Consider the following dynamical system composed of two discrete objects connected with the springs. The objects...

  

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Consider the following dynamical system composed of two discrete objects connected with the springs. The objects both have the same mass m1 = m2 = 1kg. BCS: k = 1 N/m hum 1 U(t = 0 s) = 1; U(t = 0 s) = -1; [10] a) Derive the ODE system that governs the dynamical system given above. [15] b) Considering the BCs given below solve the ODE system analytically. U(t = 12 s) = 1; U(t = 12 s) = 1; i) ii) k = 100 N/m [15] c) Discretize the ODE systems using the finite difference method. In this regard, divide the time window of 10s uniform time intervals At = 2s and discretize the ODE system and come up with a linear system of equation as a matrix-vector equation. U1,1 U2,1 U1,1-1 U2,1-1 A U1,i U2,i [1,1] U1,2 21,3 U1,4 U1, N = b U2,1 22,2 2 22,3 U2,4 [U,N] k = 1 N/m M U1,i+1 U2,i+1 U1,N [60] d) Write a Matlab code that solves numerically for the same problem. U2,N Use At = 0.6s and plot the numerical solutions u1 & u2 with the analytical solutions in the same figure. Use At = 0.06s and plot the numerical solutions u1 & u2 with the analytical solutions in the same figure. Consider the following dynamical system composed of two discrete objects connected with the springs. The objects both have the same mass m1 = m2 = 1kg. BCS: k = 1 N/m hum 1 U(t = 0 s) = 1; U(t = 0 s) = -1; [10] a) Derive the ODE system that governs the dynamical system given above. [15] b) Considering the BCs given below solve the ODE system analytically. U(t = 12 s) = 1; U(t = 12 s) = 1; i) ii) k = 100 N/m [15] c) Discretize the ODE systems using the finite difference method. In this regard, divide the time window of 10s uniform time intervals At = 2s and discretize the ODE system and come up with a linear system of equation as a matrix-vector equation. U1,1 U2,1 U1,1-1 U2,1-1 A U1,i U2,i [1,1] U1,2 21,3 U1,4 U1, N = b U2,1 22,2 2 22,3 U2,4 [U,N] k = 1 N/m M U1,i+1 U2,i+1 U1,N [60] d) Write a Matlab code that solves numerically for the same problem. U2,N Use At = 0.6s and plot the numerical solutions u1 & u2 with the analytical solutions in the same figure. Use At = 0.06s and plot the numerical solutions u1 & u2 with the analytical solutions in the same figure.

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We would typically use a method like undetermined coefficients or eigenvalueeigenvector approach for such secondorder linear homogeneous differential ... View the full answer