the class of conical motions. The initial and the terminal vectors of spacecraft angular velocity must be on the conical surface generated by arbitrary given constant conditions of the problem. The numerical example is presented. The example contain optimal reorientation of the Space Shuttle in the class of conical motions.

scholarly journals On the Geodetic Rotation of the Major Planets, the Moon and the Sun

2009 ◽  
Vol 44 (2) ◽  
pp. 43-52
Author(s):  
G. Eroshkin ◽  
V. Pashkevich
Keyword(s):  
Angular Velocity ◽  
Reference Frame ◽  
Velocity Vector ◽  
Previous Investigation ◽  
The Body ◽  
The Other ◽  
The Sun ◽  
The Moon ◽  
Angular Velocity Vector

On the Geodetic Rotation of the Major Planets, the Moon and the SunThe problem of the geodetic (relativistic) rotation of the major planets, the Moon and the Sun was studied in the paper by Eroshkin and Pashkevich (2007) only for the components of the angular velocity vectors of the geodetic rotation, which are orthogonal to the plane of the fixed ecliptic J2000. This research represents an extension of the previous investigation to all the other components of the angular velocity vector of the geodetic rotation, with respect to the body-centric reference frame from Seidelmann et al. (2005).

Investigating Rigidity of Military Vehicle Body Using Operating Deflection Shape (ODS) Analysis

2006 ◽  
Vol 49 (2) ◽  
pp. 16-24 ◽  
Author(s):  
Mark Bounds ◽  
George White
Keyword(s):  
Rigid Body ◽  
Dynamic Models ◽  
The Body ◽  
Vehicle Body ◽  
Rigid Body Dynamic ◽  
Military Vehicle ◽  
Acceleration Data ◽  
Specific Frequency ◽  
Body Dynamic ◽  
Insight Into

The Army has many rigid-body dynamic models of various vehicle platforms. The adequacy of these rigid-body models has been questioned. In an effort to gain insight into the significance of flexibility in the development of dynamic vehicle models, operating deflection shape (ODS) techniques were applied to acceleration data gathered from the body of a wheeled military vehicle. The data were analyzed in an effort to determine a specific frequency range over which the assumption of rigidity would be valid. For the particular platform examined in this study, the assumption of rigidity would apply up to approximately 14 Hz. Future efforts include using operational modal analysis (OMA) to further examine the problem.

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 SOME PROBLEMS ABOUT THE MOTION OF A RIGID BODY IN A RESISTIVE MEDIUM

2021 ◽  
Vol 3 (2) ◽  
pp. 6-17
Author(s):  
D. Leshchenko ◽  
◽  
T. Kozachenko ◽  
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Modern Technology ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
The Body ◽  
Body Motion ◽  
Body Rotation ◽  
Angular Velocity Vector ◽  
Resistive Medium

The dynamics of rotating rigid bodies is a classical topic of study in mechanics. In the eighteenth and nineteenth centuries, several aspects of a rotating rigid body motion were studied by famous mathematicians as Euler, Jacobi, Poinsot, Lagrange, and Kovalevskya. However, the study of the dynamics of rotating bodies of still important for aplications such as the dynamics of satellite-gyrostat, spacecraft, re-entry vehicles, theory of gyroscopes, modern technology, navigation, space engineering and many other areas. A number of studies are devoted to the dynamics of a rigid body in a resistive medium. The presence of the velocity of proper rotation of the rigid body leads to the apearance of dissipative torques causing the braking of the body rotation. These torques depend on the properties of resistant medium in which the rigid body motions occur, on the body shape, on the properties of the surface of the rigid body and the distribution of mass in the body and on the characters of the rigid body motion. Therefore, the dependence of the resistant torque on the orientation of the rigid body and its angular velocity can de quite complicated and requires consideration of the motion of the medium around the body in the general case. We confine ourselves in this paper to some simple relations that can qualitative describe the resistance to rigid body rotation at small angular velocities and are used in the literature. In setting up the equations of motion of a rigid body moving in viscous medium, we need to consider the nature of the resisting force generated by the motion of the rigid body. The evolution of rotations of a rigid body influenced by dissipative disturbing torques were studied in many papers and books. The problems of motion of a rigid body about fixed point in a resistive medium described by nonlinear dynamic Euler equations. An analytical solution of the problem when the torques of external resistance forces are proportional to the corresponding projections of the angular velocity of the rigid body is obtain in several works. The dependence of the dissipative torque of the resistant forces on the angular velocity vector of rotation of the rigid body is assumed to be linear. We consider dynamics of a rigid body with arbitrary moments of inertia subjected to external torques include small dissipative torques.

A Study on Mixed Kinetic-Kinematic Equations for Multibody Systems

2013 ◽  
Author(s):  
Sung-Soo Kim ◽  
Bongcheol Seo ◽  
Myungho Kim
Keyword(s):  
Angular Velocity ◽  
Velocity Vector ◽  
Reference Frames ◽  
Time Derivative ◽  
Kinetic Equations ◽  
Angular Acceleration ◽  
The Body ◽  
Integration Step ◽  
Angular Velocity Vector ◽  
The Relationship

In this paper, mixed kinetic-kinematic equations for a multibody system have been studied in order to resolve the difficulties of non-integrability of angular velocity vectors. As for the kinetic equations, the Newton-Euler equations of motion are considered. They are derived in terms of angular velocity and angular acceleration vectors expressed in the body fixed reference frames. As for the kinematic compatibility equations, two different equations are considered. One is from the relationship between the angular velocity vector and the time derivatives of Euler parameters. The other is from the relationship between the rotational orientation matrix, its time derivative, and the angular velocity vector. In order to investigate the accuracy of the solution methods using two different kinematic compatibility equations, simulations of a spherical pendulum model and a 1/6 robot vehicle model have been carried out. With different integration step-sizes, the constraint violation errors have been also investigated.

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