ENGR270 2026 Q1 (ENGR5001)

ENGR270 2026 Q1Past paperCurrent spec2:145 marks54 min

ENGR270 Summer 2026 Q1[VALID]

A second-order model for a point absorber Wave Energy Converter (WEC) captures the oscillatory motion of the device in response to wave excitation and hydrodynamic forces. By making some linearising assumptions, the model is as follows:

$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F(t) \qquad \text{(Q1-1)}$

where $x$ is the buoy displacement and $F(t)$ is the wave excitation force. Units are not stated in this question, and coefficients represent scaled values. However, in physical terms, the coefficients $m$ , $b$ and $k$ are linked to the equivalent mass, damping coefficient and hydrostatic restoring stiffness, respectively.

Formulas you may need

Generalised second order form: $\ddot x + 2\zeta\omega_n\dot x + \omega_n^2 x = Ku(t)$ (learn this)
For $m\ddot x + b\dot x + kx = F$ : $\omega_n = \sqrt{k/m}$ , $\zeta = \dfrac{b}{2\sqrt{km}}$ , steady state gain $1/k$ (learn this)
Steady state gain $G(0)$ ; poles from the characteristic equation (learn this)
A linear system driven by a sine at frequency $\omega$ responds at steady state with a sine of the SAME frequency, scaled by $M = |G(j\omega)|$ and shifted by $\phi = \mathrm{Arg}\,G(j\omega)$ (learn this)
$\omega = 2\pi f$ (learn this)
Stability shortcuts: all coefficients must exist and have the same sign; a single pole at the origin ( $a_n = 0$ ) gives marginal stability (learn this)

(a)
Convert the model (Q1-1) into the generalised second order differential equation form. Identify the input and output variables. Determine the damping ratio and natural frequency in terms of the model coefficients. Briefly comment on potential limitations of this model e.g. if used to design control systems for the WEC.
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(b)
For $m = 12048$ , $b = 4200$ and $k = 876$ , show that the Transfer Function for equation (Q1-1) takes the form given by equation (Q1-2). State any assumptions made in developing the Transfer Function. $X(s) = \frac{0.000083}{s^2 + 0.349s + 0.073}\,F(s) \qquad \text{(Q1-2)}$
[4]
(c)
Determine the steady state gain of the Transfer Function (Q1-2). If the wave force takes a constant value $F_0$ (unlikely in practice), write down an expression for the steady state displacement of the buoy.
steady state gain =
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(d)
Write down the characteristic equation associated with equation (Q1-2) and determine the poles. Sketch these poles on the complex $s$ -plane. What is the stability of the model and the response mode, i.e. the nature of the transient response? What is the steady state gain? Comment on the physical interpretation of all these results, linking to your answer to part (a) if possible.
pole real part =
pole imaginary part (+/-) =
zeta =
omega_n =rad/s
[10]
(e)
An oncoming series of waves has a frequency of 0.09 Hz. In simulation, this is represented by a sine input. What is the frequency of oscillations of the model at steady state? Sketch the Bode diagram for a generalised second order system with a resonant peak. Hint: label the axis and any important points on your sketch, but there is no need to include any numerical values in your sketch. Briefly speculate on the significance of the Bode diagram in the context of wave energy conversion.
steady state frequency =Hz
[10]
(f)
Unrelated to the above WEC model, define stability. With the help of a time response sketch, explain marginal stability. Give the stability condition (stable, unstable or marginally stable) for systems with the following characteristic equations: (i) $s^5 + s^3 + s^2 + \alpha s + \beta = 0$ , (ii) $s^3 + s^2 + \alpha s = 0$ , (iii) $s = 0$ , (iv) $s^3 + s^2 - s + 5 = 0$ and (v) $(s + 3.72)(s + 0.08)(s + 0.12) = 0$ , where $\alpha$ and $\beta$ are positive coefficients; in each case, give the reason for your answer.
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