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 = 250 kgx

A 250 kg mass hangs from a spring with a stiffness of 25 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 250 kgk = 25 kN/m
fn = Hz
2.Easy
k = 5 kN/mm = 20 kgx

A 20 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 = 20 kgk = 5 kN/m
ωn = rad/s
3.Medium
k = 150 kN/mζ = 0.35m = 8 kg

An 8 kg mass on a spring with a stiffness of 150 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 = 8 kgk = 150 kN/mζ = 0.35
fd = Hz
4.Medium
tx25 mm3 cycles2.6 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 25 mm is followed 3 cycles later by a peak of 2.6 mm. Find the damping ratio ζ.

x0 = 25 mmx3 = 2.6 mmn = 3
ζ =
5.Hard
k1​ = 15k2​ = 30m = 20 kg

A 20 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 15 kN/m and k₂ = 30 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 = 15 kN/mk2 = 30 kN/m
fn = Hz
6.Hard
k1​ = 30k2​ = 10k3​ = 12m = 5 kg

A 5 kg mass hangs from spring k₃ = 12 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 = 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 = 5 kgk1 = 30 kN/mk2 = 10 kN/mk3 = 12 kN/m
fn = Hz

Answer key

  1. 1. fn = 1.592 Hz
  2. 2. ωn = 15.81 rad/s
  3. 3. fd = 20.41 Hz
  4. 4. ζ = 0.1192
  5. 5. fn = 7.549 Hz
  6. 6. fn = 6.838 Hz

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