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 = 8 kN/mm = 2 kgx

A 2 kg mass hangs from a spring with a stiffness of 8 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 2 kgk = 8 kN/m
fn = Hz
2.Easy
k = 40 kN/mm = 50 kgx

A 50 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 50 kgk = 40 kN/m
fn = Hz
3.Medium
k = 20 kN/mζ = 0.6m = 5 kg

A 5 kg mass on a spring with a stiffness of 20 kN/m has a viscous damper giving a damping ratio of 0.6. Find the damped natural frequency fd of its free vibration in Hz.

m = 5 kgk = 20 kN/mζ = 0.6
fd = Hz
4.Medium
tx12 mm2 cycles1.8 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed 2 cycles later by a peak of 1.8 mm. Find the damping ratio ζ.

x0 = 12 mmx2 = 1.8 mmn = 2
ζ =
5.Hard
k1​ = 12k2​ = 12m = 50 kg

A 50 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 12 kN/m and k₂ = 12 kN/m. The mass stays level. Find the equivalent stiffness keq of the spring system in kN/m.

m = 50 kgk1 = 12 kN/mk2 = 12 kN/m
keq = kN/m
6.Hard
k1​ = 5k2​ = 40m = 100 kg

A 100 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 5 kN/m and k₂ = 40 kN/m. The mass stays level. Find the equivalent stiffness keq of the spring system in kN/m.

m = 100 kgk1 = 5 kN/mk2 = 40 kN/m
keq = kN/m

Answer key

  1. 1. fn = 10.07 Hz
  2. 2. fn = 4.502 Hz
  3. 3. fd = 8.053 Hz
  4. 4. ζ = 0.1493
  5. 5. keq = 24 kN/m
  6. 6. keq = 45 kN/m

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