Q$4$

$($a$)$ A car ferry, operating in pure water $($that is$,=1000k\frac{g}{m^3}),$ can be modelled as two buoyant tanks with the car deck laid across the top as shown in Fig Q$4.$ The mass of the ship is $5000$ tonnes and its pitching inertia is $1{10}^{11}kg{m}^2.$ The ship can be modelled as a cuboid $100$ m long, $10$ m broad, and $10$ m tall, with each tank being $50m10m10m,$ arranged lengthways. The centre of mass is exactly in the centre of the ship, between the two identical tanks. $g$ is $10\frac{m}{s^2}.$

You may assume that the hull shape is such that, for small deflections in pitch, the centre of buoyancy of each buoyant tank does not move, and remains $25$ m either forwards or aft of the centre of gravity. As each buoyant tank has a planar area of $50m10m,$ you may also assume that deflecting the centroid of either tank downwards by $1$ m will displace $500$ tonnes of water.

Fig. Q$4$

What is the natural pitching period of the ship?

$[8]$

$($b$)$ It has been noted that the amplitude of the pitching movement of the ship in $($a$)$ decreases by $50\%$ with each oscillation in calm water. However, our ferry has encountered sinusoidal waves such that it is being forced to pitch at $90\%$ of its natural frequency. What is the phase shift between the passage of the waves and the oscillation of the ship?

$($c$)$ The motion of such a system may be described, in general form, by the expression $x=e^{-{}_{n}t}[A{e}^{i{}_{d}t}+B{e}^{-i{}_{d}t}].$ Apply Euler's equation, $e^i=cos+isin,$ and the harmonic addition theorem to prove that $x=e^{-{}_{n}t}$Rsin$({}_{d}t+).$ It may help if you simplify with $P=A+B$ and $Q=i(A-B)$ as you work.

$($d$)$ Draw a sketch illustrating, conceptually, how the harmonic addition theorem can yield Pcos$+$Qsin$=$Rsin$(+)$ from $0$ to $2.$ Take $P=1$ and $Q=2$ to estimate values of $R$ and $.$

$($e$)$ Which part of the final expression in $($c$)$ is the exponential decay part and which is the oscillatory part?

$[2]$