Analytical Methods in Rotor Dynamics – Second Edition

Analytical Methods in Rotor Dynamics – Second Edition
Andrew D. Dimarogonas , Stefanos A. Paipetis , Thomas G. Chondros
Contents
1 Approximate Evaluation of Eigenfrequencies 1
1.1 Introduction 1
1.2 Formulation of the Eigenvalue Problem 3
1.3 Dunkerley’s Procedure 4
1.4 The Question of Accuracy 8
1.5 The Root-Squaring Process . 11
1.6 Application with Dissipative Systems 14
1.7 Applications with Continuous Systems . 16
1.8 Summary and Conclusions 20
References . 21
2 Variable Elasticity Effects in Rotating Machinery 25
2.1 Introduction 25
2.1.1 Variable Length l 27
2.1.2 Variable Stiffness EJ 27
2.1.3 Variable Mass or Moment of Inertia . 28
2.2 The Problem of Stability . 28
2.3 The Mathieu-Hill Equation . 31
2.4 The Classical Floquet Theory 32
2.5 Matrix Solution of Hill’s Equation 35
2.6 Solution by Transition into an Equivalent
Integral Equation 36
2.7 The Bwk Procedure 37
2.8 Vibrations of Different-Modulus Media . 38
References . 40
xi3 Mathematical Models for Rotor Dynamic Analysis . 43
3.1 Introduction 43
3.2 The Single Disc Model 47
3.2.1 Critical Speeds 51
3.2.2 Internal Damping 52
3.2.3 Bearing Forces 52
3.2.4 Environmental Forces . 54
3.2.5 Stability of Motion, Second-Order Equations . 54
3.3 The Discrete Model 59
3.4 Summary and Conclusions 72
References . 73
4 Flow-Induced Vibration of Rotating Shafts . 77
4.1 The Steam Whirl Problem 77
4.2 Stability Criteria . 86
4.3 Rotor Dynamics for Annular Flows . 94
4.4 Dynamics of a Hollow Rotor Partially Filled with a Liquid . 102
References . 111
5 Heat-Flow-Induced Vibration of Rotating Shafts:
The Newkirk Effect 115
5.1 Introduction 115
5.2 Analytical Model 117
5.3 Modes of the Newkirk Effect 137
Appendix: Numerical Example . 141
References . 142
6 Dynamics of Cracked Shafts 145
6.1 Introduction 145
6.2 Local Flexibility of a Cracked Shaft . 147
6.3 The Open Crack . 150
6.4 The Closing Crack . 154
References . 160
7 Identification of Cracks in Rotors and Other Structures
by Vibration Analysis . 163
7.1 Flexibility Matrix of Cracked Structural Members . 163
7.1.1 Prismatic Cracked Beam Element . 165
7.1.2 Circular Cracked Rod . 168
7.2 Direct Methods . 173
7.2.1 Rotors with a Circumferential Crack . 173
7.2.2 Beam with a Lateral Crack . 178
7.2.3 Clamped Circular Plate with a Peripheral
Surface Crack 186
xii Contents7.3 The Eigenvalue Sensitivity Problem . 189
7.3.1 Introduction 189
7.3.2 Rayleigh’s Quotient 191
7.3.3 Torsional Vibration of a Cracked Rotor 193
7.3.4 Cracked Structural Members 195
7.4 Summary and Conclusions 197
References . 198
8 Thermal Effects Due to Vibration of Shafts . 203
8.1 Heat Propagation Due to Torsional Vibration of Shafts 203
8.2 Heat Propagation in Rotating Shafts Due to Bending . 211
8.3 Summary and Conclusions 218
References . 218
9 Variational Formulation of Consistent: Continuous
Cracked Structural Members . 221
9.1 Variational Formulation of Cracked Beams and Rods . 221
9.2 Lateral Vibration of a Continuous Cracked Beam 222
9.2.1 The Variational Theorem for a Simply
Supported Beam . 222
9.2.2 The Crack Disturbance Function . 229
9.2.3 Natural Frequencies of Cracked Beams . 233
9.2.4 The Beam with Lumped Crack Flexibility . 234
9.2.5 The Finite Element Method . 239
9.2.6 Experimental Procedure 239
9.3 Torsional Vibration of a Continuous Cracked Rod . 240
9.3.1 The Variational Theorem for a Cracked Rod
in Torsion . 240
9.4 Summary and Conclusions 248
References . 249
10 The Variational Formulation of a Rod in Torsional Vibration
for Crack Identification . 251
10.1 Dynamic Behaviour of Cracked Shafts . 251
10.2 Torsional Vibration of a Continuous Cracked Shaft:
Variational Theorem 255
10.2.1 Cracked Rod-Variational Theorem 255
10.2.2 The Crack Disturbance Function . 257
10.2.3 The Differential Equation of Motion . 258
10.2.4 Boundary Conditions . 259
10.2.5 Torsional Natural Frequencies
of the Cracked Rod-Rayleigh Quotient . 260
Contents xiii10.3 Finite Element Analysis of a Vibrating Cracked Rod . 263
10.4 Summary and Conclusions 265
References . 266
Index 269
xiv Contents
Index
A
Accuracy, 8–13, 21
Adams-Moulton predictor-corrector method,
69
Alford, 77–78
Annular flows, 94–102
Asymmetry of rotating parts, 57
Asymptotic behaviour, 128, 137
Axial flow around a bucket cover, 79
Axial forces, 40, 147
B
Backward whirl, 51
Beam(s)
bending moment, 146, 163–165, 237
flexibility matrix, 163–167, 171, 178–189,
195, 230–231, 247
lateral crackwith, 165–173, 178–184
simply supported, 4, 7, 13, 17, 90, 222,
231, 233, 238–240
stiffness, 3–4, 7, 165, 171, 178, 189,
193–198
Bearing(s)
damping coefficients, 56, 67, 80, 86–94
forces, 52, 71
fraction of critical damping, 88
instabilities, 57
second stiffness coefficient, 72
seal stiffness, 78
spring constant, 72
stiffness, 106
Bearing/rotor stiffness ratio, 88
Bending
heat propagation due, 210–216, 218
mode, 237, 185
moment, 146, 148, 163–164, 198
stress, 166, 210–212, 217
Bessel function, 103, 123–127, 188
Betti’s theorem of reciprocity, 121
Biot number, 207–210, 215
Boundary conditions, 3, 30, 38–40, 60–62,
103–105, 121–122, 127–132, 180–183,
212, 221–223, 226–229, 233–247,
255–265
Buckling load of column
under pressure, 45
local flexibility, 163
BWK procedure, 36–37
C
Calibration constants, 79
Castigliano’s theorem, 165, 230–231
Centrifugal force due to imbalance, 98
Characteristic equation, 8–21, 51–59, 104,
151, 234
Characteristic exponents, 33
Characteristic polynomial, 56
Characteristic roots, 32–33
Christides and Barr, 147, 222–223
Circumferential crack, 164–174, 254, 265
Clamped circular plate with peripheral surface
crack, 185–189
Compatibility conditions, 156
Complex displacement vector, 64
Complex force vector, 65
Complex functions, 53, 61
Complex roots, 15–16, 21
Condensation, 102
C (cont.)
Constant mode, 44, 136–137
Continuity conditions, 157, 237
Continuous beam, 3, 146
Continuous cracked beam, 222–236
cantilever, 233
free-free, 234
simply supported, 233
Continuous cracked shaft, 176–178, 240–259
free-free, 261–263
Continuous systems, 16–21, 197, 210
Coordinate system, 49–52, 60–65, 120,
129–130, 150–155, 158–159
rotating, 60–65, 158–159
rotor-fixed, 51, 65, 118–121, 130–139
Crack depth versus frequency ratio, 145,
175–196
Crack detection, 153, 163–165, 251, 254
Crack disturbance function, 230–233, 240,
257–266
Crack flexibility, 146, 177–178, 197
continuous, 255
local, 163, 175
lumped, 146, 222, 235–248, 253
matrix, 197
parameter, 176–178
Crack identification, 145–146, 222–237, 248,
255, 265
direct methods, 171, 189
eigenvalue sensitivity problem, 189–197
Crack propagation, 152, 185, 239, 248
Cracked beam, 163–165, 178, 221–223, 231
continuous, 147–154, 221–229
Cracked clamped circular plate, 13–14,
185–188
Cracked rotor, 40, 146, 150, 163–178, 196,
197, 252–263
torsional vibration of, 193–195
Cracked shafts, 145–150
breathing crack, 155, 198, 233–234, 248,
253
closing crack, 145, 147, 154–159,
254, 265
dimensionless flexibility, 149, 175
local flexibility, 175–178, 188–198
open crack, 146, 150–154
responses of, 155
stability chart, 156
Cracked structural members, 163–171,
195–197
Critical speed, 1–4, 44–48, 51, 58, 68–69, 72,
86–94, 98, 102, 110, 117, 137, 153,
159, 164, 251–254
Cross-coupling
forces, 81
terms, 67, 72, 80, 253
D
Damping
bearing, 57, 93, 106
coefficients, 67, 150–154
constants, 72, 82, 109
effects, 31, 40, 65, 80, 85, 198, 262
factor, 80, 210–216
forces, 52, 80
function, 86
heat propagation, 218
internal-external, 2–3, 43–44, 52–57, 80
matrix, 4, 8, 239, 262
parameters, 82
viscous, 14, 21
De-Laval rotor, 1, 47, 141, 150–155
stiffness, 150, 158
Deformation energy, 204
Different-modulus media, 38
Differential equations, 25, 44–46, 59, 132,
233, 247
Dimensionless flexibility, 149–150, 175
Dirichlet’s conditions, 214
Disc weight, 152–155
Discrete model, 59–64
Displacement function, 29
Dissipative systems, 14–16
Dunkerley formula, 4
Dunkerley procedure, 16–20
Dynamic analysis, 43–47, 247
Dynamic behaviour, 2, 44, 146, 251
Dynamic characteristics, 116, 163, 197
Dynamic effects, 203
Dynamic equilibrium term, 225–226, 228,
244, 247
Dynamic response, 44, 137, 145–147, 165,
171, 178, 197, 239, 254, 265
E
Eccentric stage, 78–79
Eccentricity, 70, 80, 117, 141, 155
ratio, 70
Eigenfrequencies
accuracy, 8–11
approximate evaluation, 1–19
circumferentially clamped massless
circular plate, of, 14
factors affecting, 2, 30
simply supported uniform beam, 17
Eigenfunctions, 18
270 IndexEigenvalue(s), 3, 18, 59, 68, 107–109, 134,
189–192
beam, 17
change, 78, 191–193
discrete systemsfor, 5–7
Dunkerley’s, 7–8
modal, 265
problem, 17–21, 189–191, 195–197
sensitivity problem, 189–192
Elastic deformation, 63, 204–205, 211
Energy of elastic deformation, 64, 204–205,
211
External forces, 50, 54, 60–64, 229, 247
Equations of motion, 2, 25, 48, 51, 54–59,
64–68, 81, 98, 99, 106, 145–147, 150,
154, 164, 194, 222, 253–255, 265
Euler-Bernoulli beam, 221, 223, 237, 262
External damping, 2, 56–57, 109
F
Finite difference method, 103
Finite element method, 46, 59–60, 239,
253–254
Finite rotations, 49
Flexibility
coefficient, 149
influence coefficient, 149, 166
matrix, 147, 154, 165, 167, 171, 173, 178,
189, 195–196, 230
Flexural vibrations, 2–3, 26, 40
Floquet theory, 32–34
Flow around eccentric cylinders, 80
Flow-induced vibration, 77–94
Fluid(s)
bearings, 44, 53, 80, 116
dynamics, 44, 79
effects, 102
forces, 94–99
machines, 94
pressure, 94
Force-eccentricity function, 70
Forcing vector, 159
Forward whirl, 51
Fourier series, 29–31, 158
Fourier transform, 213, 214
Four-pad journal bearing, 88
Free vibration, 18, 30–31
Frequency equation, 10, 176–183, 189, 237
coefficients, 10
Frequency parameter, 176–177
Frequency shift calculation error, 194
G
Global stability, 152
Goodier’s formula, 121
Graeffe’s method, 11, 13, 20
H
Heat conduction equation, 204, 206, 212
Heat-flow-induced vibration, 126, 205
analytical model, 119–139
Heat function(s), 44, 131, 205, 213
Heat generation, 138, 203–205, 218
rate, 204, 211
Heat propagation, 117, 203, 210
bendingdue to, 211–217
damping, 203–204, 218
torsional vibrationdue to, 203–210
Heat rate density per unit area, 122
Heat transfer coefficient, 141, 207–208
Hill’s equation, 31, 33–36
matrix solution, 34–36
Hollow rotor partially filled with liquid, 99,
102–110
Hu-Washizu-Barr variational principle,
221–223, 225, 247, 254–256, 265, 267
Hurwitz-Routh determinants, 100
Hysteretic damping, 52, 65
I
Imbalance response, 66, 152–154
Inertia matrix, 194
Infinitesimal rotations, 49
Influence coefficients, 6–8, 10
Influence function, 16–18
Instability, 30
bearing, 78
factors, 79
load-controlled, 69
Newkirk, 43
speed-controlled, 55–58, 68
steam flow, 77–78
Instability
threshold, 67–68, 156
types of, 67–69
Internal damping, 2, 43, 52–56
Index 271I (cont.)
Internal friction, 56, 65
Inverse transforms, 129
J
Jumping phenomenon, 72
K
Kalman Filter, Extended, 255
Kinetic energy, 63–64, 223
Kronecker’s delta, 18, 223
L
Lagrangian equations of motion, 64
Lagrangian function, 64
Laplace transform, 123–129
Lateral crack in beam, 164–171, 178–184
Leakage flow, 78, 92
Liapunov’s first theorem, 68–69
Limit cycle, 69–71
Limit orbit, 71–72
Linear external forces, 65
Linear models, 46
Linear viscoelastic damping, 65
Load Stability Criterion (LSC), 84
Lobachevsky-Graeff’e method, 11
Local flexibility, 145–168, 175–176, 188, 193,
221–222, 230, 236–237, 247, 253,
265–266
Logarithmic decrement, 91
Loss factor, 65, 211
Lumped mass systems, 59, 203–208
M
Maclaurin expansion, 66, 71
Maclaurin series, 36
Massless elastic rotor, 47
Material damping, 44, 155, 203, 211
factor, 80, 210, 216
Material loss factor, 204
Mathematical models, 43–47
Mathieu’s equation, 31
Maxwell’s reciprocity theorem, 6
Mean thermoelastic flexural rotationof one
end relative to the other, 121
Midspan force gradients, 54
Modal eigenvalue problem, 263
Moment of inertia, 142, 150, 153, 180, 211,
237, 256
Moment vector, 49
Momentum equation, 97
N
Natural frequency versus crack depth,
175–184, 186–188, 223
Natural vibration, 51, 60, 66, 178
Newkirk effect, 44, 137–140
modes, 137–140
numerical example, 141
Newton–Raphson method, 66, 68
Newtonian cooling, 206–213
Newton’s law, 50
Nonlinear behaviour, 146
Non-linear equations, 46, 60, 68, 132
Non-linear force(s), 66, 68
functions, 69
vector, 66
Non-linear systems, 59, 155
Nusselt number, 210, 217
O
Oscillatory mode, 137–138
Out-of-balance, 155
P
Packing clearance, 77–78, 92
Packing-rotor interaction, 118
Packing-rub effect, 116–117, 129, 141
Pad bearings, 44, 53, 67–70
Paris equation, 165, 230
Penny-shaped crack, 173
Period solution, 34
Perturbation vector, 67, 82
Peturation method, 72, 222, 234
Phase angle, 3, 116–118, 137–140
Plastic deformation, 203–205
Plastic flow, 203
Plexiglas shaft, 175–176
Proportionality coefficient, 78–80
Q
Quadratic equation, 16, 30, 89, 128, 135
R
Radius of gyration, 63
Rayleigh quotient, 191–197, 222, 260
Rayleigh principle, 3
272 IndexResponse vector, 152
Reynolds number, 82, 97, 103, 217
Root-squaring process, 11–20
Rotating coordinate system, 52, 57, 58,
154–158
Rotor(s)
behaviour, 21, 27, 43–46, 65, 69
bending mode, 47, 130
bending moment, 146–147, 211, 231–232
cracked, 28, 40, 146, 150, 153, 155, 163,
193–195, 252
deflection, 52–53, 66, 86, 146
design, 45–46
dynamic model, 2, 60, 72
geometry, 2, 46
inertia moment, 204
internal damping, 80
nodes, 61
operation, 40, 65
stability, 85–86
stiffness, 86–90
Rotor-fixed coordinate system, 65, 120, 140,
150
Routh-Hurwitz criteria, 54–58, 87
Routh stability criterion, 55, 135–136
Rubbing
force, 117, 142
heat generation, 58, 119, 125, 131
instability, 116
phenomena, 44, 72, 115, 140
problems, 116
response, 119–121
whip, 117
Runge–Kutta method, 69, 136
S
Saint-Venant, 240
Second-order equations, 54
Shaft
bending, 210
continuous flexibility, 255
cracked, 176–179
flexibility matrix, 167, 168, 171, 173, 178,
189, 195, 197, 247
local flexibility, 145–147, 152, 167–178,
193–197, 221, 265
material damping, 155
seal stiffness, 78
stiffness, 50, 150, 155, 158, 159
Shear
modulus, 173, 175, 204, 210, 224, 243
stiffness, 50
stress, 147, 173, 204, 210, 240–244, 257
Shrink-fit joints, 56
Single disc model, 47–53
Slotted shaft, 155
Sommerfeld number, 71, 86
Southwell’s theorem, 7
Spiralling mode, 136–137
Spring coefficient, 67, 71, 86
Spring constant, 5, 26, 47, 57–58, 72, 82,
132, 150, 178–187
Spring fluctuation, 30
Spring stiffness, 131
Stability, 132–140, 151–152
analysis, 171
chart, 82–85, 109, 140–142, 157
concept, 46
conditions, 57, 86, 100
criteria, 54–55, 90
limits, 107–108, 134
motion of, 54
problem(s), 30
study, 81
threshold, 96
Stable mode, 139
Stable rotor orbits, 70
Stability analysis, 171
Static deflection, 91, 150, 154, 158, 197
Stationary coordinate system, 98, 145,
153–158, 239, 263
Steam flow, 72–80, 92
Steam force, 78–92
Steam force gradient, 78–89
ratio, 83, 88
Steam whirl
evaluation, 91
problem, 77–88, 90
Stepped rotors, 59
Stiffness
constants, 50
distribution, 59
effects, 85, 155, 158, 165
function, 3–4, 26, 159
matrix, 4, 171, 191–197
parameters, 82
varying, 27
Strain energy, 148
density function, 148, 165, 173
release rate, 168–173, 230, 253
Strain release function, 173
Stress disturbance function, 241, 257–265
Stress intensity factor, 148, 163, 164, 166–174,
230–231, 253, 257–258
stress field, 229–230
Superposition Principle, 46
Surface friction heating, 203
Index 273T
Tangential force variation, 92
Taylor number, 97
Taylor series, 107–108
Temperature effects, 251
Thermal bow, 116–120
Thermal conductivity, 115, 141, 204, 212–216
Thermal effects, 203–217
Thermal imbalance, 58
Thermodynamic constants, 122–126
Thomas stability criterion, 93–94
Three-dimensional problem, 117
Torque deflection number (TDN), 77, 90–93
Torsional spring constant, 186–187
parameter, 180
Torsional vibration
amplitude, 204, 218
continuous rod, 240, 258
cracked rotor, of, 193–194, 222, 240, 265
heat propagation due to, 203
Hu-Washizu theory, 254, 256
rod, 251
variational principle, 222, 255, 256
Torsional wave scattering, 173
Transfer matrix method, 59, 72
Transformation matrix, 62, 154
Transverse circumferential cracks, 173–175,
255, 257
Transverse crack at welded root of beam, 180
Transverse flexibility, 180
Transverse surface cracks. See Cracked shafts
Transverse vibration, 180–186, 188, 223
Turborotors, 88
Two-finite element rotor model, 81
U
Unstable rotor orbits, 70
V
Variable elasticity effects, 25
Variable length, member, 26
Variable mass or moment of inertia, 27
Variable stiffness, 27
Variational formulation
boundary conditions, 180–182, 211, 221,
223–240
cracked rotor, 252–254
Hu-Washizu-Barr, 221, 255, 256
prismatic beams, 222–230
rods, 251–258
strain field, 241, 254–255
stress field, 147, 164, 180, 222, 223,
229–230, 240
stress intensity functions, 230
warping, 240
warping function, 242–243, 256, 259
Variational theorem, 222–224, 235, 240, 255
Vector equation, 134
Velocity function, 29, 259
Velocity-Momentum Term, 243–244, 247
Vibrating elastic system, 8, 20
Vibrating mechanical systems, 25
Vibrating shafts, 20
Vibration frequency, 204–210, 239
Viscoelastic loss factor, 66
Viscoelastic model, 52
Viscous damping, 14–21
Volterra integral equation, 36
W
Warping function, 242–243, 259
Washizu, 254, 265
Wauer, 164, 263
Wave mechanics, 36
Whirl, 1, 44, 51, 68, 78, 154
forward-backward, 51
frequency, 72, 108, 109
speed, 68
Y
Young’s modulus, 148, 166, 180–188,
224, 230–231

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