Step response worksheet with answers

Free printable Step response worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
t (s)y(t)800.5 s?

A first-order system G(s) = 4/(2.5s + 1) receives a step input of size 20 at t = 0, starting from rest. Find the output y(0.5) at t = 0.5 s.

G(s) = 4/(2.5s + 1)U = 20t = 0.5 s
y(0.5) =
2.Easy
t (s)y(t)41 s?

A first-order system G(s) = 4/(s + 2) receives a step input of size 2 at t = 0, starting from rest. Find the output y(1) at t = 1 s.

G(s) = 4/(s + 2)U = 2t = 1 s
y(1) =
3.Medium
t (s)y(t)1

A system has the transfer function G(s) = 96/(s² + 6s + 96) and receives a unit step input. Find the damping ratio ζ.

G(s) = 96/(s² + 6s + 96)
ζ =
4.Medium
t (s)y(t)1

A system has the transfer function G(s) = 86/(s² + 11s + 86) and receives a unit step input. Find the damping ratio ζ.

G(s) = 86/(s² + 11s + 86)
ζ =
5.Hard
k = 5000 N/mc = 95 N·s/mm = 5 kgF = 5 Nx(t)

A mass m = 5 kg hangs on a spring k = 5000 N/m with a viscous damper c = 95 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 5 N is suddenly applied. Find the largest displacement xmax in mm.

m = 5 kgk = 5000 N/mc = 95 N·s/mF = 5 N
xmax = mm
6.Hard
R(s)+75s(s + 12)G(s)C(s)−

A unity negative feedback loop has G(s) = 75/(s(s + 12)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 75/(s(s + 12))
ts = s

Answer key

  1. 1. y(0.5) = 14.5
  2. 2. y(1) = 3.459
  3. 3. ζ = 0.3062
  4. 4. ζ = 0.5931
  5. 5. xmax = 1.372 mm
  6. 6. ts = 0.6667 s

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