Consider the differential equation below. m \ddot{y}=k(z(t)-y)+b \frac{d}{d t}(z(t)-y) where y(t) = Yejwt and z(t) = Zejwt Ya) Solve for the frequency response, i.e., the ratio Y/Z ) Assuming that

the damping ratio is 3 = 0.05 and that m = 1, plot the magnitude of the transfer function vs. frequency of the response of part (a), i.e., plot Y/Z vs.(W/WN). Use a logarithmic scale for the x-axis. ) Use the plot in part (b) to explain for which range of frequencies one can ignore z(t) in the original differential equation (i.e. set z(t) to zero). Assuming that the damping ratio is = 0.05 and that m = 1, plot the phase angle of the transfer function vs. frequency of the response of part (a) (i.e. the phase angle of Y/Z vs. (W/WN). Use a logarithmic scale for the x-axis. (e) Assuming that 3 = 0.1, and m = 1, use the plots in parts (b) and (d) to explain for which range of frequencies one can ignore all the derivative terms in the original differential equation. ) With the same assumptions as in part (e), use the solution for part (a) to find y(t)when z(t)10sin(3t).= Consider a car moving at a constant speed Vo on a wavy road where the "waves"have a peak-to-peak distance (wavelength) of L, and an amplitude of Z, inches.Show that this system will result in the same differential equation provided at the beginning of the problem.