The condition of rotor rotation stability with liquid autobalancing device consisting of chamber, float and incompressible homogeneous liquid filling the space between them has been obtained. The restoring force and forces of internal and external friction take effect on the rotor. The latter depend linearly respectively on strain rate and absolute velocity of rotor connection point to the shaf

Dubovik, V. A.

Pashkov, E. N.

Electronic archive of Tomsk Polytechnic University

To remove rotating bodies unbalance various autobalansing device with liquid (ADL) are used [1]. Whenusing such systems it is necessary to know critical angular velocities at which stationary rotations stability is disturbed. In a number of papers, for example in [2, 3],the approximate conditions of stability of balanced cylinder steady rotation filled partially with liquid whichare difficult to be applied to the systems of ADL are obtained. The immediate investigation of stability of rotorsrotation with liquid ADL is not described in literature.In the proposed paper by analogy with [4] for unbalanced discs at flexible shaft the stability of rotor rotationwith liquid ADL without free surface when influenced byforces of internal and external friction is studied. The nature of these forces is stated in detail in [4]. So the forcesof external friction are caused by viscous resistance ofenvironment, supports, special dampers and depend onrates of absolute displacements of rotor and shaft points;forces of internal friction are caused by material particles resistance and are taken as proportional to the shaftstrain rate to a first approximation. The investigation ofthe considered forces ratio influence on stability of rotorrotation with ADL is of some interest. The solution ofsuch problem for the unbalanced disc without ADL is given in [4]. Let rotor with ADL is fixed symmetricallywith respect to the supports of vertical flexible shaft passing through its geometric centre O1 (Figure).ADL consists of the correction chamber – 1 withthe height h, the float – 2 and homogeneous incompressible fluid – 3 filling the space between their walls. Centre of rotor mass (point P) is shifted from O1 to the distance O1P=e. Point is a projection of shift support axisto the plane of motion. At system rotation the shift sagsat a place of rotor attaching point per the value O2O1 thefloat for which geometric and material symmetry axescoincide as well in floating gyroscopes [5] is centered onrotation axis O2 due to pressure forces and fluid flows tothe side of sag. Let us suppose that at rotor disturbedmotion fluid separation from walls does not occur andfloat centering remains. In this case fluid layer centre ofmass is on the line of O2O1 centers in point G. The stated suppositions allow mapping out a hydrodynamicproblem.According to [4] let us introduce in a plane of motion of points O1, G, P two coordinate systems withcommon origin in point O2: fixed O2xy and mobileO2ξη, axis O2ξ which is parallel to the segment O1P.Laws of angular motion of rotor and system O2ξηare determined by the same rotation angle β(t) (t – time) therefore, rotor in the mobile coordinate system can moveonly steadily. Let us take the coordinates in a fixed system of point O1 which are denoted by x, y as generalized coordinates. We consider that rotor is influencedfrom the side of a shaft by elastic force F–c=–c⎯O1O2⎯, forces of internal and external friction F–k=–kV–O1r andF–χ=–χV–O1, applied in the point O1 proportional correspondingly to a shaft bending⎯O1O2⎯=x.i+y.j; rate of pointO1 in the mobile coordinate system O2ξη(1)and absolute velocity, (2)where i, jare the orts of rectangular coordinate systemO2xy. Equilibrium condition of the enumerated forcesand system inertial forces is written down in the form of(3)Figure. Cross section of rotor with ADLTherein m1, m2 and a–P, a–G are masses and acceleration of mass centers of rotor and liquid respectively. Thelatter are calculated through the coordinates of points Pand G in the fixed reference system.(4)cos , sin ,, ,P PG Gx x e y y ex r x y r yβ β= + ⋅ = + ⋅= ⋅ = ⋅1 12 1 1 20.rO O P GcO O kV V m a m a− − − χ − − =1ÎV x i y j= ⋅ + ⋅ 1( ) ( )rÎV x y i y x jβ β= + + −  Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 212UDC 621.0STATIONARY ROTATION STABILITY OF UNBALANCED ROTOR WITH AUTOBALANCING DEVICE WITH LIQUID ON FLEXIBLE SHAFT V.A. Dubovik, E.N. PashkovTomsk Polytechnic UniversityEmail: epashkov@rambler.ruThe condition of rotor rotation stability with liquid autobalancing device consisting of chamber, float and incompressible homogeneousliquid filling the space between them has been obtained. The restoring force and forces of internal and external friction take effect onthe rotor. The latter depend linearly respectively on strain rate and absolute velocity of rotor connection point to the shaft. where r=r12/(r12–r22), r1 and r2 are the radii of the chamber and the float respectively. Projecting (3) on the axesx, y subject to (1, 2) and (4) we obtain the differentialequations of motion of rotor with ADL(5)in which the notation m=m1+r.m2 is entered.Supposing in (5) that е=0, m2=0  and χ=0 we havethe equation of motion of balanced disk mass centre onan elastic shaft at internal friction influence given in [6].Comparing the equations in [6] with (5) we correlate theratio between the coefficient p and coefficient k enteringinto dependence of stress on deformation of shaft viscoelastic material σ=Eε+кε., p=ск/E. Therein there isshaft flexural stiffness for simply supported shaft withthe length l with a disk in the middle [6] с=48ЕI/l 3; Е isthe module of material elasticity; I is the inertia moment of sectional area relative to the centroidal axis ofshaft bending perpendicular plane. At β. =ω=const ofthe equation (5) the calculation is admitted(6)where А is the shaft bending, γ is the motion phase angle (the angle between rotor unbalance vector ⎯O1P⎯andbending deflection ⎯O2O1⎯).It is seen from (6) that the given calculation corresponding to the stationary rotation of rotor with FBDdoes not depend on the forces of internal friction. It isobvious as the shaft deformation does not change atsuch motion. The calculation (6) is studied in detailin [7].To study the stability of the rotation under consideration let us enter the deviations x'=x—xc, y'=y—yc.Supposing that angular velocity ω remains constant thedisturbed motion equation is obtained from (5) subjectto (6).(7)Characteristic equation for the system (7) has theform(8)whereGurvits matrix H of the coefficients of equation (8)according to [8] is writtenThe conditions of stability are of the form of inequations [8]Δi>0; i=1...4, (9)where Δi are the main diagonal minors of matrix Н.Inequations (9) are fulfilled if(10)At μ=1 we obtain the condition of stability of rotorstationary rotation without ADL given in [4]. Comparing (10) with a congruent inequation in [4] we conclude that the limiting value of angular velocity at stationary rotation of rotor with ADL ω*(μ) is less than for therotor without ADL. It is shown in [7] that fluid ADLdecreases shaft bending and system unbalance at angular velocities(11)It follows from (10, 11) that steady stationary rotationof rotor with fluid ADL decreasing vibration is possible atangular velocities meeting the condition ω*<ω<ω*.ConclusionsAutobalansing device with liquid connected to therotor decreases angular critical speed of the shaft thetransition through which results in system rotation instability. Stable system operation is realized in the rangeof angle velocities determined by the parameters of rotor and autobalancing device.*12 .1cmω ω μ> = ⋅ +*1 11(1 ) , 1.c mp m mχω ω μμ< = + ⋅ = >1 30 2 41 30 2 40 00.0 00a aa a aHa aa a a=2 20 1 22 2 23 4, 2 ( ), ( ) 2 ,2 ( ), .a m a m p a p cma c p a cχ χχ χ ω= = + = + += + = +4 3 2 10 1 2 3 4 0,a a a a aλ λ λ λ+ + + + =( ) 0;( ) 0.mx p x cx p ymy p y cy p xχ ωχ ω′ ′ ′ ′+ + + + =′ ′ ′ ′+ + + − =  2122 2 2 2cos( ) sin( );; tg ;( )c cx A t y A tm eAc mc mω γ ω γω χωγ ωω χ ω= + = += = − −− +2121( ) ( cos sin );( ) ( sin cos );mx p x cx py m emy p y cy px m eχ β β β β βχ β β β β β+ + + + = ++ + + − = −      Mathematics and mechanics. Physics13REFERENCES1. Gusarov A.A. Autobalancing devices of immediate action. – Moscow: Nauka, 2002. – 119 p.2. Epishev L.V. On dynamic instability of rotating rotor at incompletefluid filling // Scientific report of Higher school. Mashinostroenie iPriborostroenie. – 1959. – № 2. – P. 66–74.3. Derendyaev N.V., Sandaljv V.M. On stability of stationary rotationof a cylinder partially filled up with viscous incompressible fluid //Prikladnaya Matematika i Mechanika. – 1982. – Т. 46. – Issue 4. –P. 578–586.4. Dimentberg F.M. Bending oscillations of rotating shifts. – Moscow:USSR AS Press. – 1959. – 246 p.5. Andreichenko K.P. The dynamic of floating gyroscopes and accelerometers. – Moscow: Mashinostroenie, 1987. – 128 p.6. Panovko Ya. G. The basic of applied theory of oscillations and impact. – Leningrad: Mashinostroenie, 1976. – 320 p.7. Dubovik V.A., Pashkov E.N. Stationary rotation of unbalanced rotorwith fluid autobalancing device at forces of external friction influence // Bulletin of the Tomsk Polytechnic University. – 2006. – V. 309.– № 4. – P. 145–147.8. Merkin D.R. Introduction into the theory of stability of motion. –Moscow: Nauka, 1971. – 312 p.Received on 23.11.2006

Stationary rotation stability of unbalanced rotor with autobalancing device with liquid on flexible shaft

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