ABSTRACT
This chapter considers numerical techniques, applicable to multiple-degree-of-freedom structures, by considering both the finite element method and incorporating some of the techniques presented in Chapter 3 used for one-degree-of-freedom systems. The principal concepts behind finite elements are presented first. A discussion of one-dimensional (Lagrange and Hermition) interpolation (shape) formulas is presented, followed by discussions of two-dimensional interpolation formula for triangular, rectangular, tetrahedral, solid rectangular hexahedron, isoparametric, and plate elements along with a description of general element properties. Procedures for the assembly of elements into a finite element mesh are then discussed along with appropriate boundary conditions. After assembly, the free response of a finite element system (eigenvalue analysis) is presented with emphasis on the orthogonality of eigenvalues, the Rayleigh quotient, and the reduction to standard form. Methods for calculating eigenvalues and eigenvectors are presented with the focus being vector iteration methods (inverse and forward), vector iteration with shifts, and subspace iteration. Transformation methods are presented based on the generalized Jacobi method. The chapter culminates with discussions of numerical techniques for multiple-degree-of-freedom systems compatible with those presented in Chapter 3. These include the central difference, Humbolt, Newmark-β, Wilson-θ, and HHT-α methods.
