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)100.04 s?

A first-order system G(s) = 25/(s + 5) receives a step input of size 2 at t = 0, starting from rest. Find the output y(0.04) at t = 0.04 s.

G(s) = 25/(s + 5)U = 2t = 0.04 s
y(0.04) =
2.Easy
t (s)y(t)103 s9

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 10. At t = 3 s it reads 9. Find the time constant τ.

y(∞) = 10y(3) = 9
τ = s
3.Medium
t (s)y(t)1

A system has the transfer function G(s) = 5/(s² + 3s + 5) and receives a unit step input. Find the natural frequency ωn in rad/s.

G(s) = 5/(s² + 3s + 5)
ωn = rad/s
4.Medium
t (s)y(t)0.5

A system has the transfer function G(s) = 14/(s² + 3s + 28) and receives a unit step input. Find the natural frequency ωn in rad/s.

G(s) = 14/(s² + 3s + 28)
ωn = rad/s
5.Hard
k = 2000 N/mc = 54 N·s/mm = 1 kgF = 25 Nx(t)

A mass m = 1 kg hangs on a spring k = 2000 N/m with a viscous damper c = 54 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 25 N is suddenly applied. Find the largest displacement xmax in mm.

m = 1 kgk = 2000 N/mc = 54 N·s/mF = 25 N
xmax = mm
6.Hard
t (s)y(t)120 %

A second-order system without zeros must have a step response with 20 % overshoot. Find the damping ratio ζ.

%OS = 20 %
ζ =

Answer key

  1. 1. y(0.04) = 1.813
  2. 2. τ = 1.303 s
  3. 3. ωn = 2.236 rad/s
  4. 4. ωn = 5.292 rad/s
  5. 5. xmax = 13.66 mm
  6. 6. ζ = 0.4559

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