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 = 2 kN/mm = 20 kgx

A 20 kg mass hangs from a spring with a stiffness of 2 kN/m. Ignore damping. Find the natural circular frequency ωn in rad/s.

m = 20 kgk = 2 kN/m
ωn = rad/s
2.Easy
kmδst​ = 25 mmunloaded

A machine rests on spring mounts, which compress by 25 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 = 25 mmg = 9.81 m/s²
fn = Hz
3.Medium
k = 150 kN/mζ = 0.4m = 50 kg

A 50 kg mass on a spring with a stiffness of 150 kN/m has a viscous damper giving a damping ratio of 0.4. Find the damped natural frequency fd of its free vibration in Hz.

m = 50 kgk = 150 kN/mζ = 0.4
fd = Hz
4.Medium
k = 2 kN/mζ = 0.5m = 8 kg

An 8 kg mass on a spring with a stiffness of 2 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 = 8 kgk = 2 kN/mζ = 0.5
fd = Hz
5.Hard
k1​ = 50k2​ = 60k3​ = 80m = 40 kg

A 40 kg mass hangs from spring k₃ = 80 kN/m. The top of k₃ is attached to a rigid bar that hangs from two springs side by side, k₁ = 50 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 = 40 kgk1 = 50 kN/mk2 = 60 kN/mk3 = 80 kN/m
fn = Hz
6.Hard
k1​ = 5k2​ = 12m = 100 kg

A 100 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 5 kN/m and the lower one k₂ = 12 kN/m. Find the equivalent stiffness keq of the spring system in kN/m.

m = 100 kgk1 = 5 kN/mk2 = 12 kN/m
keq = kN/m

Answer key

  1. 1. ωn = 10 rad/s
  2. 2. fn = 3.153 Hz
  3. 3. fd = 7.99 Hz
  4. 4. fd = 2.179 Hz
  5. 5. fn = 5.416 Hz
  6. 6. keq = 3.529 kN/m

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