Question

A ship undergoing harmonic oscillation has an uncoupled heave motion, \(\eta_3(t)\), governed by the following equation
\[ (M + A_{33})\,\ddot{\eta}_3(t) + B_{33}\,\dot{\eta}_3(t) + K_{33}\eta_3(t) = f_3(t) \]
If \(F_3(\omega)\) is the Fourier transform of \(f_3(t)\), then the response in the frequency domain can be written as ______.

• A. \( \dfrac{F_3(\omega)}{-\omega^2(M + A_{33}) + i\omega B_{33} + K_{33}} \)
• B. \( \dfrac{iF_3(\omega)}{-\omega^2(M + A_{33}) + i\omega B_{33} + K_{33}} \)
• C. \( \dfrac{F_3(\omega)}{\omega^2(M + A_{33}) - i\omega B_{33} + K_{33}} \)
• D. \( \dfrac{F_3(\omega)}{\omega^2(M + A_{33}) + \omega B_{33} + K_{33}} \)

Hint:

Turn each time derivative into a factor of i times omega using the standard Fourier transform property.

Answer 1

The Correct Option is A

Step 1: Assume a harmonic response.
For a linear equation driven at a single frequency \(\omega\), take \(\eta_3(t) = X(\omega)\,e^{i\omega t}\) and \(f_3(t) = F_3(\omega)\,e^{i\omega t}\), where \(X(\omega)\) is the unknown heave amplitude to be found.

Step 2: Differentiate the assumed response.
Each time derivative of \(e^{i\omega t}\) brings down a factor \(i\omega\), so \(\dot{\eta}_3(t) = i\omega X(\omega)\,e^{i\omega t}\) and \(\ddot{\eta}_3(t) = (i\omega)^2 X(\omega)\,e^{i\omega t} = -\omega^2 X(\omega)\,e^{i\omega t}\), since \(i^2 = -1\).

Step 3: Substitute back into the equation of motion.
\[ (M + A_{33})(-\omega^2 X) + B_{33}(i\omega X) + K_{33} X = F_3(\omega) \]
Collecting \(X(\omega)\) as a common factor gives \(X(\omega)\left[-\omega^2(M+A_{33}) + i\omega B_{33} + K_{33}\right] = F_3(\omega)\).

Final Answer:
Divide through to isolate \(X(\omega)\), which is the frequency domain response. \[ \boxed{X(\omega) = \dfrac{F_3(\omega)}{-\omega^2(M + A_{33}) + i\omega B_{33} + K_{33}}} \]