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 = 150 kN/mm = 40 kgx

A 40 kg mass hangs from a spring with a stiffness of 150 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 40 kgk = 150 kN/m
fn = Hz
2.Easy
kmδst​ = 15 mmunloaded

A machine rests on spring mounts, which compress by 15 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 = 15 mmg = 9.81 m/s²
fn = Hz
3.Medium
tx30 mm4 cycles2.4 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 30 mm is followed 4 cycles later by a peak of 2.4 mm. Find the damping ratio ζ.

x0 = 30 mmx4 = 2.4 mmn = 4
ζ =
4.Medium
tx20 mm5 cycles0.9 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 20 mm is followed 5 cycles later by a peak of 0.9 mm. Find the damping ratio ζ.

x0 = 20 mmx5 = 0.9 mmn = 5
ζ =
5.Hard
k1​ = 50k2​ = 20k3​ = 60m = 25 kg

A 25 kg mass hangs from spring k₃ = 60 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 50 kN/m and k₂ = 20 kN/m; the bar stays level. Ignore damping and the masses of the springs and bar. Find the natural frequency fn of vertical vibration in Hz.

m = 25 kgk1 = 50 kN/mk2 = 20 kN/mk3 = 60 kN/m
fn = Hz
6.Hard
k1​ = 25k2​ = 40m = 25 kg

A 25 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 25 kN/m and k₂ = 40 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 = 25 kgk1 = 25 kN/mk2 = 40 kN/m
fn = Hz

Answer key

  1. 1. fn = 9.746 Hz
  2. 2. fn = 4.07 Hz
  3. 3. ζ = 0.09999
  4. 4. ζ = 0.09823
  5. 5. fn = 5.721 Hz
  6. 6. fn = 8.115 Hz

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