Question

1. What is the transfer function Y (s)/X(s) of the system below? dt2+0.01 ye d2x(t)dx(t) dt24()

AND THEN

Answer 1

Given differential equation is d^2 y(t)/dt^2 + 0.01 d y(t)/dt + 0.5 y(t) = 3 d^2 x(t)/dt^2 + 4 d x(t)/dt + 5 x(t) Apply laplace Transform on both sides L[d^2 y/dt^2 + 0.01 dy/dt + 0.5 y(t)] = L[3 d^2 x/dt^2 + 4 dx/dt + 5x] s^2 y(s) + 0.01 s y(s) + 0.5 y(s) = 3 s^2 x(s) + 4s x(s) + 5 x(s) because L[d^2 y(t)/dt^2] = s^2 y(s) L[d y(t)/dt] = s y(s) L[y(t)] = y(s) } with initial conditions assumed as zero y(s)[s^2 + 0.01s + 0.5] = x(s)[3s^2 + 4s + 5] y(s)/x(s) = 3s^2 + 4s + 5/(s^2 + 0.01s + 0.5) This is the required Transfer function of the system given x(t) = u(t) x(s) = 1/s y(s) = 1/s times (3s^2 + 4s + 5)/(s^2 + 0.01s + 0.5) = 1/s times (3s^2 + 4s + 5)/(s^2 + s/100 + 1/2) \r\n y(s) = 1/s times (3s^2 + 4s