Dynamics of Machinery II

5.  Finite element analysis of rotor-bearing systems   1 
 5.1  Rotor component models   1
 5.2  Kinematics of rigid body precession   3 
  5.2.1  Main reference frames   3
  5.2.2  Rigid-body precession   3
  5.2.3  Small rotations of the spin axis   4 
 5.3  Equations of motion for rotor components   7 
  5.3.1  Thin disks   7
  5.3.2  Uniform shaft elements  13
  5.3.3  Bearings and seals  31
  5.3.4  Flexible couplings  35 
 5.4  System equations of motion  36 
  5.4.1  Second order configuration space form  37
  5.4.2  First order state space form  40 
 5.5  Eigenvalue analysis  40 
  5.5.1  Right and left eigenvectors  41
  5.5.2  Reduction to the standard eigenvalue problem  42
  5.5.3  Campbell and stability diagrams  43 
 5.6  Unbalance response  47 
  5.6.1  Modal analysis solution  47
  5.6.2  Spectral analysis solution  48 
 5.7  Kinematics of elliptic motion  49 
  5.7.1  Elliptic orbits  49
  5.7.2  Decomposition into forward and backward circular motions  52
  5.7.3  Variable angular speed along the ellipse  54 
 5.8  Model order reduction  56 
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  5.8.1  Model condensation  56
  5.8.2  Model substructuring  62
  5.8.3  Stepwise model reduction methods  68 
 References  76 
6.  Fluid film bearings and seals  77 
 6.1  Fluid film bearings  77
 6.2  Static characteristics of journal bearings  78 
  6.2.1  Geometry of a plain cylindrical bearing  79
  6.2.2  Equilibrium position of journal center in bearing  80 
 6.3  Dynamic coefficients of journal bearings  83
 6.4  Reynolds equation and its boundary conditions  84 
  6.4.1  General assumptions  86
  6.4.2  Reynolds’ equation  87
  6.4.3  Boundary conditions for the fluid film pressure field  89 
 6.5  Analytical solutions of Reynolds’equation  90 
  6.5.1  Short bearing (Ocvirk) solution  90
  6.5.2  Infinitely long bearing (Sommerfeld) solution  99
  6.5.3  Finite-length cavitated bearing (Moes) solution  99  
 6.6  Physical significance of the bearing coefficients 103
 6.7  Bearing temperature 107 
  6.7.1  Approximate bearing temperature 108
  6.7.2  Viscosity-temperature relationship 109 
 6.8  Common fluid film journal bearings 112 
  6.8.1  Plain journal bearings 112
  6.8.2  Axial groove bearings 112
  6.8.3  Pressure dam bearings 114
  6.8.4  Offset halves bearings 116
  6.8.5  Multilobe bearings 117
  6.8.6  Tilting pad journal bearings 124
  6.8.7  Rayleigh step journal bearings 128
  6.8.8  Floating ring bearings 129 
 6.9  Squeeze film dampers 130 
  6.9.1  Basic principle 130 
CONTENTS   
  6.9.2  SFD design configurations 132
  6.9.3  Squeeze film stiffness and damping coefficients 133
  6.9.4  Design of a squeeze film damper 135 
 6.10  Annular liquid seals 137 
  6.10.1  Hydrostatic reaction. Lomakin effect 138
  6.10.2  Rotordynamic coefficients 139
  6.10.3  Final remarks on seals 146 
 6.11  Annular gas seals 147
 6.12  Floating contact seals 150 
  6.12.1  Design characteristics 151
  6.12.2  Seal ring lockup 154
  6.12.3  Locked-up oil seal rotordynamic coefficients 155 
 References 159 
7.  Instability of rotors 161 
 7.1  Whirling of rotating shafts 161
 7.2  Instability due to rotating damping 164 
  7.2.1  Planar rotor model 165
  7.2.2  Qualitative effect of damping 166
  7.2.3  Whirl speeds of rotor with rotating damping 168
  7.2.4  Quantitative effects of damping 172 
173
  7.3.1  Oil-whirl and oil-whip phenomena 174 
 7.3  Whirl in hydrodynamic bearings 
  7.3.2  Half frequency whirl 176
  7.3.3  Onset speed of instability 178
  7.3.4  Crandall’s explanation of journal bearing instability 179
  7.3.5  Stability of linear systems 187
  7.3.6  Instability of a simple rigid rotor 189
  7.3.7  Instability of a simple flexible rotor 194
  7.3.8  Instability of complex flexible rotors 199 
 7.4  Interaction with fluid flow forces 199 
  7.4.1  Steam whirl 199
  7.4.2  Impeller-diffuser interaction 201
    
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                                                                                           DYNAMICS OF MACHINERY 
 7.5  Dry friction backward whirl 203 
  7.5.1  Rotor-stator rubbing 203
  7.5.2  Dry friction whirl 204 
 7.6  Instability due to asymmetric factors 206 
  7.6.1  Parametric excitation 207
  7.6.2  Shaft anisotropy 207
  7.6.3  Asymmetric inertias 219
  7.6.4  Finite element analysis of asymmetric rotors 226 
 References 229 
Index 233