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 = 5 kN/mm = 40 kgx

A 40 kg mass hangs from a spring with a stiffness of 5 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 40 kgk = 5 kN/m
fn = Hz
2.Easy
k = 8 kN/mm = 5 kgx

A 5 kg mass hangs from a spring with a stiffness of 8 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 5 kgk = 8 kN/m
fn = Hz
3.Medium
k = 200 kN/mc = 550 N·s/mm = 150 kg

A 150 kg mass is supported by a spring with a stiffness of 200 kN/m and a viscous damper with a damping coefficient of 550 N·s/m. Find the damping ratio ζ.

m = 150 kgk = 200 kN/mc = 550 N·s/m
ζ =
4.Medium
k = 20 kN/mζ = 0.7m = 5 kg

A 5 kg mass on a spring with a stiffness of 20 kN/m has a viscous damper giving a damping ratio of 0.7. Find the damped natural frequency fd of its free vibration in Hz.

m = 5 kgk = 20 kN/mζ = 0.7
fd = Hz
5.Hard
k1​ = 5k2​ = 100k3​ = 50m = 10 kg

A 10 kg mass hangs from spring k₃ = 50 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 = 100 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 = 100 kN/mk3 = 50 kN/m
fn = Hz
6.Hard
k1​ = 50k2​ = 15k3​ = 12m = 25 kg

A 25 kg mass hangs from spring k₃ = 12 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 = 15 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 = 15 kN/mk3 = 12 kN/m
fn = Hz

Answer key

  1. 1. fn = 1.779 Hz
  2. 2. fn = 6.366 Hz
  3. 3. ζ = 0.05021
  4. 4. fd = 7.188 Hz
  5. 5. fn = 9.263 Hz
  6. 6. fn = 3.204 Hz

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