Vol.2, No 10, 2000 pp. 1111-1135
UDC 534.01:531.53(045)
RHEONOMIC
COORDINATE METHOD
APPLIED TO NONLINEAR
VIBRATION SYSTEMS
WITH HEREDITARY
ELEMENTS
Katica (Stevanović) Hedrih
Faculty of Mechanical Engineering University
of Niš
Ul. Vojvode Tankosića 3/22, Yugoslavia,
18000 Niš, telefax: +381 18 41-663
e-mail: katica@masfak.masfak.ni.ac.yu
Abstract. Results pointed out in this
paper, are inspired by papers of O. A. Goroshko and N. P. Puchko (see Ref.
[13] and [14]), about Lagrange's equations for the multybodies hereditary
systems, and rheological models of the bodes properties presented in the
monograph written by G.M. Savin and Ya.Ya. Ruschitsky (see Ref. [24]),
as well as a monograph on rheonimic dynamics written by V.A. Vujičić (see
[6]). By using rhelogical body models for designing deformable rheological
hereditary elements with hybrid rheological elastoviscosic and/or viscoelastic
properties (see Ref. [23], [24] and [18]), discrete oscillatory systems
with hereditary elements as constraints, are designed, as systems with
one degree of freedom as well as with many degrees of freedom. For these
oscillatory hereditary systems, the integro-differential equations of the
second and/or third kind are composed. The solutions of these integro-differential
equations are studied.
Equations of dynamics of a disrete system
with finite constraints and standard hereditary elements are composed.
Covarinat integro-differential equations
of the motion of the discrete hereditary system are composed.
The rheonomic coordinate method is applied
to dicrete hereditary systems, and the modified system of the covarint
integro-differential equations of motion of discrete hereditary systems
with rheonomic constraints are composed.
For example, the rheological pendulum
on the wool's thread with changeable length is modeled by rheonomic coordinate
as well as by rheological hereditary element. By using defined rheological
pendulum basic properties of the rheonomic coordinate in the sense of the
Vujičić's, rheonomic coordinate are introduced. The force, as well as the
power of the rate of rheological and rheonomic constraints change are determined.
For the designed discrete hereditary systems
with corresponding rheological and relaxational hereditary elements the
integro-differential equations second and/or differential equations of
the third order are composed. On the basis of the analysis of the
discrete hereditary oscillatory systems the Goroshko's definition on dynamically
determinated or indeterminated discrete hereditary systems was confirmed.
Key words: Discrete hereditary
system, standard hereditary element, oscillatory hereditary systems, rheological
elements, rheonomic coordinate, rheonomic coordinate method, rheological
pendulum,
rheological and relaxational kernels,
covariant coordinate.
METODA REONOMNE
KOORDINATE
U PRIMENI NA
NELINEARNE OSCILATORNE SISTEME
SA NASLEDNIM
ELEMENTIMA
Rezultati prikazani u ovom radu inspirisani
su radovima O. A. Goroshko i N. P. Puchko (vidi Ref. [13] i [14]), o Lagrange-ovim
jednačinama za nasledne diskretne sisteme (više tela) i reološkim modelima
tela koji su prikazani u monografiji G.M. Savin-a i Ya.Ya. Ruschitsky (vidi
Ref. [24]), kao i monografijom V.A. Vujičić-a (vidi [6]). Koristeći modele
reoloških tela za opisivanje deformabilnih reoloških, naslednih elemenata
sa hibridnim reološkim-elasto-viskoznim i/ ili visko-elastičnim svojstvima,
postavljeni su modeli diskretnih naslednih sistema sa jednim i više stepeni
slobode kretanja. Za takve oscilatorne nasledne sisteme integro-diferencijalne
jednačine drugog, i/ili diferencijalne jednačine trećeg reda su sastavljene.
Sastavljene su jednačine dinamike diskretnog
sistema sa konačnim vezama i standardrnim naslednim elementima. Sastavljene
su integro-diferencijalne jednačine kretanja diskretnog naslednog sistema
u kovarijantnom obliku. Prikazana je metoda reonomne koordinate u primeni
na diskretne nasledne sisteme. Izveden je prošireni sistem integro-diferencijalnih
jednačina kretanja, u kovarijantnom obliku, naslednog sistema sa reonomnim
vezama.
Dat je veći broj primera reološko-reonomnog
oscilatora, kao i reološko-reonomnih klatna.
Ključne reči: Diskretni nasledni
sistem, standardni nasledni element, oscilatorni nasledni sistem, reonomna
koordinata, reološka koordinata, metoda reonomne koordinate, reološko-reonomno
klatno,
reološko i relaksaciono jezgro, kovarijantne
koordinate.
