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ENGR5001 / Week 1
ENGR202 2019 Q B3Past paperOld spec ENGR2022:225 marks30 min

ENGR202 Summer 2019 Q B3[VALID] official answers

Generalised first order model: τdxdt+x=Ku(t)\tau\dfrac{dx}{dt} + x = Ku(t)

Generalised second order model: d2xdt2+2ζωndxdt+ωn2x=Ku(t)\dfrac{d^2x}{dt^2} + 2\zeta\omega_n\dfrac{dx}{dt} + \omega_n^2 x = Ku(t)

Figure B3-1: Single tank system (area A, head h, inflow Q_i, outflow Q_o through an orifice).
Figure B3-1: Single tank system (area A, head h, inflow Q_i, outflow Q_o through an orifice).
Figure B3-2: LC (inductor capacitor) circuit.
Figure B3-2: LC (inductor capacitor) circuit.
Formulas you may need
  • Generalised first order model τdxdt+x=Ku(t)\tau\dfrac{dx}{dt} + x = Ku(t); second order d2xdt2+2ζωndxdt+ωn2x=Ku(t)\dfrac{d^2x}{dt^2} + 2\zeta\omega_n\dfrac{dx}{dt} + \omega_n^2x = Ku(t) (given in the question)
  • Tank: Adhdt=Qi−QoA\dfrac{dh}{dt} = Q_i - Q_o (conservation of volume) (learn this)
  • Inductor VL=LdidtV_L = L\dfrac{di}{dt}; capacitor i=CdVCdti = C\dfrac{dV_C}{dt}; Kirchhoff's voltage law (learn this)
  • Steady state: x(t→∞)=K ux(t \to \infty) = K\,u (first order); K/ωn2K/\omega_n^2 times the input (second order) (learn this)
  1. (a)
    In the generalised first order model, what are the names of the variables τ\tau and KK? For the first order model with K=5K = 5 and a steady input of 2 units, determine the value of x(t→∞)x(t \to \infty). For the second order model, sketch the time response mode if ζ=0\zeta = 0 and explain the physical significance of this result.
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  2. (b)
    For the tank system shown in Figure B3-1, derive an ordinary differential equation expressing the outflow rate QoQ_o, as a function of the inflow rate QiQ_i and area AA. You may assume that QoQ_o is proportional to the head of water hh. List any other assumptions you have made to develop this model. Convert the model into the generalised first order form. Determine the steady state gain and comment on the physical significance of this result.
    [5]
  3. (c)
    For the LC circuit shown in Figure B3-2, derive a model expressing the voltage Vo(t)V_o(t) in terms of LL, CC and the applied voltage Vi(t)V_i(t). List the assumptions you have made. Convert the model into a generalised second order form. Determine the forcing function, output, damping ratio and natural frequency of the model.
    [7]
  4. (d)
    Using the LC circuit as an example, explain what is meant by the following terms: (i) linear system; (ii) mechanistic model; (iii) open-loop system. Briefly explain how generalised model forms can be useful when analysing dynamic systems.
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