Conservation Principles for Systems With a Varying Number of Degrees-of-Freedom

1998 ◽  
Vol 65 (3) ◽  
pp. 719-726 ◽  
Author(s):  
S. Djerassi
Keyword(s):  
Angular Momentum ◽  
Degrees Of Freedom ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Linear Momentum ◽  
Dynamical Equations ◽  
The Third ◽  
Conservation Principles

This paper is the third in a trilogy dealing with simple, nonholonomic systems which, while in motion, change their number of degrees-of-freedom (defined as the number of independent generalized speeds required to describe the motion in question). The first of the trilogy introduced the theory underlying the dynamical equations of motion of such systems. The second dealt with the evaluation of noncontributing forces and of noncontributing impulses during such motion. This paper deals with the linear momentum, angular momentum, and mechanical energy of these systems. Specifically, expressions for changes in these quantities during imposition and removal of constraints are formulated in terms of the associated changes in the generalized speeds.

Comparison between Analytical and Vectorial Methods of the Synthesis of Equations of Motion

1983 ◽  
Vol 197 (4) ◽  
pp. 265-274
Author(s):  
Z J Goraj
Keyword(s):  
Angular Momentum ◽  
Analytical Method ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Discrete Systems ◽  
Momentum Equation ◽  
Lagrange Equations ◽  
Weak Points ◽  
Complicated Geometry ◽  
Boltzmann Hamel Equations

In this paper the advantages and weak points of the analytical and vectorial methods of the derivation of equations of motion for discrete systems are considered. The analytical method is discussed especially with respect to Boltzmann-Hamel equations, as generalized Lagrange equations. The vectorial method is analysed with respect to the momentum equation and to the generalized angular momentum equation about an arbitrary reference point, moving in an arbitrary manner. It is concluded that, for the systems with complicated geometry of motion and a large number of degrees of freedom, the vectorial method can be more effective than the analytical method. The combination of the analytical and vectorial methods helps to verify the equations of motion and to avoid errors, especially in the case of systems with rather complicated geometry.

scholarly journals Taming the Beast: Diffusion Method in Nonlocal Gravity

Universe ◽  
2018 ◽  
Vol 4 (9) ◽  
pp. 95 ◽  
Author(s):  
Gianluca Calcagni
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Form Factors ◽  
Specific Form ◽  
Diffusion Method ◽  
Dynamical Equations ◽  
Nonlinear Dynamical ◽  
Gravitational Theories

We present a method to solve the nonlinear dynamical equations of motion in gravitational theories with fundamental nonlocalities of a certain type. For these specific form factors, which appear in some renormalizable theories, the number of field degrees of freedom and of initial conditions is finite.

On First-Order Decoupling of Equations of Motion for Constrained Dynamical Systems

1993 ◽  
Author(s):  
Timothy A. Loduha ◽  
Bahram Ravani
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Relaxation Method ◽  
Nonholonomic Systems ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
First Order ◽  
Constraint Relaxation ◽  
Constrained Dynamical Systems ◽  
Generalized Coordinate

Abstract In this paper we present a method for obtaining first-order decoupled equations of motion for multi-rigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupled equations of motion for holonomic systems. In the case of nonholonomic systems, or more complex dynamical systems, where the appropriate congruency transformation may be difficult to obtain, we present a constraint relaxation method based on the use of orthogonal complements. The results are illustrated using several examples. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

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 Explicit Overturning Limit of Rigid Block Using Triple and Pseudo-Triple Impulses Under Critical Near-Fault Ground Motions

2021 ◽  
Vol 7 ◽  
Author(s):  
Sae Homma ◽  
Kunihiko Nabeshima ◽  
Izuru Takewaki
Keyword(s):  
Angular Momentum ◽  
Conservation Law ◽  
Mechanical Energy ◽  
Rotational Velocity ◽  
Rigid Block ◽  
The Third ◽  
Maximum Angle ◽  
Critical Timing ◽  
The Impact ◽  
Angle Of Rotation

An explicit limit for the overturning of a rigid block is derived on the input level of the triple impulse and the pseudo-triple impulse as a modified version of the triple impulse that are a substitute of a near-fault forward-directivity ground motion. The overturning behavior of the rigid block is described by applying the conservation law of angular momentum and the conservation law of mechanical energy (kinetic and potential). The initial velocity of rotation after the first impulse and the change of rotational velocity after the impact on the floor due to the movement of the rotational center are determined by using the conservation law of angular momentum. The maximum angle of rotation after the first impulse is obtained by the conservation law of mechanical energy. The change of rotational velocity after the second impulse is also characterized by the conservation law of angular momentum. The maximum angle of rotation of the rigid block after the second impulse, which is mandatory for the computation of the overturning limit, is also derived by the conservation law of mechanical energy. This allows us to prevent from computing complex non-linear time-history responses. The critical timing of the second impulse (also the third impulse timing to the second impulse) is featured by the time of impact after the first impulse. As in the case of the double impulse, the action of the second impulse just after the impact is employed as the critical timing. It is induced from the explicit expression of the critical velocity amplitude limit of the pseudo-triple impulse that its limit is slightly larger than the limit for the double impulse. Finally, it is found that, when the third impulse in the triple impulse is taken into account, the limit input velocity for the overturning of the rigid block becomes larger than that for the pseudo-triple impulse. This is because the third impulse is thought to prevent the overturning of the rigid block by giving an impact toward the inverse direction of the vibration of the rigid block.

Coordinates Used in Range or Range-Rate Systems and Their Extension to a Dynamic Earth

1975 ◽  
Vol 26 ◽  
pp. 181-200
Author(s):  
R. R. Newton
Keyword(s):  
Angular Momentum ◽  
Coordinate System ◽  
Equations Of Motion ◽  
Internal Motion ◽  
Linear Momentum ◽  
Satellite Systems ◽  
The Earth ◽  
Range Rate

AbstractThis paper describes briefly the coordinates that have been used with satellite systems that measure range or range-rate, and it shows that the methods previously used to define them lead naturally to the coordinate system needed for a dynamic Earth. This is one in which the Earth has no linear momentum and no angular momentum, and it is the only system that allows the motion of the Earth to be separated into a translation, a rotation, and an internal motion, and that preserves the usual forms of the equations of motion. The observations needed to define this system are outlined.

Dynamics of scalar shell for rotating and charged rotating BTZ black holes

2019 ◽  
Vol 35 (02) ◽  
pp. 1950350 ◽  
Author(s):  
M. Sharif ◽  
Faisal Javed
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Angular Momentum ◽  
Thin Shell ◽  
Equations Of Motion ◽  
Scalar Fields ◽  
Massless Scalar ◽  
Massive Scalar ◽  
Matter Distribution ◽  
Dynamical Equations

This paper studies the dynamics of thin-shell for (2 + 1)-dimensional rotating and charged rotating Bañados–Teitelboim–Zanelli black holes by using Israel thin-shell formalism. We consider the matter distribution located at thin-shell associated with a scalar field and analyze its effects on the dynamics of thin-shell through equations of motion and effective potential. The corresponding dynamical equations are numerically studied for both massless as well as massive scalar fields. For rotating case, the rate of expansion and collapse increases with massless scalar shell but decreases for massive case. For charged rotating, the rate of expansion and collapse decreases by increasing angular momentum for both massless as well as massive case. We conclude that the rate of expansion and collapse of the rotating case is greater than charged rotating black hole.

A Method for Testing Numerical Integrations of Equations of Motion of Mechanical Systems

1988 ◽  
Vol 55 (3) ◽  
pp. 711-715 ◽  
Author(s):  
T. R. Kane ◽  
D. A. Levinson
Keyword(s):  
Angular Momentum ◽  
Differential Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Test Results ◽  
Conservation Of Energy ◽  
Testing Method ◽  
Conservation Principle ◽  
Numerical Integrations ◽  
Conservation Principles

It is common practice to use conservation principles, such as the principles of conservation of energy or angular momentum, to test results of numerical integrations of differential equations of motion of mechanical systems. This paper deals with a testing method that can be used even when no conservation principle is applicable.

scholarly journals An Argument Against Augmenting the Lagrangean for Nonholonomic Systems

2009 ◽  
Vol 76 (3) ◽  
Author(s):  
Carlos M. Roithmayr ◽  
Dewey H. Hodges
Keyword(s):  
Nonholonomic System ◽  
Nonholonomic Constraint ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Euler Method ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
New Approach ◽  
Constraint Equations ◽  
Kane's Method

Although it is known that correct dynamical equations of motion for a nonholonomic system cannot be obtained from a Lagrangean that has been augmented with a sum of the nonholonomic constraint equations weighted with multipliers, previous publications suggest otherwise. One published example that was proposed in support of augmentation purportedly demonstrates that an accepted method fails to produce correct equations of motion whereas augmentation leads to correct equations. This present paper shows that, in fact, the opposite is true. The correct equations, previously discounted on the basis of a flawed application of the Newton–Euler method, are verified by using Kane’s method together with a new approach for determining the directions of constraint forces.

On First-Order Decoupling of Equations of Motion for Constrained Dynamical Systems

1995 ◽  
Vol 62 (1) ◽  
pp. 216-222 ◽  
Author(s):  
T. A. Loduha ◽  
B. Ravani
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Numerical Implementation ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
First Order ◽  
Holonomic Systems ◽  
Constrained Dynamical Systems ◽  
Generalized Coordinate

In this paper we present a method for obtaining first-order decoupled equations of motion for multirigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupled equations of motion for holonomic systems. In the case of nonholonomic systems, or holonomic systems with unreduced configuration coordinates, we incorporate an orthogonal complement in conjunction with the congruency transformation. A pair of examples illustrate the results. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

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