Small Oscillations

1990 ◽  
pp. 33-64
Author(s):  
A. S. Monin
Keyword(s):  
Small Oscillations

scholarly journals The formation of vortices from a surface of discontinuity

1931 ◽  
Vol 134 (823) ◽  
pp. 170-192 ◽  
Keyword(s):  
Steady State ◽  
Plane Surface ◽  
Equations Of Motion ◽  
Steady Motion ◽  
Small Oscillations ◽  
Approximate Form ◽  
History Of ◽  
State Increase ◽  
Liquid Surfaces

Helmholtz was the first to remark on the instability of those “liquid surfaces” which separate portions of fluid moving with different velocities, and Kelvin, in investigating the influence of wind on waves in water, supposed frictionless, has discussed the conditions under which a plane surface of water becomes unstable. Adopting Kelvin’s method, Rayleigh investigated the instability of a surface of discontinuity. A clear and easily accessible rendering of the discussion is given by Lamb. The above investigations are conducted upon the well-known principle of “small oscillations”—there is a basic steady motion, upon which is superposed a flow, the squares of whose components of velocity can be neglected. This method has the advantage of making the equations of motion linear. If by this method the flow is found to be stable, the equations of motion give the subsequent history of the system, for the small oscillations about the steady state always remain “small.” If, however, the method indicates that the system is unstable, that is, if the deviations from the steady state increase exponentially with the time, the assumption of small motions cannot, after an appropriate interval of time, be applied to the case under consideration, and the equations of motion, in their approximate form, no longer give a picture of the flow. For this reason, which is well known, the investigations of Rayleigh only prove the existence of instability during the initial stages of the motion. It is the object of this note to investigate the form assumed by the surface of discontinuity when the displacements and velocities are no longer small.

Complete Accessibility of Oscillations in Robotic Systems by Orthogonal Projections

1990 ◽  
Vol 112 (2) ◽  
pp. 194-202 ◽  
Author(s):  
Sabri Tosunoglu ◽  
Shyng-Her Lin ◽  
Delbert Tesar
Keyword(s):  
Current Practice ◽  
Industrial Robot ◽  
Robotic Systems ◽  
Orthogonal Projections ◽  
Small Oscillations ◽  
Robotic Arms ◽  
Driven System ◽  
System Equations ◽  
Six Degree Of Freedom

The current practice of controller development for flexible robotic systems generally focuses on one-link robotic arms and is valid for small oscillations. This work addresses the control of n-link, serial, spatial robotic systems modeled with m1 joint and m2 link flexibilities such that n≥m1+m2. System compliance is modeled by local springs and nonactuated prismatic and revolute type pseudo joints. The coupled, nonlinear, error-driven system equations are derived for the complete model without linearization or neglecting certain terms. For this system, the complete accessibility of vibrations is studied by orthogonal projections. It is shown that under some configurations of a robotic system, the induced oscillations may not be accessible to the controller. Given accessibility, the controller developed in this work assures the global asymptotic stability of the system. Example numerical simulations are presented based on the model of a six-degree-of-freedom Cincinnati Milacron T3-776 industrial robot. One example models the system compliance in four joints, while another case study simulates four lateral link oscillations. These examples show that this controller, even under inaccurate payload description, eliminates the oscillations while tracking desired trajectories.

Microcosmos and Macrocosmos: A Look at these Two Universes in a Unified Way

1997 ◽  
Vol 12 (07) ◽  
pp. 1373-1384 ◽  
Author(s):  
P. R. Silva
Keyword(s):  
Strong Interaction ◽  
Gravitational Force ◽  
Inertial Mass ◽  
Hadronic Matter ◽  
Vacuum Pressure ◽  
Bag Model ◽  
Small Oscillations ◽  
Mit Bag Model ◽  
Two Universes ◽  
The Universe

An extension of the MIT bag model, developed to describe the strong interaction inside the hadronic matter (nucleons), is proposed as a means to account for the confinement of matter in the universe. The basic hypotheses of the MIT bag model are worked out in a very simplified way and are also translated in terms of the gravitational force. We call the nucleon "microcosmos" and the bag-universe "macrocosmos." We have found a vacuum pressure of 10-15 atm at the boundary of the bag-universe as compared with a pressure of 1029 atm at the boundary of the nucleon. Both universes are also analyzed in the light of Sciama's theory of inertia, which links the inertial mass of a body to its interaction with the rest of the universe. One of the consequences of this work is that the Weinberg mass can be interpreted as a threshold mass, namely the mass where the frequency of the small oscillations of a particle coupled to the universe matches its de Broglie frequency. Finally, we estimate an averaged density of matter in the universe, corresponding to [Formula: see text] of the critical or closure density.

SMALL OSCILLATIONS

Dynamics ◽  
2010 ◽  
pp. 247-259
Author(s):  
Arthur Stanley Ramsey
Keyword(s):  
Small Oscillations

scholarly journals On the possible ellipticity of Saturn's ring

1924 ◽  
Vol 106 (739) ◽  
pp. 526-552 ◽  
Keyword(s):  
Nineteenth Century ◽  
The Other ◽  
Gravitational Attraction ◽  
History Of Astronomy ◽  
Small Oscillations ◽  
First Approximation ◽  
Elliptical Ring ◽  
Single Ring ◽  
History Of ◽  
Concentric Circles

Clerk Maxwell’s investigation into the constitution and stability of Saturn’s Ring was based upon the fundamental supposition that the rings were composed of particles arranged in concentric circles. Although the particles could perform small oscillations about their mean positions, it is evident that the outline of the ring would be to all appearances circular. A line in Clerke’s ‘Popular History of Astronomy in the Nineteenth Century’ (page 299) suggests that some observers have regarded the outline of the ring as elliptical. It is therefore of interest to enquire under what conditions an elliptical ring could exist. The ellipse is a more general form than a circle. If, therefore, it can be shown that an elliptic ring is possible, any explanation of the origin of the rings must take account of this fact. In this paper a single ring of particles is examined. The actual Ring of Saturn probably consists of numerous rings, one within the other. By their gravitational attraction, each will affect the motions of the others. As a first approximation, however, it is sufficient to consider the case of a single ring.

Two-dimensional motion of a rolling non-circular cylinder on a flat horizontal surface

2018 ◽  
Vol 96 (6) ◽  
pp. 627-632
Author(s):  
Amir Aghamohammadi ◽  
Mohammad Khorrami
Keyword(s):  
Circular Cylinder ◽  
Energy Conservation ◽  
Cross Section ◽  
Equilibrium Configuration ◽  
Focal Point ◽  
Center Of Mass ◽  
Horizontal Surface ◽  
Two Dimensional ◽  
Small Oscillations ◽  
Equilibrium Configurations

The two dimensional motion of a generally non-circular non-uniform cylinder on a flat horizontal surface is investigated. Assuming that the cylinder does not slip, energy conservation is used to study the motion in general. Points of returns, and small oscillations around equilibrium configuration are studied. As examples, cylinders are studied for which the cross section is an ellipse, with the center of mass at the center of the ellipse or at a focal point, and the frequencies of small oscillations around their equilibrium configurations are found. The conditions for losing contact or sliding are also investigated. Finally, the motion is studied in more detail for the case of a nearly circular cylinder.

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