The Resonant Ship Hull · AI Generator #GEN-0F1CCF17

Physics

Problem

A marine engineer is investigating the vibration of a ship's hull when waves pass underneath. The hull can be modeled as a flexible beam of length

L = 120\;\text{m}

that is fixed at both ends (bow and stern). When ocean waves with wavelength

\lambda = 240\;\text{m}

and speed

v = 15\;\text{m/s}

pass under the ship, they create a time-varying pressure distribution along the hull bottom given by

p(x,t) = p_0 \sin\left(\frac{2\pi x}{\lambda}\right) \cos(\omega t)

, where

p_0 = 5000\;\text{Pa}

and

\omega

is the wave frequency. The hull responds by vibrating in its normal modes. For a beam fixed at both ends, the natural frequencies are

f_n = \frac{n^2}{4L^2}\sqrt{\frac{E}{\rho}}

, where

n = 1, 2, 3, ...

E = 2.1 \times 10^{11}\;\text{Pa}

is the effective elastic modulus, and

\rho = 7800\;\text{kg/m}^3

is the effective density. The driving force has a spatial profile that only couples to certain normal modes. Specifically, the coupling strength to the

n

-th mode is proportional to the spatial overlap integral

\int_0^L \sin\left(\frac{2\pi x}{\lambda}\right) \sin\left(\frac{n\pi x}{L}\right) dx

. (a) What is the frequency of the driving ocean waves? (b) Which normal mode of the hull (value of

n

) experiences the strongest resonance when driven by these ocean waves? (c) Calculate the numerical value of the coupling integral for this resonant mode.

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