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Problem 1 (Submit): Longitudinal vibrations of an elastic bar with zero strain and stress at both...
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the s...
parts a,b, c
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the string is initially at rest. The initial boundary value problem (IBVP) is written as u(0,t) -u(1,t) u(x,0) = f(x), 0 ut (z, 0-0, 0 < x < 1. The solution of this IBVP using the method of separation of variables is given by n-l a) Find the coefficients bn. b) Show that this wave function can be written as the...
1. Let yar, t) denote the longitudinal displacements in an elastic bar of length unity whose...
1. Let yar, t) denote the longitudinal displacements in an elastic bar of length unity whose end x = 0 is fixed and the end r = 1 a force equal to sin it. Thus, the boundary conditions are y(0,t) = 0, y (1,t) = sin it. The bar is initially unstrained and at rest. The longitudinal displacements are governed by Ytt = Yrr Find y(2,t).
Q5. Consider the Heat Equation as the following boundary-value problem, find the solution u(x, t) by...
Q5. Consider the Heat Equation as the following boundary-value problem, find the solution u(x, t) by using separation-variables methods. (25 Points) (Boundary Condition : ux0,t) = 0 ux(10,t) = 0 Heat Equation ut = 9uxx (Hint: u(xt) = X(X)T(t)) Initial Condition : u(x,0) = 0.01x(10-x)
a) Use the d'Alembert solution to solve au au - <r< ,t> 0, at2 48,2 ux,0)...
a) Use the d'Alembert solution to solve au au - <r< ,t> 0, at2 48,2 ux,0) = cos 3x, u(,0) = 21 b) Consider the heat equation диди 0<x<1, t > 0, at ax? with boundary conditions uz (0,t) = 0, uz(1,t) = 0, > 0, and initial conditions u(x,0) = { 0, 2.0, 0<r < 0.5, 0.5 <<1. Use the method of separation of variables to solve the equation.
Torsional vibration of a shaft is govened by e wave equation where e(z,t) is the anqular...
Torsional vibration of a shaft is govened by e wave equation where e(z,t) is the anqular displacement (angle of twist) along the shaft, z is the distance from the end of the shaft and t is time. For a shaft of length that is supported by frictionless bearings at each end, boundary conditions are 0(0,t) 0(4x,t) 0, t> 0. Suppose that the initial angular displacement and angular velocity are e(z,0) 3cos(2r), 0(z,0)= 4+cos(2r), 0<z< 4m, respectively You may use the...
Torsional vibration of a shaft is govened by e wave equation where e(z,t) is the anqular...
Torsional vibration of a shaft is govened by e wave equation where e(z,t) is the anqular displacement (angle of twist) along the shaft, z is the distance from the end of the shaft and t is time. For a shaft of length that is supported by frictionless bearings at each end, boundary conditions are 0(0,t) 0(4x,t) 0, t> 0. Suppose that the initial angular displacement and angular velocity are e(z,0) 3cos(2r), 0(z,0)= 4+cos(2r), 0<z< 4m, respectively You may use the...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x,...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = ut; 0 < x < 6; t> 0; B.C.: ux(0,t) = 0; uz(6,t) = 0; t>0; I. C.: u(x,0) = 12 + 5cos (6x) – 4cos(21x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann,...
4.5.6 Let g(x,t) = cos(π2/1) in Exercise 4.5.4 and obtain the solution ofthe initial and boundary...
Hello, I have the answer for 4.5.4, but I need help with
4.5.6.
4.5.6 Let g(x,t) = cos(π2/1) in Exercise 4.5.4 and obtain the solution ofthe initial and boundary value problem in that case. 4.5.4.) Apply Duhamel's principle to obtain the solution of the initial and boundary value problem ut (z, t) _ C"uze(r, t) = g(z; a(0, t) = u (l, t) = 0. t), 0 < x < l, t > 0, u(x,0) = 0,
4.5.6 Let g(x,t)...
A long steel bar (length L-100 m, elastic modulus E = 2x 1011 N/m2, density ρ...
A long steel bar (length L-100 m, elastic modulus E = 2x 1011 N/m2, density ρ 7850 kgm3) of unit cross section is held at one end while hanging freely inside a deep pit and you are asked to estimate the longitudinal stress and displacement along the length of the bar Assuming linear elastic behavior, the internal elastic energy U of the hanging bar is given by the expression where z is the distance from the top of the pit...
Consider a uniform bar of length L having an initial temperature distribution given by f(x), 0...
Consider a uniform bar of length L having an initial temperature distribution given by f(x), 0 < x < L. Assume that the temperature at the end x=0 is held at 0°C, while the end x=L is thermally insulated. Heat is lost from the lateral surface of the bar into a surrounding medium. The temperature u(x, t) satisfies the following partial differential equation and boundary conditions aluxx – Bu = Ut, 0<x<l, t> 0 u(0,t) = 0, uz (L, t)...
parts a,b, c
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the string is initially at rest. The initial boundary value problem (IBVP) is written as u(0,t) -u(1,t) u(x,0) = f(x), 0 ut (z, 0-0, 0 < x < 1. The solution of this IBVP using the method of separation of variables is given by n-l a) Find the coefficients bn. b) Show that this wave function can be written as the...
1. Let yar, t) denote the longitudinal displacements in an elastic bar of length unity whose end x = 0 is fixed and the end r = 1 a force equal to sin it. Thus, the boundary conditions are y(0,t) = 0, y (1,t) = sin it. The bar is initially unstrained and at rest. The longitudinal displacements are governed by Ytt = Yrr Find y(2,t).
Q5. Consider the Heat Equation as the following boundary-value problem, find the solution u(x, t) by using separation-variables methods. (25 Points) (Boundary Condition : ux0,t) = 0 ux(10,t) = 0 Heat Equation ut = 9uxx (Hint: u(xt) = X(X)T(t)) Initial Condition : u(x,0) = 0.01x(10-x)
a) Use the d'Alembert solution to solve au au - <r< ,t> 0, at2 48,2 ux,0) = cos 3x, u(,0) = 21 b) Consider the heat equation диди 0<x<1, t > 0, at ax? with boundary conditions uz (0,t) = 0, uz(1,t) = 0, > 0, and initial conditions u(x,0) = { 0, 2.0, 0<r < 0.5, 0.5 <<1. Use the method of separation of variables to solve the equation.
Torsional vibration of a shaft is govened by e wave equation where e(z,t) is the anqular displacement (angle of twist) along the shaft, z is the distance from the end of the shaft and t is time. For a shaft of length that is supported by frictionless bearings at each end, boundary conditions are 0(0,t) 0(4x,t) 0, t> 0. Suppose that the initial angular displacement and angular velocity are e(z,0) 3cos(2r), 0(z,0)= 4+cos(2r), 0<z< 4m, respectively You may use the...
Torsional vibration of a shaft is govened by e wave equation where e(z,t) is the anqular displacement (angle of twist) along the shaft, z is the distance from the end of the shaft and t is time. For a shaft of length that is supported by frictionless bearings at each end, boundary conditions are 0(0,t) 0(4x,t) 0, t> 0. Suppose that the initial angular displacement and angular velocity are e(z,0) 3cos(2r), 0(z,0)= 4+cos(2r), 0<z< 4m, respectively You may use the...
Q2 Given the following heat conduction initial-boundary value problem of a thin homogeneous rod, where u(x, t) represents the temperature. 9uxx = ut; 0 < x < 6; t> 0; B.C.: ux(0,t) = 0; uz(6,t) = 0; t>0; I. C.: u(x,0) = 12 + 5cos (6x) – 4cos(21x); 0<x< 6 (a) When t = 0, what would be the temperature at x = 3? (Use the initial condition.) (b) Determine whether the boundary conditions in this case is Dirichlet, Neumann,...
Hello, I have the answer for 4.5.4, but I need help with
4.5.6.
4.5.6 Let g(x,t) = cos(π2/1) in Exercise 4.5.4 and obtain the solution ofthe initial and boundary value problem in that case. 4.5.4.) Apply Duhamel's principle to obtain the solution of the initial and boundary value problem ut (z, t) _ C"uze(r, t) = g(z; a(0, t) = u (l, t) = 0. t), 0 < x < l, t > 0, u(x,0) = 0,
4.5.6 Let g(x,t)...
A long steel bar (length L-100 m, elastic modulus E = 2x 1011 N/m2, density ρ 7850 kgm3) of unit cross section is held at one end while hanging freely inside a deep pit and you are asked to estimate the longitudinal stress and displacement along the length of the bar Assuming linear elastic behavior, the internal elastic energy U of the hanging bar is given by the expression where z is the distance from the top of the pit...
Consider a uniform bar of length L having an initial temperature distribution given by f(x), 0 < x < L. Assume that the temperature at the end x=0 is held at 0°C, while the end x=L is thermally insulated. Heat is lost from the lateral surface of the bar into a surrounding medium. The temperature u(x, t) satisfies the following partial differential equation and boundary conditions aluxx – Bu = Ut, 0<x<l, t> 0 u(0,t) = 0, uz (L, t)...
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