Dynamics of Gyro-rotors: Theory and Applications

by Ljljana Veljovic (Faculty of Mechanical Engineering University of Kragujevac)

Wednesday 22 Dec 2010, 18:00 → 19:00 Europe/Belgrade

301 F (Mathematical Institute)

Brief review of realized models of gyro-rotors, applied gyr control and stabilizations and motion of ships, aircrafts, vehicles, and torpedo. With development of micro and nano- technology it feels a need for new class of gyroscopic systems. An analysis of principle of gyro-system work is presented as well as analysis of component motions. Most of realized gyro-stabilizators are realized on the basis of coupled rotations, with resultant rotation around fixed point. Gyroscopic moments are analyzed. (see References [1-2], [9-10], [15-16]).

For a model of gyro-rotor, by use vector method, based on the mass moment vectors for an axis and pole, introduced by Hedrih (Stevanovic) K. [4-8],, vector expressions for linear momentum and angular momentum of a heavy rigid body rotation around two axes without intersection are derived, as well as their derivatives. These vector expressions are used for obtaining expressions of the kinetic parameters of nonlinear dynamics of considered system dynamics or vibrations. The following vector expressions: for kinetic pressures to bearings on the self rotation axis and to the axis of precession rotation, as well as their components are pointed out and analyzed. For the special case that heavy rigid disk is eccentrically and skew positioned on the self rotation axis which rotate in the horizontal plane around vertical axis with constant angular velocity on a distance, we derived: the nonlinear differential equation of the system dynamics in the gravitational field and corresponding equations of the phase trajectory as well as kinetic pressure components on self rotation bearing and vector rotator. Series of graphical presentation are presented.

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Daje se pregled realizovanih modela girorotota koji su u upotrebi i koji se koriste za upravljanje i stabilizaciju kretanja brodova, aviona, vozila, letilica, i torpeda. Sa razvojem mikro i nano tehnologija javila se potreba za novom klasom giroskopa pa se najavljuju novi modali, kao i savremenim pravcima razvoja. Daje se analiza principa rada i vrsta kretanja. Analiziraju se odgovarajuće kinematičkih i kinetičkih paarametara dinamike postojećih izvedenih modela girostabiliz i ukazani na vrste kretanja. Većina starih uređaja je izvedena na bazi složenih spregnutih komponentnih rotac koje rezultiraju u obrtanje girost oko nepomične tačke. Ukazuje se i na izvore giroskopskih momenata kojima se vrši stabilizacija ili upravljanje kretanja. Na primeru jednog girorotora prikaѕuju se neki kinetički parametri girorotora i ukazuje se na pojavu novih modela namenjenih mikrouređajima. (Reference [1-2], [9-10], [15-16]).

Za izabrani model girorotota, koji se sastoji od teškog krutog tela, koje se obrće oko dve mimoilazne ose, vektorkom metodom, zasnovanom na korišćenju vektora momenata masa vezanih za pol i osu, koje je uvela K. Hedrih [4-8], izvedeni su vektorski izrazi za količinu kretanja i moment količine kretanja, kao i odgovarajući izvodi po vremenu. Telo je ekscentripno postavljeno u odnosu na osu sopstvene rotacije, a ni jedna glavna centrana osa inercije teškog krutog tela nije paralelna sa osom sopsvene rotacije tela, sto znači da je teko koso postavljeno u odnosu na istu. Za tako definisani maretijalni sistem u polju zemljine teže, kada je kruto telo disk ekscentrično i koso postavljen i za slučaj da je osa sopstvene rotacije u horizontalnoj ravni, a osa prenosnog kretanja vertikalna i da je ugaona brѕzina prenosnog kretanja konstantna, a ose mimoilazne izvedeni su: nelinearna diferencijalna jednačina sopstvene rotacije, jednačina faznih trajektorija, izrazi za kinetičke pritiske u vektorskom obliku, kao i odgovarajući vektori rotatori i ugaone brzine njihove ritacije oko ose sopstvene rotacije. Dara je analiza svojstava komponenata kinetičkih pritisaka na režišta ose sopstvene rotacije. Korišćenjem MathCad Software sastavljene su serije grafičkih prikaza: faznih portreta, intenzteta vektora rotatora i njegove ugaone brzine obtanja oko sopstvene ose rotacije girorotota, intenziteta komponenata kinetičkih pritisaka u zavisnosti od ekcentriciteta diska i ugla njegovog nagiba u odnosu na osu sopstvene rotacije, kao i rastojanja mimoilaznih osa prenosnog i sopstvenog obrtanja. Pokazuju se neka svojstva i identifikuju fiksne tače na tim dijagramima kada se menjaju parametri ekscentričnosti, ugla nagiba diska ili rastojanja između mimoilaznih osa (Reference).

References

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