2012
An important class of problems in dynamics concerns the free vibrations of systems. (The concept of free vibrations is important; this means that although an outside agent may have participated in causing an initial displacement or velocity–or both – of the system, the outside agent plays no further role, and the subsequent motion depends only upon the inherent properties of the system. This is in contrast to ”forced ” motion in which the system is continually driven by an external force.) We shall consider only undamped systems for which the total energy is conserved and for which the frequencies of oscillation are real. This forms the basis of the approach to more complex studies for forced motion of damped systems. We saw in Lecture 13, that the free vibration of a mass-spring system could be described as an oscillatory interchange between the kinetic and potential energy, and that we could determine the natural frequency of oscillation by equating the maximum value of these two quantities. (The natural frequency is the frequency at which the system will oscillate unaffected by outside forces. When we consider the oscillation of a pendulum, the gravitational force is considered to be an inherent part of the system.) The general behavior of a mass-spring system can be extended to elastic structures and systems experiencing gravitational forces, such as a pendulum. These systems can be combined to produce complex results, even for one-degree of freedom systems. We begin our discussion with the solution of a simple mass-spring system, recognizing that this is a model
Cited by 0
John A. Knox · Encyclopedia of Atmospheric Sciences · 2003
Cited by 10
Craig A. Tovey · 1991
Abstract : Democratic simple majority voting is perhaps the most widely used method of group decision making in our time. Current theory, based on instability theorems, predicts that a group employing this method will almost always fail to reach a stable conclusion. There is one case that the classical instability theorems do not treat: an even number of voters in 2 dimensions. We resolve this remaining case, proving that instability occurs with probability converging rapidly to 1 as the population increases. But empirical observations do not support the gloomy predictions of the instability theorems. We show that the instability theorems are themselves unstable in the following sense: if the model of voter behavior is altered however slightly to incorporate any of several plausible characteristics of decision-making, then the instability theorems do not hold and in fact the probability of stability converges to 1 as the population increases, when the population is sampled from a centered distribution. The assumptions considered are: a cost of change; bounded rationality; perceptual thresholds; a cost of uncertainty; a discrete proposal space, and others. One consequence of this work is to render precise and rigorous the solution proposed by Tullock (63,64) and refined by Arrow (2) to the impossibility problem. The stability results all hold for arbitrary dimension, and generalize to establish a tradeoff between the characteristics and the degree of noncenteredness of a population. As a by-product of the analysis, we establish the statistical consistency of the sample yolk radius.
Cited by 21
Tapan K. Sengupta · 2021
Baroclinic instability of flow is of interest in multiple branches of engineering, geophysics and astrophysics. It is considered very important in explaining motion in earth's atmosphere and in the ocean. Rayleigh–Taylor instability is an example of a baroclinic instability introduced in Chapter 1, as a special case of the inviscid Kelvin– Helmholtz instability. Rayleigh–Taylor instability, in its simple connotation, arises when a heavier fluid is made to rest on top of a lighter one, as the initial equilibrium state. If the heavier and lighter fluids are perfectly aligned with the direction of gravity, then such an equilibrium state would remain stable. If an instability misaligns this configuration, then the ensuing turning moment will unabatedly destabilize the initial state. However, if the opposite initial configuration of a lighter fluid resting on top of heavier fluid is perturbed, then the turning moment will restore the equilibrium state.
Cited by 0
Giulio Sapelli · 2023
Cited by 0
A. Br, G. Rüdiger · 2008
Cited by 0
A. G. Morris, G. A. Ward, W. J. Bateman · 2013
Cited by 0
R. Huang A B, Z. Suo · 2001
Cited by 0
Reem Khalil (10898163), Sadok Kallel (12908672), Ahmad Farhat (9191878) et al. · Figshare · 2022
Cited by 0Open access
Olivier Darrigol · 2005
Abstract In the previous chapter, we encountered a special kind of instability, now called the Kelvin–Helmholtz instability, which occurs when two fluid masses slide on each other, for instance along smoke jets or on a plane water surface under wind. Helmholtz arrived at this instability by reasoning on vortex sheets and used it to explain phenomena that seemed to elude Euler’s equations. He was neither the first nor the last theorist to emphasize the role of instabilities in fluid mechanics. The nineteenth-century interest in this question was for two reasons. Firstly, the discrepancy between actual fluid behavior and known solutions of the hydrodynamic equations suggested the instability of these solutions. Secondly, the British endeavor to reduce all physics to the motion of a perfect liquid presupposed the stability of the forms of motion used to describe matter and ether. Instability in the former case and stability in the latter case needed to be proved. In nineteenth-century parlance, kinetic instability broadly meant a departure from an expected regularity of motion. In hydrodynamics alone, this notion included unsteadiness of motion, non-uniqueness of the solutions of the fundamental equations under given boundary conditions, sensibility of these solutions to infinitesimal local perturbation, sensibility to infinitesimal harmonic perturbations, sensibility to finite perturbations, and sensibility to infinitely-small viscosity. Although this spectrum of meanings is much wider than a modern treatise on hydrodynamic stability would tolerate, it must be respected in a historical study that does not artificially separate issues that nineteenth- century writers conceived as a whole.
Cited by 0
Albrecht Böttcher, Sergei M. Grudsky · Birkhäuser Basel eBooks · 2000
Cited by 0
T. W. Körner · Cambridge University Press eBooks · 1993
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Cited by 0
Pijush K. Kundu · Fluid Mechanics · 1990
Cited by 72
E. R. Priest · 2000
Cited by 0
Pijush K. Kundu, Ira M. Cohen, David R. Dowling et al. · Fluid Mechanics · 2024
Cited by 0Open access
H.‐J. Trnka, Matthias J. Feucht, L. Willinger et al. · 2024
Cited by 0
Gerald F. Gebhart, Robert F. Schmidt · 2013
Cited by 0
Jonas Pogorzelski, Matthias J. Feucht, Sepp Braun et al. · 2024
Cited by 0
Ambrogio Rebucci, Ambrogio Cesa-bianchi, Alessandro Rebucci · 2013
Die ZBW räumt Ihnen als Nutzerin/Nutzer das unentgeltliche, räumlich unbeschränkte und zeitlich auf die Dauer des Schutzrechts beschränkte einfache Recht ein, das ausgewählte Werk im Rahmen der unter
Cited by 0
2014
Cited by 0