# MECHANICAL VIBRATION pdf Lecture notes and eBook Dowload

Ravi Chopra | 24-Feb-2016 |

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CONTENTS
0.1 INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
I MODELLINGANDANALYSIS 7
1 MECHANICAL VIBRATION OF ONE-DEGREE-OF-FREEDOM
LINEAR SYSTEMS 9
1.1 MODELLING OF ONE-DEGREE-OF-FREEDOM SYSTEM . . . . 9
1.1.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . 12
1.1.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2 ANALYSISOFONE-DEGREE-OF-FREEDOMSYSTEM . . . . . . 28
1.2.1 Free vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.2.2 Forced vibration . . . . . . . . . . . . . . . . . . . . . . . . 34
1.2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2 MECHANICAL VIBRATION OFMULTI-DEGREE-OF-FREEDOM
LINEAR SYSTEMS 66
2.1 MODELLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.1.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.1.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . 67
2.1.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.2 ANALYSISOFMULTI-DEGREE-OF-FREEDOMSYSTEM. . . . . 93
2.2.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.2.2 Modal analysis - case of small damping . . . . . . . . . . 102
2.2.3 Kinetic and potential energy functions - Dissipation
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.3 ENGINEERINGAPPLICATIONS . . . . . . . . . . . . . . . . . . . 151
2.3.1 Balancing of rotors . . . . . . . . . . . . . . . . . . . . . . 151
2.3.2 Dynamic absorber of vibrations . . . . . . . . . . . . . . 157
3 VIBRATION OF CONTINUOUS SYSTEMS 162
3.1 MODELLINGOF CONTINUOUS SYSTEMS . . . . . . . . . . . . . 162
3.1.1 Modelling of strings, rods and shafts . . . . . . . . . . . 162
3.1.2 Modelling of beams . . . . . . . . . . . . . . . . . . . . . . 166
3.2 ANALYSISOF CONTINUOUS SYSTEMS . . . . . . . . . . . . . . 168
3.2.1 Free vibration of strings, rods and shafts . . . . . . . . . 168
3.2.2 Free vibrations of beams . . . . . . . . . . . . . . . . . . . 174
3.2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.3 DISCRETEMODELOF THE FREE-FREE BEAMS . . . . . . . . . 214
3.3.1 Rigid Elements Method. . . . . . . . . . . . . . . . . . . . 214
3.3.2 Finite Elements Method. . . . . . . . . . . . . . . . . . . . 217
3.4 BOUNDARYCONDITIONS. . . . . . . . . . . . . . . . . . . . . . . 225
3.5 CONDENSATIONOF THEDISCREET SYSTEMS . . . . . . . . . 226
3.5.1 Condensation of the inertia matrix. . . . . . . . . . . . . 227
3.5.2 Condensation of the damping matrix. . . . . . . . . . . . 228
3.5.3 Condensation of the stiffness matrix. . . . . . . . . . . . 228
3.5.4 Condensation of the external forces. . . . . . . . . . . . . 228
3.6 PROBLEMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
II EXPERIMENTAL INVESTIGATION 237
4 MODAL ANALYSIS OF A SYSTEMWITH 3 DEGREES OF FREEDOM
238
4.1 DESCRIPTIONOF THE LABORATORYINSTALLATION . . . . . 238
4.2 MODELLINGOF THEOBJECT . . . . . . . . . . . . . . . . . . . . 239
4.2.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . . 239
4.2.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . 240
4.3 ANALYSISOF THEMATHEMATICALMODEL. . . . . . . . . . . 241
4.3.1 Natural frequencies and natural modes of the undamped
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
4.3.2 Equations ofmotion in terms of the normal coordinates
- transfer functions . . . . . . . . . . . . . . . . . . . . . . 241
4.3.3 Extraction of the natural frequencies and the natural
modes from the transfer functions . . . . . . . . . . . . . 242
4.4 EXPERIMENTAL INVESTIGATION . . . . . . . . . . . . . . . . . 243
4.4.1 Acquiring of the physical model initial parameters . . 243
4.4.2 Measurements of the transfer functions . . . . . . . . . . 244
4.4.3 Identification of the physical model parameters . . . . 245
4.5 WORKSHEET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246