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· MAE 384 #3 Page 1 of 2 MAE 384 Advanced Mathematical Methods for Engineers Homework #3 Due: Sept. 16, Tuesday in class Problem 1: Matlab Three masses are attached to spring, k1 = 30 N/m, 25 2 k = N/m, 20 3 k = N/m, and 15 4 k = N/m, as shown. Initially the masses are positioned such that the springs are in their natural length (not stretched or compressed); then the masses are slowly released and move downward to an equilibrium position as shown on the right. The equilibrium equations of the three masses are úúúûùêêêëé=úúúûùêêêëéúúúûùêêêëé––+–++– 3 2 1 3 2 1 4 4 3 3 4 4 1 2 3 3 0 0 W W W u u u k k k k k k k k k k where u1, u2, and u3 are the relative displacement (from the unstretched position) of each mass as shown. If the masses have true weights 20 1 W = N, 30 2 W = N, and 15 3 W = N. However, due to a bias error in the measuring device, the actual weight column vector on the right hand size used to determine the displacements is úúúûùêêêëé+úúúûùêêêëé=úúúûùêêêëé c c c W W W W W W 3 2 1 3 2 1 ‘ ‘ ‘ where c is a constant representing the bias error in the weight measurements. The effect of bias c on the error in the displacements u is studied by the error analysis equation [][][][][][][][][][b] r A A x e b r A A 1 1 1 ––££ TS Write a Matlab script to complete the following tasks: (a) Determine the true displacements using the true weights 1 W , 2 W , and 3 W . (b) Assume the bias c changes from 0 to 5 N with a step size of 0.1 N. For each value of c, determine (1) the lower error bound [][][][b] r A A 1 1 – , (2) the upper error bound [][][][b] r A A –1 , and (3) the relative error [][] TS x e . (c) Plot the lower bound vs. c, upper error bound vs. c, and relative error vs. c on the same graph (i.e., three curves in one figure). Note: Please use 1-norm for the analysis and you may use the built-in function norm for this purpose. MAE 384 Liao Homework #3 Page 2 of 2 Problem 2: By hand A dynamical system can be modeled by a system of linear equation [A][x]=[b], where [x] represents the input to the system, [b] represents the output, and the system matrix []úúúúûùêêêêëé––––––= 3 4 2 5 2 1 1 2 1 2 3 4 5 4 3 2 A (a) Determine the condition number of the system using the infity-norm. (You may use calculators to find the inverse of the matrix.) (b) If the relative error in the output measurement is about ±2%, i.e., [r][b]= 2%, estimate the range of the relative error in the solution, i.e., [][] TS e x (c) If we want the relative error in the solution to be below 5%, what would be requirement on the relative error in the output measurement?

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· MAE 384 Liao Homework #3 Page 1 of 2 MAE 384 Advanced Mathematical Methods for Engineers Homework #3 Due: Sept. 16, Tuesday in class Problem 1: Matlab Three masses are attached to spring, k1 = 30 N/m, 25 2 k = N/m, 20 3 k = N/m, and 15 4 k = N/m, as shown. Initially the masses are positioned such that the springs are in their natural length (not stretched or compressed); then the masses are slowly released and move downward to an equilibrium position as shown on the right. The equilibrium equations of the three masses are úúúûùêêêëé=úúúûùêêêëéúúúûùêêêëé––+–++– 3 2 1 3 2 1 4 4 3 3 4 4 1 2 3 3 0 0 W W W u u u k k k k k k k k k k where u1, u2, and u3 are the relative displacement (from the unstretched position) of each mass as shown. If the masses have true weights 20 1 W = N, 30 2 W = N, and 15 3 W = N. However, due to a bias error in the measuring device, the actual weight column vector on the right hand size used to determine the displacements is úúúûùêêêëé+úúúûùêêêëé=úúúûùêêêëé c c c W W W W W W 3 2 1 3 2 1 ‘ ‘ ‘ where c is a constant representing the bias error in the weight measurements. The effect of bias c on the error in the displacements u is studied by the error analysis equation [][][][][][][][][][b] r A A x e b r A A 1 1 1 ––££ TS Write a Matlab script to complete the following tasks: (a) Determine the true displacements using the true weights 1 W , 2 W , and 3 W . (b) Assume the bias c changes from 0 to 5 N with a step size of 0.1 N. For each value of c, determine (1) the lower error bound [][][][b] r A A 1 1 – , (2) the upper error bound [][][][b] r A A –1 , and (3) the relative error [][] TS x e . (c) Plot the lower bound vs. c, upper error bound vs. c, and relative error vs. c on the same graph (i.e., three curves in one figure). Note: Please use 1-norm for the analysis and you may use the built-in function norm for this purpose. MAE 384 Liao Homework #3 Page 2 of 2 Problem 2: By hand A dynamical system can be modeled by a system of linear equation [A][x]=[b], where [x] represents the input to the system, [b] represents the output, and the system matrix []úúúúûùêêêêëé––––––= 3 4 2 5 2 1 1 2 1 2 3 4 5 4 3 2 A (a) Determine the condition number of the system using the infity-norm. (You may use calculators to find the inverse of the matrix.) (b) If the relative error in the output measurement is about ±2%, i.e., [r][b]= 2%, estimate the range of the relative error in the solution, i.e., [][] TS e x (c) If we want the relative error in the solution to be below 5%, what would be requirement on the relative error in the output measurement?

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