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 = 25 kN/mm = 200 kgx

A 200 kg mass hangs from a spring with a stiffness of 25 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 200 kgk = 25 kN/m
ωn = rad/s
2.Easy
kmδst​ = 4 mmunloaded

A machine rests on spring mounts, which compress by 4 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 = 4 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 20 kN/mζ = 0.35m = 100 kg

A 100 kg mass on a spring with a stiffness of 20 kN/m has a viscous damper giving a damping ratio of 0.35. Find the damped natural frequency fd of its free vibration in Hz.

m = 100 kgk = 20 kN/mζ = 0.35
fd = Hz
4.Medium
k = 50 kN/mζ = 0.7m = 250 kg

A 250 kg mass on a spring with a stiffness of 50 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 = 250 kgk = 50 kN/mζ = 0.7
fd = Hz
5.Hard
k1​ = 60k2​ = 60m = 20 kg

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 60 kN/m and k₂ = 60 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 = 20 kgk1 = 60 kN/mk2 = 60 kN/m
fn = Hz
6.Hard
k1​ = 12k2​ = 25k3​ = 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 = 12 kN/m and k₂ = 25 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 = 12 kN/mk2 = 25 kN/mk3 = 80 kN/m
fn = Hz

Answer key

  1. 1. ωn = 11.18 rad/s
  2. 2. fn = 7.882 Hz
  3. 3. fd = 2.108 Hz
  4. 4. fd = 1.607 Hz
  5. 5. fn = 12.33 Hz
  6. 6. fn = 5.661 Hz

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