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 = 100 kN/mm = 40 kgx

A 40 kg mass hangs from a spring with a stiffness of 100 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 40 kgk = 100 kN/m
ωn = rad/s
2.Easy
k = 25 kN/mm = 100 kgx

A 100 kg mass hangs from a spring with a stiffness of 25 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 100 kgk = 25 kN/m
fn = Hz
3.Medium
k = 100 kN/mc = 4000 N·s/mm = 250 kg

A 250 kg mass is supported by a spring with a stiffness of 100 kN/m and a viscous damper with a damping coefficient of 4000 N·s/m. Find the damping ratio ζ.

m = 250 kgk = 100 kN/mc = 4000 N·s/m
ζ =
4.Medium
tx40 mm5 cycles15.6 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed 5 cycles later by a peak of 15.6 mm. Find the damping ratio ζ.

x0 = 40 mmx5 = 15.6 mmn = 5
ζ =
5.Hard
k1​ = 20k2​ = 40k3​ = 30m = 25 kg

A 25 kg mass hangs from spring k₃ = 30 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 = 40 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 = 20 kN/mk2 = 40 kN/mk3 = 30 kN/m
fn = Hz
6.Hard
k1​ = 40k2​ = 5k3​ = 15m = 10 kg

A 10 kg mass hangs from spring k₃ = 15 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 = 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 = 40 kN/mk2 = 5 kN/mk3 = 15 kN/m
fn = Hz

Answer key

  1. 1. ωn = 50 rad/s
  2. 2. fn = 2.516 Hz
  3. 3. ζ = 0.4
  4. 4. ζ = 0.02996
  5. 5. fn = 4.502 Hz
  6. 6. fn = 5.338 Hz

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