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 = 200 kgx

A 200 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 = 200 kgk = 50 kN/m
ωn = rad/s
2.Easy
kmδst​ = 25 mmunloaded

A machine rests on spring mounts, which compress by 25 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 = 25 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 150 kN/mζ = 0.7m = 20 kg

A 20 kg mass on a spring with a stiffness of 150 kN/m has a viscous damper giving a damping ratio of 0.7. Find the damped natural frequency fd of its free vibration in Hz.

m = 20 kgk = 150 kN/mζ = 0.7
fd = Hz
4.Medium
tx50 mm1 cycle19.3 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 50 mm is followed one cycle later by a peak of 19.3 mm. Find the damping ratio ζ.

x0 = 50 mmx1 = 19.3 mmn = 1
ζ =
5.Hard
k1​ = 5k2​ = 60m = 20 kg

A 20 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. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 20 kgk1 = 5 kN/mk2 = 60 kN/m
fn = Hz
6.Hard
k1​ = 50k2​ = 10m = 50 kg

A 50 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 50 kN/m and k₂ = 10 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 = 50 kgk1 = 50 kN/mk2 = 10 kN/m
fn = Hz

Answer key

  1. 1. ωn = 15.81 rad/s
  2. 2. fn = 3.153 Hz
  3. 3. fd = 9.843 Hz
  4. 4. ζ = 0.1498
  5. 5. fn = 9.073 Hz
  6. 6. fn = 5.513 Hz

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