Vibration Theory and Applications with Finite Elements and Active Vibration Control

Preface xv
Acknowledgments and Dedication xxi
About the Companion Website xxiii
List of Acronyms xxv

1 Background, Motivation, and Overview 1
1.1 Introduction 1
1.2 Background 1
1.3 Our Vibrating World 6
1.4 Harmful Effects of Vibration 9
1.5 Stiffness, Inertia, and Damping Forces 29
1.6 Approaches for Obtaining the Differential Equations of Motion 34
1.7 Finite Element Method 35
1.8 Active Vibration Control 37
1.9 Chapter 1 Exercises 37

2 Preparatory Skills: Mathematics, Modeling, and Kinematics 41
2.1 Introduction 41
2.2 Getting Started with MATLAB and MAPLE 42
2.3 Vibration and Differential Equations 50
2.4 Taylor Series Expansions and Linearization 56
2.5 Complex Variables (CV) and Phasors 60
2.6 Degrees of Freedom, Matrices, Vectors, and Subspaces 63
2.7 Coordinate Transformations 75
2.8 Eigenvalues and Eigenvectors 79
2.9 Fourier Series 80
2.10 Laplace Transforms, Transfer Functions, and Characteristic Equations 83
2.11 Kinematics and Kinematic Constraints 86
2.12 Dirac Delta and Heaviside Functions 100
2.13 Chapter 2 Exercises 101

3 Equations of Motion by Newton's Laws 103
3.1 Introduction 103
3.2 Particle Motion Approximation 103
3.3 Planar (2D) Rigid Body Motion Approximation 107
3.4 Impulse and Momentum 129
3.5 Variable Mass Systems 138
3.6 Chapter 3 Exercises 140

4 Equations of Motion by Energy Methods 143
4.1 Introduction 143
4.2 Kinetic Energy 143
4.3 External and Internal Work and Potential Energy 147
4.4 Power and Work-Energy Laws 151
4.5 Lagrange Equation for Particles and Rigid Bodies 157
4.6 LE for Flexible, Distributed Mass Bodies: Assumed Modes Approach 211
4.7 LE for Flexible, Distributed Mass Bodies: Finite Element Approach-General Formulation 267
4.8 LE for Flexible, Distributed Mass Bodies: Finite Element Approach-Bar/Truss Modes 275
4.9 Chapter 4 Exercises 306

5 Free Vibration Response 309
5.1 Introduction 309
5.2 Single Degree of Freedom Systems 309
5.3 Two-Degree-of-Freedom Systems 319
5.4 N-Degree-of-Freedom Systems 346
5.5 Infinite Dof Continuous Member Systems 390
5.6 Unstable Free Vibrations 408
5.7 Summary 418
5.8 Chapter 5 Exercises 418

6 Vibration Response Due to Transient Loading 421
6.1 Introduction 421
6.2 Single Degree of Freedom Transient Response 421
6.3 Modal Condensation of Ndof: Transient Forced Vibrating Systems 451
6.4 Numerical Integration of Ndof Transient Vibration Response 493
6.5 Summary 521
6.6 Chapter 6 Exercises 522

7 Steady-State Vibration Response to Periodic Loading 525
7.1 Introduction 525
7.2 Complex Phasor Approach 525
7.3 Single Degree of Freedom Models 527
7.4 Two Degree of Freedom Response 559
7.5 N Degree of freedom Steady-State Harmonic Response 566
7.6 Other Phasor Ratio Measures of Steady-State Harmonic Response 591
7.7 Summary 593
7.8 Chapter 7 Exercises 593

8 Approximate Methods for Large-Order Systems 595
8.1 Introduction 595
8.2 Guyan Reduction: Static Condensation 596
8.3 Substructures: Superelements 608
8.4 Modal Synthesis 609
8.5 Eigenvalue/Natural Frequency Changes for Perturbed Systems 620
8.6 Summary 633
8.7 Chapter 8 Exercises 634

9 Beam Finite Elements for Vibration Analysis 637
9.1 Introduction 637
9.2 Modeling 2D Frame Structures with Euler-Bernoulli Beam Elements 637
9.3 Three-Dimensional Timoshenko Beam Elements: Introduction 670
9.4 3D Timoshenko Beam Elements: Nodal Coordinates 672
9.5 3D Timoshenko Beam Elements: Shape Functions, Element Stiffness, and Mass Matrices 679
9.6 3D Timoshenko Beam Element Force Vectors 704
9.7 3D Frame: Beam Element Assembly Algorithm 713
9.8 2D Frame Modeling with Timoshenko Beam Elements 725
9.9 Summary 748
9.10 Chapter 9 Exercises 749

10 2D Planar Finite Elements for Vibration Analysis 751
10.1 Introduction 751
10.2 Plane Strain (Pe) 751
10.3 Plane Stress (Ps) 753
10.4 Plane Stress and Plane Strain: Element Stiffness and Mass Matrices and Force Vector 754
10.5 Assembly Procedure for 2D, 4-Node, Quadrilateral Elements 763
10.6 Computation of Stresses in 2D Solid Elements 768
10.7 Extra Shape Functions to Improve Accuracy 774
10.8 Illustrative Example 776
10.9 2D Axisymmetric Model 786
10.10 Automated Mesh Generation: Constant Strain Triangle Elements 801
10.11 Membranes 810
10.12 Banded Storage 815
10.13 Chapter 10 Exercises 820

11 3D Solid Elements for Vibration Analysis 823
11.1 Introduction 823
11.2 Element Stiffness Matrix 825
11.3 The Element Mass Matrix and Force Vector 835
11.4 Assembly Procedure for the 3D, 8-Node, Hexahedral Element Model 842
11.5 Computation of Stresses for a 3D Hexahedral Solid Element 846
11.6 3D Solid Element Model Example 856
11.7 3D Solid Element Summary 864
11.8 Chapter 11 Exercises 865

12 Active Vibration Control 867
12.1 Introduction 867
12.2 AVC System Modeling 871
12.3 AVC Actuator Modeling 874
12.4 System Model with an Infinite Bandwidth Feedback Approximation 878
12.5 System Model with Finite Bandwidth Feedback 886
12.6 System Model with Finite Bandwidth Feedback and Lead Compensation 893
12.7 Sensor/Actuator Noncollocation Effect on Vibration Stability 901
12.8 Piezoelectric Actuators 907
12.9 Summary 923
12.10 Chapter 12 Exercises 923

References 924
Appendix A Fundamental Equations of Elasticity 927
Index 941