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 = 500 kgx

A 500 kg mass hangs from a spring with a stiffness of 50 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 500 kgk = 50 kN/m
fn = Hz
2.Easy
k = 1 kN/mm = 10 kgx

A 10 kg mass hangs from a spring with a stiffness of 1 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 10 kgk = 1 kN/m
ωn = rad/s
3.Medium
k = 25 kN/mζ = 0.5m = 150 kg

A 150 kg mass on a spring with a stiffness of 25 kN/m has a viscous damper giving a damping ratio of 0.5. Find the damped natural frequency fd of its free vibration in Hz.

m = 150 kgk = 25 kN/mζ = 0.5
fd = Hz
4.Medium
k = 12 kN/mc = 1180 N·s/mm = 80 kg

An 80 kg mass is supported by a spring with a stiffness of 12 kN/m and a viscous damper with a damping coefficient of 1180 N·s/m. Find the damping ratio ζ.

m = 80 kgk = 12 kN/mc = 1180 N·s/m
ζ =
5.Hard
k1​ = 12k2​ = 40k3​ = 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 = 12 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 = 10 kgk1 = 12 kN/mk2 = 40 kN/mk3 = 15 kN/m
fn = Hz
6.Hard
k1​ = 50k2​ = 50k3​ = 15m = 25 kg

A 25 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 = 50 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 = 25 kgk1 = 50 kN/mk2 = 50 kN/mk3 = 15 kN/m
fn = Hz

Answer key

  1. 1. fn = 1.592 Hz
  2. 2. ωn = 10 rad/s
  3. 3. fd = 1.779 Hz
  4. 4. ζ = 0.6022
  5. 5. fn = 5.43 Hz
  6. 6. fn = 3.635 Hz

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