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 = 40 kgx

A 40 kg mass hangs from a spring with a stiffness of 150 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 40 kgk = 150 kN/m
ωn = rad/s
2.Easy
k = 150 kN/mm = 10 kgx

A 10 kg mass hangs from a spring with a stiffness of 150 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 10 kgk = 150 kN/m
fn = Hz
3.Medium
k = 12 kN/mζ = 0.3m = 2 kg

A 2 kg mass on a spring with a stiffness of 12 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 = 2 kgk = 12 kN/mζ = 0.3
fd = Hz
4.Medium
tx30 mm1 cycle20.6 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 30 mm is followed one cycle later by a peak of 20.6 mm. Find the damping ratio ζ.

x0 = 30 mmx1 = 20.6 mmn = 1
ζ =
5.Hard
k1​ = 25k2​ = 80k3​ = 5m = 25 kg

A 25 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 = 25 kN/m and k₂ = 80 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 = 25 kgk1 = 25 kN/mk2 = 80 kN/mk3 = 5 kN/m
fn = Hz
6.Hard
k1​ = 20k2​ = 50m = 25 kg

A 25 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 = 50 kN/m. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 25 kgk1 = 20 kN/mk2 = 50 kN/m
fn = Hz

Answer key

  1. 1. ωn = 61.24 rad/s
  2. 2. fn = 19.49 Hz
  3. 3. fd = 11.76 Hz
  4. 4. ζ = 0.05972
  5. 5. fn = 2.199 Hz
  6. 6. fn = 3.805 Hz

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