Influence coefficient method is used in the balancing calculation. The theory is based on two papers:
Since the least square method is used to solve the simultaneous equations, the Number of Speed Points times the Number of Measured Probes must be greater than or equal to the Number of Balancing Planes. (NsXNm >= Nb).
To use the influence coefficient method, no prior knowledge in rotor mode is required. However, trial weights are required to obtain the influence coefficients.
All the inputs are self-explanatory. They are briefly described below:
1. Number of Balancing Planes: Nb
The balancing planes are the planes along the rotor where the trial weights and balancing corrections are applied. Note that the trial weight can be left-in or removed after the trial run.
2. Number of Measured Probes: Nm
The measurement probes are where the vibrations are taken and recorded. The purpose of the balancing is to find the optimal balancing corrections at the balancing planes such that the vibrations at the measurement probes are minimized.
3. Number of Speeds/Cases: Ns
The number of speeds or cases allows for different speeds or cases, such as idle speed, full speed, full load, unloads, etc…
4. Runout Compensation
Runout can be included or excluded in this balancing calculation.
5. Comments
Up to 3 comment lines can be used to describe the system under study.
6. Shaft Rotation: CCW or CW
7. Phase: Lag or Lead
8. 0 degree: Up or Right.
0 degree position defines the reference mark where all the angles (phases) are measured from.
9. Weighting Factors
Weighting factor allows one to strengthen or weaken the data from the measurement probes or speeds. For example, one may use higher weighting factors for the probes where the critical components are located and/or speeds where the rotor will be operated most of the time. Weighting factor zero indicates that the specific probe data will not be included in the calculation.
Example 1:
Handbook of Rotordynamics Example 3.11, pp 3.90
Runout Compensation is included
The first trial weight is removed afterward, 2nd trial weight is left-in
***** Number of Speeds or Cases : 1
***** Number of Balancing Planes : 2
***** Number of Measurement Probes: 2
***** Runout (slow-roll vectors) *****
Probe Amplitude Phase Angle
1 0.50000 272.00
2 0.40000 123.00
====== Initial Response (Without Trails) ======
Speed Probe Amplitude Phase Angle
1 1 1.8000 148.00
1 2 3.6000 115.00
************** Trial Unbalance Run: 1 **************
Plane Amplitude Phase Angle Afterward
1 4.9000 120.00 Remove
--------- Response to Trial Unbalance ---------
Speed Probe Amplitude Phase Angle
1 1 1.1000 178.00
1 2 2.0000 98.000
************** Trial Unbalance Run: 2 **************
Plane Amplitude Phase Angle Afterward
2 4.9000 220.00 Left-In
--------- Response to Trial Unbalance ---------
Speed Probe Amplitude Phase Angle
1 1 2.1000 98.000
1 2 3.7000 102.00
*** Weighting Factors for probes and speeds ***
Probe Weighting Factor
1 1.0000
2 1.0000
Speed Weighting Factor
1 1.0000
===================================================
<<<<<<<< Total Balance Correction >>>>>>>>
Correction Required to Balance the Rotor
Plane No. Amplitude Phase Angle
1 7.4873 84.957
2 5.3209 179.73
<<<<<<<<< Trim Balance Correction >>>>>>>>>>
Correction Required if Trial Weight Left-in
Trim Balance = Total Balance - Trial Weight
Plane No. Amplitude Phase Angle
2 3.5402 116.28
===================================================
___________________________________________________
********** The Influence Coefficients ***********
Influence Coef.
Trial-Run Speed Probe Amplitude Phase
1 1 1 0.20617 175.
1 1 2 0.36446 194.
2 1 1 0.34092 182.
2 1 2 0.16986 165.
___________________________________________________
===================== Predicted Residual Response ========================
----- WithOUT Runout ----- ------ With Runout ------
Speed Probe Amplitude Phase Angle Amplitude Phase Angle
1 1 0.0000 0.0000 0.50000 272.00
1 2 0.0000 0.0000 0.40000 123.00
===========================================================================Example 2:
Example from ROTORBAL Example 5.3.2 - 70 MW Gas Turbine
2 probes at 6 speeds
No runout
***** Number of Speeds or Cases : 6
***** Number of Balancing Planes : 2
***** Number of Measurement Probes: 2
***** NO Runout Compensation
====== Initial Response (Without Trails) ======
Speed Probe Amplitude Phase Angle
1 1 1.7000 339.00
1 2 4.6000 54.000
2 1 2.8000 226.00
2 2 6.7000 10.000
3 1 3.9000 145.00
3 2 3.7000 333.00
4 1 4.5000 103.00
4 2 4.7000 302.00
5 1 5.4000 74.000
5 2 6.5000 113.00
6 1 1.9800 98.000
6 2 5.7000 114.00
************** Trial Unbalance Run: 1 **************
Plane Amplitude Phase Angle Afterward
1 20.000 359.00 Left-In
--------- Response to Trial Unbalance ---------
Speed Probe Amplitude Phase Angle
1 1 2.6000 313.00
1 2 5.9000 7.0000
2 1 3.9000 232.00
2 2 4.4000 4.0000
3 1 4.2000 160.00
3 2 2.8000 340.00
4 1 3.5000 120.00
4 2 5.4000 325.00
5 1 4.1000 73.000
5 2 3.7000 97.000
6 1 1.5000 141.00
6 2 3.1000 99.000
************** Trial Unbalance Run: 2 **************
Plane Amplitude Phase Angle Afterward
2 10.000 270.00 Left-In
--------- Response to Trial Unbalance ---------
Speed Probe Amplitude Phase Angle
1 1 1.8000 319.00
1 2 4.3000 15.000
2 1 3.1000 244.00
2 2 3.4000 9.0000
3 1 3.2000 107.00
3 2 1.9000 329.00
4 1 2.4000 122.00
4 2 4.4000 330.00
5 1 2.4000 61.000
5 2 3.5000 101.00
6 1 1.0200 170.00
6 2 3.1100 104.00
*** Weighting Factors for probes and speeds ***
Probe Weighting Factor
1 1.0000
2 1.0000
Speed Weighting Factor
1 1.0000
2 1.0000
3 1.0000
4 1.0000
5 1.0000
6 1.0000
===================================================
<<<<<<<< Total Balance Correction >>>>>>>>
Correction Required to Balance the Rotor
Plane No. Amplitude Phase Angle
1 26.867 347.83
2 20.639 267.30
<<<<<<<<< Trim Balance Correction >>>>>>>>>>
Correction Required if Trial Weight Left-in
Trim Balance = Total Balance - Trial Weight
Plane No. Amplitude Phase Angle
1 8.2159 319.71
2 10.660 264.77
===================================================
___________________________________________________
********** The Influence Coefficients ***********
Influence Coef.
Trial-Run Speed Probe Amplitude Phase
1 1 1 0.65281E-01 279.
1 1 2 0.21766 317.
1 2 1 0.57655E-01 248.
1 2 2 0.11846 202.
1 3 1 0.54915E-01 228.
1 3 2 0.49103E-01 134.
1 4 1 0.77078E-01 242.
1 4 2 0.10636 26.
1 5 1 0.65130E-01 258.
1 5 2 0.15575 313.
1 6 1 0.67568E-01 230.
1 6 2 0.14110 312.
2 1 1 0.83143E-01 210.
2 1 2 0.17475 257.
2 2 1 0.10809 105.
2 2 2 0.10554 258.
2 3 1 0.34210 118.
2 3 2 0.10027 271.
2 4 1 0.11046 26.
2 4 2 0.10867 214.
2 5 1 0.18221 359.
2 5 2 0.32108E-01 318.
2 6 1 0.78362E-01 12.
2 6 2 0.27106E-01 279.
___________________________________________________
======================= Predicted Residual Response ========================
----- WithOUT Runout ----- ------ With Runout ------
Speed Probe Amplitude Phase Angle Amplitude Phase Angle
1 1 1.0968 310.43 1.0968 310.43
1 2 2.6067 358.29 2.6067 358.29
2 1 2.8744 257.03 2.8744 257.03
2 2 1.7963 40.778 1.7963 40.778
3 1 4.8509 66.019 4.8509 66.019
3 2 0.73525 317.09 0.73525 317.09
4 1 1.6051 154.41 1.6051 154.41
4 2 4.3282 341.01 4.3282 341.01
5 1 0.55031 343.11 0.55031 343.11
5 2 2.1098 114.00 2.1098 114.00
6 1 1.6347 207.09 1.6347 207.09
6 2 2.0997 118.93 2.0997 118.93
=============================================================================
=============================================================================Example 3:
W501 Gas Turbine 108 MW
2 Speeds, 3070 RPM, 3600 RPM
4 Probes and 2 Balancing Planes
***** Number of Speeds or Cases : 2
***** Number of Balancing Planes : 2
***** Number of Measurement Probes: 4
***** NO Runout Compensation
====== Initial Response (Without Trails) ======
Speed Probe Amplitude Phase Angle
1 1 1.1000 143.00
1 2 3.7000 268.00
1 3 1.7000 287.00
1 4 0.80000 156.00
2 1 3.7000 98.000
2 2 7.5000 41.000
2 3 3.9000 1.0000
2 4 4.2000 209.00
************** Trial Unbalance Run: 1 **************
Plane Amplitude Phase Angle Afterward
1 19.500 185.00 Left-In
--------- Response to Trial Unbalance ---------
Speed Probe Amplitude Phase Angle
1 1 2.5000 70.000
1 2 1.9000 216.00
1 3 2.9000 23.000
1 4 1.9000 216.00
2 1 2.5000 2.0000
2 2 6.8000 350.00
2 3 9.4000 359.00
2 4 6.3000 202.00
************** Trial Unbalance Run: 2 **************
Plane Amplitude Phase Angle Afterward
2 7.4000 70.000 Left-In
--------- Response to Trial Unbalance ---------
Speed Probe Amplitude Phase Angle
1 1 2.3000 301.00
1 2 4.4000 216.00
1 3 2.5000 294.00
1 4 0.80000 139.00
2 1 1.7000 12.000
2 2 5.6000 355.00
2 3 5.3000 344.00
2 4 3.8000 181.00
*** Weighting Factors for probes and speeds ***
Probe Weighting Factor
1 1.0000
2 1.2000
3 1.1000
4 1.0000
Speed Weighting Factor
1 1.0000
2 1.2000
===================================================
<<<<<<<< Total Balance Correction >>>>>>>>
Correction Required to Balance the Rotor
Plane No. Amplitude Phase Angle
1 21.356 134.01
2 9.3609 13.616
<<<<<<<<< Trim Balance Correction >>>>>>>>>>
Correction Required if Trial Weight Left-in
Trim Balance = Total Balance - Trial Weight
Plane No. Amplitude Phase Angle
1 17.665 74.947
2 8.1048 324.12
===================================================
___________________________________________________
********** The Influence Coefficients ***********
-- Before application of Weighting Factors --
Influence Coef.
Trial-Run Speed Probe Amplitude Phase
1 1 1 0.12406 219.
1 1 2 0.15077 294.
1 1 3 0.18008 227.
1 1 4 0.84732E-01 56.
1 2 1 0.23984 125.
1 2 2 0.31737 95.
1 2 3 0.28226 173.
1 2 4 0.11241 3.
2 1 1 0.58558 204.
2 1 2 0.33784 146.
2 1 3 0.51293 174.
2 1 4 0.25519 350.
2 2 1 0.11851 92.
2 2 2 0.17773 78.
2 2 3 0.60743 127.
2 2 4 0.41498 338.
___________________________________________________
======================= Predicted Residual Response ========================
----- WithOUT Runout ----- ------ With Runout ------
Speed Probe Amplitude Phase Angle Amplitude Phase Angle
1 1 3.9669 49.648 3.9669 49.648
1 2 1.9065 348.46 1.9065 348.46
1 3 2.2628 79.442 2.2628 79.442
1 4 0.22558 305.47 0.22558 305.47
2 1 1.8109 9.5156 1.8109 9.5156
2 2 1.8740 232.93 1.8740 232.93
2 3 3.3169 159.79 3.3169 159.79
2 4 1.8632 30.989 1.8632 30.989
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