DR. RONALD J. ANDERSON is a Professor in the Department of Mechanical and Materials Engineering, Queen's University at Kingston, Canada. He received his B.Sc.(Eng) from the University of Alberta in 1973, his M.Sc.(Eng) from Queen's University in 1974, and his Ph.D. from Queen's University in 1977. His doctoral research was in the field of road vehicle dynamics. From 1977 to 1979, he was a Defence Scientist with the Defence Research Establishment Atlantic where he was engaged in research on the dynamics of novel ships. From 1979 to 1981 he was Senior Dynamicist with the Urban Transportation Development Corporation where he worked on rail vehicle dynamics, particularly suspension design for steerable rail vehicles. He joined Queen's University in 1981 and, while conducting research into vehicle dynamics and multibody dynamics, has been teaching undergraduate courses on dynamics and vibrations and postgraduate courses on advanced dynamics and engineering analysis. Dr. Anderson has been the recipient of several departmental and faculty-wide teaching awards. He has also served the University in the academic administrative roles of Head of Department, Associate Dean (Research), and Dean of Graduate Studies.

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• Preface xiAbout the Companion Website xvPart I Modeling: Deriving Equations of Motion 11 Kinematics 31.1 Derivatives of Vectors 31.2 Performing Kinematic Analysis 51.3 Two Dimensional Motion with Constant Length 61.4 Two Dimensional Motion with Variable Length 81.5 Three Dimensional Kinematics 101.6 Absolute Angular Velocity and Acceleration 131.7 The General Acceleration Expression 14Exercises 162 Newton’s Equations of Motion 192.1 The Study of Motion 192.2 Newton’s Laws 192.3 Newton’s Second Law for a Particle 202.4 Deriving Equations of Motion for Particles 212.5 Working with Rigid Bodies 252.6 Using F = ma in the Rigid Body Force Balance 262.7 Using F = dG/dt in the Rigid Body Force Balance 282.8 Moment Balance for a Rigid Body 302.9 The Angular Momentum Vector – HO 332.10 A Physical Interpretation of Moments and Products of Inertia 362.11 Euler’s Moment Equations 402.12 Throwing a Spiral 412.13 A Two Body System 422.14 Gyroscopic Motion 48Exercises 523 Lagrange’s Equations of Motion 553.1 An Example to Start 553.2 Lagrange’s Equation for a Single Particle 583.3 Generalized Forces 623.4 Generalized Forces as Derivatives of Potential Energy 643.5 Dampers – Rayleigh’s Dissipation Function 653.6 Kinetic Energy of a Free Rigid Body 673.7 A Two Dimensional Example using Lagrange’s Equation 703.7.1 The Kinetic Energy 703.7.2 The Potential Energy 713.7.3 The 𝜃 Equation 723.7.4 The 𝜙 Equation 733.8 Standard Form of the Equations of Motion 73Exercises 74Part II Simulation: Using the Equations of Motion 774 Equilibrium Solutions 794.1 The Simple Pendulum 794.2 Equilibrium with Two Degrees of Freedom 804.3 Equilibrium with Steady Motion 814.4 The General Equilibrium Solution 84Exercises 855 Stability 875.1 Analytical Stability 875.2 Linearization of Functions 925.3 Example: A System with Two Degrees of Freedom 955.4 Routh Stability Criterion 995.5 Standard Procedure for Stability Analysis 103Exercises 1056 Mode Shapes 1076.1 Eigenvectors 1076.2 Comparing Translational and Rotational Degrees of Freedom 1116.3 Nodal Points in Mode Shapes 1156.4 Mode Shapes with Damping 1166.5 Modal Damping 118Exercises 1227 Frequency Domain Analysis 1257.1 Modeling Frequency Response 1257.2 Seismic Disturbances 1327.3 Power Spectral Density 1337.3.1 Units of the PSD 1387.3.2 Simulation using the PSD 139Exercises 1438 Time Domain Solutions 1458.1 Getting the Equations of Motion Ready for Time Domain Simulation 1468.2 A Time Domain Example 1478.3 Numerical Schemes for Solving the Equations of Motion 1498.4 Euler Integration 1498.5 An Example Using the Euler Integrator 1518.6 The Central Difference Method: An (h2) Method 1538.7 Variable Time Step Methods 1558.8 Methods with Higher Order Truncation Error 1578.9 The Structure of a Simulation Program 159Exercises 163Part III Working with Experimental Data 1659 Experimental Data – Frequency Domain Analysis 1679.1 Typical Test Data 1679.2 Transforming to the Frequency Domain – The CFT 1699.3 Transforming to the Frequency Domain – The DFT 1729.4 Transforming to the Frequency Domain – A Faster DFT 1749.5 Transforming to the Frequency Domain – The FFT 1759.6 Transforming to the Frequency Domain – An Example 1769.7 Sampling and Aliasing 1799.8 Leakage and Windowing 1849.9 Decimating Data 1879.10 Averaging DFTs 189Exercises 189A Representative Dynamic Systems 193A.1 System 1 193A.2 System 2 193A.3 System 3 194A.4 System 4 194A.5 System 5 195A.6 System 6 195A.7 System 7 196A.8 System 8 197A.9 System 9 197A.10 System 10 198A.11 System 11 198A.12 System 12 199A.13 System 13 200A.14 System 14 200A.15 System 15 201A.16 System 16 201A.17 System 17 202A.18 System 18 202A.19 System 19 203A.20 System 20 203A.21 System 21 204A.22 System 22 204A.23 System 23 205B Moments and Products of Inertia 207B.1 Moments of Inertia 207B.2 Parallel Axis Theorem for Moments of Inertia 208B.3 Parallel Axis Theorem for Products of Inertia 210B.4 Moments of Inertia for Commonly Encountered Bodies 210C Dimensions and Units 213D Least Squares Curve Fitting 215Index 219

DR. RONALD J. ANDERSON is a Professor in the Department of Mechanical and Materials Engineering, Queen's University at Kingston, Canada. He received his B.Sc.(Eng) from the University of Alberta in 1973, his M.Sc.(Eng) from Queen's University in 1974, and his Ph.D. from Queen's University in 1977. His doctoral research was in the field of road vehicle dynamics. From 1977 to 1979, he was a Defence Scientist with the Defence Research Establishment Atlantic where he was engaged in research on the dynamics of novel ships. From 1979 to 1981 he was Senior Dynamicist with the Urban Transportation Development Corporation where he worked on rail vehicle dynamics, particularly suspension design for steerable rail vehicles. He joined Queen's University in 1981 and, while conducting research into vehicle dynamics and multibody dynamics, has been teaching undergraduate courses on dynamics and vibrations and postgraduate courses on advanced dynamics and engineering analysis. Dr. Anderson has been the recipient of several departmental and faculty-wide teaching awards. He has also served the University in the academic administrative roles of Head of Department, Associate Dean (Research), and Dean of Graduate Studies.

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• Preface xiAbout the Companion Website xvPart I Modeling: Deriving Equations of Motion 11 Kinematics 31.1 Derivatives of Vectors 31.2 Performing Kinematic Analysis 51.3 Two Dimensional Motion with Constant Length 61.4 Two Dimensional Motion with Variable Length 81.5 Three Dimensional Kinematics 101.6 Absolute Angular Velocity and Acceleration 131.7 The General Acceleration Expression 14Exercises 162 Newton’s Equations of Motion 192.1 The Study of Motion 192.2 Newton’s Laws 192.3 Newton’s Second Law for a Particle 202.4 Deriving Equations of Motion for Particles 212.5 Working with Rigid Bodies 252.6 Using F = ma in the Rigid Body Force Balance 262.7 Using F = dG/dt in the Rigid Body Force Balance 282.8 Moment Balance for a Rigid Body 302.9 The Angular Momentum Vector – HO 332.10 A Physical Interpretation of Moments and Products of Inertia 362.11 Euler’s Moment Equations 402.12 Throwing a Spiral 412.13 A Two Body System 422.14 Gyroscopic Motion 48Exercises 523 Lagrange’s Equations of Motion 553.1 An Example to Start 553.2 Lagrange’s Equation for a Single Particle 583.3 Generalized Forces 623.4 Generalized Forces as Derivatives of Potential Energy 643.5 Dampers – Rayleigh’s Dissipation Function 653.6 Kinetic Energy of a Free Rigid Body 673.7 A Two Dimensional Example using Lagrange’s Equation 703.7.1 The Kinetic Energy 703.7.2 The Potential Energy 713.7.3 The 𝜃 Equation 723.7.4 The 𝜙 Equation 733.8 Standard Form of the Equations of Motion 73Exercises 74Part II Simulation: Using the Equations of Motion 774 Equilibrium Solutions 794.1 The Simple Pendulum 794.2 Equilibrium with Two Degrees of Freedom 804.3 Equilibrium with Steady Motion 814.4 The General Equilibrium Solution 84Exercises 855 Stability 875.1 Analytical Stability 875.2 Linearization of Functions 925.3 Example: A System with Two Degrees of Freedom 955.4 Routh Stability Criterion 995.5 Standard Procedure for Stability Analysis 103Exercises 1056 Mode Shapes 1076.1 Eigenvectors 1076.2 Comparing Translational and Rotational Degrees of Freedom 1116.3 Nodal Points in Mode Shapes 1156.4 Mode Shapes with Damping 1166.5 Modal Damping 118Exercises 1227 Frequency Domain Analysis 1257.1 Modeling Frequency Response 1257.2 Seismic Disturbances 1327.3 Power Spectral Density 1337.3.1 Units of the PSD 1387.3.2 Simulation using the PSD 139Exercises 1438 Time Domain Solutions 1458.1 Getting the Equations of Motion Ready for Time Domain Simulation 1468.2 A Time Domain Example 1478.3 Numerical Schemes for Solving the Equations of Motion 1498.4 Euler Integration 1498.5 An Example Using the Euler Integrator 1518.6 The Central Difference Method: An (h2) Method 1538.7 Variable Time Step Methods 1558.8 Methods with Higher Order Truncation Error 1578.9 The Structure of a Simulation Program 159Exercises 163Part III Working with Experimental Data 1659 Experimental Data – Frequency Domain Analysis 1679.1 Typical Test Data 1679.2 Transforming to the Frequency Domain – The CFT 1699.3 Transforming to the Frequency Domain – The DFT 1729.4 Transforming to the Frequency Domain – A Faster DFT 1749.5 Transforming to the Frequency Domain – The FFT 1759.6 Transforming to the Frequency Domain – An Example 1769.7 Sampling and Aliasing 1799.8 Leakage and Windowing 1849.9 Decimating Data 1879.10 Averaging DFTs 189Exercises 189A Representative Dynamic Systems 193A.1 System 1 193A.2 System 2 193A.3 System 3 194A.4 System 4 194A.5 System 5 195A.6 System 6 195A.7 System 7 196A.8 System 8 197A.9 System 9 197A.10 System 10 198A.11 System 11 198A.12 System 12 199A.13 System 13 200A.14 System 14 200A.15 System 15 201A.16 System 16 201A.17 System 17 202A.18 System 18 202A.19 System 19 203A.20 System 20 203A.21 System 21 204A.22 System 22 204A.23 System 23 205B Moments and Products of Inertia 207B.1 Moments of Inertia 207B.2 Parallel Axis Theorem for Moments of Inertia 208B.3 Parallel Axis Theorem for Products of Inertia 210B.4 Moments of Inertia for Commonly Encountered Bodies 210C Dimensions and Units 213D Least Squares Curve Fitting 215Index 219

DR. RONALD J. ANDERSON is a Professor in the Department of Mechanical and Materials Engineering, Queen's University at Kingston, Canada. He received his B.Sc.(Eng) from the University of Alberta in 1973, his M.Sc.(Eng) from Queen's University in 1974, and his Ph.D. from Queen's University in 1977. His doctoral research was in the field of road vehicle dynamics. From 1977 to 1979, he was a Defence Scientist with the Defence Research Establishment Atlantic where he was engaged in research on the dynamics of novel ships. From 1979 to 1981 he was Senior Dynamicist with the Urban Transportation Development Corporation where he worked on rail vehicle dynamics, particularly suspension design for steerable rail vehicles. He joined Queen's University in 1981 and, while conducting research into vehicle dynamics and multibody dynamics, has been teaching undergraduate courses on dynamics and vibrations and postgraduate courses on advanced dynamics and engineering analysis. Dr. Anderson has been the recipient of several departmental and faculty-wide teaching awards. He has also served the University in the academic administrative roles of Head of Department, Associate Dean (Research), and Dean of Graduate Studies.

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• Preface xiAbout the Companion Website xvPart I Modeling: Deriving Equations of Motion 11 Kinematics 31.1 Derivatives of Vectors 31.2 Performing Kinematic Analysis 51.3 Two Dimensional Motion with Constant Length 61.4 Two Dimensional Motion with Variable Length 81.5 Three Dimensional Kinematics 101.6 Absolute Angular Velocity and Acceleration 131.7 The General Acceleration Expression 14Exercises 162 Newton’s Equations of Motion 192.1 The Study of Motion 192.2 Newton’s Laws 192.3 Newton’s Second Law for a Particle 202.4 Deriving Equations of Motion for Particles 212.5 Working with Rigid Bodies 252.6 Using F = ma in the Rigid Body Force Balance 262.7 Using F = dG/dt in the Rigid Body Force Balance 282.8 Moment Balance for a Rigid Body 302.9 The Angular Momentum Vector – HO 332.10 A Physical Interpretation of Moments and Products of Inertia 362.11 Euler’s Moment Equations 402.12 Throwing a Spiral 412.13 A Two Body System 422.14 Gyroscopic Motion 48Exercises 523 Lagrange’s Equations of Motion 553.1 An Example to Start 553.2 Lagrange’s Equation for a Single Particle 583.3 Generalized Forces 623.4 Generalized Forces as Derivatives of Potential Energy 643.5 Dampers – Rayleigh’s Dissipation Function 653.6 Kinetic Energy of a Free Rigid Body 673.7 A Two Dimensional Example using Lagrange’s Equation 703.7.1 The Kinetic Energy 703.7.2 The Potential Energy 713.7.3 The 𝜃 Equation 723.7.4 The 𝜙 Equation 733.8 Standard Form of the Equations of Motion 73Exercises 74Part II Simulation: Using the Equations of Motion 774 Equilibrium Solutions 794.1 The Simple Pendulum 794.2 Equilibrium with Two Degrees of Freedom 804.3 Equilibrium with Steady Motion 814.4 The General Equilibrium Solution 84Exercises 855 Stability 875.1 Analytical Stability 875.2 Linearization of Functions 925.3 Example: A System with Two Degrees of Freedom 955.4 Routh Stability Criterion 995.5 Standard Procedure for Stability Analysis 103Exercises 1056 Mode Shapes 1076.1 Eigenvectors 1076.2 Comparing Translational and Rotational Degrees of Freedom 1116.3 Nodal Points in Mode Shapes 1156.4 Mode Shapes with Damping 1166.5 Modal Damping 118Exercises 1227 Frequency Domain Analysis 1257.1 Modeling Frequency Response 1257.2 Seismic Disturbances 1327.3 Power Spectral Density 1337.3.1 Units of the PSD 1387.3.2 Simulation using the PSD 139Exercises 1438 Time Domain Solutions 1458.1 Getting the Equations of Motion Ready for Time Domain Simulation 1468.2 A Time Domain Example 1478.3 Numerical Schemes for Solving the Equations of Motion 1498.4 Euler Integration 1498.5 An Example Using the Euler Integrator 1518.6 The Central Difference Method: An (h2) Method 1538.7 Variable Time Step Methods 1558.8 Methods with Higher Order Truncation Error 1578.9 The Structure of a Simulation Program 159Exercises 163Part III Working with Experimental Data 1659 Experimental Data – Frequency Domain Analysis 1679.1 Typical Test Data 1679.2 Transforming to the Frequency Domain – The CFT 1699.3 Transforming to the Frequency Domain – The DFT 1729.4 Transforming to the Frequency Domain – A Faster DFT 1749.5 Transforming to the Frequency Domain – The FFT 1759.6 Transforming to the Frequency Domain – An Example 1769.7 Sampling and Aliasing 1799.8 Leakage and Windowing 1849.9 Decimating Data 1879.10 Averaging DFTs 189Exercises 189A Representative Dynamic Systems 193A.1 System 1 193A.2 System 2 193A.3 System 3 194A.4 System 4 194A.5 System 5 195A.6 System 6 195A.7 System 7 196A.8 System 8 197A.9 System 9 197A.10 System 10 198A.11 System 11 198A.12 System 12 199A.13 System 13 200A.14 System 14 200A.15 System 15 201A.16 System 16 201A.17 System 17 202A.18 System 18 202A.19 System 19 203A.20 System 20 203A.21 System 21 204A.22 System 22 204A.23 System 23 205B Moments and Products of Inertia 207B.1 Moments of Inertia 207B.2 Parallel Axis Theorem for Moments of Inertia 208B.3 Parallel Axis Theorem for Products of Inertia 210B.4 Moments of Inertia for Commonly Encountered Bodies 210C Dimensions and Units 213D Least Squares Curve Fitting 215Index 219