Centrifugal waves

1960 ◽  
Vol 7 (3) ◽  
pp. 340-352 ◽  
Author(s):  
O. M. Phillips
Keyword(s):  
Free Surface ◽  
Angular Velocity ◽  
Circular Cylinder ◽  
Rigid Body ◽  
Water Depth ◽  
Frequency Equation ◽  
Rigid Rotation ◽  
Body Motion ◽  
Small Perturbations ◽  
The Stability

When a hollow circular cylinder with its axis horizontal is partially filled with water and rotated rapidly about its axis, an almost rigid-body motion results with an interior free surface. The emotion is analysed assuming small perturbations to a rigid rotation, and a criterion is found for the stability of the motion. This is confirmed experimentally under varying conditions of water depth and angular velocity of the cylinder. The modes of oscillation (centrifugal waves) of the free surface are examined and a frequency equation deduced. Two particular modes are considered in detail, and satisfactory agreement is found with the frequencies observed.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

1985 ◽  
Vol 52 (3) ◽  
pp. 686-692 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Classical Problem ◽  
Moment Of Inertia ◽  
Model Parameters ◽  
Stability Chart ◽  
The Stability ◽  
Transition Curves ◽  
Minimum Moment ◽  
Small Angular Velocity

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

On some exact solutions in magnetohydrodynamics with astrophysical applications

1972 ◽  
Vol 51 (1) ◽  
pp. 33-38 ◽  
Author(s):  
C. Sozou
Keyword(s):  
Free Surface ◽  
Angular Velocity ◽  
Incompressible Fluid ◽  
Exact Solutions ◽  
Equilibrium Configuration ◽  
Constant Angular Velocity ◽  
Magnetohydrodynamic Equations ◽  
The Stability

Some exact solutions of the steady magnetohydrodynamic equations for a perfectly conducting inviscid self-gravitating incompressible fluid are discussed. It is shown that there exist solutions for which the free surface of the liquid is that of a planetary ellipsoid and rotates with constant angular velocity about its axis. The stability of the equilibrium configuration is not investigated.

Position and Attitude Tracking Control Law Design Using Geometric Algebra

2014 ◽  
Vol 602-605 ◽  
pp. 1113-1116
Author(s):  
Di Min Wu ◽  
Zhen Jing Li ◽  
Bin Li ◽  
Yu Xia Chen ◽  
Li Li
Keyword(s):  
Rigid Body ◽  
Geometric Algebra ◽  
Tracking Control ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Control Law ◽  
Proportional Control ◽  
Attitude Tracking ◽  
Attitude Tracking Control ◽  
The Stability

A position and attitude tracking control law is developed using geometric algebra (GA). The rigid body motion can be represented by the screw versor (or motor) in GA. Using the kinematics of the motor, the tracking control law of the rigid body motion can be formulated similar to the proportional control law. This paper provides a GA-based position and attitude tracking control law by using the negative feedback of the motor logarithm. The stability of the control law is validated by the numerical simulation.

FLUTTER STABILITY BEHAVIOR OF FREE–FREE COLUMN WITH A MOVABLE MASS SUBJECTED TO FOLLOWER DRAG FORCE

2006 ◽  
Vol 06 (02) ◽  
pp. 215-231 ◽  
Author(s):  
D. L. PRABHAKARA ◽  
M. D. DHARSHAN ◽  
U. J. UTHPAL
Keyword(s):  
Rigid Body ◽  
Frictional Resistance ◽  
Element Model ◽  
Equation Of Motion ◽  
Body Motion ◽  
Frictional Drag ◽  
Stability Behavior ◽  
Design Engineers ◽  
Governing Differential Equation ◽  
The Stability

The stability behavior of a prismatic free–free column, carrying a movable mass and subjected to the action of follower end thrust and follower frictional resistance has been investigated. A finite element model of the beam is formulated using the modified Hamilton's principle. The eigenvalues of the characteristic equation of the governing differential equation of motion are extracted using the Q-R algorithm. The parameters considered in the investigation include: (i) ratio of the movable mass to the total mass, (ii) position of the movable mass, and (iii) ratio of resultant frictional drag to the magnitude of the end thrust. Some of the interesting results conducted in this study may be useful to the design engineers. Depending on the various parameters used, the column may lose stability either by divergence or by flutter. The first two frequencies vanish when the column is subjected to only an end thrust without any extra mass and/or frictional drag, indicating the occurrence of both the rigid body translation and rotation. However, only one vanishing frequency is observed when the column is carrying a movable mass and is subjected to a frictional drag (irrespective of their values), indicating the occurrence of only one rigid body motion.

A Linear Algebra Approach to the Analysis of Rigid Body Velocity From Position and Velocity Data

1983 ◽  
Vol 105 (2) ◽  
pp. 92-95 ◽  
Author(s):  
A. J. Laub ◽  
G. R. Shiflett
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Matrix Theory ◽  
The Body ◽  
Instantaneous Velocity ◽  
Body Motion ◽  
Translational Velocity ◽  
Velocity Data ◽  
Algebra Approach ◽  
Body Velocity

The instantaneous velocity of a rigid body in space is characterized by an angular and translational velocity. By representing the angular velocity as a matrix and the translational component as a vector the velocity of any point in the rigid body may be found if the position of the point and the parameters of the angular and translational velocities are known. Alternatively, the parameters of the rigid body velocity may be determined if the velocity and position of three points fixed in the body are known. In this paper, a new matrix-theory-based method is derived for determining the instantaneous velocity parameters of rigid body motion in terms of the velocity and position of three noncollinear points fixed in the body. The method is shown to possess certain advantages over traditional vectoral solutions to the same problem.

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