An equation representing a vibrating string of one unit long, fixed at both ends, is given in Equation (T13.3). (T13.3) Equation (T13.3) can be written in the form of (T13.4) using the central finite difference approximation of the second derivative of their respective variables. The subscripts represent locations while superscripts represent time steps. The string is discretised into five nodes equally distanced, and At = 0.1. (T13.4) y = By₁₁+Cy+ Dy + Ey¹ a) Express B, C, D and E in numerical values. After that, write down three equations as a result of applying Equation (T13.4) to the domain. Subscripts must be replaced with correct node numbers, while the superscripts remain. You are not required to incorporate the boundary and initial conditions yet. (Answer: r = At/Ax = 0.1/0.25 = 0.4 B=0.16, C=1.68, D=0.16, E=-1) b) Assemble the three equations above in the form of matrix equation: {y}[4]{y}+{y}+{BC), where [4] is a square matrix, {BC) is a vector containing the boundary conditions, {y}",{y}', and {y} are vectors containing nodal variables at times +1, tand t-1, respectively. You are not required to incorporate the boundary and initial conditions yet. (Answer: [1.68 0.16 0y y's 0.16 1.68 0.16 y 2 0 0.16 1.68 c) Initially the string is at rest at the profile described in Figure T13.1. Find {y2 y3 y4} and {y2 y3 y4}. Show all the necessary calculations. (Answer: 1 1.68 (0.9488] 1.1424 and (0.9488

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Transcribed Image Text:An equation representing a vibrating string of one unit long, fixed at both ends, is given in Equation (T13.3). (T13.3) Equation (T13.3) can be written in the form of (T13.4) using the central finite difference approximation of the second derivative of their respective variables. The subscripts represent locations while superscripts represent time steps. The string is discretised into five nodes equally distanced, and At = 0.1. (T13.4) y = By₁₁+Cy+ Dy + Ey¹ a) Express B, C, D and E in numerical values. After that, write down three equations as a result of applying Equation (T13.4) to the domain. Subscripts must be replaced with correct node numbers, while the superscripts remain. You are not required to incorporate the boundary and initial conditions yet. (Answer: r = At/Ax = 0.1/0.25 = 0.4 B=0.16, C=1.68, D=0.16, E=-1) b) Assemble the three equations above in the form of matrix equation: {y}[4]{y}+{y}+{BC), where [4] is a square matrix, {BC) is a vector containing the boundary conditions, {y}",{y}', and {y} are vectors containing nodal variables at times +1, tand t-1, respectively. You are not required to incorporate the boundary and initial conditions yet. (Answer: [1.68 0.16 0y y's 0.16 1.68 0.16 y 2 0 0.16 1.68 c) Initially the string is at rest at the profile described in Figure T13.1. Find {y2 y3 y4} and {y2 y3 y4}. Show all the necessary calculations. (Answer: 1 1.68 (0.9488] 1.1424 and (0.9488

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