The third order equations of motion in the fast motion approach in general relativity

1982 ◽  
Vol 93 (1) ◽  
pp. 11-14
Author(s):  
Arnold Rosenblum
Keyword(s):  
General Relativity ◽  
Equations Of Motion ◽  
Third Order ◽  
The Third ◽  
Fast Motion

scholarly journals A new pendulum motion with a suspended point near infinity

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Degrees Of Freedom ◽  
Angular Displacement ◽  
Equations Of Motion ◽  
Circular Path ◽  
Large Parameter ◽  
Pendulum Motion ◽  
Third Order ◽  
The Third ◽  
Lagrange’S Equation ◽  
Pendulum Model

AbstractIn this paper, a pendulum model is represented by a mechanical system that consists of a simple pendulum suspended on a spring, which is permitted oscillations in a plane. The point of suspension moves in a circular path of the radius (a) which is sufficiently large. There are two degrees of freedom for describing the motion named; the angular displacement of the pendulum and the extension of the spring. The equations of motion in terms of the generalized coordinates $$\varphi$$ φ and $$\xi$$ ξ are obtained using Lagrange’s equation. The approximated solutions of these equations are achieved up to the third order of approximation in terms of a large parameter $$\varepsilon$$ ε will be defined instead of a small one in previous studies. The influences of parameters of the system on the motion are obtained using a computerized program. The computerized studies obtained show the accuracy of the used methods through graphical representations.

scholarly journals Bifurcation analysis of rotor/bearing system using third-order journal bearing stiffness and damping coefficients

2021 ◽  
Author(s):  
T. A. El-Sayed ◽  
Hussein Sayed
Keyword(s):  
Equations Of Motion ◽  
Journal Bearings ◽  
Third Order ◽  
Damping Coefficients ◽  
The Third ◽  
Rotor Bearing ◽  
Bearing Stiffness ◽  
Rotor Stiffness ◽  
Stiffness And Damping ◽  
Bearing System

AbstractHydrodynamic journal bearings are used in many applications which involve high speeds and loads. However, they are susceptible to oil whirl instability, which may cause bearing failure. In this work, a flexible Jeffcott rotor supported by two identical journal bearings is used to investigate the stability and bifurcations of rotor bearing system. Since a closed form for the finite bearing forces is not exist, nonlinear bearing stiffness and damping coefficients are used to represent the bearing forces. The bearing forces are approximated to the third order using Taylor expansion, and infinitesimal perturbation method is used to evaluate the nonlinear bearing coefficients. The mesh sensitivity on the bearing coefficients is investigated. Then, the equations of motion based on bearing coefficients are used to investigate the dynamics and stability of the rotor-bearing system. The effect of rotor stiffness ratio and applied load on the Hopf bifurcation stability and limit cycle continuation of the system are investigated. The results of this work show that evaluating the bearing forces using Taylor’s expansion up to the third-order bearing coefficients can be used to profoundly investigate the rich dynamics of rotor-bearing systems.

Elasticity in general relativity

1963 ◽  
Vol 272 (1348) ◽  
pp. 44-53 ◽  
Keyword(s):  
General Relativity ◽  
Elasticity Theory ◽  
Elastic Body ◽  
Equations Of Motion ◽  
Rigid Motion ◽  
Theory Of Elasticity ◽  
Body Motion ◽  
Einstein Tensor ◽  
Third Order ◽  
The One

This paper develops a new theory of elasticity in general relativity distinct from the one proposed by Synge (1959). In the new theory, stress is related to strain by a formula analogous to Hooke’s law, whereas Synge’s theory is formulated in terms of rates of change of stress and strain. Special features of the new theory are: (1) With any elastic body motion there is a uniquely associated rigid motion (in the Born sense), which plays a role analogous to that of the rigid body in the ordinary elasticity theory. (2) The 4-velocity of matter is an eigenvector of the Einstein tensor. (3) At each point of an elastic body there are at most 21 independent elastic coefficients. (4) The differential equations of motion of an elastic body are of the second order. (In Synge’s theory the corresponding equations are of the third order.) (5) As in Synge’s theory, shock waves travel with the same speeds as occur in ordinary elasticity theory. In the formulation of the new theory, consistent use is made of Lie derivatives.

scholarly journals Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory

2016 ◽  
Vol 38 (2) ◽  
pp. 103-122 ◽  
Author(s):  
Pham Tien Dat ◽  
Do Van Thom ◽  
Doan Trac Luat
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
The Finite Element Method ◽  
Power Law Distribution ◽  
Third Order ◽  
The Third

In this paper, the free vibration of functionally  sandwich grades plates with stiffeners is investigated by using the finite  element method. The material properties are assumed to be graded in the  thickness direction by a power-law distribution. Based on the third-order  shear deformation theory, the governing equations of motion are derived from  the Hamilton's principle. A parametric study is carried out to highlight the  effect of material distribution, stiffener parameters on the free  vibration characteristics of the plates.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

The post-Newtonian approximation

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Body Problem ◽  
Equations Of Motion ◽  
Two Body Problem ◽  
Newtonian Approximation ◽  
Actual Motion ◽  
Law Of Attraction ◽  
Universal Law ◽  
Two Bodies

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.

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