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)50.4 s?

A first-order system G(s) = 5/(0.8s + 1) receives a step input of size 1 at t = 0, starting from rest. Find the output y(0.4) at t = 0.4 s.

G(s) = 5/(0.8s + 1)U = 1t = 0.4 s
y(0.4) =
2.Easy
t (s)y(t)5t = ?95 %

A first-order system G(s) = 5/(1s + 1) receives a step input of size 1 at t = 0, starting from rest. How long does the output take to reach 95 % of its final value? Find t.

G(s) = 5/(1s + 1)U = 195 %
t = s
3.Medium
t (s)y(t)1?

A system has the transfer function G(s) = 84/(s² + 10s + 84) and receives a unit step input. Find the percent overshoot %OS of the step response.

G(s) = 84/(s² + 10s + 84)
%OS = %
4.Medium
t (s)y(t)0.5tp​ = ?

A system has the transfer function G(s) = 6.5/(s² + s + 13) and receives a unit step input. Find the peak time tp of the step response.

G(s) = 6.5/(s² + s + 13)
tp = s
5.Hard
t (s)y(t)1tp​ = 0.8 s8 %

The step response of a second-order system without zeros shows 8 % overshoot, and the first peak occurs at t = 0.8 s. Find the natural frequency ωn.

%OS = 8 %tp = 0.8 s
ωn = rad/s
6.Hard
k = 100 N/mc = 12 N·s/mm = 1 kgF = 20 Nx(t)

A mass m = 1 kg hangs on a spring k = 100 N/m with a viscous damper c = 12 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 20 N is suddenly applied. Find the time tp at which the displacement is largest.

m = 1 kgk = 100 N/mc = 12 N·s/mF = 20 N
tp = s

Answer key

  1. 1. y(0.4) = 1.967
  2. 2. t = 2.996 s
  3. 3. %OS = 12.94 %
  4. 4. tp = 0.8798 s
  5. 5. ωn = 5.039 rad/s
  6. 6. tp = 0.3927 s

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