Page 11 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Preface

and an example of static condensation for pinned joints is added. Orthogonality of
eigenvectors and the modal matrix and its orthonormal form enable concise
presentation of basic equations for the diagonal eigenvalue matrix that forms the
basis for the computation of the eigenvalue-eigenvector problem. They also pro­
vide a background for the normal mode-summation method. The chapter con­
cludes with modal damping and examples of equal roots and degenerate systems.
Chapter 7 presents the classic method of Lagrange, which is associated with
virtual work and generalized coordinates. Added to this chapter is the method of
assumed modes, which enables the determination of eigenvalues and eigenvectors
of continuous systems in terms of smaller equations of discrete system equations.
The Lagrangian method offers an all-encompassing view of the entire field of
dynamics, a knowledge of which should be acquired by all readers interested in a
serious study of dynamics.
Chapter 8, “Computational Methods,” examines the basic methods of com­
putation that are utilized by the digital computer. Most engineering and science
students today acquire knowledge of computers and programming in their fresh­
man year, and given the basic background for vibration calculation, they can easily
follow computer programs for the calculation of eigenvalues and eigenvectors.
Presented on the IBM computer disk are four basic Fortran programs that cover
most of the calculations encountered in vibration problems. The source programs
written as subroutines can be printed out by typing “.For” (for Fortran) after the
file name; i.e., “Choljac .For”. The user needs only to input the mass and stiffness
matrices and the printout will contain the eigenvalues and eigenvectors of the
problem. Those wishing additional information can modify the command instruc­
tions preceding the computation.
In Chapter 9, “Vibration of Continuous Systems,” a section on suspension
bridges is added to illustrate the application of the continuous system theory to
simplified models for the calculation of natural frequencies. By discretizing the
continuous system by repeated identical sections, simple analytic expressions are
available for the natural frequencies and mode shapes by the method of difference
equations. The method exercises the disciplines of matching boundary conditions.
Chapter 10, “Introduction to the Finite Element Method,” remains essen­
tially unchanged. A few helpful hints have been injected in some places and the
section on generalized force proportional to displacement has been substantially
expanded by detailed computation of rotating helicopter blades. Brought out by
this example is the advantage of forming equal element sections of length / = 1 (all
/’s can be arbitrarily equated to unity inside of the mass and stiffness matrices
when the elements are of equal lengths) for the compiling of the mass and stiffness
matrices and converting the final results to those of the original system only after
the computation is completed.
Chapters 9, 11 and 12 of the former edition are consolidated into new
chapter 11, “Mode-Summation Procedures for Continuous Systems,” and Chapter
12, “Classical Methods.” This was done mainly to leave undisturbed Chapter 13,
“Random Vibrations,” and Chapter 14, “Nonlinear Vibrations,” and in no way