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 = 80 kN/mm = 500 kgx

A 500 kg mass hangs from a spring with a stiffness of 80 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 500 kgk = 80 kN/m
fn = Hz
2.Easy
k = 50 kN/mm = 12 kgx

A 12 kg mass hangs from a spring with a stiffness of 50 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 12 kgk = 50 kN/m
ωn = rad/s
3.Medium
k = 80 kN/mζ = 0.7m = 25 kg

A 25 kg mass on a spring with a stiffness of 80 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 = 25 kgk = 80 kN/mζ = 0.7
fd = Hz
4.Medium
k = 25 kN/mζ = 0.3m = 50 kg

A 50 kg mass on a spring with a stiffness of 25 kN/m has a viscous damper giving a damping ratio of 0.3. Find the damped natural frequency fd of its free vibration in Hz.

m = 50 kgk = 25 kN/mζ = 0.3
fd = Hz
5.Hard
k1​ = 30k2​ = 10k3​ = 5m = 20 kg

A 20 kg mass hangs from spring k₃ = 5 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 30 kN/m and k₂ = 10 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 = 30 kN/mk2 = 10 kN/mk3 = 5 kN/m
fn = Hz
6.Hard
k1​ = 20k2​ = 30m = 50 kg

A 50 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 20 kN/m and the lower one k₂ = 30 kN/m. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 50 kgk1 = 20 kN/mk2 = 30 kN/m
fn = Hz

Answer key

  1. 1. fn = 2.013 Hz
  2. 2. ωn = 64.55 rad/s
  3. 3. fd = 6.43 Hz
  4. 4. fd = 3.395 Hz
  5. 5. fn = 2.373 Hz
  6. 6. fn = 2.466 Hz

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