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 = 8 kN/mm = 5 kgx

A 5 kg mass hangs from a spring with a stiffness of 8 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 5 kgk = 8 kN/m
fn = Hz
2.Easy
kmδst​ = 1 mmunloaded

A machine rests on spring mounts, which compress by 1 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 = 1 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 50 kN/mζ = 0.45m = 2 kg

A 2 kg mass on a spring with a stiffness of 50 kN/m has a viscous damper giving a damping ratio of 0.45. Find the damped natural frequency fd of its free vibration in Hz.

m = 2 kgk = 50 kN/mζ = 0.45
fd = Hz
4.Medium
k = 40 kN/mζ = 0.7m = 40 kg

A 40 kg mass on a spring with a stiffness of 40 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 = 40 kgk = 40 kN/mζ = 0.7
fd = Hz
5.Hard
k1​ = 25k2​ = 100m = 25 kg

A 25 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 25 kN/m and the lower one k₂ = 100 kN/m. Find the equivalent stiffness keq of the spring system in kN/m.

m = 25 kgk1 = 25 kN/mk2 = 100 kN/m
keq = kN/m
6.Hard
k1​ = 15k2​ = 25k3​ = 80m = 40 kg

A 40 kg mass hangs from spring k₃ = 80 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 = 25 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 = 15 kN/mk2 = 25 kN/mk3 = 80 kN/m
fn = Hz

Answer key

  1. 1. fn = 6.366 Hz
  2. 2. fn = 15.76 Hz
  3. 3. fd = 22.47 Hz
  4. 4. fd = 3.594 Hz
  5. 5. keq = 20 kN/m
  6. 6. fn = 4.109 Hz

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