MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering IFall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.085Quiz 1 Professor Strang October 6, 2003 Your name is: Grading 1 2 3 Total 1) (30 pts.) A system with 2 springs and nlasses is fixed-free. Co~lstantsare cl; c2. u,= 0 (a) Write down the matrices A and K = ATCA. (b) Prove by two tests (pivots, determinant,^, independence of columns of 4) that this matrix K is (positive definite) (positive semidefinite). Tell me which two tests you are using! 1 (c) hIultiply colunln times row to compute the "element matrices" Iil,K2: c2 Compute h; (column 1 of 4T)(~1)(ro~ = 1 of A) Compute h; (column 2 of 4T)(~2)(ro~ = 2 of A) Then K = K1 + 1c2.What vect,ors solve K2 For those displacements .rl and q,what is the energy in spring 2?2) (33 pts.) A network of nodes and edges and t,heir conductances cj > 0 is drawn without arrows. Arrows don't affect the answers to this problem; the edge numbers are with t,he c's. Node 5 is grounded (pot,ential us = 0). pp (a) List all posit,ions (i,j) of the 4 by 4 mat,rix K = ATC4 that will have zero entries. What is row 1of K? (b) Find as many independent solut,ions as possible to Kirchhoff's Law c6C7 -5 -4 AT?/= 0. -(c) Is ATA always positive definit,e for every matrix A? If there is a test, on A, what is it,? What is t,he trick that proves uTKu2 0 for every vector u?3) (37 pts.) ---Make the network in Problem 2 into a 7-bar truss! The grounded node 5 is now a support,ed (but turnable) pin joint, with known displacements uf= uy = 0. All angles are 45"or 90". (a) How many rows and columns in t,he (reduced) mat,rix A, after we know = O? Describe in words (or a pict,ure) all solutions to A'u = 0. If you add 1 bar can A become square and invertible? (b) Write out row 2 of A, corresponding to bar 2. Then (row 2) times t,he column u of displacements has what physical meaning? (c) What is the first equat,ion of rlT,u!= f (with right side fP)? Why does 1 TTU KU = $lITC-l?Iand what does this quantity represent physically? (More than 1word in that last answer, less than 10
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