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 = 150 kN/mm = 40 kgx

A 40 kg mass hangs from a spring with a stiffness of 150 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 40 kgk = 150 kN/m
fn = Hz
2.Easy
k = 10 kN/mm = 10 kgx

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

m = 10 kgk = 10 kN/m
fn = Hz
3.Medium
k = 2 kN/mc = 80 N·s/mm = 20 kg

A 20 kg mass is supported by a spring with a stiffness of 2 kN/m and a viscous damper with a damping coefficient of 80 N·s/m. Find the damping ratio ζ.

m = 20 kgk = 2 kN/mc = 80 N·s/m
ζ =
4.Medium
k = 100 kN/mζ = 0.7m = 10 kg

A 10 kg mass on a spring with a stiffness of 100 kN/m has a viscous damper giving a damping ratio of 0.7. Find the damped natural frequency fd of its free vibration in Hz.

m = 10 kgk = 100 kN/mζ = 0.7
fd = Hz
5.Hard
k1​ = 80k2​ = 10m = 40 kg

A 40 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 80 kN/m and k₂ = 10 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 = 40 kgk1 = 80 kN/mk2 = 10 kN/m
fn = Hz
6.Hard
k1​ = 60k2​ = 50k3​ = 15m = 20 kg

A 20 kg mass hangs from spring k₃ = 15 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 60 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 = 20 kgk1 = 60 kN/mk2 = 50 kN/mk3 = 15 kN/m
fn = Hz

Answer key

  1. 1. fn = 9.746 Hz
  2. 2. fn = 5.033 Hz
  3. 3. ζ = 0.2
  4. 4. fd = 11.37 Hz
  5. 5. fn = 7.549 Hz
  6. 6. fn = 4.089 Hz

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