Actual experiments were performed to understand motion of ship tied in a river. Apparatus consists of a long tank where flowing water and wave generator creates forces on the tied ship. The differential equation of motion of ship is given by

M d2x/d2t + C dx/dt + Kx = F + F2*exp(-2t). t >0.

(a.) Find x(t) when M = 3+0.1*your number, C = 3, K = 5, F = 2+0.1*32, F2 = 3, x(t=0)=1 and dx/dt (t=0)= 1.

(b) Find the order of X(s) and the number of poles. Find all poles and zeros of X(s). Find poles corresponding to external forces. Using poles determine the stability of system.

(c) Find damped natural frequency, steady state value of x(t) (which equals xss ) using final value theorem and from expression of x(t).



Due to addition of ice chunks and crushed ice to water, only C changes to 4C. All other parameters remain the same as given above.

(d) Find x(t).

(e) Find all poles and zeros of X(s). Find poles corresponding to external forces. Using other poles determine the stability of system.

(f) Does system in (d) oscillates under given inputs. Find steady state value of x(t) called xss using final value theorem and from expression of x(t). Compare this xss to that obtained in question (c).



(h) Find x(t) if mass is set to zero in case (a). Use all parameters same as given in case (a). Find order and poles of X(s) and steady state xss.