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)1t = ?98 %

A first-order system G(s) = 20/(s + 20) receives a step input of size 1 at t = 0, starting from rest. How long does the output take to reach 98 % of its final value? Find t.

G(s) = 20/(s + 20)U = 198 %
t = s
2.Easy
t (s)y(t)52 s3

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 5. At t = 2 s it reads 3. Find the time constant τ.

y(∞) = 5y(2) = 3
τ = s
3.Medium
t (s)y(t)0.5ts​ = ?

A system has the transfer function G(s) = 33.5/(s² + 7s + 67) and receives a unit step input. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 33.5/(s² + 7s + 67)
ts = s
4.Medium
t (s)y(t)0.5

A system has the transfer function G(s) = 14/(s² + 6s + 28) and receives a unit step input. The poles are s = σ ± jωd. Find the real part σ.

G(s) = 14/(s² + 6s + 28)
σ =
5.Hard
k = 200 N/mc = 17 N·s/mm = 4 kgF = 25 Nx(t)

A mass m = 4 kg hangs on a spring k = 200 N/m with a viscous damper c = 17 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 damping ratio ζ.

m = 4 kgk = 200 N/mc = 17 N·s/mF = 25 N
ζ =
6.Hard
t (s)y(t)1tp​ = 1.5 s8 %

The step response of a second-order system without zeros shows 8 % overshoot, and the first peak occurs at t = 1.5 s. Find the natural frequency ωn.

%OS = 8 %tp = 1.5 s
ωn = rad/s

Answer key

  1. 1. t = 0.1956 s
  2. 2. τ = 2.183 s
  3. 3. ts = 1.143 s
  4. 4. σ = −3
  5. 5. ζ = 0.3005
  6. 6. ωn = 2.687 rad/s

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