Homework Set 4

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University of Florida *

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3344

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Mechanical Engineering

Date

Oct 30, 2023

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Problem Set 4 Problems Completed: 6.24, 8.3, 8.4, 8.10, 9.5, 9.6, S1

Problem 6.24 In control systems analysis, transfer functions are developed that mathematically relate the dynamics of a systems input to its output. A transfer function for a robotic positioning system is given by ?(𝑠) = ?(𝑠) 𝑁(𝑠) = 𝑠 3 + 9𝑠 2 + 26𝑠 + 24 𝑠 4 + 15𝑠 3 + 77𝑠 2 + 153𝑠 + 90 where G(s) = system gain, C(s) = system output, N(s) =system input, and s = Laplace transform complex frequency. Use MATLAB to find the roots of the numerator and denominator and factor these into the form ?(𝑠) = (𝑠 + ? 1 )(𝑠 + ? 2 )(𝑠 + ? 3 ) (𝑠 + ? 1 )(𝑠 + ? 2 )(𝑠 + ? 3 )(𝑠 + ? 4 ) where -ai and -bi =the roots of the numerator and denominator, respectively. Solution: ▪ MATLAB Code: see file: EGM3344_HW4_624.m ▪ Output: There for the system gain function is : ?( 𝑠) = (𝑠 + 4 ) (𝑠 + 3 ) (𝑠 + 2 ) (𝑠 + 6 ) (𝑠 + 5 ) (𝑠 + 3 ) (𝑠 + 1 )

Problem 8.3 Write the following set of equations in matrix form: 50 = 5𝑥 3 − 6𝑥 2 2𝑥 2 + 7𝑥 3 + 30 = 0 𝑥 1 − 7𝑥 3 = 50 − 3𝑥 2 + 5𝑥 1 Use MATLAB to solve for the unknowns. In addition, use it to compute the transpose and the inverse of the coefficient matrix. Solution: Matrix form: [ 0 −6 5 0 2 7 −4 3 −7 ] [ 𝑥 1 𝑥 2 𝑥 3 ] = [ 50 −30 50 ] ▪ MATLAB code: see file: EGM3344_HW4_83.m ▪ Output: x 1 = -17.0192 x 2 = -9.6154 x 3 = -1.5385

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Problem 8.4 Three matrices are defined as ? = [ 6 −1 12 8 −5 4 ] , ? = [ 4 0 0.5 2 ] , ? = [ 2 −2 3 1 ] (a) Perform all possible multiplications that can be computed between pairs of these matrices. (b) Justify why the remaining pairs cannot be multiplied. (c) Use the results of (a) to illustrate why the order of multiplication is important. Solution: ▪ MATLAB Code: see file: EGM3344_HW4_84.m ▪ Output: (a) (b) B*A and C*A cannot be multiplied because their inner dimensions do not match. (c) The order of multiplication is important because they can result in different values. For instance, B*C results do not equal C*B.

Problem 8.10 An important problem in structural engineering is that of finding the forces in a statically determinate truss (Fig. P8.10). This type of structure can be described as a system of coupled linear algebraic equations derived from force balances. The sum of the forces in both horizontal and vertical directions must be zero at each node, because the system is at rest. Therefore, for node 1: ∑ ? 𝐻 = 0 = −? 1 cos 30 ° + ? 3 cos 60 ° + ? 1,ℎ ∑ ? 𝑉 = 0 = −? 1 sin 30 ° − ? 3 sin 60 ° + ? 1,𝑣 For node 2: ∑ ? 𝐻 = 0 = ? 2 + ? 1 cos 30 ° + ? 2,ℎ + ? 2 ∑ ? 𝑉 = 0 = ? 1 sin 30 ° − ? 2,𝑣 + 𝑉 2 For node 3: ∑ ? 𝐻 = 0 = −? 2 − ? 3 cos 60 ° + ? 3,ℎ ∑ ? 𝑉 = 0 = ? 3 sin 60 ° + ? 3,𝑣 + 𝑉 3 where Fi,h is the external horizontal force applied to node i (where a positive force is from left to right) and Fi,υ is the external vertical force applied to node i (where a positive force is upward). Thus, in this problem, the 1000-N downward force on node 1 corresponds to Fi,υ = −1000. For this case, all other Fi,υ’s and Fi,h’s are zero. Express this set of linear algebraic equations in matrix form and then use MATLAB to solve for the unknowns. Solution: Linear system of equations:

▪ MATLAB Code: see file: EGM3344_HW4_810.m ▪ Output: F1 = -500, F2 = 433.0127, F3 = -866.0254, H2 = 0, V2 = 250, and V3 = 750

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Problem 9.5 Given the equations 0.5x 1 – x 2 = − 9.5 1.02x 1 − 2x 2 = −18.8 (a) Solve graphically. (b) Compute the determinant. (c) On the basis of (a) and (b), what would you expect regarding the system’s condition? (d) Solve by the elimination of unknowns. (e) Solve again, but with a11 modified slightly to 0.52. Interpret your results. Solution: (a) ▪ MATLAB Code: clc; clear; x21 = @(x1) 0.5*x1 + 9.5; x22 = @(x1) 0.51*x1 + 9.4; x1 = linspace(5, 20); x2_1 = x21(x1); x2_2 = x22(x1); plot(x1, x2_1); plot(x1, x2_2); grid on ; ▪ Output: Based on the graph, the solution is around x1 = 14.5 and x2 = 10

(b) Determinate = 0.02 (c) Regarding the system ’ s condition being that there are similar slopes for each equation and the determinate being very small, the system would be expected to be ill-conditioned. (d) x1 = 10, x2 = 14.5 (e) x1 = -10, x2 = 4.3 Based on these results, the slight change in one of the coefficients producing a large difference in x1 and x2 values demonstrates that this system is very ill-conditioned.

Problem 9.6 Given the equations 10x 1 + 2x 2 – x 3 = 27 − 3x 1 −5x 2 + 2x 3 = −61.5 x 1 + x 2 + 6x 3 = −21.5 (a) Solve by naive Gauss elimination. Show all steps of the computation. (b) Substitute your results into the original equations to check your answers . Solution: (a) x1 = 0.152482, x2 = 10.0922, x3 = -5.29078 (b) ▪ MATLAB Verification Code: see file: EGM3344_HW4_96.m ▪ Output: Output of code confirms that x values give the correct results.

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Problem S1 (a) Show that the following matrix is singular. ? = [ 1 1 0 1 2 1 1 3 2 ] (b) If b = [2 4 6 ] T , how many solutions are there to the system Ax = b? Solution: (a) ▪ MATLAB Verification Code: clc; clear; A = [1 1 0; 1 2 1; 1 3 2]; x = det(A); disp( 'Determinant= ' ); disp(x); ▪ Output: (b) Infinite Solutions because too explicit