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Über dieses Buch

G. Grioli: Particular solutions in stereodynamics.- P. Hagedorn: On the converse of the Lagrange-Dirichlet stability theorem.- M. Langlois: Contribution à l’étude du mouvement du corps rigide à n dimensions autour d’un point fixe.- E. Leimanis: Some recent results concerning the motion of a rigid body about a fixed point.- H. Price: A canonical form of Euler’s equations and a method of solutions for arbitrary applied couples.- V.V. Rumyentsev: Dynamics and stability of rigid bodies.- J. Wittenburg: The dynamics of systems of coupled rigid bodies. A new general formalism with applications.

Inhaltsverzeichnis

Frontmatter

Particular Solutions in Stereodynamics

La dinamica del corpo rigido è uno dei capitoli della Meccanica razionale che ha destato per oltre due secoli il massimo interesse, sopratutto con riguardo al problema del solido pesante con un punto fisso.

Successivamente l'interesse si è un pò attenuato a vantaggio di altri settori della Meccanica, quali ad esempio, la teoria dei continui, la Meccanica analitica, la Relatività, ecc.

G. Grioli

On the Converse of Lagrange-Dirichlet's Stability Theorem

Stability questions are of considerable importance in many problems of applied mathematics and of mathematical physics. The systematic analysis of stability problems received a large impulse at the beginning of this century; this impulse was due to the russian mathematician

Liapunov

. He studied particularly the stability of solutions of ordinary differential equations relative to disturbances in the initial conditions. His methods are frequently used very successfully in mechanics for the discussion of the stability of motions.

Peter Hagedorn

Contribution a L'etude du Mouvement du Corps Rigide a n Dimensions Autour d'un Point Fixe.

L'espace pris au départ est R

n

rapporté à un repère (O, e

i

). Nous noterons (α

i

) la base duale de la base (e

i

) et nous supposerons que que R

n

est muni d'une métrique g

ij

.

Nous noterons successivement par: a)

U

1'ensemble des formes bilinéaires antisymétriques sur R

n

, ce dernier étant un espace vectoriel isomorphe a celui des n × n - matrices anti symétriques.

Michel Langlois

Some Recent Results Concerning the Motion of a Rigid Body About a Fixed Point

We will be concerned with three topics: reduction of the order of the equations of motion, kinematic interpretation of the motion, and existence of invariant relations of equations of motion.

Our starting point is a system of six equations, the so-called EULER and POISSON equations of motion of a heavy rigid body about a fixed point. The three known first integrals of this system permit us, at least in principle, to reduce the order of this system to three, or, if we eliminate the independent variable, even to two. There have been several attempts actually to carry out such a reduction of the order but in several cases the reduction remained hlfway, resulting for example in a system of order four, or being carried out subject to certain conditions on the parameters of the system.

Eugene Leimanis

A Canonical Form of Euler'S Equations and a Method of Solution for Arbitrary Applied Couples

The problem considered is that of the angular motion of a rigid body solely under the action of arbitrary time-variant couples applied about each of the three principal axes of inertia. A problem falling into this category is the control of an aircraft at high altitude when the aerodynamic damping is neglected, or alternatively the manoeuvring of a space vehicle in vacuo by the application of jet reaction couples.

The basic approach is to reduce the standard equations of motion by appropriate change of variables to a non-dimensional form in which none of the moments of inertia appears explicitly. The clarity of the resulting equations leads to the recognition of integrating factors that enable the motion to be calculated entirely by means of quadratures.

H. L. Price, G. R. Walsh

Dynamics and Stability of Rigid Bodies

Some problems of dynamics and stability of motion of rigid body and systems consisting of rigid, elastic bodies or bodies with fluid - containing cavities are discussed. It begins with a brief account of the theory of integrability of ordinary differential equa tions and formulation of main theorems on stability and instability of motion of a system with finite degrees of freedom as well as the system with parameter distribution and described by joint system of ordinary and partial differential equations. Further the problems of dynamics and stability of steady motions of rigid body, gyrostats of different types and also the stability of steady motions of satellites with spinning rotors and having fluid-containing cavities and elastic parts are successively considered.

V. V. Rumyantsev

The Dynamics of Systems of Coupled Rigid Bodies. A New General Formalism with Applications

1. Introduction : The dynamics of rotational motions of bodies as treated in the literature can be roughly divided into two parts. Part one is dealing with large angle rotations. This theory is non-linear. Part two is characterized by the a priori assumption of a small deviations from some particular state of motion which is known to be a stationary solution. This theory, called theory of gyroscopic systems, is linear. In most engineering applications systems comprising rotational dynamics are treated and can safely be treated with this linear theory. The non-linear theory has primarily been the domain of pure science although, of course, many results obtained in this field have become important for engineering applications. One of the most obvious differences between the groups of non-linear and linear problems, respectively, which have been successfully dealt with so far is the number of degrees of freedom of the systems. It has always been small in the non-linear case whereas in the linear case large systems have been considered.

The reason for this is not only the fact that systems of linear differential equations are easier to solve but also that they are much easier to formulate in the first place. The usual procedure followed when deriving non-linear equations of motion for rigid bodies is to define as variables angles (e. g. Euler angles) describing the attitude of the bodies in some reference coordinate system, then to express the Lagrangian as function of these angles and their time-derivatives and then to proceed to the differential equations. This procedure has two major disadvantages.

J. Wittenburg
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