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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
t (s)y(t)1003 s80

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 100. At t = 3 s it reads 80. Find the time constant τ.

y(∞) = 100y(3) = 80
τ = s
2.Easy
t (s)y(t)2t = ?98 %

A first-order system G(s) = 1/(2s + 1) receives a step input of size 2 at t = 0, starting from rest. How long does the output take to reach 98 % of its final value? Find t.

G(s) = 1/(2s + 1)U = 298 %
t = s
3.Medium
t (s)y(t)0.5

A system has the transfer function G(s) = 40.5/(s² + 11s + 81) and receives a unit step input. Find the damping ratio ζ.

G(s) = 40.5/(s² + 11s + 81)
ζ =
4.Medium
t (s)y(t)3ts​ = ?

A system has the transfer function G(s) = 18/(s² + 2s + 6) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 18/(s² + 2s + 6)
ts = s
5.Hard
t (s)y(t)125 %

A second-order system without zeros must have a step response with 25 % overshoot. Find the damping ratio ζ.

%OS = 25 %
ζ =
6.Hard
k = 800 N/mc = 152 N·s/mm = 20 kgF = 50 Nx(t)

A mass m = 20 kg hangs on a spring k = 800 N/m with a viscous damper c = 152 N·s/m (m·x″ + c·x′ + k·x = F). At rest in equilibrium, a constant force F = 50 N is suddenly applied. Find the damping ratio ζ.

m = 20 kgk = 800 N/mc = 152 N·s/mF = 50 N
ζ =

Answer key

  1. 1. τ = 1.864 s
  2. 2. t = 7.824 s
  3. 3. ζ = 0.6111
  4. 4. ts = 4 s
  5. 5. ζ = 0.4037
  6. 6. ζ = 0.6008

Full step-by-step solutions: practise this topic on examproblems.com.