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 = 40 kN/mm = 200 kgx

A 200 kg mass hangs from a spring with a stiffness of 40 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 200 kgk = 40 kN/m
ωn = rad/s
2.Easy
k = 25 kN/mm = 10 kgx

A 10 kg mass hangs from a spring with a stiffness of 25 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 10 kgk = 25 kN/m
fn = Hz
3.Medium
tx25 mm4 cycles0.6 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 25 mm is followed 4 cycles later by a peak of 0.6 mm. Find the damping ratio ζ.

x0 = 25 mmx4 = 0.6 mmn = 4
ζ =
4.Medium
tx40 mm1 cycle33.1 mm

In a free-vibration test of a damped mass–spring system, a displacement peak of 40 mm is followed one cycle later by a peak of 33.1 mm. Find the damping ratio ζ.

x0 = 40 mmx1 = 33.1 mmn = 1
ζ =
5.Hard
k1​ = 5k2​ = 50k3​ = 10m = 5 kg

A 5 kg mass hangs from spring k₃ = 10 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 5 kN/m and k₂ = 50 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 = 5 kN/mk2 = 50 kN/mk3 = 10 kN/m
fn = Hz
6.Hard
k1​ = 12k2​ = 60k3​ = 50m = 25 kg

A 25 kg mass hangs from spring k₃ = 50 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 = 60 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 = 12 kN/mk2 = 60 kN/mk3 = 50 kN/m
fn = Hz

Answer key

  1. 1. ωn = 14.14 rad/s
  2. 2. fn = 7.958 Hz
  3. 3. ζ = 0.1468
  4. 4. ζ = 0.03012
  5. 5. fn = 6.547 Hz
  6. 6. fn = 5.468 Hz

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