Dynamics of Machinery III

8.Rolling element bearings 1
8.1 Rolling-element radial bearings 1
8.2 Kinematics of rolling bearings 3
8.2.1 Basic assumptions 3
8.2.2 Simple kinematic relations for angular contact ball bearings 4
8.2.3 Primary rolling element bearing frequencies 6
8.2.4 Kinematic relations for tapered roller bearings 7
8.2.5 General kinematic relations 8
8.3 Structural frequencies 9
8.4 Bearing “mechanical signature” 10
8.5 Rolling element bearing damage 13
8.5.1 Primary damage 14
8.5.2 Secondary damage 14
8.5.3 Other damages 15
8.6 Time domain bearing diagnostic methods 16
8.6.1 Time-waveform indices 16
8.6.2 Crest factor 17
8.6.3 Amplitude probability density 18
8.6.4 Statistical moments 21
8.6.5 Kurtosis 22
8.7 Frequency domain bearing diagnostics methods 23
8.7.1 Band-pass analysis 24
8.7.2 Spike energy 25
8.7.3 Envelope detection 28
8.7.4 Shock Pulse Method 30
8.8 Cepstrum analysis 35
iv
FINITE ELEMENT ANALYSIS
References 36
9. Gears 39
9.1 Gear types 39
9.2 Gear tooth action 40
9.3 Gear vibrations 45
9.3.1 Tooth engagement 45
9.3.2 Effect of tooth deflection 46
9.3.3 Effect of tooth wear 47
9.3.4 Ghost components 48
9.3.5 Modulation effects 48
9.3.6 Resonance effects 53
9.4 Gear errors 54
9.5 Gear faults 55
9.5.1 Wear effects 55
9.5.2 Effects of fatigue 56
9.5.3 Tooth fracture 58
9.6 Gear condition monitoring 58
9.6.1 Vibration signal processing 59
9.6.2 Condition indicators 61
9.6.3 Oil debris analysis 67
9.7 Cepstrum analysis 69
9.8 Time-frequency analysis 72
References 72
10. Vibration measurement 75
10.1 General considerations 75
10.2 Measurement locations 76
10.2.1 General criteria 76
10.2.2 Shaft precession 77
10.2.3 Casing vibrations 78
10.3 Measured parameters 79
10.3.1 Measurement of rotor precession 80
10.3.2 Measurement on bearings 81
10.3.3 Displacement, velocity or acceleration 81
CONTENTS
10.3.4 Peak-to-peak vs. r.m.s. 82
10.4 Transducers and pickups 85
10.4.1 Transducer selection 85
10.4.2 Eddy current proximity transducers 88
10.4.3 Velocity pickups 91
10.4.4 Accelerometers 94
10.4.5 Summary about transducers 96
10.4.6 Placement of transducers 98
10.4.7 Instrumentation 100
10.5 Data reduction 101
10.5.1 Steady state vibration data 101
10.5.2 Transient vibration data 108
References 112
11 Condition monitoring and fault diagnostics 115
11.1 Machine deterioration 115
11.2 Machine condition monitoring 116
11.2.1 General considerations 116
11.2.2 Maintenance strategies 117
11.2.3 Factors influencing maintenance strategies 119
11.3 Diagnosis process 120
11.4 Fault diagnostics 121
11.4.1 Unbalance 121
11.4.2 Misalignment and radial preload 123
11.4.3 Fluid induced instabilities 127
11.4.4 Rotor-to-stator rubbing 130
11.4.5 Mechanical looseness 135
11.4.6 Cracked shafts 138
11.5 Problems of specific machines 141
11.5.1 Centrifugal equipment 141
11.5.2 Bladed machines 145
11.5.3 Electrical machines and gears 151
11.5.4 Reciprocating compressors 152
Annex 11.1 Shaft alignment 155
References 159
v
vi
FINITE ELEMENT ANALYSIS
12 Vibration limits 163
12.1 Broadband vibration standards and guidelines 163
12.2 Vibration severity charts 164
12.3 Vibration limits for nonrotating parts 168
12.3.1 General guidelines 168
12.3.2 Steam turbine sets 169
12.3.3 Coupled industrial machines 170
12.3.4 Gas turbine sets 172
12.3.5 Hydraulic machines 172
12.3.6 Reciprocating machines 174
12.4 Vibration limits for rotating parts 176
12.4.1 General guidelines 176
12.4.2 Steam turbine sets 177
12.4.3 Coupled industrial machines 178
12.4.4 Gas turbine sets 180
12.4.5 Hydraulic machine sets 181
12.4.6 Selection of measurements 183
12.5 Gear units 185
12.6 API Standards 186
12.7 Industrial buildings 187
12.7.1 Vibration intensity 188
12.7.2 Limits based on vibration velocity 190
Annexes 192
References 199
13 Balancing of rotors 203
13.1 The mass unbalance 204
13.1.1 Definitions 204
13.1.2 Static unbalance 205
13.1.3 Couple unbalance 205
13.1.4 Quasi-static unbalance 206
13.1.5 Dynamic unbalance 207
13.1.6 Static vs dynamic unbalance 207

13.2 Single plane balancing
13.2.1 Vector balancing 208
13.2.2 Influence coefficient method 209
208
CONTENTS


13.2.3 Three-trial-mass method 215
13.3 Two-plane balancing 217
13.3.1 Influence coefficient method 217
13.3.2 Resolution into static and couple unbalance 223
13.4 Unbalance tolerances 225
13.4.1 Permissible residual unbalance 225
13.4.2 Balance quality grades 225
13.4.3 Classification of rigid rotors 226
13.5 Multiplane flexible rotor balancing 229
13.5.1 Balancing in N+2 planes 229
13.5.2 Modal balancing 232
13.5.3 General remarks 234
References 235
14 Reciprocating machines 237
14.1 Single cylinder engines 237
14.1.1 Gas pressure excitation 237
14.1.2 Inertia effects 239
14.1.3 Kinematics of crank mechanism 241
14.1.4 Connecting rod and equivalent two-mass system 242
14.1.5 Unbalance of a single cylinder engine 243
14.2 Multi cylinder engines 246
14.2.1 Unbalance forces and couples 246
14.2.2 Othe vibration sources 250
14.2.3 Fault diagnosis of a diesel engine 251
14.3 Reciprocating compressors and piping systems 256
14.3.1 Compressor-manifold system 256
14.3.2 Excitation forces 258
14.3.3 Pulsation analysis 261
14.3.4 Piping vibration 274
References 284
vii
Index 287