TORSIONAL VIBRATION ANALYSIS - BASICS

Pranabesh De Choudhury, Ph.D. Pran RDA Consulting Inc.

Greensburg, PA 15601

OUTLINE Basic Concepts Torsional Analysis Procedure and Purpose Simple Torsional Systems

Single Inertia Two Inertias Three Inertias

Holzer Method Method for the analysis of multi-inertia lumped-parameter systems. Applicable for free and forced vibrations, systems with or without damping.

Geared Systems

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OUTLINE (Continued) Torsional Modeling

Commonly Used Formulae For Torsional Modeling Shaft Inertia Distribution Shaft Penetration Effect Effect of Fillet Radius

Torsional Stress Calculations Sample Output Torsional Critical Speed Analysis Typical Mode Shape Plots Using Excel Comparison of Calculated Torsional Natural

Frequencies with Test Data References

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BASIC CONCEPTS

Torsional vibration is a periodic motion of elastic shafts with circular rotors (compressor impellers, turbine discs etc.) rigidly attached to the shafts.

Torsional vibrations are controlled by the polar moments of inertias of attached rotors to the shafts and the torsional stiffness between the attached rotors.

Number of oscillations per unit of time is defined as the torsional natural frequencies.

Torsional vibrations are caused by various types of torsional excitations, e.g., gear imperfections, motor air gap torque during start up and short circuit, fluctuating torques in rotating machinery e.g., compressors and turbines.

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TORSIONAL ANALYSIS PROCEDURE AND PURPOSE

Holzer method is widely used for accurate torsional critical speed analysis of turbo-machinery systems.

The method is also used for steady state torsional response analysis around a torsional natural frequency having insufficient margin from any exciting frequency.

Stress analysis is conducted from the results obtained from steady state response analysis to assure adequate design of torsional system.

To perform a torsional analysis requires accurate modeling of torsional system.

Determination of torsional critical speeds is necessary to assure adequate margins from exciting frequencies to avoid potential torsional vibration problems. The required margins are to be obtained to satisfy API 617 specifications.

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SINGLE INERTIA SYSTEM

J IS THE INERTIA (LB-IN-SEC2) K IS THE TORSIONAL STIFFNESS (LB-IN/RAD) EQUATION OF MOTION J + K =0 (NO DAMPING) SET = 0 COS t SOLUTION -2 J 0 + K 0 = 0 = K/J RAD./SEC. IS THE TORSIONAL NATURAL FREQUENCY

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K

J

TWO INERTIA SYSTEM

EQUATIONS OF MOTION: J1 1 + K (1 - 2) = 0 J2 2 + K (2 - 1) = 0 LET 1 = 01 ejt AND 2 = 02 ejt CHARACTERISTICS (K-2 J1) 01 K 02 = 0 EQUATIONS - K 01 + (K-2 J2) 02 = 0 RESONANCE OCCURS WHEN THE SYSTEM DET = 0 DET = K - 2 J1 -K = 0 -K K - 2 J2 SOULTION = K(J1 + J2)/(J1 +J2) RAD./SEC. EXAMPLE PROBLEM (TURBINE COMPRESSOR) J1 = 138.4824 LB.IN.SEC2 J2 =26.50472 LB.IN.SEC2 K = 3.8*10E6 LB.IN./RAD

CALCULATED TORSIONAL CRITICAL SPEED = 3944 CPM Pran RDA Consulting Inc.

K

J2 J1

THREE INERTIA SYSTEM

J1 1 + K1 (1 - 2) = 0 J2 2 + K1 (2 - 1) + K2 (2 - 3) = 0 J3 3 + K2 (3 - 2) = 0 NOW WE HAVE A 3 x 3 SYSTEM DETERMINANT. THE SOLUTION IS OF THE FORM: a4 + b 2 + c = 0 SOLUTION : 2 = [ -b +/- (b2 4 ac)] / 2a WHERE a = 1, b = - K1/J1 - K2 /J2 - (K1 + K2)/J2 c =(K1K2)/(J1 J2 J3)*(J1+J2+J3 ) NOTICE THAT TWO VALUES OF EXIST. THE FIRST CRITICAL SPEED AND THE SECOND. EXAMPLE PROBLEM (TURBINE COUPLING - COMPRESSOR) J1 = 138.2064 LB.IN.SEC2 J2 =2.551244 LB.IN.SEC2 J3=25.22953 LB.IN.SEC2 K1 = 7.35*10E6 LB.IN./RAD K2 = 7.67*10E6 LB.IN./RAD CALCULATED FIRST TORSIONAL CRITICAL SPEED = 3975 CPM CALCULATED SECOND TORSIONAL CRITICAL SPEED = 23530 CPM. One can thus estimate torsional natural frequencies using these equations quickly. However to get accurate answers, different methods will be required.

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K1

J3 J1 J2

K2

N INERTIA SYSTEM (Holzer Method)

If the system considered, has N disks and Is a natural frequency then the sum of inertia torques given below must go to zero N Ji i 2 =0 i=1 If the system considered has N disks and N-1 shafts then the angular displacements are given by j-1 j = i-1 - 2 / K(j-1) Ji i 2 j= 2,3,.. N i=1 ITERATIVE PROCEDURE IS FOLLOWED, ASSUMING DIFFERENT VALUES.

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J1 J2 J3 J4 J5 J6 JN

K1 K2 K3 K5 KN

GEARED SYSTEM

MODELING PROCESS REQUIRES THE FOLLOWING

System Description Shaft Material Properties Shaft Diameters and Section Lengths Lumped Wr2 Values Associated Damping Gear Ratio(s) If used in the System Coupling Data Driver/Motor Data Driven Machine Data Gear Data

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COMMONLY USED FORMULAE FOR TORSIONAL MODELING

SHAFT INERTIA DISTRIBUTION

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J1

J2

J3

J4

J1+J2

J3+ J4

KSHAFT

KSHAFT

NO SHAFT INERTIA

SHAFT PENETRATION EFFECT

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1/3 L 2/3 L

L

EFFECT OF FILLET RADIUS ON TORSIONAL STIFFNESS

D1 = 10.0 r/R1 = 0.1 R1 = 5.0 D2/D1 = 1.5 D2 = 15.0 L / D1 = 0.067 r= 0.5 L = 0.67

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Effect of Corner Fillet Radius on Torsional Stiffness

r/R1=0.0

r/R1=0.1

r/R1=0.2

r/R1=0.3

r/R1=0.4

r/R1=0.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3D2/D1

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TORSIONAL STRESS CALCULATIONS

Unidirectional Stress is:

R = (63025 x Hpt / RPM) x (Do/2) (Do 4 Di 4)/32 Where Hpt is the transmitted power

Alternating Stress is: P = (63025 x Hpr x Ap x Pct/RPM) x (Do/2) (Do 4 Di 4)/32 Hpr = Rated Power, Ap = Amplification factor Pct = Per cent of Rated Torque

Allowable Vibratory Stress is: A = f (u , R , YP , Kts, C) Where: u = Ultimate minimum tensile strength of shaft material Kts = Stress concentration factor C= Number of stress cycles R = Unidirectional stress YP = Yield point of the shaft end material in tension

Safety Factor = A / P

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SAMPLE OUTPUT - TORSIONAL CRITICAL SPEED ANALYSIS

Copyright PRAN RDA Consulting Inc. CRITICAL SPEED SAMPLE PROBLEM PDC 4/30/12 NO. OF MAJOR INERTIAS = 12 ROTOR INPUT DATA: GEAR OUT DELTA INS. WR2VB GAMMA G STIFFNESS SR MAJ. CTRL. RATIO DIA. L DIA. *10E6 *10E6 1.000 2.76 1.80 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 1.000 4.72 6.60 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 1.000 0.00 0.00 0.00 10224.0 0.000 0.00000 0.00000 0.000 1 0 1.000 4.72 11.00 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 7.50 10.38 0.00 0.0 0.283 11.80000 0.00000 0.062 0 0 1.000 8.10 5.92 0.00 0.0 0.283 11.80000 0.00000 0.062 0 0 1.000 9.90 8.10 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 9.94 3.50 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 10.00 23.70 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 0.00 0.00 0.00 1338480.0 0.000 0.00000 0.00000 0.000 2 0 1.000 10.00 23.70 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 9.94 3.50 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 9.90 8.10 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 8.10 5.92 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 7.50 10.28 0.00 0.0 0.283 11.80000 0.00000 0.062 0 0 1.000 7.38 11.67 0.00 0.0 0.283 11.80000 0.00000 0.062 0 0 1.000 0.00 0.00 0.00 29744.5 0.000 0.00000 0.00000 0.000 3 0 1.000 0.00 0.00 0.00 0.0 0.000 0.00000 38.36070 0.000 0 0 1.000 0.00 0.00 0.00 94104.5 0.000 0.00000 0.00000 0.000 4 0 1.000 6.00 2.92 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 1.000 6.00 7.58 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 40.79 4.42 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 1.000 0.00 0.00 0.00 683178.9 0.000 0.00000 0.00000 0.000 5 0

5.290 6.94 1.37 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 5.290 7.71 5.26 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 5.290 4.00 7.36 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 5.290 2.76 3.82 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 5.290 6.12 1.00 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 5.290 2.76 1.67 0.00 0.0 0.283 11.80000 0.00000 0.000 0 0 5.290 2.76 0.83 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 0.00 0.00 0.00 765.0 0.000 0.00000 0.00000 0.000 6 0 5.290 0.00 0.00 0.00 0.0 0.000 0.00000 14.20000 0.000 0 0 5.290 0.00 0.00 0.00 765.0 0.000 0.00000 0.00000 0.000 7 0 5.290 2.50 0.50 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 2.38 0.12 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 3.75 0.93 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 3.87 3.10 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 4.00 7.54 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 4.02 1.82 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 4.12 0.62 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 4.00 0.12 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 4.13 0.40 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 4.44 0.44 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 4.38 2.06 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 6.00 1.90 0.00 0.0 0.283 11.35000 0.00000 0.320 0 0 5.290 7.50 0.67 6.00 0.0 0.283 11.35000 0.00000 0.120 0 0 5.290 6.80 1.90 6.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 0.67 7.50 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 6.94 0.25 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 8.10 0.25 7.75 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 0.31 7.75 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 1.87 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 0.00 0.00 0.00 9587.0 0.000 0.00000 0.00000 0.000 8 0 5.290 7.50 1.87 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 4.83 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 12.82 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 0.00 0.00 0.00 5805.0 0.000 0.00000 0.00000 0.000 9 0 5.290 8.10 4.83 7.50 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 1.30 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 2.35 7.50 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 7.53 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0

5.290 0.00 0.00 0.00 7625.0 0.000 0.00000 0.00000 0.000 10 0 5.290 8.10 2.35 7.50 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 1.53 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 2.44 7.50 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 8.07 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 0.00 0.00 0.00 7862.0 0.000 0.00000 0.00000 0.000 11 0 5.290 8.10 2.44 7.50 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 1.65 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 4.03 7.50 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 11.03 0.00 0.0 0.283 11.35000 0.00000 0.000 0 0 5.290 0.00 0.00 0.00 7071.0 0.000 0.00000 0.00000 0.000 12 2 5.290 8.10 4.03 7.50 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 2.67 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 9.10 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 7.50 0.11 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 6.94 0.25 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 0.11 7.75 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 0.25 7.75 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 8.10 1.24 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 6.80 1.90 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 4.38 2.06 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 4.44 0.44 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 4.13 0.40 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 4.00 0.12 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 4.12 0.62 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 4.02 1.82 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 4.00 4.68 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 3.74 1.46 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 3.19 0.74 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 3.16 1.00 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 3.00 0.12 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 3.12 1.50 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0 5.290 3.00 0.34 0.00 0.0 0.283 0.00000 0.00000 0.000 0 0

STATION WR**2 STIFFNESS NO. LB. IN2 LB. IN./RAD. 1 0.104226538E+05 0.369885975E+08 2 0.136056027E+07 0.950963281E+08 3 0.312949252E+05 0.383607000E+08 4 0.943461467E+05 0.198015492E+09 5 0.104178066E+07 0.256980342E+09 6 0.226617983E+05 0.397374220E+09 7 0.251648469E+05 0.491568569E+09 8 0.314665269E+06 0.769598372E+10 9 0.197028549E+06 0.131078134E+11 10 0.240579570E+06 0.122258378E+11 11 0.253301194E+06 0.894492396E+10 12 0.263036660E+06 TORS. CRIT. SPEED (RPM) EXCITING FREQUENCIES (CPM) 1200.00 3600.00 6348.00 7200.00 SEPARATION MARGINS 0.9499993E+03 20.83 73.61 85.03 86.81 0.2712998E+04 126.08 24.64 57.26 62.32 0.9251993E+04 671.00 157.00 45.75 28.50 0.1121899E+05 834.92 211.64 76.73 55.82

0.1267799E+05 956.50 252.17 99.72 76.08 0.2136698E+05 1680.58 493.53 236.59 196.76 0.2508798E+05 1990.67 596.89 295.21 248.44 0.4063397E+05 3286.16 1028.72 540.11 464.36 0.4235397E+05 3429.50 1076.50 567.20 488.25 THE NORMALIZED CRITICAL SPEED MODE SHAPES ARE CRITICAL SPEED NO. 1 0.9499993E+03 RPM STATION NO. NORMALIZED DEFLECTION 1 0.10000E+01 2 0.99278E+00 3 0.62578E+00 4 -0.29709E+00 5 -0.47224E+00 6 -0.55812E+00 7 -0.61284E+00 8 -0.61284E+00 9 -0.61498E+00 10 -0.61599E+00 11 -0.61677E+00 12 -0.61738E+00

CRITICAL SPEED NO. 2 0.2712998E+04 RPM STATION NO. NORMALIZED DEFLECTION 1 0.11047E+00 2 0.10396E+00 3 -0.20959E+00 4 -0.95111E+00 5 -0.10000E+01 6 -0.18997E+00 7 0.33614E+00 8 0.33614E+00 9 0.36020E+00 10 0.37320E+00 11 0.38560E+00 12 0.40026E+00 CRITICAL SPEED NO. 3 0.9251993E+04 RPM STATION NO. NORMALIZED DEFLECTION 1 0.58545E-01 2 0.18427E-01 3 -0.63831E+00 4 -0.10000E+01 5 0.88617E-01 6 0.53805E-01 7 0.23830E-01 8 0.23830E-01 9 0.19724E-01 10 0.16591E-01 11 0.12439E-01

12 0.59077E-02 CRITICAL SPEED NO. 4 0.1121899E+05 RPM STATION NO. NORMALIZED DEFLECTION 1 0.10000E+01 2 -0.75996E-02 3 -0.10720E-01 4 0.12817E-01 5 -0.44600E-02 6 0.46881E-01 7 0.70523E-01 8 0.70523E-01 9 0.60608E-01 10 0.51529E-01 11 0.38170E-01 12 0.16045E-01 CRITICAL SPEED NO. 5 0.1267799E+05 RPM STATION NO. NORMALIZED DEFLECTION 1 0.55728E-01 2 -0.15978E-01 3 0.10000E+01 4 -0.20666E+00 5 0.92054E-02 6 0.51321E-02 7 0.11615E-02 8 0.11615E-02 9 0.72227E-03

10 0.41482E-03 11 0.47910E-04 12 -0.45977E-03 CRITICAL SPEED NO. 6 0.2136698E+05 RPM STATION NO. NORMALIZED DEFLECTION 1 0.11494E-05 2 -0.30515E-05 3 0.56158E-03 4 -0.39810E-02 5 0.19741E-01 6 -0.10000E+01 7 -0.91977E+00 8 -0.91977E+00 9 -0.38884E+00 10 -0.13090E-02 11 0.41452E+00 12 0.83061E+00 CRITICAL SPEED NO. 7 0.2508798E+05 RPM STATION NO. NORMALIZED DEFLECTION 1 0.16995E-06 2 -0.68638E-06 3 0.17458E-03 4 -0.19377E-02 5 0.14162E-01 6 -0.10000E+01 7 -0.63610E+00

8 -0.63610E+00 9 -0.11505E+00 10 0.22179E+00 11 0.50489E+00 12 0.63618E+00 CRITICAL SPEED NO. 8 0.4063397E+05 RPM STATION NO. NORMALIZED DEFLECTION 1 0.59665E-09 2 -0.72897E-08 3 0.48820E-05 4 -0.16982E-03 5 0.35918E-02 6 -0.67653E+00 7 0.69344E+00 8 0.69344E+00 9 -0.67215E+00 10 -0.10000E+01 11 -0.42844E+00 12 0.92188E+00 CRITICAL SPEED NO. 9 0.4235397E+05 RPM STATION NO. NORMALIZED DEFLECTION 1 0.26282E-09 2 -0.35115E-08 3 0.25554E-05 4 -0.97344E-04 5 0.22470E-02

6 -0.46018E+00 7 0.57823E+00 8 0.57823E+00 9 -0.66939E+00 10 -0.88912E+00 11 -0.23304E+00 12 0.10000E+01

TYPICAL MODE SHAPE PLOTS USING MICROSOFT EXCEL

Critical Speed Mode Shape - Nc1= 950 CPM

-8.00E-01

-6.00E-01

-4.00E-01

-2.00E-01

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

0 2 4 6 8 10 12 14

Inertia Station Numbers

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Critical Speed Mode Shape - Nc2=2713 CPM

-1.20E+00

-1.00E+00

-8.00E-01

-6.00E-01

-4.00E-01

-2.00E-01

0.00E+00

2.00E-01

4.00E-01

6.00E-01

0 2 4 6 8 10 12 14

Inertia Numbers

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Comparison of Calculated Torsional Natural Frequencies with Test Data

System 1

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Calculated Multimass

(54 4 inertia model

Percent difference

Test Data Per Cent Per Cent

crit. Spd. Inertia) model

Multimass Vs. Dev. of Test Dev. of Test

no. (CPM) (CPM) 4 inertia model HZ CPM

from Calculated

from Calculated

Multimass 4 inertia model

1 513 522 1.754385965 8.625 517.5 0.869565217 -0.869565217

2 797 808 1.380175659 13.75 825 3.393939394 2.060606061

3 2086 2073 -

0.623202301

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Comparison of Calculated Torsional Natural Frequencies with Test Data System 2

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Calculated Multimass

(33 3 inertia model

Percent difference

Test Data Per Cent Per Cent

crit. Spd. Inertia) model

Multimass Vs. Dev. of Test Dev. of Test

no. (CPM) (CPM) 3 inertia model HZ CPM

from Calculated

from Calculated

Multimass 3 inertia model

1 689 695 0.870827286 11.5 690 0.144927536-

0.724637681

2 1418 1414 -

0.282087447 25 1500 5.466666667 5.733333333

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References

Thomson, W. T., Vibration Theory and Applications, Prentice Hall Inc., Englewood Cliffs, N.J., 1965.

Den Hartog, J.P., Mechanical Vibrations, Dover Publications Inc., New York 1985 edition. (First published 1934)

Tse, Francis S., Morse, Ivan E., Hinkle, Rolland T., Mechanical Vibrations, Allyn and Bacon, Inc., Boston, 1964.

API Standard 617, Seventh Edition, July 2002. A HANDBOOK ON TORSIONAL VIBRATION The British

Internal Combustion Research Association (B.I.C.E.R.A) Kirschbaum, H.S., Transient Electrical Torques of Turbine

Generators During Short Circuits and Synchronizing, Transactions Electrical Engineering, February 1945, Volume 64.

Wahl, A.M., Kilgore, L.A., Transient Starting Torques in Induction Motors, November, 1940, Transactions Electrical Engineering.

Godwin, G.I., The Nature of A.C. Machine Torques, IEEE , 1976.

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References (Continued) Pollard, E.I., Torsional Response of Systems, Transactions of

the ASME, July 1967. Pollard, E.I., Transient Torsional Vibration Due to Suddenly

Applied Torque, Journal of Engineering for Industry, May 1972. De Choudhury, P., Torsional System Design Relative to

Synchronous Motor Start-Up with a Variable Frequency Power Supply System, Proceedings of the International Conference on Rotordynamics, September, 1986.

De Choudhury, P., Aspects of Torsional System Design for Large Single Shaft Gas Turbine Drives with Variable Frequency Start Up Motor, Proceedings of the Sixth International Conference on Rotordynamics, September, 2002.

Milton, A. Miner, Cumulative Fatigue Damage Analysis, Journal of Applied Mechanics, September 1945.

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