Assignment 5 solutions

Assignment 5 solutions - MIT OpenCourseWare...

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MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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3 18.085 - Mathematical Methods for Engineers I Prof. Gilbert Strang Solutions - Problem Set 5 Section 2.7, Problem 1 : How many independent solutions to Au = 0. Draw them and find solutions u = ( u 1 , u 1 , . . . , u 4 , u H V H V 4 ). What shapes are A and A T A ? First rows? 6 bars n = 2(6) 4 = 8 unknown disp. So Au = 0 has 8 6 = 2 independent solutions (mechanisms). H H 1 1 u u 1 1 1 1 2 2 1 2 3 4 V V 4 0 1 u u 1 1 H H u 1 1 u 2 2 V V u 0 1 u 2 2 u = = u = = . H H 0 1 u u 3 3 V V 0 1 u u 3 3 H H 0 1 u u 4 4 V V 0 1 u u 4 4 A = 6 × 8 matrix A T = 8 × 6 matrix A T A = 8 × 8 matrix First row of A : First row of A T A : [ 1 [ 1 0 0 1 1 0 0 0 0 0 0 0 0 0 ] . 0 ] . Section 2.7, Problem 2 : 7 bars, N = 5 so n = 2 N r = 2(5) 2 = 8 unknown displacements. a) What motion solves Au = 0? b) By adding one bar, can A become square/invertible?
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