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 = 12 kgx

A 12 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 = 12 kgk = 150 kN/m
ωn = rad/s
2.Easy
k = 12 kN/mm = 100 kgx

A 100 kg mass hangs from a spring with a stiffness of 12 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 100 kgk = 12 kN/m
fn = Hz
3.Medium
tx40 mm4 cycles8.8 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed 4 cycles later by a peak of 8.8 mm. Find the damping ratio ζ.

x0 = 40 mmx4 = 8.8 mmn = 4
ζ =
4.Medium
k = 80 kN/mc = 1700 N·s/mm = 25 kg

A 25 kg mass is supported by a spring with a stiffness of 80 kN/m and a viscous damper with a damping coefficient of 1700 N·s/m. Find the damping ratio ζ.

m = 25 kgk = 80 kN/mc = 1700 N·s/m
ζ =
5.Hard
k1​ = 50k2​ = 12k3​ = 10m = 25 kg

A 25 kg mass hangs from spring k₃ = 10 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 50 kN/m and k₂ = 12 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 = 50 kN/mk2 = 12 kN/mk3 = 10 kN/m
fn = Hz
6.Hard
k1​ = 100k2​ = 10k3​ = 12m = 5 kg

A 5 kg mass hangs from spring k₃ = 12 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 100 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 = 5 kgk1 = 100 kN/mk2 = 10 kN/mk3 = 12 kN/m
fn = Hz

Answer key

  1. 1. ωn = 111.8 rad/s
  2. 2. fn = 1.743 Hz
  3. 3. ζ = 0.06014
  4. 4. ζ = 0.601
  5. 5. fn = 2.954 Hz
  6. 6. fn = 7.404 Hz

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