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 = 40 kN/mm = 12 kgx

A 12 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 12 kgk = 40 kN/m
ωn = rad/s
2.Easy
kmδst​ = 0.5 mmunloaded

A machine rests on spring mounts, which compress by 0.5 mm under its weight. Treat it as an undamped single-degree-of-freedom system and use g = 9.81 m/s². Find the natural frequency fn of vertical vibration in Hz.

δst = 0.5 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 12 kN/mζ = 0.35m = 2 kg

A 2 kg mass on a spring with a stiffness of 12 kN/m has a viscous damper giving a damping ratio of 0.35. Find the damped natural frequency fd of its free vibration in Hz.

m = 2 kgk = 12 kN/mζ = 0.35
fd = Hz
4.Medium
tx20 mm1 cycle14.6 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 20 mm is followed one cycle later by a peak of 14.6 mm. Find the damping ratio ζ.

x0 = 20 mmx1 = 14.6 mmn = 1
ζ =
5.Hard
k1​ = 15k2​ = 60m = 10 kg

A 10 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 15 kN/m and the lower one k₂ = 60 kN/m. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 10 kgk1 = 15 kN/mk2 = 60 kN/m
fn = Hz
6.Hard
k1​ = 40k2​ = 60m = 20 kg

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 40 kN/m and k₂ = 60 kN/m. The mass stays level. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 20 kgk1 = 40 kN/mk2 = 60 kN/m
fn = Hz

Answer key

  1. 1. ωn = 57.74 rad/s
  2. 2. fn = 22.29 Hz
  3. 3. fd = 11.55 Hz
  4. 4. ζ = 0.05003
  5. 5. fn = 5.513 Hz
  6. 6. fn = 11.25 Hz

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