Engineering Mathematics and Basic Sciences
This domain tests the mathematical and physical toolkit behind mechatronic models: linear algebra, calculus, differential equations, complex numbers, probability, and basic mechanics and electricity. Expect questions that require choosing a model, applying a rule, and interpreting units rather than memorising long derivations.
Core rules
- A first-order system written as τ dy/dt + y = K u has time constant τ and steady-state gain K; its step response is y(t)=K(1−e^{−t/τ}).
- For a matrix A, eigenvalues solve det(A−λI)=0; they govern stability of ẋ=Ax.
- Laplace transforms convert linear ODEs into algebra: L{dy/dt}=sY(s)−y(0). The final-value theorem applies only if poles of sY(s) lie in the left half-plane.
- In probability, independent events multiply; for a continuous uniform variable on [a,b], mean=(a+b)/2 and variance=(b−a)^2/12.
Worked example
A sensor obeys 0.2 dy/dt + y = 5, y(0)=0. Rewrite as τ=0.2 s, gain K=5. The step response is y(t)=5(1−e^{−5t}). At t=0.2 s, one time constant, y=5(1−e^{−1})=3.16. The initial slope is K/τ=25 units/s; the final value is 5.
Traps
- Mixing seconds and milliseconds in τ, or treating τ as the settling time; 4τ is about 98% of final value.
- Using the final-value theorem when a pole lies on or right of the imaginary axis.
- Degree/radian errors in trigonometric models and sign errors when taking moments.
- Reporting a stress or current without units or with mm² treated as m².