1 24-311 NUMERICAL METHODS Fall 03 Carnegie Mellon University PROBLEM SET 4 Issued: 9/20/03 Due: 9/25/03 10:30am Weight: 4% of total grade PS4-1 Formulation of a matrix equation and the usage of Mathcad An engineer supervises the production of three types of electrical components. Three kind of material—metal, plastic, and rubber—are required for production. The amounts needed to produce each component are shown in the following table. -------------------------------------------------------------------------- component Metal Plastic Rubber g/component g/component g/component -------------------------------------------------------------------------- 1 15 0.25 1.0 2 17 0.33 1.2 3 19 0.42 1.6 -------------------------------------------------------------------------- If totals of 2.12, 0.0434, and 0.164kg of metal, plastic, and rubber, respectively, are available each day, how many component can be produced per day? Write down a matrix equation and solve it using Mathcad. Name this Mathcad file ps4-1.mcd. In your hand-in directory on AFS, make a new directory called ps4 (in lower case) and copy your Mathcad file. Copy your Mathcad file in the ps4 directory in your hand-in directory on AFS. Also, hand in the printout of the Mathcad file. (Include the title of this problem and your name, e.g., “Metal, plastic and rubber by Kenji Shimada,” at the top of your Mathcad file.) PS4-2 Gauss elimination (1) Solve the following matrix equation using Gauss elimination. Show all the intermediate steps. −=−642116215114321xxx (2) Solve the following matrix equation using Gauss elimination. Find all the solutions, ),,(321xxx , that satisfy the set of equations. Show all the intermediate steps. 1422213321321321=++=++=−−xxxxxxxxx2 PS4-3 Consider a system of three springs with spring constants 11N mk=, 21N mk = and 32N mk = as shown below. The stable positions of the two right-hand-side ends of the springs are 11x = , 22x = and 33x=. Suppose three external forces 12Nf=, 21Nf =− and 32Nf = are applied, what are the new positions of the ends of the springs,1x, 2xand 3x? Initial state After three forces are applied (1) The condition of force balance gives three linear algebraic equations to be satisfied. What is the matrix equation to be solved in order to find the positions of the ends of the springs,1x, 2xand 3x, after three external forces, 1f , 2f , and 3f , are applied? (2) Solve the matrix equation for external forces using Gauss elimination. Show the complete derivation process for full credit. (3) Solve the same matrix equation for external forces using Gauss elimination with partial pivoting. Show the complete derivation process for full credit. 1x2x1k2k3k3x1f2f1x2x1k2k3x3f3k3 PS4-4 Formulation of a matrix equation and the usage of Mathcad Finding the forces associated with a statically determinate truss is an important problem in structural engineering. An overhanging roof-support structure is designed as a simple truss to hold a load of F = 5kN. Computer the reactions at A and B and the force in each member. Also, state whether the members are in tension or in compression. Write down a matrix equation and solve it using Mathcad. Name this Mathcad file ps4-4.mcd and copy the file to the ps4 directory. Also, hand in the printout of the Mathcad file. (Include the title of this problem and your name, e.g., “Truss Problem by Kenji Shimada,” at the top of your Mathcad file.) PS4-5 Gauss-Jordan elimination (1) Solve the following matrix equation using Gauss-Jordan elimination −=−541113225112321xxx (2) Solve the same matrix equation using Gauss-Jordan elimination with partial pivoting4 PS4 The first letter of __________________________ ____________________________ your LAST name First Name Last Name PS4-1 (20 pts) PS4-2 (20 pts) PS4-3 (20 pts) PS4-4 (20 pts) PS4-5 (20 pts) Total (100 pts) 24-311 NUMERICAL METHODS Fall 03 Carnegie Mellon University PROBLEM SET 4 Issued: 9/20/03 Due: 9/25/03 10:30am Weight: 4% of total