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 = 15 kN/mm = 100 kgx

A 100 kg mass hangs from a spring with a stiffness of 15 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 100 kgk = 15 kN/m
ωn = rad/s
2.Easy
kmδst​ = 25 mmunloaded

A machine rests on spring mounts, which compress by 25 mm under its weight. Treat it as an undamped single-degree-of-freedom system and use g = 9.81 m/s². Find the natural frequency fn of vertical vibration in Hz.

δst = 25 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 50 kN/mc = 4000 N·s/mm = 500 kg

A 500 kg mass is supported by a spring with a stiffness of 50 kN/m and a viscous damper with a damping coefficient of 4000 N·s/m. Find the damping ratio ζ.

m = 500 kgk = 50 kN/mc = 4000 N·s/m
ζ =
4.Medium
k = 80 kN/mζ = 0.45m = 25 kg

A 25 kg mass on a spring with a stiffness of 80 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 = 25 kgk = 80 kN/mζ = 0.45
fd = Hz
5.Hard
k1​ = 40k2​ = 5k3​ = 20m = 100 kg

A 100 kg mass hangs from spring k₃ = 20 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 = 100 kgk1 = 40 kN/mk2 = 5 kN/mk3 = 20 kN/m
fn = Hz
6.Hard
k1​ = 10k2​ = 80m = 100 kg

A 100 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 10 kN/m and k₂ = 80 kN/m. The mass stays level. Find the equivalent stiffness keq of the spring system in kN/m.

m = 100 kgk1 = 10 kN/mk2 = 80 kN/m
keq = kN/m

Answer key

  1. 1. ωn = 12.25 rad/s
  2. 2. fn = 3.153 Hz
  3. 3. ζ = 0.4
  4. 4. fd = 8.04 Hz
  5. 5. fn = 1.873 Hz
  6. 6. keq = 90 kN/m

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