Computational Dynamics P2

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...
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Computational Dynamics P2CHAPTER 2LINEAR ALGEBRAVector and matrix concepts have proved indispensable in the development ofthe subject of dynamics. The formulation of the equations of motion using theNewtonian or Lagrangian approach leads to a set of second-order simultaneousdifferential equations. For convenience, these equations are often expressed invector and matrix forms. Vector and matrix identities can be utilized to providemuch less cumbersome proofs of many of the kinematic and dynamic relation-ships. In this chapter, the mathematical tools required to understand the devel-opment presented in this book are discussed briefly. Matrices and matrix oper-ations are discussed in the first two sections. Differentiation of vector functionsand the important concept of linear independence are discussed in Section 3. InSection 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 ofparallelism are used in this book to define the kinematic constraint equationsof many joints in the three-dimensional analysis. Computer methods for solv-ing algebraic systems of equations are presented in Sections 5 and 6. Amongthe topics discussed in these two sections are the Gaussian elimination, piv-oting and scaling, triangular factorization, and Cholesky decomposition. Thelast two sections of this chapter deal with the QR decomposition and the sin-gular value decomposition. These two types of decompositions have been usedin computational dynamics to identify the independent degrees of freedom ofmultibody systems. The last two sections, however, can be omitted during afirst reading of the book.22 2.1 MATRICES 232.1 MATRICESAn m × n matrix A is an ordered rectangular array that has m × n elements.The matrix A can be written in the form a  a 11 a12 a22 ··· ··· a 1n a 2n    21  A (aij )  . . .. .  (2.1)  . . . . . . .  a m1 am2 ··· amn The matrix A is called an m × n matrix since it has m rows and n columns. Thescalar element aij lies in the ith row and jth column of the matrix A. Therefore,the index i, which takes the values 1, 2, . . . , m, denotes the row number, whilethe index j, which takes the values 1, 2, . . . , n denotes the column number. A matrix A is said to be square if m n. An example of a square matrix is A  6..0  3 3 − 2.0 0.0 12.0 0.95    9.0 3.5 1.25 In this example, m n 3, and A is a 3 × 3 matrix. The transpose of an m × n matrix A is an n × m matrix denoted as AT anddefined as a  a 11 a21 a22 ··· ...