Dynamics of Rigid Rotor in Elastic Bearing

8/13/2019 Dynamics of Rigid Rotor in Elastic Bearing

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Dynamics of a rigid rotor in the

elastic bearings

Inga M. Arkhipova

Theoret. Appl. Mech., Vol.31, No.1, pp.73-83, Belgrade 2004

Abstract

As a rule in the studies of a rigid rotor in the elastic bearings theauthors consider the linear system corresponding to the plane-parallel motion and the effect of self-centring under unlimitedgrowth of the rotation frequency. In the present paper rotor is

considered as a mechanical system with four degrees of freedom.Different motions of a statically and dynamically unbalancedvertical rotor supported in the non-linear bearings are studied.

Keywords: unbalanced rotor, precession motion, self-inducedvibrations.

1 IntroductionThe absolute rigid rotor of mass Mand lengthLis supported verticallyin two immovable non-linear bearings in such a way that the center ofmass of the rotor is placed symmetrically with respect to the bearings(Fig. 1). The rotor is dynamically symmetric; A is the moment ofinertia about the axis of symmetry and B is the equatorial moment ofinertia. The distance between the center of mass of the rotor and theaxis of revolution (static eccentricity) is equal to e, the angle between

Department of Theoretical and Applied Mechanics, St.Petersburg StateUniversity, 28 University Ave., St.Petersburg, 198504, Russia (e-mail:

[email protected])

73


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74 I. M. Arkhipova

the axis of dynamical symmetry and the axis of revolution (dynamicaleccentricity) is equal to. The angle between the plane containing theaxis of revolution and the mass center and the plane containing theangle is denoted by.

Figure 1: Unbalanced rotor in elastic bearings

It is assumed that the elastic bearings are centrally symmetric andthe reactions in the bearings have only radial components. We onlyconsider the case of the hard characteristic of the restoring forces withcubic non-linearity:

Pj =Sj

a0+a1|Sj|2

, j= 1, 2. (1)

Here Sj is the vector describing the displacement of the center of abearing from the equilibrium position, a0 and a1 are the positive realconstants, characterising the elasticity of the bearings.

The rotor rotates with an ideal engine (an engine with unlimited

power), so that the angular velocity is supposed to be constant. The


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Dynamics of a rigid rotor in the elastic bearings 75

rotor is under external friction forces Rej , which are proportional to theabsolute velocity of the bearing centers:

Rej =e Sj , j= 1, 2. (2)

If one neglects the displacement along the axis of rotation, then theposition of the rotor is determined by the co-ordinates of the center ofbearings and the rotor is a system with four degrees of freedom. Theequations of the motion can be derived by using the theorem of themotion of the center of mass and the theorem of the moments.

We consider vibrations near the equilibrium (vertical) position,when the co-ordinates of the bearings and their velocities and eccen-tricitiese and are considered to be small. The system of differentialequations in complex variables S1 and S2 is the following:

M

2(S1

+S2

) + e(S1

+ S2

) +a0

(S1

+S2

)+

a1(|S1|2 S1+ |S2|2 S2) =M e 2 exp(I t),

B(S2 S1) I A(S2 S1) + e L2

2(S2 S1)+

L2a02

(S2 S1+ a1a0

(|S2|2 S2 |S1|2 S1)) =

(B A)2

Lexp(I(t )). (3)This is the linear system, but with the non-linear cubic terms in therestoring forces.

Forced vibrations due to either static or dynamical eccentricity orboth of them have the form of the direct synchronous precession:

Sj = Rjexp(I t)exp(I j), j= 1, 2, (4)

where Rj and j are the real constants characterizing the amplitudesand phases of the bearings displacements. Depending on a type of the

surface traced by the axis of a rotor, the rotor motion could be either


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76 I. M. Arkhipova

cylindrical, or conical or hyperboloidal precession. The characteristicfeature for a cylindrical precession is the equality of the amplitudes andphases. For a conical precession the phases are either equal or differedby. In the first case the axis of the rotor traces the truncated coneand in the second case the rotor axis traces the cone with the coneapex between the bearings. In the case of a hyperboloidal precessionthe amplitudes and phases relate arbitrarily. We call the precessionsas symmetric if both bearings in their plane motions make the circlesof equal radii.

2 Symmetric hyperboloidal precessions of

completely unbalanced rotor

Here we analyze hyperboloidal precessions for a rotor with two disbal-

ances (e= 0, = 0). Symmetric precessions of statically and dynam-ically unbalanced rotor may only occur for a system without externalfriction forces Re and when = /2. Otherwise non-symmetric hy-perboloidal precessions take place. Consider symmetric hyperboloidalprecessions defined as

y=

x

2

1

(1 +cy x)2 + d2

(k(1 +cy) x)21/2

,

tan 1= d 1 + cy x

k(1 +cy) x, 1=

2.

(5)

Here for convenience we introduce the amplitude-frequency responsein the plane (x, y) (x= 2,y = R2), where the dimensionless variablesand parameters are:

R= R/(2e), 2 =2/(2a0/M),

c= 4e2a1a0

, k= ML2

4B(1 A/B) , d=L

2e.

(6)

For the limiting values R and , when x is large enough we

obtain


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y= R= 1/2

1 +d2, tan 1,= d. (7)

If we introduce a minimal radius of the hyperboloid of rotationr = R cos 1 and the angle of the deflection of the rotor axis from the

vertical = 2R sin 1/L (L = L/(2e)), then the limiting values forthese quantities arer= 1/2 and= respectively. The last resultmeans that the self-centring regime takes place. The rotor rotates insuch a way that the center of mass remains stationary and the axis ofdynamic symmetry takes the equilibrium position.

If the rotor is a dynamically prolate body ( 0) thenthe resonance set consists of the lines 1 +cyx= 0 andk(1+cy)x=0, which are the skeleton lines for the modified amplitude-frequencyresponse curve (AFR) in the plane (x, y) (Fig. 2). The cylindricaland conical precessions resonate near the first and the second lines,respectively.

y

6

5

4

3

2

1

0 1 2 3 4

A

B

C

D

x

Figure 2: AFR of symmetric hyperboloidal precession

One can see in Fig. 2 that for one frequency there can be either one,

three or five different regimes of symmetric hyperboloidal precessions.


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78 I. M. Arkhipova

It can be shown that in the first approximation the segments AB andCD of AFR for the values k < 1/3 and k > 3 correspond to theunstable symmetric hyperboloidal precessions.

If rotor is dynamically oblate ( >1 andk 0) in the

plane (x, y) has the skeleton line k(1 + cy) x = 0 (Fig. 3). For suf-ficiently large x the limiting value for the amplitude is R = 1/4 andthe axis of dynamical symmetry of the rotor tends to take the equilib-rium position. This motion corresponds to the self-centring regime ofthe rotor.

Depending on the parameters of the system there may be one orthree regimes of symmetric conical precessions. To study stability ofthese regimes in the linear approximation the characteristic equationmay be represented as a product of two fourth order polynomials incharacteristic numberp(M N= 0). The stability conditions have beenreduced to the inequalities m4 >0, n4 > 0, where m4 and n4 are the

absolute values of the polynomial M and N.


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y

3

2

1

0 1 2 3 4 5x

n4=0

m4=0

Figure 3: AFR of symmetric conical precession

The setsm4= 0 andn4= 0 are the bifurcation sets determining thebounds of the stability domains for the symmetric conical precessions,which have the form of hyperbolas in the plane (x, y) (Fig. 3). Inthe bifurcation points on the bound, where m4= 0, the regime of thesymmetric conical precessions (when the center of mass of the rotoris motionless) may appear or disappear. Instability of the segment,where inequalitym4< 0 is satisfied is quite common for the theory of

non-linear vibrations. In the bifurcation points on the bound, wheren4= 0, the regime of the nonsymmetric hyperboloidal precessions dueto the motion of the center of mass may appear or disappear. Thatmeans that the symmetric conical precessions are unstable in 3D andthe domain of instability in 3D appears near the resonance set forthe cylindrical precessions of the rotor with two disbalances. Suchinstability may exist only for the mechanical system with three andmore degrees of freedom. In domain n4 < 0 the symmetric conicalprecessions may transform to the hyperboloidal precessions or someother regimes (for example, strange attractor).

Among the steady-state motions of statically unbalanced rotor there


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80 I. M. Arkhipova

may be cylindrical precessions, non-symmetric conical precessions andsymmetric and non-symmetric hyperboloidal precessions. For the cylin-drical precession the effect of self-centring is typical; the dynamicallyprolate rotor may be unstable in 3D and for the resulting regimes theaxis of rotation of the rotor swings.

4 Non-linear equations of the rotor

In the analysis of the rotor motion with the second order terms theequations are nonsymmetric and can not be represented in the complexvariables as we did it in the previous sections. The coefficients at thesecond derivatives of the co-ordinates of the bearings in their plane ofmotion depend on the same co-ordinates.

If for a linear system the main motion is the symmetric preces-sion, then for a nonlinear system the symmetric motions for the rotorwith either static or dynamic disbalance or both of them can onlybe symmetric cylindrical precessions for a statically unbalanced rotor.Symmetric cylindrical precessions are described by the same formulasas in the linear approximation.

5 The influence of the internal friction

force on the rotor motion

Consider the motion of the rotor under external friction forces Rej and

internal friction forces Rij. Internal friction forces are proportional to

the relative velocity of motion of the center of bearings and may existin the bearings due to the oil film that is partly entrained by the rotor:

Rij =i

Sj ISj

, j = 1, 2. (9)

The forces Ri do not affect the parameters of the direct synchronousprecessions of rotor, but make the domain of the stable regimes signifi-cantly more narrow and prevent the self-centring effect. For symmetricconical precession of the dynamically unbalanced rotor and cylindricalprecessions of statically unbalanced rotor the regimes induced are the

self-induced vibration which can be represented in the form


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sj =Rjexp(I)exp(Ij) +rjexp(I1), (10)

where Rj and j are the amplitudes and phases of forced vibrationswith the frequency ,rj and 1 are the amplitudes and the frequency

of self-induced vibrations [3].For dynamically unbalanced rotor self-induced vibrations are asso-

ciated with the motion of the center of mass. When the mass cen-ter outlines the circle the resulting motion can be represented as asuperposition of symmetric conical precessions (R1 = R2 = R andexp(I2) =exp(I1) = exp(I)) and symmetric self-induced vibra-tions (r1= r2= r).

Using the harmonic balance method we obtain the following ap-proximate equations for R, r, 1 and :

r(1 +c(r2 +R2) 21) = 0,

r(1(e+i) i) = 0,

R(1 )((k(1 +c(R2 + 2r2)) 2)sin +ekcos ) = 0,

(1 )(R(k(1 +c(R2 + 2r2)) 2)cos ekR sin 14

2) = 0.

(11)

Equations (10) permit to find the amplitudes of self-induced vibra-

tions and forced vibrations as the functions of the rotation frequencyand the frequency bound of the soft inducing for the self-induced vi-brations:

s= (1 + e/i)

1 + 2cR2. (12)

In Fig. 4 AFR curves for the amplitudes squares (Y = R2, y =r2) vs. frequency square (x = 2) are represented. The dotted linecorresponds to the self-induced vibrations. Note that if the angularvelocity increases then the amplitude of the self-induced vibrationsalso goes up. That is a cause of the bearings destruction and rotor

collapse.


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82 I. M. Arkhipova

Y,y

xs

2 6420

2

4

6

8

Figure 4: AFR of forced vibrations and self-induced vibrations

If the rotor has only two degrees of freedom then one should seekthe modes for self-induced vibrations in the form (9), where s1=s2=s. Such regimes may exist only for a dynamically prolate rotor, when 0. Although such type of solutions satisfy formally thesystem of differential equations of dynamically unbalanced rotor withfour degrees of freedom, numerical integration results in the modesrelated to the motion of the mass center.

For a statically unbalanced rotor (e= 0,= 0) with four degrees offreedom the developing self-induced vibrations form a planar motion.

The resistance force Rm =m SSS proportional to the velocityof the radial displacements of the centers of bearings, which appears ina rigid rotor due to the deformation of the balls in the rolling bearing[4], has the same destabilizing effect on the symmetric precessions and

generates the self-induced vibrations.


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References

[1] A.S. Kelzon, Yu.P. Tsimanskii, V.I. Yakovlev, Dynamics of Ro-tor in Elastic Bearings, Nauka, Moscow, 1982.

[2] I.M. Arkhipova, Steady Motions of a Rigid Unbalanced Rotor inNon-linear Elastic Bearings, J. Appl. Maths Mechs, Vol. 66, No.4, pp. 539-545, 2002.

[3] A. Tondl, Some Problems of Self-Excited Vibrations of Rotors,Praha: Publ. SNTL, 1974.

[4] D.R. Merkin,On the Stability of Stationary Motion of the Axis ofa Rotating Shaft with Non-linear Supports, J. Sov. Appl. Math.,vol. 47, No. 3, 1983.

Submitted on May 2004, revised on July 2004.

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Dynamics of Rigid Rotor in Elastic Bearing