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 = 4 kN/mm = 40 kgx

A 40 kg mass hangs from a spring with a stiffness of 4 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 40 kgk = 4 kN/m
fn = Hz
2.Easy
kmδst​ = 10 mmunloaded

A machine rests on spring mounts, which compress by 10 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 = 10 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 100 kN/mζ = 0.5m = 10 kg

A 10 kg mass on a spring with a stiffness of 100 kN/m has a viscous damper giving a damping ratio of 0.5. Find the damped natural frequency fd of its free vibration in Hz.

m = 10 kgk = 100 kN/mζ = 0.5
fd = Hz
4.Medium
tx12 mm5 cycles1 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed 5 cycles later by a peak of 1 mm. Find the damping ratio ζ.

x0 = 12 mmx5 = 1 mmn = 5
ζ =
5.Hard
k1​ = 15k2​ = 30m = 10 kg

A 10 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 15 kN/m and k₂ = 30 kN/m. The mass stays level. Find the equivalent stiffness keq of the spring system in kN/m.

m = 10 kgk1 = 15 kN/mk2 = 30 kN/m
keq = kN/m
6.Hard
k1​ = 15k2​ = 30k3​ = 10m = 20 kg

A 20 kg mass hangs from spring k₃ = 10 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 15 kN/m and k₂ = 30 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 = 20 kgk1 = 15 kN/mk2 = 30 kN/mk3 = 10 kN/m
fn = Hz

Answer key

  1. 1. fn = 1.592 Hz
  2. 2. fn = 4.985 Hz
  3. 3. fd = 13.78 Hz
  4. 4. ζ = 0.07885
  5. 5. keq = 45 kN/m
  6. 6. fn = 3.219 Hz

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