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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
k = 10 kN/mm = 12 kgx

A 12 kg mass hangs from a spring with a stiffness of 10 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 12 kgk = 10 kN/m
ωn = rad/s
2.Easy
kmδst​ = 2 mmunloaded

A machine rests on spring mounts, which compress by 2 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 = 2 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 40 kN/mζ = 0.5m = 15 kg

A 15 kg mass on a spring with a stiffness of 40 kN/m has a viscous damper giving a damping ratio of 0.5. Find the damped natural frequency fd of its free vibration in Hz.

m = 15 kgk = 40 kN/mζ = 0.5
fd = Hz
4.Medium
k = 5 kN/mζ = 0.7m = 25 kg

A 25 kg mass on a spring with a stiffness of 5 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 = 25 kgk = 5 kN/mζ = 0.7
fd = Hz
5.Hard
k1​ = 10k2​ = 40k3​ = 25m = 50 kg

A 50 kg mass hangs from spring k₃ = 25 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 10 kN/m and k₂ = 40 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 = 50 kgk1 = 10 kN/mk2 = 40 kN/mk3 = 25 kN/m
fn = Hz
6.Hard
k1​ = 15k2​ = 15m = 40 kg

A 40 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 15 kN/m and k₂ = 15 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 = 15 kN/mk2 = 15 kN/m
fn = Hz

Answer key

  1. 1. ωn = 28.87 rad/s
  2. 2. fn = 11.15 Hz
  3. 3. fd = 7.118 Hz
  4. 4. fd = 1.607 Hz
  5. 5. fn = 2.906 Hz
  6. 6. fn = 4.359 Hz

Full step-by-step solutions: practise this topic on examproblems.com.