net1 - Application of Matrix Algebra in Mechanics Matrix...

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Application of Matrix Algebra in Mechanics Matrix notation is the most convenient for analyzing complicated problems, it is applied in many areas in engineering and social sciences. A little bit of patience is required to condition yourself to like the notation, but once you’ve familiarized yourself with the basic operations, you can analyze simple or complicated problems in the same way without introducing separate theories for each situation. Matrix notation is used heavily today in advanced theory and undoubtedly will be the way of the future because it is (1) compact and convenient to write, (2) easily converted from mathematics to computer programs, and (3) in an ideal form for concurrent processing. In CE 325, there will be two basic type of notations used: (a) Abstract Vector Notation – The subject, “Linear Algebra,” deals with abstract vectors and their transformation operators. The results of linear algebraic expressions are also abstract quantities and they are applicable in any coordinate systems, for example, ~ F = m~a is an abstract vector expression, ~ F and ~a are abstract vectors and m is a scalar. The above equation, i.e., Newton’s Second Law, is valid in any coordinate system. (b) Matrix Notation – The subject, “Matrix algebra,” deals with numbers because a matrix is nothing more than a set of numbers arranged in a rectangular manner. For example, the matrix [ A ]=[ a ij ]= ± a 11 a 12 a 13 a 21 a 22 a 23 ² is a “two-by-three” matrix, meaning 2 rows and 3 columns. The subscript integers, i and j ,of the element a ij are the row and column indices, respectively. Matrix algebra is useful in many disciplines inside and outside of engineering. It is quite effective because only numbers are dealt with, the actual meaning of a particular element in a matrix is implied only by its position in the matrix. For example, in dynamics, an acceleration vector p =3 . ı +2 . - 7 . 2 ˆ k can be written in matrix form as p = 3 . 4 2 . 9 - 7 . 2 where the number 2.9 is implied to be the y -component because it is the second number in the “column vector,” it is no longer necessary to identify the components with the basis vectors ˆ ı , ˆ and ˆ k . In dynamics, there are many vectors to deal with, e.g., forces, moments, displacements, velocities, accelerations, impulse, momentum, angular impulse, angular momentum, angular velocities and angular accelerations. All these vectors can be written in the form of a column vector. The square matrix is used for the moment of inertia matrix and various vector transformation matrices (rotation or translation). –1/1–
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Linear Algebra Operations The following are some of the frequently used (not complete) linear algebra operations: ~u + ~v = + ( a ) ( + )+ ~ w = ~ u +( ~ v + ~ w )( b ) ~ u = - ~ v ( c ) ~ w = α~v ( d ) α = · ( e ) ~w = × ( f ) In the expressions (d) and (e), α is a scalar. There is no multiplication of vectors.
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This note was uploaded on 04/16/2008 for the course CE 325 taught by Professor Docwong during the Spring '08 term at USC.

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net1 - Application of Matrix Algebra in Mechanics Matrix...

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