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ENGR5001 / Week 3
ENGR202 2023 Q B1Past paperOld spec ENGR2022:225 marks30 min

ENGR202 Summer 2023 Q B1[VALID] official answers

This question concerns first and second order models. Using standard notation, the general forms of these models are stated below.

τdxdt+x=Kuandd2xdt2+2ζωndxdt+ωn2x=Ku(t)\tau\frac{dx}{dt} + x = Ku \quad \text{and} \quad \frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = Ku(t)

The unit step response of a first order system is as follows: x(t)=K(1−e−t/τ)x(t) = K\left(1 - e^{-t/\tau}\right)

Formulas you may need
  • Generalised forms: τdxdt+x=Ku\tau\dfrac{dx}{dt} + x = Ku and d2xdt2+2ζωndxdt+ωn2x=Ku(t)\dfrac{d^2x}{dt^2} + 2\zeta\omega_n\dfrac{dx}{dt} + \omega_n^2 x = Ku(t) (given in the question)
  • First order unit step response: x(t)=K(1−e−t/τ)x(t) = K\left(1 - e^{-t/\tau}\right) (given in the question)
  • Second order steady state gain: K/ωn2K/\omega_n^2 (learn this)
  • Complex poles: s=−ζωn±jωn1−ζ2s = -\zeta\omega_n \pm j\omega_n\sqrt{1 - \zeta^2} (learn this)
  1. (a)
    Briefly explain the difference between the data-based and mechanistic approaches to modelling.
    [3]
  2. (b)
    For an engineering system of your choice (e.g. the simple thermal or hydraulic systems from the lectures), develop a first order, linear, mechanistic model, stating all the notation and assumptions made.
    [4]
  3. (c)
    Sketch the unit step response of a first order system and use it to fully define the following terms: (i) initial conditions, (ii) steady state gain and (iii) time constant. Link these definitions to your answer from part (b). Comment on how this information could be used to estimate a model from experimental data, briefly explaining the likely experimental protocol.
    [8]
  4. (d)
    The following model represents the power take off component of a wave energy convertor, in which pp is the generator reaction torque and f(t)f(t) is the force of the damper (arbitrary scaled units for the purpose of this question). 2d2pdt2+0.8dpdt+0.46p=f(t)(B1-1)2\frac{d^2p}{dt^2} + 0.8\frac{dp}{dt} + 0.46p = f(t) \qquad \text{(B1-1)} Convert this model into the appropriate generalised form and hence determine the damping and natural frequency of the model.
    [3]
  5. (e)
    Develop the Transfer Function form of the power take off model (B1-1), stating the assumption made. Write down the characteristic equation. Determine the poles and steady state gain of the system. Plot the position of the poles on the complex ss-plane. What is the stability of the model? Link your answer to the damping and natural frequency as appropriate.
    [7]