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 = 80 kgx

An 80 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 = 80 kgk = 80 kN/m
ωn = rad/s
2.Easy
k = 15 kN/mm = 20 kgx

A 20 kg mass hangs from a spring with a stiffness of 15 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 20 kgk = 15 kN/m
fn = Hz
3.Medium
tx10 mm5 cycles2.1 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 10 mm is followed 5 cycles later by a peak of 2.1 mm. Find the damping ratio ζ.

x0 = 10 mmx5 = 2.1 mmn = 5
ζ =
4.Medium
k = 12 kN/mζ = 0.35m = 80 kg

An 80 kg mass on a spring with a stiffness of 12 kN/m has a viscous damper giving a damping ratio of 0.35. Find the damped natural frequency fd of its free vibration in Hz.

m = 80 kgk = 12 kN/mζ = 0.35
fd = Hz
5.Hard
k1​ = 20k2​ = 50k3​ = 20m = 5 kg

A 5 kg mass hangs from spring k₃ = 20 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 20 kN/m and k₂ = 50 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 = 20 kN/mk2 = 50 kN/mk3 = 20 kN/m
fn = Hz
6.Hard
k1​ = 40k2​ = 30k3​ = 25m = 10 kg

A 10 kg mass hangs from spring k₃ = 25 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 40 kN/m and k₂ = 30 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 = 10 kgk1 = 40 kN/mk2 = 30 kN/mk3 = 25 kN/m
fn = Hz

Answer key

  1. 1. ωn = 31.62 rad/s
  2. 2. fn = 4.359 Hz
  3. 3. ζ = 0.04962
  4. 4. fd = 1.826 Hz
  5. 5. fn = 8.877 Hz
  6. 6. fn = 6.831 Hz

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