Systems with Many Degrees of Freedom—Small Oscillations and Normal Coordinates

1965 ◽  
pp. 416-477
Author(s):  
JERRY B. MARION
Keyword(s):  
Degrees Of Freedom ◽  
Small Oscillations ◽  
Normal Coordinates

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.

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.

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.

scholarly journals The Vibrations of a Particle about a Position of Equilibrium

1920 ◽  
Vol 39 ◽  
pp. 34-57 ◽  
Author(s):  
Bevan B. Baker ◽  
E. B. Ross
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Large Intensity ◽  
Normal Coordinates ◽  
Small Intensity ◽  
In Series ◽  
Image Position

1. In obtaining a solution of the differential equations corresponding to the motion of a particle about a position of equilibrium, it is usual to express the displacements in terms of a series of periodic terms, each sine or cosine having for its coefficient a series of powers of small quantities. Korteweg has discussed the general form of such solutions, and, from the developments in series which he has obtained, has deduced certain features of interest. In particular, he has shown that, under certain circumstances, it is possible that certain vibrations of higher order, which are normally of small intensity compared with the principal vibrations, may acquire an abnormally large intensity. Considering the oscillations of a dynamical system having a number of degrees of freedom, and supposing to be the frequencies corresponding to infinitesimal oscillations in the different normal coordinates, Korteweg has shown that these cases of interest arise only whenis zero or very small, where p1, p2,… are small integers, positive or negative; the most important cases occur when

scholarly journals Dynamic Reductions for the Nonlinear Soil-Foundation-Structure System Interactions

2018 ◽  
Vol 28 (1) ◽  
pp. 146-158
Author(s):  
Volodymyr O. Sakharov
Keyword(s):  
Nonlinear Analysis ◽  
Degrees Of Freedom ◽  
Reduction Method ◽  
Automated System ◽  
Explicit Method ◽  
Normal Coordinates ◽  
High Rise ◽  
Inhomogeneous Systems ◽  
Soil Foundation ◽  
Foundation Structure

Abstract The article presents new dynamic superelements for DOF (degrees of freedom) reduction. Proposed spectral superelements (SES) are based on the method of normal coordinates. The developed superelements with basis nodes (SEB) for the inhomogeneous systems are also shown. Proposed superelements ensure to obtain the diagonal matrix of masses which allow to use the efficiency of explicit method. The results of nonlinear analysis using proposed reduction method for the group of high-rise building are presented. The calculations were performed by explicit method with the help of Automated System of Scientific Research (ASSR) “VESNA-DYN”.

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.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

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