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)y(∞) = ?

A first-order system G(s) = 1/(s + 2) receives a step input of size 10 at t = 0, starting from rest. Find the final value y(∞) of the output.

G(s) = 1/(s + 2)U = 10
y(∞) =
2.Easy
t (s)y(t)200.5 s18

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 20. At t = 0.5 s it reads 18. Find the time constant τ.

y(∞) = 20y(0.5) = 18
τ = s
3.Medium
t (s)y(t)1ts​ = ?

A system has the transfer function G(s) = 82/(s² + 4s + 82) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 82/(s² + 4s + 82)
ts = s
4.Medium
t (s)y(t)0.5ts​ = ?

A system has the transfer function G(s) = 21.5/(s² + 3s + 43) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 21.5/(s² + 3s + 43)
ts = s
5.Hard
R(s)+125s(s + 10)G(s)C(s)−

A unity negative feedback loop has G(s) = 125/(s(s + 10)). Find the 2 % settling time ts of the closed-loop step response, using ts ≈ 4/(ζωₙ).

G(s) = 125/(s(s + 10))
ts = s
6.Hard
k = 200 N/mc = 40 N·s/mm = 8 kgF = 100 Nx(t)

A mass m = 8 kg hangs on a spring k = 200 N/m with a viscous damper c = 40 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 100 N is suddenly applied. Find the largest displacement xmax in mm.

m = 8 kgk = 200 N/mc = 40 N·s/mF = 100 N
xmax = mm

Answer key

  1. 1. y(∞) = 5
  2. 2. τ = 0.2171 s
  3. 3. ts = 2 s
  4. 4. ts = 2.667 s
  5. 5. ts = 0.8 s
  6. 6. xmax = 581.5 mm

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