Computational Dynamics, 3rd Edition

Chapter 2. LINEAR ALGEBRA

Vector and matrix concepts have proved indispensable in the development of the subject of dynamics. The formulation of the equations of motion using the Newtonian or Lagrangian approach leads to a set of second-order simultaneous differential equations. For convenience, these equations are often expressed in vector and matrix forms. Vector and matrix identities can be utilized to provide much less cumbersome proofs of many of the kinematic and dynamic relationships. In this chapter, the mathematical tools required to understand the development presented in this book are discussed briefly. Matrices and matrix operations are discussed in the first two sections. Differentiation of vector functions and the important concept of linear independence are discussed in Section 3. In Section 4, important topics related to three-dimensional vectors are presented. These topics include the cross product, skew-symmetric matrix representations, Cartesian coordinate systems, and conditions of parallelism. The conditions of parallelism are used in this book to define the kinematic constraint equations of many joints in the three-dimensional analysis. Computer methods for solving algebraic systems of equations are presented in Sections 5 and 6. Among the topics discussed in these two sections are the Gaussian elimination, pivoting and scaling, triangular factorization, and Cholesky decomposition. The last two sections of this chapter deal with the QR decomposition and the ...