Page 13 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

10 ContEntS
ChAPtEr 2 2.10 Free Vibration with Hysteretic Damping 225
Free Vibration of Single-Degree-of-Freedom 2.11 Stability of Systems 231
Systems 153 2.12 Examples Using MATLAB 235
Chapter Summary 241
2.1 Introduction 155 References 242
2.2 Free Vibration of an Undamped Review Questions 242
Translational System 158 Problems 247
2.2.1 Equation of Motion Using Newton’s Design Projects 294
Second Law of Motion 158
2.2.2 Equation of Motion Using Other ChAPtEr 3
Methods 159
2.2.3 Equation of Motion of a Spring-Mass harmonically Excited Vibration 297
System in Vertical Position 161 3.1 Introduction 299
2.2.4 Solution 162 3.2 Equation of Motion 299
2.2.5 Harmonic Motion 163 3.3 Response of an Undamped System Under
2.3 Free Vibration of an Undamped Harmonic Force 301
Torsional System 176 3.3.1 Total Response 305
2.3.1 Equation of Motion 177 3.3.2 Beating Phenomenon 305
2.3.2 Solution 178 3.4 Response of a Damped System Under Harmonic
2.4 Response of First-Order Systems Force 309
and Time Constant 181 3.4.1 Total Response 312
2.5 Rayleigh’s Energy Method 183 3.4.2 Quality Factor and Bandwidth 316
2.6 Free Vibration with Viscous Damping 188 3.5 Response of a Damped System Under
2.6.1 Equation of Motion 188 F1t2 = F 0 e iVt 317
2.6.2 Solution 189 3.6 Response of a Damped System Under
2.6.3 Logarithmic Decrement 198 the Harmonic Motion of the Base 320
2.6.4 Energy Dissipated in Viscous 3.6.1 Force Transmitted 322
Damping 199 3.6.2 Relative Motion 323
2.6.5 Torsional Systems with Viscous 3.7 Response of a Damped System Under Rotating
Damping 201 Unbalance 326
2.7 Graphical Representation of Characteristic Roots 3.8 Forced Vibration with Coulomb Damping 332
and Corresponding Solutions 207 3.9 Forced Vibration with Hysteresis Damping 337
2.7.1 Roots of the Characteristic Equation 207 3.10 Forced Motion with Other Types
2.7.2 Graphical Representation of Roots and of Damping 339
Corresponding Solutions 208 3.11 Self-Excitation and Stability Analysis 340
2.8 Parameter Variations and Root Locus 3.11.1 Dynamic Stability Analysis 340
Representations 209 3.11.2 Dynamic Instability Caused by Fluid
2.8.1 Interpretations of v n , v d , z, and t in the Flow 344
s-plane 209 3.12 Transfer-Function Approach 352
2.8.2 Root Locus and Parameter 3.13 Solutions Using Laplace Transforms 356
Variations 212 3.14 Frequency Transfer Functions 359
2.9 Free Vibration with Coulomb Damping 218 3.14.1 Relation between the General Transfer
2.9.1 Equation of Motion 219 Function T(s) and the Frequency Transfer
2.9.2 Solution 220 Function T1iv2 361
2.9.3 Torsional Systems with Coulomb 3.14.2 Representation of Frequency-Response
Damping 223 Characteristics 362