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Consider the forced but undamped system described by the initial value problem 3cosuwt, (0) 0, (0...
3. Consider the forced but undamped system described by the initial value problem u" +u =...
3. Consider the forced but undamped system described by the initial value problem u" +u = 3 cos(wt), 4(0) = 0, 1'0) = 0. a. Find solution for u(t) when w 1. b. Plot the solution u(t) versus t for w = 0.7, 0.8, and 0.9. Describe how the response u(t) changes as w varies. What happens when w gets close to 1? Note that the natural frequency of the system is wo = 1.
10. Consider the forced but undamped system described by the initial value prob- lem u" +...
10. Consider the forced but undamped system described by the initial value prob- lem u" + 25u = sin(wt), (0) = 0, ta (0) =1. a. Find solution for u(t) when w #5. b. Plot the solution u(t) versus t for w = : 4, 4.5, and 4.9. Describe how the response u(t) changes as w varies. What happens when w gets close to 5? Note that the natural frequency of the system is wo = 5. (Hint: Make the...
Problem 1.Consider the harmonically forced undamped oscillator described by the following ODE:mx′′+kx=F0cosωt, k >0, m >0,...
Problem 1.Consider the harmonically forced
undamped oscillator described by the following ODE:mx′′+kx=F0cosωt,
k >0, m >0, ω >0, F0∈R.
Problem 1. Consider the harmonically forced undamped oscillator described by the following ODE: mx" + kx = Fo cos wt, k > 0, m > 0,w > 0, F0 E R. (1) a) Suppose wa #k/m. Find the general solution of the ODE ). b) Consider the initial value problem of the ODE () with initial conditions x(0) = 0 and...
Problem 2. Recall that any undamped spring-mass system is described by an initial value problem of...
Problem 2. Recall that any undamped spring-mass system is described by an initial value problem of the form m" + ky= 0, (0) = 0, v(0) = to, where m is the mass and k is the spring constant. Since there is no damping, we would expect that no energy is lost as the mass moves. That is, the total energy (potential plus kinetic) in the system at any time I should equal the initial amount of energy in the...
6. Undamped Vibrations: Solve the initial value problem for y(t). y" +y = cos(wt); w2 #1;...
6. Undamped Vibrations: Solve the initial value problem for y(t). y" +y = cos(wt); w2 #1; y(0) = 0; y'(0) = 0. (8) Plot y(t) versus t, for w= -0.2, 0.9 and 6 to observe beats and resonance.
9imereal Condltion and note the behavior of the system. xercise 5. For this exercise you will app...
using improved eulers method using excel
9imereal Condltion and note the behavior of the system. xercise 5. For this exercise you will approximate the solution of an undamped periodically forced mass-spring system with mass m and spring constant k = 1.The second order equation for this system is given by x(t)" + x(t) = Cos(at). Use the initial conditions x(0) = 0 and x'(0) =y(0)-0. Use the improved Euler's method and Excel to generate plots of x(t) versus time for...
Consider a damped forced mass-spring system with m = 1, γ = 2, and k =...
Consider a damped forced mass-spring system with m = 1, γ = 2, and k = 26, under the influence of an external force F(t) = 82 cos(4t). a) (8 points) Find the position u(t) of the mass at any time t, if u(0) = 6 and u 0 (0) = 0. b) (4 points) Find the transient solution uc(t) and the steady state solution U(t). How would you characterize these two solutions in terms of their behavior in time?...
rx2 has 0 coefficient in the first equation QUESTION 2 Consider the linear system 11 +...
rx2 has 0 coefficient in the first equation
QUESTION 2 Consider the linear system 11 + 0.5X1 T1 12 0.5x2 + 13 0.25x3 X3 0.2 -1.425 2 = whose solution is (0.9, -0.8,0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using x(0) = (0,0,0)t as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme,...
21.6 A,B,C,D result given in part c of this exercise. 21.6. Consider a damped mass/spring system...
21.6 A,B,C,D
result given in part c of this exercise. 21.6. Consider a damped mass/spring system given by m dy gdy tr dt + ky = Fo cos(nt) where m. y. K and Fo are all positive constants. (This is the same as equation (214) a. Using the method of educated guess, derive the particular solution given by equation ser (21.10) on page 409. genelaidi b. Then show that the solution in the previous part can be rewritten as described...
3. Consider the initial value problem y(0) 0-105z(t Clearly, the solution to the system is y(t) e...
Ignore crossed out questions, thanks
3. Consider the initial value problem y(0) 0-105z(t Clearly, the solution to the system is y(t) e and(t) e-10t. Suppose we tried solving the system using forward Euler. This would give us with to- 0, y(to) 1, and z(to) 1. 2.10-5 c. In general, why would you expect forward Euler to require smaller time-steps than backward Euler?
3. Consider the initial value problem y(0) 0-105z(t Clearly, the solution to the system is y(t) e and(t)...
3. Consider the forced but undamped system described by the initial value problem u" +u = 3 cos(wt), 4(0) = 0, 1'0) = 0. a. Find solution for u(t) when w 1. b. Plot the solution u(t) versus t for w = 0.7, 0.8, and 0.9. Describe how the response u(t) changes as w varies. What happens when w gets close to 1? Note that the natural frequency of the system is wo = 1.
10. Consider the forced but undamped system described by the initial value prob- lem u" + 25u = sin(wt), (0) = 0, ta (0) =1. a. Find solution for u(t) when w #5. b. Plot the solution u(t) versus t for w = : 4, 4.5, and 4.9. Describe how the response u(t) changes as w varies. What happens when w gets close to 5? Note that the natural frequency of the system is wo = 5. (Hint: Make the...
Problem 1.Consider the harmonically forced
undamped oscillator described by the following ODE:mx′′+kx=F0cosωt,
k >0, m >0, ω >0, F0∈R.
Problem 1. Consider the harmonically forced undamped oscillator described by the following ODE: mx" + kx = Fo cos wt, k > 0, m > 0,w > 0, F0 E R. (1) a) Suppose wa #k/m. Find the general solution of the ODE ). b) Consider the initial value problem of the ODE () with initial conditions x(0) = 0 and...
Problem 2. Recall that any undamped spring-mass system is described by an initial value problem of the form m" + ky= 0, (0) = 0, v(0) = to, where m is the mass and k is the spring constant. Since there is no damping, we would expect that no energy is lost as the mass moves. That is, the total energy (potential plus kinetic) in the system at any time I should equal the initial amount of energy in the...
6. Undamped Vibrations: Solve the initial value problem for y(t). y" +y = cos(wt); w2 #1; y(0) = 0; y'(0) = 0. (8) Plot y(t) versus t, for w= -0.2, 0.9 and 6 to observe beats and resonance.
using improved eulers method using excel
9imereal Condltion and note the behavior of the system. xercise 5. For this exercise you will approximate the solution of an undamped periodically forced mass-spring system with mass m and spring constant k = 1.The second order equation for this system is given by x(t)" + x(t) = Cos(at). Use the initial conditions x(0) = 0 and x'(0) =y(0)-0. Use the improved Euler's method and Excel to generate plots of x(t) versus time for...
rx2 has 0 coefficient in the first equation
QUESTION 2 Consider the linear system 11 + 0.5X1 T1 12 0.5x2 + 13 0.25x3 X3 0.2 -1.425 2 = whose solution is (0.9, -0.8,0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using x(0) = (0,0,0)t as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme,...
21.6 A,B,C,D
result given in part c of this exercise. 21.6. Consider a damped mass/spring system given by m dy gdy tr dt + ky = Fo cos(nt) where m. y. K and Fo are all positive constants. (This is the same as equation (214) a. Using the method of educated guess, derive the particular solution given by equation ser (21.10) on page 409. genelaidi b. Then show that the solution in the previous part can be rewritten as described...
Ignore crossed out questions, thanks
3. Consider the initial value problem y(0) 0-105z(t Clearly, the solution to the system is y(t) e and(t) e-10t. Suppose we tried solving the system using forward Euler. This would give us with to- 0, y(to) 1, and z(to) 1. 2.10-5 c. In general, why would you expect forward Euler to require smaller time-steps than backward Euler?
3. Consider the initial value problem y(0) 0-105z(t Clearly, the solution to the system is y(t) e and(t)...
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