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 = 5 kgx

A 5 kg mass hangs from a spring with a stiffness of 25 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 5 kgk = 25 kN/m
fn = Hz
2.Easy
kmδst​ = 8 mmunloaded

A machine rests on spring mounts, which compress by 8 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 = 8 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 20 kN/mζ = 0.4m = 5 kg

A 5 kg mass on a spring with a stiffness of 20 kN/m has a viscous damper giving a damping ratio of 0.4. Find the damped natural frequency fd of its free vibration in Hz.

m = 5 kgk = 20 kN/mζ = 0.4
fd = Hz
4.Medium
k = 200 kN/mc = 550 N·s/mm = 150 kg

A 150 kg mass is supported by a spring with a stiffness of 200 kN/m and a viscous damper with a damping coefficient of 550 N·s/m. Find the damping ratio ζ.

m = 150 kgk = 200 kN/mc = 550 N·s/m
ζ =
5.Hard
k1​ = 20k2​ = 5m = 10 kg

A 10 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 20 kN/m and k₂ = 5 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 = 10 kgk1 = 20 kN/mk2 = 5 kN/m
fn = Hz
6.Hard
k1​ = 5k2​ = 25m = 10 kg

A 10 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 5 kN/m and the lower one k₂ = 25 kN/m. Ignore damping and the masses of the springs. Find the natural frequency fn of vertical vibration in Hz.

m = 10 kgk1 = 5 kN/mk2 = 25 kN/m
fn = Hz

Answer key

  1. 1. fn = 11.25 Hz
  2. 2. fn = 5.573 Hz
  3. 3. fd = 9.225 Hz
  4. 4. ζ = 0.05021
  5. 5. fn = 7.958 Hz
  6. 6. fn = 3.249 Hz

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