DEV 08 Problem 1 The mathematical model of

ÖDEV 08 Problem 1: The mathematical model of a system is given as below : k=2. At t=0, x=0, v=20 and c 1, is the step input with value 2. Find V(s) Answer: D(s)=s 2+2 s+0. 1 s 1=-0. 0513, s 2=-1. 9487

Problem 2: The mathematical model of a system is: T(t) is input, θ(t) and ω(t) are the output. a) Find the eigenvalues of a system. Is the system stationary or not? b) At t=0 θ=0. 1 and ω=-6. T(t), is the step input with value -5. Find the solution of ω(t) with Laplace transform depending on initial conditions. Find Laplace transform of solutions for θ(t) input. c) If T(t) = 3 e-tcos(4 t+2. 7) , find θ(t). Answer: a) s 1, 2=-1± 2 i, stationary s=-1+4 i; hs=0. 5/(s^2+2*s+5); abs(hs), angle(hs) θ(t) = (0. 0417) (3) e-tcos(4 t+2. 7 -3. 1416)

Problem 3: The mathematical model of a system is: a) V(t) input, q 1(t) ve q 2(t) ouput Write the euations interms of variables form. b) Write Mat. LAB codes to find the eigenvalues of system. c) Find the matrix [s. I-A]-1 by writing Mat. LAB codes. d) The results for (b) and (c) obtained with computer are given below: s 1, 2=-0. 75± 1. 7854 i, s 3=0. and V(t), is the step function with magnitude 2. Find the solution of Laplace transform based on initial conditions for Find the Laplace transform solution depending on q 1 input Answer : b) a=[-1. 5, 0; 0, 0, 1; 3. 75, -3. 75, 0]; eig(a) c) a=[-1. 5, 0; 0, 0, 1; 3. 75, -3. 75, 0]; syms s; i 1=eye(3); a 1=inv(s*i 1 -a); pretty(a 1) d)

Problem 4: Answer: Find eigenvalues of matrix. Write the program codes to find eigenvalues by using Mat. LAB. a) 0. 2984, 6. 7016 b) a=[2, -4; -2, 5]; eig(a) Problem 5: For the system the matematical model is given below. Find the eigen values of a system by using Mat. LAB codes. Answer: Variables: x, θ, v, ω. At Mat. LAb : a=[0, 0, 1, 0; 0, 0, 0, 1; -20, -10, -2, -1; -8, -10, -2, -4]; eig(a) Eigenvalues: -1. 8± 4. 3163 i, -1. 19± 2. 015 i