MIT2_007s09_lec21

MIT2_007s09_lec21 - MIT OpenCourseWare http:/ocw.mit.edu...

Info iconThis preview shows pages 1–16. Sign up to view the full content.

View Full Document Right Arrow Icon
MIT OpenCourseWare http://ocw.mit.edu 2.007 Design and Manufacturing I Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2.007 –Design and Manufacturing I Optimization and Solution of Systems Dan Frey 28 APR 2009 -2 -1 0 1 2 3 -4 -3 0 1 f g h a b c d e 0 x 1 x 2 x * x
Background image of page 2
Today’s Agenda • Seeding and impounding procedures • Methods for Solving Systems – Newton-Raphson – Secant – Bisection • Examples related to mechanism design
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Seeding • Run on the table unopposed • Timing and set-up as in the actual contest • Three tries – best of three counts • Your “seeding card” is essential – Get your scores recorded and initialed – Don’t lose your card • “In-lab” competition – Basically a way to get round 1 partly finished – Same as next Weds but not broadcast
Background image of page 4
Impounding • A way to bring the work to an end • Your machine is checked –Safety – Wiring – Rules issues • Your “seeding card” is essential – Your impound checks are recorded – Your card goes in the WOODEN BOX
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Linear Systems (Back Solving) A=[1 1 1; 0 2 3; 0 0 6]; b=[3; 1; 4]; x(3)=b(3)/A(3,3) x(2)=(b(2)-x(3)*A(2,3))/A(2,2) x(1)=(b(1)-x(2)*A(1,2)-x(3)*A(1,3))/A(1,1); norm(b-A*x') What will happen when I run this code?
Background image of page 6
Linear Systems (Solving) A=[1 1 1; 1 2 3; 1 3 6]; b=[3; 1; 4]; x=A\b b=[5; 0; -10]; x=A\b What will happen when I run this code?
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Linear Systems (Existence of Soln) A=[1 1 1; 1 2 3; 1 3 6; -1 -1 1]; b=[3; 1; 4; 7]; x=A\b; norm(b-A*x) What will happen when I run this code?
Background image of page 8
Linear Systems (Existence of Soln) A=[1 1 1; 1 2 3; 1 3 6; -1 -1 1]; b=[3; 1; 4; 6 ]; x=A\b; norm(b-A*x) What will happen when I run this code?
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Linear Systems (Multiple Solutions) A=[1 1 1; 1 2 3; 1 3 6; -1 -1 1]; b1=[3; 1; 4; 7]; x1=A\b1; norm(b1-A*x1) b2=[5; 0; -10; -15]; x2=A\b2; norm(b2-A*x2) What will happen when I run this code? b3=5*b1-2*b2; x3=A\b3; norm(b3-A*x3) norm(x3-(5*x1-2*x2))
Background image of page 10
Comparisons Linear Systems • Sometimes solved sequentially • # of equations = # of unknowns • # of equations > # of unknowns • When we can find two solutions Nonlinear systems •?
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Newton-Raphson Method • Make a guess at the solution • Make a linear approximation of a function by e.g., finite difference • Solve the linear system • Use that solution as a new guess • Repeat until some criterion is met Initial guess 0 Next estimate
Background image of page 12
Newton-Raphson Method ) ( ) ( 1 k k k k x f x f x x + = + 0 x 1 x 2 x ( )( ) ) ( 1 k k k k F x F x x x J = + If one equation in one variable Solve this system for x k+ 1 Generalizing to systems of equations * x
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
A Fundamental Difficulty x guess 10 := x root root y x guess () x guess , := 5051 0 1 5 40 20 0 20 yx root xx root , • If there are many solutions, which solution you find will depend on the initial guess x guess 3 := x root root y x guess x guess , := 0 1 5 40 20 0 20 root root ,
Background image of page 14
If you seek to find a root of a function f ( x ) , and you use the Newton-Raphson method.
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 16
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 46

MIT2_007s09_lec21 - MIT OpenCourseWare http:/ocw.mit.edu...

This preview shows document pages 1 - 16. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online