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ENGR5001 / Week 3
ENGR201 Laplace deck example, railway buffer hit by a 5 s pulseTutorialOld spec ENGR2012:120 min

ENGR201 (old spec) Laplace transforms deck, railway buffer example (pages 45-53)

A damped spring constrained to move in one direction, such as a railway buffer, is subjected to a force of 1 unit applied for 5 seconds (a rectangular pulse starting at t=0t = 0). The equation of motion is d2xdt2+4dxdt+20x=F(t)\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 20x = F(t) and the buffer starts with no displacement and no velocity.

Formulas you may need
  • L[u(t−c)]=e−css\mathcal{L}[u(t - c)] = \dfrac{e^{-cs}}{s}; shift theorem L−1[e−csF(s)]=f(t−c)u(t−c)\mathcal{L}^{-1}[e^{-cs}F(s)] = f(t - c)u(t - c) (learn this)
  • L−1[s−a(s−a)2+b2]=eatcos⁡bt\mathcal{L}^{-1}\left[\dfrac{s - a}{(s - a)^2 + b^2}\right] = e^{at}\cos bt, L−1[b(s−a)2+b2]=eatsin⁡bt\mathcal{L}^{-1}\left[\dfrac{b}{(s - a)^2 + b^2}\right] = e^{at}\sin bt (learn this)
  1. (a)
    Write the pulse force F(t)F(t) using unit step (Heaviside) functions.
  2. (b)
    Find X(s)X(s).
  3. (c)
    Solve for the displacement x(t)x(t).
  4. (d)
    Find the static deflection under the 1 unit force, and compare it with the maximum deflection (and when it occurs). Evaluate x(6)x(6).