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

A step is applied to a first-order process at t = 0. The output starts at 0 and settles at 50. At t = 5 s it reads 25. Find the time constant τ.

y(∞) = 50y(5) = 25

τ = s

2.Easy

A first-order system G(s) = 0.5/(2.5s + 1) receives a step input of size 5 at t = 0, starting from rest. How long does the output take to reach 99 % of its final value? Find t.

G(s) = 0.5/(2.5s + 1)U = 599 %

t = s

3.Medium

A system has the transfer function G(s) = 82/(s² + 6s + 82) and receives a unit step input. Find the peak time t_p of the step response.

G(s) = 82/(s² + 6s + 82)

t_p = s

4.Medium

A system has the transfer function G(s) = 68/(s² + 4s + 34) and receives a unit step input. Find the 2 % settling time t_s of the step response, using t_s ≈ 4/(ζωₙ).

G(s) = 68/(s² + 4s + 34)

t_s = s

5.Hard

A mass m = 2.5 kg hangs on a spring k = 1000 N/m with a viscous damper c = 40 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 percent overshoot %OS of the displacement.

m = 2.5 kgk = 1000 N/mc = 40 N·s/mF = 25 N

%OS = %

6.Hard

The step response of a second-order system without zeros shows 4.3 % overshoot, and the first peak occurs at t = 0.5 s. Find the natural frequency ω_n.

%OS = 4.3 %t_p = 0.5 s

ω_n = rad/s

Answer key

1. τ = 7.213 s
2. t = 11.51 s
3. t_p = 0.3677 s
4. t_s = 2 s
5. %OS = 25.38 %
6. ω_n = 8.893 rad/s

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