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 = 20 kgx

A 20 kg mass hangs from a spring with a stiffness of 5 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 20 kgk = 5 kN/m
ωn = rad/s
2.Easy
kmδst​ = 15 mmunloaded

A machine rests on spring mounts, which compress by 15 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 = 15 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 200 kN/mζ = 0.45m = 250 kg

A 250 kg mass on a spring with a stiffness of 200 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 = 250 kgk = 200 kN/mζ = 0.45
fd = Hz
4.Medium
k = 150 kN/mc = 390 N·s/mm = 100 kg

A 100 kg mass is supported by a spring with a stiffness of 150 kN/m and a viscous damper with a damping coefficient of 390 N·s/m. Find the damping ratio ζ.

m = 100 kgk = 150 kN/mc = 390 N·s/m
ζ =
5.Hard
k1​ = 15k2​ = 5m = 10 kg

A 10 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 15 kN/m and k₂ = 5 kN/m. The mass stays level. Find the equivalent stiffness keq of the spring system in kN/m.

m = 10 kgk1 = 15 kN/mk2 = 5 kN/m
keq = kN/m
6.Hard
k1​ = 50k2​ = 60m = 25 kg

A 25 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 50 kN/m and the lower one k₂ = 60 kN/m. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 25 kgk1 = 50 kN/mk2 = 60 kN/m
fn = Hz

Answer key

  1. 1. ωn = 15.81 rad/s
  2. 2. fn = 4.07 Hz
  3. 3. fd = 4.02 Hz
  4. 4. ζ = 0.05035
  5. 5. keq = 20 kN/m
  6. 6. fn = 5.257 Hz

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