MIT2_72s09_lec05

# MIT2_72s09_lec05 - MIT OpenCourseWare http:/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 2.72 Elements of Mechanical Design Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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2.72 Elements of Mechanical Design Lecture 05: Structures
chedule and reading assignment © Martin Culpepper, All rights reserved Quizzes ± Quiz – None Topics ± Finish fatigue ± Finish HTMs in structures Reading assignment ± None ± Quiz next time on HTMs 2

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Matrix Review
hat is a Matrix? b A matrix is an easy way to 1 represent a system of linear b equations 2 Linear algebra is the set of “Vector” rules that governs matrix and vector operations a 1 a 2 a 3 a 4 “Matrix” © Martin Culpepper, All rights reserved 4

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atrix Addition/Subtraction You can only add or subtract matrices of the same dimension Operations are carried out entry by entry b b b b + a a a a 1 2 1 2 1 1 2 2 + (2 x 2) (2 x 2) (2 x 2) + = + b b b b + a a a a 3 4 3 4 3 3 4 4 b b b b a a a a 1 2 1 2 1 1 2 2 b b b b a a a a 3 4 3 4 3 3 4 4 (2 x 2) (2 x 2) (2 x 2) = © Martin Culpepper, All rights reserved 5
atrix Multiplication An matrix times an matrix produces an matrix m x n n x p m x p b b b b b b + + a a a a a a 1 2 1 2 1 1 2 3 1 2 2 4 b b b b b b + + a a a a a a 3 4 3 4 3 1 4 3 3 2 4 4 (2 x 2) (2 x 2) (2 x 2) = © Martin Culpepper, All rights reserved 6

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atrix Properties © Martin Culpepper, All rights reserved Notation: A, B, C = matrix , c = scalar Cumulative Law: A + B = B + A Distributive Law: c(A + B) = cA + cB C(A + B) = CA + CB Associative Law: A + (B – C) = (A + B) – C A(BC) = (AB)C NOTE that AB does not equal BA !!!!!!! 7
atrix Division © Martin Culpepper, All rights reserved To divide in linear algebra we multiply each side by an inverse matrix: AB = C A -1 AB = A -1 C B = A -1 C Inverse matrix properties: A -1 A = AA -1 = I (The identity matrix) ( AB ) -1 = B -1 A -1 8

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Structures
Structure = backbone = affects everything Satisfies a multiplicity of needs ± Enforcing geometric relationships (position/orientation) ± Material flow and access ± Reference frame Requires first consideration and serves to link modules: ± Joints (bolted/welded/etc…) ± Bearings ± Shafts Image removed due to copyright restrictions. Please see

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## This note was uploaded on 02/24/2012 for the course MECHANICAL 2.72 taught by Professor Martinculpepper during the Spring '09 term at MIT.

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MIT2_72s09_lec05 - MIT OpenCourseWare http:/ocw.mit.edu...

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