Vol.3, No 11, 2001 pp. 71-79
UDC 531.15:519.614(045)
ON CONDITIONS OF EXISTENCE OF PERMANENT ROTATIONS OF THE CONNECTED RIGID BODIES SYSTEM ABOUT THE VERTICAL VECTOR
D. Chebanov, N. Khlistunova
Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R.Luxemburg Str.74, Donetsk 83114, Ukraine
Abstract. In the classical problem of motion of a heavy rigid body about a fixed point the permanent rotations are well known and completely investigated as the most simple and good visually demonstrated type of motions. Numerous properties of these motions are established and their theoretical and applied significance is commonly known (here the list of scientific references is so extensive that O. Staude's paper [1] must be singled out at first). In multibody mechanics, where under a increasing of the quantity of the system bodies the quantity of mechanical parameters and the order of differential motion equations are increasing too, the studying of conditions of existence of such motions is a complicated problem. This, apparently, is a reason in a view of which the problem on permanent rotations of coupled rigid bodies system does not have a exhaustive solution up to present time.
The success of analytical investigations in different mechanics problems, especially in multibody system dynamics, is often caused by a good choice of a form of motion equations for studied object. In 1st section of this paper the new form of motion equations of the considered mechanical system is suggested. It is derived from P.V. Kharlamov's equations [2,3] under the using of the mechanical parameters of the augmented bodies [4-6] in these equations. The obtained equations have a more compact form suitable for its studying.
In second section for the system of n heavy rigid bodies which are sequentially jointed in a chain by ideal spherical joints the conditions of existence for the motions are determined when the each of the bodies permanently rotates about the vertical vector. Section 4 contains the analysis of these conditions in a general case when the bodies angular velocities are different. Under the investigation a prior conditions on the mass distribution of the bodies and a way of their jointing are not used. The most simple case of two bodies is studied in 3rd section in detail.
O USLOVIMA POSTOJANJA NEPRESTANE ROTACIJE SISTEMA NEPOKRETNIH  KRUTIH TELA OKO VERTIKALNOG VEKTORA
U klasičnom problemu kretanja teškog krutog tela oko nepokretne talke, neprestane rotacije su dobro poznate i potpuno istražene kao najprostiji i vizuelno dobro predstavljeni tip kretanja. Mnogobrojne osobine ovih kretanja su utvrđene i njihov teoretski i primenjeni značaj je uopšteno poznat (ovde je lista naučnih referenci tako iscrpna da rad Staude-a mora biti istaknut pre svih). U mehanici sistema tela gde pri porastu broja tela sistema raste i broj mehaničkih parametara kao i red diferencijalnih jednačina kretanja, proučavanje uslova postojanja takvih kretanja je jedan komplikovan problem.
Uspeh analitičkog istraživanja u različitim mehaničkim problemima, naročito u dinamici sistema mnogostrukih tela je često prouzrokovan dobrim izborom oblika jednačina kretanja za proučavani objekat. U prvom delu ovog rada razmatra se novi oblik jednačina kretanja razmatra-nog mehaničkog sistema. On je izveden iz jednačina P.V. Kharlamova korišćenjem mehaničkih parametara uvećanih tela ?4,6? u ovim jednačinama. Dobijene jednačine imaju mnogo kompaktniji oblik koji je pogodan za njihovo proučavanje.
U drugom delu rada su određeni uslovi za postojanje kretanja za sistem od n teških krutih tela koji su jedan za drugim u nizu povezani u lanac idealnim sfernim zglobovima, te su ti uslovi određeni kada svako od ovih tela neprekidno rotira oko vertikalnog vektora. Odeljak broj 4 sadrži analizu ovih uslova u opštem slučaju kada su ugaone brzine ovih tela različite. U istraživanje a priori uslova o raspodeli mase i načinu njihovog spajanja se nije ulazilo. Najprostiji slučaj dva tela je proučen u trećem odeljku do detalja.