Introduction to Structural Dynamics

by Bruce K. Donaldson

• ISBN13: 9780521865746
• ISBN10: 0521865743
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-10-23
• Publisher: Cambridge University Press

Summary

This textbook provides the student of aerospace, civil, and mechanical engineering with all the fundamentals of linear structural dynamics analysis. It is designed for an advanced undergraduate or first year graduate course. This textbook is a departure from the usual presentation in two important respects. First, descriptions of system dynamics are based on the simpler to use Lagrange equations. Second, no organizational distinctions are made between multi-degree of freedom systems and single-degree of freedom systems. The textbook is organized on the basis of first writing structural equation systems of motion, and then solving those equations mostly by means of a modal transformation. The text contains more material than is commonly taught in one semester so advanced topics are designated by an asterisk. The final two chapters can also be deferred for later studies. The text contains numerous examples and end-of-chapter exercises.

Author Biography

Bruce K. Donaldson became a professor of aerospace engineering and then a professor of civil engineering at the University of Maryland.

Table of Contents

Preface for the Student	xi
Preface for the Instructor	xv
Acknowledgments	xvii
List of Symbols	xix
The Lagrange Equations of Motion	1	(45)
Introduction	1	(1)
Newton's Laws of Motion	2	(3)
Newton's Equations for Rotations	5	(3)
Simplifications for Rotations	8	(4)
Conservation Laws	12	(1)
Generalized Coordinates	12	(3)
Virtual Quantities and the Variational Operator	15	(4)
The Lagrange Equations	19	(6)
Kinetic Energy	25	(4)
Summary	29	(17)
Chapter 1 Exercises	33	(4)
Endnote (1): Further Explanation of the Variational Operator	37	(4)
Endnote (2): Kinetic Energy and Energy Dissipation	41	(1)
Endnote (3): A Rigid Body Dynamics Example Problem	42	(4)
Mechanical Vibrations: Practice Using the Lagrange Equations	46	(53)
Introduction	46	(1)
Techniques of Analysis for Pendulum Systems	47	(6)
Example Problems	53	(13)
Interpreting Solutions to Pendulum Equations	66	(5)
Linearizing Differential Equations for Small Deflections	71	(1)
Summary	72	(1)
**Conservation of Energy versus the Lagrange Equations**	73	(7)
**Nasty Equations of Motion**	80	(2)
**Stability of Vibratory Systems**	82	(17)
Chapter 2 Exercises	85	(8)
Endnote (1): The Large-Deflection, Simple Pendulum Solution	93	(1)
Endnote (2): Divergence and Flutter in Multidegree of Freedom, Force Free Systems	94	(5)
Review of the Basics of the Finite Element Method for Simple Elements	99	(58)
Introduction	99	(1)
Generalized Coordinates for Deformable Bodies	100	(3)
Element and Global Stiffness Matrices	103	(9)
More Beam Element Stiffness Matrices	112	(11)
Summary	123	(34)
Chapter 3 Exercises	133	(5)
Endnote (1): A Simple Two-Dimensional Finite Element	138	(8)
Endnote (2): The Curved Beam Finite Element	146	(11)
FEM Equations of Motion for Elastic Systems	157	(56)
Introduction	157	(1)
Structural Dynamic Modeling	158	(5)
Isolating Dynamic from Static Loads	163	(2)
Finite Element Equations of Motion for Structures	165	(7)
Finite Element Example Problems	172	(14)
Summary	186	(7)
**Offset Elastic Elements**	193	(20)
Chapter 4 Exercises	195	(10)
Endnote (1): Mass Refinement Natural Frequency Results	205	(1)
Endnote (2): The Rayleigh Quotient	206	(4)
Endnote (3): The Matrix Form of the Lagrange Equations	210	(1)
Endnote (4): The Consistent Mass Matrix	210	(1)
Endnote (5): A Beam Cross Section with Equal Bending and Twisting Stiffness Coefficients	211	(2)
Damped Structural Systems	213	(50)
Introduction	213	(1)
Descriptions of Damping Forces	213	(17)
The Response of a Viscously Damped Oscillator to a Harmonic Loading	230	(9)
Equivalent Viscous Damping	239	(3)
Measuring Damping	242	(1)
Example Problems	243	(4)
Harmonic Excitation of Multidegree of Freedom Systems	247	(1)
Summary	248	(15)
Chapter 5 Exercises	253	(7)
Endnote (1): A Real Function Solution to a Harmonic Input	260	(3)
Natural Frequencies and Mode Shapes	263	(71)
Introduction	263	(2)
Natural Frequencies by the Determinant Method	265	(8)
Mode Shapes by Use of the Determinant Method	273	(6)
**Repeated Natural Frequencies**	279	(10)
Orthogonality and the Expansion Theorem	289	(4)
The Matrix Iteration Method	293	(7)
**Higher Modes by Matrix Iteration**	300	(7)
Other Eigenvalue Problem Procedures	307	(4)
Summary	311	(4)
**Modal Tuning**	315	(19)
Chapter 6 Exercises	320	(3)
Endnote (1): Linearly Independent Quantities	323	(1)
Endnote (2): The Cholesky Decomposition	324	(2)
Endnote (3): Constant Momentum Transformations	326	(3)
Endnote (4): Illustration of Jacobi's Method	329	(3)
Endnote (5): The Gram-Schmidt Process for Creating Orthogonal Vectors	332	(2)
The Modal Transformation	334	(68)
Introduction	334	(1)
Initial Conditions	334	(3)
The Modal Transformation	337	(3)
Harmonic Loading Revisited	340	(2)
Impulsive and Sudden Loadings	342	(9)
The Modal Solution for a General Type of Loading	351	(2)
Example Problems	353	(10)
Random Vibration Analyses	363	(3)
Selecting Mode Shapes and Solution Convergence	366	(5)
Summary	371	(2)
**Aeroelasticity**	373	(15)
**Response Spectrums**	388	(14)
Chapter 7 Exercises	391	(5)
Endnote (1): Verification of the Duhamel Integral Solution	396	(2)
Endnote (2): A Rayleigh Analysis Example	398	(1)
Endnote (3): An Example of the Accuracy of Basic Strip Theory	399	(1)
Endnote (4): Nonlinear Vibrations	400	(2)
Continuous Dynamic Models	402	(49)
Introduction	402	(1)
Derivation of the Beam Bending Equation	402	(4)
Modal Frequencies and Mode Shapes for Continuous Models	406	(25)
Conclusion	431	(20)
Chapter 8 Exercises	438	(1)
Endnote (1): The Long Beam and Thin Plate Differential Equations	439	(3)
Endnote (2): Derivation of the Beam Equation of Motion Using Hamilton's Principle	442	(3)
Endnote (3): Sturm--Liouville Problems	445	(1)
Endnote (4): The Bessel Equation and Its Solutions	445	(4)
Endnote (5): Nonhomogeneous Boundary Conditions	449	(2)
Numerical Integration of the Equations of Motion	451	(32)
Introduction	451	(1)
The Finite Difference Method	452	(8)
Assumed Acceleration Techniques	460	(3)
Predictor-Corrector Methods	463	(5)
The Runge-Kutta Method	468	(6)
Summary	474	(1)
**Matrix Function Solutions**	475	(8)
Chapter 9 Exercises	480	(3)
Appendix I. Answers to Exercises	483	(48)
Chapter 1 Solutions	483	(3)
Chapter 2 Solutions	486	(8)
Chapter 3 Solutions	494	(4)
Chapter 4 Solutions	498	(11)
Chapter 5 Solutions	509	(7)
Chapter 6 Solutions	516	(3)
Chapter 7 Solutions	519	(6)
Chapter 8 Solutions	525	(4)
Chapter 9 Solutions	529	(2)
Appendix II. Fourier Transform Pairs	531	(6)
Introduction to Fourier Transforms	531	(6)
Index	537