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 = 200 kN/mm = 80 kgx

An 80 kg mass hangs from a spring with a stiffness of 200 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 80 kgk = 200 kN/m
fn = Hz
2.Easy
k = 4 kN/mm = 10 kgx

A 10 kg mass hangs from a spring with a stiffness of 4 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 10 kgk = 4 kN/m
ωn = rad/s
3.Medium
k = 50 kN/mζ = 0.6m = 8 kg

An 8 kg mass on a spring with a stiffness of 50 kN/m has a viscous damper giving a damping ratio of 0.6. Find the damped natural frequency fd of its free vibration in Hz.

m = 8 kgk = 50 kN/mζ = 0.6
fd = Hz
4.Medium
k = 200 kN/mζ = 0.3m = 100 kg

A 100 kg mass on a spring with a stiffness of 200 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 = 100 kgk = 200 kN/mζ = 0.3
fd = Hz
5.Hard
k1​ = 5k2​ = 10m = 10 kg

A 10 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 = 10 kN/m. Find the equivalent stiffness keq of the spring system in kN/m.

m = 10 kgk1 = 5 kN/mk2 = 10 kN/m
keq = kN/m
6.Hard
k1​ = 10k2​ = 10k3​ = 15m = 40 kg

A 40 kg mass hangs from spring k₃ = 15 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 10 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 = 40 kgk1 = 10 kN/mk2 = 10 kN/mk3 = 15 kN/m
fn = Hz

Answer key

  1. 1. fn = 7.958 Hz
  2. 2. ωn = 20 rad/s
  3. 3. fd = 10.07 Hz
  4. 4. fd = 6.79 Hz
  5. 5. keq = 3.333 kN/m
  6. 6. fn = 2.33 Hz

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