lecture03-1

# lecture03-1 - INTRODUCTION TO NUMERICAL SIMULATION L DANIEL...

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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T. L ECTURE 3. Equation Formulation – Nodal Analysis 1 T ODAY S O UTLINE : Matrix Construction from Schematics Nodal “Stamping Procedure” Struts and Joints Circuits Comparing Node-Branch vs. Nodal Solution of Linear Systems Existence and Uniqueness M ATRIX C ONSTRUCTION FROM S CHEMATICS Nodal “Stamping Procedure” ± Struts and Joints Two struts aligned with the X axis Conservation Law At node 1: 0 * , 2 * , 1 = + x x f f At node 2: 0 * , 2 = + L x f f Constitutive Equations () 2 1 0 2 1 2 1 * , 2 1 0 1 1 * , 1 0 0 0 x x L x x x x f x L x x f x x ε = ε = Reduced (Nodal) Equations 0 * 2 * 1 = + x x f f 0 * , 2 * , 1 2 1 0 2 1 2 1 1 0 1 1 = ε + ε 4 4 4 43 4 4 4 42 1 4 4 1 x x f f x x L x x x x x L x x 0 * 2 = + L x f f 0 * , 2 2 1 0 2 1 2 1 = + ε L f f x x L x x x x x 4 4 4 4 4 4 1 L f r * 1 f r * 2 f r x 1 y 1 = 0 x 2 y 2 = 0 ( ) * * * , y x r = r ( ) y x r , = r ( ) r r L r r r r f r r r r r r r ε = * 0 * * * = r r y y r r x x l r r r r * * * * , ˆ

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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T. L ECTURE 3. Equation Formulation – Nodal Analysis 2 Example.) 1 ˆ 10 e f L = r (force in positive x direction) Solution of Nodal Equations ε + + = ε + = 10 10 0 1 2 0 1 L x x L x Notice the signs of the forces: direction negative in force 10 direction positive in force 10 * , 1 * , 2 x f x f x x = = ± Circuits (1) Number the nodes with one node as 0. (2) Write a conservation law at each node except (0) in terms of the node voltages! i s A i s B i s C 0 1 2 3 4 () 0 1 1 2 1 2 = + + + V R V V R i i E B s s C B 0 1 1 2 1 1 = + + V V R V R i B A s A 0 1 1 3 4 4 = + + V V R V R i i C D s s B A 4 3 1 V V R i C s C + R E R A R D R C R B L f r * 1 f r * 2 f r x 1 y 1 = 0 x 2 y 2 = 0 Two struts aligned with the X axis
I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T. L ECTURE 3. Equation Formulation – Nodal Analysis 3 4 43 4 42 1 4 4 4 4 4 4 4 43 4 4 4 4 4 4 4 42 1 s B A C C B A I s s s s s s G D C C C C E B B B B A i i i i i i V V V V R R R R R R R R R R R KCL + = + + + 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 equations Notice that the contributions are positive on the diagonal and negative on the off-diagonal. G is square. Node-Branch Matrix N + B α 0 A A I T N + B Nodal Matrix [ ] G N N i s A i s B i s C 0 R E R A V 1 R B V 2 V 3 V 4 R D R C R n 2 One row per node, One column per node For each resistor n 1 i b n 1 n 2 R k ( ) 1 2 1 n n k b V V R i = at n 1 : at n 2 : ( ) = s n n k other i V V R i 1 2 1 ( ) = + s n n k other i V V R i 1 2 1

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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T.
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## This note was uploaded on 09/26/2010 for the course MECHANICAL 2.001 taught by Professor Prof.carollivermore during the Spring '06 term at MIT.

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lecture03-1 - INTRODUCTION TO NUMERICAL SIMULATION L DANIEL...

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