lecture02

lecture02 - INTRODUCTION TO NUMERICAL SIMULATION - L....

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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T. L ECTURE 2. Equation Formulation & Node-Branch Stamping 1 T ODAY S O UTLINE : Formulating Equations Circuit Example Struts and Joints Example Matrix Construction From Schematics Node-Branch “Stamping Procedure” Circuits Struts and Joints F ORMULATING E QUATIONS FROM S CHEMATICS Circuit Example Step 1: Identifying Unknowns i s A i s B i s C 0 1 2 3 4 Assign each element a current. i A i B i E i D i C i s A i s B i s C 0 1 2 3 4 Assign each node a voltage, with one node as 0. i s A i s B i s C
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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T. L ECTURE 2. Equation Formulation & Node-Branch Stamping 2 Step 2: Conservation Laws Sum of currents = 0 (Kirchhoff’s Current Law) Step 3: Constitutive Equations Use Constitutive Equations to relate branch currents to node voltages (Currents flow from plus node to minus node) R C i C = V 3 – V 4 R D i D = V 4 – 0 R E i E = 0 – V 2 i s A i s B i s C 0 1 2 3 4 i A i B i E i D i C R E R A R C R D R A i A = 0 – V 1 R B i B = V 1 – V 2 R B i s A i s B i s C 0 1 2 3 4 i A i B i E i D i C i A + i E i D = 0 i s B + i s C i B i E = 0 i s A i A + i B = 0 I D i s A i s B i C = 0 i C i s C = 0
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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T. L ECTURE 2. Equation Formulation & Node-Branch Stamping 3 Struts Example ( ) * * * , 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 ( x 1 , y 1 ) ( x 2 , y 2 ) F load * 2 f r * 1 f r 1 2 Will the solutions be the same? Will the set of conservation law equations be different? 0 0 * , 2 * , 1 * , 2 * , 1 = + = + y y x x f f f f ( x 1 , y 1 ) ( x 2 , y 2 ) F load * 2 f r * 1 f r 1 2 Conservation laws for the two examples will be exactly the same. The perceived force “direction” is inconsequential, it is the adjacent forces that matter.
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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T. L ECTURE 2. Equation Formulation & Node-Branch Stamping 4 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 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 = = L f r * 1 f r * 2 f r x 1 y 1 = 0 x 2 y 2 = 0 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 ( ) * * * , y x r = r ( ) y x r , = r ε = 3 2 1 r r r r r r r L r r L r r r r f * 0 * * *
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I NTRODUCTION TO N UMERICAL S IMULATION - L. D ANIEL M.I.T.
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lecture02 - INTRODUCTION TO NUMERICAL SIMULATION - L....

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