Mechanical vibrations worksheet with answers

Free printable Mechanical vibrations 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
k = 50 kN/mm = 20 kgx

A 20 kg mass hangs from a spring with a stiffness of 50 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 20 kgk = 50 kN/m
ωn = rad/s
2.Easy
k = 20 kN/mm = 40 kgx

A 40 kg mass hangs from a spring with a stiffness of 20 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 40 kgk = 20 kN/m
fn = Hz
3.Medium
tx12 mm1 cycle10.6 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed one cycle later by a peak of 10.6 mm. Find the damping ratio ζ.

x0 = 12 mmx1 = 10.6 mmn = 1
ζ =
4.Medium
tx50 mm2 cycles14.1 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 50 mm is followed 2 cycles later by a peak of 14.1 mm. Find the damping ratio ζ.

x0 = 50 mmx2 = 14.1 mmn = 2
ζ =
5.Hard
k1​ = 5k2​ = 60m = 5 kg

A 5 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 5 kN/m and k₂ = 60 kN/m. The mass stays level. Find the equivalent stiffness keq of the spring system in kN/m.

m = 5 kgk1 = 5 kN/mk2 = 60 kN/m
keq = kN/m
6.Hard
k1​ = 80k2​ = 20m = 25 kg

A 25 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 80 kN/m and k₂ = 20 kN/m. The mass stays level. Find the equivalent stiffness keq of the spring system in kN/m.

m = 25 kgk1 = 80 kN/mk2 = 20 kN/m
keq = kN/m

Answer key

  1. 1. ωn = 50 rad/s
  2. 2. fn = 3.559 Hz
  3. 3. ζ = 0.01974
  4. 4. ζ = 0.1002
  5. 5. keq = 65 kN/m
  6. 6. keq = 100 kN/m

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