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ENGR5001 / Week 2
Old spec 2.4 exerciseExercise sheetCurrent specThird10 min

ENGR202 (old spec) Control 2.4 Time Response, exercise

A vehicle of mass mm has displacement yy from a reference point, so its speed is x=y˙x = \dot y. The engine imparts a force u(t)u(t), friction is proportional to velocity (force by˙b\dot y) and the rotational inertia of the wheels is negligible.

In an experiment, the engine force is stepped from 0% (zero force) to 100% (full power) at t=10t = 10 s. The resulting vehicle speed, in miles per hour, is shown in the figure.

Vehicle speed (mph) against time (s) after a step from 0 to 100% engine power at t = 10 s.
Vehicle speed (mph) against time (s) after a step from 0 to 100% engine power at t = 10 s.
Formulas you may need
  • Newton's second law; friction force by˙b\dot y proportional to velocity (learn this)
  • Generalised first order form τx˙+x=Ku(t)\tau\dot x + x = Ku(t) (learn this)
  • K=x(t→∞)u(t→∞)K = \dfrac{x(t \to \infty)}{u(t \to \infty)}; at t=τt = \tau the output has completed 63% of its change (learn this)
  1. (a)
    Develop a linear first order mechanistic model for the vehicle speed and express it in generalised first order form.
  2. (b)
    Estimate the steady state gain and time constant of the model from the graph, and hence write down the estimated model.