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 = 40 kN/mm = 20 kgx

A 20 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 20 kgk = 40 kN/m
ωn = rad/s
2.Easy
kmδst​ = 0.5 mmunloaded

A machine rests on spring mounts, which compress by 0.5 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 = 0.5 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 12 kN/mc = 490 N·s/mm = 20 kg

A 20 kg mass is supported by a spring with a stiffness of 12 kN/m and a viscous damper with a damping coefficient of 490 N·s/m. Find the damping ratio ζ.

m = 20 kgk = 12 kN/mc = 490 N·s/m
ζ =
4.Medium
k = 4 kN/mζ = 0.45m = 8 kg

An 8 kg mass on a spring with a stiffness of 4 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 = 8 kgk = 4 kN/mζ = 0.45
fd = Hz
5.Hard
k1​ = 12k2​ = 12k3​ = 100m = 5 kg

A 5 kg mass hangs from spring k₃ = 100 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 12 kN/m and k₂ = 12 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 = 5 kgk1 = 12 kN/mk2 = 12 kN/mk3 = 100 kN/m
fn = Hz
6.Hard
k1​ = 12k2​ = 5k3​ = 80m = 100 kg

A 100 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 = 12 kN/m and k₂ = 5 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 = 100 kgk1 = 12 kN/mk2 = 5 kN/mk3 = 80 kN/m
fn = Hz

Answer key

  1. 1. ωn = 44.72 rad/s
  2. 2. fn = 22.29 Hz
  3. 3. ζ = 0.5001
  4. 4. fd = 3.178 Hz
  5. 5. fn = 9.902 Hz
  6. 6. fn = 1.885 Hz

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