Vol.6, No 1, 2007 pp. 1-22
UDC 531.3:531.3.011:532.511(045)=111
THE CONSTRUCTION OF THE LAGRANGE MECHANICS OF THE DISCRETE HEREDITARY SYSTEMS
Oleg Aleksandrovich Gorosko1, Katica (Stevanović) Hedrih2
1Ukrainian Higher Education Academy of Sciences Kiev National University named Taras Shevchenko, KIEV (Kiïv) 2166 Vul. Lísoviy prospekt No 22. kv. 32. (Lesnoy prospekt No 22. kv. 32.) Tel: (044) 5181508 e-mail: p_natalia@univ.kiev.ua
2Faculty of Mechanical Engineering University of Niš Mathematical Institute SANU Belgrade Yu-18 000- Niš, ul. Vojvode Tankosiċa 3/22, Telefax: 381 18 241 663, Mobile 063 8 75 75 99 e-mail: katica@masfak.ni.ac.yu * khedrih@eunet.yu
Abstract. The research results in the area of mechanics of hereditary discrete systems, obtained by the authors of this paper, are generalized and presented in the monograph [4] which contains the first completed presentation of the analytical dynamics of hereditary discrete systems. Two classes of dynamically defined and undefined hereditary systems are defined and considered by introducing corresponding restrictions. The main results of mechanics of hereditary discrete systems are presented with new applications important to engineering.
The approximation of expressions for the coefficients of damping and corresponding decrements as well as for the circular frequency of oscillations of hereditary oscillatory systems are obtained with high accuracy in the first and second approximations.
The analogy between hereditary interactions and reactive forces in the systems of automatic control is identified and a possibility to extend the theory of analytical dynamics of hereditary systems to mechanical systems with automatic control is pointed out.
The Lagrange's mechanics of hereditary systems is extended and generalized to thermo-rheological and piezo-rheological discrete mechanical systems as well as to discrete mechanical systems with standard light creep elements.
Key words: hereditary system, rheological element, rheological and relaxational kernels, standard hereditary element, integro-differential equation, fractional order derivative, material particles, rheonomic coordinate, rheological pendulum, rheological coordinate, covariant coordinate, thermo-rheological and piezo-rheological hereditary elements
KONSTRUKCIJA LAGRANGE-OVE MEHANIKE DISKRETNIH NASLEDNIH SISTEMA
Istraživački rezultazi u oblasti naslednih diskretnih sistema, koje su dobili autori ovog rada, su uopšteni i predstavljeni u monografiji [4], koja sadrži prvu kompletnu predstavu annalitičke dinamike naslednih diskretnih sistema. Dve klase dinamički određenih i dinamički neodređenih sistema su definisane i razmotrene u svetlu određenih ograničenja. Glavni rezultati analitičke dinamike diskretnih naslednih sistema su provereni na novim primerima značajnim za inženjersku praksu.
Aproksimacije izraza za koeficijent prigušenja i odgovarajući dekrement, kao i za kružnu frekvenciju oscilovanja naslednog oscilatornog sistema su dobijene sa visokom tačnošću u prvoj i drugoj aproksimaciji.
Analogija između nasledne interakcije i reaktivnih sila u sistemu automatskog upravljanja je otkrivena kao i mogućnost proširenja teorije analitičke dinamike diskretnih naslednih sistema na mehaničke sisteme sa automatskim upravljanjem.
Lagrange-ova mehanika diskretnih naslednih sistema je proširena i uopštena na termo-reološke [4, 8] i piezo-reološke [4, 9] diskretne sisteme, kao i na diskretne mehaničke sisteme sa puzećim standardnim lakim elementima.
Ključne reči: Nasledni system, reološki element, reološko i relaksaciono jezgro, standardni kali nasledni element, integro-diferencijalna jednačina, izvod necelog reda, materijalna tačka, reonomna koordinata, reološka koordinata, reološko klatno, kovarijantne coordinate, termo-reološki i piezo-reološki standardni nasledni laki element