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 = 40 kN/mm = 12 kgx

A 12 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 12 kgk = 40 kN/m
ωn = rad/s
2.Easy
k = 150 kN/mm = 500 kgx

A 500 kg mass hangs from a spring with a stiffness of 150 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 500 kgk = 150 kN/m
fn = Hz
3.Medium
k = 100 kN/mζ = 0.45m = 100 kg

A 100 kg mass on a spring with a stiffness of 100 kN/m has a viscous damper giving a damping ratio of 0.45. Find the damped natural frequency fd of its free vibration in Hz.

m = 100 kgk = 100 kN/mζ = 0.45
fd = Hz
4.Medium
k = 25 kN/mc = 180 N·s/mm = 8 kg

An 8 kg mass is supported by a spring with a stiffness of 25 kN/m and a viscous damper with a damping coefficient of 180 N·s/m. Find the damping ratio ζ.

m = 8 kgk = 25 kN/mc = 180 N·s/m
ζ =
5.Hard
k1​ = 20k2​ = 10k3​ = 80m = 20 kg

A 20 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 = 20 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 = 20 kgk1 = 20 kN/mk2 = 10 kN/mk3 = 80 kN/m
fn = Hz
6.Hard
k1​ = 30k2​ = 30m = 50 kg

A 50 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 30 kN/m and k₂ = 30 kN/m. The mass stays level. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 50 kgk1 = 30 kN/mk2 = 30 kN/m
fn = Hz

Answer key

  1. 1. ωn = 57.74 rad/s
  2. 2. fn = 2.757 Hz
  3. 3. fd = 4.495 Hz
  4. 4. ζ = 0.2012
  5. 5. fn = 5.257 Hz
  6. 6. fn = 5.513 Hz

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