The Practice of Engineering Dynamics

Preface xiii 1 Kinematics 3 1.1 Derivatives of Vectors 3 1.2 Performing Kinematic Analysis 8 1.3 2-D Motion with Constant Length 9 1.4 2-D Motion with Variable Length 14 1.5 Three Dimensional Kinematics 16 1.6 Absolute Angular Velocity and Acceleration 21 1.7 The General Acceleration Expression 23 Exercises 28 2 Newton's Equations of Motion 2.1 The Study of Motion 33 2.2 Newton's Laws 34 2.3 Newton's Second Law for a Particle 35 2.4 Deriving Equations of Motion for Particles 36 2.5 Working with Rigid Bodies 45 2.6 Using F = ma in the Rigid Body Force Balance 46 2.7 Using F = dG-- in the Rigid Body Force Balance 51 2.8 Moment Balance for a Rigid Body54 2.9 The Angular Momentum Vector - H O 59 2.10 A Physical Interpretation of Moments and Products of Inertia 65 2.11 Euler's Moment Equations 73 2.12 Throwing a Spiral76 2.13 A Two Body System77 2.14 Gyroscopic Motion 89 Exercises 97 3 Lagrange's Equations of Motion 101 3.1 An Example to Start 102 3.2 Lagrange's Equation for a Single Particle 106 3.3 Generalized Forces 114 3.4 Generalized Forces as Derivatives of Potential Energy 117 3.5 Dampers - Rayleigh's Dissipation Function 120 3.6 Kinetic Energy of a Free Rigid Body 123 3.7 A 2-D Example using Lagrange's equation 128 3.8 Standard Form of the Equations of Motion 133 Exercises 135 4 Equilibrium Solutions 139 4.1 The Simple Pendulum 139 4.2 Equilibrium with Two Degrees of Freedom 141 4.3 Equilibrium with Steady Motion 144 4.4 The General Equilibrium Solution 149 Exercises 150 5 Stability 153 5.1 Analytical Stability 154 5.2 Linearization of Functions 163 5.3 Example: A System with Two Degrees of Freedom 170 5.4 Routh Stability Criterion 177 5.5 Standard Procedure for Stability Analysis 183 Exercises 187 6 Mode Shapes 189 6.1 Eigenvectors 190 6.2 Comparing Translational and Rotational Degrees of Freedom 196 6.3 Nodal Points in Mode Shapes 207 6.4 Mode Shapes with Damping 208 6.5 Modal Damping 211 Exercises 218 7 Frequency Domain Analysis 223 7.1 Modeling Frequency Response 223 7.2 Seismic Disturbances 235 7.3 Power Spectral Density 238 7.3.1 Units of the PSD 248 7.3.2 Simulation using the PSD249 Exercises 256 8 Time Domain Solutions 259 8.0.1 Getting the Equations of Motion Ready for Time Domain Simulation 260 8.1 A time domain Example 262 8.2 Numerical Schemes for Solving the Equations of Motion 266 8.3 Euler Integration 267 8.4 An Example Using the Euler Integrator 272 8.5 The Central Difference Method - an O(h2) method 274 8.6 Variable Time-Step Methods 276 8.7 Methods with Higher Order Truncation Error 282 8.8 The Structure of a Simulation Program 285 Exercises 291 9 Experimental Data - Frequency Domain AnalysisF 295 9.1 Typical Test Data296 9.2 Transforming to the Frequency Domain - The CFT 298 9.3 Transforming to the Frequency Domain - The DFT 304 9.4 Transforming to the Frequency Domain - A Faster DFT 307 9.5 Transforming to the Frequency Domain - The FFT 309 9.6 Transforming to the Frequency Domain - An Example 310 9.7 Sampling and Aliasing 315 9.8 Leakage and Windowing 324 9.9 Decimating Data 328 9.10 Averaging DFTs 331 Exercises 332 A Representative Dynamic Systems 337 B Moments and Products of Inertia 361 B.1 Moments of Inertia 362 B.2 Parallel Axis Theorem for Moments of Inertia 364 B.3 Parallel Axis Theorem for Products of Inertia 367 B.4 Moments of Inertia for Commonly Encountered Bodies 368 C Dimensions and Units 369 D Least Squares Curve Fitting 373 Index