ME or MEng Master of Engineering Control Systems and Dynamics 2 — Questions and Answers
Question 1: For a second-order system to be classified as underdamped, the damping ratio ζ must satisfy:
- ζ > 1
- ζ = 1
- 0 < ζ < 1 (Correct answer)
- ζ = 0
Correct answer: 0 < ζ < 1
An underdamped system has 0 < ζ < 1, producing an oscillatory transient response that decays exponentially to the steady-state value.
Question 2: What is the natural frequency ωn of a simple spring-mass system with mass m and spring constant k?
- √(k/m) (Correct answer)
- k/m
- √(m/k)
- m/k
Correct answer: √(k/m)
From the equation of motion mẍ + kx = 0, the natural frequency is ωn = √(k/m) rad/s.
Question 3: Using the 2% criterion, the settling time Ts of a second-order system is approximately:
- π/ωd
- 4/(ζωn) (Correct answer)
- 1/(2ζωn)
- ωn/ζ
Correct answer: 4/(ζωn)
The 2% settling time is approximated as Ts ≈ 4/(ζωn), derived from the time constant of the exponential decay envelope.
Question 4: By standard definition, rise time is the time for the step response to travel from:
- 0% to 100% of its final value
- 10% to 90% of its final value (Correct answer)
- 5% to 95% of its final value
- 0% to 50% of its final value
Correct answer: 10% to 90% of its final value
Rise time is conventionally defined as the time for the response to increase from 10% to 90% of its final steady-state value.
Question 5: Critical damping in a mass-spring-damper system occurs when:
- The spring force equals the damping force at all times
- The damping ratio ζ equals 1 (Correct answer)
- The system oscillates indefinitely
- The natural frequency equals the damping coefficient numerically
Correct answer: The damping ratio ζ equals 1
Critical damping (ζ = 1) gives the fastest return to equilibrium without oscillation, representing the boundary between underdamped and overdamped responses.
Question 6: The impulse response h(t) of a linear system is related to its transfer function H(s) by:
- H(s) is the Fourier transform of h(t)
- h(t) is the inverse Laplace transform of H(s) (Correct answer)
- h(t) is the derivative of the step response divided by H(s)
- H(s) is the convolution of h(t) with the input
Correct answer: h(t) is the inverse Laplace transform of H(s)
Since the Laplace transform of the unit impulse δ(t) is 1, the output transform equals H(s)·1 = H(s), so h(t) = L⁻¹{H(s)}.
Question 7: Convolution in the time domain corresponds to which operation in the Laplace domain?
- Addition of transforms
- Differentiation of transforms
- Multiplication of transforms (Correct answer)
- Integration of transforms
Correct answer: Multiplication of transforms
The convolution theorem states that time-domain convolution of two signals corresponds to multiplication of their Laplace transforms in the s-domain.
ME or MEng Master of Engineering Control Systems and Dynamics 2
For a second-order system to be classified as underdamped, the damping ratio ζ must satisfy: