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 = 150 kN/mm = 10 kgx

A 10 kg mass hangs from a spring with a stiffness of 150 kN/m. Ignore damping. Find the natural frequency fn in Hz.

m = 10 kgk = 150 kN/m
fn = Hz
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 = 8 kN/mζ = 0.4m = 40 kg

A 40 kg mass on a spring with a stiffness of 8 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 = 40 kgk = 8 kN/mζ = 0.4
fd = Hz
4.Medium
k = 5 kN/mζ = 0.7m = 40 kg

A 40 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 = 40 kgk = 5 kN/mζ = 0.7
fd = Hz
5.Hard
k1​ = 40k2​ = 25m = 10 kg

A 10 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 40 kN/m and k₂ = 25 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 = 10 kgk1 = 40 kN/mk2 = 25 kN/m
fn = Hz
6.Hard
k1​ = 20k2​ = 50m = 10 kg

A 10 kg mass hangs from two springs connected end to end (in series): the upper spring has k₁ = 20 kN/m and the lower one k₂ = 50 kN/m. Find the equivalent stiffness keq of the spring system in kN/m.

m = 10 kgk1 = 20 kN/mk2 = 50 kN/m
keq = kN/m

Answer key

  1. 1. fn = 19.49 Hz
  2. 2. fn = 11.15 Hz
  3. 3. fd = 2.063 Hz
  4. 4. fd = 1.271 Hz
  5. 5. fn = 12.83 Hz
  6. 6. keq = 14.29 kN/m

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