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 = 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 circular frequency ωn in rad/s.

m = 500 kgk = 150 kN/m
ωn = rad/s
2.Easy
kmδst​ = 10 mmunloaded

A machine rests on spring mounts, which compress by 10 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 = 10 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 100 kN/mζ = 0.45m = 5 kg

A 5 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 = 5 kgk = 100 kN/mζ = 0.45
fd = Hz
4.Medium
k = 40 kN/mζ = 0.6m = 25 kg

A 25 kg mass on a spring with a stiffness of 40 kN/m has a viscous damper giving a damping ratio of 0.6. Find the damped natural frequency fd of its free vibration in Hz.

m = 25 kgk = 40 kN/mζ = 0.6
fd = Hz
5.Hard
k1​ = 30k2​ = 30k3​ = 100m = 20 kg

A 20 kg mass hangs from spring k₃ = 100 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 30 kN/m and k₂ = 30 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 = 30 kN/mk2 = 30 kN/mk3 = 100 kN/m
fn = Hz
6.Hard
k1​ = 10k2​ = 100m = 20 kg

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 10 kN/m and k₂ = 100 kN/m. The mass stays level. Find the equivalent stiffness keq of the spring system in kN/m.

m = 20 kgk1 = 10 kN/mk2 = 100 kN/m
keq = kN/m

Answer key

  1. 1. ωn = 17.32 rad/s
  2. 2. fn = 4.985 Hz
  3. 3. fd = 20.1 Hz
  4. 4. fd = 5.093 Hz
  5. 5. fn = 6.892 Hz
  6. 6. keq = 110 kN/m

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