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)2010 s16

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 20. At t = 10 s it reads 16. Find the time constant τ.

y(∞) = 20y(10) = 16
τ = s
2.Easy
t (s)y(t)y(∞) = ?

A first-order system G(s) = 1/(0.2s + 1) receives a step input of size 20 at t = 0, starting from rest. Find the final value y(∞) of the output.

G(s) = 1/(0.2s + 1)U = 20
y(∞) =
3.Medium
t (s)y(t)2ts​ = ?

A system has the transfer function G(s) = 116/(s² + 12s + 58) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 116/(s² + 12s + 58)
ts = s
4.Medium
t (s)y(t)1

A system has the transfer function G(s) = 24/(s² + 3s + 24) and receives a unit step input. Find the natural frequency ωn in rad/s.

G(s) = 24/(s² + 3s + 24)
ωn = rad/s
5.Hard
k = 100 N/mc = 38 N·s/mm = 10 kgF = 100 Nx(t)

A mass m = 10 kg hangs on a spring k = 100 N/m with a viscous damper c = 38 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 = 10 kgk = 100 N/mc = 38 N·s/mF = 100 N
xmax = mm
6.Hard
t (s)y(t)1tp​ = 0.5 s15 %

The step response of a second-order system without zeros shows 15 % overshoot, and the first peak occurs at t = 0.5 s. Find the natural frequency ωn.

%OS = 15 %tp = 0.5 s
ωn = rad/s

Answer key

  1. 1. τ = 6.213 s
  2. 2. y(∞) = 20
  3. 3. ts = 0.6667 s
  4. 4. ωn = 4.899 rad/s
  5. 5. xmax = 1094 mm
  6. 6. ωn = 7.34 rad/s

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