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 = 50 kN/mm = 100 kgx

A 100 kg mass hangs from a spring with a stiffness of 50 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 100 kgk = 50 kN/m
ωn = rad/s
2.Easy
k = 12 kN/mm = 8 kgx

An 8 kg mass hangs from a spring with a stiffness of 12 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 8 kgk = 12 kN/m
ωn = rad/s
3.Medium
k = 10 kN/mc = 360 N·s/mm = 20 kg

A 20 kg mass is supported by a spring with a stiffness of 10 kN/m and a viscous damper with a damping coefficient of 360 N·s/m. Find the damping ratio ζ.

m = 20 kgk = 10 kN/mc = 360 N·s/m
ζ =
4.Medium
tx40 mm4 cycles14.6 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 14.6 mm. Find the damping ratio ζ.

x0 = 40 mmx4 = 14.6 mmn = 4
ζ =
5.Hard
k1​ = 80k2​ = 100m = 20 kg

A 20 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 80 kN/m and the lower one k₂ = 100 kN/m. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 20 kgk1 = 80 kN/mk2 = 100 kN/m
fn = Hz
6.Hard
k1​ = 60k2​ = 30k3​ = 80m = 40 kg

A 40 kg mass hangs from spring k₃ = 80 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 60 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 = 40 kgk1 = 60 kN/mk2 = 30 kN/mk3 = 80 kN/m
fn = Hz

Answer key

  1. 1. ωn = 22.36 rad/s
  2. 2. ωn = 38.73 rad/s
  3. 3. ζ = 0.4025
  4. 4. ζ = 0.04007
  5. 5. fn = 7.503 Hz
  6. 6. fn = 5.179 Hz

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