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 = 8 kgx

An 8 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 = 8 kgk = 5 kN/m
ωn = rad/s
2.Easy
kmδst​ = 0.5 mmunloaded

A machine rests on spring mounts, which compress by 0.5 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 = 0.5 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 8 kN/mζ = 0.6m = 15 kg

A 15 kg mass on a spring with a stiffness of 8 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 = 15 kgk = 8 kN/mζ = 0.6
fd = Hz
4.Medium
k = 8 kN/mζ = 0.3m = 25 kg

A 25 kg mass on a spring with a stiffness of 8 kN/m has a viscous damper giving a damping ratio of 0.3. Find the damped natural frequency fd of its free vibration in Hz.

m = 25 kgk = 8 kN/mζ = 0.3
fd = Hz
5.Hard
k1​ = 20k2​ = 10k3​ = 5m = 25 kg

A 25 kg mass hangs from spring k₃ = 5 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 20 kN/m and k₂ = 10 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 = 20 kN/mk2 = 10 kN/mk3 = 5 kN/m
fn = Hz
6.Hard
k1​ = 50k2​ = 40k3​ = 5m = 20 kg

A 20 kg mass hangs from spring k₃ = 5 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 = 40 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 = 50 kN/mk2 = 40 kN/mk3 = 5 kN/m
fn = Hz

Answer key

  1. 1. ωn = 25 rad/s
  2. 2. fn = 22.29 Hz
  3. 3. fd = 2.94 Hz
  4. 4. fd = 2.716 Hz
  5. 5. fn = 2.084 Hz
  6. 6. fn = 2.449 Hz

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