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 = 100 kN/mm = 500 kgx

A 500 kg mass hangs from a spring with a stiffness of 100 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 500 kgk = 100 kN/m
fn = Hz
2.Easy
kmδst​ = 8 mmunloaded

A machine rests on spring mounts, which compress by 8 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 = 8 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 80 kN/mc = 890 N·s/mm = 10 kg

A 10 kg mass is supported by a spring with a stiffness of 80 kN/m and a viscous damper with a damping coefficient of 890 N·s/m. Find the damping ratio ζ.

m = 10 kgk = 80 kN/mc = 890 N·s/m
ζ =
4.Medium
k = 50 kN/mζ = 0.35m = 200 kg

A 200 kg mass on a spring with a stiffness of 50 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 = 200 kgk = 50 kN/mζ = 0.35
fd = Hz
5.Hard
k1​ = 100k2​ = 20k3​ = 40m = 40 kg

A 40 kg mass hangs from spring k₃ = 40 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 100 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 = 40 kgk1 = 100 kN/mk2 = 20 kN/mk3 = 40 kN/m
fn = Hz
6.Hard
k1​ = 15k2​ = 5m = 50 kg

A 50 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 = 5 kN/m. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 50 kgk1 = 15 kN/mk2 = 5 kN/m
fn = Hz

Answer key

  1. 1. fn = 2.251 Hz
  2. 2. fn = 5.573 Hz
  3. 3. ζ = 0.4975
  4. 4. fd = 2.357 Hz
  5. 5. fn = 4.359 Hz
  6. 6. fn = 1.378 Hz

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