Problem Overview

This problem involves using Laplace transforms to solve a second-order differential equation. The first part involves finding a Laplace transform pair using the residue theorem and the inversion formula. The second part involves solving a second-order differential equation using the results from part (i).

(i) Finding the Laplace Transform Pair

You are given:

$\bar{\theta}(s) = \frac{\omega_n^2}{s(s^2 + 2\xi \omega_n s + \omega_n^2)}, \quad 0 < \xi < 1$ and need to find the inverse Laplace transform $\theta(t)$ using the residue theorem and the inversion formula:

$f(t) = \frac{1}{2\pi i} \int_{\gamma - i\infty}^{\gamma + i\infty} \bar{f}(s)e^{st}ds$

The given form suggests solving for $\theta(t)$ as:

$\theta(t) = 1 - e^{-\xi \omega_n t} \left( \frac{\xi}{\sqrt{1 - \xi^2}} \sin(\omega_d t) + \cos(\omega_d t) \right)$ where $\omega_d = \omega_n \sqrt{1 - \xi^2}$.

This expression corresponds to the inverse Laplace transform of the transfer function, which describes an underdamped system (since $0 < \xi < 1$).

(ii) Solving the Second Order Equation

You are asked to solve the second-order differential equation:

$\frac{d^2x}{dt^2} + 2\xi \omega_n \frac{dx}{dt} + \omega_n^2 x = \frac{1}{m} F(t)$

where $F(t)$ is piecewise-defined: