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 = 12 kN/mm = 20 kgx

A 20 kg mass hangs from a spring with a stiffness of 12 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 20 kgk = 12 kN/m
fn = Hz
2.Easy
kmδst​ = 2 mmunloaded

A machine rests on spring mounts, which compress by 2 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 = 2 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 20 kN/mζ = 0.7m = 20 kg

A 20 kg mass on a spring with a stiffness of 20 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 = 20 kN/mζ = 0.7
fd = Hz
4.Medium
tx40 mm1 cycle29.2 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed one cycle later by a peak of 29.2 mm. Find the damping ratio ζ.

x0 = 40 mmx1 = 29.2 mmn = 1
ζ =
5.Hard
k1​ = 60k2​ = 40k3​ = 15m = 20 kg

A 20 kg mass hangs from spring k₃ = 15 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 60 kN/m and k₂ = 40 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 = 60 kN/mk2 = 40 kN/mk3 = 15 kN/m
fn = Hz
6.Hard
k1​ = 30k2​ = 5m = 20 kg

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 30 kN/m and k₂ = 5 kN/m. The mass stays level. Find the equivalent stiffness keq of the spring system in kN/m.

m = 20 kgk1 = 30 kN/mk2 = 5 kN/m
keq = kN/m

Answer key

  1. 1. fn = 3.898 Hz
  2. 2. fn = 11.15 Hz
  3. 3. fd = 3.594 Hz
  4. 4. ζ = 0.05003
  5. 5. fn = 4.064 Hz
  6. 6. keq = 35 kN/m

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