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 = 200 kN/mm = 15 kgx

A 15 kg mass hangs from a spring with a stiffness of 200 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 15 kgk = 200 kN/m
ωn = rad/s
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
tx30 mm4 cycles2.4 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 30 mm is followed 4 cycles later by a peak of 2.4 mm. Find the damping ratio ζ.

x0 = 30 mmx4 = 2.4 mmn = 4
ζ =
4.Medium
k = 150 kN/mζ = 0.45m = 200 kg

A 200 kg mass on a spring with a stiffness of 150 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 = 200 kgk = 150 kN/mζ = 0.45
fd = Hz
5.Hard
k1​ = 100k2​ = 5m = 5 kg

A 5 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 100 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 = 5 kgk1 = 100 kN/mk2 = 5 kN/m
fn = Hz
6.Hard
k1​ = 100k2​ = 100m = 25 kg

A 25 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 100 kN/m and the lower one k₂ = 100 kN/m. Find the equivalent stiffness keq of the spring system in kN/m.

m = 25 kgk1 = 100 kN/mk2 = 100 kN/m
keq = kN/m

Answer key

  1. 1. ωn = 115.5 rad/s
  2. 2. fn = 5.573 Hz
  3. 3. ζ = 0.09999
  4. 4. fd = 3.892 Hz
  5. 5. fn = 23.06 Hz
  6. 6. keq = 50 kN/m

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