Applied numerical methods lec7
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This document discusses three engineering applications of linear algebraic equations: 1) Analysis of a statically determinate truss by setting up equations for the forces at each node. An example truss is shown. 2) Modeling a spring-mass system using equations relating the positions, velocities, forces, and properties of 3 connected masses and springs. The equations of motion are derived. 3) Modeling a chemical system using a stiffness matrix relating the concentrations of chemical species to reaction rates and equilibrium constants.
Applied numerical methods lec7
- 1. 1 Engineering Applications: Linear Algebraic Equations
- 2. 2 Analysis of a Statically Determinate Truss v h v h v h V V H F F F V V H F F F , , , , , , . . . . . . . . 3 3 2 2 1 1 3 2 2 3 2 1 1 0 0 866 0 0 0 0 0 0 5 0 1 0 0 1 0 0 0 5 0 0 0 1 0 1 866 0 0 0 0 866 0 0 5 0 0 0 0 5 0 0 866 0 External forces: F1,h, F1,v, F2,h, F2,v, F3,h, F3,v h o o H F F F F , cos cos 1 3 1 60 30 0 v o o V F F F F , sin sin 1 3 1 60 30 0 Node 1: Node 2 and 3: similarly ……
- 4. 4 Example of Statically Determinate Truss 1000 2 1 h h F F , , Wind blows east Wind blows west 1000 3 1 h h F F , ,
- 6. 6 Steady state: all the speeds are zero 3kx1 - 2kx2 = m1g -2kx1 + 3kx2 - kx3 = m2g - kx2 + kx3 = m3g Spring-Mass Systems (cont.) 1 1 1 2 1 2 2 1 2 kx g m x x k m dt x d U D dt x d F F m 2 2 Mass m1: FU = kx1; FD = 2k(x2 – x1) + m1g 2 3 3 3 2 3 2 x x k g m m dt x d 1 2 2 2 3 2 2 2 2 2 x x k g m x x k m dt x d Mass m3: g m g m g m x x x k k k k k k k 3 2 1 3 2 1 0 3 2 0 2 3 Stiffness matrix Mass m2
- 8. 8 Example of Chemical System
- 9. 9