ENGR201 2022 A2 (ENGR5001)

ENGR201 2022 A2Past paperOld spec ENGR2012:225 marks30 min

ENGR201 Engineering Analysis 2021-22 Part A, A2[VALID]

An impact protection device is subjected to a constant unit impact force $P(t)$ from $t = 0$ . The response of the device is described by the equation of motion

$\frac{d^2x(t)}{dt^2} + 2\frac{dx(t)}{dt} + 50x(t) = P(t) \qquad \text{(A2)}$

where $x(t)$ is the displacement of the device from its initial position and $t$ is time in seconds. The device has no initial displacement or velocity.

Formulas you may need

$\mathcal{L}[x'] = sX - x(0)$ , $\mathcal{L}[x''] = s^2X - sx(0) - x'(0)$ ; unit step $\to 1/s$ (learn this; printed in the ENGR201 Laplace table)
$e^{at}\cos\omega t \leftrightarrow \dfrac{s - a}{(s - a)^2 + \omega^2}$ , $e^{at}\sin\omega t \leftrightarrow \dfrac{\omega}{(s - a)^2 + \omega^2}$ (learn this; printed in the ENGR201 Laplace table)
Step function $H_a(t) \leftrightarrow e^{-as}/s$ (learn this; printed in the ENGR201 Laplace table)
Generalised second order form $\ddot x + 2\zeta\omega_n\dot x + \omega_n^2 x = Ku$ (learn this)

(a)
Find the Laplace transform of the displacement of the device.
[15]
(b)
Find the displacement as a function of time.
final displacement =-
peak displacement =-
[8]
(c)
If the impact force lasts only for 3 seconds, write down and plot the force pulse using the step (Heaviside) function.
[2]Third