Page 15 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

12 ContEntS
6.8 Equations of Motion of Undamped Systems in 7.8 Examples Using MATLAB 728
Matrix Form 624 Chapter Summary 731
6.9 Eigenvalue Problem 626 References 731
6.10 Solution of the Eigenvalue Problem 628 Review Questions 733
6.10.1 Solution of the Characteristic (Polynomial) Problems 735
Equation 628 Design Projects 744
6.10.2 Orthogonality of Normal Modes 634
6.10.3 Repeated Eigenvalues 637
6.11 Expansion Theorem 639 ChAPtEr 8
6.12 Unrestrained Systems 639 Continuous Systems 745
6.13 Free Vibration of Undamped Systems 644
6.14 Forced Vibration of Undamped Systems Using 8.1 Introduction 746
Modal Analysis 646 8.2 Transverse Vibration of a String or Cable 747
6.15 Forced Vibration of Viscously Damped 8.2.1 Equation of Motion 747
Systems 653 8.2.2 Initial and Boundary Conditions 749
6.16 Self-Excitation and Stability Analysis 660 8.2.3 Free Vibration of a Uniform String 750
6.17 Examples Using MATLAB 662 8.2.4 Free Vibration of a String with Both Ends
Chapter Summary 670 Fixed 751
References 670 8.2.5 Traveling-Wave Solution 755
Review Questions 671 8.3 Longitudinal Vibration of a Bar or Rod 756
Problems 675 8.3.1 Equation of Motion and Solution 756
Design Projects 696 8.3.2 Orthogonality of Normal Functions 759
8.4 Torsional Vibration of a Shaft or Rod 764
ChAPtEr 7 8.5 Lateral Vibration of Beams 767
Determination of natural Frequencies 8.5.1 Equation of Motion 767
Initial Conditions 769
8.5.2
and Mode Shapes 699 8.5.3 Free Vibration 769
7.1 Introduction 700 8.5.4 Boundary Conditions 770
7.2 Dunkerley’s Formula 701 8.5.5 Orthogonality of Normal
7.3 Rayleigh’s Method 703 Functions 772
7.3.1 Properties of Rayleigh’s Quotient 704 8.5.6 Forced Vibration 776
7.3.2 Computation of the Fundamental Natural 8.5.7 Effect of Axial Force 778
Frequency 706 8.5.8 Effects of Rotary Inertia and Shear
7.3.3 Fundamental Frequency of Beams and Deformation 780
Shafts 708 8.5.9 Beams on Elastic Foundation 785
7.4 Holzer’s Method 711 8.5.10 Other Effects 788
7.4.1 Torsional Systems 711 8.6 Vibration of Membranes 788
7.4.2 Spring-Mass Systems 714 8.6.1 Equation of Motion 788
7.5 Matrix Iteration Method 715 8.6.2 Initial and Boundary Conditions 790
7.5.1 Convergence to the Highest Natural 8.7 Rayleigh’s Method 791
Frequency 717 8.8 The Rayleigh-Ritz Method 794
7.5.2 Computation of Intermediate Natural 8.9 Examples Using MATLAB 797
Frequencies 718 Chapter Summary 800
7.6 Jacobi’s Method 723 References 800
7.7 Standard Eigenvalue Problem 725 Review Questions 802
7.7.1 Choleski Decomposition 726 Problems 805
7.7.2 Other Solution Methods 728 Design Project 818