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 = 5 kgx

A 5 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 5 kgk = 40 kN/m
fn = Hz
2.Easy
k = 80 kN/mm = 200 kgx

A 200 kg mass hangs from a spring with a stiffness of 80 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 200 kgk = 80 kN/m
ωn = rad/s
3.Medium
k = 1 kN/mζ = 0.7m = 10 kg

A 10 kg mass on a spring with a stiffness of 1 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 = 10 kgk = 1 kN/mζ = 0.7
fd = Hz
4.Medium
tx12 mm1 cycle9.3 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 12 mm is followed one cycle later by a peak of 9.3 mm. Find the damping ratio ζ.

x0 = 12 mmx1 = 9.3 mmn = 1
ζ =
5.Hard
k1​ = 80k2​ = 60k3​ = 60m = 100 kg

A 100 kg mass hangs from spring k₃ = 60 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 80 kN/m and k₂ = 60 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 = 80 kN/mk2 = 60 kN/mk3 = 60 kN/m
fn = Hz
6.Hard
k1​ = 12k2​ = 40m = 20 kg

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 12 kN/m and k₂ = 40 kN/m. The mass stays level. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 20 kgk1 = 12 kN/mk2 = 40 kN/m
fn = Hz

Answer key

  1. 1. fn = 14.24 Hz
  2. 2. ωn = 20 rad/s
  3. 3. fd = 1.137 Hz
  4. 4. ζ = 0.04053
  5. 5. fn = 3.262 Hz
  6. 6. fn = 8.115 Hz

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