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 = 12 kgx

A 12 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 = 12 kgk = 80 kN/m
ωn = rad/s
2.Easy
k = 10 kN/mm = 5 kgx

A 5 kg mass hangs from a spring with a stiffness of 10 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 5 kgk = 10 kN/m
ωn = rad/s
3.Medium
k = 50 kN/mc = 770 N·s/mm = 12 kg

A 12 kg mass is supported by a spring with a stiffness of 50 kN/m and a viscous damper with a damping coefficient of 770 N·s/m. Find the damping ratio ζ.

m = 12 kgk = 50 kN/mc = 770 N·s/m
ζ =
4.Medium
tx40 mm2 cycles31.1 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed 2 cycles later by a peak of 31.1 mm. Find the damping ratio ζ.

x0 = 40 mmx2 = 31.1 mmn = 2
ζ =
5.Hard
k1​ = 30k2​ = 10k3​ = 10m = 100 kg

A 100 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 = 30 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 = 100 kgk1 = 30 kN/mk2 = 10 kN/mk3 = 10 kN/m
fn = Hz
6.Hard
k1​ = 5k2​ = 5k3​ = 10m = 10 kg

A 10 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 = 5 kN/m and k₂ = 5 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 = 5 kN/mk2 = 5 kN/mk3 = 10 kN/m
fn = Hz

Answer key

  1. 1. ωn = 81.65 rad/s
  2. 2. ωn = 44.72 rad/s
  3. 3. ζ = 0.497
  4. 4. ζ = 0.02002
  5. 5. fn = 1.424 Hz
  6. 6. fn = 3.559 Hz

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