Vol.2, No 9, 1999 pp. 973-982
UDC: 531.134
EXTENSION
OF THE BERNOULLI'S CASE
OF A BRACHISTOCHRONIC
MOTION
TO THE MULTIBODY
SYSTEM IN THE FORM
OF A CLOSED KINEMATIC
CHAIN
Vukman M. Čović, Mirjana M. Lukačević
Faculty of Mechanical Engineering, University
of Belgrade, Yugoslavia
Abstract. Considering the brachistochronic
motion in a homogeneous field of gravity of the multibody system given
in the form of a closed kinematic chain, free from external constraints,
we prove that the trajectory of the system in the part of the configuration
space (this part being the one which includes all the generalized coordinates
of the system except the Cartesian coordinate yC of the system's centre
of inertia referring to the vertical axis) is a geodesic. Having in mind
that this geodesic is completely determined by the known initial and terminal
conditions given for the brachistochronic motion considered, as well as
by the nature of the mentioned part of the configuration space, and using
the fact that the configuration of our multibody system in this part of
the space is determined by the position of a representative point on the
geodesic, we further define the position of the whole system using two
coordinates only. One of them is the arc-lenght *, which determine the
position of a representative point on the geodesic, and another is the
coordinate yC, determining the one-dimensional subspace of the configuration
space. This sub-space, together with the subspace containing the geodesic,
constitute the complete configuration space of our system. Considering
the motion of the system in such a way, we obtain the result referring
to the trajectory of the system which is completely analogous to the famous
Bernoulli's result, found in the case of a single particle. Obtained result
is illustrated by the numerical example.
Keywords: Brachistochronic motion,
Closed kinematic chain, Analogy with the Bernoulli's case of a brachistochronic
motion
UOPŠTENJE
BERNULIJEVOG SLUČAJA BRAHISTOHRONOG KRETANJA NA SISTEM KRUTIH TELA
U OBLIKU ZATVORENOG
KINEMATIČKOG LANCA
Rešava se problem brahistohronog kretanja
zatvorenog kinematičkog lanca, slobodnog od spoljašnjih veza u homogenom
polju teže. Geometrizacijom problema ovo se kretanje razmatra u konfiguracionom
prostoru koji se može razdvojiti na dva potprostora - jedan je jednodimenzioni
i određen Dekartovom koordinatom središta masa sistema koja odgovara vertikalnoj
osi, yC, a drugi obuhvata sve ostale generalisane koordinate sistema. Pokazuje
se da je u tome drugom potprostoru trajektorija sistema geodezijska linija.
Birajući dalje za koordinate koje određuju položaj sistema luk te geodezijske
linije i koordinatu yC, dobija se rezultat koji se potpuno poklapa sa poznatim
Bernulijevim rezultatom koji se odnosi na slučaj brahistohronog kretanja
jedne materijalne tačke. Rezultat rada ilustrovan je primerom.
