Small oscillations of systems with several degrees of freedom

2020 ◽  
pp. 191-244
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Degrees Of Freedom ◽  
Forced Oscillations ◽  
Free Oscillations ◽  
Small Oscillations ◽  
Symmetry Properties ◽  
Gyroscopic Forces ◽  
Simple Molecules ◽  
Simple Systems ◽  
Three Degrees Of Freedom

This chapter addresses the free and forced oscillations of simple systems (with two or three degrees of freedom), the free oscillations of systems with the degenerate frequencies, and the eigen-oscillations of the electromechanical systems. This chapter also studies the oscillations of more complex systems using orthogonality of eigenoscillations and the symmetry properties of the system, the free oscillations of an anisotropic charged oscillator moving in a uniform constant magnetic field, and the perturbation theory adapted for the small oscillations. Finally, the chapter addresses oscillations of systems in which gyroscopic forces act and the eigen-oscillations of the simple molecules.

Small oscillations of systems with several degrees of freedom

2020 ◽  
pp. 29-40
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Degrees Of Freedom ◽  
Forced Oscillations ◽  
Free Oscillations ◽  
Small Oscillations ◽  
Symmetry Properties ◽  
Gyroscopic Forces ◽  
Simple Molecules ◽  
Simple Systems ◽  
Three Degrees Of Freedom

This chapter addresses the free and forced oscillations of simple systems (with two or three degrees of freedom), the free oscillations of systems with the degenerate frequencies, and the eigen-oscillations of the electromechanical systems. This chapter also studies the oscillations of more complex systems using orthogonality of eigenoscillations and the symmetry properties of the system, the free oscillations of an anisotropic charged oscillator moving in a uniform constant magnetic field, and the perturbation theory adapted for the small oscillations. Finally, the chapter addresses oscillations of systems in which gyroscopic forces act and the eigen-oscillations of the simple molecules.

Harmonic Oscillations of Nonlinear Two-Degree-of-Freedom Systems

1955 ◽  
Vol 22 (1) ◽  
pp. 107-110
Author(s):  
T. C. Huang
Keyword(s):  
Nonlinear System ◽  
Degrees Of Freedom ◽  
Forced Oscillations ◽  
Viscous Damping ◽  
Free Oscillations ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Restoring Force ◽  
Nonlinear Restoring Force ◽  
Two Degree Of Freedom

Abstract In this paper an investigation is made of equations governing the oscillations of a nonlinear system in two degrees of freedom. Analyses of harmonic oscillations are illustrated for the cases of (1) the forced oscillations with nonlinear restoring force, damping neglected; (2) the free oscillations with nonlinear restoring force, damping neglected; and (3) the forced oscillations with nonlinear restoring force, small viscous damping considered. Amplitudes of oscillations and frequency equations are derived based on the mathematically justified perturbation method. Response curves are then plotted.

The Vibrations of the Engine with Neo-Hookean Suspension

2013 ◽  
Vol 430 ◽  
pp. 53-59 ◽  
Author(s):  
Nicolae Doru Stanescu ◽  
Dinel Popa
Keyword(s):  
Degrees Of Freedom ◽  
Stable Equilibrium ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Harmonic Force ◽  
Small Oscillations ◽  
Excited System ◽  
Beat Phenomenon ◽  
Three Degrees Of Freedom

Our paper realizes a study of the vibrations of an engine excited by a harmonic force and sustained by four identical neo-Hookean springs of negligible masses. The considered model is one with three degrees of freedom (one translation and two rotations) and we obtain for it the equations of motion. Using these equations, we determine for the unexcited system the equilibrium positions and their stability. We also study the small oscillations about the stable equilibrium positions and we find the fundamental eigenpulsations of the system. For the case of the excited system we perform a numerical study considering the situation when the pulsation of the excitation is far away from the eigenpulsations and the situation when the pulsation of the excitation is closed to one eigenpulsation, highlighting the beat phenomenon.

Interactions Among Friction, Wear, and System Stiffness—Part 2: Vibrations Induced by Dry Friction

1984 ◽  
Vol 106 (1) ◽  
pp. 59-64 ◽  
Author(s):  
V. Aronov ◽  
A. F. D’Souza ◽  
S. Kalpakjian ◽  
I. Shareef
Keyword(s):  
Dry Friction ◽  
Degrees Of Freedom ◽  
Structural Model ◽  
Normal Load ◽  
Companion Paper ◽  
Random Surface ◽  
Small Oscillations ◽  
Surface Irregularities ◽  
Frictional Forces ◽  
Three Degrees Of Freedom

Different types of vibrations induced by dry friction are investigated by means of a model apparatus described in Part 1. The structural model is obtained from the measurement of the modal frequencies and damping ratios of three degrees of freedom. The oscillations in the normal and frictional forces, as well as the slider vibrations, have been measured and analyzed. As the normal load is increased, four different regions of vibrations are observed corresponding to the four friction regimes discussed in a companion paper. Small oscillations are encountered at low values of the normal load and they are possibly caused by random surface irregularities. The vibration characteristics are changed when transition occurs from steady state friction. When the normal load is further increased, self-excited periodic vibrations are produced. The spectra of the oscillations are related to the modal frequencies. Self-excited vibrations are analyzed on the basis of the experimental data.

Simple Systems

2017 ◽  
Author(s):  
Jochen Rau
Keyword(s):  
Magnetic Field ◽  
Low Temperatures ◽  
General Framework ◽  
Gibbs State ◽  
Macroscopic System ◽  
Basic Model ◽  
System A ◽  
Paramagnetic Salt ◽  
High And Low Temperatures ◽  
Simple Systems

Even though the general framework of statistical mechanics is ultimately targeted at the description of macroscopic systems, it is illustrative to apply it first to some simple systems: a harmonic oscillator, a rotor, and a spin in a magnetic field. These applications serve to illustrate how a key function associated with the Gibbs state, the so-called partition function, is calculated in practice, how the entropy function is obtained via a Legendre transformation, and how such systems behave in the limits of high and low temperatures. After discussing these simple systems, this chapter considers a first example where multiple constituents are assembled into a macroscopic system: a basic model of a paramagnetic salt. It also investigates the size of energy fluctuations and how—in the case of the paramagnet—these fluctuations scale with the number of constituents.

scholarly journals Active Disturbance Rejection for a Three Degrees of Freedom Gyroscope

2018 ◽  
Vol 51 (13) ◽  
pp. 372-377 ◽  
Author(s):  
Juan E. Andrade García ◽  
Alejandra Ferreira de Loza ◽  
Luis T. Aguilar ◽  
Ramón I. Verdés
Keyword(s):  
Disturbance Rejection ◽  
Degrees Of Freedom ◽  
Active Disturbance Rejection ◽  
Three Degrees Of Freedom
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