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