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4.41. Use modal analysis to calculate the solution of 3 -1 x(t) = 0 010mm/s). for the initial conditions x(0)=[0 (mm) and x(0)=[0 4.41. Use modal analysis to calculate the solution of 3 -1 x...
Using the Runge-Kutta fourth-order method, obtain a solution to dx/dt=f(t,x,y)=xy^3+t^2; dy/dt=g(t,x,y)=ty+x^3 for t= 0 to t= 1 second. The initial conditions are given as x(0)=0, y(0) =1. Use a time...
Using the Runge-Kutta fourth-order method, obtain a solution to dx/dt=f(t,x,y)=xy^3+t^2; dy/dt=g(t,x,y)=ty+x^3 for t= 0 to t= 1 second. The initial conditions are given as x(0)=0, y(0) =1. Use a time increment of 0.2 seconds. Do hand calculations for t = 0.2 sec only.
1. (10 pts) For the system below, write the 4th modal equation f(t) Base Fixed x,o (t) Assume: に...
1. (10 pts) For the system below, write the 4th modal equation f(t) Base Fixed x,o (t) Assume: に100000-, m=5kg, and,f(1)-1000 sin(50m)×1の Modal damping ratio is 1% Zero initial conditions . Use orthonormal modes, eg. {nr[M]1.- .
1. (10 pts) For the system below, write the 4th modal equation f(t) Base Fixed x,o (t) Assume: に100000-, m=5kg, and,f(1)-1000 sin(50m)×1の Modal damping ratio is 1% Zero initial conditions . Use orthonormal modes, eg. {nr[M]1.- .
B-9-15 Hint: Transform into modal coordinates to find the solution in modal coordinates then transform the...
B-9-15 Hint: Transform into modal coordinates to find the solution in modal coordinates then transform the modal coordinates solution back into physical coordinates (See CW6 and CW7) Problem B-9-15 Consider the mechanical system shown in Figure 9-51. Determine the natural frequen- cies and modes of vibration. In the diagram, the displacements x and y are measured from their respective equilibrium positions. Assume that m = 1 kg, M = 10 kg, ki = 10 N/m, k2 = 100 N/m Determine...
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velo...
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. di+10 dı+15di+20 (b) Check for the wave equation in (a) that if f(xtct) (use appropriate value...
Given initial conditions x1(0) = 1 and x2(0) = 0, determine solution components x1(t) and x2(t)....
Given initial conditions x1(0) = 1 and x2(0) = 0, determine
solution components x1(t) and x2(t).
7. Consider the following differential equation system for 11(t), 12(t), where x = (*1). x = (1 %)* (a) (7 points) Find the general solution.
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solu...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1.
Let...
d1= 3 & d2= 2 Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram bel...
d1= 3 & d2= 2
Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and 1, somewhat similar to fex) on page 8s of the Notes Part 2. 2 d2+5 r-0 di+10 di+15 di+20 3...
Let u be the solution to the initial boundary value problem for the Heat Equation an(t,r)-301a(t, z), te(0,00), z E (0,3); with initial condition 3 0 and with boundary conditions 6xu(t,0)-0, u(t, 3)...
Let u be the solution to the initial boundary value problem for the Heat Equation an(t,r)-301a(t, z), te(0,00), z E (0,3); with initial condition 3 0 and with boundary conditions 6xu(t,0)-0, u(t, 3) 0 Find the solution u using the expansion with the normalization conditions vn (0)-1, wn(0) 1 a. (3/10) Find the functionsw with index n1 b. (3/10) Find the functions vn with index n1 Un c. (4/10) Find the coefficients cn, with index n 1
Let u be...
A third order system of ODE's is defined by the block diagram of Fig. 1. Initial conditions on th...
A third order system of ODE's is defined by the block diagram of Fig. 1. Initial conditions on the states are: x1(0) 1; x2(0)-1; 3(0) 10 1. u 0 Ху X1 -2 -10 .9 Fig. 1. Third Order System of ODE's Derive the system matrix, A, spectral matrix, S, and modal matrix, M, and hence determine the states, x,(t), x2(t), and x3 (C) in response to the initial conditions given above. a) (30 Marks) Select a suitable set of initial...
Solve the equation yu- xui = u, t > 0,x >0 with the initial conditions u(x,...
Solve the equation yu- xui = u, t > 0,x >0 with the initial conditions u(x, 0) =1 + x2 using the method of characteristics. Find the u(x, y). Substitute your found solution u(x, y) in the equation and verify that it satisfies the equation. solution explicitly in the form u =
1. (10 pts) For the system below, write the 4th modal equation f(t) Base Fixed x,o (t) Assume: に100000-, m=5kg, and,f(1)-1000 sin(50m)×1の Modal damping ratio is 1% Zero initial conditions . Use orthonormal modes, eg. {nr[M]1.- .
1. (10 pts) For the system below, write the 4th modal equation f(t) Base Fixed x,o (t) Assume: に100000-, m=5kg, and,f(1)-1000 sin(50m)×1の Modal damping ratio is 1% Zero initial conditions . Use orthonormal modes, eg. {nr[M]1.- .
B-9-15 Hint: Transform into modal coordinates to find the solution in modal coordinates then transform the modal coordinates solution back into physical coordinates (See CW6 and CW7) Problem B-9-15 Consider the mechanical system shown in Figure 9-51. Determine the natural frequen- cies and modes of vibration. In the diagram, the displacements x and y are measured from their respective equilibrium positions. Assume that m = 1 kg, M = 10 kg, ki = 10 N/m, k2 = 100 N/m Determine...
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. di+10 dı+15di+20 (b) Check for the wave equation in (a) that if f(xtct) (use appropriate value...
Given initial conditions x1(0) = 1 and x2(0) = 0, determine
solution components x1(t) and x2(t).
7. Consider the following differential equation system for 11(t), 12(t), where x = (*1). x = (1 %)* (a) (7 points) Find the general solution.
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1.
Let...
d1= 3 & d2= 2
Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and 1, somewhat similar to fex) on page 8s of the Notes Part 2. 2 d2+5 r-0 di+10 di+15 di+20 3...
Let u be the solution to the initial boundary value problem for the Heat Equation an(t,r)-301a(t, z), te(0,00), z E (0,3); with initial condition 3 0 and with boundary conditions 6xu(t,0)-0, u(t, 3) 0 Find the solution u using the expansion with the normalization conditions vn (0)-1, wn(0) 1 a. (3/10) Find the functionsw with index n1 b. (3/10) Find the functions vn with index n1 Un c. (4/10) Find the coefficients cn, with index n 1
Let u be...
A third order system of ODE's is defined by the block diagram of Fig. 1. Initial conditions on the states are: x1(0) 1; x2(0)-1; 3(0) 10 1. u 0 Ху X1 -2 -10 .9 Fig. 1. Third Order System of ODE's Derive the system matrix, A, spectral matrix, S, and modal matrix, M, and hence determine the states, x,(t), x2(t), and x3 (C) in response to the initial conditions given above. a) (30 Marks) Select a suitable set of initial...
Solve the equation yu- xui = u, t > 0,x >0 with the initial conditions u(x, 0) =1 + x2 using the method of characteristics. Find the u(x, y). Substitute your found solution u(x, y) in the equation and verify that it satisfies the equation. solution explicitly in the form u =
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