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 = 80 kgx

An 80 kg mass hangs from a spring with a stiffness of 150 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 80 kgk = 150 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 = 100 kN/mζ = 0.35m = 5 kg

A 5 kg mass on a spring with a stiffness of 100 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 = 5 kgk = 100 kN/mζ = 0.35
fd = Hz
4.Medium
k = 40 kN/mc = 1010 N·s/mm = 40 kg

A 40 kg mass is supported by a spring with a stiffness of 40 kN/m and a viscous damper with a damping coefficient of 1010 N·s/m. Find the damping ratio ζ.

m = 40 kgk = 40 kN/mc = 1010 N·s/m
ζ =
5.Hard
k1​ = 5k2​ = 60m = 100 kg

A 100 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 5 kN/m and the lower one k₂ = 60 kN/m. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 100 kgk1 = 5 kN/mk2 = 60 kN/m
fn = Hz
6.Hard
k1​ = 5k2​ = 5m = 25 kg

A 25 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 5 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 = 25 kgk1 = 5 kN/mk2 = 5 kN/m
fn = Hz

Answer key

  1. 1. fn = 6.892 Hz
  2. 2. fn = 5.573 Hz
  3. 3. fd = 21.08 Hz
  4. 4. ζ = 0.3992
  5. 5. fn = 1.081 Hz
  6. 6. fn = 1.592 Hz

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