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Research Papers: Gas Turbines: Structures and Dynamics

# Predicted Rotordynamic Behavior of a Labyrinth Seal as Rotor Surface Speed Approaches Mach 1

[+] Author and Article Information
Manish R. Thorat, Dara W. Childs

Turbomachinery Laboratory, Texas A&M University, College Station, TX 77843

J. Eng. Gas Turbines Power 132(11), 112504 (Aug 11, 2010) (8 pages) doi:10.1115/1.4000895 History: Received August 07, 2009; Revised October 06, 2009; Published August 11, 2010; Online August 11, 2010

## Abstract

Prior one-control-volume (1CV) models for rotor-fluid interaction in labyrinth seals produce synchronously reduced (at running speed), frequency-independent stiffness and damping coefficients. The 1CV model, consisting of a leakage equation, a continuity equation, and a circumferential-momentum equation (for each cavity), was stated to be invalid for rotor surface speeds approaching the speed of sound. However, the present results show that while the 1CV fluid-mechanic model continues to be valid, the calculated rotordynamic coefficients become strongly dependent on the rotor’s precession frequency. A solution is developed for the reaction-force components for a range of precession frequencies, producing frequency-dependent stiffness and damping coefficients. They can be used to define a Laplace-domain transfer-function model for the reaction-force/rotor-motion components. Calculated results are presented for a simple Jeffcott rotor model acted on by a labyrinth seal. The model’s undamped natural frequency is 7.6 krpm. The fluid properties, seal radius $Rs$, and running speed $ω$ cause the rotor surface velocity $Rsω$ to equal the speed of sound $c0$ at $ω=58 krpm$. Calculated synchronous-response results due to imbalance coincide for the synchronously reduced and the frequency-dependent models. For an inlet preswirl ratio of 0.5, both models predict the same log-dec out to $ω≈14.5 krpm$. The synchronously reduced model predicts an onset speed of instability (OSI) at 10 krpm, but a return to stability at 48 krpm, with subsequent increases in log-dec out to 70 krpm. The frequency-dependent model predicts an OSI of 10 krpm and no return to stability out to 70 krpm. The frequency-dependent models predict small changes in the rotor’s damped natural frequencies. The synchronously reduced model predicts large changes. The stability-analysis results show that a frequency-dependent labyrinth seal model should be used if the rotor surface speed approaches a significant fraction of the speed of sound. For the present example, observable discrepancies arose when $Rsω=0.26c0$.

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## Figures

Figure 3

(a) Radial dynamic-stiffness coefficient versus excitation frequency (preswirl ratio=0) and (b) circumferential dynamic-stiffness coefficient versus excitation frequency (preswirl ratio=0)

Figure 4

Predicted first acoustic damped natural frequency versus running speed (preswirl ratio=0)

Figure 5

Magnitude and phase plot of calculated and curve-fitted D transfer function (zero-preswirl, rotor speed 15,200 rpm)

Figure 6

Magnitude and phase plot of calculated and curve-fitted E transfer function (zero-preswirl, rotor speed 15,200 rpm)

Figure 7

Simple Jeffcott rotor

Figure 8

Magnitude plot for frequency-independent model (synchronously reduced coefficients) and frequency-dependent model (transfer-function model) for u0(0)=0.5

Figure 9

Log-dec versus u0(0) (rotor speed 15.2 krpm)

Figure 10

Log-dec versus u0(0) (rotor speed 40 krpm)

Figure 11

Log-dec versus u0(0) (rotor speed 70 krpm)

Figure 12

Log-dec versus rotor speed for u0(0)=0.5

Figure 13

Damped natural frequencies versus rotor speed for u0(0)=0.5

Figure 1

Axial view of labyrinth seal cavity and control volume

Figure 2

Radial view of labyrinth seal cavity and control volume

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