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 = 100 kN/mm = 10 kgx

A 10 kg mass hangs from a spring with a stiffness of 100 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 10 kgk = 100 kN/m
ωn = rad/s
2.Easy
kmδst​ = 15 mmunloaded

A machine rests on spring mounts, which compress by 15 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 = 15 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 8 kN/mc = 30 N·s/mm = 10 kg

A 10 kg mass is supported by a spring with a stiffness of 8 kN/m and a viscous damper with a damping coefficient of 30 N·s/m. Find the damping ratio ζ.

m = 10 kgk = 8 kN/mc = 30 N·s/m
ζ =
4.Medium
k = 15 kN/mc = 150 N·s/mm = 150 kg

A 150 kg mass is supported by a spring with a stiffness of 15 kN/m and a viscous damper with a damping coefficient of 150 N·s/m. Find the damping ratio ζ.

m = 150 kgk = 15 kN/mc = 150 N·s/m
ζ =
5.Hard
k1​ = 12k2​ = 60k3​ = 10m = 50 kg

A 50 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 = 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 = 50 kgk1 = 12 kN/mk2 = 60 kN/mk3 = 10 kN/m
fn = Hz
6.Hard
k1​ = 15k2​ = 10k3​ = 12m = 100 kg

A 100 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 = 15 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 = 100 kgk1 = 15 kN/mk2 = 10 kN/mk3 = 12 kN/m
fn = Hz

Answer key

  1. 1. ωn = 100 rad/s
  2. 2. fn = 4.07 Hz
  3. 3. ζ = 0.05303
  4. 4. ζ = 0.05
  5. 5. fn = 2.109 Hz
  6. 6. fn = 1.433 Hz

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