Rotordynamic Force Coefficients of Pocket Damper Seals

Ahmed Gamal

Rotordynamic Force Coefficients of Pocket Damper Seals

Ahmed Gamal

2006, Journal of Turbomachinery

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Abstract

This paper presents measured frequency dependent stiffness and damping coefficients for 12-bladed and 8-bladed pocket damper seals (PDS) subdivided into four different seal configurations. Rotating experimental tests are presented for inlet pressures at 69 bar (1000 psi), a frequency excitation range of 20–300 Hz, and rotor speeds up to 20,200 rpm. The testing method used to determine direct and cross-coupled force coefficients was the mechanical impedance method, which required the measurement of external shaker forces, system accelerations, and motion in two orthogonal directions. In addition to the impedance measurements, dynamic pressure responses were measured for individual seal cavities of the eight-bladed PDS. Results of the frequency dependent force coefficients for the four PDS designs are compared. The conclusions of the tests show that the eight-bladed PDS possessed significantly more positive direct damping and negative direct stiffness than the 12-bladed seal. The resu...

Figures (419)

Figure 1. Annular gas Seals in a Straight-Through Compressor dynamic behavior of machines can be predicted.

the top in between the last blade of the PDS and the journal.

Figure 4. Ransom’s Test rig (Ransom 1998) journal speeds, indicating the benefits of low destabilizing forces in the PDS. Further tests were conducted by Laos (1999), which involved both non-rotating and

Figure 5. Rotating Test Rig (Laos 1999) coefficients that possessed the same sign.

Figure 6. Ball Bearing Test Rig (Ertas and Vance 2004) by one-third the response without the PDS.

Figure 7. Twelve Bladed PDS Configurations

Figure 8. Eight Bladed PDS Configurations

Figure 9. Conventional Six Bladed PDS Configuration

Figure 10. New Fully Partitioned PDS Design

Figure 12. Conventional Six Bladed PDS Blade Modification

Table 1. Summarized PDS Geometry and Parameters

Figure 13. High Pressure Annular Gas Seal Test Rig inlet air is supplied at the center plenum (between the test seals) of the stator assembly.

Figure 14. High Pressure Test Rig Cross Section (Dawson, 2000)

Figure 15. Stator Assembly Isometric View

Figure 16. Stator Assembly Standard Views

Also note that each force equation has a contribution of direct coefficients and cross- part (determines stiffness) and an imaginary part (determines damping). direction equations of motion. The left hand side of Equations 3.1-3.4 is the effective force

Figure 18. Pseudorandom Force Excitation: 6 Bladed PDS Configurations tests are shown in the Appendix.

and Y directions are calculated at each test frequency from the four independent equations. Before determining the test seal coefficients, one must realize that the impedance

coupling to be destabilizing. The effective damping is defined in Equation 3.19-3.20. Past research on PDS has shown that the cross-coupled stiffness coefficients in the hat for circular orbits the Ky, and Ky, term must be have opposite for stiffness cross- stiffness cross-coupling force component F’, is tangential to the whirl orbit. This implies

fhe principle axis shift is illustrated in Figure 23 and Figure 24 shows the resultant conical

radial component F, is a maximum at points 2, 4, 6, and 8 and zero at points 1, 3, 5, and 7.

Figure 26. Same Sign Stiffness Cross-Coupling Force Components

Figure 27. Orbit Distortion from Same Sign Stiffness Cross-Coupling

generated cavity forces into the X and Y directions. Next, the matrix shown in Equatior 3.70-3.71), where F, is the pressure force. It is also important to decompose each pressure L

pressure induced dynamic cavity forces and the vibratory motion (Equations 3.72-3.75). The positive X and Y direction motion is illustrated in Figure 28, and also shown is the 3.69 can be solved for the seal coefficients in the frequency domain as a function of the

Figure 28. Dynamic Pressure Response to Vibratory Motion

Table 2. Data Acquisition Parameters direction. cavities at the 90 degree location. A shake test at 80 Hz was also performed in the >

Figure 29. Experimental Measurements: Pressure Response Method

Figure 30. Structural Baseline Test Set-up two configurations in Table 3.

Table 3. Baseline Test Matrix — Impact Hammer Tests curve fitting the real and imaginary components (Figure 32) of the dynamic stiffness. through the squirrel cages. The second method used to extract stiffness and damping wa:

Figure 31. Structural Compliance of Test Rig Stator Assembly did the damping for certain frequency ranges.

Figure 33. Test Rig Vibration Decay summary of the results for the impact hammer tests are shown in Table 4. Table 4. Structural Baseline Coefficients-Impact Hammer Tests

Table 5. Baseline Impedance Test Matrix The coefficients for these pressurized impedance tests are not only generated from without the squirrel cage is not expected. structural components, but also the fluid pressure forces generated between stator and rotor

Figure 34. Baseline Direct Coefficients Y Direction-Impedance Method as frequency is increased.

Figure 35. Baseline Cross-coupled Coefficients-Impedance Method

Table 6. Test Matrix-12 Bladed PDS Impedance Method increase with excitation frequency.

Figure 36. Twelve Bladed PDS 10,200 rpm: Direct Coefficients Impedance Method Inlet Plenum Pressure 1,000 psi (68.9 bar)

Figure 37. Twelve Bladed PDS 15,200 rpm: Direct Coefficients Impedance Method rotor-bearing system.

Figure 38. Twelve Bladed PDS: Cross-Coupled Stiffness Coefficients Impedance Method

Figure 39. Twelve Bladed PDS— Cross-coupled Damping Coefficients Impedance Method inability of cross-coupled damping to directly influence rotordynamic stability. 4.3 Eight Bladed PDS Force Coefficients Experimental results presented in this section are for the eight bladed sea

Table 7. Test Matrix-8 Bladed PDS Impedance Method Figures 40-42 display the direct damping and stiffness for all four seals at 10,200

Figure 40. Eight Bladed PDS-Direct Coefficients 10,200 rpm Impedance Method

Figure 41. Eight Bladed PDS-Direct Coefficients 15,200 rpm Impedance Method

Figure 42. Eight Bladed PDS-Direct Coefficients 20,200 rpm Impedance Method

Figure 43. Eight Bladed PDS-Cross-Coupled Stiffness Coefficients Impedance Method

Table 8. Test Matrix-Conventional 6 Bladed PDS Impedance Method

Figure 44. Conventional Six Bladed PDS-Direct Coefficients Impedance Method Inlet Plenum Pressure 1,000 psi (68.9 bar)

Figure 45. Conventional Six Bladed PDS-Cross-Coupled Coefficients Impedance Method

Inlet Plenum Pressure 1,000 psi (68.9 bar) Table 9. Test Matrix-Fully Partitioned 6 Bladed PDS Impedance Method

Figure 46. FP Six Bladed PDS-Direct Coefficients 10,200 rpm Impedance Method

Figure 47. FP Six Bladed PDS-Direct Coefficients 15,200 rpm Impedance Method

Figure 48. FP Six Bladed PDS-Cross-Coupled Coefficients 4.6 Comparison of Impedance Test Results

Table 10. Configurations Tested Using the Impedance Method

Figure 49. Direct Damping Coefficients: Impedance Method

Figure 50. Direct Stiffness Coefficients: Impedance Method

Figure 52. Pront Control Panel: NI 4472-16 Channel Simultaneous DAQ Board

Figure 53. LABVIEW Front Panel-8 Bladed PDS

Figure 54. Phasor Plot for Diverging and Straight-Through Cavities

Figure 55. Static Seal Cavity Pressure Measurements the following results were linear unidirectional vibration tests based on a single degree of

Figure 56. Peak Pressures and Vibration Amplitudes 8 Bladed PDS 1:1 Clr Ratio

Figure 57. Stiffness and Damping 8 Bladed 1:1 Clr Ratio

Figure 58. Pressure Phase and Force Density: 8 Bladed PDS 1:1 Clr Ratio

Table 12. Cavity Flow Parameters: 8 Bladed 1:1

Figure 59. Asynchronous Pressure Frequencies 8 Bladed PDS 1:1 Clr Ratio One major difference between the two cavities is the static pressure, which alsc

Figure 60. Stiffness and Damping 8 Bladed 1:1.5 Clr Ratio ~1.2 mils pk-pk amplitudes (1.2 mils and 1.8 mils) are almost identical in magnitude and trend. Vibration and pressure amplitudes for each of the diverging seal test are shown in Figure 62.

Figure 61. Stiffness and Damping 8 Bladed PDS 1:1.5 Clr Ratio ~1.8 mils pk-pk The stiffness results also show significant change due to the notch. whereas the phase in Figures 63-64 exhibits a weaker dependence on increasing frequency.

Figure 62. Peak Pressures and Vibration 8 Bladed 1:1.5 Clr Ratio ~1.2 mils pk-pk

Figure 63. Phase and Force Density 8 Bladed PDS 1:1.5 Clr Ratio ~1.2 mils pk-pk

Figure 64. Phase and Force Density 8 Bladed PDS 1:1.5 Clr Ratio ~1.8 mils pk-pk

Figure 65. Pressure Spectrum for Diverging 8 Bladed PDS (200 Hz

Figure 66. Conventional Six Bladed PDS Single Row of Cavities: Transducer Installation static pressure measurements aided in determining discharge coefficients. For determining seal coefficients only the dynamic measurements were used; whereas, the

Figure 67. Front Panel Dynamic Measurements Display: 0 Degree Angle X Direction Excitation 40 Hz

Figure 69. Dynamic Pressure Phase: 45 Degree Location

Figure 70. Direct Stiffness and Damping: Cavity Coefficients Conventional Six Bladed PDS

Figure 71. Cavity Coefficients: 0 Degree Position Cross-Coupled Stiffness and Damping

Figure 72. Row Coefficients: 45 Degree Position Cross-Coupled and Direct Stiffness

Figure 73. Row Coefficients: 45 Degree Position Cross-Coupled and Direct Damping

Figure 74. Conventional 6 Bladed PDS Y Dir. Direct Coefficients- Pressure Method

Figure 75. Conventional 6 Bladed PDS X Dir. Direct Coefficients- Pressute Method

Figure 76. Conventional 6 Bladed PDS Y Dir. Cross-Coupled Coefficients-Pressure Method

Figure 77. Conventional 6 Bladed PDS X Dir. Cross-Coupled Coefficients-Pressure Method

Figure 78. Local Pressure Modulation-Inactive Plenums 2 and 4, 90 and 0 Degrees

Figure 79. Local Pressure Modulation-Inactive Plenums 2 and 4, 270 Degrees

Figure 80. X Direction Excitation- 0 Degree Position 100 Hz

Figure 81. X Direction Excitation- 0 Degree Position 200 Hz

Figure 82. X Direction Excitation- 45 Degree Position 100 Hz

Table 13. Inactive Plenum Pressure Magnitudes

Figure 86. Single Row of Pockets FP 6 Bladed PDS: Pressure Transducer Installation from Figure 80 to Figure 87. higher pressure modulations that are approximately 10 times larger, when comparing results

Figure 87. Pully Partitioned Seal: Y Direction Excitation 100 Hz -90 Degree Angle Position

Rotordynamic Coefficients Row Values Y Direction Rotordynamic Coefficients Row Values X Direction Figure 88. Front Panel: Data Reduction Fully Partitioned Six Bladed PD

Figure 89. Stand Alone Cavities- Diverging and Converging Configurations of the five cavities in the 0 degree location.

Figure 90. Phasor Diagram- Stand Alone Cavities

direction force coefficients for both the FP design and the conventional design.

Figure 92. Direct Damping-Cavity Coefficients Fully Partitioned Seal 0 Degree Position

Figure 93. Direct Stiffness-Cavity Coefficients Fully Partitioned Seal 0 Degree Position

Figure 94. Direct Force Coefficients Y Dir: FP PDS vs. Conventional PDS

Figure 95. Direct Force Coefficients X Dir: FP PDS vs. Conventional PDS

Figure 96. Cross-Coupled Force Coefficients Y Dir: FP PDS vs. Conventional PDS

Figure 97. Cross-Coupled Force Coefficients X Dir: FP PDS vs. Conventional PDS become positive. Cross-coupled damping coefficients are positive and also possess the sami In the previous sections showing the pressure signals in the time domain, there exists

Table 14. Gamal’s Results for Static Direct and Cross-coupled Stiffness (2003) directions. Table 14 summarizes his results.

Figure 99. Benckert and Wachter’s Test Rig and Experimental Results (1980

Figure 100. Static Pressure Measurements Displacement Pattern: 15,200 rpm

Table 15. Cavity Pressure Measurements: Rotating Displacement Test Results

Figure 101. Static Pressure Measurements: Rotor Offset Displacement Tests bottom pocket, Ph is the high inlet pressure and Pb is the lower back pressure.

Figure 103. Stand-Alone Cavities vs. Sequential Cavities must utilize a coupled analysis between the conservation of mass equations.

Figure 104. Single Row of Three Pressurized Pockets rotor the flow through the seal is in a steady state condition and can be described using

ras, the enthalpy (Equation 6.9) can be manipulated using the ideal gas law, definitions for

A useful parameter to note from Equation 6.19 is the non-dimensional flow coefficient (y) the ideal gas law (6.18) can be written as Equation 6.19. defined in Equation 6.20. The flow coefficient is helpful when determining the choked flow condition across a constriction. Choking through a nozzle or an area constriction occurs

6.2 Control Volume Analysis for Compressible Flow cavity are calculated using (6.24-6.27) in an iterative method, where C, is the discharge

function of arbitrary rotor displacement.

Figure 107. PDS Cavity Clearance Area for Arbitrary Rotor Displacement

Figure 108. Area of a Sector Using Polar Coordinates which is a function of 8, one can calculate the area within the sector using Equation 6.53

Figure 109. PDS Clearance Area for Cavity-Centered Rotor Displacement and the cavity centered displacement amplitude X. Now consider a rotor displacement where the displacement 6 is centered (Equation 6.61)

The total flow rate (steady state and dynamic) can be found by expanding Equations 6.64-

The resultant dynamic cavity pressures are composed of a sine and a cosine component, as rewrite Equations 6.75-6.77 in the form shown in (6.90-6.92). Substituting the results from Equations 6.84-6.89 into Equations 6.90-6.92 results in Equations 6.93-6.95. Note that

The remaining terms from 6.75-6.77 are defined in 6.80-6.83.

Figure 110. Dynamic Cavity Pressures-Theoretical Predications Fully Partitioned PDS

Table 16. Cases for Fully Partitioned PDS-Variable Pitch Ratio

Figure 111. Direct Damping Coefficients for Varying Pitch Ratios * Axial Pitch Ratio=(Diverging Cavity Axial Pitch) /(Converging Cavity Axial Pitch)

Figure 112. Direct Stiffness Coefficients for Varying Pitch Ratios positive direct stiffness at higher frequencies.

Figure 113. Theory vs. Impedance Method: 12 Bladed PDS Direct Damping however do not agree well with the theory.

Figure 114. Theory vs. Impedance Method: 12 Bladed PDS Direct Stiffness

Figure 115. Theory vs. Impedance Method: 8 Bladed PDS Stiffness and Damping where the experiment starts to deviate away from the theory.

Figure 116. Theory vs. Experiments: Conventional 6 Bladed PDS Stiffness and Damping coefficients. the impedance method has higher damping coefficients and lower valued stiffnes:

Figure 117. Theory vs. Experiments: FP 6 Bladed PDS Stiffness and Damping

Figure 118. Six Bladed PDS Experimental Results

Figure 119. Static Cavity Pressure Measurements vs Theory: FP PDS

Table 17. Leakage Results for 6 Bladed PDS Configurations

Table 18. Inlet Blade and Exit Blade Discharge Coefficients

Curve Fitting Program for Modal Vibration Tests

Phase Conversion and Complex Number Generation

Direct and Cross-Coupled Impedance Calculations (90, 270 Degree position)

Direct and Cross-Coupled Impedance Calculations (45, 135, 225, 315 Degree position)

Direct and Cross-Coupled Impedance Calculations (0, 180 Degree position)

Conversion of Time Data to FFT Magnitudes and Phases (0, 45, 90 degree)

Conversion of Time Data to FFT Magnitudes and Phases (135 degree)

Conversion of Time Data to FFT Magnitudes and Phases (180, 225, 270 degree)

Solution Matrix-Dynamic Pressure Components P:

Converging Cavities: Dynamic Cavity Pressure and Rotor Vibration

Cavity Pressure Phase Respect to X Direction

Cavity Pressure Phase Respect to Y Direction

Y Direction Cavity Direct Coefficients & X Direction Cavity Cross-coupled Coefficients

X Direction Cavity Direct Coefficients & Y Direction Cavity Cross-coupled Coefficients

Total Row Coefficients- Y Direction Direct and X Direction Cross-coupled

Total Row Coefficients- X Direction Direct and Y Direction Cross-coupled

Cavity Phase Respect to X direction Excitation

Cavity Phase Respect to Y direction Excitation

Y Direction Direct Damping & X-Direction Cross-coupled Damping

Y Direction Direct Stiffness & X-Direction Cross-coupled Stiffness

X Direction Direct Damping & Y-Direction Cross-coupled Damping

X Direction Direct Stiffness & Y-Direction Cross-coupled Stiffness

Y Direction Direct Coefficients & X-Direction Cross-coupled Coefficients

X Direction Direct Coefficients & Y-Direction Cross-coupled Coefficients

Y Direction Direct Damping & X-Direction Cross-coupled Damping 6 Bladed Fully Partitioned Seal Cavity Coefficients: 135 Degree Position

80 Hz 8 Bladed No Divergence 60 Hz 8 Bladed No Divergence

40 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.2 mils pk -pk)

80 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.2 mils pk -pk) 60 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.2 mils pk -pk)

120 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.2 mils pk -pk) 100 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.2 mils pk -pk)

160 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.2 mils pk -pk) 140 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.2 mils pk -pk)

20 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk) 200 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.2 mils pk-pk)

40 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk)

60 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk)

80 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk)

100 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk)

140 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk) 120 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk)

180 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk) 160 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk)

200 Hz 8 Bladed Diverging Seal 1:1.5 Clr Ratio (Vibration amp. 1.8 mils pk-pk)

20 Hz 0 Degree Position X Direction Excitation

40 Hz 0 Degree Position X Direction Excitation

80 Hz 0 Degree Position X Direction Excitation

100 Hz 0 Degree Position X Direction Excitation

140 Hz 0 Degree Position X Direction Excitation

160 Hz 0 Degree Position X Direction Excitation

200 Hz 0 Degree Position X Direction Excitation

40 Hz 0 Degree Position Y Direction Excitation

80 Hz 0 Degree Position Y Direction Excitation

100 Hz 0 Degree Position Y Direction Excitation

140 Hz 0 Degree Position Y Direction Excitation

160 Hz 0 Degree Position Y Direction Excitation

200 Hz 0 Degree Position Y Direction Excitation

20 Hz 45 Degree Position X Direction Excitation

40 Hz 45 Degree Position X Direction Excitation

80 Hz 45 Degree Position X Direction Excitation

100 Hz 45 Degree Position X Direction Excitation

140 Hz 45 Degree Position X Direction Excitation

160 Hz 45 Degree Position X Direction Excitation

200 Hz 45 Degree Position X Direction Excitation

20 Hz 45 Degree Position Y Direction Excitation

40 Hz 45 Degree Position Y Direction Excitation

80 Hz 45 Degree Position Y Direction Excitation

100 Hz 45 Degree Position Y Direction Excitation

140 Hz 45 Degree Position Y Direction Excitation

160 Hz 45 Degree Position Y Direction Excitation

200 Hz 45 Degree Position Y Direction Excitation

20 Hz 90 Degree Position X Direction Excitation

40 Hz 90 Degree Position X Direction Excitation

80 Hz 90 Degree Position X Direction Excitation

100 Hz 90 Degree Position X Direction Excitation

140 Hz 90 Degree Position X Direction Excitation

160 Hz 90 Degree Position X Direction Excitation

200 Hz 90 Degree Position X Direction Excitation

20 Hz 90 Degree Position Y Direction Excitation

40 Hz 90 Degree Position Y Direction Excitation

80 Hz 90 Degree Position Y Direction Excitation

100 Hz 90 Degree Position Y Direction Excitation

140 Hz 90 Degree Position Y Direction Excitation

160 Hz 90 Degree Position Y Direction Excitation

200 Hz 90 Degree Position Y Direction Excitation

20 Hz 0 Degree Position Y Direction Excitation

20 Hz 45 Degree Position Y Direction Excitation 200 Hz 45 Degree Position X Direction Excitation

ROTOR REMOVED: TES RIG SIDE VIEW ROTOR REMOVED: 12 BLADED ASSEMBLY

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