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The 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery

Honolulu, Hawaii, February 10-14, 2002

NON-LINEAR EFFECT OF AXIAL ROTOR OSCILLATIONS ON ROTORDYNAMIC STABILITY

HANS E. KIMMEL Vice President

Research & Development Ebara International Corporation

Cryodynamics Division Sparks, Nevada, USA

ABSTRACT

Axial rotor oscillations in thrust-balanced rotating machinery have a periodic effect on the stiffness of ball bearings. In practical technical applications, these axial rotor oscillations are forced oscillations due to changes in the electromagnetic field strength of the electric motor or generator or caused by flow induced pressure oscillations.

In the presence of axial rotor oscillations, the stiffness of the ball bearings is not constant and becomes a function of the time. Matthieu Equations, which are non-linear differential equations with periodic coefficients, are used to describe the non-linear effect of axial rotor oscillations on rotordynamic stability.

INTRODUCTION

The predominant design in liquefied gas fluid machinery is the multi-stage centrifugal pump with submerged electric motor and the multi-stage turbine expander with submerged electric generator, both equipped with a thrust-balancing device to eliminate the high axial hydraulic thrust. These pumps and turbine expanders are equipped with ball bearings, which are submerged, lubricated, and cooled by the cryogenic fluid. It is therefore important to balance the axial thrust properly in a controlled manner to increase the life of the cryogenic bearings.

In addition to centrifugal pumps, power recovery from the expansion of high-pressure cryogenic liquids has significantly advanced in recent years, using hydraulic reaction turbines for the thermodynamic expansion of liquefied gases.

Consistent with the early experience on cryogenic centrifugal pumps, the prevailing design for cryogenic turbines is that of a multi-stage reaction turbine with submerged electric generator and thrust balancing device (Kimmel, 1998).

Both centrifugal pumps and reaction turbines operated without thrust balancing devices generate a high axial hydraulic thrust, which can reach several kilo-Newtons on multistage turbines, resulting in heavy loads with significantly reduced hydraulic efficiency and bearing life (Evrensel et al.,1999). It is therefore an important requirement in cryogenic fluid machinery

to furnish the centrifugal pumps and reaction turbines with a thrust-balancing device.

AXIAL ROTOR OSCILLATIONS

The integration of these thrust-balancing devices into the design of pumps and turbines generates axial rotor oscillations, observed and reported by Evrensel et al., 1999 and 2002, and Finley et al., 2002. Axial rotor oscillations are transmitted to the ball bearings through the shaft, shifting the inner race of the ball bearing in relation to the outer race periodically in the axial direction. These axial displacements are of very small magnitude as reported by Evrensel et al. 1999.

Figure 1 demonstrates the axial displacement of the inner bearing race relative to the outer bearing race for the neutral and the two peripheral positions. The stiffness characteristic of the ball bearing changes with the axial loads generated by the axial displacement.

Figure 2 shows the typical stiffness characteristic ][sR as a function of the displacement ][ts . The stiffness characteristic can be approximated to sufficient accuracy by a quadratic function.

20 ])[(][ tsrRsR += (1)

The axial displacement is a multi-frequency periodic

function as reported by Evrensel et al. 1999 and 2002, and to estimate the effect of axial rotor oscillations on rotordynamic stability, a simplified periodic function with a primary frequency and its first harmonic is introduced.

]2sin[]sin[][ tbtats ωω += (2)

Substituting equation (2) into equation (1), simplifying and

renaming all constants, presents the periodic stiffness function of the ball bearing:

]4cos[]3cos[]2cos[ ]cos[2

tDtCtBtAkR

ωωωω

++++= (3)

2

The rotordynamic equation without exciting forces and

damping describes the natural frequency of the system. In the case of bearings with periodically time dependent

stiffness, the rotordynamic equation is transferred into a Mathieu Equation, a non-linear differential equation with periodic coefficients (Rao, 1995).

0][][][2

2

=+ tZtRdt

tZd (4)

Equation (4) describes the coupled oscillator consisting of

axial and lateral rotor oscillations under the simplifying assumptions of no damping and no exciting forces. The solution of this Mathieu Equation presents the natural frequency of the coupled oscillator.

Figure 1: Axial displacement in ball bearings

Figure 2: Ball bearing stiffness

MATHIEU EQUATION NATURAL FREQUENCY

Mathieu Equations are second order non-linear differential equations with time dependent periodic coefficients that cannot be solved by elementary functions.

To calculate the natural frequency of the coupled oscillator, the Ritz-Galerkin method offers good approximations for steady state solutions (Rao, 1995).

By using a one-term approximation for ][tZ as

]sin[][ 0 ftZtZ = (5) Equation (4) transforms into

],[)][](sin[ 2

0 tfEftRftZ =− (6)

],[ tfE is the periodically time dependent error between the approximation and the correct solution. The Ritz-Galerkin method is a least square approximation and requires the minimization of the total squared error expressed by the definite

integral of ][2 tE between t1 = 0 and f

Nt π22 = , with N as

a selected natural number.

( )( )∫ −=2

1

220 ][]sin[][

t

t

dtftRftZfI (7)

The evaluation of this integral leads to a function ][ fI of the following type:

][][][][

fffNfI

ΩΨ+Φ= (8)

In equation (8) ][ fΨ are harmonic functions of f and

][ fΦ are polynomial functions of f . The denominator ][ fΩ is the product of all resonant frequencies. With increasing value of N the harmonic part ][ fΨ becomes very small compared to the polynomial part ][ fNΦ and for non-resonant cases the total squared error is primarily dependent on ][ fΦ , under the condition that 0][ ≠Ω f

To minimize the total error ][2 fE for 1>>N the polynomial function ][ fΦ must be a minimum and must satisfy the condition:

dffd ][Φ

(9)

3

Substituting ][tR from equation (3) the minimum condition of the polynomial function ][ fΦ can be found as

( ) 0246 22224224 =++++−− DCBAkkff (10)

The natural frequency f can be calculated from equation (10) and represents the least squares approximation of the natural frequency for the described Mathieu Equation.

2383333

3

422222 kDCBAkf +++++= (11)

The natural frequency f of the rotordynamic equation is

larger if axial oscillations are present. For the first approximation, the shift to higher frequency values depends only on the amplitudes of the axial oscillations. This statement is true under the condition

fN >>ω (12) which follows from the application of the Ritz-Galerkin method.

Axial rotor oscillations increase the stiffness of the ball bearings and consequently increase the natural frequency of the lateral rotor oscillations. STABILITY OF THE COUPLED OSCILLATOR

The application of the Ritz-Garlekin method also offers information on stable and unstable solutions of the coupled oscillator. The integrated error function ][ fI indicates unstable solutions, if for certain frequencies f the denominator ][ fΩ is equal to zero.

For the case of axial rotor oscillations as described by equation (2) the unstable solutions of the coupled oscillator are

ω=f

ω2=f

ω3=f

ω4=f

2ω=f

ω23=f

ω25=f

ω27=f

These unstable solutions are described as parametric

resonances, with ω=f as the primary resonance and ω2=f as the principal parametric resonance (Nayfeh, 1993).

The subharmonic resonance 2ω=f occurs in addition with its

third, fifth and seventh superharmonics.

CONCLUSIONS The non-linear effect of axial rotor oscillations on

rotordynamic stability can be described by two phenomena.

The first effect is the increase of the natural frequency value caused by the periodically time dependent increase of the ball bearing stiffness.

The second effect is the appearance of parametric resonance frequencies, which determine the stable and unstable solutions of the governing non-linear Mathieu Equation without damping and with no exciting forces.

REFERENCES Kimmel, H.E., 1998, Power Generation using Thrust Balanced Hydraulic Turbines, Proceedings of the American Power Conference, Vol.60, pg.200 ff, Illinois Institute of Technology, Chicago Evrensel, C.E., Kimmel, H.E. and Cullen, D.M., 1999, “Axial Rotor Oscillations in Cryogenic Fluid Machinery”, Proceedings of the 3rd ASME/JSME Joint Fluids Engineering Conference, San Francisco, California Kimmel, H.E., 2000, “Cryogenic Francis Turbines”, Proceedings of the 8th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii Evrensel, C.E., Finley, C.D., 2002, “Verification of Axial Rotor Oscillations in Cryogenic Turbine Generators”, Proceedings of the 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii. Finley, C.D., Evrensel, C.E., 2002, “Observation of Jump Pheno-mena in Nonlinear Axial Rotor Oscillations” Proceedings of the 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii. LeGoy, P., Fuelling, S., 2002, “The Kimmel Equation for Hydraulically Damped Axial Rotor Oscillations”, Proceedings of the 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii. Nayfeh, A.H.,1993, “Method of Normal Forms”, Wiley Series in Nonlinear Science, John Wiley & Sons, Inc. New York, 1993 Rao, S.S., 1995, “Mechanical Vibrations” Addison-Wesley Publishing Company, Inc. ,USA, ISBN 0-201-52686-7 Childs, D.,1993,"Turbomachinery Rotordynamics" John Wiley & Sons, Inc., New York, ISBN 0-471-53840-X